Recent Advances in Control Problems of Dynamical Systems and Networks [1st ed.] 9783030491222, 9783030491239

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Studies in Systems, Decision and Control 301

Ju H. Park   Editor

Recent Advances in Control Problems of Dynamical Systems and Networks

Studies in Systems, Decision and Control Volume 301

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

More information about this series at http://www.springer.com/series/13304

Ju H. Park Editor

Recent Advances in Control Problems of Dynamical Systems and Networks

123

Editor Ju H. Park Department of Electrical Engineering Yeungnam University Kyongsan, Korea (Republic of)

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-49122-2 ISBN 978-3-030-49123-9 (eBook) https://doi.org/10.1007/978-3-030-49123-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my parents

Preface

In the last decade, there have been rapid developments and changes in the field of control engineering. Departing from the study of classical linear/nonlinear control systems, tremendous efforts have been made to tackle more complex systems or networks and many real-world application problems. In particular, with the more powerful IT technology, the components of the control systems are connected to a wireless network, and the concept of a cyber-physical system in which physical layers and cyberspaces coexist has also emerged. Besides, under the influence of the 4th Industrial Revolution, advanced technologies such as artificial intelligence, machine learning, Internet of Things, virtual reality, wearable devices, and big data are grafting with control systems. In addition to the progress of this innovative technology, the field of control theory has been creating a stepping stone to solve the limitations of reality one by one that has not been addressed in the past. From this point of view, even in the same research subject in control engineering, the intention of planning this book is to cover with issues such as (i) control theories that weren’t developed, (ii) constraints of systems that weren’t addressed, or (iii) control systems of new structures that could not be dealt with when readers look at books or papers published 10 years ago.

Aim of the Book The aim of this book is (1) to provide an introduction to recent research trend on various topics on control problems of dynamical systems and networks and (2) to present novel analysis and synthesis frameworks for investigating the stability and designing controllers of the systems and networks. To show the progress of research on these topics from various perspectives, in this book, several dynamic system models in the continuous or discrete-time domain such as traditional linear and nonlinear systems, networked control systems, fractional-order systems, switched systems, large-scale interconnected systems, vii

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neural networks, and complex networks have been considered for various control problems. In the system analysis, certain system phenomena including uncertainties, time delays, actuator faults, switching, asynchronous sampling, event-triggered scheme, impulse, and cyber-attacks are considered to deal with the stability and stabilization problems in the systems and networks. In order to show applicability to real problems, it is noted that some examples are chosen from practical industrial models. From a detailed view of the contents of the book, it collects works carried out recently by promising experts in this field. By their contribution, this book covers the problems on observer/estimator design, observer-based control, finite-time stability and stabilization, eigenvector-centrality-based consensus protocol, sampled-data control, fuzzy-based control, sliding mode control, fault alarm, fault detection/estimation/reconstruction, fault-tolerant control, event-triggered control, asynchronous quantized control, synchronization, positivity, stochastic stability, and so on. Some preliminary results of the material presented in the book may have been published in eminent international journals in recent years by contributors. Therefore, if readers want to know additional basic knowledge and advanced one related to the contents of each chapter, please refer to the previously published results of each contributor, which can be easily found through academic DB search. It is noted that this book attempts to consolidate the recent new results or extended results of previous studies from contributors on the topics mentioned above. In this respect, the book as a tutorial reference as well as an advanced monograph is likely to be of use for the wide and heterogeneous group of graduate students and doctoral researchers in science and engineering who are focusing on control theory and corresponding real applications for various dynamical systems and networks.

Main Features of This Book This book presents both theoretical development and applications in the real world in the important research areas in control engineering including linear systems, nonlinear systems, time-delay systems, positive systems, multi-agent systems, fractional-order systems, networked control systems, hybrid and switching systems, stochastic systems, fuzzy systems, descriptor systems, parabolic systems, impulsive systems, complex networks, and neural networks. As can be seen from the system classification mentioned above, this book covers the latest control problems for almost all systems studied in control engineering. For dealing with these systems and networks, some new control strategies are applied based on the following control schemes including sampled-data control, fault-tolerant control, event-triggered control, sliding mode control, observer-based control, and H1 control.

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Appropriate features suggested by this book include the following: 1. Providing various mathematical and engineering basic and advanced knowledge. 2. Introduction of various control systems and networks. 3. Presenting recent research trends on control theory. 4. State-of-the-art controller design techniques. 5. Constraints in the control system to be considered in the real environment. 6. Examples to help readers understand how the theories and methods presented apply. 7. Accompanying detailed simulation results.

Prerequisites and Intended Audience This book should be of interest to both graduate students and researchers in the area of control theory and applications, as well as those new to the area. The reporting on the latest developments from various points of view in this area is certainly to attract intense interest to relevant researchers who are concerned with the newest trends and topics. To understand the research flow and contents on the various topics covered in this book, the background required of the readers is basic and advanced knowledge of linear algebra, calculus, linear and nonlinear control system theory, Lyapunov stability theory, probability theory, and simulation techniques. In particular, for the fractional-order systems discussed in Part, additional knowledge of the Lyapunov function theory associated with fractional derivatives is required.

Outline of the Book The book is composed of four parts, with the system classification ranging from Linear and Nonlinear Dynamic Systems, Switched Systems and Hybrid Control, Fractional-Order Systems, and Dynamical Networks. In soliciting the chapter contributions from experts on related fields, maximum effort has been attempted to cover the hottest topics in four above research categories. Part, “Linear and Nonlinear Dynamic Systems”, starts in chapter “Partial State Observers for Linear Time-Delay Systems” and ends with chapter “Fault Estimator for Parabolic Systems with Distributed Inputs and Outputs”, which introduces the recent control problems on a variety of topics on linear and nonlinear dynamic systems.

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As we know well, a dynamic system is a system in which the states (or variables) are expressed as a function of time in physical space. Many people believe that the concept of such dynamic systems was invented by French mathematician Henri Poincare in the late 1800s. The most basic classification of dynamic systems is to distinguish between linear and nonlinear systems. A linear dynamic system is a mathematical model of a system based on the use of a linear operator. Nonlinear systems refer to the type of dynamic systems where the output from the system does not vary directly with respect to input to the system. Starting from the nineteenth century, control problems of linear or nonlinear dynamic systems in various areas we encounter result from system modeling to stability analysis and controller design. First, there has been a tremendous amount of research into the stability of dynamic systems over two centuries. Various stability concepts including asymptotic stability, exponential stability, bounded-input bounded-output stability, Nyquist stability, uniform stability, stochastic stability, and absolute stability have been introduced through stability theories. In addition, control theories for stabilizing such systems have evolved. For example, PID control, optimal control, adaptive control, robust control, fuzzy control, neural control, stochastic control, model-predictive control, and so on. In general, systems are analyzed in the time domain or in the frequency domain by considering the characteristics of the system to deal with control problems. As is well known, in the frequency domain analysis of classical control theory, the Fourier transform, Laplace transform, or Z-transform is used to deal with the various analyses of linear systems in depth. On the other hand, in the time domain for modern control theory, the analysis of real-world nonlinear systems including linear systems is carried out using state equations, another form of differential equations. For stability and stabilization of dynamic systems, in this Part, for various linear systems and nonlinear systems having impulse, fault, delay, or uncertain factors in continuous-time or discrete-time space, the controller design techniques of a number of ways for the stabilization of closed-loop systems are introduced. Part, “Switched Systems and Hybrid Control”, starts in chapter “Event-Triggered Control of Discrete-Time Switched Linear Systems Via Communication Channels with Limited Capacity” and ends with chapter “Asynchronous Quantized Control for Markov Switching Systems with Channel Fading”, which introduces hottest topics on switched systems and hybrid control. A hybrid system is capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system like continuous-time dynamics, discrete-time dynamics, jump phenomena, operating mode changes, and logic-based switching. From the mathematical standpoint, these hybrid systems have mathematical models of different forms, such as algebraic equations, difference equations, ordinary differential equations, and/or partial differential equations. Switched systems are an important class of hybrid dynamical systems, which are widely used to model many real physical systems with multi-mode switching, such as, in electronic circuits field, power electronics, chemical processes, mechanical systems, automotive industry, aircraft and air traffic control, and so on. In general, a switched system consists of a finite number of continuous (or discrete)-time

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subsystems and a logical rule that orchestrates switching between them. The switching rule, determined by time or system state, both time and state, other supervisory logic decision, or even stochastic processes, yields different switching signals. The switching signals, which have a great influence on the system performances, can be classified into two categories: arbitrary and constrained. It is known that the constrained switching compared with arbitrary switching arises quite naturally from the physical constraints of a real switched system. Naturally, stability and stabilization are very important issues in the study of switched systems. A great number of efficient methods have been proposed to solve these problems for switched systems in the past few decades. It is known that the stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of switching signals. When all the subsystems are stable, it usually cannot ensure the stability of the switched system under arbitrary switching. There is always the possibility, however, that the stability of the whole system is maintained if the switching is sufficiently slow, which means that the interval between consecutive switchings is large enough. In fact, it really does not matter if the system occasionally has fast switchings, provided this does not occur too frequently. The idea is proved to be reasonable and captured by the concept of average dwell-time switching proposed by J. P. Hespanha and A. S. Morse from the paper “Stability of switched systems with average dwell-time, in: Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 3, pp. 2655–2660, 1999” which has showed that switching among stable linear systems results in a stable system provided that the average dwell time is sufficiently large. Until now, several approaches have been proposed for the control problems for switched systems, such as arbitrary switching, restricted switching (like dwell time and average dwell time), multiple Lyapunov functions method, and piecewise quadratic Lyapunov functions. Especially, the average dwell-time approach has been acknowledged to be more flexible and efficient in the analysis of switched systems. While there are many books and papers that can be consulted on these switched systems, the following two references (R1. D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003; R2. H. Lin, P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automatic Control, Vol. 54, pp. 308–322, 2009) are recommended for readers to get the basic knowledge of the system. Markov jump systems are a type of stochastic hybrid systems with finite operation modes, where the jumps among these modes are governed by a Markov process taking values in a finite set. Markov jump systems have the advantage of modeling dynamical systems subject to random abrupt changes in their parameters or structures, and have been applied to many practical plants, such as powers systems, manufactory processes, fault-tolerant systems, telecommunications systems, immunology, etc.

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When dealing with this system, transition probabilities determining the transition between different modes have a crucial influence on system stability and performance. In some cases, the transition probability obeys the exponential distribution, making the sojourn time memoryless and resulting in a constant index distribution for Markov jump systems. However, in the actual process, the sojourn time does not follow the exponential distribution, which describes semi-Markov jump systems of more general cases. Meanwhile, in recent years, a remarkable development of networked control systems in which components are networked by wire or wireless has been progressed. In addition to this, a change in the control structure is taking place for efficient control of the system, away from the classical control method. For instance, a hybrid-triggered scheme consisting of time-triggered scheme and event-triggered scheme is developed recently to alleviate the load of network bandwidth. The representative new type of control method is the event-triggered control method. This method can save resources and cost by changing the control input only when a certain trigger condition is satisfied. As another special control method, the hybrid-triggered scheme contains a random switch between the time-triggered scheme and the event-triggered scheme. For a good application of this hybrid control scheme, refer to a recent paper by J. Liu, Y. Gu, X. Xie, D. Yue, J. H. Park, Hybrid-driven-based H1 control for networked cascade control systems with actuator saturations and stochastic cyber attacks, IEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol. 49, No. 12, pp. 2452–2463, 2019. In any case, the purpose of Part, “Switched Systems and Hybrid Control”, is to explore the latest trends in research on hybrid characteristics of systems or controllers. Part, “Fractional-Order Systems”, starts in chapter “Observer-Based Controller Design for Fractional-Order Neutral-Type Systems” and ends with chapter “New Results on Stability of Coupled Impulsive Fractional-Order Systems on Networks”, which presents some recent hot topics on fractional-order dynamic systems or networks. As an extension and generalization of the traditional integer-order differential and integral calculus, it is known that the fractional calculus is a mathematical branch of the analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator and of the integration operator. Since its first appearance in 1695, the fractional calculus was introduced in one of Niels Henrik Abel’s early papers in 1823. Especially, from the twenty-first century, the fractional calculus becomes a dynamic research topic in applied science and engineering and has appealed more consideration because the fractional-order characterization is established to be more enough to explain some real-time problems than the integer-order representation. Some notable contributions on fractional calculus have been generated to the implementation of fractional differential equations. Compared to the integer-order models, the main desirability of fractional models is that it provides a valuable implementation for the account of

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remembrance and hereditary effects of many resources and procedures. Therefore, fractional differential equations occur in many notable applications such as viscoelasticity, electrochemistry, control, electromagnetic theory, aerodynamics, polymer rheology, etc. Notably, due to the memory effect, the fractional calculus has been intensively used in neural networks and complex dynamic systems in the last decade because the feature of the fractional calculus plays a crucial role in the description of neural networks having memory. For instance, it was reported that a generalized capacitor can be more accurately modeled by fractional operators and the circuit systems were developed with the use of fractional calculus. As an application of fractional-order systems, let us look at it in the field of neural networks. A model of fractional Hopfield neural network was proposed by E. Kaslik and S. Sivasundaram in the paper titled “Nonlinear dynamics and chaos in fractional-order neural networks,” Neural Networks, Vol. 32, pp. 245–256, 2012, which has the following description: C a t0 Dt xðtÞ

¼ AxðtÞ þ Bf ðxðtÞÞ þ d

where x ¼ ðx1 ; x2 ; . . .; xn ÞT 2 Rn is the state vector, the diagonal matrix A ¼ diagða1 ; a2 ; . . .; an Þ with 0\ai is the state feedback coefficients, B ¼ ðbij Þnn is the network’s interconnection matrix, f ðxÞ is the activation function, and d denotes the constant external input. This neural network has the memory and storage information from initial time t0 to t. Also, very recently, a fractional discrete-time neural network is proposed by L. Huang, J. H. Park, G.-C. Wu, and Z.-W. Mo from the paper, “Variable-order fractional discrete-time recurrent neural networks,” Journal of Computational and Applied Mathematics, Vol. 370, Article ID 112622, 2020, which are described by C

Dma xðtÞ ¼ Axðt þ m  1Þ þ Bf ðxðt þ m  1ÞÞ; 0\m  1; xðaÞ ¼ xa ; t 2 Na þ 1m

n where f is a continuous function and f : ‘1 a ! R . It is clear that the above equation has a zero solution if f ð0Þ ¼ 0. For m ¼ 1, it can be reduced to a standard recurrent neural network by the use of Euler method x_ ðtÞ  DxðtÞ:

xðk þ 1Þ ¼ ðI  AÞxðkÞ þ Bf ðxðkÞÞ þ d if d ¼ 0. Further, in control engineering, the stability and stabilization problems on linear or nonlinear fractional-order systems in continuous- or discrete-time domain become one of the hottest topics in recent 5 years. Some real applications such fractional dynamic systems include chaotic systems with applications to image encryption, color image denoising, enhancing and decomposition, secure communications, pattern recognition applications, battery external short circuit fault

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diagnosis, tumor-immune system interaction, financial systems, population dynamic models, human respiratory syncytial virus infection, and so on. Part, “Dynamical Networks”, starts with the positivity and stability problems in neural networks in chapter “Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks” and ends with a synchronization problem in chapter “Finite-Time Synchronization Control for Markovian Jump Memristive Neural Networks with Reaction-Diffusion Terms”. This part introduces recent progress of some spotlighting topics of dynamic networks including neural networks and complex networks which are the areas of research that are of most interest in the era of the 4th industrial revolution along with cyber-physical systems, artificial intelligence, and big data. Artificial neural networks are computing systems vaguely inspired by the biological neural networks that constitute animal brains. Usually, a practical neural network that can perform some complex tasks has a very large number of neurons and layers, since more neurons and layers bring higher accuracy output results. There are some popular models of neural networks including Hopfield neural networks, cellular neural networks, convolutional neural networks, CohenGrossberg neural networks, and bidirectional associative memory neural network, which have found many successful applications in various fields, such as signal processing, pattern recognition, image processing, parallel computation, and combinatorial optimization. Especially, convolutional neural networks have particular strengths in these areas such as image and video recognition, image classification, medical image analysis, and natural language processing. Thus, in the recent study of intelligent autonomous vehicles, convolutional neural networks are widely used to find patterns to recognize objects, faces, and scenes in images because the networks learn features directly, extract features automatically, and then provide the highest level of recognition results. For the research on neural networks from the control engineering perspective, the problems on mathematical modeling and their stability of developed neural networks have been active in the last two decades since the developed neural network is regarded as a dynamic system in the model-based control problems. Since the past two decades, complex networks have drawn increasing attention from various scientific fields, such as mathematics, engineering, physics, economics, ecology, and even sociology. A complex network is a large set of interconnected nodes, in which every node represents an individual agent of the system, while edges represent relations between nodes. In this regard, complex dynamical networks have garnered wide-scale attention from many fields due to their broad applications in various fields, such as electrical power grids, sensor networks, communication networks, Internet, coupled oscillators, World Wide Web, epidemic spreading networks, a platoon of unmanned aerial vehicles or mobile robots, and so on. In particular, in recent years, several types of flu or epidemics have occurred in various regions and spread to the whole world, which has a great negative effect on the health of many people. The spread of these viruses and infectious diseases is also a problem of a complex network.

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The nature of complex networks, such as topological structures, dynamical evolution, and node diversities, has been fully investigated in the existing literature. Also, the study on geometric features, collective behaviors, and control and synchronization of complex networks has received another significant concern. Among these topics, a lot of interest and related research has been recently focused on synchronization by control engineering researchers. Some types of synchronization have been discovered, such as complete synchronization, cluster synchronization, impulsive synchronization, phase synchronization, lag synchronization, quasi-synchronization, and cluster synchronization. It is noted that the cluster synchronization needs the coupled complex networks split into several subgroups, called cluster, and the systems in the same clusters should be synchronized. That is, each cluster has its own synchronization goal which could be distinct for different clusters. In this Part, we will look at stability, fault detection, control, and synchronization issues in various types of dynamic networks.

Linear and Nonlinear Dynamic Systems Chapter “Partial State Observers for Linear Time-Delay Systems”. This chapter considers the design problem of state observers for linear systems with time delay. It is well known that state estimation is an essential part of any model-based control systems. In particular, partial state estimation is very useful when the full set of the states of a system is not observable or detectable, or whenever the system is large scale and only a few states of the system need to be monitored. Recent advancements in functional observer design algorithms for linear time-invariant systems will be reviewed first in this chapter. Then, delay-free and delay-dependent functional observer design schemes for linear time-delay systems with state and input delays are illustrated. Chapter “Finite-Time Stability and Control of Impulsive Positive Systems with Interval Uncertainty”. This chapter addresses the problem of finite-time stability and stabilization of positive dynamic systems including impulses and interval uncertainties. By considering a few types of impulsive effects, the finite-time stability conditions are proposed for interval impulsive positive systems on the basis of a time-varying copositive Lyapunov function and an average impulsive interval method. Also, the finite-time stability guaranteeing L1 -gain performance is further studied as an application work. Chapter “An Eigenvector-Centrality Based Consensus Protocol Design for Discrete-Time Multi-agent Systems with Communication Delays”. This chapter investigates a leader-follower consensus protocol design problem for a class of multi-agent systems in the discrete-time domain with time-varying communication delay. A consensus protocol is proposed by applying the concept of centrality measure for the agent which is determined by the structure of information flow among multiple agents. In order to find the most central node within the

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connected graph, the various centrality measures have been attempted in the field of social network analysis. To improve the consensus performance in two aspects of H1 performance and the stability in consideration of the time-delay, a novel weighted consensus protocol is proposed by calculating the eigenvector centrality for agents in a network by building simple Lyapunov-Krasovskii functional combined with a linear matrix inequality framework. Chapter “Fault-Tolerant Sampled-Data Synchronization of Chaotic Systems with Random Occurring Uncertainties: A Semi-Markov Jump Model Approach”. This chapter focuses on addressing the issue of sampled-data synchronization for nonlinear chaotic systems with uncertainties by fault-tolerant control techniques, where a semi-Markov jump chain is utilized to describe the switching characteristics of the system. In this approach, it is assumed that the randomly occurring uncertainties in the chaotic systems are subject to the Bernoulli distributed white noise sequences. A mean-square synchronization condition of the system by means of employing appropriate Lyapunov-Krasovskii functional and reasonable integral inequality methods is presented. Chapter “Fuzzy-Based Sliding Mode Control Design for Stability Analysis of Nonlinear Interconnected Systems”. This chapter investigates the problem of the stability and stabilization of interconnected large-scale systems. The nonlinear large-scale interconnected system is equivalently expressed into a T-S fuzzy-model-based system with time delays. Via a sliding mode control scheme having a suitable sliding surface combined with the techniques such as Wirtinger inequality approach, Lyapunov stability theory, and linear matrix inequality framework, delay-dependent stability and stabilization conditions for guaranteeing asymptotic stability of the system are derived. Chapter “Observer-Based Fault-Tolerant Control for Non-Infinitely Observable Descriptor Systems”. This chapter is concerned with a fault-tolerant control for a class of non-infinitely observable descriptor systems with disturbances. The proposed control scheme consists of a sliding mode observer to estimate the states and unknown inputs and a controller to perform fault-tolerant stabilization. Then, the closed-loop system can be expressed as an infinitely observable reduced-order system by treating some states as unknown inputs. A linear matrix inequality optimization technique is used to design the sliding mode observer and stabilizing controller such that the estimates from the observer and the output of the closed-loop system converge asymptotically. A practical system is considered so that the proposed method can demonstrate its effectiveness. Chapter “Fault Estimator for Parabolic Systems with Distributed Inputs and Outputs”. This chapter investigates the problem of the H1 fault estimation of a class of parabolic systems with distributed inputs and outputs. The fault in the considered parabolic system is characterized by a minimal state-space description of a low-pass filter, which incorporates fault’s prior knowledge. First, the fault estimator is designed to estimate both the fault and the states of the considered system. Then, exponential stability is considered for the system without external disturbance.

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Next, the results are extended to guarantee the improved robustness and ensure that the system is the H1 performance or mixed H1 and passivity performance in terms of an optimization problem based on the Lyapunov theory with the framework of the linear matrix inequality approach.

Switched Systems and Hybrid Control Chapter “Event-Triggered Control of Discrete-Time Switched Linear Systems via Communication Channels with Limited Capacity”. This chapter considers a state feedback control problem for discrete-time switched linear systems. The switched system with a control input is assumed to be connected via a limited capacity communication channel and the controller can only access the successfully event-triggered transmitted information of system state and mode. Considering the signal transmission delays and packet losses, respectively, mode-dependent event-triggered transmission schemes are proposed to mitigate unnecessary data transmission. Based on the augmented switching signal, which is generated by merging the switching signal with its transmitted switching signal, the closed-loop systems are modeled as switched systems with the delayed state. Then, by using the multiple Lyapunov functional and average dwell-time methods, sufficient exponential stability conditions for the closed-loop systems are obtained. Chapter “Fault Alarm-Based Hybrid Control Design for Periodic Piecewise Time-Delay Systems”. This chapter presents a novel fault alarm-based hybrid control scheme which is about to stabilize the periodic piecewise systems with time delay and disturbances. The hybrid controller has devised to handle both the normal and faulty systems. By an alarm signal in the system, the controller automatically switches to the reliable control mode from conventional control mode. The asymptotic stability of the concerned system is ensured with a minimum disturbance attenuation level by the controller. Chapter “Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems”. This chapter examines the design problem of sliding mode controller for the stabilization of stochastic Markov jump systems via a novel event-triggered communication scheme as a tool of saving computational resources in the system. Through stochastic Lyapunov functional and observer design theories, a stabilizing sliding mode control law is constructed to guarantee that the closed-loop system’s trajectories can arrive at the specified desired sliding surface. To demonstrate the effectiveness and applicability of the proposed control method, an example of a practical model is utilized. Chapter “Asynchronous Quantized Control for Markov Switching Systems with Channel Fading”. This chapter deals with an asynchronous control problem for Markov switching systems with channel fading, where a more general class of discrete-time Markov

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switching system is developed by exploring the channel fading. To analyze the system as a hidden Markov model, the original systems and designed controller are characterized by stochastic variables. As an effective method to reduce energy consumption in the controller, a mode-dependent event-triggered communication mechanism for guaranteeing stochastic stability of a closed-loop system is proposed.

Fractional-Order Systems Chapter “Observer-Based Controller Design for Fractional-Order Neutral-Type Systems”. This chapter studies the design problem of the observer-based controller for a class of fractional-order neutral-type systems. Then, based on the Lyapunov stability theory, an observer-based guaranteed cost control problem of the fractional-order systems is formulated. A stability criterion for guaranteeing the stability of the closed-loop system is given, and the design of the observer and the feedback controller is attributed to the feasible solution problem of a class of linear matrix inequalities. Chapter “Adaptive Synchronization of Fractional-Order Delayed Memristive Neural Networks”. This chapter deals with an adaptive synchronization problem for fractional-order memristor-based neural networks with a constant time delay. It is noted that this work is the first attempt to study the synchronization of fractional-order delayed memristive neural networks using a novel adaptive delay feedback controller. Chapter “New Results on Stability of Coupled Impulsive Fractional-Order Systems on Networks”. This chapter is devoted to researching the stability issue of fractional-order impulsive coupled systems over networks. Five classes of criteria on stability, uniform stability, uniform asymptotic stability, finite-time stability, and Mittag-Leffler stability, respectively, of the fractional-order impulsive coupled systems are derived by the graph theory and Lyapunov second technique.

Part Dynamical Networks Chapter “Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks”. This chapter deals with the problems of positivity and stability of nonlinear time-delay systems in the area of neural network models including Hopfield neural networks, bidirectional associative memory neural networks, and inertial neural networks. A systematic approach involving comparison techniques via differential and integral inequalities in combination with nonlinear analysis has been presented.

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Then, linear-programming-based conditions for the existence and exponential stability of positive equilibriums have been derived. Chapter “Fault Detection and Isolation Methodology in Finite Frequency Domain for Constrained Networked systems”. This chapter investigates the transmission-dependent fault detection and isolation strategy in a finite frequency domain for networked systems with constrained communication capacities. A switched system with multiple stochastic parameters is firstly derived to model the whole system. Then, drawing support from the geometric method and the finite frequency stochastic performance indices, sufficient conditions are established to guarantee the desired performances and characterize the filter gains based on the proposed transmission-dependent fault detection and isolation scheme. Chapter “Mean-Square Stochastic Stability of Delayed Hybrid Stochastic Inertial Neural Networks”. This chapter considers the problem of the mean-square stochastic stability analysis for hybrid stochastic inertial neural networks with time delays. The hybrid stochastic inertial neural networks are represented as the combination of a two-level system in which the first level is directed by the system of second-order differential equations and the second level is directed by the discrete set representing switching jump nodes. By considering a few types of time delays, sufficient delay-dependent mean-square stability conditions for the networks are established using variable transformation techniques and some integral inequalities based on the Lyapunov theory and linear matrix inequality framework. Chapter “Exponential Stability and Stabilization of Stochastic Neural Network Systems via Switching and Impulsive Control”. This chapter deals with the exponential stability and stabilization problem of stochastic neural network systems via a new switching and impulsive control scheme. The main characteristics of this proposed controller lie in the asynchronous behavior between switchings and impulses. Also, the concepts of dwell time and impulsive interval are employed to analyze the system. Then, based on multiple Lyapunov functions approach, sufficient conditions for mean-square exponential stability are established for stability and controller design. Chapter “Hybrid-Triggered Synchronization of Delayed Complex Dynamical Networks Subject to Stochastic Cyber-Attacks”. This chapter deals with a synchronization problem for a class of delayed complex dynamical networks with randomly occurring cyber-attacks. A hybrid event-triggered communication scheme, which includes a new event-triggered scheme and the conventional time-triggered scheme in one framework, is proposed to reduce the transmission load of networks and control resources. Then, a stability criterion of an error dynamic system for the synchronization problem is obtained such that the synchronization error is asymptotically stable in the mean-square sense. Chapter “Cluster Synchronization on Derivative Coupled Lur’e Networks: Impulsive Pinning Strategy”.

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This chapter is concerned with the global and exponential synchronization issue for a class of nonidentically delay coupled Lur’e networks with stochastic disturbance and derivative couplings. By thinking the cluster topological structure of the Lur’e networks, a novel impulsive pinning controller is proposed. Then, the stability criteria and estimate of the exponential convergence velocity for the cluster synchronization are derived. Chapter “Finite-Time Synchronization Control for Markovian Jump Memristive Neural Networks with Reaction-Diffusion Terms”. This chapter investigates the issue of finite-time synchronization for a class of memristive neural networks, where the reaction-diffusion items and Markovian jump parameters are both considered. In the analysis, it is assumed that the drive and response systems are set to obey inconsistent Markovian jumping laws, which is more practical in real systems. A discontinuous controller is designed so that the influence of the inconsistent parameters can be avoided by combining the novel Lyapunov-Krasovskii functional with a canonical Bessel-Legendre inequality and free weighted matrix method. Kyongsan, Korea (Republic of) April 2020

Ju H. Park Chumna Chair Professor

Acknowledgements

The editor of this edited book, I, Ju H. (Jessie) Park, has been studying various topics on nonlinear dynamical systems and networks for about 25 years. In particular, in recent years, my research team named “Laboratory for Advanced Nonlinear Dynamics and Control” in Yeungnam University, Republic of Korea has made remarkable achievements in the areas of control theory, signal processing, and artificial intelligence. So, in connection with these recent achievements, I have found consensus with my members of the team that the latest work on various hot topics on dynamic systems and networks needs to be introduced to as many readers as possible. In addition, I invited some of the experts in the related research field to increase the completeness of this project. This led to the high quality and diversity of this book, which consisted of 21 chapters. The material presented in this book has been the recent outcome of research efforts by contributors. Hence, I would like to thank all contributors for their dedicated participation to complete each chapter. I as the editor of this edited volume would like to express my gratitude to the Editor of the book series Studies in Systems, Decision and Control of Springer-Nature, Prof. Dr. Janusz Kacprzyk (Polish Academy of Sciences); Executive Editor of the series, Dr. Thomas Ditzinger; and Project Coordinator, Ms. Saranya Kalidoss, Project Manager, Viju Falgon at the Springer-Nature for their professional and powerful handling of this book. Without their editorial comments and detailed examination, the publication of the book would not have gone so smoothly. Personally, there are no words that suffice to thank my acquaintances, Eunkyung, Jaeho, Hoyoul, Michelle, Claude, Melvin, Jeonghee, Hyunjik, William, Jinguk, Jiyeon, Maurice, Woocheol, George, Brian, Johanne, Achim, Kevin, James, Johnny, and Helena for their constant encouragement, support, and sacrifice. Finally, this work of writing the book was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B02002002). Kyongsan, Korea (Republic of)

Ju H. Park

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Contents

Linear and Nonlinear Dynamic Systems Partial State Observers for Linear Time-Delay Systems . . . . . . . . . . . . . Reza Mohajerpoor Finite-Time Stability and Control of Impulsive Positive Systems with Interval Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mengjie Hu and Ju H. Park An Eigenvector-Centrality Based Consensus Protocol Design for Discrete-Time Multi-agent Systems with Communication Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Park, S. H. Lee, and O. M. Kwon Fault-Tolerant Sampled-Data Synchronization of Chaotic Systems with Random Occurring Uncertainties: A Semi-Markov Jump Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yude Xia, Yudong Wang, Jing Wang, and Hao Shen

3

37

61

83

Fuzzy-Based Sliding Mode Control Design for Stability Analysis of Nonlinear Interconnected Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A. Manivannan, G. Bhuvaneshwari, R. Krishnasamy, V. Parthiban, and S. Dhanasekar Observer-Based Fault-Tolerant Control for Non-Infinitely Observable Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Joseph Chang Lun Chan, Tae H. Lee, and Chee Pin Tan Fault Estimator for Parabolic Systems with Distributed Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 K. Mathiyalagan

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Contents

Switched Systems and Hybrid Control Event-Triggered Control of Discrete-Time Switched Linear Systems via Communication Channels with Limited Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Xiaoqing Xiao, Lei Zhou, and Ju H. Park Fault Alarm-Based Hybrid Control Design for Periodic Piecewise Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 R. Sakthivel and S. Harshavarthini Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Wenhai Qi Asynchronous Quantized Control for Markov Switching Systems with Channel Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Jun Cheng, Yaonan Shan, and Ju H. Park Fractional-Order Systems Observer-Based Controller Design for Fractional-Order Neutral-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Yongxia Qu, Youggui Kao, and Cunchen Gao Adaptive Synchronization of Fractional-Order Delayed Memristive Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Haibo Bao, Ju H. Park, and Jinde Cao New Results on Stability of Coupled Impulsive Fractional-Order Systems on Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Li Zhang, Youggui Kao, and Cunchen Gao Dynamical Networks Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Le Van Hien Fault Detection and Isolation Methodology in Finite Frequency Domain for Constrained Networked Systems . . . . . . . . . . . . . . . . . . . . . 387 Yue Long and Ju H. Park Mean-Square Stochastic Stability of Delayed Hybrid Stochastic Inertial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 R. Krishnasamy, A. Manivannan, and Raju K. George

Contents

xxv

Exponential Stability and Stabilization of Stochastic Neural Network Systems via Switching and Impulsive Control . . . . . . . . . . . . . . . . . . . . 435 Ticao Jiao and Ju H. Park Hybrid-Triggered Synchronization of Delayed Complex Dynamical Networks Subject to Stochastic Cyber-Attacks . . . . . . . . . . . . . . . . . . . . 457 Xiaojian Yi and Yajuan Liu Cluster Synchronization on Derivative Coupled Lur’e Networks: Impulsive Pinning Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Ze Tang, Dong Ding, and Ju H. Park Finite-Time Synchronization Control for Markovian Jump Memristive Neural Networks with Reaction-Diffusion Terms . . . . . . . . . 499 Xiaona Song, Jingtao Man, and Shuai Song Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Editor and Contributors

About the Editor Ju H. Park (Jessie Park) received the Ph.D. degree in Electronics and Electrical Engineering from Pohang University of Science and Technology (POSTECH), Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate Fellow in Engineering Research Center—Automation Research Center, POSTECH. He joined Yeungnam University, Kyongsan, Republic of Korea, in March 2000, where he is currently the Chuma Chair Professor. From 2006 to 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. His research interests include robust control and filtering, neural networks, complex networks, fuzzy systems, multi-agent systems, chaotic systems, embedded systems, signal processing, intelligent vehicles, and applied mathematics. He has published a number of papers more than 600 in top leading journals and two books titled Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals and Dynamic Systems with Time Delays: Stability and Control in these areas. Professor Park serves as an Editor of the International Journal of Control, Automation and Systems, Springer-Nature. He is also a Subject Editor/Advisory Editor/Associate Editor/Editorial Board member for several international journals, including IET Control Theory and Applications, Applied Mathematics and Computation, Journal of The Franklin Institute, Nonlinear Dynamics, Cogent Engineering, Engineering Reports, IEEE Transactions on Fuzzy Systems, IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Cybernetics, and so on.

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Editor and Contributors

Since 2015, he is a recipient of Highly Cited Researcher Award by Clarivate Analytics (formerly, Thomson Reuters) and listed in three fields, engineering, computer sciences, and mathematics in 2019. He is a Fellow of the Korean Academy of Science and Technology (KAST). Further information about Prof. Park and his laboratory is available from the web page: http://yu.ac.kr/*rcl/.

Contributors Haibo Bao School of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China G. Bhuvaneshwari Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology-Chennai Campus, Chennai, India Jinde Cao School of Mathematics, Southeast University, Nanjing, People’s Republic of China Joseph Chang Lun Chan Division of Electronic Engineering, Jeonbuk National University, Jeonju, Republic of Korea Jun Cheng College of Mathematics and Statistics, Guangxi Normal University, Guilin, People’s Republic of China S. Dhanasekar Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology-Chennai Campus, Chennai, India Dong Ding Engineering Research Center of Internet of Things Technology Applications (Ministry of Education), Jiangnan University, Wuxi, People’s Republic of China Cunchen Gao School of Mathematical Science, Ocean University of China, Qingdao, People’s Republic of China Raju K. George Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala, India S. Harshavarthini Department of Applied Mathematics, Bharathiar University, Coimbatore, India Le Van Hien Faculty of Mathematics and Informatics, Hanoi National University of Education, Hanoi, Vietnam Mengjie Hu Department of Electrical Engineering, Yeungnam University, Gyeongsan, Republic of Korea Ticao Jiao School of Electrical and Electronic Engineering, Shandong University of Technology, Shandong, People’s Republic of China

Editor and Contributors

xxix

Youggui Kao Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, People’s Republic of China R. Krishnasamy Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore, Tamil Nadu, India O. M. Kwon School of Electrical Engineering, Chungbuk National University, Seowon-gu, Cheongju, Republic of Korea S. H. Lee School of Electrical Engineering, Chungbuk National University, Seowon-gu, Cheongju, Republic of Korea Tae H. Lee Division of Electronic Engineering, Jeonbuk National University, Jeonju, Republic of Korea Yajuan Liu School of Control and Computer Engineering, North China Electric Power University, Beijing, People’s Republic of China Yue Long School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, People’s Republic of China Jingtao Man School of Information Engineering, Henan University of Science and Technology, Luoyang, People’s Republic of China A. Manivannan School of Advanced Sciences - Division of Mathematics, Vellore Institute of Technology-Chennai Campus, Chennai, India K. Mathiyalagan Department of Applied Mathematics, Bharathiar University, Coimbatore, India Reza Mohajerpoor Data 61, CSIRO, Sydney, Australia Ju H. Park Department of Electrical Engineering, Yeungnam University, Gyeongsan, Republic of Korea M. J. Park Center for Global Converging Humanities, Kyung Hee University, Giheung-gu, Yongin, Republic of Korea V. Parthiban Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology-Chennai Campus, Chennai, India Wenhai Qi School of Engineering, Qufu Normal University, Rizhao, People’s Republic of China Yongxia Qu School of Mathematical Science, Ocean University of China, Qingdao, People’s Republic of China R. Sakthivel Department of Applied Mathematics, Bharathiar University, Coimbatore, India Yaonan Shan University of Electronic Science and Technology of China, Chengdu, Sichuan, People’s Republic of China

xxx

Editor and Contributors

Hao Shen School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan, People’s Republic of China Shuai Song School of Automation, Nanjing University of Science and Technology, Nanjing, People’s Republic of China Xiaona Song School of Information Engineering, Henan University of Science and Technology, Luoyang, People’s Republic of China Chee Pin Tan School of Engineering and Advanced Engineering Platform, Monash University Malaysia, Selangor, Malaysia Ze Tang Engineering Research Center of Internet of Things Technology Applications (Ministry of Education), Jiangnan University, Wuxi, People’s Republic of China Jing Wang School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan, People’s Republic of China Yudong Wang School of Metallurgy Engineering, Anhui University of Technology, Ma’anshan, People’s Republic of China Yude Xia School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan, People’s Republic of China Xiaoqing Xiao School of Information Science and Technology, Nantong University, Jiangsu, People’s Republic of China Xiaojian Yi School of Mechanical Engineering, Beijing Institute of Technology, Beijing, People’s Republic of China Li Zhang School of Mathematical Science, Ocean University of China, Qingdao, People’s Republic of China Lei Zhou School of Information Science and Technology, Nantong University, Jiangsu, People’s Republic of China

List of Figures

Partial State Observers for LinearTime-Delay Systems Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

The errors due to estimating functions zðtÞ ¼ La xðtÞ using: (i) presented solution in Sect. 2.1 (left figure) and (ii) the solution proposed in [19] (right figure) . . . . . . . . . . . . . . . . . . The block diagram of the robust closed-loop functional observer-based controlled system. Observer parameters, system parameters, and functional processes are highlighted in gray, yellow, and orange, respectively . . . . . . . . . . . . . . . . . . . . . . . Estimation of the functions with insufficient estimated upper-bound on the disturbance term eðt; ðtÞ; xðtÞÞ (qðtÞ ¼ 0:1). zi and ^zi indicate the ith element of za and ^za , respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of the functions with sufficiently large estimated upper-bound on the disturbance term eðt; ðtÞ; xðtÞÞ (qðtÞ ¼ 5). zi and ^zi indicate the ith element of za and ^za , respectively . . . Estimation results of the delay-free observer obtained from omitting the delayed terms and removing the sliding term mðtÞ. zi and ^zi indicate the ith element of za and ^za , respectively . . .

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Finite-Time Stability and Control of Impulsive Positive Systems with Interval Uncertainty Fig. Fig. Fig. Fig. Fig. Fig. Fig.

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Drug metabolism model . . . . . . . . . . . . . . . . . . . . . . . . . State response in Example 1 . . . . . . . . . . . . . . . . . . . . . Evolution of xT ðtÞrðtÞ in Example 1 . . . . . . . . . . . . . . . The bound of ¿ a with varying b of Example 1 . . . . . . . The bound of ¿ a with varying tf of Example 1 . . . . . . . State evolution of the open-loop system of Example 2 . Evolution of xT ðtÞrðtÞ of the open-loop system of Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 8 Fig. 9 Fig. Fig. Fig. Fig.

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Fig. 14 Fig. 15 Fig. 16 Fig. 17

State evolution of the closed-loop system of Example 2 Evolution of xT ðtÞrðtÞ of the closed-loop system of Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The bound of ¿ a with varying b of Example 2 . . . . . . . The bound of ¿ a with varying tf of Example 2 . . . . . . . State evolution of the open-loop system of Example 3 . Evolution of xT ðtÞrðtÞ of the open-loop system of Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State evolution of the closed-loop system of Example 3 Evolution of xT ðtÞrðtÞ of the closed-loop system of Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The bound of ¿ a with varying b of Example 3 . . . . . . . The bound of ¿ a with varying tf of Example 3 . . . . . . .

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An Eigenvector-Centrality Based Consensus Protocol Design for Discrete-Time Multi-agent Systems with Communication Delays Fig. 1 Fig. 2 Fig. 3 Fig. Fig. Fig. Fig. Fig.

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Fig. 9

Structure example for information flow . . . . . . . . . . . . . . . . . . 3-Aircrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of the consensus protocol based on the eigenvector-centrality . . . . . . . . . . . . . . . . . . . . . . . . . . The position trajectories of the three aircrafts. . . . . . . . . . . . . . The velocity error trajectories of the three aircrafts . . . . . . . . . The trajectories of rmseðtk Þ with each protocol . . . . . . . . . . . . The position trajectories of the three aircrafts. . . . . . . . . . . . . . 5-nodes network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P5 ðp ðt Þp0 ðtk ÞÞ2 i¼1 i k The trajectories of rmseðtk Þ ¼ with each 5

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Fault-Tolerant Sampled-Data Synchronization of Chaotic Systems with Random Occurring Uncertainties: A Semi-Markov Jump Model Approach Fig. Fig. Fig. Fig.

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The chaotic behavior of the hyperchaotic Rossler system . . . . Uncontrolled and controlled error signals . . . . . . . . . . . . . . . . . Uncontrolled and controlled state response . . . . . . . . . . . . . . . The sampled-data control input with three sampling intervals .

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Fuzzy-Based Sliding Mode Control Design for Stability Analysis of Nonlinear Interconnected Systems Fig. Fig. Fig. Fig.

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State State State State

response response response response

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subsystem-1 . . subsystem-1 . . subsystem-2 . . subsystem-2 . .

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List of Figures

Fig. Fig. Fig. Fig.

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Sliding surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . State under different initial conditions . . . . . . . . . . State under different initial conditions . . . . . . . . . . Sliding motion under different initial conditions. . .

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States and unknown inputs (solid) and their estimates (dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation errors for the states and unknown inputs . . . . . . . . . . Inputs into system from the fault-tolerant controller . . . . . . . . . . .

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Observer-Based Fault-Tolerant Control for Non-Infinitely Observable Descriptor Systems Fig. 1 Fig. 2 Fig. 3

Fault Estimator for Parabolic Systems with Distributed Inputs and Outputs Fig. Fig. Fig. Fig. Fig. Fig.

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Response of wðx; tÞ . . . . . . . . . . . . . . . . ^ tÞ . . . . . . . . . . . . . . . . Response of wðx; Fault signal f ðtÞ . . . . . . . . . . . . . . . . . . . Estimation of fault signal ^f ðtÞ . . . . . . . . Temperature profile wðx; tÞ . . . . . . . . . . ^ tÞ Estimation of temperature profile wðx;

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Event-Triggered Control of Discrete-Time Switched Linear Systems via Communication Channels with Limited Capacity Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8

The event-triggered control framework for switched system (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The simulation of the closed-loop system state with ¿ a ¼ 7 . . ^ðkÞ of the The switching signals rðkÞ of the system and r controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Length of event-triggered intervals and packet dropout instants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event-triggered control framework for switched system (1) with network-induced delays . . . . . . . . . . . . . . . . . . . . . . . . . . The simulation of the closed-loop system state with ¿ a ¼ 4:7 ^ðkÞ of the The switching signals rðkÞ of the system and r controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Length of event-triggered intervals and network-induced delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Evolution of PPTV positive-definite matrix PðtÞ . . . . . . . . . . . . .

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Fault Alarm-Based Hybrid Control Design for Periodic Piecewise Time-Delay Systems Fig. 1

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Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6

List of Figures

^ ^ ^ Variations of jjKðtÞjj; jjMðtÞjj and jjNðtÞjj over one period based on Theorem 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Variations of jjKðtÞjj; jjMðtÞjj and jjNðtÞjj over one period based on Theorem 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of observer gain matrix over one period . . . . . . . . . State responses of the closed-loop system (4) with fault under robust control (17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State responses of the closed-loop system (4) with fault under hybrid control (34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems Fig. Fig. Fig. Fig. Fig. Fig. Fig.

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Boost converter circuit . System mode . . . . . . . . Release instants . . . . . . System state xðtÞ . . . . . Observer state xðtÞ . . . . Sliding variable sðtÞ . . . SMC law uðtÞ . . . . . . .

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Asynchronous Quantized Control for Markov Switching Systems with Channel Fading Fig. Fig. Fig. Fig. Fig.

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System mode uðkÞ . . . . . . . . . . . . . . . . . Control mode /ðkÞ. . . . . . . . . . . . . . . . . Triggered instants and release intervals . State response . . . . . . . . . . . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . .

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Observer-Based Controller Design for Fractional-Order Neutral-Type Systems Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7

The trajectory of state x1 and trajectory of state estimation ^x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of state x2 and trajectory of state estimation ^x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of errors . . . . . . . . . . . . . . . . . . . . . The trajectory of state x1 and trajectory of state estimation ^x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of state x2 and trajectory of state estimation ^x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of errors . . . . . . . . . . . . . . . . . . . . . The trajectory of state x1 and trajectory of state estimation ^x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Figures

Fig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12

The trajectory of state x2 and trajectory of state estimation ^x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of errors . . . . . . . . . . . . . . . . . . . . . The trajectory of state x1 and trajectory of state estimation ^x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of state x2 and trajectory of state estimation ^x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The trajectory of errors . . . . . . . . . . . . . . . . . . . . .

xxxv

........... ...........

284 284

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286

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286 287

Adaptive Synchronization of Fractional-Order Delayed Memristive Neural Networks Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

1 2 3 4 5 6 7 8 9 10

Chaotic attractor of FMNN (19) . . . . . . . . . . . . . . . . . . . . . . . . The synchronization error of ui ðtÞ ¼ ~ui ðtÞ  ui ðtÞ; i ¼ 1; 2 . . . Time evolution of u1 ðtÞ; u2 ðtÞ; ~u1 ðtÞ; ~u2 ðtÞ . . . . . . . . . . . . . . . . Trajectories of ai ðtÞ; i ¼ 1; 2 . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of bi ðtÞ; i ¼ 1; 2 . . . . . . . . . . . . . . . . . . . . . . . . . . Chaotic attractor of FMNN (21) . . . . . . . . . . . . . . . . . . . . . . . . The synchronization error of ui ðtÞ ¼ ~ui ðtÞ  ui ðtÞ; i ¼ 1; 2 . . . Trajectories of u1 ðtÞ; u2 ðtÞ; ~u1 ðtÞ; ~u2 ðtÞ . . . . . . . . . . . . . . . . . . . Trajectories of ai ðtÞ; i ¼ 1; 2 . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of bi ðtÞ; i ¼ 1; 2 . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

304 305 306 306 307 308 308 309 309 310

..

329

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330 330 331 331

.......

350

....... .......

351 363

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364

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New Results on Stability of Coupled Impulsive Fractional-Order Systems on Networks Fig. 1

Fig. 2

Fig. 3 Fig. 4 Fig. 5

Times-series of three-dimension system with initial value X1 ¼ ½0:6; 0:2; 0:5, Y1 ¼ ½0:4; 0:6; 0:3, Z1 ¼ ½0:4; 0:5; 0:2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Times-series of three-dimension system with initial value X1 ¼ ½0:6; 0:2; 0:5, Y1 ¼ ½0:4; 0:6; 0:3, Z1 ¼ ½0:4; 0:5; 0:2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a ¼ 0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a ¼ 0:75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a ¼ 0:98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks Fig. 1 Fig. 2 Fig. 3 Fig. 4

State trajectories of the system with ¿ðtÞ ¼ 5j sinð0:1tÞj. Convergence of the system to the EP xe ¼ ð2:0059; 2:1552; 0:8057Þ> . . . . . . . . . . . . . . . . . . . . . . . Convergence of positive solutions to x . . . . . . . . . . . . . Convergence of positive solutions of (58) to a unique positive EP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxvi

Fig. 5

Fig. 6

List of Figures

Convergence of state trajectories to the unique positive EP v for the input vector J ¼ ð1:0; 2:0; 2:0; 1:0Þ> and delays ¿ i ðtÞ ¼ 1 þ 3j sinð10ptÞj, rj ðtÞ ¼ 4j cosð5ptÞj . . . . . . . . . . . . . . . . Convergence of solutions to the positive EP v in phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381 382

Fault Detection and Isolation Methodology in Finite Frequency Domain for Constrained Networked Systems Fig. 1 Fig. 2 Fig. 3 Fig. 4

Fig. 5

Fig. 6 Fig. 7

Fig. 8

Structure of the Networked Systems . . . . . . . . . . . . . . . . . . . . Illustration of the transmission . . . . . . . . . . . . . . . . . . . . . . . . . The accessed node and hk at each transmission instant in CASE-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The residuals and the corresponding evaluation function for CASE-1-Fmode1 (red line—by our method, blue line—by existing method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The residuals and the corresponding evaluation function for CASE-1-Fmode2 (red line—by our method, blue line—by existing method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The accessed node and hk at each transmission instant in CASE-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The residuals and the corresponding evaluation function for CASE-2-Fmode1 (red line—by our method, blue line—by existing method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The residuals and the corresponding evaluation function for CASE-2-Fmode2 (red line—by our method, blue line—by existing method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. ..

389 390

..

404

..

405

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405

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406

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406

..

407

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426 427 427 429 430 430

. . . . .

. . . . .

449 449 450 451 452

Mean-Square Stochastic Stability of Delayed Hybrid Stochastic Inertial Neural Networks Fig. Fig. Fig. Fig. Fig. Fig.

1 2 3 4 5 6

State trajectories of subsystem 1 . . State trajectories of subsystem 2 . . Markovian jump modes . . . . . . . . . State trajectories of subsystem 1 . . State trajectories of subsystem 2 . . Markovian jump modes . . . . . . . . .

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Exponential Stability and Stabilization of Stochastic Neural Network Systems via Switching and Impulsive Control Fig. Fig. Fig. Fig. Fig.

1 2 3 4 5

Time variation of system states . . . . . . . . . . . . . . . The evolutionary process of switching signal rðtÞ . Times-series of impulsive instants . . . . . . . . . . . . . Time variation of the first subsystem states . . . . . . Time variation of the second subsystem states . . . .

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List of Figures

Fig. 6 Fig. 7 Fig. 8

xxxvii

Times-trajectories of system states . . . . . . . . . . . . . . . . . . . . . . . . The evolutionary process of switching signal rðtÞ . . . . . . . . . . . . Times-series of impulsive instants . . . . . . . . . . . . . . . . . . . . . . . .

452 453 453

Hybrid-Triggered Synchronization of Delayed Complex Dynamical Networks Subject to Stochastic Cyber-Attacks Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

1 2 3 4 5 6 7 8

State response of eðtÞ . . . . . . . . . . . . . . . . . . . . . . . The signal aðtÞ in Example 1 . . . . . . . . . . . . . . . . . The signal bðtÞ in Example 1 . . . . . . . . . . . . . . . . . Event-triggered instants and released intervals . . . . State response of eðtÞ . . . . . . . . . . . . . . . . . . . . . . . The signal aðtÞ in Example 2 . . . . . . . . . . . . . . . . . The signal bðtÞ in Example 2 . . . . . . . . . . . . . . . . . Event-triggered instants and released intervals . . . .

. . . . . . . .

469 470 470 470 473 473 473 474

The phase graph of two different Lur’e systems in (27) . . . . . . . . The synchronization error curves of three states in two different clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

494

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Cluster Synchronization on Derivative Coupled Lur’e Networks: Impulsive Pinning Strategy Fig. 1 Fig. 2

495

Finite-Time Synchronization Control for Markovian Jump Memristive Neural Networks with Reaction-Diffusion Terms Fig. 1 Fig. 2 Fig. 3 Fig. Fig. Fig. Fig. Fig.

4 5 6 7 8

The Markov stochastic processes of the drive and response systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dynamic behaviors of the error system without controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dynamic behaviors of the error system with the controller (6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The chaotic behaviors of the drive system (18) . . . . . . . . . . . . PRNG produced by reaction-diffusion IMNNs (18) . . . . . . . . . Original signals oðtÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Encrypted signal generated by oðtÞ and xðtÞ. . . . . . . . . . . . . . . Decrypted signal generated by eðtÞ and xðtÞ. . . . . . . . . . . . . . .

..

516

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517

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518 519 520 520 521 521

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List of Tables

Finite-Time Stability and Control of Impulsive Positive Systems with Interval Uncertainty Table 1 Table 2

The bounds of ¿ a and tf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The correlations among key parameters. . . . . . . . . . . . . . . . . . . .

44 45

An Eigenvector-Centrality Based Consensus Protocol Design for Discrete-Time Multi-agent Systems with Communication Delays Table 1 Table Table Table Table

2 3 4 5

Local information about each node of unknown 10-node graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplacian matrices . . . . . . . . . . . . . . . . . . . . . . . . . Minimum values of  with lM ¼ 3 . . . . . . . . . . . . Maximum bounds . . . . . . . . . . . . . . . . . . . . . . . . . Maximum bounds on lM . . . . . . . . . . . . . . . . . . . .

. . . . .

64 74 74 77 77

The minimum value of f for different stochastic variable ^ d. . . .

97

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Fault-Tolerant Sampled-Data Synchronization of Chaotic Systems with Random Occurring Uncertainties: A Semi-Markov Jump Model Approach Table 1

Asynchronous Quantized Control for Markov Switching Systems with Channel Fading Table 1 Table 2

The meaning of g, R, l, MðuðtÞÞ and JðuðtÞÞ . . . . . . . . . . . . . . . The values of MðuðtÞÞ and JðuðtÞÞ . . . . . . . . . . . . . . . . . . . . . . .

258 258

Fault Detection and Isolation Methodology in Finite Frequency Domain for Constrained Networked Systems Table 1 Table 2

Comparison of b for different H and /i . . . . . . . . . . . . . . . . . . . b corresponded to different  . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 403

xxxix

Linear and Nonlinear Dynamic Systems

Partial State Observers for Linear Time-Delay Systems Reza Mohajerpoor

Abstract State estimation is an essential component of any model-based control system. In particular, partial state estimation is favorable when the full set of the states of a system is not observable or detectable, or for large-scale systems that need only a few functions of the states to be monitored. This chapter overviews recent advancements in functional observer design algorithms for linear time-invariant systems. The concept of functional observability is explained for systems with and without perturbations, as a generalization of observability or detectability perceptions. Moreover, delay-free and delay-dependent functional observer design schemes for linear time-delay systems with state and input delays are illustrated. Since the exact delay values are often not measurable in practice, a methodology is demonstrated to deal with uncertain state delays in designing delay-dependent functional observers. In light of that, an auxiliary time-varying delay with arbitrary bounds is exploited in the observer, which effectively relaxes the conservative nature of delay-free observers. Keywords State estimation · Time-delay · Functional observability · Observer

1 Introduction A broad range of nonlinear dynamic systems can be linearized around desired set points that maximize the performance of the system. To maintain the operating condition of the system around a desired set point, linear control laws can be constructed by real-time feedback from the states of the system. However, monitoring/measuring all the states of a system is often not possible or extremely costly. To lift this burden the un-measured states of the system can be estimated using model-based state observers such as Luenberger observers [1]. If the linearized dynamics of the system

R. Mohajerpoor (B) Data 61, CSIRO, Sydney, Australia e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_1

3

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R. Mohajerpoor

is accurate, then the error due to estimation is zero amid the steady-state operations. Otherwise, bounded errors can be expected depending on the accuracy of the modeled dynamics. Another problem facing with model-based state estimators is that there might be a number of states that do not have any connection with the measured states, making the system un-observable. If the un-observable states decay to zero asymptotically over time, then the system is detectable. Traditional Luenberger-type observers can only estimate the states of the system if the system is observable or detectable. Otherwise, asymptotic state estimation cannot be achieved for aims such as system monitoring, fault detection and isolation, and control of the observable states of the system. Moreover, in large-scale systems, such as interconnected power systems or chemical reactors [2–4], it is often required to estimate a small set of linear functions of the states to achieve system monitoring, while estimating the whole states due to the implied numerical costs and complexities. To resolve the issues related to conventional observers, functional observers (FOs) were introduced [5, 6] that aim to estimate a single or multiple linear functions of the states of the system, instead of the full set of the states. FO design for timedelay systems has been studied since 1999. Trinh [7] treated state delay terms as unknown inputs and designed a delay-free observer for the system. Later, Darouach [8] designed delay-dependent FOs for linear time-invariant (LTI) systems with constant and known state delays. The state delay is employed in the observer structure proposed in the paper. Several contributions were made thereafter to address delaydependent FO design for state delay systems, though, the majority made the same limiting assumption of accurately measuring/estimating the state delays in real-time [9–12]. Seuret et al. [13] addressed the unknown state delay issue for full-order observers by proposing a sliding mode observer. Mohajerpoor et al. [14], later on, proposed a new robust functional observer structure and design algorithm to relax the problem of uncertain state delays. This chapter provides an overview of the structure and design procedures of functional observers for LTI systems. We first introduce functional observers for LTI systems, and layout the existence conditions for FOs in Sect. 2. Thereafter, recent advancements on the design of functional observer for systems with input-delay and state-delay given that the state-delay is accurately measurable in real-time are comprehended in Sects. 3 and 4, respectively. Then, the restrictive assumption on the delay measurement for delay-dependent observer structure is relaxed in Sect. 5. Finally, the chapter is summarized and a few foreseeable directions are sketched in Sect. 6. Notations: In this chapter, R is the space of real numbers; C+ is the space of complex numbers with positive real values; Rn×m is the space of n × m matrices; Sn×n is the space of n × n symmetric matrices; Rn is the n-dimensional Euclidean space; T  [X 1 ; X 2 ] is equivalent to X 1T , X 2T ; rows(X ) shows the number of rows of the matrix X ; Ik the k × k identity matrices, and 0 is the zero matrix with appropriate dimension. Furthermore, ⊗ is the Kronecker product; ∗ in a symmetric matrix stands for the symmetric element; sym (X ) = X + X T ; X † indicates the pseudo-inverse or

Partial State Observers for Linear Time-Delay Systems

5

the generalized inverse of matrix X ; X ⊥ represents the right orthogonal matrix of X , wherein X X ⊥ = 0; X ≺ 0 (X  0) means that matrix X is negative definite (negative semi-definite), and X  0 (X  0) indicates that X is positive definite (positive semidefinite). In addition, R(X ) and N (X ) are the row space and the null space of matrix X , respectively. [[S]] demonstrates a matrix constructed from the row basis vectors for subspace S. Finally, Cn (Ω) pinpoints the space of continuous functions mapping from Ω to Rn with the topology of uniform convergence.

2 Functional Observer Design for LTI Systems Consider the following LTI dynamic system: x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t), z(t) = L x(t),

(1)

where x(·) ∈ Rn is the vector of the states of the system, u(·) ∈ Rm is the control input, and y(·) ∈ R p and z(·) ∈ Rl are the vector of measurement outputs and the desired functions to be constructed. To add, A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n , and L ∈ Rl×n . Moreover, we assume rank([C; L]) = p + l ≤ n, i.e. matrices C and L are full row rank and span independent spaces, and the number of functions to be estimated is not greater than the detectable number of states n − p. The aim of the functional observer is to construct zˆ (t) ∈ Rl in a way that the error e(t) = zˆ (t) − z(t) approaches to zero asymptotically. To this aim, the below lth order dynamic structure can be considered for the observer: ω(t) ˙ = Fω(t) + Gu(t) + H y(t), zˆ (t) = ω(t) + V y(t),

(2)

where ω(·) ∈ Rl is the internal states of the observer, and F ∈ Rl×l , G ∈ Rl×m , H ∈ Rl× p , and V ∈ Rl× p are observer design parameters. The following theorem summarizes the conditions required for the FO error dynamics to be (globally) asymptotically stable. Theorem 1 ([15]) The observer error e(t) converges to zero asymptotically if and only if the following conditions are satisfied: I. there exists a matrix T ∈ Rl×n that complies with the following matrix equations: F T − T A + H C = 0,

(3a)

G = T B,

(3b)

L − T − V C = 0.

(3c)

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R. Mohajerpoor

II. Matrix F is stable, i.e. F ≺ 0. Nonlinear Eq. (3a) is called Sylvester equation and appears in numerous practical problems. Designing matrices F, T , H , V , and G is not straightforward due to the nonlinear term F T in the Sylvester equation. There are a few algorithms that solve (3) in a way that F is Hurwitz. It has been shown that each method implies numerical conservatism and carries limitations in solving the equation effectively and efficiently [6]. Transformation based frameworks employ matrix transformations to linearize the nonlinear equation, and they are found to be more effective than the other solution methods. There are multiple transformation based solution algorithms in the literature. A recent less conservative solution method in this category proposed in [6] is presented in the coming section.

2.1 A FO Design Algorithm It is obtained from (3c) that 

 T V = L M † + Z (In+ p − M M † )

(4)

where M  [In ; C] ∈ R(n+ p)×n , and Z ∈ Rl×(n+ p) is an arbitrary matrix that works as a design parameter. By partitioning M † and In+ p − M M † as [ M1 M2 ]  M † and [ M¯ 1 M¯ 2 ] := In+ p − M M † , where M1 ∈ Rn×n , M2 ∈ Rn× p , M¯ 1 ∈ R(n+ p)×n , and M¯ 2 ∈ R(n+ p)× p . Therefore, (4) can be written as T = L M1 + Z M¯ 1 ,

(5a)

V = L M2 + Z M¯ 2 .

(5b)

Moreover, post-multiplying (3a) by C¯  C(In − L † L), one gets H = −F T C † + T AC † .

(6)

F T C ⊥ = T AC ⊥ .

(7)

F L M1 C ⊥ + F Z M¯ 1 C ⊥ = L M1 AC ⊥ + Z M¯ 1 AC ⊥ .

(8)

Now, from (5a) and (7) we have

Since it can be shown that M¯ 1 and C ⊥ are orthogonal, i.e. M¯ 1 C ⊥ = 0 [6], (8) can be written as   F, −Z Ω = Φ, (9)

Partial State Observers for Linear Time-Delay Systems

7

where Ω  [L M1 C ⊥ ; M¯ 1 AC ⊥ ] ∈ R(n+ p+l)×(n− p) and Φ  L M1 AC ⊥ ∈ Rl×(n− p) . If there is a solution for (9), one obtains F = N11 + Z˜ N21 ,

(10)

Z = −N12 − Z˜ N22 ,

(11)

    N21 N22 = N2  (In+ p+l − N11 N12 = N1  ΦΩ † ∈ Rl×(n+ p+l) , where ΩΩ † ) ∈ R(n+ p+l)×(n+ p+l) , N11 ∈ Rl×l , N12 ∈ Rl×(n+ p) , N21 ∈ R(n+ p+l)×l , N22 ∈ R(n+ p+l)×(n+ p) , and Z˜ ∈ Rl×(n+ p+l) is a free matrix that is designed in a way that matrix F is Hurwitz. Note that Z˜ can be designed using multiple numerical methods such as poleplacement [16], provided that the pair (N11 , N21 ) is observable or detectable.

2.2 Existence of a Solution Solving the set of matrix equations (3) in a way that conditions of Theorem 1 are fulfilled implies that (9) can be solved and the pair (N11 , N21 ) is at least detectable. These two conditions are necessary and sufficient for the presence of an stable functional observer (2) for System (1), and it has been proven that they can be translated into the following conditions [17]: ⎛⎡

⎤⎞ ⎛⎡ ⎤⎞ LA CA ⎜⎢ C A ⎥⎟ ⎢ ⎥⎟ ⎝⎣ C ⎦⎠ , rank ⎜ ⎝⎣ C ⎦⎠ = rank L L ⎤⎞ ⎛⎡ ⎤⎞ sL − L A CA rank ⎝⎣ C A ⎦⎠ = rank ⎝⎣ C ⎦⎠ ∀s ∈ C+ . C L

(12)

⎛⎡

(13)

It can be shown that Conditions (12) and (13) are less conservative than conventional observability/detectability conditions. A beautiful aspect of functional observers is their potential flexibility, such that even if Conditions (12) and (13) are not realized for a system, there is still an opportunity to design a functional observer to estimate z(t) = L x(t). This can be achieved via augmenting extra functions to z(t) to generate a new functional z a (t) = L a x(t), where L a = [L; R1 ; R2 ]. Functional vector R1 x(t) is added to fulfill Condition (12) and R2 x(t) is added to satisfy (13) in a way that (12) is not violated. Prior constructing additional row vectors R1 and R2 , we should ensure that there exist an stable functional observer for System (1). The answer lies in the concept of functional observability/detectability defined in theorem below.

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Theorem 2 ([18]) The triple (A, C, L) is functional detectable if and only if ⎛⎡

⎤⎞   s In − A s In − A , ∀s ∈ C+ rank ⎝⎣ C ⎦⎠ = rank C L

(14)

To add, System (1) is functional observable if and only if (14) holds for all s ∈ C. Equivalently, the triple (A, C, L) is functional observable if and only if ⎛⎡

C CA .. .

⎤⎞

⎥⎟ ⎜⎢ ⎥⎟ ⎜⎢ ⎛⎡ ⎤⎞ ⎥⎟ ⎜⎢ C ⎥⎟ ⎜⎢ ⎟ ⎥ ⎜⎢ ⎜⎢ C A ⎥⎟ ⎜⎢ C An−1 ⎥⎟ ⎢ ⎥⎟ ⎢ ⎥⎟ = rank ⎜ rank ⎜ ⎜⎢ .. ⎥⎟ ⎜⎢ L ⎥⎟ ⎝⎣ . ⎦⎠ ⎥⎟ ⎜⎢ ⎜⎢ L A ⎥⎟ C An−1 ⎥⎟ ⎜⎢ ⎜⎢ . ⎥⎟ ⎝⎣ .. ⎦⎠ L An−1

(15)

Remark 1 Condition (14) implies that the complex Rosenbrock matrices on both sides of the equation have the same invariant zeros. This condition can be readily testified using numerical matrix libraries such as Python libraries or MATLAB.

2.3 Constructing Minimal Order Functional Observers Establishing the row vectors R1 and R2 such that minimum possible order FO (minimal FO) can be designed is not straightforward. A powerful recursive algorithm called MOI-FO (minimum order increase for the functional observer) algorithm is presented below to achieve this aim (refer to [6] for further insights). MOI-FO Algorithm: 1. If L 0 = L fulfills (12), define L β = L 0 , and proceed to Step 3. Otherwise, set i = 0 proceed to the next step. H1i [[R([C; C A; L i ])]], 2. Define Πi  [[R(Θi )⊥ ]]Φi H2i , where       j j i i , and Θi Γi  H2  [[R([H1 ; L i A])]], Φi Ψi  N H2 A; H2    j j−1 . For j = {1, . . . , rows(Πi )} define L i = [L i ; q j ] with N Φi H2i ; H1i j L i1 = [L i ; q1 ], where q j is the jth row of Πi . If (12) is complied by L i , then j L β = L i and proceed to Step 4. Otherwise, apply j ← j + 1 and repeat this step. If augmenting the whole Πi does not satisfy condition (12), then move to the next step.

Partial State Observers for Linear Time-Delay Systems

9

 ⊥   , define L i+1 = [L i ; Πi ; Λi ], set 3. Establish Λi  R(H2i ) ∩ R H1i ; Πi i ← i + 1, and return to Step 2. 4. If (13) is satisfied by L β , the algorithm is terminated successfully by L a = L β .      , where Ξ has p columns. Otherwise, define Ξ Δ  N C A2 ; H1β j j−1 Furthermore, define Π  R(Ξ C A), and L β = [L β ; q j ] for j = {1, . . . , j rows(Π )}, where L 1β = [L β ; q1 ] and q j is the jth row of Π . If L β satisfies (13), j then L a ← L β , and the algorithm is terminated. Otherwise, the step is repeated by j ← j + 1. Example 1 Consider the following system ⎡ ⎤ ⎤ 1 −5 −2 5 0 1 ⎢ −3 ⎥ ⎢ 2 −6 1 0 −3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ A = ⎢ 0 −2 −8 0 0 ⎥ , B = ⎢ ⎢ −5 ⎥ , ⎣ 5 ⎦ ⎣ −6 5 7 −5 −5 ⎦ −2 −2 0 −4 0 0      01 00100 C= , 02×3 , L = . 11 00010 ⎡

The pair (A, C) is not observable, nor detectable. Moreover, the triple (A, C, L) is not functional observable, however, it is functional detectable according to Condition (14) in Theorem 2. It can be realized that L meets Condition (12), whereas Condition (13) is not satisfied indicating that the system is not functional detectable under this setting.  Therefore, following the MOI-FO algorithm a row vector q = −3 −8 6 0 −2 is obtained to be augmented to L. Hence, L a = [L; q] meets both Conditions (12)  and (13). The invariant zeros of the matrix pencil S = s In − A; C; L is λ = −5, which is the undetectable eigenvalue of the observer and should be incorporated as desired FO’s eigenvalue. Assigning the FO’s poles at {−10.2648, −5, −9} ({−10.2648, −9} are arbitrary and adjust the performance of the observer) the observer parameters were calculated based on the presented method in Sect. 2.1 and another transformation-based framework proposed in [19]. Thereafter, the performance of the designed FOs were examined through numerical simulation of the system excited with an arbitrary input u(t) = 2 + 10e−0.4t cos(2t), (t ≥ 0), as shown in Fig. 1. The results indicate that the presented observer design algorithm is performing strong in estimating functions [z 1 ; z 2 ; z 3 ] = L a x(t), while the solution algorithm proposed in [19] fails to estimate the functions with a desirable accuracy. The reason for such failure lies in the numerical issues confronted with the majority of solution algorithms for the nonlinear Sylvester equation.

20

20

15

15

Observer errors using

10 5 0 -5 -10 -15 -20

0

0.5

1

1.5

2

2.5

the methodology suggested in [19]

Observer errors using

R. Mohajerpoor

the presented solution scheme

10

3

10 5 0 -5 -10 -15 -20 0

0.5

1

1.5

2

2.5

3

Time [s]

Time [s]

Fig. 1 The errors due to estimating functions z(t) = L a x(t) using: (i) presented solution in Sect. 2.1 (left figure) and (ii) the solution proposed in [19] (right figure)

3 Delay-Free FO Design for Systems with Unknown Input Delay Time delay can appear in the states (internal delay), the input (actuator lag), or even the output (measurement lag) of a system [20]. This section focuses on LTI systems with input delay, and the delay terms are treated as unknown-inputs and a delay-free FO structure with minimal order is designed for the system. Consider an LTI system with time-varying input delay described below: x(t) ˙ = Ax(t) + Bu(t) + Bd u d (t − du (t)), y(t) = C x(t), z(t) = L x(t),

(16)

where A, B, C, and L are as defined in (1), Bd ∈ Rn×m d , and du (·) is the time-varying input delay. We design an observer with structure (2) for the system such that the error e(t) = zˆ (t) − z(t) asymptotically converges to zero, irrespective of the delay term Bd u d (t − du (t)). Indeed, the observer parameters are designed to decouple the delay term from the error dynamics. This aim can be achieved if conditions given in Proposition 1 below are assured. Proposition 1 The lth order FO (2) is an asymptotically stable FO for system (16) if and only if conditions I and II of Theorem 1 are satisfied, and in addition parameter T satisfies the following equality constraint: T Bd = 0.

(17)

Partial State Observers for Linear Time-Delay Systems

11

Additional condition (17) is added to the conditions of Theorem 2 to decouple the observer dynamics from the input delay terms. Similar to the FO design procedure, solving (3) and (17) such that matrix F is stable is obscured and faces numerical difficulties. Therefore, a solution algorithm adopted from [21] is suggested in the next section that is less conservative than a number of other algorithms. This class of observers are categorized as unknown-input functional observers (UIFOs) in the literature [22–24].

3.1 A Solution Algorithm for UIFOs   ˜  Let us first define a non-singular transformation matrix U  C † C ⊥ , and x(t) U −1 x(t). Thus, system (16) is transformed into ˙˜ = A˜ x(t) ˜ x(t) ˜ + Bu(t) + B˜ d u d (t − du (t)), y(t) = C˜ x(t), z(t) = L˜ x(t),

(18)

where A˜  U −1 AU , B˜  U −1 B, B˜ d  U −1 Bd , C˜  CU , and L˜  LU . As a result of the system transformation, conditions (3) and (17) also undergo the transformation, which results in the following set of equations (here T˜  T U ): F T˜ + H C˜ − T˜ A˜ = 0,

(19a)

T˜ + V C˜ − L˜ = 0,

(19b)

T˜ B˜ d = 0,

(19c)

˜ G = T˜ B.

(19d)

Combining (19b) and (19b), one obtains: T˜ = L˜ m M˜ 1 + Z u M¯˜ 1 ,

(20)

V = L˜ m M˜ 2 + Z u M¯˜ 2 ,

(21)

      ˜ 0 p×m d , L˜ m  L, ˜ 0l×m d , where M˜  In , B˜ d ; C, M˜ 1 M˜ 2  M˜ † , and   † (n+m d )×n ˜ , M2 ∈ R(n+m d )× p , M¯˜ 1 ∈ R(n+ p)×n , M¯˜ 1 M¯˜ 2  In+ p − M˜ M˜ , with M˜ 1 ∈ R M¯˜ ∈ R(n+ p)× p , and Z ∈ Rl×(n+ p) is an arbitrary matrix to be designed. 2

u

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R. Mohajerpoor

  Now, post-multiplying Sylvester equation (19a) by C˜ † C˜ ⊥ , we get H = −F T˜ C˜ † + T˜ A˜ C˜ † .

(22)

F T˜ C˜ ⊥ = T˜ A˜ C˜ ⊥ .

(23)

Putting T˜ from (20) into (23) yields 

 F, −Z u Ωu = Φu ,

(24)

  where Ωu  L˜ m M˜ 1 C˜ ⊥ ; M¯˜ 1 A˜ C˜ ⊥ , and Φu  L˜ m M˜ 1 A˜ C˜ ⊥ . Equation (24) is similar to (9) when designing ordinary FOs. Assuming that (24) can be solved, the following can be obtained F = N˜ 11 + Z˜ u N˜ 21 ,

(25)

Z u = − N˜ 12 − Z˜ u N˜ 22 ,

(26)

    where N˜ 1  Φu Ωu† , N˜ 2  (In+ p+l − Ωu Ωu† ), N˜ 11 N˜ 12  N˜ 1 , and N˜ 21 N˜ 22  N˜ 2 , with N˜ 11 ∈ Rl×l , N˜ 12 ∈ Rl×(n+ p) , N˜ 21 ∈ R(n+ p+l)×l , and N˜ 22 ∈ R(n+ p)×(n+ p+l) . Moreover, Z˜ u ∈ Rl×(n+ p+l) is a free matrix that should be designed in a way that matrix F is Hurwitz. This is doable if and only if the pair ( N˜ 11 , N˜ 21 ) is observable or detectable.

3.2 Existence of a Solution for UIFOs The following theorem explains easy to check conditions for the existence of an asymptotically stable and delay-free FO (2) for System (16). Theorem 3 ([22]) There exists an asymptotically stable lth order and proper FO with structure (2) for System (16) if and only if the below conditions are satisfied: ⎛⎡

⎤⎞ ⎛⎡ ⎤⎞ L A L Bd C A C Bd ⎜⎢ C A C Bd ⎥⎟ ⎢ ⎥⎟ ⎝⎣ C 0 ⎦⎠ . rank ⎜ ⎝⎣ C 0 ⎦⎠ = rank L 0 L 0

(27)

⎛⎡

⎤⎞ ⎛⎡ ⎤⎞ s L − L A L Bd C A C Bd rank ⎝⎣ C A C Bd ⎦⎠ = rank ⎝⎣ C 0 ⎦⎠ ∀s ∈ C+ . C 0 L 0

(28)

Partial State Observers for Linear Time-Delay Systems

13

It can be shown that condition (27) ensures that there is a solution for (24), and condition (28) guarantees the detectability of the pair ( N˜ 11 , N˜ 21 ). It can be shown that conditions of Theorem 3 can be relaxed by adding extra auxiliary rows to matrix L, if one or both of conditions (27) and (28) are not satisfied, provided that the quadruple (A, C, L , Bd ) is unknown-input functional observable or unknown-input functional detectable. Theorem 4 ([25]) The quadruple (A, C, L , Bd ) is unknown-input functional detectable if and only if the following conditions are attained, ⎛⎡

⎤⎞   s In − A Bd s In − A Bd ⎝ ⎣ ⎦ ⎠ C 0 , ∀s ∈ C+ rank = rank C 0 L 0  rank(C Bd ) = rank

C Bd L Bd

(29)

 (30)

The quadruple is unknown-input functional observable if and only if condition (30) is realized, and condition (29) holds for all s ∈ C. Remark 2 Condition (30) is called the matching condition for a proper UIFO. It is dictated by this condition that R(L Bd ) must lie in the row space spanned by C Bd . However, this condition is still less conservative than the matching condition for designing unknown-input observers, where it is essential for the number of perturbations to be less than or equal to the number of outputs implied by rank(C Bd ) = rank(Bd ) [21]. By the way, the matching condition can be further relaxed, if the derivatives of the outputs are employed in the observer structure, resulting in improper observers [26]. In addition, condition (29) insinuates that L must not generate additional invariant zeros in the right half complex plane than what C imposes on the system’s matrix. A systematic and recursive algorithm to find minimum number of additional row vectors to be appended to L to satisfy conditions (27) and (28) is presented below [21]. The algorithm is called MOI-UIFO (minimum order increase for unknowninput functional observer), and it essentially builds minimum number of auxiliary functions R1 x(t) and R2 x(t), such that L a = [L; R1 ; R2 ] fulfills conditions (27) and (28). This enables the design of a minimal order UIFO for the delayed system (16). R1 literally is added to assure (27), and R2 is added to satisfy (28) without breaching (27). Unknown-input functional detectability guarantees the existence or nonexistence of the row vectors R1 and R2 . Derivation of the algorithm may look intricate, though it follows the same philosophy of the MOI-FO algorithm. MOI-UIFO Algorithm: 1. If L satisfies (27), define L β = L, β = 0, and proceed to Step 3. Otherwise, set i = 0, L 0 = L, and proceed to the next step.

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  2. Define Πid  Wi R(Θid )⊥ Φ¯ id Φid H¯ 2d,i , where       d,i ; H1d,i  R C A, C Bd ; C, 0; L i , 0 ; H2  R H1d,i ; L i A, L i Bd      d   ; Φi , Ψid  N H¯ 2d,i A, Bd ; H2d,i  d      d,i  d,i d d ¯ ¯ ¯ ; Φi , Ψi  N Φi H2 In 0n×m d ; H2     d d   d d,i d,i ¯ ¯ Θi Γi  N ; κ  rows(Φid ); and Φi Φi H2 , 0κ×m d ; H1        Wi Gi  N . Moreover, H¯ 1d,i and R(Θid )⊥ Φ¯ id Φid H¯ 2d,i Bd ; C Bd d,i d,i d,i H¯ 2 are any n independent columns of H1 and H2 , respectively. Define j j−1 L i = [L i ; q j ] for j = {1, . . . , rows(Πid )}, with L i1 = [L i ; q1 ], where q j is the j j jth row of Πid . If (27) is met by L i , then L β = L i , β = i, and proceed to Step 4. Contrarily, if (27) is not realized even by augmenting the whole Πid to L i , then the next step should be carried forward.  ⊥   d ¯ and 3. Define Λi  R( H2d,i ) ∩ R H¯ 1i ; Πid     d  ¯ i Λd ], ¯ i G¯i  N Λi Bd ; C Bd . Then, set L i+1 ← [L i ; Πid ; W W i i ← i + 1, and return to Step 2. 4. If the detectability condition (28) is fulfilled by L = L β , then L a = L β and   d ˜ C A), where Ξ d Δd   R( WΞ the algorithm stops. Otherwise, define Π           ˜ G˜  N Ξ d C ABd ; C Bd , and W . N C A A Bd ; H d,β 1

j

j−1

Now, for j = {1, . . . , rows(Π d )} define L β = [L β ; q j ], with L 1β = [L β ; q1 ]. j j If (28) is satisfied by L = L β , then we have L a = L β . Otherwise, this step is repeated after an increment in j ← j + 1. Example 2 Consider a system with the following parameters: ⎡

⎤ −2.51 0.33 0.68 1.12 −0.25 ⎢ 0.14 −0.23 −0.31 0.91 0.36 ⎥ ⎢ ⎥ ⎥ A=⎢ ⎢ 0.51 −1.18 0.41 0.63 −0.77 ⎥ , ⎣ 0.22 0.33 0.46 0.65 −0.77 ⎦ 0.23 0.33 3.97 0.06 0.69 ⎡ ⎡ ⎤ ⎤ 0.43 1 0 ⎢ 0 ⎥ ⎢ −3 −1 ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ , Bd = ⎢ 0 0.5 ⎥ , 0.92 B=⎢ ⎢ ⎢ ⎥ ⎥ ⎣ 1.2 ⎦ ⎣ 0.45 0 ⎦ 0 0 −1.27     1 0 0 0 0.6 C= , L = 2 0 0 9 0.3 . 0001 0 After checking conditions (29) and (30), it is acknowledged that the quadruple (A, C, L , Bd ) is unknown-input functional detectable. Nevertheless,  the system is not unknown-input functional observable, as the system’s matrix s In − A, Bd ; C, 0 , has two invariant zeros at −2.1622 ± 2.0253i,   while the augmented system’s matrix s In − A, Bd ; C; L , 0 has no invariant

Partial State Observers for Linear Time-Delay Systems

15

zeros. The two invariant zeros correspond to the stable un-observable modes of the system that should be included in the set of desired observer’s eigenvalues. In addition, it can be simply found that rank(C Bd ) = rank(Bd ). Conclusively, there is no proper unknown-input observer for the system. Condition (27) is not satisfied by L, thus, the order of observer is increased by applying the MOI-UIFO algorithm that results in an additional row vector   q1 = 0.0603 0.1620 0.3239 −0.0439 −0.01 to be appended to L. In addition, Condition (28) is not met  by L β = [L; q1 ].  Accordingly, q2 = −0.0045 0.1041 0.2082 0.2082 −0.2050 is obtained from Step 4 of the MOI-UIFO algorithm, such that L a = [L; q1 ; q2 ] fulfills both conditions of Theorem 4. Adhering to the framework outlined in Sect. 3.1 appropriate observer parameters are south to assign the poles of F at (−2.1622 ± 2.0253i, −5). However, due to a near-singular numerical composition involved in this particular example and numerous matrix operations such as inverse and multiplications that propagate numerical errors, the algorithm proposed in [22] fails to address appropriate parameters. This highlights the numerical superiority of the proposed algorithm in this chapter over comparable frameworks.

4 Delay-Dependent FO Design for Systems with Measurable State Delay When a number of the states of system (1) are lagged, instead of the actuator inputs, the dynamics of the system is written as x(t) ˙ = Ax(t) + Ad x(t − d(t)) + Bu(t), y(t) = C x(t), z(t) = L x(t), x(θ ) = φ(θ ) ∀θ ∈ [−d2 , 0],

(31)

where Ad ∈ Rn×n , φ(·) ∈ Cn ([−d2 , 0]) is the initial function, and A, B, C, and L are as defined in (1). The delay is assumed to be time-varying, bounded and with arbitrary upper-bound on its derivative [27]: 0 < d1 ≤ d(t) ≤ d2 ,

˙ ≤ μ, d(t)

(32)

where d2 > d1 ≥ 0, and μ ≥ 0 is the upper-bound of the rate of change of the delay. Let us denote d12  d2 − d1 . The aim of the FO design algorithm is to achieve the asymptotic stability in the sense formulated below: lim

sup |e(t)| = 0

t→∞ −d ≤θ≤0 2

∀θ ∈ [−d2 , 0].

(33)

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The following lth order delay-dependent proper functional observer structure is appraised that incorporates the real-time delay value: ω(t) ˙ = Fω(t) + Fd ω(t − d(t)) + Gu(t) + H y(t) + Hd y(t − d(t)) zˆ (t) = ω(t) + V y(t) ω(θ ) = 0 ∀θ ∈ [−d2 , 0],

(34)

where Fd ∈ Rl×l , G d ∈ Rl×m , and Hd ∈ Rl× p . A number of lemmas that are employed in deriving the stability criterion for the observer are given below. Lemma 1 (Wirtinger-based single integral inequality [28]) Given Y  0 ∈ Sn×n and any function f : [a, b] → Rn , if f (·) can be integrated on [a, b], the following inequality holds: 

b

2 χT f (s)Y f (s)ds ≥ b−a 1



T

a

2Y −3 Y b−a

−3 Y b−a 6 Y 2 (b−a)

 χ1 ,

(35)

    where χ1T  ab f T (s)ds, ab ub f T (s)dsdu . Jensen’s inequality that is a more conservative estimation of the upper-limit of the integral can be written as 

b

1 f (s)Y f (s)ds ≥ b−a



T

a

b





f (s)ds Y T

a

b

 f (s)ds .

a

Lemma 2 ([29]) The following inequalities hold for a differentiable function f : [a, b] → Rn , given real scalars a, b, and c, such that a < c and b − a > b − c, and Y  0 ∈ Sn×n ,    c b 2ζ3 (a,b) Y − Y (a, b) 6ζ 4 ζ4 (a,b) χ2T f T (s)Y f (s)dsdu ≥ (36) χ2 , (a,b) (a,b) − 2ζζ43(a,b) Y 6ζζ42(a,b) Y ζ˜ (a, b) a u  c a

b

u

6ζ4 (a, b) T χ3 f (s)Y f˙(s)dsdu ≥ ζ˜ (a, b) ˙T



(a,b) − 2ζζ43(a,b) Y (a,b) (a,b) − 2ζζ43(a,b) Y 6ζζ42(a,b) Y

Y

 χ3 ,

(37)

where    cbb c b T f (s)dsdu, a u u 1 f T (s)dsdu 1 du , a u    cb T T , χ3T  (c − a) f T (b) − ac f T (s)ds, ζ2 (a,b) f (b) − f (s)dsdu a u 2

χ2T 

ζi (a, b)  (b − a)i − (b − c)i , and ζ˜ (a, b)  3ζ2 (a, b)ζ4 (a, b) − 2ζ32 (a, b). Lemma 3 (Reciprocally convex combination lemma [30]) Given f 1 , f 2 , . . . , f N are functions mapping from Rn to R, and positive in an open subset D of Rn , then

Partial State Observers for Linear Time-Delay Systems

17

the reciprocally convex combination of the functions over D fulfills the following equation  1  min f i (t) = f i (t) + max gi, j (t),  gi, j (t) {αi |αi >0, i αi =1} αi i i i= j subject to:

 gi, j (t) : Rn → R,

 ! f i (t) gi, j (t) 0 . gi, j (t) f j (t)

Lemma 4 ([29]) Suppose G : [a, b] → Rn×n is a convex function. Then, G(t) is negative for all t ∈ [a, b], if and only if G(a) ≺ 0 and G(b) ≺ 0. Let us define a supplementary error vector (t)  ω(t) − T x(t). The following theorem discusses the stability conditions of the observer error dynamics. Theorem 5 ([31]) Functional observer (34) for System (31) is globally asymptotically stable, if and only if i. the auxiliary error dynamics comprehended below is asymptotically stable ˙ (t) = F(t) + Fd (t − d(t)) (θ ) = −T φ(θ ) ∀θ ∈ [−d2 , 0]

(38)

ii. there exist a matrix T that satisfies the following constrained Sylvester equations F T − T A + H C = 0,

(39a)

Fd T − T Ad + Hd C = 0,

(39b)

T + V C − L = 0,

(39c)

G = T B.

(39d)

The delay-dependent observer design algorithm thus confronts with two steps: (i) finding a sufficient condition on the asymptotic stability of the delayed (and therefore infinite-dimensional) dynamics (38) (studied in Sect. 4.1), and (ii) solving the set of interconnected equations (39) in a way to comply with the stability condition established in the former step (addressed in Sect. 4.2). Remark 3 It is possible to design a delay-free observer for system (31) in a similar way that we designed the observer for the system with input delays, (16), as demonstrated in Sect. 3, by treating the delay term as an unknown-input. This study has been covered in the literature to some extent (see e.g. [7]). Moreover, LTI systems with multiple state and input delays can be dealt with in a similar way as comprehended in this section (see [10, 11]).

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4.1 Developing a Stability Criterion Establishing delay-dependent sufficient conditions for stability of delay differential equation (38) has been broadly studied in the literature. The following theorem summarizes one of the latest stability criterion for the system. A sketch of the proof is also given, though a more comprehensive proof can be found in [31]. Theorem 6 Observer error dynamics (38) is asymptotically stable for any delay function that complies with (32), if there exist l × l symmetric matrices Pi  0, i = {1, 4, 6}, R j  0, j = {1, . . . , 4}, Q m  0, m = {1, 2, 3}, and Sk  0, k = {1, 2}, and any l × l matrices Pi , i = {2, 3, 5} and U j , j = {1, . . . , 4} such that the following matrix inequalities are satisfied Ψi ≺ 0 i = {1, 2},

(40)

where Ψ1  Ψ (d(t))|d(t)=d1 , Ψ2  Ψ (d(t))|d(t)=d2 , and ⎡

Γ (t) Γ9 Γ10 (t) Γ11 (t) ζ22 (d1 ,d2 ) ⎢ ∗ −72 S 0 0 ⎢ ζ˜ (d1 ,d2 ) 2 ⎢ Ψ (d(t))  ⎢ ∗ 0 ∗ −12R2 ⎢ ⎣ ∗ ∗ ∗ −12R2 ∗ ∗ ∗ ∗

⎤ Γ12 ⎥ 0 ⎥ ⎥ . 0 ⎥ ⎥ 0 ⎦ Γ˜12

In addition, Γ (t) = [Γi, j ]8×8 consisting non-zero elements as below (d1 ,d2 ) 4 Γ1,1  sym(P3 ) + Q 1 + d12 R1 + d12 R2 − 6d12 S1 − 12 ζζ˜2(d η S − ,d ) 11 2

4 d1

R3 T + sym(U1 F1 ), Γ2,1  FdT U1 , Γ2,2  −(1 − μ)Q 3 − 4(d2 − d(t))2 R4 − 4(d(t) − d1 )2 R4 , Γ3,1  P2T − P3T − d21 R3 , Γ3,2  −2(d(t) − d1 )2 R4 , Γ3,3  Q 2 − Q 1 + Q 3 − d41 R3 − 4(d(t) − d1 )2 R4 , Γ4,1  −P2T , Γ4,2  −2(d2 − d(t))2 R4 , Γ4,4  −Q 2 − 4(d(t) − d1 )2 R4 , Γ5,1  P6 + d62 R3 + U3T F1 , Γ5,2  U3T Fd , Γ5,3  −P6 + d62 R3 + P5T , 1 1 Γ5,4  −P5T , Γ5,5  −4R1 − 12 3 R3 − 12S1 , d1 S, Γ6,1  12S1 , Γ6,5  d61 R1 + 24 d1 1 12 72 2 (d1 ,d2 ) η17 S2 + P5 + U2T F1 , Γ6,6  − d 2 R1 − d 2 S1 , Γ7,1  − 12ζ ζ˜ (d1 ,d2 ) 1 1 Γ7,2  6(d2 − d(t))R4 + U2T Fd , Γ7,3  P4 − P5 , 1

Γ7,4  −P4 + 6(d2 − d(t))R4 , 2 )ζ4 (d1 ,d2 ) S2 , Γ7,7  −12R4 − 4(d2 − d(t))2 R2 − 12 ζ2 (d1ζ,d ˜ (d ,d ) 1

2

2

2 (d1 ,d2 ) η17 S2 + P5 + U2T F1 , Γ8,2  6(d(t) − d1 )R4 + U2T Fd , Π8,1  − 12ζ ζ˜ (d ,d ) 1

2

Γ8,3  6(d(t) − d1 )R4 + P4 − P5 , Γ8,4  −P4 , (d1 ,d2 ) ζ (d , d )S , Γ8,7  −12 ζζ˜2(d ,d ) 4 1 2 2 1

2

Partial State Observers for Linear Time-Delay Systems

19

2 )ζ4 (d1 ,d2 ) Γ8,8  −12 ζ2 (d1ζ,d S2 − 12R4 − 4(d(t) − d1 )2 R2 , ˜ (d1 ,d2 ) T  (d1 ,d2 ) Γ9  12 ζζ˜2(d η S , 0, 0, 0, 0, 0, 24 ζ2 (d1ζ˜,d(d2 )ζ,d3 (d)1 ,d2 ) S2 , 24 ζ2 (d1ζ˜,d(d2 )ζ,d3 (d)1 ,d2 ) S2 , ,d ) 19 2 1

2

1

2

1

2

Γ10 (t)  [0, 0, 0, 0, 0, 0, 6(d2 − d(t))R2 , 0]T , Γ11 (t)  [0, 0, 0, 0, 0, 0, 0, 6(d2 − d(t))R2 ]T , T  Γ12  P1 − U1 + U4T F1 , U4T Fd , 0, 0, P3 − U3 , 0, P2 − U2 , P2 − U2 , and 4 R4 + d14 S1 + ζ22 (d1 , d2 )S2 − sym(U4 ). Γ˜12  d1 R3 + d12 Furthermore, 2 ζ4 (d1 , d2 ) − 2d12 ζ2 (d1 , d2 )ζ3 (d1 , d2 ) + 1.5ζ23 (d1 , d2 ), η11  d12 η17  ζ2 (d1 , d2 )ζ3 (d1 , d2 ) − d12 ζ4 (d1 , d2 ), and η19  3ζ22 (d1 , d2 ) − 2d12 ζ3 (d1 , d2 ). Proof Consider the following Lyapunov–Krasovskii functional candidate 4 

V (t, d(t)) =

Vi (t),

(41)

i=1

where ˜ V1 (t, d(t)) = ˜ T (t)E P (t),  t  T  (s)Q 1 (s)ds + V2 (t, d(t)) = 

t−d1

+ 

t−d(t) 0  t

−d1 3 + d12 3 + d12

 V4 (t, d(t)) = 2d12 + with

 T (s)Q 2 (s)ds

t−d2

t−d1

V3 (t, d(t)) = h 1

t−d1

 T (s)Q 3 (s)ds,   T (s)R1 (s)dsds1 +



t+s1 −d1



−d2  −d1

t+s1  t

t

2(d22

−



−d1

t

˙ T (s)R3 ˙ (s)dsds1

t+s1

 T (s)R2 (s)dsds1

t+s1 −d2 0  0 t

−d1

0

˙ T (s)R4 ˙ (s)dsds1 ,

˙ T (s)S1 ˙ (s)dsds2 ds1

s1

d12 )

t+s2  −d1 −d2



0 s1



t

t+s2

˙ T (s)S2 ˙ (s)dsds2 ds1 ,

20

R. Mohajerpoor

 T  t−d1  t (t) ˜ =  T (t),  T (s)ds,  T (s)ds, ˙ T (t) , t−d2 t−d1 ⎤ ⎡ P1 P2 P3 0 ⎢ ∗ P4 P5 0 ⎥ ⎥ P⎢ ⎣ ∗ ∗ P6 0 ⎦ , U1 U2 U3 U4   E  I3l , 0; 0, 0 . Moreover, let us define the following variables with omitted arguments (t, d(t)): z 1  x(t), z 2  x(t − d(t)), − d2 ),  0 z 30  x(t − d1 ), z 4  x(t  −d(t) t z 5  t−d1 x(s)ds, z 6  −d1 s x(t + u)duds, z 7  −d2 x(t + u)du,  −d1  −d  0  −d(t)  −d(t) z 8  −d(t) x(t + u)du, z 9  −d21 s x(t + u)duds, z 10  −d2 s x(t+u)  −d1  −d1 3 ˙ and δ(t)  (d2 − d(t)) + (d(t) duds, z 11  −d(t) s x(t + u)duds, z 12  x(t), − d1 )3 .  t−d(t)  t−d Differentiating (41) along the solution of (38), noting that t−d21 f (s)ds = t−d2  t−d1 f (s)ds + t−d(t) f (s)ds holds for any integrable function f : [t − d2 , t − d1 ] → R, 3 using Lemmas 1, 2 and 3, and the fact that δ(t) ≤ d12 , results in the sequel expressions ˜ (arguments of ζi (·, ·), i = {2, 3, 4}, and ζ (·, ·) are removed for simplicity): ⎤T z1 ⎜⎢ z 7 + z 8 ⎥ ⎥ ⎢ V˙1 (t, d(t)) = sym ⎜ ⎝⎣ z 5 ⎦ z 12 ⎛⎡

⎡

P1 ⎢ ∗ ⎢ ⎣ ∗ U1

P2 P4 ∗ U2

P3 P5 P6 U3

⎤T 0 0 ⎥ ⎥ 0 ⎦ U4

⎡

⎤⎞ z 12 ⎢ ⎥⎟ z3 − z4 ⎢ ⎥⎟ , ⎣ ⎦⎠ z1 − z3 F z 1 + Fd z 2 − z 12 (42)

V˙2 (t, d(t)) ≤ z 1T Q 1 z 1 − z 3T Q 1 z 3 + z 3T Q 2 z 3 − z 4T Q 2 z 4 + z 3T Q 3 z 3 − (1 − μ)z 2T Q 3 z 2 ,

(43)

T 4 T 4 T R3 z 12 + d12 z 1 R2 z 1 + d12 z 12 R4 z 12 V˙3 (t, d(t)) ≤ d12 z 1T R1 z 1 + d1 z 12     T  −3 2 d1 z z −2 5 ⊗ R1 5 − 2˜z 1T Υ (t) ⊗ R2 z˜ 1 −3 6 z6 z 6 d1 d12   T    −3 2 d1 2 z1 − z3 z1 − z3 − ⊗ R −3 6 3 d1 z 1 − z 5 d1 d1 z 1 − z 5 d1 d 2

(44)

1

V˙4 (t, d(t))

− 2˜z 2T Υ (t) ⊗ R4 z˜ 2 , T T ≤ d14 z 12 S1 z 12 + ζ22 z 12 S2 z 12 

− 12

d1 z 1 − z 5 d12 z − z6 2 1

T 

1 −2 d1 −2 6 d1 d12



 ⊗ S1

d1 z 1 − z 5 d12 z − z6 2 1



(45)

Partial State Observers for Linear Time-Delay Systems

 T   ζ2 d12 z 1 − z 7 − z 8 ζ4 −2ζ3 ⊗ S2 − 12 ζ2 −2ζ3 6ζ2 z − z9 ζ˜ 2 1   d z − z7 − z8 , × 12 ζ21 z − z9 2 1

21

(46)

where z˜ 1  [z 7 ; z 10 ; z 8 ; z 11 ], z˜ 2  [(z 2 − z 4 ); ((d2 − d(t))z 2 − z 7 ); (z 3 − z 2 ); ((d(t) − d1 )z 3 − z 8 )], Θ(a, b)  2(a − b)2 , −3(a − b); −3(a − b), 6 ,   and Υ (t)  Θ(d2 , d(t)), 0; 0, Θ(d(t), d1 ) . Accordingly, it is observed from (46)–(42) that V˙ (t, d(t)) ≤ z T Ψ (d(t))z,

(47)

  T . where z T  z 1T , z 2T , . . . , z 12 Since Ψ (·) is constructed by weighted sum and vector composition of convex functions d(t) − d1 , d2 − d(t), and (d(t) − di )2 , i = {1, 2}, it is a convex function of d(t) [32]. Therefore, according to the convex combination technique (Lemma 4), criterion (47) is satisfied if and only if matrix inequalities (40) are achievable. The proof thus follows from applying the Lyapunov–Krasovskii stability theorem [33]. 

4.2 Observer Design Scheme The observer design procedure can be called an extension of the framework outlined in Sect. 2.1. The aim is to solve equations (39), while adhering to the stability criterion developed in Theorem 6. Retaining (4) to (8), it is followed from the same process that Fd T C ⊥ = T Ad C ⊥ .

(48)

Fd L M1 C ⊥ + Fd Z M¯ 1 C ⊥ = L M1 Ad C ⊥ + Z M¯ 1 Ad C ⊥ .

(49)

Conclusively, we have



 F, Fd , −Z Ωd = Φd ,

(50)

  where Ωd  L M1 C ⊥ , 0; 0, L M1 C ⊥ ; M¯ 1 AC ⊥ , M¯ 1 Ad C ⊥ ∈ R(n+ p+2l)×2(n− p) and   Φd  L M1 AC ⊥ , L M1 Ad C ⊥ ∈ Rl×2(n− p) . If there is a solution for (50), one obtains F = Nd,11 + Z˜ Nd,21 ,

(51)

22

R. Mohajerpoor

Fd = Nd,12 + Z˜ Nd,22 ,

(52)

Z = −Nd,13 − Z˜ Nd,23 ,

(53)

    where Nd,11 Nd,12 Nd,13  Φd Ωd† , Nd,21 Nd,22 Nd,23  (In+ p+2l − Ωd Ωd† ) ∈ R(n+ p+2l)×(n+ p+2l) , Nd,11 ∈ Rl×l , Nd,12 ∈ Rl×l , Nd,13 ∈ Rl×(n+ p) , Nd,21 ∈ R(n+ p+2l)×l , Nd,22 ∈ R(n+ p+2l)×l , Nd,23 ∈ R(n+ p+2l)×(n+ p) , and Z˜ ∈ Rl×(n+ p+2l) is a free matrix. The necessary and sufficient condition for the existence of a solution for (50) can be expressed as [31] ⎤⎞ ⎛⎡ ⎤⎞ L M1 A L M1 A d L M1 0 ⎥⎟ ⎜⎢ L M1 0 ⎜⎢ 0 L M1 ⎥⎟ ⎥⎟ ⎜⎢ ⎜⎢ ⎟ ⎜⎢ 0 ⎥⎟ L M1 ⎥ ⎢ M¯ 1 A M¯ 1 Ad ⎥⎟ ⎟ ⎢ ⎥ ⎜ = rank ⎜ rank ⎜⎢ ¯ ⎢ ⎜ ⎟ ⎥ ⎥⎟ ¯ ⎜⎢ M1 A M1 Ad ⎥⎟ ⎝⎣ C 0 ⎦⎠ ⎠ ⎦ ⎝⎣ C 0 0 C 0 C ⎛⎡

(54)

Given that condition (54) is met, the free matrix Z˜ should be found in a way that matrix inequalities (40) are satisfied. The inequalities are nonlinear due to the multiplicative terms UiT Z˜ , i = {1, . . . , 4}, which make it hard to solve the matrix inequalities as the problem is non-convex and thus N P-hard. Various algorithms are proposed in the literature to resolve this prevalent obstacle. A resolution to this dilemma is simplifying the problem by defining Ui = αi U4 , i = {1, 2, 3}, where αi are free scalar parameters. Next, the following corollary provides a guideline to design Z˜ assuming that αi are given. Corollary 1 Given scalars αi , i = {1, 2, 3}, functional observer (34) is a globally asymptotically stable functional state estimator for System (31) with delay function complying with (32), if condition (54) is realized, and there exist symmetric positive definite l × l matrices Pi , i = {1, 4, 6}, Q j , j = {1, 2, 3}, Rk , k = {1, . . . , 4}, and Sm , m = {1, 2}, and any matrices Pi ∈ Rl×l , i = {2, 3, 5}, U4 ∈ Rl×l , and Λ ∈ Rl×(2l+n+ p) , such that the following linear matrix inequalities (LMIs) are feasible Ψ˜ i ≺ 0 i = {1, 2}, (55) where Ψ¯ 1  Ψ¯ (t)|d(t)=d1 , Ψ¯ 2  Ψ˜ (t)|d(t)=d2 , and ⎡

Γ¯ (t) Γ9 Γ10 (t) Γ11 (t) ⎢ ζ2 0 0 ⎢ ∗ −72 ζ˜2 S2 ⎢ Ψ˜ (t) = ⎢ ∗ 0 ∗ −12R2 ⎢ ⎣ ∗ ∗ ∗ −12R2 ∗ ∗ ∗ ∗

⎤ Γ¯12 ⎥ 0 ⎥ ⎥ , 0 ⎥ ⎥ 0 ⎦ Γ˜12

Partial State Observers for Linear Time-Delay Systems

23

wherein Γ9 , Γ10 , Γ11 , and Γ˜12 are as defined in Theorem 6,  Γ¯12 = P1 − α1 U4 + U4T Nd,11 + ΛNd,21 , U4T Nd,12 + ΛNd,22 , 0, 0, P3 − α3 U4 , 0, P2 − α2 U4 , P2 − α2 U4 ]T , and all the components of Γ¯ (t) = [Γ¯i, j ]8×8 are identical to Γ (t) defined in Theorem 6 other than the following elements: 4 R2 − 6d12 S1 − 12 ζζ˜2 η11 S2 − d41 R3 + α1 sym Γ¯1,1 = sym(P3 ) + Q 1 + d12 R1 + d12 T T (U4 Nd,11 + ΛNd,21 ), Γ¯2,1 = α1 Nd,22 ΛT + α1 Nd,12 U4 , Γ¯5,1 = P6 + 62 R3 + α3 d1

2 U4T Nd,11 +α3 ΛNd,21 , Γ¯5,2 =α3 U4T Nd,12 + α3 ΛNd,22 , Γ¯7,1 = − 12ζ η17 S2 + P5 + ζ˜ T T α2 U4 Nd,11 + α2 ΛNd,21 , Γ¯7,2 =6(d2 − d(t))R4 +α2 U4 Nd,12 + α2 ΛNd,22 , Γ¯8,1 2 = − 12ζ η17 S2 + P5 + α2 U4T Nd,11 + α2 ΛNd,21 , and Γ¯8,2 = 6(d(t) − d1 )R4 + α2 ζ˜

U4T Nd,12 + α2 ΛNd,22 . In addition,

Z˜ = U4−T Λ.

(56)

Proof The proof follows from Theorems 5 and 6. Firstly, Condition (ii) of Theorem 5 is met, only upon the fulfillment of rank condition (54). Secondly, by substituting for F and Fd from (51) and (52), setting Ui = αi U4 , i = {1, 2, 3}, and defining Λ  U4−1 Z˜ , it can be shown that inequalities (55) and (40) are equivalent. Therefore, from Theorem 6 Condition (i) of Theorem 5 is also satisfied.  The final dilemma is how to find the scalar weightings αi , i = {1, 2, 3}, that appear in nonlinear forms αi U4 . To resolve this issue an artificial intelligence (AI) based approach introduced in [31] is summarized below that is named GABT (genetic algorithm based tuning) scheme. GABT Algorithm: 1. Consider f = tmin as the fitness function constrained by Ψ¯ i ≺ tmin i = {1, 2},

(57)

where tmin < 0 is the value to be minimized, and αi are the variables to be tuned. Moreover, select arbitrary and feasible bounds for the variables as αi ∈ [αimin , αimax ]. 2. Generate N p arbitrary sets of αi s to be the initial population’s chromosomes, j j j let α¯ j  [α1 , α2 , α3 ], j = {1, . . . , N p }, and define an arbitrarily large integer number N g > 0. 3. Employ the genetic algorithm to efficiently reach a chromosome that complies with LMIs (57). If even after N g generations the GA algorithm was unable to find an appropriate set of values α¯ j , then the algorithm terminates without any solution.

24

R. Mohajerpoor

5 Addressing Uncertainties in State Delays The main shortcoming of the delay-dependent FO structure and design framework comprehended in Sect. 4 is using the actual real-time values of the state delay, which generally can be erroneously estimated in the majority of practical problems. To resolve the sustainability problem of FO (34), an alternative observer structure that employs an auxiliary delay function instead of the actual delay values is demonstrated in this section. A sliding mode robust scheme to design the observer parameters is also illustrated. Let us augment the output vector and the functional z(t) as z a (t) = L a x(t), where L a  [L; C]. Consider the following FO structure ˆ ˆ + Gu(t) + H y(t) + Hd y(t − d(t)) ω(t) ˙ = Fω(t) + Fd ω(t − d(t))       X1 R X2 ˆ  (t − d(t))  (t) + ν(t) − − Fd,22 2 F22 − G  2 −I p

(58)

zˆ a (t) = ω(t) + V y(t) ω(θ ) = 0

∀θ ∈ [−dˆ2 , 0],

where ω(·) ∈ Rl+ p is the states of the observer dynamics; zˆ a (·) is the estimation of z a (·); X 1 ∈ Rl× p , X 2 ∈ Rl× p , R ∈ Rl× p , and G  ∈ R p× p are additional observer parameters; and F and Fd are partitioned as [F11 , F12 ; F21 , F22 ]  F and  Fd,11 , Fd,12 ; Fd,21 , Fd,22  Fd , respectively. Moreover, for a free parameter T ∈ R(l+ p)×n , (·)  ω(·) − T x(·) ∈ Rl+ p , [1 (·); 2 (·)]  (·), 2 (·) ∈ R p , and ν(t) is the norm-bounded uncertainty vector function defined as " ˜ 2 (t) ρ(t) | P ˜ 2 (t)| 2 (t)  = 0 P (59) ν(t) = 0 2 (t) = 0 where P˜ ∈ S p× p is a positive-definite matrix and ρ(t) is a time-dependent upperˆ is the ancillary delay term that complies bound of the perturbation term. To add, d(t) with the below conditions ˙ˆ ≤ μˆ < 1, ˆ ≤ dˆ2 , d(t) 0 ≤ dˆ1 ≤ d(t)

(60)

where dˆ1 , dˆ2 , and μˆ are constant scalars. Note that the delay bounds of the auxiliary delay function can be set at the actual delay bounds d1 , d2 , and μ. However, being capable of adjusting these parameters empowers the FO design algorithm to come up with less conservative stability criterion for the observer error dynamics, and at the same time it mitigates the structural restrictions that face delay-free observers.

Partial State Observers for Linear Time-Delay Systems

25

Fig. 2 The block diagram of the robust closed-loop functional observer-based controlled system. Observer parameters, system parameters, and functional processes are highlighted in gray, yellow, and orange, respectively

Figure 2 depicts the block diagram of a closed-loop control system, wherein the system behaves according to (31) and the observer dynamics complies with (58). The observed functions here could be for instance z(t) = u(t) = K p x(t), where K p is the proportional control gain. The following theorem outlines the necessary and sufficient conditions for the stability of FO (58) for System (31) in the sense of (33), where e(·) = zˆ a (·) − z a (·). Theorem 7 ([14]) Consider System (31). FO (58) is a globally asymptotically stable functional observer for the system in the sense of (33), if and only if i. for any initial function φ(·) ∈ Cn ([−d2 , 0]), the following error dynamics is globally asymptotically stable  t−d(t)  t−d(t) ˙ (t) = F(t) + Fd (t − d(t)) − Hd C t−d(t) x(s)ds ˙ − Fd t−d(t) ω(s)ds ˙ ˆ   ˆ    X2 X R 1 ˆ  (t − d(t))  (t) + ν(t), − − Fd,22 2 F22 − G  2 −I p (θ ) = −T φ(θ ) ∀θ ∈ [−d2 , 0], (61) ii. there exists a matrix T that satisfies constrained equations (39) with L in (39c) replaced by L a .

26

R. Mohajerpoor

Let us partition T , V , and ω(·) as [T1 ; T2 ]  T , [V1 ; V2 ]  V , and [ω1 (·); ω2 (·)]  ω(·). It is obtained from T + V C − L a = 0 that T2 = (I p − V2 )C. Replacing ω(·)   by (·) + T x(·), conducting a system transformation using T  Il , R; 0, I p and dictating X 1 and X 2 to be fixed at X 1 = −F11 R + F12 + RG  − R F21 R,

(62)

X 2 = −Fd,11 R − R Fd,21 R + Fd,12 ,

(63)

in an attempt to decouple the dynamics of 1 (·) and 2 (·), error dynamics (61) can be represented as follows: ˆ ˙¯1 (t) = F¯11 ¯1 (t) + F¯d,11 ¯1 (t − d(t))



− (Hd,1 C + Fd,1 T + R Hd,2 C + R Fd,2 T )

(64) t−d(t) ˆ t−d(t)

x(s)ds, ˙

¯ x(t)) − ν(t), ˙¯2 (t) = (G  − F21 R)¯2 (t) + ε(t, (t),

(65)

where Fd,1  [Fd,11 , Fd,12 ], Fd,2  [Fd,21 , Fd,22 ], ¯ (t) = T (t), [¯1 (·); ¯2 (·)]  ¯ (·), F¯11  F11 + R F21 , F¯d,11  Fd,11 + R Fd,21 , and ε(·, ·, ·) is the perturbation term defined as follows ˆ ˆ − Fd,21 R ¯2 (t − d(t)) ε(t, (t), ¯ x(t)) F21 ¯1 (t) + Fd,21 ¯1 (t − d(t))  t−d(t) − (Hd,2 C + Fd,2 T ) x(s)ds. ˙ ˆ t−d(t)

(66)

  Conclusively, after appropriately partitioning [H1 , H2 ]  H and Hd,1 , Hd,2  Hd , Theorem 7 can be re-written as below. Corollary 2 FO (58) is a globally asymptotically stable functional observer for system (31) if and only if Condition (i) of Theorem 7 is appeased, and a. for any initial function φ(·) ∈ Cn ([−d2 , 0]), the following error dynamics is globally asymptotically stable ˆ ˙¯1 (t) = F¯11 ¯1 (t) + F¯d,11 ¯1 (t − d(t)),

(67a)

¯ x(t)) − ν(t), ˙¯2 (t) = (G  − F21 R)¯2 (t) + ε(t, (t),

(67b)

¯ (θ ) = −T T φ(θ ). b. the following constraint equation holds Hd,1 C + Fd,11 T1 + Fd,12 T2 + R Hd,2 C + R Fd,21 T1 + R Fd,22 T2 = 0.

(68)

Partial State Observers for Linear Time-Delay Systems

27

5.1 Establishing the Stability Criteria It is deduced from (66) that the perturbation term ε(·, ·, ·) is essentially bounded if ˆ are bounded. The the error terms 1 (·) and 2 (·), as well as x(t − d(t)) − x(t − d(t)) dynamic upper-bound of ε(·, ·, ·) is expressed as ε(t, (t), ¯ x(t)) ≤ ρ(t).

(69)

We first prove the finite time stability of the error dynamics ¯2 (t) (67b) in Theorem 8, and then establish sufficient conditions for the asymptotic stability of ¯1 (t) in Theorem 9. Theorem 8 The ancillary error function ¯2 (·) comprehended in (67b) is finite time stable subject to applying the sliding control (59). Proof Consider the following Lyapunov candidate function: Vs (t) = ¯2T (t) P˜ ¯2 (t),

(70)

where P˜ is any symmetric positive definite matrix. Moreover, let us define G˜  ≺ 0 as an arbitrary matrix with negative eigenvalues, and determine G  = G˜  + F21 R. Now, differentiating (70) along (67b), taking the upper-bound of the perturbation term into account according to (69), and substituting from (59) we have    V˙s (t) = 2¯2T (t) P˜ G˜  ¯2 (t) + ε(t, (t), ¯ x(t)) − ρ(t)sign P˜ ¯2 (t) ≤ 2  P˜ 2 ¯2T (t)G˜  ¯2 (t) ≤ −2ρV ¯ s (t),

(71)

where sign(·) is the signum function, and ρ¯ > 0 is a scalar that fits the following criterion ˜  P˜ 2 G˜  ≤ −ρ¯ P. Accordingly, Vs (t) is exponentially stable, and this results in the ideal sliding of ¯2 (t) to the origin [34]. This concludes the finite time convergence of the error vector ¯2 (t).  Theorem 9 The error dynamics (67) with auxiliary delay function complying with (60) is globally asymptotically stable, if there exist l × l symmetric positive definite matrices Pii , Q i , i = {1, 2, 3}, R j , j = {1, . . . , 4}, and arbitrary l × l real matrices P12 , P13 , P23 , and Di , i = {1, . . . , 4} that satisfy the following matrix inequality:

28

R. Mohajerpoor

⎡

⎤ ∗ ∗ ∗ ∗ ∗ ∗ Ω˜ 1,1 T − PT + R ⎢ P12 Ω˜ 2,2 ∗ ∗ ∗ ∗ ∗ ⎥ 3 13 ⎢ ⎥ T ⎢ −P R −Q − R ∗ ∗ ∗ ∗ ⎥ 4 2 4 12 ⎢ ⎥ T ⎢ P33 + D T F¯11 P T − P33 −P23 −R1 ∗ ∗ ∗ ⎥ 3 23 ⎢ ⎥ ≺ 0 (72) ⎢ P23 + D T F¯11 P22 − P23 −P22 0 −R2 ∗ ∗ ⎥ ⎢ ⎥ 2 T D T D T ⎣ F¯d,11 0 0 F¯d,11 ˆ 3 ∗ ⎦ 1 3 F¯d,11 D2 −(1 − μ)Q Ω˜ 7,1 0 0 P13 − D3 P12 − D2 D4T F¯d,11 Ω˜ 7,7 2 where Ω˜ 1,1  sym(P13 ) + Q 1 + dˆ12 R1 + dˆ12 R2 + sym(D1T F¯11 ), Ω˜ 2,2  Q 2 + Q 3 − Q 1 − R3 − R4 , Ω˜ 7,1  D4T F¯11 + P11 − D1 , and Ω˜ 7,7  −sym(D4 ) + dˆ12 R3 + 2 dˆ12 R4 .

Proof The proof follows a similar procedure undertaken for Theorem 6, considering the Lyapunov–Krasovskii functional described below and employing Jensen’s inequality instructed in Lemma 1. For further details the reader can consult with [14]. V (t) =

3 

Vi (t),

(73)

i=1

V1 (t) = ˜1T (t)E P ˜1 (t),  t  V2 (t) = ¯1T (s)Q 1 ¯1 (s)ds + t−dˆ1



+ V3 (t) = dˆ1



t−dˆ1 ˆ t−d(t)

0

−dˆ1

+ dˆ1



where ˜1 (t)= and ⎡ P11 P12 ⎢ ∗ P22 P=⎢ ⎣ ∗ ∗ D1 D2

¯1T (s)Q 2 ¯1 (s)ds

¯1T (s)R1 ¯1 (s)dsds1 + dˆ12

t+s1  0  t

+ dˆ12

t−dˆ2

¯1T (s)Q 3 ¯1 (s)ds,

t

−dˆ1



(74) t−dˆ1

t+s1

−dˆ1



−dˆ2

(75) 

−dˆ1 −dˆ2



t

t+s1

¯1T (s)R2 ¯1 (s)dsds1

¯˙1T (s)R3 ˙¯1 (s)dsds1

t

t+s1

¯˙1T (s)R4 ˙¯1 (s)dsds1 ,

(76)

 T  t−dˆ t   ¯1T (t), t−dˆ 1 ¯1T (s)ds, t−dˆ1 ¯1T (s)ds, ˙¯1T (t) , E  I3l , 0; 0, 0 , P13 P23 P33 D3

⎤

0 0 ⎥ ⎥. 0 ⎦ D4

2



Partial State Observers for Linear Time-Delay Systems

29

5.2 Observer Design Algorithm The observer design policy is similar to the strategy explained in Sect. 2.1. Considering (4)–(5b) with L a instead of L, M¯ 1 C ⊥ = 0, and a system transformation using [C † , C ⊥ ], Sylvester equations (39a) and (39b) can be broken into the following equations: (77a) H1 = T1 AC † − F12 T2 C † − F11 T1 C † . H2 = T2 AC † − F22 T2 C † − F21 T1 C † .

(77b)

Hd,1 = T1 Ad C † − Fd,12 T2 C † − Fd,11 T1 C † .

(78a)

Hd,2 = T2 Ad C † − Fd,22 T2 C † − Fd,21 T1 C † .

(78b)



 F11 F12 Fd,11 Fd,12 Z 1 Ω˜ d = Φ˜ 1,d ,

(79)



 F21 F22 Fd,21 Fd,22 Z 2 Ω˜ d = Φ˜ 2,d ,

(80)

where [Z 1 , Z 2 ]  Z ∈ R(l+ p)×(n+ p) is a free parameter, ⎡

⎤ L M1 C ⊥ 0 ⎢ C M1 C ⊥ ⎥ 0 ⎢ ⎥ ⊥ ⎥ ⎢ ˜ 0 L M1 C ⎥ , Ωd  ⎢ ⎣ 0 C M1 C ⊥ ⎦ ⊥ ¯ − M1 AC − M¯ 1 Ad C ⊥     Φ˜ 1,d  L M1 AC ⊥ L M1 Ad C ⊥ , and Φ˜ 2,d  C M1 AC ⊥ C M1 Ad C ⊥ . In addition, it can be shown that the constraint equation (68) is equivalent to 

   M¯ 1 Ad Z1 R Z2 = −(L + RC)M1 Ad . M¯ 1 Ad

(81)

Furthermore, it can be verified that (80) has at least one nontrivial solution [14], which can be expressed as 

 F21 F22 Fd,21 Fd,22 Z 2 = N˜ d,1,2 + Z˜ 2 N˜ d,2 ,

(82)

where N˜ d,1,2  Φ˜ 2,d Ω˜ d† , N˜ d,2  (I − Ω˜ d Ω˜ d† ), Z˜ 2 ∈ R p× is an arbitrary matrix, and   2(l + p) + n + p. Unlike (80) that always has a solution, Eq. (79) has a solution if and only if the following rank condition is satisfied [14]:

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R. Mohajerpoor

⎛⎡

⎤⎞ ⎛⎡ ⎤⎞ L M1 A L M1 A d L M1 0 ⎟ ⎜⎢ L M 1 ⎥ 0 ⎜⎢ C M1 ⎜⎢ ⎥⎟ ⎥⎟ 0 ⎜⎢ ⎜⎢ C M 1 ⎥⎟ ⎥⎟ 0 ⎜⎢ 0 ⎜⎢ ⎥⎟ ⎥⎟ L M 1 ⎥⎟ ⎢ ⎜ ⎟ ⎜⎢ 0 ⎥ L M1 ⎥⎟ ⎜⎢ 0 ⎢ ⎥⎟ C M = rank rank ⎜ 1 ⎥⎟ . ⎜⎢ ⎟ ⎜⎢ 0 C M1 ⎥ ⎜⎢ − M¯ 1 A − M¯ 1 Ad ⎥⎟ ⎜⎢ ⎥⎟ ⎜⎢ ⎜⎢ − M¯ 1 A − M¯ 1 Ad ⎥⎟ ⎥⎟ ⎜⎢ ⎥⎟ ⎝⎣ C ⎦⎠ 0 ⎠ ⎝⎣ C ⎦ 0 0 C 0 C

(83)

The solution of (79) is expressed as 

 F11 F12 Fd,11 Fd,12 Z 1 = N˜ d,1,1 + Z˜ 1 N˜ d,2 ,

(84)

where N˜ d,1,1  Φ˜ 1,d Ω˜ d† , and Z˜ 1 ∈ Rl× is an arbitrary matrix. Furthermore, given that rank([L; C]) = l + p, it can be shown that (81) has a solution if and only if the following matching condition is realized [14]:  rank

L Ad C Ad

 = rank(C Ad ).

(85)

However, if L is not in the orthogonal space spanned by C, matrix R can be designed to reduce the rank of (L + RC)M1 Ad , and in such a circumstance a less conservative condition than matching condition (85) can be achieved for the existence of a solution for (81):   M¯ 1 Ad = rank( M¯ 1 Ad ). (86) rank (L + RC)M1 Ad It highlights that the design parameter R can contribute as a leverage to relax the restrictive  (85) to a less conservative  condition. Next, via parti matching condition ˜ ˜ ˜ tioning Nd,1,i,1 , . . . , Nd,1,i,5  Nd,1,i , i = {1, 2}, and N˜ d,2,1 , . . . , N˜ d,2,5  N˜ d,2 , (82) and (84) read as (i, j = {1, 2}) Fi j = N˜ d,1,i, j + Z˜ i N˜ d,2, j , Fd,i j = N˜ d,1,i, j+2 + Z˜ i N˜ d,2, j+2 , Z i = N˜ d,1,i,5 + Z˜ i N˜ d,2,5 .

(87) (88) (89)

Accordingly, from (89) and assuming that (86) holds, (81) reads Z˜ 1 + R Z˜ 2 = W + J W¯ ,

(90)

Partial State Observers for Linear Time-Delay Systems

31

 †  where W  −(L + RC)M1 Ad − ( N˜ d,1,1,5 + R N˜ d,1,2,5 ) M¯ 1 Ad N˜ d,2,5 M¯ 1 Ad ,   †   ˜ ¯ ˜ ¯ ¯ , and J ∈ Rl× is an arbitrary paramW  I − Nd,2,5 M1 Ad Nd,2,5 M1 Ad eter. Hence, one can find from (87), (88), and (90) that F¯11 = W f + J W¯ N˜ d,2,1 ,

(91)

F¯d,11 = Wn + J W¯ N˜ d,2,3 ,

(92)

where W f  N˜ d,1,1,1 + R N˜ d,1,2,1 + W N˜ d,2,1 and Wn  N˜ d,1,1,3 + R N˜ d,1,2,3 + W N˜ d,2,3 . Now, the whole procedure of the FO design scheme has ended in finding an appropriate matrix J that fulfills the matrix inequality in Theorem 9. However, apparently the inequality suffers from nonlinear terms DiT J , similar to what occurred in Theorem 6. Therefore, an analogous trick to linearize matrix inequality (72) is to first assume that Di = αi D4 , i = {1, 2, 3}, where αi are scalar parameters, secondly fix the scalar parameters and derive an LMI equivalent to (72), and finally adjusting those fixed parameters using an AI empowered framework like the GABT algorithm. The following corollary that follows from Corollary 2 and Theorem 9 comprehends on finding the key parameter J . Corollary 3 Given αi , i = {1, 2, 3}, functional observer (58) for System (31) is globally asymptotically stable with any ancillary delay function complying with (60), if rank Conditions (83) and (86) are satisfied, and there exist l × l symmetric matrices Pii  0, Q i  0, i = {1, 2, 3}, and R j , j = {1, . . . , 4}, and arbitrary l × l real matrices P12 , P13 , P23 , D4 , and Λ ∈ Rl× , such that the following LMI is feasible: ⎡

⎤ ∗ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ≺ 0, ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎦ Ω¯˜ 7,7 (93) ¯ 2 2 T ˆ ˆ ¯ ˜ ˜ where Ω1,1  sym(P13 )+Q 1 +d1 R1 +d12 R1 +α1 sym(D4 W f )+α1 sym(ΛW Nd,2,1 ), T T − P13 +R3 , Ω¯˜ 2,2  Q 2 +Q 3 − Q 1 − R3 − R4 , Ω¯˜ 4,1  P33 + α3 D4T W f Ω¯˜ 2,1  P12 + α ΛW¯ N˜ , Ω¯˜  P + α D T W + α ΛW¯ N˜ , Ω¯˜  α W T E + α Ω¯˜ ∗ ∗ ∗ ∗ ∗ ⎢ ¯ 1,1 ¯˜ ⎢ Ω˜ 2,1 Ω ∗ ∗ ∗ ∗ 2,2 ⎢ ⎢ −P T R −Q − R ∗ ∗ ∗ 4 2 4 12 ⎢ ⎢ ¯˜ T T −R1 ∗ ∗ ⎢ Ω4,1 P23 − P33 −P23 ⎢ ¯ ⎢ Ω˜ 5,1 P22 − P23 −P22 0 −R2 ∗ ⎢ ⎢ ¯˜ ¯˜ ¯˜ Ω Ω 0 0 Ω −(1 − μ)Q ˆ 3 ⎣ 6,1 6,4 6,5 ¯ ¯ Ω˜ 7,1 Ω˜ 7,6 0 0 P13 − α3 D4 P12 − α2 D4

3

d,2,1

5,1

23

2

4

f

2

d,2,1

6,1

1

n

4

1

T T T N˜ d,2,3 W¯ T ΛT , Ω˜¯ 6,4  α3 WnT D4 + α3 N˜ d,2,3 W¯ T ΛT , Ω¯˜ 6,5  α2 WnT D4 + α2 N˜ d,2,3 W¯ T ΛT , Ω¯˜ 7,1  D4T W f + ΛW¯ N˜ d,2,1 + P11 − α1 D4 , Ω¯˜ 7,6  D4T Wn + ΛW¯ N˜ d,2,3 , and 2 R4 . Ω˜¯ 7,7  −sym(D4 ) + dˆ12 R3 + dˆ12

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R. Mohajerpoor

In addition, we have

J = ΛD4−T .

(94)

Example 3 Consider a delayed LTI system (31) with parameters described below: ⎡

−10 ⎢ −41 A=⎢ ⎣ 0 2

1 −2 0 0

0 4 −2 −20

⎡ ⎤ 2 0 ⎢0 0 ⎥ ⎥ , Ad = ⎢ ⎣0 1 ⎦ 0 −6

0 0 0 0

0 0 0 0

⎤ −5   9 ⎥ ⎥ , C = 1.5 0 0 0 , 0 ⎦ 0 −2 0 0 0

   T B = 1 2 −1 0.2 , and L = 02 I2 . The actual time-varying unknown delay   function is assumed to behave as d(t) = t with delay bounds arbitrarily fixed at d1 = 0.5(d2 + d1 ) + 0.5(d2 − d1 ) sin d22μ −d1 0.1[s], d2 = 3[s], and μ = 1.2. It can be examined that Conditions (83) and (86) are satisfied. Interestingly, it can ˆ ≡ 0, i.e. a delay-free FO, a necessary condition for the be shown that by setting d(·) stability of the observer dynamics outlined below is not met [14]:  rank

s Il − W f − Wn W¯ ( N˜ d,2,1 + N˜ d,2,3 )



= l ∀s ∈ C+ .

(95)

This phenomenon emphasizes that adding auxiliary delay terms into the observer dynamics can effectively reduce from conservatism structurally implied by unknowninput functional observers (see Sect. 3). We further assume that only the upper-bound of delay d2 is known. Hence, dˆ2 = d2 = 3[s], dˆ1 = 0 and μˆ = 0.4 are arbitrarily assigned, and  the ancillary ˆ ˆ ˆ ˆ ˆ delay function is chosen as d(t) = 0.5(d2 + d1 ) + 0.5(d2 − d1 ) cos dˆ 2−μˆdˆ t . Fixing 2 1 αi = 1, i = {1, 2, 3}, returns infeasible  LMI (93). Therefore, the GABT algorithm is used to obtain [α1 , α2 , α3 ] = 25 4 5 as a feasible set that results in appropriate observer parameters. Simulations were carried out in the MATLAB/Simulink environment using an arbitrary input signal that reads as u(t) = 5 + 10e−0.4t cos(2t), and assuming φ(θ ) =  T −2 0.01 0 10 , θ ∈ [−3, 0], and P˜ = I2 . Two alternative values were assigned for the sliding gain to emphasize the crucial role of the estimation of the upper-bound of disturbance terms: (i) ρ(t) = 0.1 and (ii) ρ(t) = 5. The results are depicted in Figs. 3 and 4. It is clear from Fig. 3 that ρ(t) = 0.1 is not an upper-bound of the disturbance term ε(t, ¯ (t), x(t)), thus finitetime convergence of the outputs estimation errors does not occur, which causes deviations in the estimation of the desired functional z(t). On the other hand, Fig. 4 demonstrates that sufficiently enlarging the sliding gain to ρ¯ = 5 the outputs estimations converge in a finite time, and the estimated functionals converge asymptotically.

Partial State Observers for Linear Time-Delay Systems

33

12 10

e 1 (t)

8

e 2 (t)

6

e 3 (t)

4

e 4 (t)

2 0 -2

0

1

2

3

4

5

6

7

8

9

10

Time [s] 0.5

actual values estimation

0 -0.5

10

actual values estimation

8 6

-1

4

-1.5

2

-2

0

-2.5 0

2

4

6

8

10

Time [s]

0

2

4

6

8

Time [s]

Fig. 3 Estimation of the functions with insufficient estimated upper-bound on the disturbance term ε(t, ¯ (t), x(t)) (ρ(t) = 0.1). z i and zˆ i indicate the ith element of z a and zˆ a , respectively

To emphasize the effectiveness of the delayed terms and the robustifying sliding term, the estimation were carried out by dropping parameters Fd , Hd , X 2 , and function ν(·) from the observer structure, thus replacing observer (58) with a conventional FO similar to (2). The results shown in Fig. 5 clarifies that the simplified FO structure fails to satisfy the stability requirements.

6 Summary and Future Directions Fundamental concepts and recent advancements in partial state estimation of LTI systems with time-varying delays have been reviewed in this chapter. The chapter sheds light on the import aspects of studying (unknown-input) functional observers and the main challenges confronting the current state of the practice. It has been shown this class of generic Luenberger observers can relax the observability/detectability requirement of the system, as well as the matching condition that is invoked for designing unknown-input observers. Sophisticated order-increase algorithms illustrated in this chapter can be readily practiced using matrix programming languages available as Phyton/R libraries or MATLAB language. It has been shown that delay-free and delay-dependent FOs can

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R. Mohajerpoor 12

e 1 (t)

10

0

8

-0.2

e 2 (t)

6

-0.4

e 3 (t)

4

-0.6 0

2

0.2

e 4 (t)

0.4

0 -2 0

0.5

1

1.5

2

2.5

3

3.5

4

Time [s]

0.5

10

actual values estimation

0 -0.5

6

-1

4

-1.5

2

-2

0

-2.5 0

-2

4

2

6

8

actual values estimation

8

10

0

2

4

Time [s]

6

8

10

Time [s]

Fig. 4 Estimation of the functions with sufficiently large estimated upper-bound on the disturbance term ε(t, ¯ (t), x(t)) (ρ(t) = 5). z i and zˆ i indicate the ith element of z a and zˆ a , respectively 10 8

e 1 (t)

6

e 2 (t)

4

e 3 (t)

2

e 4 (t)

0 0

5

10

15

20

25

30

Time [s] 0.5

10

0

8

-0.5

6

true values estimation

4

-1

true values estimation

-1.5

2 0

-2

-2

-2.5 0

5

10

15

Time [s]

20

25

30

0

5

10

15

20

25

30

Time [s]

Fig. 5 Estimation results of the delay-free observer obtained from omitting the delayed terms and removing the sliding term ν(t). z i and zˆ i indicate the ith element of z a and zˆ a , respectively

Partial State Observers for Linear Time-Delay Systems

35

be designed for time-delay systems with measurable or uncertain input and state timevarying delays. Lyapunov–Krasovskii stability analysis techniques have empowered the delay-dependent observer design schemes. There are numerous fields that the developed theories for functional observers can be employed to cope with restrictions implied by conventional full-order observers. This includes system monitoring, fault detection and isolation, and feedback control of large-scale dynamic systems, such as power systems [35], chemical processes [2], and traffic networks [36]. On the theoretical side of this field, the most interesting challenge to the author’s mind is designing optimum FO-based control algorithms for (delayed) dynamic systems that are not observable nor detectable.

References 1. Luenberger, D.: Observers for multivariable systems. IEEE Trans. Autom. Control 11(2), 190– 197 (1966) 2. Nazmi, S., Mohajerpoor, R., Abdi, H.: Functional observer design with application to pre-compensated multi-variable systems. In: IEEE Conference on Control Applications (CCA2015), Sydney, Australia (2015) 3. Pham, T.N., Trinh, H., Hien, L.V.: Load frequency control of power systems with electric vehicles and diverse transmission links using distributed functional observers. IEEE Trans. Smart Grid 7(1), 238–252 (2016) 4. Ashraf-Modarres, A., Momeni, H., Yazdizadeh, A.: A new decentralized sliding mode observer design for interconnected power systems with unknown and time-varying delay. In: 4th Conference on Thermal Power Plants (CTPP2012), Tehran, Iran, pp. 1–6 (2012) 5. Sirisena, H.R.: Minimal-order observers for linear functions of a state vector. Int. J. Control 29(2), 235–254 (1979) 6. Mohajerpoor, R., Abdi, H., Nahavandi, S.: A new algorithm to design minimal multi-functional observers for linear systems. Asian J. Control 18(3), 842–857 (2016) 7. Trinh, H.: Linear functional state observer for time-delay systems. Int. J. Control 72(18), 1642– 1658 (1999) 8. Darouach, M.: Linear functional observers for systems with delays in state variables. IEEE Trans. Autom. Control 46(3), 491–496 (2001) 9. Mohajerpoor, R., Leong, W.Y., Abdi, H., Nahavandi, S.: A delay-dependent functional observer for linear time-invariant systems with input delay. In: Proceedings of the 13th International Conference on Control, Automation, Robotics and Vision (ICARCV2014). IEEE, Singapore, pp. 746–751 (2014) 10. Mohajerpoor, R., Abdi, H., Nahavandi, S.: Reduced-order functional observers with application to partial state estimation of linear systems with input-delays. J. Control Decis. 2(4), 233–256 (2015) 11. Mohajerpoor, R., Abdi, H., Shanmugam, L., Nahavandi, S.: Minimum-order filter design for partial state estimation of linear systems with multiple time-varying state delays and unknown input delays. Int. J. Robust Nonlinear Control 27(3), 393–409 (2017) 12. Mohajerpoor, R., Abdi, H., SHanmugam, L., Nahavandi, S.: Partial state estimation of lti systems with multiple constant time-delays. J. Franklin Inst. 353(2), 541–560 (2016) 13. Seuret, A., Floquet, T., Richard, J.P., Spurgeon, S.K.: A sliding mode observer for linear systems with unknown time varying delay. In: American Control Conference (ACC2007), New York, USA, pp. 4558–4563 (2007) 14. Mohajerpoor, R., Shanmugam, L., Abdi, H., Nahavandi, S., Park, J.H.: Delay-dependent functional observer design for linear systems with unknown time-varying state delays. IEEE Trans. Cybern. 48(7), 2036–2048 (2018)

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15. Mohajerpoor, R., Abdi, H., Nahavandi, S.: Partial state estimation: a new design approach. In: 2014 13th International Conference on Control Automation Robotics Vision (ICARCV), Singapore, pp. 752–757 (2014) 16. Brogan, W.L.: Modern Control Theory. Pearson Education India, Noida (1982) 17. Darouach, M.: Existence and design of functional observers for linear systems. IEEE Trans. Autom. Control 45(5), 940–943 (2000) 18. Jennings, L.S., Fernando, T.L., Trinh, H.M.: Existence conditions for functional observability from an eigenspace perspective. IEEE Trans. Autom. Control 56(12), 2957–2961 (2011) 19. Trinh, H., Nahavandi, S., Tran, T.D.: Algorithms for designing reduced-order functional observers of linear systems. Int. J. Innov. Comput. Inf. Control 4(2), 321–334 (2008) 20. Park, J.H., Lee, T.H., Liu, Y., Chen, J.: Dynamic Systems with Time Delays: Stability and Control. Springer-Nature, Berlin (2019) 21. Mohajerpoor, R., Abdi, H., Nahavandi, S.: Minimal unknown-input functional observers for multi-input multi-output lti systems. J. Process Control 35, 143–153 (2015) 22. Darouach, M.: Functional observers for systems with unknown inputs. In: 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Leuven, Belgium (2004) 23. Hou, M., Pugh, A.C., Muller, P.C.: Disturbance decoupled functional observers. IEEE Trans. Autom. Control 44(2), 382–386 (1999) 24. Bejarano, F.J., Fridman, L., Poznyak, A.: Unknown input and state estimation for unobservable systems. SIAM J. Control Optim. 48(2), 1155–1178 (2009) 25. Mohajerpoor, R., Abdi, H., Nahavandi, S.: On unknown-input functional observability of linear systems. In: American Control Conference (ACC2015), Chicago, USA 26. Angulo, M.T., Moreno, J.A., Fridman, L.: On functional observers for linear systems with unknown inputs and hosm differentiators. J. Franklin Inst. 351(4), 1982–1994 (2014) 27. Chen, J., Park, J.H., Xu, S.: Stability analysis of continuous-time systems with time-varying delay using new lyapunov-krasovskii functionals. J. Franklin Inst. 355(13), 5957–5967 (2018) 28. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49(9), 2860–2866 (2013) 29. Mohajerpoor, R., Shanmugam, L., Abdi, H., Rakkiyappan, R., Nahavandi, S., Shi, P.: New delay range-dependent stability criteria for interval time-varying delay systems via wirtinger-based inequalities. Int. J. Robust Nonlinear Control 28(2), 661–677 (2018) 30. Park, P., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with timevarying delays. Automatica 47(1), 235–238 (2011) 31. Mohajerpoor, R., Shanmugam, L., Abdi, H., Nahavandi, S., Saif, M.: Functional observer design for retarded system with interval time-varying delays. Int. J. Syst. Sci. 48(5), 1060–1070 (2017) 32. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) 33. Hale, J.K.: Introduction to Functional Differential Equations, vol. 99. Springer, Berlin (1993) 34. Edwards, C., Spurgeon, S.: Sliding Mode Control, Theory and Applications. CRC Press, Boca Raton (1998) 35. Emami, K., Ariakia, H., Fernando, T.: A functional observer based dynamic state estimation technique for grid connected solid oxide fuel cells. IEEE Trans. Energy Convers. 33(1), 96–105 (2018) 36. Mohajerpoor, R., Saberi, M., Vu, H.L., Garoni, T.M., Ramezani. M.: H∞ robust perimeter flow control in urban networks with partial information feedback. Transportation Research Part B: Methodological (2019)

Finite-Time Stability and Control of Impulsive Positive Systems with Interval Uncertainty Mengjie Hu and Ju H. Park

Abstract This chapter addresses finite-time stability and control issues of impulsive positive systems (IPSs) with interval uncertainty. In terms of different types of impulsive effects, a time-varying copositive Lyapunov function is constructed and the average impulsive interval method is applied for studying the finite-time stability of interval IPSs. The relevance between the impulsive effect and the finite time period is disclosed by theoretical results and simulations. Sufficient conditions for the existence of the finite-time controller are established to ensure the finite-time stability of the closed-loop system while maintaining the positivity. The conditions in terms of vector inequalities are easily solvable. Numerical examples are presented to show the usefulness of the obtained theorems. Keywords Positive systems · Finite-time stability · Interval uncertainty · Robustness

1 Problem Formulation and Preliminaries In nature and engineering applications, there exists a class of dynamic systems which can keep the system state and output variables non-negative under a non-negative initial state [1]. It is widely used in many fields, such as the metabolic kinetics of drugs in the biological field [2], the industrial process of distillation columns in the chemical field [3], and the probability problem of stochastic models in economics and M. Hu (B) · J. H. Park Department of Electrical Engineering, Yeungnam University, Gyeongsan 38541, Republic of Korea e-mail: [email protected] J. H. Park e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_2

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sociology [4]. It is worth noting that the state of positive systems is constrained within the convex polyhedron in the first quadrant instead of the entire state space, which makes the method for studying general dynamic systems often not applicable to the analysis of positive systems. In 1912, Frobenius proposed the Perron–Frobenius theorem and a series of results on non-negative matrices [5], which provides a powerful mathematical tool for later research works. Since then, positive systems have attracted attentions of many scholars and achieved rapid development [6–10]. In practice, the system dynamic evolution often exhibits the transient state, that is, impulsive phenomenon. Dynamic systems with positive constraints and impulsive effects are called impulsive positive systems (IPSs). IPSs include positive continuoustime dynamics and positive discrete-time dynamics, and possess typical hybrid characteristics and rich dynamic behaviors. In recent years, some researches have been conducted on IPSs, please see [11–13] as references. As is well known, Lyapunov stability is defined in limitless time intervals and utilized to indicate the steady-state characteristics. From the perspective of practical applications, some systems may be defined only within a finite time interval, or their states need to be restricted to a specific range. If the system state does not overstep a preset threshold within a specified time interval under a prescribed initial condition, then the system is finite-time stable (FTS). [14] summarized the development of finite-time stability research, and pointed out that the difference between the concepts of finite-time stability and Lyapunov stability is mainly in two aspects: (1) The finite-time stability means the system will be analysed within a limited time interval. (2) A threshold for the system state is needed to be set in advance. Results on the finite-time stability of general impulsive systems can refer to [15, 16], and this problem has also made much progress in positive systems [17–20]. Among them, [19] established sufficient and necessary conditions for the finite-time stability of linear positive systems. By the average dwell time method, [20] discussed the finite-time L1 control issue of positive switched systems with delay. Based on the above discussion, although there are some interesting results on the finite-time stability for positive systems and impulsive systems, there is a lack of research on the finite-time stability of interval IPSs. This chapter extends the definition of finite-time stability to IPSs. The finite-time stability and control issues of interval IPSs are studied. With the help of the time-varying co-positive Lyapunov function and the average impulsive interval method, under different impulsive effects (disturbance impulses, stabilizing impulses and “neutral” impulses), a finite-time stability criterion for interval IPSs is established. The relationship between the impulses and the finite time interval is disclosed. The finite-time controller is designed to achieve the positive stabilization of interval IPSs. Numerical examples are presented to show the usefulness of the obtained theorems. Notations: Rn is the n-dimensional Euclidean space. N is the natural number set, Rn+ denotes the positive quadrant of Rn . We denote 1 = [1, 1, . . . , 1]T . A matrix

Finite-Time Stability and Control of Impulsive Positive Systems …

39

A  0 (≺ 0,  0,  0) denotes all entries of A are positive (negative, nonnegative, non-positive). Consider the impulsive system with interval uncertainty: ⎧ ˙ = Ax(t), t = tk , ⎨ x(t) x(tk ) = (I + H)x(t − ), k ∈ N, ⎩ x(t0 ) = x0 , t0 = 0,

(1)

where x(t) ∈ Rn is state vector. A and H are constant matrices with appropriate dimensions. Define the set of interval uncertainty matrices as follows: A ∈ [Am , A M ], H ∈ [Hm , H M ]. x(tk− ) = lim− x(tk + h). x0 is the initial state. h→0

Assume x(tk ) = x(tk+ ) = lim+ x(tk + h), k ∈ N. The impulse time sequence σ = h→0

{t1 , t2 , t3 ...} satisfies 0 < t1 < t2 < t3 ... < tk−1 < tk < ... with limk→∞ tk → ∞.

Remark 1 Note that the uncertainty refers to changes within a possible range, which is unavoidable in practice. There are usually two types of system uncertainty. One is the integral quadratic constraint description. The other is a structured description, which usually involves interval uncertainty. Before presenting the main results, the following definitions and lemmas are provided. Definition 1 ([1]) If for x0  0, the system state x(t)  0 holds ∀t ≥ t0 , then system (1) is positive. Definition 2 ([11]) If there exist τa > 0 and integer N0 ≥ 0, such that T2 − T1 T2 − T1 − N0 ≤ Nσ (T2 , T1 ) ≤ + N0 , ∀T2 ≥ T1 ≥ 0 τa τa

(2)

holds, where Nσ (T2 , T1 ) refers to the number of impulsive instants over (T1 , T2 ), then the average impulsive interval of impulse time sequence σ is τa . Lemma 1 ([12]) If Am is Metzler matrix and (I + Hm )  0, then system (1) is positive.

2 Finite-Time Stability Analysis In this section, we will establish sufficient conditions to make interval impulsive positive systems (IPSs) finite-time stable (FTS), specially, different impulsive effects are considered.

40

M. Hu and J. H. Park

In order to ensure the positivity of system (1), suppose the system matrices satisfy Am is Metzler and (I + Hm )  0. First, the definition of finite-time stability is presented below. Definition 3 ([21]) For a vector v  0, and a vector-valued function r (•) : t ∈ [t0 , t f ] → Rn+ with r (t0 ) ≺ v, IPS (1) is FTS w.r.t. (t0 , t f , v, r (·)) if x0T v ≤ 1 ⇒ x T (t)r (t) < 1, ∀t ∈ [t0 , t f ].

(3)

Now, we have the first main result. Theorem 1 For prescribed positive parameters τa , t f , λ1 and λ2 , an integer N0 ≥ 0, a vector v ∈ Rn+ , and a vector function r (t)  0 with r (t0 ) ≺ v, set φ= λλ21 β −N0 and ϕ= λλ21 β N0 , if there exist parameters β > 0 and ε, and a continuously differentiable vector function δ(t)  0 such that the following inequalities hold, ˙ + AT δ(t) − εδ(t)  0, δ(t) M ((1 − β)I + HTM )δ(tk )  0, k ∈ N, λ1r (t) ≺ δ(t), δ(t0 )  λ2 v,

(4) (5) (6) (7)

(a) When β ≥ 1, (i) if τa and t f satisfy ln φ τa , ln β + ετa lnβ + ετa > 0,

0 < tf ≤

(8) (9)

then system (1) is FTS w.r.t. (t0 , t f , v, r (·)). (ii) if τa satisfies lnβ + ετa ≤ 0,

(10)

and φ ≥ 1, then system (1) is FTS w.r.t. (t0 , t f , v, r (·)) ∀t f > t0 . (b) When 0 < β < 1, (iii) if τa and t f satisfy 0 < tf ≤

ln ϕ τa , ln β + ετa

(11)

and τa satisfies (9), then system (1) is FTS w.r.t. (t0 , t f , v, r (·)). (iv) if τa satisfies (10) and ϕ ≥ 1, then system (1) is FTS w.r.t. (t0 , t f , v, r (·)) ∀t f > t0 .

Finite-Time Stability and Control of Impulsive Positive Systems …

41

Remark 2 Note that in the two special cases of Theorem 1, namely, Case (ii) and Case (iv), under certain conditions, IPS (1) is FTS w.r.t. (t0 , t f , v, r (·)) regardless of the value of t f . Proof Define a time-varying copositive Lyapunov function: V (t) = x T (t)δ(t),

(12)

where δ(t)  0 is to be determined. When t ∈ [tk−1 , tk ), k ∈ N, we have ˙ D + V (t) = x˙ T (t)δ(t) + x T (t)δ(t) T T ˙ = x (t)(A δ(t) + δ(t)).

(13)

By (4), we can derive ˙  δ(t) ˙ + AT δ(t)  εδ(t), AT δ(t) + δ(t) M

(14)

˙ + AT δ(t) − εδ(t)) ≤ 0. D + V (t) − εV (t) = x T (t)(δ(t)

(15)

then,

Integrating (15) from tk−1 to t, t ∈ [tk−1 , tk ), k ∈ N, we can obtain V (t) ≤ eε(t−tk−1 ) V (tk−1 , x(tk−1 )).

(16)

V (tk , x(tk )) = x T (tk )δ(tk ) = x T (tk− )(I + H)T δ(tk ).

(17)

For t = tk ,k ∈ N,

From (5), we have (I + H)T δ(tk )  (I + H M )T δ(tk )  βδ(tk ),

(18)

V (tk ) ≤ βV (tk− ).

(19)

V (t) ≤ β Nσ (t,t0 ) eε(t−t0 ) V (t0 , x(t0 )), t ≥ t0 ,

(20)

then,

Then, we can obtain

where Nσ (t, t0 ) is the number of impulsive instants in [t0 , t). In what follows, two cases of β ≥ 1 and 0 < β < 1 are considered respectively.

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M. Hu and J. H. Park

(A) For β ≥ 1: From (2), we derive lnβ

V (t) ≤ β N0 e( τa +ε)(t−t0 ) V (t0 , x(t0 )).

(21)

By (6), (7) and (21), we have ln β

λ1 x T (t)r (t) = V (t) ≤ β N0 e( τa

+ε)(t−t0 ) T

x (t0 )δ(t0 )

ln β N0 ( τa +ε)(t−t0 ) T

≤ λ2 β e

x (t0 )v.

(22)

(i) When lnβ + ετa > 0, by (8), we have the following inequality λ2 N0 ( ln β +ε)(t−t0 ) λ2 N0 ( ln β +ε)(t f −t0 ) β e τa ≤ β e τa ≤ 1, t ∈ [t0 , t f ], λ1 λ1

(23)

holds, then from (22), we can derive x T (t)r (t)
κ2 • t f has no bounds if τa ≤ κ2 τa has no bounds t f is has no bounds

ln ϕ ε

and υ3 =

ln ϕ ln β+ετa τa

Finite-Time Stability and Control of Impulsive Positive Systems … Table 2 The correlations among key parameters ε β β≥1 ε≥0

0 < β < 1 (ε = 0)

τa ↑ when β ↑. τa ↑ when t f ↑.

τa ↓ when β ↑. τa ↓ when t f ↑ if τa > κ2 and t f > υ2 . t f ↓ when τa ↑ if τa > κ2 . \

t f ↑ when τa ↑. ε 0, γ > 0, η > 0 and ω(t) satisfies (33). Without loss of generality, with the help of Theorem 1, we present the theoretical criterion for the case of disturbance impulses. Theorem 2 For prescribed positive constants τa , t f , λ1 , λ2 , λ and h, an integer N0 ∈ N , a vector v ∈ Rn+ , and a vector function r (t)  0 defined over [t0 , t f ] with 1 β −N0 , if there exist parameters ε > 0, β > 1 and 0 < γ ≤ λ, r (t0 ) ≺ v, set ψ= λ2λ+λh and a continuously differentiable vector function δ(t)  0 such that ˙ + AT δ(t) − εδ(t) + C T 1q  0, δ(t) M T FM δ(t)

((1 −

T + EM 1q − γ1 p β)I + HTM )δ(tk )

 0,  0, k ∈ N,

λ1r (t) ≺ δ(t), δ(t0 )  λ2 v, hold and τa satisfies

(36) (37) (38) (39) (40)

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M. Hu and J. H. Park

τa ≥ max

 ln β ε

,

ln βt f  , ln ψ − εt f

(41)

then system (32) is finite-time bounded under L1 -gain performance w.r.t. (t0 , t f , v, r (·), h). Proof The proof is conducted by two steps: (1) Finite-time boundedness. Consider the Lyapunov function in (12). When t ∈ [tk−1 , tk ), k ∈ N, we derive ˙ D + V (t) = x˙ T (t)δ(t) + x T (t)δ(t) T T ˙ = x (t)(A δ(t) + δ(t)) + ω T (t)F T δ(t).

(42)

By (36) and (37), we obtain D + V (t) − εV (t) ≤ λω T (t)1 p .

(43)

Integrating (43) from tk−1 to t, t ∈ [tk−1 , tk ), k ∈ N, we can obtain V (t) ≤ eε(t−tk−1 ) V (tk−1 , x(tk−1 )) + λ



t

eε(t−s) ω(t) ds.

(44)

tk−1

For t = tk , k ∈ N, by (38), we obtain V (tk , x(tk )) ≤ βV (tk− , x(tk− )).

(45)

Combining (44) and (45), we derive V (t) ≤ β Nσ (t,t0 ) eε(t−t0 ) V (t0 , x(t0 ))  t +λ eε(t−s) β Nσ (t,s) ω(t) ds, t ≥ t0 .

(46)

t0

Note that t ≤ t f and

tf t0

ω(t) ds ≤ h, by (2), we have

V (t) ≤ β Nσ (t,t0 ) eε(t−t0 ) (V (t0 , x(t0 )) + λh) ln β

≤ β N0 e( τa

+ε)(t−t0 )

(V (t0 , x(t0 )) + λh).

(47)

By (39)–(41), we have x T (t0 )v ≤ 1 ⇒ x T (t)r (t) < 1. Then, system (32) is finitetime bounded w.r.t. (t0 , t f , v, r (·), h). (2) L1 -gain performance of system (32). Substituting (36) and (37) to (42), we have

Finite-Time Stability and Control of Impulsive Positive Systems …

D + V (t) ≤ εV (t) + γ ω(t) − y(t) .

47

(48)

Let Γ (t) = γ ω(t) − y(t). Integrating (48) from tk−1 to t, t ∈ [tk−1 , tk ), k ∈ N, we can derive 

V (t) ≤ eε(t−tk−1 ) V (tk−1 , x(tk−1 )) +

t

eε(t−s) Γ (s)ds.

(49)

tk−1

For t = tk , k ∈ N, by (38), we have (45) holds, then 

V (t) ≤ β Nσ (t,t0 ) eε(t−t0 ) V (t0 , x(t0 )) +

t

eε(t−s) β Nσ (t,s) Γ (s)ds, t ≥ t0 .

(50)

eε(t−s) β Nσ (t,s) ω(t) ds,

(51)

eε(t−s) β −Nσ (s,0) ω(t) ds.

(52)

t0

Since the zero initial condition, 

t



eε(t−s) β Nσ (t,s) y(s) ds ≤ γ

t0

t

t0

holds, then, 

t

e

ε(t−s) −Nσ (s,0)

β

 y(s) ds ≤ γ

t0

t

t0

Substituting (2) and (41) to (52), we obtain 

t

e

ε(t−s)−εs

β

−N0



t

y(s) ds ≤ γ

t0

eε(t−s) ω(t) ds.

(53)

t0

Thus, β −N0



tf

e−2εs y(s) ds ≤ γ

t0

 

tf

e−εs ω(t) ds

t0 tf

≤γ

ω(t) ds.

(54)

t0

From Definition 5, we can conclude that system (32) is finite-time bounded under L1 -gain performance w.r.t. (t0 , t f , v, r (·), h). This completes the proof. 

3 Finite-Time Stabilization of Interval IPSs In this section, the finite-time control problem of interval IPSs is addressed.

48

M. Hu and J. H. Park

Based on the finite-time stability result of Theorem 1, the state feedback controller for system (1) is designed. Consider the following system: ⎧ ˙ = Ax(t) + Du(t), t = tk , ⎨ x(t) Δx(t) = Hx(t − ), t = tk , k ∈ N, ⎩ x(t0 ) = x0 , t0 = 0,

(55)

u(t) = K x(t),

(56)

where matrix K  0 is to be determined. System matrices are belong to the set of interval uncertainty matrices as follows: A ∈ [Am , A M ], H ∈ [Hm , H M ], D ∈ [Dm , D M ], in which Am is Metzler, I + Hm  0 and Dm  0. Lemma 2 ([7]) For matrix A ∈ Rn×n , A is Metzler iff there exists a constant ς such that A + ς I  0 holds. Theorem 3 For preset positive scalars τa , t f , λ1 and λ2 , an integer N0 ∈ N , vectors v ∈ Rn+ , δ˜ ∈ Rr+ , and a vector function r (t)  0 with r (t0 ) ≺ v, set φ= λλ21 β −N0 , if there exist scalars β > 0, ε and ς, and vectors δ ∈ Rn+ and w ∈ Rn such that the following inequalities hold, ˜ T + ς I  0, δ˜ T DmT δAm + Dm δw

(57)

ATM δ − εδ + w  0, ((1 − β)I + HTM )δ  0, λ1r (t) ≺ δ  λ2 v,

(58) (59) (60)

(A) When β ≥ 1, (i) if τa and t f satisfy ln φ τa , ln β + ετa ετa + ln β > 0,

0 < tf ≤

(61) (62)

then with the controller u(t) = K x(t) =

1 ˜ T δw x(t). ˜δ T D T δ m

Then, system (55) is positive and FTS w.r.t. (t0 , t f , v, r (·)). (ii) if φ ≥ 1 and τa satisfies

(63)

Finite-Time Stability and Control of Impulsive Positive Systems …

ετa + ln β ≤ 0,

49

(64)

then with the controller (63), system (55) is positive and FTS w.r.t. (t0 , t f , v, r (·)) ∀t f > t0 . (B) When 0 < β < 1, (iii) if τa satisfies (9) and t f satisfies 0 < tf ≤

ln ϕ τa , ln β + ετa

(65)

then with the controller (63), system (55) is positive and FTS w.r.t. (t0 , t f , v, r (·)). (iv) if ϕ ≥ 1 and τa satisfies (10), then with the controller (63), system (55) is FTS w.r.t. (t0 , t f , v, r (·)) ∀t f > t0 . Remark 4 In Theorem 3, inequality (57) makes the closed-loop system positive. Inequalities (58)–(61) and (64) ensure the closed-loop system be FTS. Remark 5 The relationships between τa /t f and K are discussed here. (1) From (61) or (64), when τa decreases, the parameter ε may be smaller, then w will decrease by (58), which will cause matrix K to decrease from (63), i.e., in order to ensure the finite-time stability of the closed-loop system, the intensity of the controller should increase as the disturbance impulses occur more frequently. (2) From (61), we have ε may be smaller when t f increases, then w will decrease by (58), which will cause matrix K to decrease from (63), i.e., when the expected finite time interval is expanded, the intensity of the controller need to increase. Proof Since δ˜  0, Dm  0, δ˜ ∈ Rr , Dm ∈ Rn×r and δ ∈ Rn , then we have δ˜ T DmT δ is a positive constant. Divide (57) by δ˜ T DmT δ, we can derive Am +

1 ς ˜ T+ D δw I  0. ˜δ T D T δ m ˜δ T D T δ m m

(66)

˜ T is Metzler. From Lemma 2, we have that Am + δ˜ T D1 T δ Dm δw m From (63), we also have the matrix Am + Dm K is Metzler. Then, we conclude that system (55) is positive. By Theorem 1, let the vector function δ(t) be a constant vector δ. Then, the finite-time stability conditions are presented as follows: ATM δ − εδ + K T DmT δ  0,

(67)

((1 − β)I + HTM )δ  0, λ1r (t) ≺ δ  λ2 v.

(68) (69)

50

M. Hu and J. H. Park

It is easy to see that inequalities (58)–(60) are equivalent to inequalities (67)–(69). In addition, Cases (i)–(iv) of Theorem 3 are the same as Cases (i)–(iv) of Theorem 1. Then, system (55) is FTS. This completes the proof. 

4 Illustrative Example In this section, three examples are given to exhibit the usefulness of obtained results. The first one based on the drug metabolism is for the finite-time stability of IPSs, and the other two are for the positive finite-time stabilization. Example 1 Consider the drug metabolism model in Fig. 1 [22], which is widely applied to analyze the drug distribution in the body after injections. This example is to show that the concentrations of drugs in plasma and extravascular space begin from a specified range, then they are kept within a permissible range over a finite time period after impulse injection. The system model is ⎧ 

a12 −(a11 + a21 ) ⎪ ⎪ x(t), t = tk , x(t) ˙ = ⎪ ⎪ ⎨

a21  −(a22 + a12 ) b11 0 x(t − ), t = tk , k ∈ N, Δx(t) = ⎪ ⎪ 0 b22 ⎪ ⎪ ⎩ x(t0 ) =x0 , t0 = 0, where system state x1 and x2 denote the concentration of drugs in blood plasma and extravascular space respectively. a11 > 0 and a22 > 0 are the ratios of drug excretion in the two spaces respectively. a12 and a21 are the ratios of drug transmission between the two compartments. b11 and b22 are the amounts of drug injection in the two compartments respectively. We consider the system parameter uncertainty, the following system matrices are taken into consideration:

Fig. 1 Drug metabolism model

Finite-Time Stability and Control of Impulsive Positive Systems …

51

0.8

x 1 (t)

0.6 0.4 0.2 0

0

1

2

3

4

5

6

4

5

6

t(s) 0.8

x 2 (t)

0.6 0.4 0.2 0

0

1

2

3

t(s)

Fig. 2 State response in Example 1 Fig. 3 Evolution of x T (t)r (t) in Example 1

1 0.9 0.8

x T(t)r(t)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

t(s)





−0.15 0.45 0.1 0 , HM = , 0.35 −0.25 0 0.15

−0.2 0.4 0.05 0 , Hm = . Am = 0.3 −0.3 0 0.15

AM =

Suppose that the concentration of drugs starts from 6x1 + 6x2 ≤ 1. After impulse injection, the permissible range is 0.8x1 + 0.7x2 < 1 over a finite time period [0, 4s]. Then, we have vectors r (t) = [0.8, 0.7]T and v = [6, 6]T .

52

M. Hu and J. H. Park

Fig. 4 The bound of τa with varying β of Example 1 The lower bound of a

12 10 8 6 4 2 0

Fig. 5 The bound of τa with varying t f of Example 1

1

1.5

2

2.5

3

3.5

4

4.5

5

4

5

6

7

8

8

The lower bound of

a

7 6 5 4 3 2 1 0

0

1

2

3

t f (s)

The initial state is given as [0.09, 0.05]T . The parameters are chosen as τa = 0.9s, ε = 0.2, β = 1.15, λ1 = 1 and λ2 = 0.24. By applying the results in Theorem 1, the feasible solution of inequalities (4)–(7) is

1.3412 − 0.0817 ∗ t δ(t) = . 1.3689 − 0.0748 ∗ t As simulation results, Fig. 2 presents state response of system (1). Figure 3 depicts the curve of x T (t)r (t) of system (1). It is easy to see that the IPS (1) with disturbance impulses is FTS w.r.t. (t0 , t f , v, r (·)). When other parameters being settled, the relationships between β and τa , and between τa and t f are discussed here.

Finite-Time Stability and Control of Impulsive Positive Systems …

53

2

x 1 (t)

1.5 1

0.5 0

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

t(s) 2

x 2 (t)

1.5 1

0.5 0

0

1

2

3

4

5

t(s)

Fig. 6 State evolution of the open-loop system of Example 2

From Case (i) in Theorem 1, the lower bound of τa with varying β is depicted in Fig. 4. We can observe that the lower bound of τa will be larger if β increases. By (8), the relationship between τa and t f is shown in Fig. 5. We can see the lower bound of τa increases as t f increases. The simulations verify the analysis in Table 2. Example 2 This example is to show the case of finite-time control with stabilizing disturbance. Consider system (55) with matrices





−0.1 0.6 −0.17 0.13 1.1 AM = , HM = , DM = , 0.5 −0.1 0.06 −0.3 1.1



−0.12 0.6 −0.2 0.1 1 , Hm = , DM = . Am = 0.45 −0.1 0.05 −0.3 1 The initial state is x(t0 ) = [0.09 0.05]T . Parameters are t f = 9, v = [6.5 6.5]T , r (t) = [0.7 0.7]T , and τa = 0.9. The state response of the open-loop system is in Fig. 6. We can observe from Fig. 7 that x T (t)r (t) is larger than 1 when t ≥ 7.917, which suggests that the system is not FTS over [t0 , t f ]. Choose scalars λ1 = 1, λ2 = 0.23, β = 0.89, ε = 0.28, ς = 0.4 and N0 = 0, and vector δ˜ = 1. By Theorem 3, one feasible solution of inequalities (57)–(60) is

54

M. Hu and J. H. Park 3

2.5

x T(t)r(t)

2

1.5

1

0.5

0

0

2

4

6 t(s)

8

10

12

Fig. 7 Evolution of x T (t)r (t) of the open-loop system of Example 2

x 1 (t)

0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

t(s)

x 2 (t)

0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

t(s)

Fig. 8 State evolution of the closed-loop system of Example 2





0.8072 −0.1344 δ= , w= . 0.8033 −0.2082 Further, from (63), we have the matrix   K = −0.0834 −0.1293 .

Finite-Time Stability and Control of Impulsive Positive Systems … Fig. 9 Evolution of x T (t)r (t) of the closed-loop system of Example 2

55

0.5 0.45 0.4

x T(t)r(t)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

6

7

8

9

10

t(s)

Fig. 10 The bound of τa with varying β of Example 2

30 The lower bound of a The upper bound of a

25

a

20

15

10

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 8 presents the state trajectory of the closed-loop system. The curve of x T (t)r (t) of the closed-loop system is in Fig. 9. We can observe that the closed-loop system (55) is positive and FTS w.r.t. (t0 , t f , v, r (·)) under the proposed controller. When other parameters being selected, the relationships between β and τa , and between τa and t f are discussed. From Case (iii) in Theorem 3, the bounds of τa with varying β is in Fig. 10. We can see that the upper bound of τa will decrease if β increases. By (65), the relationship between τa and t f is shown in Fig. 11. We can see the upper bound of τa will decrease if t f increases. The simulation results can confirm the analysis of Table 2.

56

M. Hu and J. H. Park

Fig. 11 The bound of τa with varying t f of Example 2

The upper bound of

a

15

10

5

0

4

6

8

10

12

14

t f (s)

Example 3 This example is to present the case of finite-time control with disturbance impulses. Consider system (55) with





−0.3 0.15 0.05 0.25 0.5 , HM = , DM = . 0.1 −0.3 0.2 −0.1 0.5





−0.3 0.15 0.04 0.25 0.4 , Hm = , Dm = . Am = 0.08 −0.32 0.2 −0.1 0.4 AM =

The initial state is set as x(t0 ) = [0.2 0.3]T . Parameters are chosen as t f = 10, v = [2 1.9]T , r (t) = [0.4 0.4]T , and τa = 0.5. The state response of the open-loop system is in Fig. 12. We can observe from Fig. 13 that x T (t)r (t) is larger than 1 when t ≥ 1.834, so the open-loop system is not FTS over [t0 , t f ]. Let λ1 = 2.3, λ2 = 0.6, β = 1.23, ε = −0.28, ς = 10, N0 = 0, and δ˜ = 1. By Theorem 3, the inequalities (57)–(60) are feasible with the solution: δ=



1.1334 −0.1209 ,w = . 1.0096 −0.2381

Further, by simple calculation, we have   K = −0.1410 −0.2777 . Figure 14 presents the state trajectory of the closed-loop system. The curve of x T (t)r (t) of the closed-loop system (55) is shown in Fig. 15. We can observe that the

Finite-Time Stability and Control of Impulsive Positive Systems …

57

x 1 (t)

3

2

1

0

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

t(s)

x 2 (t)

3

2

1

0

0

1

2

3

4

5

t(s)

Fig. 12 State evolution of the open-loop system of Example 3 Fig. 13 Evolution of x T (t)r (t) of the open-loop system of Example 3

2 1.8 1.6

x T(t)r(t)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

12

t(s)

closed-loop system (55) is positive and FTS w.r.t. (t0 , t f , v, r (·)) under the proposed controller. When other parameters being settled, the relationships between β and τa , and between τa and t f are discussed. From Case i) in Theorem 3, the bounds of τa with varying β is shown in Fig. 16. We can see that the lower bound of τa will increase if β increases. The relationship between τa and t f is shown in Fig. 17. We can observe that the lower bound of τa will increase if t f increases. The simulations can support the analysis in Table 2.

58

M. Hu and J. H. Park

x 1 (t)

1.5

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7

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9

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t(s)

x 2 (t)

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3

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t(s)

Fig. 14 State evolution of the closed-loop system of Example 3 Fig. 15 Evolution of x T (t)r (t) of the closed-loop system of Example 3

1 0.9 0.8

x T(t)r(t)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

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5 t(s)

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5 Conclusion In this chapter, the finite-time stability and control issues of interval IPSs have been addressed. For different impulsive effects, the finite-time stability criterion has been proposed for interval IPSs by applying a time-varying copositive Lyapunov function. The association between the frequency of impulses and the finite time interval has been disclosed thoroughly by theoretical and simulation analysis. The finite-time controller of interval IPSs has been designed with the help of the finite-time stability

Finite-Time Stability and Control of Impulsive Positive Systems … Fig. 16 The bound of τa with varying β of Example 3

59

4 The uppper bound of

3.5

The lower bound of

a a

3

a

2.5 2

1.5 1 0.5 0

Fig. 17 The bound of τa with varying t f of Example 3

1

1.2

1.4

1.6

1.8

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2.6

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20 18

The upper bound of t f

16 14 12 10 8 6 4 2 0

0

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a

results. Finally, numerical examples have been demonstrated to show the usefulness of the obtained results. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (Grant No. 2019R1A5A 808029011).

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References 1. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, Hoboken (2011) 2. Hernandez-Vargas, E., Colaneri, P., Middleton, R., et al.: Discrete-time control for switched positive systems with application to mitigating viral escape. Int. J. Robust Nonlinear Control 21(10), 1093–1111 (2011) 3. Silva-Navarro, G., Alvarez-Gallegos, J.: On the property sign-stability of equilibria in quasimonotone positive nonlinear systems. In: Proceedings of IEEE 33rd Conference on Decision and Control, Lake Buena Vista, USA, 14–16 Dec 1994, pp. 4043–4048 4. Johnson, C.R.: Sufficient conditions for D-stability. J. Econ. Theory 9(1), 53–62 (2006) 5. Berman, A., Neumann, M., Stern, R.J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York (1989) 6. Rami, M.A., Tadeo, F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II Express Briefs 54(2), 151–155 (2007) 7. Liu, X., Yu, W., Wang, L.: Stability analysis for continuous-time positive systems with timevarying delays. IEEE Trans. Autom. Control 55(4), 1024–1028 (2010) 8. Zhao, X., Zhang, L., Shi, P., et al.: Stability of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012) 9. Qi, W., Park, J.H., Cheng, J., Chen, X.: Stochastic stability and L1 -gain analysis for positive nonlinear semi-Markov jump systems with time-varying delay via TS fuzzy model approach. Fuzzy Sets Syst. 371, 110–122 (2019) 10. Qi, W., Park, J.H., Cheng, J., Kao, Y., Gao, X.: Exponential stability and L1 -gain analysis for positive time-delay Markovian jump systems with switching transition rates subject to average dwell time. Inf. Sci. 424, 224–234 (2018) 11. Hu, M., Wang, Y., Xiao, J.: On finite-time stability and stabilization of positive systems with impulses. Nonlinear Anal.: Hybrid Syst. 31, 275–291 (2019) 12. Hu, M., Xiao, J., Xiao, R., Chen, W.: Impulsive effects on the stability and stabilization of positive systems with delays. J. Franklin Inst. 354(10), 4034–4054 (2017) 13. Yang, H., Zhang, Y.: Impulsive control of continuous-time homogeneous positive delay systems of degree one. Int. J. Robust Nonlinear Control 29(11), 3341–3362 (2019) 14. Dorato P (2006) An overview of finite-time stability. Current Trends in Nonlinear Systems and Control, pp. 185–194 15. Amato, F., Ambrosino, R., Ariola, M., et al.: Robust finite-time stability of impulsive dynamical linear systems subject to norm-bounded uncertainties. Int. J. Robust Nonlinear Control 21(10), 1080–1092 (2011) 16. Amato, F., De Tommasi, G., Pironti, A.: Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems. Automatica 49(8), 2546–2550 (2013) 17. Qi, W., Zong, G., Cheng, J., et al.: Robust finite-time stabilization for positive delayed semiMarkovian switching systems. Appl. Math. Comput. 351, 139–152 (2019) 18. Han, Z., Wu, H., Zhang, J.: Robust finite-time stability and stabilisation of switched positive systems. IET Control. Theory Appl. 8(1), 67–75 (2013) 19. Chen, G., Yang, Y.: Finite-time stability of switched positive linear systems. Int. J. Robust Nonlinear Control 24(1), 179–190 (2014) 20. Xiang, M., Xiang, Z.: Finite-time L1 control for positive switched linear systems with timevarying delay. Commun. Nonlinear Sci. Numer. Simul. 18(11), 3158–3166 (2013) 21. Amato, F., Ambrosino, R., Cosentino, C., De Tommasi, G.: Finite-time stabilization of impulsive dynamical linear systems. Nonlinear Anal.: Hybrid Syst. 5(1), 89–101 (2011) 22. Haddad, W.M., Chellaboina, V.S., Hui, Q.: Nonnegative and Compartmental Dynamical Systems. Princeton University Press, Princeton (2010)

An Eigenvector-Centrality Based Consensus Protocol Design for Discrete-Time Multi-agent Systems with Communication Delays M. J. Park, S. H. Lee, and O. M. Kwon

Abstract The leader-follower consensus protocol design problem for a class of discrete-time multi-agent systems (DT-MASs) with the communication delay is addressed in this work, wherein it is assumed that the communication delay is timevarying function. The proposed consensus protocol is substituted by applying the concept of network-centrality for agent, from now on, the protocol will be called to as the substitute consensus protocol. Here, the network-centrality is determined by the flow structure of the information among multiple agents. So, in analyzing the flow structure of the information between each agent, the graph representation is used. In order to find the most influential node, which means agent, within the graph, the various network-centralities have been attempted among the methods of analyzing the social networks. Through this work, we try to take the eigenvectorcentrality which is one of network-centralities. Here, its advantage is that the node’s influence is determined by the network-centrality of the node as well as the number of other nodes to which the node is connected. The substitute consensus protocol based on the eigenvector-centrality will be applied to the leader-follower consensus problem for second-order DT-MASs with time-varying communication delay. To do this, by utilizing the Lyapunov method and some mathematical techniques, sufficient conditions for the concerned problem will be proposed in terms of linear matrix inequality. Finally, a point-mass kinematics for aircraft modeled with the DT-MASs M. J. Park Center for Global Converging Humanities, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin 17104, Republic of Korea e-mail: [email protected] S. H. Lee · O. M. Kwon (B) School of Electrical Engineering, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju 28644, Republic of Korea e-mail: [email protected] S. H. Lee e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_3

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and its simulation results are given to verify the advantages of the proposed consensus protocol from a variety of perspective; as an example, the stability in consideration of time-delay. Keywords Eigenvector centrality · Discrete-time multi-agent system · Lyapunov method

1 Introduction Multi-agent systems have received considerable attention for their extensive application in many areas, including biology, physics, robotics, control engineering, and so on [1–3]. The major concern in the multi-agent systems is the issue of the consensus problem between the leader and the followers. In recent years, this issue has been used in many research areas: for examples, vehicle systems [4], intelligent decision support system for power grid dispatching [5] and networked control systems [6]. Because of the finite speed of signal processing in the flow of information between the two agents in the multi-agent systems, the multi-agent systems are used in the issue of the leader-follower consensus problem on communication time-delay. It is well known that time-delay often causes unwelcome dynamic behaviors such as poor performance and instability of the system. It should be pointed out that the issue of consensus between the two agents in the multi-agent systems with time-delay can be addressed by investigating the stability of the multi-agent systems. In addition, the issue of consensus between the two agents is one of prime research topics on the applications of the multi-agent systems. In this regard, various approaches to the issue of consensus problem on the multi-agent systems with timedelay are in the literature [7–14]. However, the above-mentioned literature mainly addressed the terms of consensus in the continuous-time multi-agent systems. In other words, the issue of consensus on the DT-MASs with time-delay has yet to be fully investigated. In this work, motivated by the above mentioned, a new delay-dependent consensus protocol design method for the DT-MASs with time-delay is studied. The main contribution of this work lies in: • The main question in this work is How can we weight the node to improve the stability criterion?. We pay attention to the eigenvector-centrality for each node as one answer to the question. By building a suitable Lyapunov–Krasovskii functional and using the matrix theory, the consensus criteria are derived from the aspect of of linear matrix inequality (LMI). There are some numerical simulations that show the effectiveness of the proposed method.

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1.1 Graph Theory and Notations In the multi-agent systems , the flow of information between each agent is portrayed by employing the directed graph G = (V, E), where it represents the sets of edge E = {(i, j) : i, j ∈ V} ⊂ V × V and node V = {1, 2, . . . , N }. From the graph G, the matrix containing elements that satisfy aii = 0 and ai j ≥ 0 is called as the adjacency matrix A = [ai j ] N ×N . If there is the flow of information between node i and node j, the elements of matrix A are described by ai j > 0 ⇔ (i, j) ∈ E. Ni = { j ∈ V : (i, j) ∈ E} denotes the set of neighbors of agent i. ⎤ ⎡ d1  ⎥ ⎢ The diagonal matrix defined as D = ⎣ . . . ⎦, where di = k∈Ni aik is the dN degree of node i, is the degree matrix of the graph G. The Laplacian matrix L of the graph G is obtained by L = D − A. More details can be seen in [15]. Throughout the work, the used notations are standard. Rn and Rm×n are, respectively, the sets of n-vectors and m × n matrices. In , and 0 are n × n identity matrix, and zero scalar or zero matrix of appropriate dimension. X ⊥ denotes a basis for the null space of X .

1.2 Advantage Claimed for the Use of Network-Centrality In the well-used simple consensus protocol: u i (t) =

N

ai j [ p j (t) − pi (t)],

(1)

j=1, j=i

nodes directly connected by the largest number of nodes are considered to be more important. The consensus protocol based on (1) applies to almost all existing work on the multi-agent systems; hereinafter, the protocol (1) should be referred to as the existing protocol. However, having more nodes in real networks in itself does not guarantee that someone is important. For example, from Fig. 1, the betweenness-centrality of edge between the two nodes is proportional to the thickness of the edges, while node colors indicate the degrees (white is the minimum degree). There, node 6 has the largest center-to-edge value from betweenness-centrality of edge between nodes 5 and 6, and nodes 6 and 7, while node 5 has the largest value of degree. According to the work [16], by reflecting the betweenness-centrality of the two nodes, the substitute protocol:

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Fig. 1 Structure example for information flow Table 1 Local information about each node of unknown 10-node graph Node 1 (1, 4) and (1, 5) Node 6 (6, 5) and (6, 7) Node 2

(2, 5)

Node 7

Node 3 Node 4 Node 5

(3, 1) and (3, 5) Node 8 (4, 2) and (4, 5) Node 9 (5, 1), (5, 3) and (5, 4) Node 10

(7, 6), (7, 8) and (7, 10) (8, 7) and (8, 9) (9, 8) and (9, 10) (10, 7)

(i, j) corresponds to the information flow: i → j

u ,i (t) =

N

 j ai j [ p j (t) − pi (t)]

(2)

j=1, j=i

can be used not only by the local information for the interconnection between each node, but also by the effect,  j , as the intermediary between each agent of edges. When viewing network-centrality, the protocol (1) considers only the flow structure of the local information between each node because the protocol (1) is obtained by the degree-centrality, that is, the number of nodes adjacent to the node. In degreecentrality, we consider that direct interconnection places more importance on information from more nodes. However, considering more important nodes in the actual networks provides stronger information for nodes. To obtain the value of the network-centrality of all the nodes, the overall structure of the flow of information between each node in the network should be identified beforehand. The answer is very simple: if we have the information on the value of ai j used in the existing protocol (1), we can easily draw the flow structure of the information between each node in the network that is needed for the substitute protocol (2). For example, suppose we only know the flow of information between each node listed in Table 1. By considering the flow structure of the local information for the interconnection between each node listed in Table 1, the overall structure of the flow of information between each node in the network can be easily constructed

An Eigenvector-Centrality Based Consensus Protocol Design …

65

as shown in Fig. 1. Therefore, knowing the flow structure of the local information for the interconnection between each node means that the value of the networkcentrality can be computed from the overall structure of the flow of information between each node in the network. In this sense, the next step is the procedure of forming the substitute protocol. From the above-mentioned point of view, the steps are as follows: • Step 1: The flow structure of the local information between each node created by the use of the local information about the direct neighbors of each node can be drawn to the overall flow structure of the information between each node G. • Step 2: From the graph G representing the local information, we can obtain the value of the network-centrality of each node. This value is used for the flow structure of the global information between each node because the information from both direct and indirect neighbors of each node is used to obtain the node and is defined as  j . • Step 3: By multiplying ai j and  j , the graph G is recreated as the weighted graph G reflecting the global flow of information between each node. Based on the weighted graph G , the existing protocol (1) can be converted to the substitute protocol (2). At that time, the weight  j corresponds to the value of the network-centrality of each node. Thus, because the design concept presented in this work does not alter the flow structure of the information between each node, the substitute protocol does not harm the distributed design. In addition, the value of ai j as well as the networkcentrality  j is easily obtained using NodeXL, which is one of the social network analysis tools. Therefore, not only the flow structure of the local information for the interconnection between each node but also its global flow of information reflected in ai j and  j , respectively, can be considered. In this respect, by adding the main role of each node based on the eigenvectorcentrality, a new consensus protocol will be proposed to improve the consensus performance in two aspects of the stability in consideration of time-delay and H∞ performance. The advantage of using the eigenvector-centrality centered is that the centrality is proportional to the sum of the centralities of the adjacent nodes, in other words, the eigenvector-centrality incorporates the importance of adjacent nodes to generalize the degree. This is the major contribution of this work. Noted that the concept of the network-centrality can be imposed in [17].

2 Problem Formation The DT-MASs to be focused in this work is chosen in the form given below: pi (tk+1 ) = pi (tk ) + τ vi (tk ), vi (tk+1 ) = vi (tk ) + τ u i (tk ) + wi (tk ), i ∈ I = {1, 2, . . . , N }.

(3)

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In the above equations, N stands for the number of agents, pi (tk ) ∈ Rn and vi (tk ) ∈ Rn are the position and the velocity vectors, respectively. u i (tk ) ∈ Rn is the consensus control input vector (the consensus protocol). wi (tk ) ∈ Rn is the external disturbance which belongs to l2 [0, ∞). τ is sampling period with assumption that 0 < τ < 1 between instant tk and tk+1 . The leader is considered as follows: p0 (tk+1 ) = p0 (tk ) + τ v0 (tk ), v0 (tk+1 ) = v0 (tk ) + τ u 0 (tk ),

(4)

where p0 (tk ) ∈ Rn and v0 (tk ) ∈ Rn represent the position and velocity vectors, and u 0 (tk ) ∈ Rn is the given control input vector. The output vector is defined as follows:

z i (tk ) =

pi (tk ) − vi (tk ) −

1 N 1 N

 N p j (tk ) j=1 N , i ∈ I. j=1 v j (tk )

(5)

This means the relative displacement mean for all agents. Before giving the substitute consensus protocol based the eigenvector-centrality, let us introduce the consensus protocol of the existing form: u i (tk ) = u 0 (tk ) + u i (tk ) + u iv (tk ) −bi [ pi (tk ) − p0 (tk ) + vi (tk ) − v0 (tk )] p

(6)

for i ∈ I, where p

u i (tk ) =

N

ai j [ p j (tk − μ(tk )) − pi (tk − μ(tk ))],

j=1, j=i

u iv (tk ) =

N

ai j [v j (tk − μ(tk )) − vi (tk − μ(tk ))].

j=1, j=i

Here, the interconnection weights ai j and bi are defined as:

ai j = 1 if agent i is connected to agent j, ai j = 0 otherwise, bi = 1 if leader is connected to agenti, bi = 0 otherwise.

And, the flow of the information is considered to be delayed by μ(tk ) with 1 ≤ μ(tk ) ≤ μ M , where μ M is the provided scalar value. Then, the considered system (3) can be transformed as

(7)

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ρi (tk+1 ) = ρi (tk ) + τ νi (tk ), N νi (tk+1 ) = νi (tk ) − τ li j [ρ j (tk − μ(tk )) + ν j (tk − μ(tk ))] j=1

−τ bi [ρi (tk ) + νi (tk )] + wi (tk ),

(8)

for i ∈ I, where ρi (tk ) = pi (tk ) − p0 (tk ), νi (tk ) = vi (tk ) − v0 (tk ) and li j defined as N li j = −ai j (i = j) and lii = − k=1, j=i lik is the elements of the Laplacian matrix L = [li j ] N ×N . Then, the matrix forms of system (8) and output (5) can be written by x(tk+1 ) = Ax(tk ) + B|L x(tk − μ(tk )) + Ww(tk ), z(tk ) = Dx(tk )

(9)

with ⎤ ρ1 (tk ) ρ(tk ) ⎥ ⎢ , ρ(tk ) = ⎣ ... ⎦ , x(tk ) = ν(tk ) ρ N (tk ) ⎤ ⎤ ⎡ ⎡ ν1 (tk ) w1 (tk ) ⎥ ⎥ ⎢ ⎢ ν(tk ) = ⎣ ... ⎦ , w(tk ) = ⎣ ... ⎦ ,

⎡



ν N (tk )

w N (tk )

where

A=



⎡

τ IN IN ⎢ ⊗ In , B = ⎣ −τ B I N − τ B



b1

⎤ ..

⎥ ⎦,

.



bN

0 0 0 B|L = ⊗ In , ⊗ In , W = −τ L −τ L IN 

0 I − N1 1 N 1TN ⊗ In . D= N 0 I N − N1 1 N 1TN Here, B|L means to include the matrix L. The objective of this work is to configure the weighted consensus protocol for the DT-MASs (9). That is, by the property of the eigenvector-centrality, the substitute protocol will be weighted to investigate the leader-follower consensus problem for any initial condition if and only if lim ( pi (tk ) − p0 (tk )) = 0,

tk →∞

lim (vi (tk ) − v0 (tk )) = 0, i ∈ I.

tk →∞

(10)

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Note that for any initial condition, the consensus protocol is known to resolve the leader-follower consensus problem asymptotically if and only if the positions and velocities of agents satisfy (10).

3 Main Results In this section, the substitute protocol is proposed by weighting the main role of each agent based on eigenvector-centrality.

3.1 Existing Leader-Follower Consensus Protocol In this subsection, the leader-follower consensus condition for the multi-agent systems (9) with the external disturbance is proposed with the existing protocol (6) first. Now, the following theorem is given as the first result. Proposition 1 Through the protocol (6), for the provided integer μ M equal to or greater than 1, the multi-agent systems (9) is asymptotically stable with H∞ attenuation γ if positive definite matrices P, Q, R ∈ R2N ×2N fulfilling the following LMI: T (Ξ + Σ)(Υ |L + We5T )⊥ < 0, (Υ |L + We5T )⊥

(11)

where ei = [02ν×(i−1)2ν , I2ν , 02ν×(4−i)2ν , 0ν ]T , i = 1, 2, . . . , 4, e5 = [02ν×8ν , Iν ]T , ν = N n, Ξ = e1 (P ⊗ In )e1T + e4 (P ⊗ In )e1T + [e4 (P ⊗ In )e1T ]T +e1 (Q ⊗ In )e1T − e3 (Q ⊗ In )e3T +μ2M e4 (R ⊗ In )e4T − (e1 − e2 )(R ⊗ In )(e1 − e2 )T −(e2 − e3 )(R ⊗ In )(e2 − e3 )T , Σ = e1 DT De1T − γ 2 e5 e5T , Υ |L = (A − I2N ⊗ In )e1T + B|L e2T + We5T − I2ν e4T , are present. Then, consensus of the multi-agent systems (9) can be made with H∞ disturbance attenuation γ under the existing protocol (6). Proof The Lyapunov–Krasovskii functional candidate is chosen as follows:

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V (tk ) = x T (tk )(P ⊗ In )x(tk ) tk −1 x T (s)(Q ⊗ In )x(s) + s=tk −μ M −1

+μ M

tk −1

Δx T (u)(R ⊗ In )Δx(u).

(12)

s=−μ M u=tk +s

The ΔV (tk ) defined as V (tk+1 ) − V (tk ), which is called as the forward difference of V (tk ), is calculated as ΔV (tk ) = x T (tk+1 )(P ⊗ In )x(tk+1 ) − x T (tk )(P ⊗ In )x(tk ) +x T (tk )(Q ⊗ In )x(tk ) − x T (tk − μ M )(Q ⊗ In )x(tk − μ M ) +μ2M Δx T (tk )(R ⊗ In )Δx(tk ) −μ M

tk −1

Δx T (s)(R ⊗ In )Δx(s).

(13)

s=tk −μ M

Its upper bound is obtained as by Jensen’s inequality ΔV (tk ) ≤ Δx T (tk )(P ⊗ In )Δx(tk ) + 2Δx T (tk )(P ⊗ In )x(tk ) +x T (tk )(Q ⊗ In )x(tk ) − x T (tk − μ M )(Q ⊗ In )x(tk − μ M ) +μ2M Δx T (tk )(R ⊗ In )Δx(tk ) ⎛ ⎞T ⎛ tk −1 tk −1 ⎝ ⎠ ⎝ − Δx(s) (R ⊗ In ) s=tk −μ(tk )

t

k −μ(tk )−1

−



Δx(s)

T

t

⎞ Δx(s)⎠

s=tk −μ(tk )

k −μ(tk )−1



(R ⊗ In )

s=tk −μ M



Δx(s) .

(14)

s=tk −μ M

As a result, a new upper bound of ΔV (tk ) is ΔV (tk ) ≤ ζ T (tk )Ξ ζ(tk ),

(15)

where ζ(tk ) = [x T (tk ), x T (tk − μ(tk )), x T (tk − μ M ), Δx T (tk ), w T (tk )]T . It should be noted that if (15)< 0 is maintained, then there is a scalar  > 0 such that ΔV (tk ) + z T (tk )z(tk ) − γ 2 w T (tk )w(tk ) < −x T (tk )x(tk ). From the above inequality, the following relationship is obtained:

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ΔV (tk ) + z T (tk )z(tk ) − γ 2 w T (tk )w(tk ) < 0.

(16)

According to [18], the multi-agent systems (9) with an initial condition of zero and a given constant γ > 0 should be asymptotically stable in view of H∞ attenuation γ if the following inequality is met: Ψ =

∞ (z T (tk )z(tk ) − γ 2 w T (tk )w(tk )) < 0.

(17)

tk =0

In this regard, taking summation from 0 to ∞ on both sides of (16) yields ∞ V (∞) − V (0) + (z T (tk )z(tk ) − γ 2 w T (tk )w(tk )) < 0. tk =0

Because of V (0) = 0 with the initial condition of zero and V (∞) → 0, we obtain the inequality (17). Hence, it can be ensured by the Lyapunov stability theory that the system (3) is stable. The Ψ can be represented as follows: Ψ =

∞ 

 z T (tk )z(tk ) − γ 2 w T (tk )w(tk ) + ΔV (tk ) − V (tk+1 ).

(18)

tk =0

Considering V (tk+1 ) ≥ 0, the bound of Ψ is Ψ ≤

∞ 

x T (tk )DT Dx(tk ) − γ 2 w T (tk )w(tk ) + ΔV (tk )



tk =0

≤

∞

ζ T (tk )(Σ + Ξ )ζ(tk ).

(19)

tk =0

Here, the Ψ < 0 is equivalent to Ξ + Σ < 0.

(20)

Therefore, if (20) satisfies, then system (9) is asymptotically stable in view of H∞ attenuation γ. Moreover, the following zero equality can be obtained from the system equation (9) (Υ |L + We5T )ζ(tk ) = 0.

(21)

From Finsler’s lemma [19], if (20) subject to (21) holds, then the LMI (11) satisfies. It can be concluded that if (11) holds, then system (9) is asymptotically stable. This completes our proof. 

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From Proposition 1, we make the stability condition for the system (9). The system without the external disturbance is described here as x(tk+1 ) = Ax(tk ) + B|L x(tk − μ(tk )).

(22)

Now, the second criterion is given as follows: Proposition 2 Through the protocol (6), for the provided integer μ M equal to or greater than 1, the multi-agent systems (22) is asymptotically stable with H∞ disturbance attenuation γ if positive definite matrices P, Q, R ∈ R2N ×2N fulfilling the following LMI: T Ξ (Υ |L )⊥ < 0, (Υ |L )⊥

(23)

are present. Then, consensus of the multi-agent systems (22) can be made under the existing protocol (6). Proof By taking the place of ei (i = 1, 2, . . . , 4) in Proposition 1 with [0ν×(i−1)ν , Iν , 0ν×(4−i)ν ]T , 

the LMI (23) can be easily induced.

3.2 Good Substitute Leader-Follower Consensus Protocol Based on the Eigenvector-Centrality By weighting the values of the eigenvector-centrality of each agent, the substitute protocol is introduced in this subsection. The substitute protocol based on the eigenvector-centrality to be proposed in this work is constructed as the form given below: u eigen,i (tk ) = u 0 (tk ) + u eigen,i (tk ) + u veigen,i (tk ) −bi [ pi (tk ) − p0 (tk ) + vi (tk ) − v0 (tk )], p

where p

u eigen,i (tk ) =

N

ceigen, j ai j [ p j (tk − μ(tk )) − pi (tk − μ(tk ))],

j=1, j=i

u veigen,i (tk ) =

N j=1, j=i

ceigen, j ai j [v j (tk − μ(tk )) − vi (tk − μ(tk ))].

(24)

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Here, the eigenvector of the adjacency ⎤ of the structure of information flow ⎡ matrix A ceigen,1 ⎥ ⎢ of network G obtained as Ceigen = ⎣ ... ⎦ , where ceigen, j is the value of the ceigen,N eigenvector-centrality of agent j. Remark 1 The graph G should be strongly connected to give weight to the eigenvector-centrality. Perron-Frobenius theorem [20] allows us to calculate the eigenvalues of A and then select the largest eigenvalues. Then, by Perron-Frobenius theorem, all elements of the eigenvector Ceigen are positive, and the element of this vector corresponds to the value of eigenvector-centrality for the graph G. As a result, the weight of each edge in the substitute protocol (24) will be the tail agent’s eigenvector-centrality value. For the substitute protocol (24), the following results are obtained from Propositions 1 and 2. Theorem 1 Through the protocol (24), for the provided integer μ M equal to or greater than 1, the multi-agent systems (9) is asymptotically stable with H∞ attenuation γ if positive definite matrices P, Q, R ∈ R2N ×2N fulfilling the following LMI: T (Ξ + Σ)(Υ |Leigen + We5T )⊥ < 0. (Υ |Leigen + We5T )⊥

(25)

are present. Then, consensus of the multi-agent systems (9) can be made with H∞ disturbance attenuation γ under the substitute protocol (24). Proof By taking the place of L in Proposition 1 with Leigen = [leigen,i j ] N ×N , where  leigen,i j = −ceigen, j ai j for i = j and leigen,ii = − Nj=1, j=i leigen,i j , the condition (25) can be easily derived.  Theorem 2 Through the protocol (24), for the provided integer μ M equal to or greater than 1, the multi-agent systems (22) is asymptotically stable if positive definite matrices P, Q, R ∈ R2N ×2N fulfilling the following LMI: T Ξ (Υ |Leigen )⊥ < 0. (Υ |Leigen )⊥

(26)

are present. Then, consensus of the multi-agent systems (22) can be made under the substitute protocol (24). Proof By taking the place of L in Proposition 2 with Leigen , the condition (26) can be easily derived. 

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4 Numerical Simulations In this section, the advantages of the eigenvector-centrality based consensus protocol are shown by comparing Propositions (the existing case) and Theorems (the eigenvector-centrality-based case). Consider the network consisting of 3-aircrafts with the flow of the information shown in Fig. 2a. For each aircraft, its point-mass kinematics can be formulated from the two frames [21]: the inertial frame fixed to inertial space (subscript I ) and the body frame fixed to body (subscript b), shown in Fig. 2b. By choosing the former, the point-mass kinematics of each aircraft can be expressed by p˙ I = v I , 1 v˙ I = F I , m

(27)

    where p I = col px,I , p y,I , pz,I and v I = col vx,I , v y,I , vz,I are, respectively, the translational position and the translational velocity, F I and m, respectively, the force and the mass of each aircraft. With the multi-agent systems (3) of the discrete-time form, the above kinematics of each aircraft can be reformed as follows: p I,i (tk+1 ) = p I,i (tk ) + 0.1v I,i (tk ), v I,i (tk+1 ) = v I,i (tk ) + 0.1F I,i (tk ) + wi (tk ), i ∈ I,

(28)

where the values of m and τ are assumed as m = 1 and τ = 0.1. In order to apply the consensus protocol, F I,i (tk ) is replaced with u i (tk ) in (6) or u eigen,i (tk ) in (24). From Fig. 2a, the adjacency matrix can be obtained as

(a) Information flow.

Fig. 2 3-Aircrafts

(b) Two frames.

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Table 2 Laplacian matrices L

⎡

Leigen

⎡

⎤

⎤ 1.2959 -0.6479 -0.6479 ⎢ ⎥ ⎢ -0.4004 0.8009 -0.4004 ⎥ ⎣ ⎦

2 -1 -1 ⎢ ⎥ ⎢ -1 2 -1 ⎥ ⎣ ⎦ -1 0 1

-0.6479

Table 3 Minimum values of γ with μ M = 3 Methods Proposition 1 γmin

0

0.6479

Theorem 1

36.4153

19.7765

⎡

⎤ 011 A = ⎣1 0 1⎦. 100 Then, the eigenvector to obtain the values of the eigenvector-centrality of all aircrafts is computed as ⎡ ⎤ 0.6479 Ceigen = ⎣ 0.4004 ⎦ . 0.6479 Thus, the adjacency matrix with the eigenvector-centrality can be obtained as ⎡

Aeigen

⎤⎡ ⎤ ⎡ ⎤ 0.6479 0 0 011 0 0.6479 0.6479 = ⎣ 0 0.4004 0 ⎦ ⎣ 1 0 1 ⎦ = ⎣ 0.4004 0 0.4004 ⎦ . 0 0 0.6479 100 0.6479 0 0    A

In addition, the corresponding Laplacian matrices are listed in Table 2. To rule out the impact of choosing the connection between the leader any agent, the leader ⎡ and⎤ 100 assumes that it is connected to all agents, that is, B = ⎣ 0 1 0 ⎦. 001 The followings are assumed as conditions for simulation:

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Agent 1 u1 (t k)

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Fig. 3 Trajectories of the consensus protocol based on the eigenvector-centrality

Fig. 4 The position trajectories of the three aircrafts

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v 1 (t k)-v0(t k)

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Fig. 5 The velocity error trajectories of the three aircrafts

⎧ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −8 0 1 ⎪ ⎪ ⎪ ⎪ ⎣ 10 ⎦ , v I,1 (0) = ⎣ 0 ⎦ , w1 (tk ) = ⎣ 1 ⎦ p (0) = ⎪ I,1 ⎪ ⎪ ⎪ 8 0 1 ⎪ ⎪ ⎪ ⎪ when t ∈ [70, 90], ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ 14 0 1 ⎪ ⎪ ⎨ p I,2 (0) = ⎣ 6 ⎦ , v I,2 (0) = ⎣ 0 ⎦ , w2 (tk ) = ⎣ 1 ⎦ Agents : 4 0 1 ⎪ ⎪ ⎪ ⎪ when t ∈ [30, 60], ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ 10 0 1 ⎪ ⎪ ⎪ ⎪ ⎣ 4 ⎦ , v I,3 (0) = ⎣ 0 ⎦ , w3 (tk ) = ⎣ 1 ⎦ p (0) = ⎪ I,3 ⎪ ⎪ ⎪ 12 0 1 ⎪ ⎪ ⎩ when tk ∈ [50, 80], ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 1 Leader : p I,0 (0) = ⎣ 1 ⎦ , v I,0 (0) = ⎣ 1 ⎦ , u 0 (0) = ⎣ 0.5 ⎦ . 1 1 0 The advantages of the substitute protocol (24) based on the eigenvector-centrality are illustrated from the following two perspectives:

An Eigenvector-Centrality Based Consensus Protocol Design … Table 4 Maximum bounds Methods μM

Proposition 2

Theorem 2

3

5

Table 5 Maximum bounds on μ M Methods Proposition 2 μM

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Theorem 2

2

4

• H∞ performance The minimum values of γ for H∞ performance are given in Table 3 and computed by programming the following optimization problem: for given μ M = 3, if min

P,Q,R

γ s.t. LMI (3.11)

has solutions with P, Q, R and γ, then, the protocol (6) is the leader-follower H∞ consensus protocol for the multi-agent systems (9) and H∞ disturbance attenuation is minimized by γ. Thus, from a view of performing H∞ in Table 3, the substitute protocol plotted by Fig. 3 can be more effective than the existing protocol. Moreover, Figs. 4 and 5 show the positions and velocities of three aircraft follow the leader’s trajectory, even if through having the external disturbance. • Stability in consideration of time-delay In Table 4, the value of maximum bound of time-delay obtained using the eigenvector-centrality based consensus protocol (24) is greater than the one with the existing protocol (6). This means that the value of maximum bound of timedelay under the substitute protocol based on the eigenvector-centrality for each agent that guarantees the stability of multi-agent systems can be further increased. To verify the results listed in Table 4, the root mean square of position error between the leader and each follower defined as in an equation: ! rmse(tk ) =

3

i=1 ( pi (tk )

3

− p0 (tk ))2

,

according to each protocol are drawn in Fig. 6. The detail is that when there is a time-varying delay that satisfies 1 ≤ μ(tk ) ≤ 5 in this figure, the substitute protocol based on the eigenvector-centrality ensures the stability of the multi-agent system, but the existing protocol does not. Fig. 7 shows the position trajectories of the three aircrafts system (28) under the substitute protocol. We can see three aircraft are following the leader’s trajectory. In addition, consider the network consisting of 5-nodes with the flow of the information shown in Fig. 8. From this graph, the adjacency matrix can be known as

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9 8

rmse(t k)

7 6 5 4 3 2 1 0

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⎡

0 ⎢1 ⎢ A=⎢ ⎢1 ⎣0 1

1 0 0 0 0

1 1 0 1 1

0 0 1 0 0

⎤ 1 1⎥ ⎥ 1⎥ ⎥. 0⎦ 0

Then, the eigenvector to calculate the values of the eigenvector-centrality of all nodes is obtained as

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Fig. 8 5-nodes network

Fig. 9 The trajectories of rmse(t k) = "

60

with

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i=1 ( pi (tk )− p0 (tk ))

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each protocol rmse(t k)

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⎤ 0.5251 ⎢ 0.2081 ⎥ ⎢ ⎥ ⎥ =⎢ ⎢ 0.5917 ⎥ . ⎣ 0.2345 ⎦ 0.5251

From this, the corresponding Laplacian matrix can be computed as ⎡

Leigen

⎤ 1.5754 −0.5251 −0.5251 0 −0.5251 ⎢ −0.2081 0.6244 −0.2081 0 −0.2081 ⎥ ⎢ ⎥ ⎢ 0 1.7751 −0.5917 −0.5917 ⎥ = ⎢ −0.5917 ⎥. ⎣ ⎦ 0 0 −0.2345 0.2345 0 −0.5251 0 −0.5251 0 1.0503

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From the perspective of the stability in consideration of time-delay, the maximum bounds of time-delay under two protocols are listed in Table 5. From Fig. 9, the results listed in Table 5 can be confirmed when 1 ≤ μ(tk ) ≤ 4. Also, the results of Table 5 and Fig. 9 show that the substitute protocol has better performance. Note that the simulation conditions, such as the initial value of the leader, were randomly set.

5 Conclusion The issue of the leader-follower consensus problem for the DT-MASs was been investigated in this work. To improve the consensus performance in two aspects of H∞ performance and the stability in consideration of time-delay, the weighted consensus protocols were proposed by calculating the eigenvector-centrality for agents in a network. To achieve this, by building simple Lyapunov–Krasovskii functional, sufficient conditions for such problems for the proposed protocols were derived from the LMI perspective. Some numerical simulations were provided to demonstrate the effectiveness of the proposed protocols. Moreover, for the secure communication between the leader and any the selected agent, our future works will focus on grafting a new pinning control concept onto this work. Acknowledgements This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT; Ministry of Science and ICT) (NRF-2017R1C1B5076 878).

References 1. Saber, R.O., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95, 215–233 (2007) 2. Ren, W., Beard, R.W.: Consensus seeking in multi-agent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50, 655–661 (2005) 3. Wang, J., Cheng, D., Hu, X.: Consensus of multi-agent linear dynamic systems. Asian J. Control 10, 144–155 (2008) 4. Ren, W.: Consensus strategies for cooperative control of vehicle formations. IET Control Theory Appl. 1, 505–512 (2007) 5. Qiong, W., Wenyin, L., Yihan, Y., Chuan, Z., Yong, L.: Intelligent decision support system for power grid dispatching base on multi-agent system. In: International Conference on Power System Technology (PowerCon 2006), pp. 1–5 (2006) 6. DeLellis, P., di Bernardo, M., Garofalo, F., Liuzza, D.: Analysis and stability of consensus in networked control systems. Appl. Math. Comput. 217, 988–1000 (2010) 7. Yu, W., Ghen, G., Gao, M.: Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica 46, 1089–1095 (2010) 8. Song, Q., Cao, J., Yu, W.: Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. Syst. Control Lett. 59, 553–562 (2010) 9. Xue, D., Yao, J., Wang, J., Guo, Y., Han, X.: Formation control of multi-agent systems with stochastic switching topology and time-varying communication delays. IET Control Theroy Appl. 7, 1689–1698 (2013)

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10. Lu, X., Austin, F., Chen, S.: Formation control for second-order multi-agent systems with time-varying delays under directed topology. Commun. Nonlinear Sci. Numer. Simulat. 17, 1382–1391 11. He, P., Li, Y., Park, J.H.: Noise tolerance leader-following of high-order nonlinear dynamical multi-agent systems with switching topology and communication delay. J. Frankl. Inst. 353, 108–143 12. Liu, X., Dou, L., Sun, J.: Consensus for networked multi-agent systems with unknown communication delays. J. Franklin Inst. 353, 4176–4190 (2016) 13. Ding, L., Guo, G.: Sampled-data leader-following consensus for nonlinear multi-agent systems with Markovian switching topologies and communication delay. J. Franklin Inst. 352, 369–383 (2015) 14. Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M., Cha, E.J.: Randomly changing leader-following consensus control for Markovian switching multi-agent systems with interval time-varying delays. Nonlinear Anal.-Hybrid Syst. 12, 117–131 (2014) 15. Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001) 16. Park, M.J., Lee, S.H., Kwon, O.M., Park, J.H., Choi, S.G.: Betweenness centrality-based consensus protocol for second-order multiagent systems with sampled-data. IEEE Trans. Cybern. 47(8), 2067–2078 (2017) 17. Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010) 18. Anton, S.: The H∞ Control Problem. Prentice Hall, New York (1992) 19. de Oliveira, M.C., Selton, R.E.: Stability Tests for Constrained Linear Systems. Springer, Berlin (2001) 20. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambride University Press, New York (1990) 21. Stengel, R.F.: Flight Dynamics. Princeton University Press, New Jersey (2004)

Fault-Tolerant Sampled-Data Synchronization of Chaotic Systems with Random Occurring Uncertainties: A Semi-Markov Jump Model Approach Yude Xia, Yudong Wang, Jing Wang, and Hao Shen

Abstract This work focuses on addressing the issue of sampled-data synchronization for nonlinear chaotic systems with uncertainties by fault-tolerant control techniques, where a semi-Markov jump chain is utilized to describe the switching characteristic. The random occurring uncertainties in chaotic systems are subject to the Bernoulli distributed white noise sequences therein. For such a synchronization control problem, the sampled-data control strategy, capable of alleviating the bandwidth pressure is applied, in which the random time-varying delays are adequately cogitated. Then, by establishing a proper Lyapunov–Krasovskii functional, some constrained criteria are obtained via the rational use of integral inequality technique. The validity and effectiveness of the developed methods are, at last, substantiated by a numerical example. Keywords Chaotic systems · Sampled-data fault-tolerant synchronization · Semi-Markov jump model · Randomly occurring uncertainties

1 Introduction Stochastic jumping systems (SJSs), revealing the hybrid dynamic properties of systems, have gotten extensive applications in many practical situations for its great capability in modeling [1–5]. The SJSs consist of some subsystems and the switching among them are governed by some stochastic processes, such as Markov process (MP) and semi-Markov process (SMP) [6, 7]. What is important in the long and arduous process of the analysis and synthesis on Markov jump systems (MJS) is an ideal Y. Xia · J. Wang · H. Shen (B) School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, People’s Republic of China e-mail: [email protected] Y. Wang School of Metallurgy Engineering, Anhui University of Technology, Ma’anshan 243002, People’s Republic of China © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_4

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hypothesis that the sojourn-time subject to exponential distribution in MP. More precisely, in this hypothesis, the transition rate is represented by a time-invariant matrix and is not only known but also memoryless. Subsequently, two questions arise, one is that the transition rate is too hard to acquire in advance, and the other one is that memoryless transition rate is untoward to accommodate the practical situations. The features mentioned above bring many restrictions to the applications of MJSs. Compared with the MP, the SMP is with the property of a transition rate matrix and a probability density function (PDF), which are changeable. By right of relaxing the restrictions on the condition that the sojourn-time obeys to an exponential distribution, the SMP has been widely used in numerous situations. Consequently, the researches on semi-Markov jump systems have enchanted a crowd of scholars close attention [8, 9]. In parallel, the discovery of chaotic phenomena is one of the most important achievements in nonlinear disciplines. Chaos is considered to be the complex form of motion in the dynamic systems, which contains various states of dynamic systems [10, 11]. As an important research direction of chaos dynamics, chaos synchronization theories have become the key to open the gate of theoretical researches in chaotic systems, and its research progress has become a hot spot in many fields [12–14]. The work in [15] studied the synchronization condition of chaotic systems. As stated in [16], the pertinence of chaos synchronization in physiology, nonlinear optics and dynamics has been concerned. However, on account of the interference of external factors including imprecise modeling and changeable operating point, which result in the performance of the system may become deteriorating degenerate and even the instability of the system may occur. Hence, a good many researchers focus on the study of robustness analysis [17, 18]. Recently, the authors in [19] investigated the random occurring uncertainties (ROUs) on account of a fact that during the system operation may be suffered from probabilistic uncertainties. In this regard, the issue for synchronization of semi-Markov jump chaotic systems (SMJCSs) with ROUs is worth exploring. When designing controllers to realize synchronization of forced-unforced systems, the conventional control scheme commonly considered that system information at any time will be transferred to the controller, and a large number of network resources are occupied in this case. This is, whereas, unreasonable due to limited network resources. Facing this problem, the controller in the sampled-data techniques was designed, in which the control signals can only be changeable at the sampling instant, while remaining constant otherwise. Therefore, on this occasion, the limited network resources could effectively economize. Relatively speaking, the sampled-data controller has the advantages of high accuracy, effective suppression of interference and favorable commonality. By the aid of these merits, some accomplishments have been acquired [20, 21]. As concerned in [22] , the neural networks reached the synchronous state driven by a fault-tolerant controller using sampled-data. Furthermore, influenced by some practical conditions, time delays phenomena [29], which often makes the system property decline or even causes instability, constantly appear in industrial processes and engineering systems [23–25]. Meanwhile,

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the delay parameters in the systems are usually unknown or time-varying. And the existence of the time delays verge to be random, which may make the comprehension and analysis of the system more difficult. Therefore, the researches of random time-varying delays (TVDs) systems belonging to the main domain of control engineering, have been conducted for various systems in recent years. To name just a few, networked control systems [26], neural networks [27, 28], etc. It should be pointed out that the works aforesaid do not with a view to the condition that the actuator may fail during the execution. Howbeit, in the actual control process, these failures, which may result in the instability of the systems, are sometimes difficult to avoid. Based on this reason, the strategy of fault-tolerance control is introduced in this chapter, which allows or tolerates the occurrence of errors within certain limits in the systems, and making the systems more stable and reliable. In the regard, how to handle the fundamental but essential problem for time-varying delayed systems becomes the motivation of this work. Inspired by the above discussions, the attention is focused on disposing of the fault-tolerant synchronization issue of SMJCSs with ROUs, where the sampled-data control scheme is concerned to make the proposed method more pragmatic. To extract the core works from three aspects: (1) As the first attempt, the fault-tolerant sampled-data synchronization of switched chaotic systems with ROUs is investigated, and the switching among these subsystems is governed by the SMP. (2) By utilizing applicable integral inequalities to acquire the less conservative results, some constrained criteria are proposed to force the error system to be synchronously stable. (3) The cost-efficient sampled-data controller, which involves the randomness of TVDs, is designed to reduce a mass of load pressure on transferring data in the SMJCSs. Notations: The notations employed can refer to [30], hence it is omitted here.

2 Preliminaries In this section, by modifying the model in [30], a kind of unforced and forced SMJCSs with ROUs are concerned as follow   ˆ ˆ x˙ (t) = ( Aˆ (ς (t)) + δ(t)Δ A(t)) x(t) ˆ + ( Bˆ (ς (t)) + δ(t)Δ B(t)) fˆ x(t) ˆ , (1) ˆ ˆ y˙ (t) = ( A (ς (t)) + δ(t)Δ A(t)) yˆ (t)   ˆ fˆ yˆ (t) + u(t), (2) +( Bˆ (ς (t)) + δ(t)Δ B(t))

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in which xˆ (t) ∈ Rn and yˆ (t) ∈ Rn are state vectors of unforced and forced systems, separately; u(t) ∈ Rn expresses the control input. Aˆ (ς (t)) and Bˆ (ς (t)) are system matrices. ς (t) stands for the SMP which take values from a denumerable aggregate Sˇ  {1, 2, . . . , r }. Denote Aˆ v  Aˆ (ς (t)) for each ς (t)  v ∈ Sˇ and the others are similarly denoted. Furthermore, its transition rate matrix Π ()  {πvn ()} is formulated as  Pr {ς (t + ) = n|ς (t) = v} =

πvn ()  + o () , v = n, 1 + πvv ()  + o () , v = n,

(3)

where  > 0 is the sojourn time, lim→0 (o () /) = 0 and πvn () ≥ 0, (for v = n),  t+ is the transition rate from mode v −−→ n and πvv () = − πvn (). t

ˇ =v n∈ S,n

And, fˆ : R → R is assumed that as a nonlinear vector which satisfies the global Lipschitz condition, that is n

n

          ˆ  f a` − fˆ b`  ≤ l` a` − b`  ,

(4)

` with respect to a positive scalar l. ˆ and Δ B(t) ˆ Meanwhile, Δ A(t) are the uncertainties satisfying

ˆ [E a E b ] , ˆ Δ B(t) ˆ Δ A(t) = Dˆ F(t)

(5)

ˆ is restrained by ˆ E a and E b are given real matrices and F(t) where D, ˆ Fˆ T (t)F(t) ≤ I, ∀t ≥ 0.

(6)

For the synchronization purpose, the synchronization error can be marked as e(t)  yˆ (t) − x(t) ˆ and the following error system is embodied as follow naturally ˆ ˆ ˆ ˆ A(t))e(t) + ( Bˆ v + δ(t)Δ B(t)) f˜ (e(t)) + u(t), e˙ (t) = ( Aˆ v + δ(t)Δ

(7)

in which f˜ (e(t))  fˆ(y(t)) − fˆ(x(t)). ˆ obeying Bernoulli disTo illustrate the ROUs, ones adopt a stochastic variable δ(t) tribution and complying with the following stochastic distribution rules   ˆ = 1 = δ, ˆ Pr δ(t) ˆ = 0 = 1 − δ, ˆ Pr δ(t) where 0 ≤ δˆ ≤ 1. Then, the error system (7) can be expressed as

(8)

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

⎧ ˆ (t) , ⎨ e˙ (t) = Aˆ v e(t) + Bˆ v f˜ (e(t)) + u(t) + δ(t)Dp p (t) = F (t) q (t) , ⎩ q (t) = E a e(t) + E b f˜ (e(t)) .

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(9)

It is worthy noting that the sampled-data controller plays a key role in modern networked systems. In this regard, it is of great importance to show solicitude for sample-data control method. Hence, in this work, the sampled signal of feedback controller is generated by making use of a zero-order hold function with a set of hold-times 0 = t0 < t1 < · · · < tk < · · · , and its concrete expression is shown as follow (10) u(t) = K e(tk ), tk ≤ t < tk+1 , k = 0, 1, 2, . . . , N , here, K is the gain parameter to build a sampled-data controller and tk is the instantaneous time which updates by the zero order holder. Meanwhile, e(tk ) is the measured value of e(t) at sampling instantaneous time tk , limk→∞ tk = +∞. Defining tk+1 − tk  dk , and dk > 0 means the sampling interval. Assume that there exist m sampling intervals d0 , d1 , . . . , dm with 0 < d0 < d1 < · · · < dm . Then, setting τ (t)  t − tk , tk ≤ t < tk+1 , expression (10) is rewritten as follow u(t) = K e(tk ) = K e(t − τ (t)), tk ≤ t < tk+1 ,

(11)

and the time-varying delays τ (t) contents τ˙ (t) = 1. For the purpose of designing a controller owns the property of random TVDs , the stochastic variable αi (t) can be defined as follow  αi (t) =

1 di−1 ≤ τi (t) ≤ di , i = 1, 2, . . . , m. 0 otherwise.

Here, one supposes the stochastic variable αi (t) subjects to the Bernoulli distribution as well. Thus, we have Pr {αi (t) = 1} = αi , Pr {αi (t) = 0} = 1 − αi , i = 1, 2, . . . , m. Remark 1 TVDs are ubiquitous in the practical systems, which interferes with the system performance. Therefore, in this chapter, the Bernoulli distribution is employed to govern the occurrence of the TVDs, which reflects the randomness of the TVDs. Remark 2 As far as our information goes, when studying the controller design of the SMJCSs, the majority of results are based on an assumption that the complete mode information of the systems can be obtained by the mode-dependent controller on time. However, this assumption may sometimes be unsatisfied with actual applications. So as to surmount the defect of the mode-dependent controllers, we introduce the mode-

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independent one, which is more powerful and has a wide range of applications when the jump mode is not applied to the controller. Under some practical conditions, it is frequent that the actuator fails on account of the chaotic systems. Taking this feature into account, we derive the actuator fault model from [31]. For the control input, more concretely, let u f (t) be the signal sent from actuators, obeying (12) u f (t) = Fv u(t), in which

  Fv = diag f v1 , f v2 , . . . , f vn ,

l l and f v+ and satisfying 0 ≤ is actuator faults matrix, with the known constants f v− l l l f v− ≤ f v ≤ f v+ ≤ 1, l = 1, 2, . . . , n. By setting l l l f l + f v+ f l − f v− f l − f 0v l , m lv = v , j l = v+ f 0v = v− , (13) l l 2 2 f v+ + f v−

then, it is clear that Fv = F0v (I + Mv ) , |Mv | ≤ Jv ≤ I,

(14)

where      1    n , . . . , f 0v , |Mv |  diag m 1v  , . . . , m nv  , Jv  diag jv0 , . . . , jvn . F0v  diag f 0v Thus, the system (9) with m sampling intervals can be expressed as ⎧ m  ⎪ ⎪ e˙ (t) = Aˆ v e(t) + Bˆ v f˜ (e(t)) + αi (t) F0v (I + Mv ) K e(t − τi (t)) ⎪ ⎪ ⎨ i=1 ˆ + δ(t)Dp (t) , ⎪ ⎪ p = F (t) q (t) , ⎪ (t) ⎪ ⎩ q (t) = E a e(t) + E b f˜ (e(t)) ,

(15)

in which di−1 ≤ τi (t) ≤ di . Remark 3 In theoretical research, it is generally considered that the system is operating in an ideal environment. On the contrary, the operation of the system will be interfered by external factors such as parameter errors and component aging. Hence, it is spontaneous to hypothesize system uncertainties for the sake of overcoming this situation. Moreover, the uncertainties sometimes exist in a random way during the system’s operation, such as, random failures caused by the network and sudden environmental disturbances. Therefore, it is worthy to concern about ROUs for reality.

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

89

Remark 4 The random variable δˆ (t) is applied to modeling the probability distri  2  ˆ ˆ ˆ ˆ = bution of ROUs, which meets conditions that E δ (t) = δ and E δ (t) − δ   δˆ 1 − δˆ . The description like this can trace to [19] and has never been concerned in the control problem of SMJCSs.

Lemma 1 ([32]) For any given differentiable function σ in α, ´ β´ → Rn , and a symmetric matrix > 0, it is confirmed that: 

β´ α´

σ˙ T (s) σ˙ (s) ds ≥

W β´ − α´

+

3W˜ , β´ − α´

(16)

where     T      ˚ ˚ T W, W  σ β´ − σ α´ σ β´ − σ α´ , W˜  W  β´     ˚  σ β´ + σ α´ − 2 σ (s) ds T . W β´ − α´ α´

3 Stability Analysis This part is aimed at designing a sampled-data feedback controller to realize the synchronization of the unforced and forced systems. Thereinto, the switching among aforesaid two systems are described by the SMP. Now, we have the following main result. ˜ Theorem 1 For given positive constants γ , δˆ ∈ [0, 1], ε λ and known matrices F, ˆ A, ˆ Bˆ with appropriate dimensions, if there exists matrices Pv > 0, ˜ E a , E b , D, S, Q i > 0, Ri > 0, Z iv > 0, Ti > 0, any matrices Si , S˜i , and a symmetric matrices G satisfying the inequalities for i = 1, 2, . . . , m, v ∈ Sˇ ⎡

⎤ Ξ˜ 1v Φ1 Φ2 Ξ1v  ⎣  Φ3 Φ4 ⎦ < 0,   Φ5   Ri Si < 0, Ξ2  −  Ri   Ziv S˜i Ξ3v  − < 0,  Ziv  Ξ4v  h i π˚ vn Z in − Ti < 0, n∈ Sˇ

where

(17)

(18) (19) (20)

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πvn () Pn + Q 1 − 4R1 − 4Z 1v + λl 2 +G Aˆ v + εE aT E a + Aˆ vT G T ,

n∈ Sˇ

⎧ ⎪ (1, 1)  −2R1 − S11 − S13 − S12 − S14 − 2Z 1v − S˜11 − S˜13 ⎪ ⎪ ⎪ ⎪ − S˜12 − S˜14 + G F0v (I + m v ) K , ⎪ ⎪ ⎪ ⎪ ⎨ (1, 2)  S11 + S13 − S12 − S14 + S˜11 + S˜13 − S˜12 − S˜14 , Φ1  (1, 3)  6R1 + 6Z 1v , ⎪ ⎪ ⎪ (1, 4)  2S12 + 2S14 + 2 S˜12 + 2 S˜14 , ⎪ ⎪ ⎪ ⎪ (1, i)  αi G F0v (I + m v ) K , i = 5, 9, 13, . . . , 4m + 1, ⎪ ⎪ ⎩ other wise  0,       Ri 0 Z iv 0 , Ziv  , Φ2  Φ21 Φ31 δG Dˆ , Ri  0 3Ri 0 3Z iv Φ21  G B + εE aT E b , Φ31  −G + γ Aˆ vT G T + Pv , h i  di − di−1 ,  T T G , i = 1, 5, . . . , 4m − 3, αi γ K T (I + m v )T F0v Φ4  [0 Φ42 0] , Φ42 (i, 1)  other wise  0,   ⎧ ⎪ i+3 + Z i+3 i)  sym{−4 R + S i+3 + S i+3 − S i+3 − S i+3 (i, ⎪ v ⎪ 4 4 4 1 4 2 4 3 4 4 ⎪ ⎪ ⎪ ˜ ˜ ˜ ˜ + S i+3 + S i+3 − S i+3 − S i+3 , i = 1, 5, . . . , 4m − 3, ⎪ ⎪ 4 1 4 2 4 3  4 4 ⎪ ⎪ ⎪ ⎪ (i, i + 1)  −S i+3 1 + S i+3 3 − 2 R i+3 + Z i+3 v + S i+3 2 − S i+3 4 ⎪ 4 4 4 4 4 4 ⎪ ⎪ ⎪ ˜ i+3 1 + S˜ i+3 3 + S˜ i+3 2 − S˜ i+3 4 , i = 1, 5, . . . , 4m − 3, ⎪ − S ⎪ ⎪ 4 4 4 4 ⎪ ⎪ T T ˜T ˜T ⎪ + 2S (i, i + 2)  6R i+3 i+3 + 2S i+3 + 6Z i+3 v + 2 S i+3 + 2 S i+3 , ⎪ 3 4 4 ⎪ 4 4 4 4 3 4 4 ⎪ ⎪ ⎪ i = 1, 5, . . . , 4m − 3, ⎪ ⎪ ⎪ ⎪ + 2S i+3 + 6R1 − 2 S˜ i+3 + 2 S˜ i+3 + 6Z i+3 , (i, i + 3)  −2S i+3 ⎪ ⎪ 4 2 4 4 4 2 4 4 4 v ⎪ ⎪ ⎪ i = 1, 5, . . . , 4m − 3, ⎪ ⎪ ⎪ ⎪ (i, i)  −4R i+2 − 4R i+2 +1 + Q i+2 +1 − Q i+2 − 4Z i+2 v − 4Z ( i+2 +1)v , ⎪ ⎪ 4 4 4 4 4 4 ⎪ ⎪ i = 2, 6, . . . , 4m − 2, ⎪ ⎪ ⎪ ⎪ (i, i + 1)  −2S Ti+2 + 2S Ti+2 − 2 S˜ Ti+2 + 2 S˜ Ti+2 , ⎪ ⎪ ⎪ 4 3 4 4 4 3 4 4 ⎪ ⎪ ⎨ i = 2, 6, . . . , 4m − 2, Φ3  (i, i + 2)  6R i+2 + 6Z i+2 , i = 2, 6, . . . , 4m − 2, ⎪ 4   4 ⎪ ⎪ ⎪ i+2 − S i+2 + Z − S i+2 − S i+2 (i − 4, i − 1)  −2 R i+2 ⎪ v ⎪ 4 4 4 1 4 3 4 2 ⎪ ⎪ ⎪ ˜ ˜ ˜ ˜ ⎪ i+2 − S i+2 − S i+2 − S i+2 − S i+2 , − S ⎪ 4 4 4 1 4 3 4 2 4 4 ⎪ ⎪ ⎪ i = 6, 10, . . . , 4m − 2, ⎪ ⎪ ⎪ ⎪ ⎪ + S i+2 − S i+2 − S i+2 + S˜ i+2 + S˜ i+2 (i − 4, i)  S i+2 ⎪ 4 1 4 3 4 2 4 4 4 1 4 3 ⎪ ⎪ ⎪ ˜ i+2 4 , i = 6, 10, . . . , 4m − 2, − S˜ i+2 − S ⎪ 2 ⎪ 4 4 ⎪ ⎪ ⎪ + 6Z i+2 , i = 6, 10, . . . , 4m + 2, (i − 4, i + 1)  6R i+2 ⎪ ⎪ 4 4 v ⎪ ⎪ (i − 4, i + 2)  2S i+2 + 2S i+2 + 2 S˜ i+2 + 2 S˜ i+2 , ⎪ ⎪ 4 2 4 4 4 2 4 4 ⎪ ⎪ ⎪ i = 6, 10, . . . , 4m + 2, ⎪ ⎪ ⎪ ⎪ ⎪ − 12Z i+1 , i = 3, 7, . . . , 4m − 1, (i, i)  −12R i+1 ⎪ 4 4 v ⎪ ⎪ ⎪ ˜ − 4 S i+1 , i = 3, 7, . . . , 4m − 1, (i, i + 1)  −4S i+1 ⎪ ⎪ 4 4 4 4 ⎪ ⎪ ⎪ i i i)  −12R − 12Z , i = 4, 8, . . . , 4m, (i, ⎪ ⎪ 4 4v ⎩ other wise  0,

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

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⎡

⎤ Φ52 0 ˆ ˆ D, Φ53 Φ54 ⎦ , Φ51  εE bT E b − λI, Φ52  γ Bˆ vT G T , Φ54  γ δG  −ε I ! " 2 di2 − di−1 2 2 h i Z iv + Ti + h i Ri − γ G − γ G T , 2 i=1  ∞  E {πvn ()}  πvn () f v () d.

Φ51 Φ5  ⎣    m  Φ53  π˚ vn

0

Then, the robust synchronization between the system (1) and system ( 2) is guaranteed by the sampled-data controller (12). Proof For the error system (15), the following Lyapunov–Krasovskii functional can be considered (21) V (et ) = V1 (et ) + V2 (et ) + V3 (et ) , where V1 (et ) = eT (t) Pv e (t) , m #  t−di−1  eT (s) Q i e (s) ds V2 (et ) = i=1



+h i V3 (et ) =

m # 

t−di

−di

hi



−di−1

$ e˙ T (s) Ri e˙ (s) dsdθ ,

t t+θ



i=1

+



−di−1



−di−1 −di



−di

0

r



t

e˙ T (s) Z iv e˙ (s) dsdβ

t+β t

$ e˙ T (s) Z iv e˙ (s) dsdβdr .

t+β

Regard L as the weak infinitesimal operator and f i () is the PDF of sojourn time  staying at mode i [33], then we defined L of V (et ) as follow: L {V (et )} = lim+ h→0

1 {E {V (et+h ) /et } − V (et )} . h

Then, taking (21) and (22) into account, ones can be achieved that    L {V1 (et )} = sym eT (t) Pv e˙ (t) + eT (t) πvn () Pn e (t) , n∈ Sˇ

L {V2 (et )} =

m 

(eT (t − di−1 ) Q i e (t − di−1 ) − eT (t − di ) Q i e (t − di )

i=1

−h i



t−di−1 t−τi (t)

 e˙ T (s) Ri e˙ (s) ds − h i

t−τi (t) t−di

e˙ T (s) Ri e˙ (s) ds)

(22)

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+h i2 e˙ T (t) Ri e˙ (t) , ! m 2  di2 − di−1 2 T T (h i e˙ (t) Z iv e˙ (t) + e˙ (t) L {V3 (et )} = e˙ (t) Ti 2 i=1  t−τi (t)  t−di−1 −h i e˙ T (s) Z iv e˙ (s) ds − h i e˙ T (s) Z iv e˙ (s) ds)  +h i

t−di −di−1 −di



t−τi (t)

t

e˙ T (s) Ξ4v e˙ (s) dsdβ.

t+β

In the light of Lemma 1 and the results in [34], One may get that  −h i

t−di−1

t−τi (t)



t−τi (t)

e˙ (s) Ri e˙ (s) ds − h i T

e˙ T (s) Ri e˙ (s) ds)

t−di

hi hi ≤− Ψ T (t) Ri Ψi (t) Θ T (t) Ri Θi (t) − τi (t) − di−1 i di − τi (t) i ≤ −Λi (t)T Ξ2 Λi (t) ,  t−di−1  t−τi (t) −h i e˙ T (s) Z iv e˙ (s) ds − h i e˙ T (s) Z iv e˙ (s) ds) t−τi (t)

t−di

hi hi ≤− Ψ T (t) Ziv Ψi (t) ΘiT (t) Ziv Θi (t) − τi (t) − di−1 di − τi (t) i ≤ −Λi (t)T Ξ2 Λi (t) , where  Θi (t) = 

e (t − di−1 ) − e (t − τi (t)) e (t − di−1 ) + e (t − τi (t)) −

e (t − τi (t)) − e (t − di ) e (t − τi (t)) + e (t − di ) − T  T Λi (t) = Θi (t) ΨiT (t) . Ψi (t) =

" 2 τi (t)−di−1

2 di −τi (t)

% t−di−1 t−τi (t)

% t−τi (t) t−di

e (s) ds "

e (s) ds

,

,

According to the property of nonlinear function fˆ, one naturally find that λ f˜T (e(t)) f˜ (e(t)) ≤ λeT (t)l 2 e(t).

(23)

From the error system (15), for G, which is an appropriately dimensioned symmetrical matrix, satisfies the following equation   sym{ eT (t) G + γ e˙ T (t) G × [−˙e (t) + Aˆ v e (t) + Bˆ v f˜ (e(t)) +F0v (I + Mv ) K e (t − τi (t)) + δˆ Dˆ p (t)]} = 0.

(24)

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

93

Also, from (5) and (6), one can get p T (t) p (t) ≤ q T (t) q (t) . Then, ε, as a positive constant, satisfies the following equation   ε Γ − p T (t) p (t) ≥ 0,

(25)

where T E bT E a e1 Γ  ζ T (t) (e1T E aT E a e1 + e1T E aT E b e4m+2 + e4m+2 T +e4m+2 E bT E b e4m+2 )ζ (t) , T  ζ (t)  eT (t) emT (t) f˜T (e(t)) e˙ T (t) p T (t) ,

with  em (t)  eT (t − τ1 (t)) eT (t − d1 ) eT (t − τ2 (t)) eT (t − d2 ) eT (t − τm (t)) eT (t − dm )

%t t−τ1 (t)

% t−τ1 (t)

eT (s)ds

eT (s)ds d1 −τ1 (t)

t−d1

τ1 (t)

% t−d1

eT (s)ds τ2 (t)

t−τ2 (t)

% t−dm−1

% t−τ2 (t)

eT (s)ds τm (t)

t−τm (t)

eT (s)ds d2 −τ2 (t)

t−d2

% t−τm (t)

... T

eT (s)ds dm −τm (t)

t−dm

,

T  and ei (i = 1, 2, . . . , 4m + 3) ∈ R(4mn+4n)∗n , for example, e3 = 0 0 I 04mn∗n . Based on the above derivations, adding left sides of (23), (24) and (25) to L {V (et )}, the following new upper bound of L {V (et )} can be acquired L {V (et )} ≤ ζ T (t) Ξ1v ζ (t) . Therefore, in view of the explanation for the globally mean square stability in [30], the error system (15) is mean square stable if conditions in Theorem 1 hold. This illustrates that the synchronization between the drive system (1) and response system (2) is implemented by the designed sampled-data controller (12).  It is noted that some unknown matrix coupling terms exist in the conditions of Theorem 1, which lead to the difficulty of getting the controller gains. To solve such a problem, we present the theorem below. ˜ Theorem 2 For given positive constants γ , δˆ ∈ [0, 1], ε, λ and known matrices F, ˆ Aˆ v , Bˆ v , Jv with appropriate dimensions, if there exists matrices Pv > ˜S, E a , E b , D, 0, Q i > 0, Ri > 0, Z iv > 0, Ti > 0, ηv > 0, any matrices Si , S˜i , H ,and a symmetric matrices G satisfying the conditions (18)–(20) and the following inequalities for i = 1, . . . , m, v ∈ Sˇ

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⎡

Ξˇ 1v

⎤ Ξˆ 1v Φˆ 1 Φˆ 2  ⎣  Φ3 Φˆ 4 ⎦ < 0,   Φˆ 5

(26)

where Ξˆ 1v 



πvn pn + Q 1 − 4(R1 + Z 1v ) + λl 2 +sym{G A}

n∈ Sˇ

Φˆ 1

Φˆ 2 Φˆ 4 Φˆ 42 (i, 1) Φ44 (i, 1) Φˆ 5

T +εE aT E a + ηv F0v Jv JvT F0v , ⎧ ⎪ (1, 1)  −2R1 − S11 − S13 − S12 − S14 − 2Z 1v − S˜11 − S˜13 ⎪ ⎪ ⎪ ⎪ − S˜12 − S˜14 + F0v H, ⎪ ⎪ ⎪ ⎪ ⎨ (1, 2)  S11 + S13 − S12 − S14 + S˜11 + S˜13 − S˜12 − S˜14 ,  (1, 3)  6R1 + 6Z 1v , ⎪ ⎪ ⎪ (1, 4)  2S12 + 2S14 + 2 S˜12 + 2 S˜14 , ⎪ ⎪ ⎪ ⎪ (1, i)  αi F0v H, i = 5, 9, 13, . . . , 4m + 1, ⎪ ⎪ ⎩ other wise  0,   T ˆ D + ηv γ F0v Jv JvT F0v  Φ21 Φ31 δG 0 ,    04mn×n Φˆ 42 04mn×n Φ44 ,  T , i = 1, 5, . . . , 4m − 3, αi γ H T F0v  other wise = 0,  αi H T , i = 1, 5, . . . , 4m − 3,  other wise = 0, ⎤ ⎡ 0 Φ51 Φ52 0 ⎢  Φ53 Φ54 0 ⎥ ⎥ ⎢ ⎣   Φ55 0 ⎦ ,    −ηv I

T Φ55  −ε I + ηv γ 2 F0v Jv JvT F0v .

Then, the robust synchronization between the system (1) and system ( 2) is guaranteed by the sampled-data controller (12). In this regard, the expression of the controller gain (12) is presented as K = G −1 H . Proof Letting H  G K , it is clear that Ξ1v < 0 can be rewritten as the following form ⎤ ⎡ Ξ˜ 1v Φ˜ 1 Φ2 ⎣  Φ3 0 ⎦ + RMv N T + N MvT RT < 0, (27)   Φ5 where

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

95

⎧ ⎪ ⎪ (1, 1)  −2R1 − S11 − S13 − S12 − S14 − 2Z 1v ⎪ ⎪ ⎪ − S˜11 − S˜13 − S˜12 − S˜14 , ⎪ ⎪ ⎨ (1, 2)  S11 + S13 − S12 − S14 + S˜11 + S˜13 − S˜12 − S˜14 , Φ˜ 1  ⎪ (1, 3)  6R1 + 6Z 1v , ⎪ ⎪ ⎪ ⎪ (1, 4)  2S12 + 2S14 + 2 S˜12 + 2 S˜14 , ⎪ ⎪ ⎩ other wise  0, ⎧ ⎨ (1, 1)  F0v , R  (4m + 3, 1)  γ F0v , ⎩ other wise  0,  (i, 1)  αi H T , i = 2, 6, . . . , 4m − 2, N  other wise  0. In addition, one readily has RMv N T + N MvT RT ≤ ηv RJv JvT RT + ηv−1 N N T .

(28)

After that, combining (27) and (28), and using Schur complement for (27), the conditions (17)–(20) can be obtained easily. This completes the proof. Remark 5 In practical situations, it is necessary to economize the limited network bandwidth resources due to the limitation of the communication load. Based on this, this chapter designs a sampling controller, which is used for SMJCSs. In addition, the randomly occurring time delays are considered in the controller which is tally with reality. Furthermore, control signals with the sampled-data process can be transmitted at the sampling instant, which decreases the amount of data transmission such that, the bandwidth resource can be economized efficiently.

4 Numerical Example This section explores a numerical example, which for the sake of demonstrating the feasibility of the main results expressed above. It is about the synchronization of hyperchaotic [35] which is described by SMP with the following parameters with two modes (v = 1, 2) ⎡

0 ⎢2 Aˆ 1 = ⎢ ⎣0 0

−2 0.5 0 0

−2 0 0 −1

⎡ ⎤ ⎤ 0 0 −1 −1 0 ⎢ 2 ⎥ 1 ⎥ ⎥ , Aˆ 2 = ⎢ 1 0.25 0 ⎥, ⎣ ⎦ 0 0 0 0 0 ⎦ 0.1 0 0 −0.5 0.05

Bˆ 1 = diag{1, 1, 1, 1}, Bˆ 2 = diag{2, 2, 2, 2}, F01 = diag{0.5, 0.5, 0.5, 0.5}, F02 = diag{0.7, 0.7, 0.7, 0.7}, H1 = diag {0, 0, 0, 0} ,

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Fig. 1 The chaotic behavior of the hyperchaotic Rossler system

T    H2 = diag {0.5, 0.5, 0.5, 0.5} , fˆ x(t) ˆ = 0 0 3 + xˆ1 (t) xˆ3 (t) 0 . Let the Lipschitz constant l=1 in the nonlinear function fˆ and choose initial values  T  T of each system as x (0) = −7.2 −3.6 1.35 8.1 , y (0) = −5.5 −6.5 5 −10 . Then, the sampling intervals are considering as d1 = 0.04, d2 = 0.08, d3 = 0.15, respectively. In view of the conditions in Theorem 2, the mode-independent controller gain matrix may be calculated according to the procedure given below. In addition, the switching between the two modes is described by a semi-Markov process, which obeys the transfer rate matrix shown below:  1  − 21  2 . Π () = 3()2 −3()2 

On the basis of Weibull distribution [33], the probability density functions of 2 3 sojourn time are concerned as f 1 () = 21 () e−0.25() and f 2 () = 3()2 e−() . Therefore, the mathematical expectation of Π () can be computed easily:

Fault-Tolerant Sampled-Data Synchronization of Chaotic … 12

e1

97 e2

e3

e4

10 8 6

value

4 2 0 -2 -4 -6 -8 -10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Step

(a) uncontrolled error signals 4

e1

e2

e3

e4

3

value

2

1

0

-1

-2

-3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Step

(b) controlled error signals Fig. 2 Uncontrolled and controlled error signals Table 1 The minimum value of f˘− for different stochastic variable δˆ 0.1 0.3 0.5 0.7 δˆ f˘− 0.1328 0.1354 0.1381 0.1408

0.9 0.1436

98 Fig. 3 Uncontrolled and controlled state response

Y. Xia et al. 10 0 -10 -20 0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

20 0 -20

6 4 2 0

25 20 15 10 5

(a) Uncontrolled state responses 10 0 -10 0

0.5

1

1.5

2

0.5

1

1.5

2

2.5

3

3.5

4

4.5

10 0 -10 2.5

3

3.5

4

4.5

6 4 2 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

40 20 0

(b) Controlled state responses

 ε {Π ()} =

 −0.8862 0.8862 . 2.7082 −2.7082

The other parameters are listed as: E a = 0.5In , Dˆ = In , Fˆ (t) = 0.4 + 0.2 sin t, E b = 0.2In and δˆ = 0.3. Three sampling intervals are taken into account as d1 , d2 , d3 in this example.

Fault-Tolerant Sampled-Data Synchronization of Chaotic …

99

40 u1

u2

u3

u4

30 20

value

10 0 -10 -20 -30 -40 -50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Step

Fig. 4 The sampled-data control input with three sampling intervals

Figure 1 manifests the chaotic behavior. According to the above simulation setl l = f 2− = f˘− . Then, from the information obtained in tings, we can assume that f 1− Table 1 which is calculated by the Theorem 2. We can know that as the δˆ decrease, the f˘− also decrease. After that, Remark 6 can be verified. ⎡

−8.7681 ⎢ 0.0874 K =⎢ ⎣ 2.9331 −0.4818

0.0874 −10.4061 0.1717 −2.8332

2.9331 −0.1717 −8.5967 1.3147

⎤ −0.4818 −2.8332 ⎥ ⎥. 1.3147 ⎦ −7.8816

Remark 6 For this example, the system becomes unstable as the uncertainty magnitude increases. And it is not difficult to gain the information from thetable that when  the uncertainty increases, the fault tolerance of the system decreases f˘− increases . Then, the minimum value of f˘− which implies the maximum fault tolerance of the system is obtained. The uncontrolled and controlled error signals are revealed in Fig. 2a,b, respectively. Comparing these two figures, one can find that the error signals controlled by the sampled-data controller tend to zero when time goes on. Meanwhile, as desired, the error signals do not converge to zero for the uncontrolled system, which means that the controller we proposed realizes the synchronization of SMJCSs.

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Fig. 3a shows the open-loop system, which expresses that the error system (15) is not synchronization. And, by observing the curve trace of Fig. 3b, we can comprehend the error system (15) becomes synchronization, which means the controller we designed is valid. Finally, Fig. 4 shows the sampled-data control input with three sampling intervals.

5 Conclusion The design of a fault-tolerant sampled-data controller, in this work, has been put forward to accomplish the robust synchronization of SMJCSs. On the basis of a semi-Markov jump model, some available conditions have been established, which guarantee the mean-square synchronization of chaotic systems by means of employing appropriate Lyapunov–Krasovskii functional and reasonable integral inequality methods. Furthermore, ROUs have been taken into account in normal situations. In addition, by solving a group of LMIs, the anticipated controller gain can be acquired. An explained example has been employed to elucidate the preponderance of the method proposed. Note that, the aforementioned results can be generalized to the fuzzy model with extended dissipative performance, which is worthy of exploring. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 61703004, 61873002, the Natural Science Foundation of Anhui Province under grant 1808085QA18.

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Fuzzy-Based Sliding Mode Control Design for Stability Analysis of Nonlinear Interconnected Systems A. Manivannan, G. Bhuvaneshwari, R. Krishnasamy, V. Parthiban, and S. Dhanasekar

Abstract In general, the nonlinear interconnected systems are highly interconnected with N number of sub-systems, therefore, it is difficult to a design decentralized control scheme for analyzing the stability of the system. This study is mainly concerned with fuzzy based sliding mode control design to analyze the stability and stabilization of nonlinear interconnected systems. In order to derive the stabilization condition, we proposed a fuzzy-based- sliding model controller and the sufficient condition is derived based on suitable Lyapunov function and LMI approach. The sufficient condition contains the information on the sliding motions, some tuning parameters and slack variables. The condition ensures that the asymptotic stability of the states of nonlinear interconnected systems under the fuzzy-based sliding mode controller. Finally, the numerical example and their simulation results are given to demonstrate the effectiveness of the proposed conditions. Keywords Interconnected systems · Lyapunov stability · T-S fuzzy · Sliding mode control · Linear matrix inequality

A. Manivannan (B) · G. Bhuvaneshwari · V. Parthiban · S. Dhanasekar Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology-Chennai Campus, Chennai 600127, India e-mail: [email protected] G. Bhuvaneshwari e-mail: [email protected] V. Parthiban e-mail: [email protected] S. Dhanasekar e-mail: [email protected] R. Krishnasamy Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore, Tamil Nadu, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_5

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1 Introduction The large-scale interconnected system plays a significant role in the various industrial applications such as power systems, transportation systems, socio-economic systems, haptic-enabled systems, and wind turbine systems. The main advantage of large-scale systems is that it communicates the viable information between the subsystems. For an instance, the wind turbines are connected over a tie-lines, in which through interconnections the data available in each wind turbine system can be shared over a network that results in better optimization of large-scale wind farms. Therefore, many researchers have paid great attention on the stability and stabilization issues of large-scale interconnected systems in the recent years [1–11]. In general, the control setup for each sub-system, say, centralized control does not reflect the overall stable performance of the complete large-scale system. To an alternative, a decentralized controller [12], a shared controller among the subsystems will provide stable performance as well as user reliability. Notice that, different types of decentralized control strategies are frequently encountered in the literature to guarantee the stabilization of large-scale systems. To list a few, adaptive neural network control [13, 14], fault-tolerant control [15], sampled-data control [16] and integral sliding mode control (ISMC) [17, 18]. Among these controllers, ISMC has the ability to cope with the system having incomplete information and uncertainties. To an evident, authors in [17] have proposed a linear sliding variable-based decentralized sliding mode control strategy for interconnected systems. In [18], a reduced-order observer-based integral sliding mode control scheme has been proposed by the authors to stabilize a class of interconnected systems. Hence, this study focuses on designing the ISMC for nonlinear large-scale interconnected systems. Technically, the industrial problems that can be framed into nonlinear differential systems cannot be solved directly. To solve those nonlinear systems, different types of theoretical approaches are utilized such as the Jacobian method, pseudo-linearization method and Takagi-Sugeno (T-S) fuzzy approach. Among these techniques, T-S fuzzy is considered to be a convenient method to express the nonlinear system into T-S fuzzy-based model [19–22]. In the view of literature, significant works have published regarding the T-S fuzzy-model based large scale interconnected system. To list a few, authors in [23] proposed the decentralized fault-tolerant control for a class of large-scale fuzzy systems with actuator faults containing stuck, outage and loss of effectiveness. Also, the authors in [24] investigated the large-scale system involving the certainties. Based on fuzzy SMC, the global stability of the closedloop system was ensured. Besides, as each subsystem is connected over the network, the communication between the subsystems will realize the time-delay over transmission. The ignorance of these types of communication time-delays may degrade the overall efficacy of the system. Therefore, the consideration of time-delays while formulating the large-scale system becomes necessary. For instance, authors in [25] proposed a new method to stabilize the system with control design. In [26], authors have been discussed

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the decentralized control of nonlinear interconnected time-delayed systems using T-S fuzzy mixed H2 /H∞ optimization with smith predictor is established. While investigating of stability of the system, the exogenous disturbances become a part of the analysis. In order to diminish the effect of the disturbance, H2 /H∞ performance came into the picture. Theoretically, based on Lyapunov stability theory, remarkable stability and stabilization results have proposed on nonlinear dynamical systems [26–30]. In this chapter, we utilized the Lyapunov stability theory to derive the sufficient stability conditions in terms of LMIs, which can be solved with the help of MATLAB LMI control toolbox. Inspired by the above works, the overall contributions of the present study can be listed in the following. (i) The nonlinear large-scale interconnected is equivalently expressed into a T-S fuzzy-model-based system. We have utilized the double and triple integral terms with information of time-delays. (ii) Wirtinger inequality approach is used to estimate the double integral terms and utilizing a slack variable with a tuning parameter, then delay-depended stability and stabilization conditions are derived. (iii) The suitable controller is designed under the fuzzy integral sliding surface and their reachability condition is validated (iv) A numerical example is provided to validate the effectiveness of the proposed theoretical conditions. Notations. Rn and Rn×m denote n-dimensional Euclidean space and the set of all n × m matrices with real entries respectively; P1 > 0 (P1 ≥ 0) represent the positive definite (positive semi-definite) matrix. P2 < 0 (P2 ≤ 0) represent the negative definite (negative semi-definite) matrix. P T indicates the transpose and the symbol ∗ denotes the transposed elements in the symmetric matrix. I denotes an identity matrix;  ·  is Euclidean norm.

2 Preliminaries Now, let us consider the T-S fuzzy model of a nonlinear interconnected system with N -number of subsystems [10], as given below, Plant rule l: q ql q 1l 1 (t) is Mi1 and · · · and θi1 (t) is Mi1 and · · · and θi1j (t) is Mi1lj and · · · and θi j (t) IF θi1 ql

is Mi j

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THEN N 

x˙i (t) = Ail xi (t) + Dil xi (t − τi (t)) + Bil u i (t) +

E i j,l x j (t) + Fil wi (t),

j=1, j=i

yi (t) = Cil xi (t),

i = 1, 2, . . . , N . l = 1, 2, . . . , r. (1)

Here, N is the total number of subsystems, xi (t), x j (t) ∈ Rn represent the i-th and jth subsystem respectively, u i (t) ∈ Rm is control input, and wi (t) ∈ Rm stands for external disturbance. The coefficient matrices with appropriate dimensions are given as Ail , Bil , Dil , Fil , ql E i j,l , and Cil ; Mi j , (i, j, p, l) ∈ {I N := {1, 2, . . . , N }} × {I J := {1, 2, . . . , J }} × p {Iq := {1, 2, . . . , q}} × {Ir := {1, 2, . . . , r }} is the fuzzy set for θi j (t); I N × I J p denotes all pairs (i, j) ∈ Z>0 × Z>0 with 1 ≤ i, j ≤ N and i = j; θi j (t) =  T  1 q q θi1 (t), · · · θi1 (t) · · · θi1j (t) · · · θi j (t) are fuzzy premise variables. Moreover, τi (t) denotes the time-varying delays for the ith subsystem, which needs to satisfy the following condition 0 ≤ τi (t) ≤ τi τ˙i (t) ≤ μi < ∞, for real constants τi > 0, μi > 0. ϕi (t) denotes the initial value on [−τi , 0]. Finally, T-S fuzzy model is used to represent the each subsystem with r plant rules. Then, the overall fuzzy inferred system is derived as in (2) x˙i (t) =

r 

 h li (θi (t)) Ail xi (t) + Dil xi (t − τi (t)) + Bil u i (t)

l=1

+

N 

 E i j,l x j (t) + Fil wi (t) ,

j=1, j=i

yi (t) =

r 

h li (θi (t))Cil xi (t),

i = 1, 2, . . . , N ,

(2)

l=1

where vl (θi (t)) h li (θi (t)) = r i l , l=1 vi (θi (t)) pl

vil (θi (t)) =

q 

pl

p

Mi (θi ).

p=1 p

and Mi : Uθip ⊂ R → R[0,1] denotes the membership function of θi j on the compact set Uθip .

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Remark 1 The number of fuzzy plant rules depends on the number of fuzzy premise variables. For instance, for two fuzzy premise variables, it requires four fuzzy membership rules to express the nonlinear model into linear local submodels. In general, for n− premise variables, 2n fuzzy membership rules are required. The sliding mode operation based on fuzzy sliding surface function and the reachability of the sliding motion is discussed in the following sections to ensure the stabilization of the considered system.

3 Sliding Surface Design Sliding surface design is chosen as t si (t) = Fi xi (t) −

Fi

r 

h li (θi (t)) (Ail + Bil K il )xi (σ)

l=1

0

N

  + Dil xi (σ − τi (σ)) + E i j,l x j (σ) dσ.

(3)

j=1, j=i

Here, if the state trajectories reach the sliding mode, then the sliding surface function must satisfy the conditions si (t) = 0, s˙i (t) = 0. In this aspect, the derivative of the Eq. (3) with respect to time yields s˙i (t) = Fi

r 



h li (θi (t)) Bil u i (t) + Fil wi (t) − Bil K il xi (t) .

l=1

Now, the equivalent controller is derived from the above equation as follows u ie (t) = (Fi

r  l=1



h li (θi (t))Bil )−1 Fi

r 

h li (θi (t))

l=1

× − Fil wi (t) + Bil K il xi (t) .

(4)

If the system state enters into sliding mode dynamics, then the state trajectories can be controlled through a designed equivalent controller (4). In this regard, substituting the equation (4) into (2) can lead to the sliding mode dynamical state equations, which are given below

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x˙i (t) =

r 

 h li (θi (t)) (Ail + Bil K il )xi (t) + Dil xi (t − τi (t)) + Bil Fi wi (t)

l=1 N 

+

 E i j,l x j (t) ,

(5)

j=1, j=i

yi (t) =

r 

h li (θi (t))Cil xi (t),

i = 1, 2, . . . , N ,

l=1

where Bil = In − Bil (Fi Bil )−1 Fi is the transition matrix and In denotes the identity matrix. Also, the given condition must hold det (Fi Bil ) = 0, and Fi ∈ Rm i ×ni and K il are control gain matrices (i = 1, 2, . . . , N , and l = 1, 2, . . . , r ). The useful lemmas required for the analysis are given in the following: Lemma 1 ([31]) For a given matrix M1 > 0, the following inequality holds for all continuously given scalars a and b satisfying a < b,the following inequality holds for all continuously differentiable function x(t) : [a, b] → R n   (b − a)2 b b T x (u)M1 x(u)duds 2 a s  b  b

 b  T ≥ x (u)duds M1 a

where Θ = −

s

bb a

s

a

x(u)duds +

3 b−a

s

x(u)duds + 2Θ T M1 Θ

s

bbb a

b

u

x(v)dvduds.

Lemma 2 ([32]) For any constant matrix M2 > 0, any scalars a and b with a < b and a vector function x(t) : [a, b] −→ Rn such that the integrals concerned are well defined, the following holds 



T

b

x(s)ds

b

M2

a

 x(s)ds ≤ (b − a)

a

b

x T (s)M2 x(s)ds .

a

Lemma 3 (Schur Complement) [33] Given constant matrices Ω1 , Ω2 and Ω3 with appropriate dimensions, where Ω1T = Ω1 and Ω2T = Ω2 > 0, the inequality Ω1 + Ω3T Ω2−1 Ω3 < 0 holds, if and only if

Ω1 ∗

Ω3T −Ω2

< 0, or

−Ω2 ∗

Ω3 Ω1

< 0.

The following section discusses about the derivation of sufficient stability conditions with respect to the derived closed-loop system (5).

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4 Main Results This section comprises the stability and stabilization of derived closed-loop system (5) under ISMC and suitable Lyapunov functionals. The following section discusses the stability conditions which contain the nonconvex terms that cannot be solved directly. In this regard, through simple calculations, the stabilization conditions are derived in terms of solvable LMIs.

4.1 Stability and Stabilization Results The sufficient condition for stability of the system (5) is summarized in the following theorem. Theorem 1 For given gain matrices K il , if there exist the symmetric positive definite matrices Pi , R1i , R2i , Q i , and Ti , which satisfy the following LMI, then the system (5) is said to be asymptotically stable ⎤ (φli j )1,1 τi Q i G i Dil G i E i j,l (φli j )1,5 (φli j )1,6 (φli j )1,7 (φli j )1,8 ⎢ ∗ (φli j )2,2 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ ∗ 0 (φli j )3,5 0 0 0 ⎥ ∗ (φli j )3,3 ⎥ ⎢ ⎢ ∗ 0 0 0 ⎥ ∗ ∗ (φli j )4,4 (φli j )4,5 ⎥ ⎢ l Γi j = ⎢ ⎥ < 0, 0 0 ⎥ ∗ ∗ ∗ (φli j )5,5 (φli j )5,6 ⎢ ∗ ⎥ ⎢ ⎢ ∗ 0 0 ⎥ ∗ ∗ ∗ ∗ (φli j )6,6 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ (φli j )7,7 (φli j )7,8 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (φli j )8,8 (6) where ⎡

(φli j )1,1 = R1i + R2i − τi2 Ti − τi Q i − 2

τ2

i

2

Ti + 2G i Ail

+ 2G i Bil K il + Mi + CilT Cil (φli j )1,5 = Pi − G i + b1 (G i Ail )T + b1 (G i Bil K il )T , (φli j )1,6 = G i Bil Fil , (φli j )1,7 = −τi Ti , (φli j )1,8 = 3Ti , (φli j )2,2 = −R1i − τi Q i , (φli j )3,3 = −(1 − μi )R2i , (φli j )3,5 = b1 (G i Dil )T ,

T (φli j )4,4 = −M j , (φli j )4,5 = b1 G i E i j,l , (φli j )5,5 = τi2 Q i +

τ 2 2 i

2

Ti − 2b1 G i , (φli j )5,6 = b1 G i Bil Fil ,

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(φli j )6,6 = −γ 2 I, (φli j )7,7 = −3Ti , (φli j )7,8 =

6Ti −18Ti , (φli j )8,8 = . τi τi2

Proof Consider the following LKF V (xi (t)) =

N 

V1i (xi (t)) + V2i (xi (t)) + V3i (xi (t))

i=1

+ V4i (xi (t)) + V5i (xi (t))

(7)

(8)

where V1i (xi (t)) = xiT (t)Pi xi (t),  t xiT (s)R1i xi (s)ds, V2i (xi (t)) = t−τi t

 V3i (xi (t)) =

xiT (s)R2i xi (s)ds,

t−τi (t)  0  t

V4i (xi (t)) = τi V5i (xi (t)) =

−τi  τi2 t

2

t+θ

x˙iT (s)Q i x˙i (s)dsdθ,

 t

t−τi

s

u

t

x˙iT (v)Ti x˙i (v)dvduds.

Evaluating the time derivative of V (xi (t) on the state trajectories of the Eq. (5), one can obtain V˙ (xi (t)) =

N 

V˙1i (xi (t)) + V˙2i (xi (t)) + V˙3i (xi (t)) + V˙4i (xi (t)) + V˙5i (xi (t))



i=1

where V˙1i (xi (t)) = 2xiT (t)Pi x˙iT (t),

V˙2i (xi (t)) = xiT (t)R1i xi (t) − xiT (t − τi )R1i xi (t − τi ) ,

V˙3i (xi (t)) = xiT (t)R2i xi (t) − (1 − μi )xiT (t − τi (t))R2i xi (t − τi (t)) ,  t x˙iT (s)Q i x˙i (s)ds, V˙4i (xi (t)) = τi2 x˙iT (t)Q i x˙i (t) − τi V˙5i (xi (t)) =

τ2 

t

2

t−τi



i

s

t−τi

t

x˙iT (t)Ti x˙i (t)duds −

τi2 2



t t−τi



t s

x˙iT (u)Ti x˙i (u)duds .

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From V˙4i (t), it can be seen that the single integral term cannot be solved. In this regard, the most frequently used Jensen’s inequality (Lemma 1) can be t utilized and integral term −τi t−τi x˙iT (s)Q i x˙i (s)ds is approximated as V˙4i (xi (t)) ≤ τi2 x˙iT (t)Q i x˙i (t)    

− τi xiT (t) − xiT (t − τi ) Q i xiT (t) − xiT (t − τi ) .

(9) (10)

Similarly, the Wirtinger-based inequalities are employed to solve the complex double integral term. For more details, refer the Lemma 2. In detail, with lower bound of t τ2  t double integral, one can solve the integral − 2i t−τi s x˙iT (s)Ti x˙i (s)duds into the following inequality V˙5i (xi (t)) : τ4 V˙5i (xi (t)) ≤ i x˙iT (t)Ti x˙i (t) 4  t  T   xi (s)ds Ti τi xi (t) − − τi xi (t) − −2

τ

i

2

t−τi t

 x(t) −

× Ti

τ

i

2

t−τi

x(t) −

xi (s)ds + 

t t−τi

3 τi



t



t

xi (s)ds

t−τi



T

t

x(u)duds t−τi

3 xi (s)ds + τi



s t t−τi



t

 x(u)duds .

(11)

s

Instead of accompanying the closed-loop system (3), the most convenient way is to employ the closed-loop system through the zero-equation approach. In this regard, for any matrix G i and a scalar b1 , the following zero equation can be derived r    h li (θi (t)) Ail + Bil K il xi (t) 0 = 2 xiT (t) + b1 x˙iT ]G i − x˙i (t) + l=1 N

  + Dil xi (t − τi (t)) + Bil Fi wi (t) + E i j,l x j (t) .

(12)

j=1, j=i

Obviously, we can get N N   i=1 i=1, j=i

xiT (t)Mi xi (t) =

N N  

xiT (t)M j xi (t).

(13)

i=1 j=1, j=i

Now, H∞ attenuation performance for a given scalar γ under the zero initial condition can be considered as

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∞

N  

∞

yiT (τ )yi (τ )dτ

0, and T p× p = G n2 > 0. G n2 ∈ R

3.1 Observer Error Analysis Substituting (17) into (16) and rearranging gives   ¯ + G l C¯ x¯ + G n ν + V y˙ . x˙¯ˆ = R A¯ − G l C¯ xˆ¯ + R Bu

(20)

Pre-multiply (13) with R and add V y˙ , and rearrange using (15) to get   ¯ + R M¯ f¯ + V y˙ . R E¯ x˙¯ + V y˙ = R E¯ + V C¯ x¯ = R A¯ x¯ + R Bu

(21)

  e  n¯ − p , and subtract (21) from (20) to obtain the Define e = xˆ¯ − x¯ = 2 ey  p error equation (which characterizes the performance of the observer):   e˙ = R A¯ − G l C¯ e − R M¯ f¯ + G n ν.

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  ¯ M, ¯ C¯ from (13)–(14), Define A¯ 2 = A2 + R24 T44 A54 . By substituting for A, R, G l , and G n from (19), and T p from (12), Eq. (22) becomes          e2 0 0 0 e˙2 0 0 A¯ 2 ¯ = − ν. f + e˙ y T p M2 T p A 4 T p Q 2 G n2 T p A5 −Ao e y

(23)

¯ p)×(n− ¯ p) Proposition 2 Define a matrix P2 ∈ R(n− = P2T > 0, and a positive scalar ¯ γ. Let P2 = P2 R24 . Suppose for a given γ there exists a set of values for P2 and P¯2 that satisfies the following LMI:

 T P2 A2 + P¯2 T44 A54 + P2 A2 + P¯2 T44 A54 + γ P2 < 0.

(24)

Also, let ρ from (18) satisfy  −1 ρ ≥ G −1 n2 T p A5 α + G n2 T p M2 A4 Q 2 β + η, γt

where α ≥ e2 (0)e− 2



(25)

λmax (P2 ) , λmin (P2 )

β >  f max + x1 max + ξmax , and η ∈ R+ . ¯ = 0} in Then, an ideal sliding motion would take place on surface S = {e : Ce finite time. 

Proof The proof consists of two parts: the first portion aims to show how e2 is bounded by α. Define the Lyapunov candidate function W2 = e2T P2 e2 > 0, and differentiate it with respect to time to obtain   W˙ 2 = e2T P2 A2 + P2 R24 T44 A54 + (P2 A2 + P2 R24 T44 A54 )T e2 .

(26)

Substituting for P¯2 = P2 R24 into (24) yields P2 A2 + P2 R24 T44 A54 + (P2 A2 + P2 R24 T44 A54 )T + γ P2 < 0.

(27)

It can be seen from comparing (26)–(27) that satisfying LMI (24) implies W˙ 2 <  γt

(P2 ) −γW2 . This in turn implies e2 (t) ≤ e2 (0)e− 2 λλmax ≤ α [29], completing min (P2 ) the first part of the proof. The next and final part of the proof shows how setting ρ to satisfy (25) would result in sliding motion on S in finite time. Define another Lyapunov candidate function W y = e Ty G −1 n2 e y , and differentiate it with respect to time to obtain

  T −1 W˙ y = −2e y G −1 n2 Ao + Ao G n2 e y    −1 ¯ +2e Ty G −1 n2 T p A5 e2 − G n2 T p M2 A4 Q 2 f + ν . Since G −1 n2 Ao > 0 is symmetrical and e2  ≤ α, (28) can be re-expressed as

(28)

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   −1 W˙ y ≤ −2e y  ρ − G −1 n2 T p A5 α − G n2 T p M2 A4 Q 2 β .

(29)

Also, notice that T      −1 −1 e y  = e y G n2 G n2 e y G n2 2

2     −1  e ≥ λmin (G n2 )  G n2  = λmin (G n2 ) W y .  y Therefore, by setting ρ to satisfy (25), Eq. (29) becomes  √ W˙ y ≤ −2ηe y  ≤ −2η λmin (G n2 ) W y ,

(30)

which is the reachability condition [28]. This results in e y = 0 (and sliding motion taking place) in finite time, thus completing the proof.  After sliding motion takes place, e y , e˙ y = 0, and error system (23) becomes e˙2 = A¯ 2 e2 ,



(31)



0 = T p A5 e2 − T p M2 A4 Q 2 f¯ + G n2 νeq ,

(32)

where νeq is the equivalent output error injection which is needed to maintain the sliding motion on S. Then, define a measurable signal ν¯ = G n2 νeq , and substitute for A5 and Q 2 from (11) and T p from (12) to obtain ⎡ ⎤ ⎡ ⎤ ⎡ ν1 A51 Iq 0 ⎢ ν2 ⎥ ⎢ A52 ⎥ ⎢ 0 In−n¯ ⎥ ⎢ ⎥ ⎢ ν¯ = ⎢ ⎣ν3 ⎦ = − ⎣ A53 ⎦ e2 + ⎣ 0 0 ν4 A54 0 0

⎤ 0 ⎡ ⎤ f 0⎥ ⎥ ⎣x1 ⎦ , Ih ⎦ ξ 0

(33)

¯ . where ν4 ∈ R p−q−n+n−h Define the estimates for f, x1 , and ξ as

fˆ = ν1 + L f ν4 , xˆ1 = ν2 + L 1 ν4 , ξˆ = ν3 + L x ν4 ,

(34)

¯ p−q−n+n−h) ¯ ¯ , L 1 ∈ R(n−n)×( , respectively, where L f ∈ Rq×( p−q−n+n−h) h×( p−q−n+n−h) ¯ . and L x ∈ R Next, define the estimation errors e f = fˆ − f, e1 = xˆ1 − x1 , and eξ = ξˆ1 − ξ. Then, from (31) and (33) it follows that

Observer-Based Fault-Tolerant Control for Non-Infinitely …

e˙2 = A¯ 2 e2 , ⎡ ⎤ ⎡ ⎤ e1 −A52 − L 1 T44 A54 ⎢ e2 ⎥ ⎢ ⎥ In− ¯ p ⎢ ⎥=⎢ ⎥ ⎣e f ⎦ ⎣−A51 − L f T44 A54 ⎦ e2 . eξ −A53 − L x T44 A54

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From Proposition 2, e2 is asymptotically stable (i.e., as t → ∞, e2 → 0 and xˆ2 → x2 ), which implies e1 , e f , and eξ are also asymptotically stable, and therefore xˆ1 → x1 , fˆ → f , and ξˆ → ξ. Hence, one possible choice for L 1 , L f , and L x is A51 + L f T44 A54 ≈ 0, A52 + L 1 T44 A54 ≈ 0, A53 + L x T44 A54 ≈ 0.

(37)

Thus, the design of the SMO to estimate states and unknown inputs in system (1)–(2) has been completed.

4 The Controller for Fault-Tolerant Control In this section, the state and unknown input estimates obtained from the SMO are fed into a controller, which would generate the control input u. The gains of the controller would be designed such that the states of the system converge asymptotically, rejecting the influence of the unknown inputs. Let the input u be given by ˆ u = −K x xˆ0 − K f fˆ − K ξ ξ,

(38)

where K x ∈ Rm×n , K f ∈ Rm×q , and K ξ ∈ Rm×h are the gains of the controller (which are to be designed), and xˆ0 are state estimates for the system in the coordinates of (1)–(2).

4.1 Error Analysis and Controller Design Define ex = xˆ0 − x0 , and suppose K f and K ξ designed to satisfy K f = B † M, K ξ = B † Q,   where B B † M Q = M Q . Applying u onto system (1)–(2) yields

(39)

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⎡ ⎤ ex  E x˙0 = (A − B K x ) x0 + −B K x −M −Q ⎣e f ⎦ , eξ y = C x0 .

(40) (41)

⎡ ⎤ x1 Recall from (6) that x = Tc Ta x0 = ⎣x2 ⎦, where Tc Ta is non-singular. Then, y ⎡ ⎤ e1   define exca = xˆ − x = Tc Ta xˆ0 − x0 = ⎣e2 ⎦. ey ⎡ ⎤ e1 After sliding motion takes place, ex = (Tc Ta )−1 ⎣e2 ⎦. 0 Then, using (36), ex , e f , and eξ can be expressed in terms of e2 as follows: ⎤ ⎡ ⎤ ⎡ e1 −A52 − L 1 T44 A54 ⎡ ⎤ ⎥  ⎢ e2 ⎥  ⎢  In− ex ¯ p −1 −1 ⎥ ⎥ 0 ⎢ 0 ⎢ ⎥ e2 . ⎢ 0 ⎥ = (Tc Ta ) ⎢ ⎣e f ⎦ = (Tc Ta ) 0 ⎥ ⎥ ⎢ ⎢ 0 Iq+h ⎣ ⎦ 0 Iq+h ⎣ eξ ef −A51 − L f T44 A54 ⎦ eξ −A53 − L x T44 A54    Bx2

(42) Substituting ex , e f , and eξ from (42) into (40) yields  E x˙0 = (A − B K x ) x0 + −B K x −M −Q Bx2 e2 ,   

(43)

Bx

y = C x0 .

(44)

The aim is to now design the controller gain K x such that system (43)–(44) is asymptotically stable. The following lemma and theorem presents the design for the controller: Lemma 2 (Lemma 2 [7]) All non-singular Z ∈ Ra×a satisfying Z T Y T = Y Z ≥ 0 can be parameterized as Z = Z 1 Y T + NY Z 2 , where Z 1 ∈ Ra×a > 0 and Z 2 ∈ R(a−b)×a , and NY ∈ Ra×(a−b) is a matrix satisfying Y NY = 0 and rank (NY ) = a − rank(Y ) = a − b.  Theorem 1 Define matrices H ∈ Rn×n > 0, J ∈ R(n−r )×n , and S ∈ Rm×n . Suppose there exists a set of values for H, J , and S satisfying the following LMI:  T AH E T + AN E J − B S + AH E T + AN E J − B S < 0, and K f and K ξ are designed to satisfy (39), and

(45)

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 −1 Kx = S H E T + NE J ,

(46)

where E N E = 0. Then, system (1)–(2) is asymptotically stable.



Proof The proof is divided into two portions: the first part re-expresses LMI (45) into a form that facilitates analysis. Using Lemma 2, there exists a non-singular matrix U0 ∈ Rn×n satisfying U0 = H E T + N E J, U0T E T = EU0 ≥ 0.

(47)

Substitute for S from (46) and U0 into LMI (45) to obtain (A − B K x ) U0 + U0T (A − B K x )T < 0, U0T E T = EU0 ≥ 0.

(48)

Define U = U0−1 , and then pre-multiply (48) with U T and post-multiply with U to obtain Δ0 = U T (A − B K x ) + (A − B K x )T U < 0,

(49)

E U = U E ≥ 0.

(50)

T

T

LMI (45) has been re-expressed in a form that is similar to that for Theorem 2 in [7], concluding the first part of the proof. The next and final portion of the proof shows how asymptotic stability is achieved by satisfying LMI (45). Define the Lyapunov candidate functions W0 = x0T U T E x0 , W2 = e2T N e2 , Wc = W0 + δW2 , ¯ p)×(n− ¯ p) = N T > 0 and δ ∈ R+ . where N ∈ R(n−

Using (51), Wc can be re-expressed as Wc = implies



x0T

e2T

(51)

   UT E 0 x0 , which 0 δ N e2

 T   2  x0  U E 0   Wc ≤ λmax  e2  . 0 δN   

(52)

θ1

where θ1 ∈ R+ . Differentiating W0 and W2 with respect to time, substituting for E x˙0 from (43), and using (35) and (49)–(50) yields   W˙ 0 = x0T Δ0 x0 + 2x0T U T Bx e2 , W˙ 2 = e2T N A¯ 2 + A¯ 2T N e2 .    Δ2

(53)

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It can be seen that LMI (45) implies (through (49)–(50)) that Δ0 < 0. Furthermore, since A¯ 2 has been designed to be stable in Proposition 6, Δ2 < 0 as well. Therefore, define μ0 = λmin (−Δ0 ) and μ2 = λmin (−Δ2 ), where μ0 , μ2 ∈ R+ . Differentiating Wc with respect to time yields W˙ c = x0T Δ0 x0 + 2x0T U T Bx e2 + δe2T Δ2 e2 ≤ −μ0 x0 2 + 2U T Bx  x0 e2  − δμ2 e2 2 .   

(54)

μx

  √ μ2 Then, set δ > μ0 μx 2 , and inequality − 21 μ0 x0 2 + δμ2 e2 2 ≤ − μ0 μ2 δx0  e2  implies (54) can be re-expressed as μ0 μ2 W˙ c ≤ − x0 2 − x e2 2 . 2 2μ0

(55)

 2  x0   ˙ , which implies Wc ≤ −θ2  Let θ2 = min  e2  . From (52), W˙ c can then be re-expressed as W˙ c ≤ − θθ21 Wc , which in turn implies that as t → ∞, x0 → 0 [7]. Hence, the satisfaction of LMI (45) results in the asymptotic  stability of x0 , completing the proof. 

μ0 μ2x , 2 2μ0



The controller for state and unknown input feedback has also been designed, concluding the error analysis of the FTC scheme.

5 Design of the Fault-Tolerant Control Scheme Several assumptions were made during the analysis of the FTC scheme in previous sections. These assumptions are made in terms of design matrices, and are difficult to verify without undertaking the design process. Hence, it is important to re-express them in terms of the original system matrices, so that the designer knows from the outset if the proposed scheme is applicable to their system.

5.1 Existence Conditions Theorem 2 The proposed FTC scheme can be applied onto system (1)–(2) such that it is asymptotically stable if and only if the following conditions hold:

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⎡

⎤ E A M Q ⎢C 0 0 0 ⎥ ⎥ A1. rank ⎢ ⎣ 0 E 0 0 ⎦ = n + n¯ + q + h. 0 C 0 0 A2. p ≥ q + n − n ¯ + h. A3. rank B M Q = rank(B). The following condition is necessary, and also sufficient if p > q + n − n¯ + h:   sE − A M A4. rank = n + q ∀ s ∈ C+ . C 0 If p = q + n − n¯ + h, the sufficient condition is that A5. The eigenvalues of A2 as defined in (9) are stable. Proof The constructive proof is presented by the remainder of this subsection.

 

Several assumptions were made during the prior analysis of the FTC scheme:  B1. There exists a non-singular matrix Td2 that transforms Mc2 Ac4 Q c2 to have the structures in (8). B2. The matrix T44 has the dimensions given in (12). B3. A matrix B † exists satisfying (39). B4. Error system (31)–(32) is stable. The following propositions show the equivalence of A1–A5 and B1–B4. Proposition 3 Condition A1 is necessary and sufficient to satisfy B1.     Mc1 Ac1 Q c1 = rank Mc2 Ac4 Q c2 . The Proof Assumption B1 implies rank Mc2 Ac4 Q c2 left-hand side of A1 can be simplified using the structures in (5) and (7) into ⎡

E ⎢C rank ⎢ ⎣0 0

A 0 E C

M 0 0 0

⎤ Q  0⎥ ⎥ = rank Mc2 Ac4 Q c2 + 2n. ¯ 0⎦ 0

(56)

 Condition A1 therefore implies rank Mc2 Ac4 Q c2 = n + n¯ + q + h. To show q˜ < q+ n − n¯ + the necessity of A1, suppose it is not met: rank Mc2 Ac4 Q c2 =   I 0 h. Hence, performing SVD on Te1 yields Te1 Mc2 Ac4 Q c2 Te2 = q˜ , where Te1 0 0 and Te2 are non-singular. This implies that the structure in (8) cannot be obtained, i.e., B1 is not satisfied, showing the necessity of A1.  On the being satisfied i.e., rank Mc2 Ac4 Q c2 = n + n¯ + q + h,  other hand, A1 implies Mc2 Ac4 Q c2 is full-column rank. Hence Td1 can be obtained by applying   I ¯ QR decomposition on Mc2 Ac4 Q c2 : Td1 Mc2 Ac4 Q c2 = q+n−n+h , satisfying 0 B1. Therefore, the sufficiency of A1 is shown, and the proof completed. 

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Proposition 4 Assumption B2 is satisfied if and only if A2 holds.

¯ p−q−h) Proof Matrix T44 has the dimensions T44 ∈ R( p−q−n+n−h)×( . Therefore, if A2 is not satisfied (i.e., p < q + n − n¯ + h), then T44 would not exist, and B2 not be satisfied, proving the necessity of A2. On the other hand, if A2 is satisfied (i.e., p ≥ q + n − n¯ + h), since n − n¯ > 0 then T44 will have its assigned dimensions, satisfying B2 and proving the sufficiency of A2. Thus, the proof is complete. 

Proposition 5 Condition A3 is necessary and sufficient for B3 to hold. 

Proof From [23], B3 implies that the vector space spanning the columns of M Q is a subset of the vector space spanning the columns of B, i.e.,  span M Q ⊆ span(B).



(57)

 Suppose  A3 is not satisfied, i.e., rank B M Q > rank(B); this would imply that span M Q  span(B), contradicting (57), resulting in B3 not holding and proving the necessity of A3. †   hand, if A3 is satisfied, there exists a matrix  B satisfying   On the †other † In − B B M Q = 0 [27], which in turn satisfies B B M Q = M Q , showing the sufficiency of A3. Thus the proof is complete.  Proposition 6 For B4 to be satisfied, A4 is necessary. If p ≥ q + n − n¯ + h, then A4 is also sufficient; otherwise (for p = q + n − n¯ + h), A5 is sufficient.  Proof Substituting for the structures of (E, A, M, C) in (9)–(11) into the left-hand side of A4 gives ⎡

⎤ 0 s In− 0 ¯ p − A2 s E 1 − A3 0 sE − A M Q −A5 s E 2 − A 6 M2 Q 2 ⎦ rank = rank ⎣−A4 C 0 0 0 0 Ip 0 0   s In− ¯ p − A2 + p + q + n − n¯ + h. = rank A54    



(58)

K 0 (s)

Then, by applying the Popov-Hautus-Rosenbrock (PHR) rank test [30], if the values of s that make K 0 (s) lose rank (the unobservable modes of A2 ) are stable, then the pair (A2 , A54 ) is said to be detectable. Hence, recast A4 as: (A2 , A54 ) is detectable. The necessity of A4 is shown as follows: B4 implies A2 + R24 T44 A54 is stable, i.e., λ ( A2 + R24 T44 A54 ) < 0, which implies (A2 , T44 A54 ) is also detectable. Hence, B4 can be recast as: there exists T44 such that (A2 , T44 A54 ) is detectable. Using the PHR test, the values that make K 1 (s) are stable, where

Observer-Based Fault-Tolerant Control for Non-Infinitely …

K 1 (s) =

137

     I¯ p 0 s In− s In− ¯ p − A2 ¯ p − A2 = n− , T44 A54 A54 0 T44   

(59)

K 0 (s)

which implies rank (K 1 (s)) ≤ rank (K 0 (s)), i.e., a value of s making K 0 (s) lose rank would also make K 1 (s) lose rank, showing the necessity of A4 for the detectability of (A2 , T44 A54 ). To show the sufficiency of A4 when p ≥ q + n − n¯ + h, let N be a matrix containing the right-eigenvectors of A2 ; then N −1 A2 N is a diagonal matrix containing the eigenvalues of A2 .   −1 0 N and N , Pre-multiply and post-multiply K 1 (s) with N0 = 0 I p−q−n+n−h ¯ respectively, to obtain 

I¯ p 0 N0 K 1 (s)N = n− 0 T44



 N −1 A2 N . A54 N

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modes of (A2 , T44 A54 ) would also be unobservable modes of  Unobservable N −1 A2 N , A54 N , and would appear as elements of N −1 A2 N with their corresponding columns in A54 (and therefore T44 A54 ) are zero. If A4 is satisfied however, T44 can always be chosen such that T44 A54 corresponding to elements within N −1 A2 N are non-zero, satisfying B4 and showing the sufficiency of A4 when p > q + n − n¯ + h.  0 , and R24 does If p = q + n − n¯ + h, then T p in (12) becomes T p = Iq+n−n+h ¯ not exist. Therefore it is sufficient that A2 has stable eigenvalues (i.e., A5 is satisfied) for error system (31)–(32) to be stable, completing the proof.  Remark 2 The presented method is also applicable to regular state-space sys tems (where E = In ) and infinitely observable descriptor systems where rank      E E = n . This can be shown by setting E = In , and E and C such that rank = C C n, respectively, and applying the analysis as shown. The proposed scheme is therefore even more general than those applicable only to regular state-space systems [1, 6, 19–23, 27–29], or to infinitely observable descriptor systems [7, 8, 10, 15, 17, 18]. 

5.2 Design Algorithm A summarized design procedure for the FTC scheme is given as follows: Step 1.

Check that A1–A4 (and A5 if necessary) hold. If these conditions are not satisfied, do not continue as the scheme is not applicable.

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Step 2.

Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Step 9.

Calculate Ta from (5), Tb and Tc from before (6), and Td from (8). Then, apply the state equation transformation Td Tb and state transformation Tc Ta onto system (1)–(2) to obtain the structures in (9)–(11). Choose T44 such that it is full-row rank, and (A2 , T44 A54 ) is detectable. Apply the state equation transformation T given in (12). Construct the reduced-order system (13)–(14). Select a value for γ, and use a LMI solver to determine P2 and P¯2 from (24). Calculate R24 , and then R and V from (19). Pick values for Ao and G n2 such that (19) is satisfied. Determine A51 , A52 , A53 , and T44 A54 and choose L 11 , L f 1 , and L x1 to satisfy (37), and set ρ in (18) to satisfy (25). Estimate x2 and y from (17), and f, x1 , and ξ from (33). Obtain H, J , and S from LMI (45) using a LMI solver, and then calculate K x from (46), and K f and K ξ from (40).

6 Simulation Example A modified version of the model describing the movement of the endocrine disruptor diethylstilbestrol (DES) through the human body in [12] is used to show the effectiveness of the proposed scheme. The actuator signals u 1 and u 3 are considered separately in this example. The input u 1 is generated by a faulty first-order device u˙ 1 = −4u 1 + 4u 1,e + f,

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where u 1,e is the reference signal, and f is the fault. Incorporating the dynamics of u 1 into the system yields the system matrices ⎡

−1.6 0.2 0.5 0.2 0 ⎢ 0.5 −0.2 0 ⎡ ⎤ 0 0 ⎢ I4 0 0 ⎢ 0.6 0 −0.5 0 0 E = ⎣ 0 0 0⎦ , A = ⎢ ⎢ 0.5 0 0 −0.5 0 ⎢ 0 01 ⎣ 1 1 1 1 1 0 0 0 0 0  T M= 000001 , for the system variables

⎤ ⎡ ⎤ 0 0 1 ⎢0 0 ⎥ 0⎥ ⎥ ⎢ ⎥ ⎢0 3 ⎥ ⎥ 0⎥ ⎥ ⎢ , B = ⎢0 0 ⎥ , 0⎥ ⎥ ⎢ ⎥ ⎣0 −3⎦ −1⎦ 4 0 −4

(62)

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⎤ quantity of DES in the circulation system, xa ⎢quantity of DES in the reproductive system, xb ⎥ ⎥ ⎢ ⎢ quantity of DES in the digestive system, xc ⎥ ⎥ x0 = ⎢ ⎢ quantity of DES in the excretion system, xd ⎥ , ⎥ ⎢ ⎦ ⎣ quantity of DES outside the body, xe quantity of drug one injected into blood, u 1   reference for u 1 , u 1,e u= . quantity of drug two injected into liver, u 3 ⎡

(63)

The drug intake into the liver causes DES quantities to fluctuate by ξ. Additionally, assume that measurements are only available for xa , xc , xd , and u 1 . Hence, Q and C have the structures ⎡ 10 0 0  T Q = 0 0 1 0 −1 0 , C = ⎣0 0 I2 0 00 0 0

⎤ 0 0⎦ . 1

(64)

Since n¯ (5) < n (6), system (62)–(64) is non-infinitely observable. The FTC scheme would now be designed according to the steps outlined in Sect. 5.2. Step 1. It can be seen that ⎡

⎤ E A M Q ⎢C 0 0 0 ⎥ ⎥ rank ⎢ ⎣ 0 E 0 0 ⎦ = n + n¯ , p > q + n − n¯ + h, 0 C 0 0   sE − A M rank = n + q∀ s ∈ C+ , C 0  rank B M Q = rank(B) = 2,

(65)

that is, A1–A4 are satisfied, guaranteeing the existence of the proposed FTC scheme. Step 2. The transformations Td Tb and Tc Ta are calculated to be ⎡

0 ⎢0 ⎢ ⎢0 Td Tb = ⎢ ⎢0 ⎢ ⎣1 0

1 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 0 1

⎤ ⎡ 00 00 0 1⎥ ⎥ ⎢0 1 ⎢ 1 0⎥ ⎥ , Tc Ta = ⎢1 0 ⎢ 0 0⎥ ⎥ ⎣0 0 0 0⎦ 00 00

0 0 0 I2 0

1 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥. 0⎦ 1

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Step 3. The following partitions are obtained: A2 = −0.2, A54 = ⎡ ⎤ I4 0 0  Hence, choose T44 = −1 1 to get T = ⎣ 0 −1 1⎦. 0 0 1

  0.2 . 0

Step 4. The reduced-order system (13)–(14) therefore has the matrices ⎡

1 ⎢0 ⎢ E¯ = ⎢ ⎢0 ⎣0 0

0 0 0 0 −1

0 0 1 1 0

0 0 0 0 1

⎡ ⎤ −0.2 0.5 0 ⎢ 0 0 1⎥ ⎢ ⎥ ¯ ⎢ 0⎥ ⎥ , A = ⎢ 1 1.6 ⎣ 0 0.6 0⎦ −0.2 2.1 0  T M¯ = 0 I3 0 ,

⎡ ⎤ ⎤ 00 0 0 0 ⎢4 0 ⎥ 0 0 −4⎥ ⎢ ⎥ ⎥ ¯ ⎢ ⎥ 0.5 1 −1⎥ ⎥ , B = ⎢0 0⎥ , ⎣0 3⎦ −0.5 0 0 ⎦ 00 −0.5 −0.7 −1  C¯ = 0 I4 .

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Step 5. The parameter γ is chosen as γ = 0.001. Using the SeDuMi solver for YALMIP within MATLAB on LMI (24), the following are obtained: P2 = 1, P¯2 = 1.502, R24 = P2−1 P¯2 = 1.502. ⎡ ⎤ 1.502 0 −1.502 0 ⎡ ⎤ ⎢ 1 1 0 1.502 0 0 −1⎥ ⎢ ⎥ 0 0 0 0⎥ R = ⎣0 I3 0 ⎦ , V = ⎢ ⎢ ⎥. ⎣ 0 −1 0 0 1 1 0⎦ 1 0 −1 1

(68)

Steps 6–8. The following partitions are obtained: A51 = 0, A52 = 1, A53 = 0, T44 A54 = −0.2. Hence Ao , G n2 , L f , L 1 , L x , and ρ are chosen as Ao = 2I4 , G n2 = I4 , L f = L x = 0, L 1 = 5, and ρ = 6. Step 9. Using the SeDuMi solver for YALMIP within MATLAB on LMI (45), the following are obtained: ⎡ ⎤ ⎤ −3003 537.6 491.6 447.7 435.8 457.9 491.4 ⎢ −4246 ⎥ ⎢626.1 1930 436.4 549.9 457.9 409.1⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢277.0 182.4 505.2 283.6 457.9 553.3⎥ ⎥ , J = ⎢ −1547 ⎥ , H =⎢ ⎢ −3230 ⎥ ⎢572.2 738.5 403.4 994.5 457.9 357.7⎥ ⎥ ⎥ ⎢ ⎢ ⎣ −1652 ⎦ ⎣457.9 457.9 457.9 457.9 457.9 457.9⎦ −687.8 76.56 47.41 134.7 83.60 457.9 186.8   99.48 58.47 142.1 101.9 457.9 146.4 S= , 203.0 164.3 268.2 197.6 457.9 145.6     0.25 0 Kf = , Kξ = , 0 0.3333   −0.9983 −0.1708 −0.2798 −0.4975 −0.2771 5.224 Kx = . −0.9017 −0.1724 0.7694 −0.4520 −0.2771 1.774 ⎡

(69)

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The design of the FTC scheme is now complete. To demonstrate the effectiveness of the proposed scheme, the disturbance signal is set as ξ = 0.1 sin (0.5t + 3π/4) + 0.2. The initial condition of the system is set to be {0.8, 0.45, 0.68, 0.7, −20.10, 0.65}, while the observer is set to have zero initial conditions. The fault signal is f˙ = −2 f + 0.4 sin (1.2t + π/5) + 1 + 0.8h(t − 14) − 1.2h (t − 31), where h(·) is the Heaviside unit step function. Figure 1 shows the states and unknown inputs, and their estimates. It can be seen that sliding motion takes place close to the start of the simulation. The estimates

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for the non-infinitely observable state xe , output states xa , xc , xd , and u 1 , fault f , and disturbance ξ converge onto the actual values very quickly, as per the gain design in Step 7. This convergence is also reflected in Fig. 2, which shows the estimation errors for the observer. The estimate for xb (which is a non-output, infinitely observable state), however, experiences an initial mismatch due to the system and observer having different initial conditions. Figure 2 however shows that eb goes to zero asymptotically, in accordance with Proposition 2. Hence, the effectiveness of the SMO is shown. Finally, Fig. 3 shows the control inputs fed into the system. As can be seen from

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Fig. 1, all states of the system converge to the reference of zero, demonstrating the efficacy of the proposed FTC scheme.

7 Conclusion An FTC scheme for NIODS was presented in this chapter. The FTC scheme consists of an SMO to estimate the states and unknown inputs in the system, and a controller to negate the effects of the unknown inputs. The NIODS is first re-expressed as an infinitely-reduced order system by treating some of its states as unknown inputs. The SMO is then designed based on this reduced-order system to estimate the states and unknown inputs. The controller is then utilized to generate the control input signal such that it counters the unknown inputs while driving the system states to the reference. LMIs were used to design the SMO and controller such that the SMO estimates and system states converge asymptotically. Necessary and sufficient conditions for the feasibility of the scheme were presented in terms of the original system matrices. A simulation example was then used to verify the efficacy of the FTC scheme. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1012707). The authors would like to thank Prof. H. Trinh for his invaluable input on the development of the observer utilized in this scheme.

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Fault Estimator for Parabolic Systems with Distributed Inputs and Outputs K. Mathiyalagan

Abstract This chapter is concerned with the H∞ fault estimation of a class of parabolic systems with distributed inputs and outputs. The fault in the considered parabolic system is characterized by a minimal state-space description of a low-pass filter, which incorporates fault’s prior knowledge. The fault estimator is designed by using the Lyapunov theory and linear matrix inequalities (LMIs) approach. The sufficient conditions in the form of LMIs are developed to ensure the resulting error system is asymptotically stable with an optimized H∞ disturbance attenuation level. The designed estimator gain matrices can be easily determined by using the standard optimization toolboxes. Finally, two numerical examples and simulation results are provided to illustrate the effectiveness and advantages of the obtained results. Keywords Fault detection · Low pass filter · H∞ analysis · Exponential stability

1 Introduction Generally, partial differential equations (PDEs) exhibits its special features, which usually mirror the physical phenomena, many of the fundamental theories of physics and engineering are expressed by means of systems of PDEs. Analyzing the existence and stability of PDEs are far more difficult and form a key topic of contemporary mathematical research. There is no common hypothesis known concerning the solvability of all PDEs. Such a hypothesis is greatly impossible to exist, given the wealthy variety of physical, probabilistic and geometric phenomena that can be modeled by PDEs. Moreover, many research concentrates on different specific PDEs that are essential for applications in the real world scenario, for example see [1–4]. In PDEs, the typical additional constraint is so-called boundary condition, in which specified values are imposed at points on the boundary of the domain where K. Mathiyalagan (B) Department of Applied Mathematics, Bharathiar University, Coimbatore 641 046, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_7

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the solution is supposed to be defined. PDEs are usually built around the equations of classical mathematical physics. They are used to describe the evolution of gases in fluid dynamics, the formation of galaxies, describing the nature of quantum mechanics and so on [5–7]. PDEs are used to mathematically express, and thus aid the solution of physical and other problems involving functions of several variables. Parabolic PDEs which describe diffusion processes such as heat equation, chemical concentration, etc. These equations play a substantial role in understanding problems related to flow and heat motion of free fall body. Solutions are smooth in space but may have singularities. Dynamical frameworks may be a massive field, diverse sorts of tributes and PDEs that emerge as models in material science, physics, etc [8–12] and references therein. Boskovic et al., [13] discussed about the problem of nonlinear parabolic PDEs have been discussed and constructed certain Lyapunov structures to prove stability. Stability is the fundamental requirement in all dynamical systems it is mainly inspired by differential equations of motion and flows. Stability in motion means, the system is evolving in a direction such that it will reach a minimum energy state. Stability of solutions is important in physical problems if slight deviations from the mathematical model caused by unavoidable errors in measurement do not have a correspondingly slight effect on the solution, so the mathematical equations describing the problem will not accurately predict the future outcome [14]. In real-world, situations uncertainties and disturbances like noise in sensors and actuators can unstabilize the system [15–18]. Since most PDEs are express the real control systems, so they may encounter different kinds of faults due to malfunctions or imperfect performances exist. These faults are caused because of the unexpected variations, sudden changes in signals, environments, external disturbances or parameter shifting and so on [19, 20]. So, the fault detection problem becomes an important topic among the research community [21, 22]. The objective in the fault detection problem is to detect a fault value when it happens to take protection measures, or reconfiguring the running control scheme. Ferdowsi and Jagannathan [23] discussed the fault diagnosis problem for a class of distributed parameter systems modeled by parabolic partial differential equations. Demetriou [24] has discussed about model-based fault detection and diagnosis scheme for distributed parameter system. Ferdowsi et al., [23] designed fault detection schemes for distributed parameter parabolic system has been investigated. It is noted that an optimization problem is one of the most commonly implemented ways to deal the fault detection. In literature, many results related to fault estimation/detection can be found for varies types of practical systems including parabolic type ones [25, 26]. Baruh [27] studied the actuator failure detection in distributed systems. Among the many approaches in the literature, the common way is the observerbased approach [28]. This approach is based on generating residuals by designing proper state observers or filters and using these residuals to set a threshold to detect the fault. The problem of H∞ fault estimation for the discrete-time switched nonlinear system is considered by using the multiple-Krasovskii functional and average dwelltime approach in [29]. Yuan et al., [30] derived some sufficient conditions for H∞

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fault estimator for switched linear systems to be asymptotically stable and determined L2 gain bound. Zhang et al., [31] discussed about observer-based fault diagnosis techniques for faulty systems. It is well known that, an observer or filter should be designed to guarantee the improved robustness against unknown inputs and sensitivity to faults [32, 33]. There are increasing awareness to ensure robustness in performance requires simpler and stable adaptive observer schemes. The main objective of the optimization problem is to introduce a performance index on the residual signal between the robustness to disturbance and the sensitivity to faults and formulate the fault detection and isolation as an optimization problem. Moreover, it is well known that the design problem of H∞ -based observer or filter expresses the design procedure as a mathematical optimization problem for finding some required solutions [34]. The H∞ -norm measures robustness against the unknown inputs to achieving results of guaranteed performance H∞ methods are utilized to achieve stabilization with ensured performance, and to constrict the external disturbance with the reason for minimizing the effect of the disturbance of the system performance [35]. Baniamerian et al., [36] discussed about dissipative parabolic PDE and designed geometric fault detection obtained from the Galerkin method. These techniques are receiving considerable attention in research fields. Lien et al., discussed about discrete switched system then the results are extended for exponential stability and robust H∞ performance. Fridman et al., [37] discussed about H∞ boundary control for semilinear parabolic and hyperbolic systems in 2009. Fischer et al., [38] discussed about fault detection approach for parabolic systems using modulation function. However, to the best of the authors’ knowledge, the H∞ fault estimation of a class of parabolic systems with distributed inputs and outputs has not been fully investigated, which motivates this present investigation. Motivated by the above, in this chapter, the observer design for a class of distributed parameter system of parabolic type with fault inputs is discussed. The sufficient conditions are derived by using the Lyapunov functional method together with the LMI approach under which the error system is robustly stable with a guaranteed H∞ performance index. Moreover, the sufficient conditions to design the parameters of the observer gains are established in terms of LMIs, which can be easily checked by using the standard numerical toolbox. Also, the results are expressed in the form of an optimization problem that can guarantee the minimum H∞ performance index. Finally, numerical examples are presented to validate the proposed results. Now, the problem under investigation can be summarized as • Study the exponential stability conditions for the distributed parameter system of a parabolic type system. • Design a suitable H∞ fault estimator that can guarantee the minimum H∞ performance index.

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2 Problem Description 2.1 System Description Consider the parabolic partial differential equation ∂t w(z, t) = ∂z2 w(z, t) + β(z)w(z, t) + a(z) f (t) + b(z)d(t),

(1)

where (z, t) ∈ (0, 1) × R+ with β(z) ∈ C[0, 1], a(z), b(z) are piecewise continuous, f (t) is the unknown fault signal the unknown disturbance d(t) ∈ R pd . The initial condition for the above system is w(z, 0) = w0 (z), ∀ w0 ∈ L 2 (0, 1). The distributed output y(t) ∈ Rm is assumed as 

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w(z, t)dz.

(2)

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2.2 Low Pass Filter Most of the dynamical systems typically need to deal with noise, in such cases, low pass filters (LPFs) are used to remove noise from the measured signals. In order to achieve a better fault estimation results, incorporating the faults prior knowledge into the design process is very much useful, which is much more suitable and reasonable in many practical situations. So, using an LPF, we define a minimal state-space description of the fault f (t) as ∂t w f (z, t) = ∂z2 w f (z, t) + a f (z)w f (z, t) + b f (z) f 0 (t), w f (z 0 , t0 ) = 0,

 (3)

where w f (z, t) ∈ Rn f is the LPF state, f 0 (t) is a fictitious signal in L 2 (0, 1) and a f (z), b f (z) and c f are weight functions obtained from the prior knowledge of the fault. In this work, the fault is considered as the output of the above model and described as  1 w f (z, t)dz, (4) f (t) = c f 0

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2.3 Observer Description Consider the following observer to estimate the fault in (1) ˆ t) = ∂z2 w(z, ˆ t) + β(z)w(z, ˆ t) + a(z) fˆ(t) ∂t w(z, +k1 (y(t) − yˆ (t)),

(5)

with an output of the form 

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where w(z, ˆ t), yˆ (t) are the estimates of w(z, t), y(t). The goal is to find fˆ(t), such that the following ratio is satisfied under zero initial condition      f (t) − fˆ(t) ≤ γ w(t)2 , 2



T

where γ > 0, w(t) = d T (t) f 0T (t) . Now, we assume the fault estimation fˆ(t) as fˆ(t) = c f



1

wˆ f (z, t)dz,

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where wˆ f (z, t) is the state vector of the system having the dynamics ∂t wˆ f (z, t) = ∂z2 wˆ f (z, t) + a f (z)wˆ f (z, t) + k2 (y(t) − yˆ (t)), in which k1 and k2 are observer gains to be designed. Let the error signals be e(z, t) = w(z, t) − w(z, ˆ t), e f (z, t) = w f (z, t) − wˆ f (z, t) and

r f (t) = f (t) − fˆ(t).

Then, the error systems can be represented as

(6)

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∂z2 e(z, t) 

+ β(z)e(z, t) + a(z)c f

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(7)

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(8)

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and the residual function is given by  r f (t) = c f

1

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(9)

0



 e(z, t) Next, denote ξ(z, t) = , then the augmented system can be rewritten as e f (z, t) ∂t ξ(z, t) = A∂z2 ξ(z, t) + C(z)ξ(z, t) + B(z)w(t) − K

1 0

ξ(z, t)dz,

(10)

where    1 0 β(z) 0 , , C(z) = 0 1 0 a f (z)     b(z) 0 k −a(z)c f B(z) = , K = 1 , k2 0 0 b f (z) 

A=

and the residual function turns of the form  r f (t) = C f

1

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0

with

  Cf = 0 cf ,

and the periodic boundary conditions for (10) are taken as ξ(0, t) = 0, ξ(1, t) = 0. Now, denote that



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ξ(z, t)dz − ξ(z, t),

(11)

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which implies that



1

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ξ(z, t)dz = g(z, t) + ξ(z, t).

0

Now, the problem under investigation can be summarized as • to obtain the sufficient conditions for the exponential stability of the system (10) when w(t) = 0. • to obtain the sufficient conditions for the existence of a suitable H∞ fault estimator that can guarantee the H∞ performance for all nonzero w(t) ∈ l2 [0, ∞).

3 Preliminaries In this section, we are introducing some basic definitions and lemma. Lemma 1 Given constant matrices G 11 , G 12 , G 22 with appropriate dimensions, T T = G 11 and G 22 = G 22 , then where G 11 T G −1 G 11 + G 12 22 G 12 < 0

if and only if



T G 11 G 12 ∗ −G 22

 < 0.

Lemma 2 (Vector Poincare-Wirtinger inequality) Let ξ(z, t) ∈ H be a vector function. Then, the following inequality holds for any matrix 0 ≤ M ∈ Rn×n , we have  Ω

g T (z, t)Mg(z, t)dz ≤ 4φs π −2

where g(z, t) = (l2 − l1 )−1



l2

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ξzT (z, t)Mξz (z, t)dz

(12)

ξ(z, t)dz − ξ(z, t),

l1

the limits satisfying α1 ≤ l1 < l2 ≤ α2 and φs = max{(l2 − α1 )2 , (α2 − l1 )2 }. Definition 1 System (10) with w(t) = 0 is said to be exponentially stable, if there exist scalars ρ > 0, α > 0 such that the solution ξ(z, t) of the system (10) satisfies the following condition ξ(·, t0 ) ≤ ρe−α(t−t0 ) θ(·) , ∀t ≥ t0 where ξ(z, t0 ) = θ(z), α > 0 is called the decay rate.

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Definition 2 The distributed parameter system (10) is said to be exponentially stable with given disturbance attenuation level γ > 0, if it is robustly stable and the response r f (t) under zero initial condition satisfies, ||r f (t)||2 ≤ γ||w(t)||2 for every non-zero w(t) ∈ L 2 [0, ∞). The following specified notations are used throughout this chapter. – – – – – – – – –

Rn denotes the n-dimensional Euclidean space. Rn×m is the set of all n × m matrices with real entries. The superscript T stands for matrix transposition. L 2 denotes the space of square integrable functions on (0, 1) with Euclidean norm

∞ 1  ·  = 0  · 2 dt 2 . I and 0 are identity matrix and zero matrix respectively. The spatial detection domain Ω = [z 0 , z 1 ] ⊂ [0, 1] where z 0 < z 1 . For a real matrix P, P > 0 means P is Positive definite. The symmetric term in a symmetric matrix is denoted by asterisk (∗). diag{·} denotes the diagonal matrix.

4 Exponential Stability and H∞ Performance For known k1 and k2 , first we prove exponential stability for the system (10) and the result extended to ensure the H∞ performance. Theorem 1 For given scalars γ > 0, α > 0 and known k1 , k2 , there exist a positivedefinite symmetric matrix P such that the following LMI holds ⎡

⎤ Ω11 −P K + C Tf C f P B(z) ⎣ ∗ Ω22 0 ⎦ < 0, ∗ ∗ −γ 2

(13)

where Ω11 = PC(z) + C T (z)P T + Pα − P K − K T P T + C Tf C f , Ω22 = −

Π2 (P A + A T P T ) + C Tf C f , 4φs

then the system (10) is exponentially stable with H∞ performance level γ. Proof Now, we choose a Lyapunov function as follows  V (z, t, ξ(z, t)) =

Ω

ξ T (z, t)Pξ(z, t)dz,

(14)

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and taking the time derivative of V (z, t, ξ(z, t)) along the trajectory of the system (10) yields  ∂t V (z, t, ξ(z, t)) = 2 =2

Ω Ω

ξ T (z, t)P∂t ξ(z, t)dz, ξ T (z, t)P{A∂z2 ξ(z, t) + C(z)ξ(z, t) + B(z)w(t)

− K [g(z, t) + ξ(z, t)]}dz,  =2 ξ T (z, t)(P A)∂z2 ξ(z, t)dz Ω  +2 ξ T (z, t)P{C(z)ξ(z, t) Ω

+ B(z)w(t) − K [g(z, t) + ξ(z, t)]}dz. Here, using integration by parts that leads to ∂t V (z, t, ξ(z, t)) = 2(ξ T (z, t)(P A)∂z ξ(z, t))Ω  −2 ∂z ξ(z, t)(P A)∂z ξ T (z, t)dz Ω  +2 ξ T (z, t)P{C(z)ξ(z, t) + B(z)w(t) Ω

− K [g(z, t) + ξ(z, t)]}dz and applying boundary conditions, we get  ∂t V (z, t, ξ(z, t)) = −2 +2

Ω



∂z ξ T (z, t)(P A)∂z ξ(z, t)dz

Ω

ξ T (z, t)P{C(z)ξ(z, t)

+ B(z)w(t) − K [g(z, t) + ξ(z, t)]}dz. By using vector Poincare-Wirtinger inequality, above equation takes the form,  Π2 g T (z, t)(P A)g(z, t)dz 4φs Ω  ξ T (z, t)P{C(z)ξ(z, t) +2

∂t V (z, t, ξ(z, t)) ≤ −2

Ω

+ B(z)w(t) − K [g(z, t) + ξ(z, t)]}dz. First, we consider the stability of the system (10), for this case if w(t) = 0, we obtained the following inequality.

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1

∂t V (z, t, ξ(z, t)) + αV (z, t, ξ(z, t)) ≤

η T (z, t)Φη(z, t)dz,

(15)

0

  where η T (z, t) = ξ T (z, t) g T (z, t) and  Φ=

 PC(z) + C T (z)P T + Pα − P K − K T P T −P K , Π2 ∗ − 4φ (P A + A T P T ) s

if the Φ < 0 holds, then it implies that LMI (15) holds. Then, we have ∂t V (z, t, ξ(z, t)) + αV (z, t, ξ(z, t)) < 0 for all η(z, t) = 0. Integrating this inequality over [t0 , t] gives V (z, t, ξ(z, t)) ≤ e−α(t−t0 ) V (z, t0 , ξ(z, t0 )), for arbitrary initial condition ξ(z, t0 ) = θ(z) and  Ω

ξ T (z, t)Pξ(z, t)dz = V (z, t, ξ(z, t)), ≤ e−α(t−t0 ) V (z, t0 , ξ(z, t0 )),  1 −α(t−t0 ) ≤e θ T (z)Pθ(z)dz. 0

Then, it is easy to show that ξ(., t) ≤ e−α(t−t0 ) θ(·), ∀t ≥ t0 .

(16)

Therefore, by Definition 1, we conclude that the system (10) with w(t) = 0 is exponentially stable. Now, we build H∞ performance for the augmented system (10),  1 ∂t V (z, t, ξ(z, t)) + αV (z, t, ξ(z, t))+ J (t)dz 0  1 η1T (z, t)η1 (z, t)dz, ≤ 0

 by setting η1T (z, t) = ξ T (z, t) g T (z, t) w T (t) , then we easily obtain (13). By ensuring (13)< 0, for all η1 (z, t) = 0, we get 

∂t V (z, t, ξ(z, t)) + αV (z, t, ξ(z, t)) + J (t) < 0.

(17)

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follows from by denoting J (·) = r Tf (t)r f (t) − w T (t)γ 2 I w(t). Integrating (17) over [t0 , t] gives V (z, t, ξ(z, t)) ≤ e

−α(t−t0 )



t

V (z, t0 , ξ(z, t0 )) +

e−α(t−s) J (s)ds.

(18)

t0

In the same way to prove the exponential stability, (18) produces V (z, t, ξ(z, t)) ≤ e−λ(t−t0 ) V (z, t0 , ξ(z, t0 )) +



t

e−λ(t−s) J (s)ds.

(19)

t0

Taking t0 = 0, (19) implies 

t

e−λ(t−s) J (s)ds ≤ 0.

(20)

0

Integrating (20) over [0, ∞) to obtain 

∞ 0



t

e−λ(t−s) J (s)dsdt ≤ 0.

0

changing the order of integration which yields that 

∞ 0



∞

J (s) s

  1 ∞ e−λ(t−s) dt ds = J (s)ds ≤ 0. λ 0

Therefore, from the above it is easy to verify ||r f (t)||2 ≤ γ||w(t)||2 , then by Definition 2, we conclude that the system (10) is exponentially stable with

H∞ performance level γ. Suppose that the observer gains k1 and k2 are not known, now it can be designed using the results obtained in the above theorem. Theorem 2 For given scalars γ > 0, α > 0, the system (7) is exponentially stable with H∞ performance level γ, there exist a positive-definite symmetric matrix diag{ p1 , p2 }> 0 such that the following LMI holds

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⎡

Σ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Σ12 Σ22 ∗ ∗ ∗ ∗

⎤ −κ1 p1 a(z)c f p1 b(z) 0 −κ2 c Tf c f 0 p2 b f (z) ⎥ ⎥ ⎥ Σ33 0 0 0 ⎥ 0, if it is robustly stable and the response r f (t) under zero initial condition satisfies,  0

∞

(γ −1 δr Tf (t)r f (t) − 2(1 − δ)r Tf (t)w(t) ≤ γw T (t)w(t))dt,

for every non-zero w(t) ∈ L 2 [0, ∞). The result can be deduced by letting in (17) as J (·) = γ −1 δr Tf (t)r f (t) − 2(1 − δ)r Tf (t)w(t) − γw T (t)w(t),

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we can easily deduce the the following theorem based on mixed H∞ and passivity performance index. Theorem 3 For given scalars γ > 0, α > 0 if there exist a positive-definite symmetric matrix diag{ p1 , p2 }> 0 such that ⎡

Σˆ 11 ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Σˆ 12 Σˆ 22 ∗ ∗ ∗ ∗ ∗

−κ1 −κ2 Σˆ 33 ∗ ∗ ∗ ∗

√ T Σˆ 15 δc f p1 a(z)c f 0 T 0 0 p2 b f (z) − (1 − δ)c f 0 √ T 0 0 0 δc f Π2 T T − 4φs ( p2 + p2 ) 0 −(1 − δ)c f 0 ∗ −γ 0 0 ∗ ∗ −γ 0 ∗ ∗ ∗ −γ

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0 (22) ⎥ ⎥ ⎥ ⎦

where Σˆ 11 = p1 β(z) + β T (z) p1T + p1 α − κ1 − κ1T , Σˆ 12 = p1 a(z)c f (z) − κ2T , Σˆ 15 = p1 b(z) Σˆ 22 = p2 a f (z) + a Tf (z) p2T + p2 α, Σˆ 33 = −

Π2 ( p1 + p1T ), 4φs

then system (7) is exponentially stable with mixed H∞ and passivity performance level γ and observer gains are given by k1 = p1−1 κ1 , k2 = p2−1 κ2 . Proof The proof of the above theorem is immediately follows from (17) that ⎡

⎤ Σˇ 11 −P K + γ −1 δC Tf C f P B(z) − (1 − δ)C Tf ⎢ ⎥ Π2 (P A + A T P T ) + γ −1 δC Tf C f (1 − δ)C Tf ⎣ ∗ − 4φ ⎦ 0, gains k1 and k2 are known. The system (10) is exponentially stable with H∞ performance level γ, there exist a positive-definite symmetric matrix P such that the optimization problem: min γ

(23)

subject to the LMI ⎤ Ω¯ 11 −P K + C Tf C f P B(z) ⎣ ∗ Ω¯ 22 0 ⎦ < 0, ∗ ∗ −γ 2 ⎡

is solvable, Ω¯ 11 = PC(z) + C T (z)P T + Pα − P K + C Tf C f , Ω¯ 22 = −

Π2 (P A + A T P T ) + C Tf C f . 4φs

Corollary 2 Let the scalar α > 0 is known. The system (10) is exponentially stable with H∞ performance level γ, there exist a positive-definite symmetric matrix diag{ p1 , p2 } such that the optimization problem: min γ

(24)

subject to the LMI ⎡ ¯ Σ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Σ¯ 12 Σ¯ 22 ∗ ∗ ∗ ∗

⎤ p1 a(z)c f p1 b(z) 0 c Tf c f 0 p2 b f (z) ⎥ ⎥ ⎥ 0 0 0 ⎥ 0. 1 , ∀ k2 ≥ k1 ≥ 0 (2) τa is the average dwell time of σ(k) if Nσ (k1 , k2 ) ≤ N0 + k2τ−k a for a positive number N0 , and N0 is called as the chatter bound of σ(k). For simplicity, let S[τa , N0 ] be the set of the switching signals whose average dwell time and chatter bound are τa and N0 respectively. The following assumption for the σ(k) in the system (1) is necessary. Assumption 2 The dwell time of σ(k) is τd and σ(k) belongs to the set  S[τa , N0 ]. Then, based on Assumptions 1 and 2 and the classical results for switched system, exponential stability of the closed-loop system of (1) under a proper feedback controller u(k) = K σ(k) x(k) is guaranteed if τd or τa of the switching signal is large enough. In this chapter, the primary objective is how to maintain the stability when system (1) is controlled over limited capacity channel, where the switching signal and system state need to be detected and transmitted remotely. To this end, we rstly present some preliminary results of the merging technique on switching signal, which might be viewed as the counterpart of that in [41] for the discrete-time case. Suppose that σ(k) ˆ = σ(k − τ (k)) is the delayed signal of σ(k), where the timevarying delay τ (k) ∈ [τ1 , τ2 ). Then, for the switching signal σ(k) satisfying Assumption 2, we de ne the augmented signal θ(k) = (σ(k), σ(k)) ˆ by merging σ(k) with its delayed signal σ(k). ˆ Based on the similar analysis as that in Lemmas 1 and 2 in [41], we can get the following conclusion.

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Lemma 1 Suppose that the switching signal σ(k) belongs to the set S[τa , N0 ]. If  1 the delay τ (k) ∈ [τ1 , τ2 ), then σ(k) and θ(k) = ˆ = σ(k − τ (k)) ∈ S τa , N0 + τ2τ−τ a   τa τ2 −τ1 (σ(k), σ(k)) ˆ ∈ S 2 , 2N0 + τa . Suppose that Ts (k1 , k2 ) denotes the total synchronous time in any given time interˆ Then, Tas (k1 , k2 ) = k2 − k1 − Ts (k1 , k2 ) denotes the val [k1 , k2 ) for σ(k) and σ(k). asynchronous time for σ(k) and σ(k) ˆ during [k1 , k2 ). In addition, [k¯i , k¯i+1 ) denotes any switching interval of σ(k) and τ¯a be the upper bound of the asynchronous time length for σ(k) and σ(k) ˆ during [k¯i , k¯i+1 ). Similar to Lemma 3 of [41], we have the conclusion as follows. Lemma 2 Given scalar constants η > 0, ρ ∈ (0, 1) and β¯ ∈ (1 − ρ, 1). Suppose that (1 + η)τ¯a (1 − ρ)τa −τ¯a ≤ β¯ τa , then the inequality (1 − ρ)Ts (k1 ,k2 ) (1 + η)Tas (k1 ,k2 ) ≤  N0 τ¯a  . α¯ β¯ (k2 −k1 ) holds, where α¯ = 1+η 1−ρ

3 Event-Triggered Control of Discrete-Time Switched Linear Systems with Package Dropout The event-triggered control framework for the considered switched systems (1) is illustrated in Fig. 1, in which the event generator is proposed to determine transmission instants of the switching signal and system state. The corresponding eventtriggered transmission time sequence is denoted by {ks }s≥0 with k0 = 0. The event-triggered transmission scheme in this section is designed as follows:   ks+1 = min ks + τd , kˆs+1 , s ≥ 1,  kˆs+1 = min k|(x(k) − x(ks ))T Φσ(ks ) (x(k) − x(ks )) k>ks ≥ νx T (ks )Φσ(ks ) x(ks ) ,

(2)

where the event-trigger parameters, to be determined, ν ∈ (0, 1) is a positive scalar and Φi is a positive de ned matrix. Let Ωs = [ks , ks+1 ) denote the holding interval generated by the proposed eventtriggered scheme (2). Now, we propose the following state feedback controller generated by the zero-order holder (ZOH): u(k) = K σ(ks ) x(ks ), k ∈ Ωs .

(3)

It is obvious that the mode of controllers in (3) remains constant within each holding interval.

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Fig. 1 The event-triggered control framework for switched system (1)

In addition, the event-triggering scheme (2) indicates that ks+1 − ks ≤ τd , which combining Assumption 2 implies that the system might switch once within the holding interval Ωs . Therefore, the switching modes of the controller and the system become mismatch after the system switches. ˆ = For k ∈ Ωs , let e(k) = x(k) − x(ks ) and τ1 (k) = k − ks ∈ [0, τd ), then σ(k) ˆ and

Bθ(k) = Bσ(k) K σ(k) σ(k − τ1 (k)). In addition, denote θ(k) = (σ(k), σ(k)) ˆ , then the closed-loop system of (1) and (3) can be represented as follows: Bθ(k) (x(k) − e(k)) . x(k + 1) = Aσ(k) x(k) +

(4)

Furthermore, the event-triggering condition in (2) implies that ν(x(k) − e(k))T Φσ(ks ) (x(k) − e(k)) − e T (k)Φσ(ks ) e(k) > 0, k ∈ Ωs .

(5)

3.1 Stability Analysis Without Pack Dropout The rst main result is obtained as follows. Theorem 1 Assume that there exist positive definite matrices Pi j and Φ j such that the matrix inequalities (6) and (7) hold for the given scalars μ ≥ 1, η > 0, ρ ∈ (0, 1), ν ∈ (0, 1) and state feedback gains K j . Then, under event-triggered scheme (2), the closed-loop system (4) is exponentially stable if the switching signal satisfies the average dwell time condition (8). μ−1 Pii ≤ Pi j ≤ μP j j , Pii ≤ μP j j , ∀i, j ∈ M,

(6)

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⎡

⎤ −λi2j Pi j + νΦ j −νΦ j (AiT + B˜ iTj )Pi j ⎣ ∗ (ν − 1)Φ j − B˜ iTj Pi j ⎦ < 0, ∀i, j ∈ M, ∗ ∗ −Pi j

(7)

λi j = δi j (1 − ρ) + (1 − δi j )(1 + η), −τa ln(1 − ρ) > ln μ + τd ln

1+η . 1−ρ

(8)

Proof To obtain the conclusion in Theorem 1, we construct a Lyapunov function as follows: (9) Vθ(k) (k) = x T (k)Pθ(k) x(k). Let k¯ be an any switching instant of θ(k). In other words, k¯ is some transmission instant or switching instant of σ(k). From (6) we have ¯ ≤ μVθ(k−1) ¯ (k). Vθ(k) ¯ (k) ¯

(10)

Then, for k ∈ Ωs , considering the event-triggering condition (5), we compute the difference Vθ(k) (k + 1) − λ2θ(k) Vθ(k) (k) along the closed-loop system (4) as: Vθ(k) (k + 1) − λ2θ(k) Vθ(k) (k) = x T (k + 1)Pθ(k) x(k + 1) − λ2θ(k) x T (k)Pθ(k) x(k) ≤ (Aσ(k) x(k) +

Bθ(k) (x(k) − e(k)))T Pθ(k) (Aσ(k) x(k) +

Bθ(k) (x(k) − e(k))) − λ2θ(k) x T (k)Pθ(k) x(k) − e T (k)Φσ(k) ˆ e(k)

(11)

+ ν(x(k) − e(k))T Φσ(k) ˆ (x(k) − e(k)) = ζ T (k)Ωθ(k) ζ(k), where T  ζ(k) = x T (k) e T (k) ,  1 + η, k ∈ Tas (Ωs ); λθ(k) = 1 − ρ, k ∈ Ts (Ωs ),   2  T   −λθ(k) Pθ(k) 0 T + ν I −I Φ j I −I + A˜ θ(k) Pθ(k) A˜ θ(k) , Ωθ(k) = ∗ −Φσ(k) ˆ   A˜ θ(k) = Aσ(k) + B˜ θ(k) − B˜ θ(k) . Note that at most one switching exists in Ωs = [ks , ks+1 ) for θ(k), therefore, we consider the following two cases respectively. Firstly, one switching exists for θ(k) in [ks , ks+1 ). In general, let the switching ¯ Therefore, θ(k) = ( j, j) and ¯ = i, σ(k) = j for k ∈ [ks , k). instant be k¯ and σ(k)

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¯ θ(k) = (i, j) and Ωθ(k) = Ωi j when k¯ ≤ k < ks+1 . Ωθ(k) = Ω j j when ks ≤ k < k; Secondly, if θ(k) has no switch in the time interval [ks , ks+1 ), we have θ(k) = ( j, j) and Ωθ(k) = Ω j j for ks ≤ k < ks+1 . Now, applying Schur complement lemma to the matrix inequality (7), we obtain Ωi j < 0 for i, j ∈ M. Therefore, from (11), we have Vθ(k) (k + 1) − λ2θ(k) Vθ(k) (k) < 0, k ∈ Ωs .

(12)

To prove that the closed-loop system (4) is exponentially stable, we should analyze the jump behavior of the Lyapunov function (9) after each switching instant of the signal θ(k). In general, assume that k¯1 , . . . , k¯ Nθ (0,k) are the switching instants of θ(k) during the arbitrary interval (0, k) and denote k¯0 = 0 and k¯ Nθ (0,k)+1 = k. Then, it follows from (10) and (12) that Vθ(k) (k) ≤ λ2θ(k) Vθ(k) (k − 1) ≤ λ4θ(k) Vθ(k) (k − 2) ≤ ··· 2(k−k¯ Nθ (0,k) )

≤ λθ(k) ≤

Vθ(k) (k¯ Nθ (0,k) )

2(k−k¯ N (0,k) ) λθ(k) θ μVθ(k¯ N (0,k) −1) (k¯ Nθ (0,k) ) θ 2(k−k¯ Nθ (0,k) ) 2 λθ(k¯ N (0,k)−1 ) Vθ(k¯ N (0,k)−1 ) (k¯ Nθ (0,k) θ θ

≤ μλθ(k)

− 1)

(13)

≤ ··· 2(k−k¯ Nθ (0,k) ) 2(k¯ Nθ (0,k) −k¯ Nθ (0,k)−1 ) λθ(k¯ N (0,k)−1 )

k2 −k1 ) · · · λ2( Vθ(k¯0 ) (k¯1 ) θ(k¯ )

2(k−k¯ Nθ (0,k) ) 2(k¯ Nθ (0,k) −k¯ Nθ (0,k)−1 ) λθ(k¯ N (0,k)−1 )

k1 −k0 ) · · · λ2( Vθ(k˜0 ) (k¯0 ) θ(k¯ )

≤ μ Nθ (0,k) λθ(k) ≤ μ Nθ (0,k) λθ(k)

θ

θ

¯

¯

1

¯

¯

0

≤ (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) μ Nθ (0,k) Vθ(0) (0). In addition, note that τ1 (k) ∈ [0, τd ), then it follows from Lemma 1 that σ(k) ˆ ∈ . S[τa , N0 + ττda ] and θ(k) ∈ S[ τ2a , 2N0 + ττda ]. Therefore, Nθ (0, k) ≤ 2N0 + ττda + 2k τa   ττd 1 1−ρ a 1 Furthermore, the inequality (8) yields μ τa < 1−ρ . 1+η     ττd 1 1 a Then, for any β¯ ∈ (1 − ρ) 1+η , μ− τa , we have μ τa β¯ ∈ (0, 1) with β¯ ∈ 1−ρ (1 − ρ, 1). Now combining with Lemma 2, it can be obtained from (13) that Vθ(k) (k) ≤ (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) μ Nθ (0,k) Vθ(0) (0)   τ 2N0 + τda + τ2ka 2 ¯ 2k

α¯ β Vθ(0) (0) 2k ˜ = α˜ β Vθ(0) (0),

≤μ

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τ

1 d where α˜ = μ2N0 + τa · α¯ 2 , β˜ = μ τa · β¯ ∈ (0, 1). Thus, the exponential stability of closed-loop system (4) when there is no packet dropout is obtained. 

3.2 Stability Analysis with Packet Dropout The in uence of packet dropout on the stability of the considered switched system during the network transmission is considered in this subsection. It is assumed that the transmitted signal x(ks ) and σ(ks ) might be lost because of the limited network capacity in the event-triggered control framework shown in Fig. 1. Suppose that the maximum number of allowable successive packet dropouts is τ D . To this end, we propose a new event-triggered scheme based on (2) as follows:   ks+1 = min ks + τd , kˆs+1 , s ≥ 1, kˆs+1 = min{k|(x(k) − x(ks ))T Φσ(ks ) (x(k) − x(ks )) ≥ νx ˜ T (ks )Φσ(ks ) x(ks )},

(14)

k>ks

where Φi is the event-trigger parameter de ned as (2) and ν˜ > 0 is an unknown scalar. Considering the event-triggered mechanism (14), we will give a criterion for the exponential stability of the closed-loop system (4) with packet dropout based on a set of matrix inequalities. Theorem 2 Assume that there exist positive definite matrices Pi j and Φ j such that the matrix inequalities (6) and (7) hold for the given scalars μ ≥ 1, η > 0, ρ ∈ (0, 1), ν ∈ (0, 1), ν˜ > 0 and state feedback gain K j . Then, the closed-loop system (4) is exponentially stable under event-triggered scheme (14) when the average dwell time τa satisfies (8) and τ D satisfies λm ψ(ν, ˜ τ D ) ≤ ν, λn where ψ(ν, ˜ τD ) =



α−

√

 2 √ ν˜ + 1 φ(τ D ) + νφ ˜ (τ D + 1) + τ D β ,

β(1 − ατ D ) 1 − ατ D + β − , 1−α (1 − α)2 √ ˜ + α1 ), α = (1 + ν)(1   α1 = max Ai − I , β = max  Bi K j  ,

φ(τ D ) =

i

i, j

λm = max λ(Φi ), λn = min λ(Φi ). i

i

(15)

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Proof In the case of packet dropout, we only need to show that the event-triggered mechanism (14) implies the event-triggered mechanism (2). Then, it follows from Theorem 1 that the closed-loop system (4) is exponentially stable. Suppose that [ks , ks+1 ) is a triggered interval. Without loss of generality, we assume that there are τ D unsuccessfully transmitted packets during [ks , ks+1 ) and denote ks = p0 < p1 < p2 < · · · < pτ D < pτ D +1 = ks+1 . For any i = 1, · · · , τ D , let ks + l ∈ ( pi , pi+1 ), then x(ks + l) − x(ks ) ≤ x(ks + l) − x( pi ) + x(ks ) − x( pi ) √ ≤ x( pi ) − x(ks ) + ν x( ˜ pi ) . Furthermore, one has x( pi ) ≤ x( pi−1 ) + x( pi − 1) − x( pi−1 ) + x( pi ) − x( pi − 1) √ ≤ x( pi ) − x( pi − 1) + (1 + ν) x( ˜ pi−1 ) .

(16)

It is assumed that no switching occurs in the interval [ pi − 1, pi ], then x( pi ) = Ai x( pi − 1) + Bi K j x(ks ). Then, we have x( pi ) − x( pi − 1) ≤ Bi K j · x(ks ) + Ai − I · x( pi − 1) .

(17)

In addition, it follows that x( pi − 1) − x( pi−1 ) ≤ Therefore,

√ ν x( ˜ pi−1 ) .

x( pi − 1) ≤ x( pi−1 ) + x( pi − 1) − x( pi−1 ) √ ≤ (1 + ν) x( ˜ pi−1 ) .

(18)

It follows from (17) and (18) that √ x( pi ) − x( pi − 1) ≤ Ai − I (1 + ν) x( ˜ pi−1 ) + Bi K j x(ks ) √ = α1 (1 + ν) x( ˜ pi−1 ) + β x(ks ) .

(19)

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Substitute (19) into (16), and we obtain √ x( pi ) ≤ (1 + ν)(1 ˜ + α1 ) x( pi−1 ) + β x(ks ) = α x( pi−1 ) + β x(ks ) .

(20)

Then, we can obtain from (19) by recursion process that |x( pi )| ≤ α(α x( pi−2 ) + β x(ks ) ) + β x(ks ) = α2 x( pi−2 ) + (1 + α)β x(ks ) ··· ≤ αi x( p0 ) + (1 + α + · · · + αi−1 )β x(ks )   (1 − αi )β x(ks ) . = αi + 1−α Therefore, we have x( pi ) − x(ks ) ≤ x( p1 − 1) − x( p0 ) + x( p1 ) − x( p1 − 1) + · · · + x( pi − 1) − x( pi−1 ) + x( pi ) − x( pi − 1) =

i 

x( p j − 1) − x( p j−1 ) +

i 

j=1

≤

x( p j ) − x( p j − 1)

j=1

i √  ν x( ˜ p j−1 ) + β x(ks ) ) j=1

+

i 

(α1 (1 +

√

ν) x( ˜ p j−1 ) .

j=1

(21) Furthermore, from (21) we can get x(k) − x(ks ) ≤ x( pτ D ) − x(ks ) + x(k) − x( pτ D ) =

τ D +1

x( p j − 1) − x( p j−1 ) +

j=1

≤

τ D +1

j=1

√ ν x( ˜ p j−1 ) + β x(ks ) )

j=1

+ ≤

τD  √ (α1 (1 + ν) x( ˜ p j−1 )



τD 

j=1

ψ(ν, ˜ τ D ) x(ks ) .

x( p j ) − x( p j − 1)

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Finally, we obtain (x(k) − x(ks ))T Φσ(ks ) (x(k) − x(ks )) ≤ λm (x(k) − x(ks ))T (x(k) − x(ks )) ≤ λm ψ(ν, ˜ τ D )x T (ks )x(ks ) λm ψ(ν, ˜ τ D )x T (ks )Φσ(ks ) x(ks ) ≤ λn ≤ νx T (ks )Φσ(ks ) x(ks ).

(22)

Inequality (22) indicates that event-triggered mechanism (14) can guarantee that the event-triggering condition (2) holds. Therefore, by Theorem 1, we can obtain that based on new event-triggered mechanism (14), the closed-loop system (4) is exponentially stable in the case of packet dropouts. 

3.3 Controller Design Based on the stability analysis result in Theorem 2, a co-design approach for the controller gain matrices in (3) and event-trigger parameters in (14) will be given in this subsection. Let Pi j = P j and P j = P j−1 , then pre-and post-multiply both sides of (7) with diag(P j , P j , P j ). Let Θj = PjΦjPj, Yj = K jPj, then by applying Schur complement lemma to (6) and (7) in Theorem 1, we can obtain the following linear matrix inequality based co-design method for controller gains and event-trigger parameters. Theorem 3 Assume that there exist positive definite matrices P j , Θ j such that the following linear matrix inequalities hold for the given scalars μ ≥ 1, η > 0, ρ ∈ (0, 1), ν ∈ (0, 1) and ν˜ > 0. ⎡ ⎢ ⎣

P j ≤ μPi , −λi2j P j

+ νΘ j

−νΘ j

P j AiT + Y jT BiT

∗

−(1 − ν)Θ j

−Y jT BiT

∗

∗

−P j

⎤ ⎥ ⎦ < 0, ∀i, j ∈ M.

(23)

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Then, exponential stability of the closed-loop system (4) under the event-triggered mechanism (14) is obtained when the average dwell time τa satisfies (8), and τ D satisfies (15). The corresponding controller gains K j = Y j P −1 j and mode dependent −1 −1 event-trigger parameters Φ j = P j Θ j P j .

3.4 Numerical Example In this section, a numerical example is given to show the effectiveness of the above approach. A switched system with two subsystems is considered, and the corresponding system parameters are chosen as 

1.01 A1 = 0.03  0.95 A2 = 0

   0 0.5 0.3 , B1 = , 0.95 −0.6 0.8    0.03 0.5 −0.2 , B2 = . 1.01 −0.3 0.9

Assume the dwell time τd = 2. Then, for the given constants η = 0.02, ρ = 0.01, ν = 0.85 and μ = 1.01, we obtain the following results by solving the linear matrix inequalities (23) for i, j = 1, 2.     −0.1239 0.0289 −0.1180 0.0139 , K2 = , K1 = −0.0596 −0.0685 −0.0478 −0.0777     0.0126 0 0.0117 −0.0001 Φ1 = , Φ2 = . 0 0.0136 −0.0001 0.0127 Suppose that the maximum number of allowable successive packet dropout is ˜ τ D ) = 0.8265 < ν = 0.85 from (15). In τ D = 2 and ν˜ = 0.01, then we get λλmn ψ(ν, addition, according to (8), if average dwell time τa > 6.9307, then the exponential stability of closed-loop system can be guaranteed. The simulation of the closed-loop system state trajectories is illustrated in Fig. 2, which shows that the both system states converge to zero asymptotically. The switchˆ are simulated in Fig. 3. Figure 4 illustrates ing signal σ(k) with τa = 7 and σ(k) the interval length of two successive event-triggered instants, where packet dropout instants are also labelled. It shows that there are 47 event-triggered transmission and 10 packet losses within the time interval [0, 60], which implies that the eventtriggered mechanism proposed in this paper is effective.

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6

System states

4 2 0 -2 -4 -6 -8 0

10

20

30

40

50

60

Time k

Fig. 2 The simulation of the closed-loop system state with τa = 7 2.5

Switching signal σ(k)

2 1.5 1 0.5 0

10

20

30

40

50

60

k 2.5

Transmitted switching signal σ ˆ (k)

2 1.5 1 0.5 0

10

20

30

40

50

k

Fig. 3 The switching signals σ(k) of the system and σ(k) ˆ of the controller

60

Event-Triggered Control of Discrete-Time Switched Linear …

185

Event-triggered instants and holding intervals

2.5 Event-triggered instants and holding intervals Packet loss instants

2

1.5

1

0.5

0 0

10

20

30

40

50

60

Time k Fig. 4 Length of event-triggered intervals and packet dropout instants

4 Event-Triggered Control of Discrete-Time Switched Linear Systems with Network-Induced Delays 4.1 Problem Formulation We will focus on the event-triggered control analysis and design of system (1) when there exist transmission delays in this section. The primary objective is to propose a proper event-triggered scheme for the considered system (1) when only delayed signal of the system state and the system mode is accessible to the remote controller. More specially, the proposed framework is shown in Fig. 5, where ks is the event-triggered instant generated by the event generator, which should be properly designed. Moreover, both the system state x(ks ) and the system mode σ(ks ) are transmitted via the communication channel, and the transmitted information is subjected to transmission delay d(ks ) ∈ [d1 , d2 ], where d1 and d2 are the lower and upper bound of d(ks ) respectively. The upper bound d2 of d(ks ) satisfies the following assumption. Assumption 3 Assume that the maximum value d2 of the transmission delay d(ks )  satisfies d2 ≤ τd .

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Fig. 5 Event-triggered control framework for switched system (1) with network-induced delays

We propose the following event-triggered mechanism to generate the data transmitting instants {ks }s≥1 : k0 = 0, ks+1 = kˆs+1 I{kˆs+1 0,  kˆs+1 = min k|(x(k) − x(ks ))T Φσ(ks ) (x(k) − x(ks )) k>ks ≥ x T (ks )Ψσ(ks ) x(ks ) , k˜s+1 = min{k|σ(k) = σ(ks )},

(24)

k>ks

where I{·} ∈ {0, 1} is the indicator function, whose value is 1 if and only if the condition {·} holds; positive de nite matrices Φi and Ψi are event-trigger parameters to be designed. Obviously, it follows from (24) that the transmission will only be triggered when the system switches and the current state is comparatively larger than the previously transmitted one. In addition, we have ks+1 + d(ks+1 ) ≤ k˜s+1 + τd .

(25)

In fact, when kˆs+1 < k˜s+1 , we can get ks+1 + d(ks+1 ) = kˆs+1 + d(ks+1 ) ≤ kˆs+1 + τd < k˜s+1 + τd . Similarly, ks+1 + d(ks+1 ) ≤ k˜s+1 + τd − d2 + d(ks+1 ) ≤ k˜s+1 + τd when kˆs+1 ≥ k¯s+1 . Therefore, it follows from (25) and Assumption 3 that at most one switching occurs during every holding interval Ωs = [ks + d(ks ), ks+1 + d(ks+1 )). Based on the transmitted system state x(ks ) and switching signal σ(ks ), the state feedback controller can be generated in each Ωs by ZOH as u(k) = K σ(ks ) x(ks ), k ∈ Ωs .

(26)

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187

For k ∈ Ωs = [ks + d(ks ), ks+1 + d(ks+1 )), de ne e(k) = x(k − τ1 (k)) − x(ks ), where τ1 (k) ∈ [d1 , d2 ] is given as follows. When d(ks+1 ) − d(ks ) ∈ (ks − ks+1 , 0], de ne τ1 (k) = d(ks ).

(27)

When d(ks+1 ) − d(ks ) ∈ (0, ks+1 − ks ], de ne  τ1 (k) =

d(ks ), k ∈ [ks + d(ks ), ks + d(ks+1 )), d(ks+1 ), k ∈ [ks + d(ks+1 ), ks+1 + d(ks+1 )).

(28)

When ks+1 − ks < d(ks+1 ) − d(ks ), we assume κ be the nonnegative integer such that 2κ+1 (ks+1 − ks ) ≥ d(ks+1 ) − d(ks ) > 2κ (ks+1 − ks ). In this case, we de ne  κ+1 2i −2i d(ks+1 ) + 2 2κ+1 d(ks ), k ∈ Λ1i , 2κ+1 τ1 (k) = (29) κ+1 2 −(2i−1) 2i−1 d(ks+1 ) + d(ks ), k ∈ Λ2i , 2κ+1 2κ+1 where   i 2κ − i i 2κ − i d(ks ), ks+1 + κ d(ks+1 ) + d(ks ) , Λ1i = ks + κ d(ks+1 ) + κ κ 2 2 2 2  i −1 2κ − (i − 1) Λ2i = ks+1 + κ d(ks+1 ) + d(ks ), 2 2κ  κ i 2 −i d(ks ) , ks + κ d(ks+1 ) + 2 2κ  2κ   2κ     Ωs = Λ1i Λ2i , i = 0, . . . , 2κ . i=0

i=1

In summary, when k ∈ Ωs , it follows from (27), (28) and (29) that τ1 (k) ∈ [d1 , d2 ] and k − τ1 (k) ∈ (ks , ks+1 ). ˆ = σ(k − Furthermore, for k ∈ Ωs , denote τ2 (k) = k − ks , then we get σ(k) τ2 (k)). Then we can rewrite the feedback controller (26) as follows u(k) = K σ(k) (x(k − τ1 (k)) − e(k)) , k ∈ Ωs . ˆ

(30)

Consequently, under the state feedback controller (26), the closed-loop system of (1) is formulated as Bθ(k) (x(k − τ1 (k)) − e(k)) , x(k + 1) = Aσ(k) x(k) +

with

Bθ(k) = Bσ(k) K σ(k) ˆ , τ1 (k) ∈ [d1 , d2 ], and τ2 (k) ∈ [d1 , 2τd ).

(31)

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Remark 1 To de ne e(k) according to the event-triggered transmission scheme (24), we present a explicit mathematical de nition for τ1 (k), k ∈ Ωs in (27), (28) and (29), which is simpler and easier to understand than that in [24]. In addition, combining σ(k) ˆ = σ(k − τ2 (k)) and τ2 (k) ∈ [d1 , 2τd ), it follows from 1 and θ(k) belongs to the Lemma 1 that σ(k) ˆ belongs to the set S τa , N0 + 2τdτ−d a   τa 2τd −d1 set S 2 , 2N0 + τa . Remark 2 Since only the transmitted information of switching signal is available to the controllers, then there exists a time lag between the controller mode σ(ks ) and system mode σ(k), therefore, the controllers might switch asynchronously with the system. It can be derived from the transmission scheme (24) that the asynchronous time can be estimated by τd .

4.2 Stability Analysis The exponential stability of the closed-loop system (31) will be investigated in this subsection. A proper quadratic Lyapunov functional will be constructed to obtain a suf cient stability condition in the form of matrix inequality. Theorem 4 If there exist positive definite matrices Pi j , Q 1i j , Q 2i j , Q 3i j , Z 1i j , Z 2i j , Ψ j , Φ j , and matrices N1i j , N2i j , N3i j such that matrix inequalities (32), (33) (34) and (35) hold for the scalars μ ≥ 1, η > 0, ρ ∈ (0, 1) and state feedback gain matrices K j , and the switching signal satisfies (36), then the closed-loop system (31) is exponentially stable. μ−1 Pii ≤ Pi j ≤ μP j j , ∀i, j ∈ M, Q ιii ≤ μQ ιi j , Z νii ≤ μZ νi j ,

¯ λ¯ i2jdι Q ιi j ≤ μQ ι j j , ι = 1, 2, 3, ∀i, j ν −1) λ¯ i2(d Z νi j ≤ μZ ν j j , ν = 1, 2, ∀i, j

(32) ∈ M,

(33)

j ∈ M,

(34)

−1 Ω˜ i j + (d2 − d1 )NliTj Z 2i j Nli j < 0, l = 2, 3, ∀i, j ∈ M,

  1+η , −τa ln(1 − ρ) > ln μ + τd ln 1−ρ

(35) (36)

where   Ψ˜ j Ωi j ˜ + 1i j (N1iT j E 1 + E 1T N1i j ) + 2i j (N2iT j E 2 + E 2T N2i j ) Ωi j = ∗ Ψj − Φj +2i j (N3iT j E 3 + E 3T N3i j ) + d1 N1iT j Z 1i−1j N1i j + AiTj Pi j Ai j +d1 A˜ iTj Z 1i j A˜ i j + (d2 − d1 )A˜ iTj Z 2i j A˜ i j ,

Event-Triggered Control of Discrete-Time Switched Linear …

189

Ωi j = diag{Ω11i j , Ω22i j , Ω33i j , Ω44i j }, Ω11i j = −λi2j Pi j + Q 1i j + (d2 − d1 + 1)Q 3i j , 2(δi j d2 +(1−δi j )d1 )

Ω22i j = −λi j Ω33i j =

−λi2dj 1 Q 1i j

Ω44i j =

2 −d1 ) −λi2(d Q 2i j , j

Q 3i j + Ψ j ,

+ Q 2i j ,

T  Ψ˜ j = 0 −Ψ j 0 0 ,     Ai j = Ai Bi K j 0 0 −Bi K j , A˜ i j = Ai − I Bi K j 0 0 −Bi K j ,       E 1 = I 0 −I 0 0 , E 2 = 0 I 0 −I 0 , E 3 = 0 −I I 0 0 , d¯1 = d1 − 1, d¯2 = d2 − d1 − 1, d¯3 = d2 − 1, λi j = δi j (1 − ρ) + (1 − δi j )(1 + η), δi j +(1−δi j )d1 , λ¯ i j = λi j λ−1 j j , 1i j = λi j δ (d1 +1)+(1−δi j )d2

2i j = λi ji j

.

Proof A quadratic Lyapunov functional Vθ(k) (k) is constructed as follows Vθ(k) (k) = V1θ(k) (k) + V2θ(k) (k) + V3θ(k) (k), where

(37)

V1θ(k) (k) = x T (k)Pθ(k) x(k), k−1 

V2θ(k) (k) =

λ2(k−1−m) x T (m)Q 1θ(k) x(m) θ(k)

m=k−d1

+

k−1−d 1

1 −m) T λ2(k−1−d x (m)Q 2θ(k) x(m) θ(k)

m=k−d2

+

k−1  m=k−τ1 (k)

+

−d1 

λ2(k−1−m) x T (m)Q 3θ(k) x(m) θ(k) k−1 

n=−d2 +1 m=k+n

V3θ(k) (k) =

−1 

k−1 

λ2(k−1−m) x T (m)Q 3θ(k) x(m), θ(k)

λ2(k−1−m) Δx T Z 1θ(k) Δx θ(k)

n=−d1 m=k+n

+

−1−d 1

k−1 

n=−d2 m=k+n

with Δx = x(m + 1) − x(m).

λ2(k−1−m) Δx T Z 2θ(k) Δx, θ(k)

(38)

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Let k¯ ∈ Ωs be any switching instant of switching signal θ(k). Since θ(k) is the augment of σ(k) and σ(k), ˆ then k¯ must be either a transmitting instant or some switching time of σ(k). Case 1. Assume that k¯ be some switching instant of σ(k) in Ωs . In general, denote ¯ ¯ = i. σ(k − 1) = j and σ(k) ¯ and σ( ˆ k) ˆ k¯ − 1) are equal to j, Since there exists at most one switch in Ωs , then both σ( ¯ ¯ = therefore, θ(k) = (i, j) and θ(k − 1) = ( j, j). Then we have Pθ(k) ¯ = Pi j , Pθ(k−1) ¯ = Q ι j j , Z νθ(k) = Z ν j j , and λθ(k) P j j , Q ιθ(k) ¯ = Q ιi j , Q ιθ(k−1) ¯ ¯ = Z νi j , Z νθ(k−1) ¯ ¯ = = 1 − ρ. 1 + η, λθ(k−1) ¯ According to (32), (33) and (34), we can obtain the following estimation of Vθ(k) ¯ ¯ at the switching instant k: ¯ = x T (k)P ¯ i j x(k) ¯ ≤ μx T (k)P ¯ j j x(k) ¯ = μV1θ(k−1) ¯ (k), V1θ(k) ¯ (k) ¯ and

¯ k−1 

¯ = V2θ(k) ¯ (k)

¯

λi2(j k−1−m) x T (m)Q 1i j x(m)

¯ 1 m=k−d ¯ k−1−d 1

+

¯

λi2(j k−1−d1 −m) x T (m)Q 2i j x(m)

¯ 2 m=k−d

+

¯ k−1 

¯

λi2(j k−1−m) x T (m)Q 3i j x(m)

¯ 1 (k) ¯ m=k−τ

+

¯ k−1 

−d1 

¯

λi2(j k−1−m) x T (m)Q 3i j x(m)

¯ n=−d2 +1 m=k+n ¯ k−1 

≤

¯

λ2(j jk−1−m) x T (m)μQ 1 j j x(m)

¯ 1 m=k−d ¯ k−1−d 1

+

¯

λ2(j jk−1−d1 −m) x T (m)μQ 2 j j x(m)

¯ 2 m=k−d

+

¯ k−1 

¯

λ2(j jk−1−m) x T (m)μQ 3 j j x(m)

¯ 1 (k) ¯ m=k−τ

+

−d1 

¯ k−1 

¯ n=−d2 +1 m=k+n

¯ = μV2θ(k−1) (k), ¯

¯

λ2(j jk−1−m) x T (m)μQ 3 j j x(m)

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and nally, ¯ V3θ(k) ¯ (k) =

¯ −1 k−1  

¯

λi2(j k−1−m) (x(m + 1) − x(m))T Z 1i j (x(m + 1) − x(m))

¯ n=−d1 m=k+n

+

−1−d 1

¯ k−1 

¯

λi2(j k−1−m) (x(m + 1) − x(m))T Z 2i j (x(m + 1) − x(m))

¯ n=−d2 m=k+n

≤

¯ k−1 

−1 

¯

λ2(j jk−1−m) (x(m + 1) − x(m))T μZ 1 j j (x(m + 1) − x(m))

¯ n=−d1 m=k+n

+

−1−d 1

¯ k−1 

¯

λ2(j jk−1−m) (x(m + 1) − x(m))T μZ 2 j j (x(m + 1) − x(m))

¯ n=−d2 m=k+n

¯ = μV3θ(k−1) (k). ¯ ¯ ≤ μVθ(k−1) ¯ Therefore, we have Vθ(k) (k). ¯ (k) ¯ Case 2. Assume that k¯ = ks+1 + d(ks+1 ), while k¯ is not a switching instant of σ(k). ¯ = σ(ks+1 ) = i, then, σ(k) ¯ = ˆ k) In general, suppose σ( ˆ k¯ − 1) = σ(ks ) = j, σ( ¯ = (i, i), θ(k¯ − σ(k¯ − 1) = i and one switch occurs in Ωs . Therefore, we have θ(k) = Pi j , Q ιθ(k−1) = Q ιi j , Q ιθ(k) 1) = (i, j). Consequently, Pθ(k) ¯ = Pii , Pθ(k−1) ¯ ¯ ¯ = = Z , Z = Z , λ = 1 + η, λ = 1 − ρ. Q ιii , Z νθ(k−1) ¯ ¯ ¯ ¯ νi j νii νθ(k) θ(k−1) θ(k) Then, similar to the analysis as Case 1, we can obtain from (32), (33) and (34) that ¯ ≤ μVθ(k−1) ¯ (k). (39) Vθ(k) ¯ (k) ¯ The following estimation of Lyapunov functional (37) along the closed-loop system (31) can be computed for k ∈ Ωs : V1θ(k) (k + 1) − λ2θ(k) V1θ(k) (k) = x T (k + 1)Pθ(k) x(k + 1) − λ2θ(k) x T (k)Pθ(k) x(k), and

(40)

V2θ(k) (k + 1) − λ2θ(k) V2θ(k) (k) ≤ x T (k)(Q 1θ(k) + (d2 − d1 + 1)Q 3θ(k) )x(k) 2(δ

− λθ(k)θ(k) +x −

d2 +(1−δθ(k) )d1 ) T

x (k − τ1 (k))Q 3θ(k) x(k − τ1 (k))

1 (k − d1 )(−λ2d θ(k) Q 1θ(k) + Q 2θ(k) )x(k 2 −d1 ) T λ2(d x (k − d2 )Q 2θ(k) x(k − d2 ), θ(k)

T

− d1 )

(41)

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and nally, V3θ(k) (k + 1) − λ2θ(k) V3θ(k) (k) = d1 (x(k + 1) − x(k))T Z 1θ(k) (x(k + 1) − x(k)) −

k−1 

λ2(k−m) (x(m + 1) − x(m))T Z 1θ(k) (x(m + 1) − x(m)) θ(k)

m=k−d1

+ (d2 − d1 )(x(k + 1) − x(k))T Z 2θ(k) (x(k + 1) − x(k)) −

k−1−τ 1 (k)

(42)

λ2(k−m) (x(m + 1) − x(m))T Z 2θ(k) (x(m + 1) − x(m)) θ(k)

m=k−d2

−

k−1−d 1 m=k−τ1 (k)

λ2(k−m) (x(m + 1) − x(m))T Z 2θ(k) (x(m + 1) − x(m)). θ(k)

Furthermore, it follows from event-triggering condition (24) that T e T (k)Φσ(k) ˆ e(k) < [x(k − τ1 (k)) − e(k)] Ψσ(k) ˆ ×[x(k − τ1 (k)) − e(k)], k ∈ Ωs .

(43)

Combining (31), (40), (41), (42) and (43), we obtain Vθ(k) (k + 1) − λ2θ(k) Vθ(k) (k) ≤ ξ T (k)Υθ(k) (τ1 (k))ξ(k),

(44)

 T where ξ(k) = x T (k) x T (k − τ1 (k)) x T (k − d1 ) x T (k − d2 ) e T (k) and −1 T Υθ(k) (τ1 (k)) = Ω˜ θ(k) + (d2 − τ1 (k))N2θ(k) Z 2θ(k) N2θ(k) −1 T + (τ1 (k) − d1 )N3θ(k) Z 2θ(k) N3θ(k) .

For given scalars η > 0 and 0 < ρ < 1, de ne λˆ θ(k) = 1 − λ2θ(k) and  λθ(k) =

1 + η, k ∈ Tas (Ωs ); 1 − ρ, k ∈ Ts (Ωs ).

Then, we can obtain from (44) that ΔVθ(k) (k) along the switched system (31) satis es the following estimation if Υθ(k) (τ1 (k)) < 0. ΔVθ(k) (k) + λˆ θ(k) Vθ(k) (k) ≤ 0, k ∈ Ωs .

(45)

Now, we show that Υθ(k) (τ1 (k)) < 0 can be guaranteed by (35). Note that Υθ(k) (τ1 (k)) is linearly dependent on τ1 (k) and τ1 (k) ∈ [d1 , d2 ]. Thus, Υθ(k) (τ1 (k)) < 0 if and only if Υθ(k) (d1 ) < 0 and Υθ(k) (d2 ) < 0 hold simultaneously.

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193

Case 1. The θ(k) has one switch in Ωs , then σ(k) switches one time in Ωs . The ¯ corresponding switching instant is supposed to be k. ¯ = i, then we have Υθ(k) (τ1 (k)) = Without loss of generality, denote σ(k) ¯ ¯ ks+1 + d(ks+1 )), we have Υ j j (τ1 (k)) when k ∈ [ks + d(ks ), k). For k ∈ [k, Υθ(k) (τ1 (k)) = Υi j (τ1 (k)). Case 2. The θ(k) has no switch in Ωs , then we have Υθ(k) (τ1 (k)) = Υ j j (τ1 (k)), for k ∈ Ωs . Therefore, Υθ(k) (d1 ) < 0 and Υθ(k) (d2 ) < 0 are equivalent to Υi j (d1 ) < 0 and Υi j (d2 ) < 0 for all i, j ∈ M, which can be guaranteed by (35). Finally, if the switching signal satis es (36), then the exponential stability of the closed-loop system (31) is formulated as follows. During any time interval (0, k), the switching signal θ(k) has Nθ (0, k) switching instants k˜1 , . . . , k˜ Nθ (0,k) . In general, we suppose k˜ Nθ (0,k)+1 = k and k˜0 = 0. From (39) and (45), we can obtain that Vθ(k) (k) ≤ λ2θ(k) Vθ(k) (k − 1) ≤ ··· 2(k−k˜ Nθ (0,k) )

≤ λθ(k)

μVθ(k˜ N

θ (0,k)

˜

−1) (k Nθ (0,k) )

≤ ··· ≤ μ Nθ (0,k) (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) Vθ(0) (0). Note that σ(k) ∈ S[τa , N0 ], then, according to Lemma 1, we have 1 1 ], therefore, Nθ (0, k) ≤ 2N0 + 2τdτ−d + 2k . θ(k) ∈ S[ τ2a , 2N0 + τd τ−d τa a a τd −d2   1 τa 1−ρ 1 Furthermore, from (36) we obtain that μ τa < 1−ρ . 1+η    ττd  1 1 a Then for any β¯ ∈ (1 − ρ) 1+η , μ− τa , we have 0 < μ τa · β¯ < 1. 1−ρ By Lemma 2, we get Vθ(k) (k) ≤ (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) μ Nθ (0,k) Vθ(0) (0) ≤ μ(2N0 +

2τd −d1 τa

+ τ2ka ) 2 ¯ 2k

α¯ β Vθ(0) (0)

= αβ ˜ 2k Vθ(0) (0), 2τ −d

1 d 1 ¯ Therefore, exponential stability of the where α˜ = μ2N0 + τa · α¯ 2 , β = μ τa · β. closed-loop system (31) is obtained. 

Remark 3 The average dwell time condition (36) indicates that switched systems can maintain stability when the overall activation time of the unstable subsystem (with λθ(k) = 1 + η, k ∈ Tas (Ωs )) is comparatively smaller than that of the stable

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subsystem (with λθ(k) = 1 − ρ, k ∈ Ts (Ωs )). We mention that condition (36) can be viewed as a counterpart of the following stability condition obtained in [5] τa > −[Tmax (ln(1 + η) − ln(1 − ρ)) + ln μ]/ ln(1 − ρ),

(46)

where Tmax represents the maximal delay of asynchronous switching. Tmax (corresponding to τd in (36)) is assumed to be a priori in [5]. However, the dwell time τd in our proposed condition (36) can be measured from σ(k) directly. Remark 4 It should be mentioned that the obtained condition (36) has some conservativeness, since we estimate the mismatch time τd between two switching signals σ(k) and σ(k) ˆ in the worst-case scenario. On the other hand, when consider the time-triggered transmission and the delay d(k) = 0, then σ(k) and σ(k) ˆ always switch synchronously. In this case, the asynchronous time is zero. Then, the average dwell time condition (36) reduces to τa > −ln μ/ ln(1 − ρ), which recovers to the classic stability condition of discretetime switched systems obtained in [6].

4.3 Controller Design In the following, we give a co-design approach for the stabilizing controller according to the obtained stability analysis conclusion in previous subsection. To this end, we assume Pi j is invariant with respect to i in Theorem 4, that is Pi j = P j , ∀i ∈ M. Let us denote P j = P j−1 , Q1i j = P j−1 Q 1i j P j−1 , Q2i j = P j−1 Q 2i j P j−1 , −1 Q3i j = P j−1 Q 3i j P j−1 , Z1i j = Z 1i−1j , Z2i j = Z 2i j,

Ξ j = P j−1 Ψ j P j−1 , Θ j = P j−1 Φ j P j−1 , Y j = K j P j−1 ,  P j = diag P j , P j , P j , P j , P j . After pre-and post-multiplying both sides of matrix inequality (35) with P j , a codesign approach for event-trigger parameters and state feedback controllers is formulated as follows by utilizing Schur complement lemma. Theorem 5 If there exist μ ≥ 1, η > 0, ρ ∈ (0, 1), constants αi j , βi j , positive definite matrices P j , Q1i j , Q2i j , Q3i j , Z1i j , Z2i j , Θ j , Ξ j , and matrices M1i j , M2i j , M3i j , Y j satisfying linear matrix inequalities as follows. ¯ P j ≤ μPi , Qκii ≤ μQκi j , λ¯ i2jdκ Qκi j ≤ μQκ j j , κ = 1, 2, 3, ∀i, j ∈ M,

Zνii ≤ μZνi j ,

ν −1) λ¯ i2(d Zνi j j

≤ μZν j j , ν = 1, 2, ∀i, j ∈ M,

(47) (48)

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⎡

⎤ 1 1 Ω¯ i j A¯ iTj d12 Aˆ iTj Ω˜ 1i j d12 M1iT j Ω˜ 2i j ⎢ ⎥ ⎢ ∗ −P j 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −Z1i j 0 0 0 ⎥ < 0, l = 2, 3, ∀i, j ∈ M, ⎢ ⎥ ⎢ ∗ ∗ 0 0 ⎥ ∗ −Z2i j ⎢ ⎥ ⎣ ∗ ∗ 0 ⎦ ∗ ∗ Υ˜55 ∗ ∗ ∗ ∗ ∗ Υ˜66

(49)

where ⎡

⎤ Ω¯ 11i j 0 0 0 0 ⎢ ∗ Ω¯ 22i j 0 0 −Ξ j ⎥ ⎢ ⎥ ⎢ ⎥ ¯ ¯ Ωi j = ⎢ ∗ ∗ Ω33i j 0 0 ⎥ ⎣ ∗ ⎦ ¯ 0 ∗ ∗ Ω44i j ∗ ∗ ∗ ∗ Ξj − Θj + 1i j (M1iT j E 1 + E 1T M1i j ) + 2i j

3  (MliT j El + ElT Mli j ), l=2

Ω¯ 11i j = −λi2j P j + Q1i j + (d2 − d1 + 1)Q3i j , 2(δ d +(1−δi j )d1 ) Ω¯ 22i j = −λi j i j 2 Q3i j + Ξ j ,

Ω¯ 33i j = −λi2dj 1 Q1i j + Q2i j , 2 −d1 ) Q2i j , Ω¯ 44i j = −λi2(d j       E 1 = I 0 −I 0 0 , E 2 = 0 I 0 −I 0 , E 3 = 0 −I I 0 0 ,   A¯ i j = Ai P j Bi Y j 0 0 −Bi Y j ,   Aˆ i j = Ai P j − P j Bi Y j 0 0 −Bi Y j , 1 1 Ω˜ 1i j = (d2 − d1 ) 2 Aˆ iTj , Ω˜ 2i j = (d2 − d1 ) 2 MliT j ,

Υ˜55 = −2αi j P j + αi2j Z1i j , Υ˜66 = −2βi j P j + βi2j Z2i j , λi j = (1 − ρ)δi j + (1 + η)(1 − δi j ), λ¯ i j = λi j λ−1 , jj

d¯1 = d1 − 1, d¯2 = d2 − d1 − 1, d¯3 = d2 − 1, δ +(1−δi j )d1

1i j = λi ji j

δ (d1 +1)+(1−δi j )d2

, 2i j = λi ji j

.

Then, exponential stability of the closed-loop system (31) can be guaranteed for the switching signal σ(k) whose average dwell time τa satisfies the constraint (36). Furthermore, the feedback gain matrices are computed as K j = Y j P −1 j and event−1 −1 −1 Ξ P , Φ = P Θ P . trigger parameters are Ψ j = P −1 j j j j j j j

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4.4 Numerical Example To show the effectiveness of the design method, the switched system with two subsystems will be considered and system parameters are given as follows:     1.05 0.1 0.6 0 , B1 = , A1 = 0 0.1 0.2 0.4     0.6 0.1 0.1 0.3 , B2 = . A2 = 0.5 0.9 0.2 0.2 For the given dwell time τd = 4 and the transmission delay d(k) ∈ [0, 2], let μ = 1.2, ρ = 0.1, η = 0.05, αi j = βi j = 0.5, i, j = 1, 2. It can be veri ed that LMIs (47), (48) and (49) are feasible, the corresponding controller gain matrices are computed as  K1 =

   −0.4268 −0.0498 −0.3877 0.0666 , K2 = . 0.2068 −0.0104 −0.3027 −0.3921

The event-trigger parameters are 

   1.1965 −0.1422 0.4391 −0.0843 Ψ1 = , Ψ2 = , −0.1422 0.3701 −0.0843 0.0350 Φ1 =

    18.7865 2.3630 13.7775 5.9591 , Φ2 = . 2.3630 0.8000 5.9591 3.7998

Fig. 6 The simulation of the closed-loop system state with τa = 4.7

Event-Triggered Control of Discrete-Time Switched Linear … 2.5

197 Switching signal σ(k)

2 1.5 1 0.5

0

5

10

15 k

2.5

20

25

30

Transmitted switching signal σˆ(k)

2 1.5 1 0.5

5

0

10

15 k

20

25

30

Fig. 7 The switching signals σ(k) of the system and σ(k) ˆ of the controller Inter−event intervals

3

2

1

0

0

5

10

15 k

20

25

30

0

5

10

15 k

20

25

30

Transmission delay

2.5 2 1.5 1 0.5 0 −0.5

Fig. 8 Length of event-triggered intervals and network-induced delays

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Then, it follows from (36) that exponential stability of the closed-loop system can be guaranteed if the average dwell time τa > 4.6566. The state responses of the closed-loop system in both the event-triggered case and time-triggered case when τa = 4.7 are shown in Fig. 6, which converge to zero asymptotically. The switching signal σ(k) of the system and σ(k) ˆ of the controller are simulated in Fig. 7. Event-triggered intervals and network-induced delays are illustrated in Fig. 8. In the simulation, there are 23 transmissions during the whole time interval [0, 30], which means that the proposed event-triggered scheme can reduce about 7/30 = 23.33% of the data transmission.

5 Conclusion In this chapter, considering network induced packet dropouts and transmission delays, new mode dependent event-triggered schemes with dwell time constraints are given to solve the networked control problem for discrete-time switched linear systems. Utilizing multiple Lyapunov functional approach and switching signal merging technique, exponential stability conditions for the closed-loop systems are established, co-design methods for event-trigger parameters and feedback stabilizing controllers are obtained. The proposed method extends the event-triggered transmission scheme of non-switched systems, which can also be adopted to solve eventtriggered H∞ control/ ltering and other related event-triggered design problems of switched linear systems. Acknowledgements The authors would be very grateful for the support of Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents, Six Talent Peaks Project of Jiangsu (No. DZXX-044) and Nantong 226 High-level Talents Project. The work of Xiaoqing Xiao was also supported by China Scholarship Council (CSC NO. 201908320096), National Natural Science Foundation of China(Nos. 61374061, 61403216 and 61573201). Also, the work of J.H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. 2019R1A5A808029011).

References 1. Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003) 2. Zhao, X.D., Kao, Y.G., Niu, B., Wu, T.T.: Control synthesis of switched systems. Studies in Systems, Decision and Control, vol. 80. Springer (2017) 3. Lin, H., Antsaklis, P.J.: Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54, 308–322 (2009) 4. Zhu, F., Antsaklis, P.J.: Optimal control of hybrid switched systems: a brief survey. Discret. Event Dyn. Syst. 25(3), 345–364 (2015) 5. Zhang, L.X., Gao, H.J.: Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)

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Fault Alarm-Based Hybrid Control Design for Periodic Piecewise Time-Delay Systems R. Sakthivel and S. Harshavarthini

Abstract In this chapter, fault alarm-based hybrid control scheme is developed for periodic piecewise systems with time delay and disturbances. Precisely, the hybrid controller is proposed to handle both the normal and faulty systems. Moreover, a switching among the robust and reliable controller is accomplished with respect to the alarm signal. Based on the threshold value of the alarm signal, the timely alert will be encountered. Further, the linear matrix inequality based stability conditions are derived by considering suitable Lyapunov–Krasovskii functional and employing some advanced inequalities. With the aid of delay-dependent suf cient conditions, the asymptotic stability of the addressed system is ensured with a minimum disturbance attenuation level. In particular, the gain matrices of the control scheme is obtained by solving the developed stability conditions. At last, the impact and significance of the proposed hybrid controller is validated by presenting simulation results of a numerical example. Keywords Periodic piecewise time-invariant systems · Time-varying delay · Uncertainties and disturbances · Hybrid control · Lyapunov stability method

1 Introduction In recent years, the system with periodic characteristics receives considerable research attention because of its potential applications in various engineering and science elds. To mention a few, DC-DC converters, vehicle suspension, transmission lines with distinct loads, rotor-bearing systems, helicopter rotors and power electronic equipments with multi-function converters. More speci cally, the periodic systems are a special class of switched systems with xed dwell time. In particular, R. Sakthivel (B) · S. Harshavarthini Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India e-mail: [email protected] S. Harshavarthini e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_9

201

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the periodic systems are subdivided into a nite number of subsystems to accurately analyze the spectrum of the considered periodic systems in each instant of time, which is referred to as periodic piecewise systems (PPSs). Thus, the study on stability and stabilization of PPSs become an active research topic in which various control protocol is designed to stabilize the system dynamics [1–5]. Xie et al. [1] examined the stabilization problem of uncertain PPSs by Lyapunov stability theory. Notably, in [2], stability and L1 gain analysis of positive PPSs with time-invariant subsystems is studied by constructing continuous Lyapunov function. With the aid of polynomial Lyapunov function, a linear inequality based on suf cient conditions are obtained in [3] for the exponential stability of PPSs. On the other hand, the time-delays [6] are often encountered in many practical systems that are inexorable and it may lead to oscillation, poor performance and even instability of the control systems [7]. Besides, the external disturbances also may have a consequence in the performance of dynamical control systems [8–13]. However, the exact modeling of physical systems is not always guaranteed due to variation in system parameters, modeling errors and neglected nonlinearities. The differences or errors between the mathematical model and the actual physical systems are commonly referred to as model uncertainties [14–17]. Thus, the signi cance of system performance motivates the researchers in the study on stability analysis of uncertain control systems with time-delay and external disturbances. As a result, a countable number of work is reported in the stability and stabilization of PPSs with uncertainties, time-delay, and disturbances. Precisely, the stabilization of delayed PPSs subject to disturbances is retained with the aid of various control schemes such as H∞ control [8], L2 − gain analysis [9] and guaranteed cost control [10] addressed in the existing literature. On another research frontier, the occurrence of faults in system components is unavoidable and will happen at any instant of time. In such circumstances, the system trajectories may attain unanticipated performance degradation and substantial damage to the plant. Thus, it is necessary to tolerate the effects of actuator faults and also to retain the robust performance of the system dynamics. The reliable control strategy is an effective technique to tackle the impact of faults in system components and it is widely reported for various dynamical systems [18–22]. Moreover, the hybrid control design comprised of both the robust and reliable control, where the fault alarm is designed to timely alert the controller once the fault is occurred in system components. Based on the alarm-signal, the controller automatically switched to the reliable control from conventional control scheme. Precisely to switch between the controllers we design a switching method by selecting a suitable threshold value of faults. However, implementing the reliable controller, where the system does not undergo failures will cause conservative in such cases the robust controller is activated [23–27]. Despite the advantages of above-mentioned studies and based on fault-alarm approach, the hybrid control is designed for delayed PPSs with uncertainties and disturbances. In this chapter, we mainly focused on the alarm design to timely alert the controller with the aid of the appropriate threshold value. To be precise, the fault is detected

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203

with the aid of state residual observer and by simple manipulation, the threshold value is calculated. Consequently, the delay-dependent conditions are derived by constructing periodic piecewise Lyapunov–Krasovskii functional with time-varying matrices. Further, the periodically time-varying controller gain matrices are obtained by solving the developed linear matrix inequality based conditions. More speci cally, the asymptotic stability of the considered PPSs is ensured by the aid of the Lyapunov stability theory. Notations: The superscripts (−1) and T stand for matrix inverse and matrix transposition, respectively; Rn denotes the set of all n-dimensional Euclidean space; L 2 [0, ∞) denotes the space of all square integrable function on [0, ∞); We use an asterisk “∗” to represent the term induced by symmetry and we denote Sym(A) = A T + A; P > 0 represents P is real, symmetric and positive de nite; N denotes the set of all natural numbers; I represents unit matrix with compatible dimension; Moreover, the matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.

2 Problem Formulation and Preliminaries Consider a class of periodic systems with uncertainties, time-varying delay and exogenous disturbances in the following form; ⎧ x(t) ˙ ⎪ ⎪ ⎪ ⎨

= (A(t) + ΔA(t))x(t) + (Aτ (t) + ΔAτ (t))x(t − τ (t)) + B(t)u G (t) + C(t)v(t), ⎪ y(t) = D(t)x(t), ⎪ ⎪ ⎩ x(t0 ) = φ(t0 ), ∀t0 ∈ [−τ , 0],

(1)

where x(t) ∈ Rx is the state vector; u G (t) ∈ Ru is the control input vector; v(t) ∈ L 2 [0, ∞) is the exogenous disturbance input; τ (t) is the time-varying delay which satis es 0 ≤ τ (t) ≤ τ and τ˙ (t) ≤ υ < 1; y(t) ∈ R y represents output vector. A(t) = A(t + ν F p ), Aτ (t) = Aτ (t + ν F p ), B(t) = B(t + ν F p ), C(t) = C(t + ν F p ) and D(t) = D(t + ν F p ) are known matrices with fundamental period F p , ν = 0, 1, . . .. Further, the time interval [ν F p , (ν + 1)F p ) is partitioned into Q subintervals [ν F p + Q ti−1 , ν F p + ti ), i ∈ Q = [1, 2, . . . , Q], i=1 Fi = F p , t0 = 0 and t Q = F p . In this connection, the system (1) is rewritten as periodic piecewise systems (PPSs) in the following form; ⎧ x(t) ˙ ⎪ ⎪ ⎪ ⎨

= (Ai + ΔAi (t))x(t) + (Aτ i + ΔAτ i (t))x(t − τ (t)) + Bi u G (t) + Ci v(t), ⎪ y(t) = Di x(t), ⎪ ⎪ ⎩ x(t0 ) = φ(t0 ), ∀t0 ∈ [−τ , 0],

(2)

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Fig. 1 Evolution of PPTV positive-de nite matrix P(t)

where Ai , Aτ i , Bi , Ci and Di , i ∈ M are the appropriate dimensioned constant matrices of ith subsystem; ΔAi (t) and ΔAτ i (t) represent the time-varying parameter uncertainties of the form [ ΔAi (t) ΔAτ i (t) ] = Ui Δi (t)[ V Ai V Aτ i ], in which Ui and Vi are known constant matrices and Δi (t) is an unknown time-varying matrix with ΔiT (t)Δi (t) < I . On the other hand, the periodic piecewise control input in (2) is considered as follows; (3) u G (t) = K˜ˆ i (t)x(t) = (K i + ΔK i (t))x(t), where K i is the controller gain matrix of ith subsystem and ΔK i (t) represents the controller gain variation of the form ΔK i (t) = U K i Δi (t)VK i , here U K i and VK i are known constant matrices with appropriate dimensions and Δi (t) is a time-varying matrix with ΔiT (t)Δi (t) < I . Now, by substituting the designed control input (3) in system (4), we obtain the following closed-loop system; ⎧ x(t) ˙ ⎪ ⎪ ⎪ ⎨

= (Ai + ΔAi (t))x(t) + (Aτ i + ΔAτ i (t))x(t − τ (t)) +Bi (K i + ΔK i (t))x(t) + Ci v(t), ⎪ y(t) = Di x(t), ⎪ ⎪ ⎩ x(t0 ) = φ(t0 ), ∀t0 ∈ [−τ , 0].

(4)

Moreover, the following lemma is more useful for deriving our main results. Lemma 1 ([15]) For given symmetric matrices Q > 0 and D > 0, scalars a and b satisfying a < b, the following inequalities hold for all continuously differentiable function κ : [a, b] → R , such that

Fault Alarm-Based Hybrid Control Design …

 (b − a)

⎡

⎤ 36 60 Q (b−a) 9Q − b−a 2Q ⎢ ⎥ 192 360 κT (s)Qκ(s)ds ≥ π T ⎣ ∗ (b−a) 2 Q − (b−a)3 Q ⎦ π, 720 ∗ ∗ Q (b−a)4

b

a

(b − a) 2

2



205

⎡



b

s

ds a

a

6D ⎢ κT (u)Dκ(u)du ≥ πˆ T ⎣ ∗ ∗

30 D (b−a) 720 D 4 (b−a)

∗

⎤

60 D (b−a)2 ⎥ 480 − (b−a) ˆ 3 D ⎦ π, 1200 D (b−a)4

where πT =



b a

κT (s)ds

b

ds

a

s a

b

κT (u)du

a

ds

s a

du

u a

κT (v)dv

T

and  πˆ =



b

ds a



a b

s





a



s

ds

du a



b

κT (u)du

ds a



u

dv a

u

du a

v



s

κT (v)dv

a

κT (w)dw

T .

a

3 Main Results In this section, H∞ based hybrid control scheme is designed for PPSs with timevarying delay, uncertainties and exogenous disturbances. To be precise, this section consists of the following three theorems. First, linear matrix inequality based suf cient stability conditions are derived by constructing Lyapunov–Krasovskii functional with periodic piecewise time-varying (PPTV) positive-de nite matrix for known gain matrix which is presented in Theorem 1. Further, the result will be extended to the case of an unknown gain matrix with gain uctuations. Finally, Theorem 3 provides a design of a non-fragile reliable control with unknown gain and unknown fault matrix. Before proceeding, we construct a time-varying piecewise periodic positivede nite matrix P(t), ∀t > 0 with fundamental period F p and it satis es limt→ν Fp +ti P(t) = P(t + ν F p ). Further, the sub-interval [ti−1 , ti ) of each piecewise subsystems in each period F p is divided into Ji segments with length σi = Fi /Ji , where Ji ∈ N. Let ωi, j = ν F p + ti−1 + jσi , where j ∈ J = 0, 1, . . . , Ji − 1. Now, the periodic piecewise positive-de nite matrix Pi (t) is de ned as Pi (t) = Pi, j + ρi, j (t)(Pi, j+1 − Pi, j ), i ∈ Q, t ∈ [ωi, j , ωi, j+1 ) ∈ Fi , where ρi, j (t) =

Ji (t−(ν F p +ti−1 + jσi )) Fi

∈ [0, 1) and Pi,Ji = Pi+1,0 PQ,Ji = Pi,0 .

(5)

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For brevity, the continuous time-varying matrix Pi (t) is rewritten as Pi (t) = (1 − ρi, j (t))Pi, j + ρi, j (t)Pi, j+1 . Moreover, the schematic representation of PPTV positive-de nite matrix P(t) is given in Fig. 1.

3.1 H∞ Performance Analysis In this subsection, a delay-dependent stability condition is derived by considering Lyapunov–Krasovskii functional with the PPTV matrix for the design of a robust controller with known gain matrix which can ensure the asymptotic stabilization of the considered PPS (1) with prescribed H∞ performance index. Theorem 1 For given control gain K and scalars αi , μ, τ , the closed-loop PPS (4) with H∞ performance index γ > 0 is asymptotically stable if there exist matrices Pi, j > 0, Q 1 > 0, Q 2 > 0, Q 3 > 0, Q 4 > 0, j ∈ J, i ∈ S and positive scalars ε1 > 0, ε2 > 0, such that the following constraints hold:  [Ψ 1 ]8×8 [Ψ 2 ]8×5 < 0, Ψ = ∗ [Ψ 3 ]5×5   1 [Φ ]8×8 [Φ 2 ]8×5 < 0, Φ= ∗ [Φ 3 ]5×5 

(6) (7)

Pi,Ji = Pi+1,0 , PS,Ji = P1,0 ,

(8)

where 1 Ψ1,1 = Sym(Pi, j Ai + Pi, j Bi K i ) + αi Pi, j +

+Q 1 + Q 2 + τ 2 Q 3 +

Ji (Pi, j+1 − Pi, j) Fi

τ4 Q4, 4

1 Φ1,1 = Sym(Pi, j+1 Ai + Pi, j+1 Bi K i ) + αi Pi, j+1 −

τ4 Q4, 4 = Pi, j+1 Aτ i ,

+Q 1 + Q 2 + τ 2 Q 3 + 1 1 Ψ1,2 = Pi, j Aτ i , Φ1,2

1 1 Ψ1,8 = Pi, j Ci , Φ1,8 = Pi, j+1 Ci , 1 1 1 1 Ψ2,2 = Φ2,2 = −(1 − μ)Q 1 , Ψ3,3 = Φ3,3 = −Q 2 , 36 1 1 1 1 Q3, = Φ4,4 = −9Q 3 , Ψ4,5 = Φ4,5 = Ψ4,4 τ −60 1 1 Ψ4,6 = Φ4,6 = 2 Q3, τ −192 1 1 Ψ5,5 = Φ5,5 = Q 3 − 6Q 4 , τ2

Ji (Pi, j+1 − Pi, j ) Fi

Fault Alarm-Based Hybrid Control Design …

207

360 30 Q4, Q3 + 3 τ τ 60 = − 2 Q4, τ 210 720 = − 2 Q4 − 4 Q3, τ τ 480 = 3 Q4, τ 1200 1 1 = − 4 Q 4 , Ψ8,8 = Φ8,8 = −γ I, τ 2 = DiT , Ψ1,2 = ε1 Pi, j Ui ,

1 1 Ψ5,6 = Φ5,6 = 1 1 Ψ5,7 = Φ5,7 1 1 Ψ6,6 = Φ6,6 1 1 Ψ6,7 = Φ6,7 1 1 Ψ7,7 = Φ7,7 2 2 Ψ1,1 = Φ1,1

2 2 2 T = ε1 Pi, j+1 Ui , Ψ1,3 = Φ1,2 = V Ai , Φ1,2 2 2 Ψ1,4 = ε2 Pi, j Ui , Ψ1,4 = ε2 Pi, j+1 Ui , 2 2 T = Φ2,4 = V Aτ Ψ2,5 i,

Ψ 3 = Φ 3 = diag{−γ I, −ε1 I, −ε1 I, −ε2 I, −ε2 I }. Proof To establish the required stability constraints, the piecewise periodic Lyapunov–Krasovskii functional with time-varying positive-de nite matrices for the close-loop system (4) is considered as follows; V (x, t) =

3 

Vi (x, t),

(9)

i=1

where V1 (x, t) = x T (t)Pi (t)x(t),  t  x T (s)Q 1 x(s)ds + V2 (x, t) = V3 (x, t) = τ

t−τ (t)  0

t

x T (u)Q 3 x(u)du

t+s  s

ds −τ

x T (s)Q 2 x(s)ds,

t−τ

ds

−τ 2  0

τ + 2



t



t

du −τ

x T (v)Q 4 x(v)dv.

t+u

Now, the time derivative of the above de ned Lyapunov–Krasovskii functional V (x, t) together with the trajectories of closed-loop system (4) is calculated as follows: ˙ + x T (t) P˙i (t)x(t) V˙1 (x, t) = 2x T (t)Pi (t)x(t) = 2x T (t)Pi (t)((Ai + ΔAi (t))x(t) + (Aτ i + ΔAτ i (t))x(t − τ (t)) Pi, j+1 − Pi, j x(t), (10) + Bi K i (t)x(t) + Ci w(t) + x T (t) σi

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V˙2 (x, t) ≤ x T (t)(Q 1 + Q 2 )(t)x(t) − (1 − μ)x T (t − τ (t))Q 1 x(t − τ (t)) − x T (t − τ )Q 2 x(t − τ ),    t τ4 T 2 ˙ x T (s)Q 3 x(s)ds V3 (x, t) = x (t) τ Q 3 + Q 4 x(t) − τ 4 t−τ  t+s  τ2 t ds x T (u)Q 4 x(u)du. − 2 t−τ t−τ

(11)

(12)

By applying Lemma 1 to the single and double integral terms in Eq. (12), we get 

0

x T (t + s)Q 3 x(t + s)ds ⎤ ⎡ −9Q 3 36 Q 3 −60 Q3 τ τ2 ˆ ≤ ϕˆ T (t) ⎣ ∗ −192 Q 3 360 Q 3 ⎦ ϕ(t), τ2 τ3 −720 ∗ ∗ Q 3 4 τ  s  τ2 0 T − ds x (t + u)Q 4 x(t + u)du 2 −τ −τ ⎡ ⎤ −6Q 4 30 Q 4 −60 Q4 τ τ2 ˜ ≤ ϕ˜ T (t) ⎣ ∗ −720 Q 4 480 Q 4 ⎦ ϕ(t), τ4 τ3 −1200 ∗ ∗ Q 4 τ4

−τ

−τ

(13)

(14)

where ϕˆ T (t) = and



t t−τ

x T (s)ds

0 −τ

ds

 t+s t−τ

x T (u)du

0 −τ

ds

 t+s t−τ

du

 t+u t−τ

x T (v)dv

T

T  0  t+s  t+u  t+v ϕ˜ T (t) = ϕˆ T (t) −τ ds t−τ du t−τ dv t−τ x T (w)dw .

Further, by combining the inequalities (10)–(14), we obtain ¯ V˙ (x, t) + αV (x, t) ≤ ϕ¯ T (t)[Θ]8×8 ϕ(t) where T  ϕ¯ T (t) = x T (t) x T (t − τ (t)) x T (t − τ ) ϕ˜ T (t) w T (t) , Ji (Pi, j+1 − Pi, j) Θ1,1 = Sym(Pi (t)(Ai + Bi K i + ΔAi (t)) + αi Pi, j + Fi 4 τ + Q1 + Q2 + τ 2 Q3 + Q4, 4 Θ1,2 = Pi (t)(Aτ i + ΔAτ i (t)), Θ1,2 = Pi Ci , Θ2,2 = −(1 − μ)Q 1 , Θ3,3 = −Q 2 , Θ4,4 = −9Q 3 ,

(15)

Fault Alarm-Based Hybrid Control Design …

209

36 −60 Q 3 , Θ4,6 = 2 Q 3 , τ τ −192 360 30 Q4, = Q 3 − 6Q 4 , Θ5,6 = 3 Q 3 + τ2 τ τ 60 210 720 = − 2 Q 4 , Θ6,6 = − 2 Q 4 − 4 Q 3 , τ τ τ 480 1200 = 3 Q 4 , Θ7,7 = − 4 Q 4 , Θ8,8 = −γ I. τ τ

Θ4,5 = Θ5,5 Θ5,7 Θ6,7

Now, by using the Eq. (5) and by simple manipulation, the matrix Θ in above inequality (15) can be expressed in the following form ˜ Θ = (1 − σi, j (t))Ψ˜ + σi, j (t)Φ,

(16)

where Ψ˜ = [Ψ 1 ]8×8 + (υ1 Δi (t)υ2 ) + (υ1 Δi (t)υ2 )T + (υ1 Δi (t)υ4 ) + (υ1 Δi (t)υ4 )T , Φ˜ = [Φ 1 ]8×8 + (υ˜ 1 Δi (t)υ2 ) + (υ˜ 1 Δi (t)υ2 )T + (υ˜ 1 Δi (t)υ4 ) + (υ˜ 1 Δi (t)υ4 )T , with   T T υ1 = (Pi, j Ui )T 01×7 , υ˜ 1 = (Pi, j+1 Ui )T 01×7 ,    T  01×7 , υ4 = 0 V ATτ i 01×6 , υ2 = V Ai 1 1 1 1 in which Ψ1,1 = Ψ1,1 + γ −1 D T i Di , and Φ1,1 = Φ1,1 + γ −1 D T i Di .

Here, by employing S-procedure lemma and Schur complement in the above mentioned inequality (16), we obtain the LMIs stated in theorem statement. If the constraints (6)–(8) hold, we get Θ < 0. Hence, by Lyapunov stability theory, the closed-loop form of PPS (4) is asymptotically stable. Thus, the proof is completed. 

3.2 Hybrid Control Design This subsection will be focused on deriving the suf cient condition for ensuring the asymptotic stability of PPSs (1) through the robust and reliable control design. More speci cally, a robust control law is designed with an unknown gain matrix and without any actuator faults. Further, the controller gain and actuator faults are considered as unknown in reliable control design. Case (i) (Robust controller design): In this case, we aim to design robust controller with gain uctuations and without actuator fault. Thus, the control gain matrix in (3) takes the form K˜ˆ i (t) = Kˆ i (t) + Δ Kˆ i (t) in which Δ Kˆ i (t) = Uˆ K i Δi (t)Vˆ K i

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where Uˆ K i and Vˆ K i are known constant matrices with appropriate dimensions and Δi (t) is the time-varying matrix satisfying ΔiT (t)Δi (t) < I . In this connection, the robust control scheme with gain uctuation is considered as; (17) u G (t) = ( Kˆ i (t) + Δ Kˆ i (t))x(t). Then, we have the following theorem. Theorem 2 For given scalars αi , μ, τ , if there exist matrices Mˆ i, j > 0, Ni, j , Q˜ 1 > 0, Q˜ 2 > 0, Q˜ 3 > 0, Q˜ 4 > 0, j ∈ J, i ∈ S and positive scalars ε1 > 0, ε2 > 0, ε3 > 0, such that the following LMIs hold: [Υ ]15×15 < 0, [Ω]15×15 < 0,

(18) (19)

Mˆ i,Ji = Mˆ i+1,0 , Mˆ S,Ji = Mˆ 1,0 ,

(20)

Nˆ i,Ji = Nˆ i+1,0 , Nˆ S,Ji = Nˆ 1,0 ,

(21)

where Ji ( Mˆ i, j+1 − Mˆ i, j ) Υ1,1 = Sym(Ai Mˆ i, j + Bi Nˆ i, j ) + αi Mˆ i, j − Fi 4 τ + Q˜ 1 + Q˜ 2 + τ 2 Q˜ 3 + Q˜ 4 , 4 Ji ( Mˆ i, j+1 − Mˆ i, j ) Ω1,1 = Sym(Ai Mˆ i, j+1 + Bi Nˆ i, j+1 ) + αi Mˆ i, j+1 − Fi 4 τ + Q˜ 1 + Q˜ 2 + τ 2 Q˜ 3 + Q˜ 4 , 4 ˆ Υ1,2 = Aτ i Mi, j , Ω1,2 = Aτ i Mˆ i, j+1 , Υ1,8 = Ci , Ω1,8 = Ci , Υ1,9 = Mˆ i, j DiT , Ω1,9 = Mˆ i, j+1 DiT , T , Υ1,10 = Ω1,10 = ε1 Ui , Υ1,11 = Mˆ i, j V Ai T Ω1,11 = Mˆ i, j+1 V Ai , Υ1,12 = Ω1,12 = ε2 Ui ,

Υ1,14 = Ω1,14 = ε3 Bi G0 Uˆ K i , Υ1,15 = Mˆ i, j Vˆ KTi , Ω1,15 = Mˆ i, j+1 Vˆ KTi , Υ2,2 = Ω2,2 = −(1 − μ) Q˜ 1 , T T , Ω2,13 = Mi, j+1 VBi , Υ3,3 = Ω3,3 = − Q˜ 2 , Υ2,13 = Mˆ i, j VBi 36 ˜ Υ4,4 = Ω4,4 = −9 Q˜ 3 , Υ4,5 = Ω4,5 = Q3, τ −60 −192 ˜ Υ4,6 = Ω4,6 = 2 Q˜ 3 , Υ5,5 = Ω5,5 = Q 3 − 6 Q˜ 4 , τ τ2 360 30 ˜ 60 Υ5,6 = Ω5,6 = 3 Q˜ 3 + Q 4 , Υ5,7 = Ω5,7 = − 2 Q˜ 4 , τ τ τ

Fault Alarm-Based Hybrid Control Design …

211

210 ˜ 720 480 Q 4 − 4 Q˜ 3 , Υ6,7 = Ω6,7 = 3 Q˜ 4 , 2 τ τ τ 1200 ˜ Υ7,7 = Ω7,7 = − 4 Q 4 , τ Υ8,8 = Ω8,8 = −γ I, Υ9,9 = Ω9,9 = −γ I, Υ10,10 = Ω10,10 = Υ11,11 = Ω11,11 = −ε1 I,

Υ6,6 = Ω6,6 = −

Υ12,12 = Ω12,12 = Υ13,13 = Ω13,13 = −ε2 I, Υ14,14 = Ω14,14 = Υ15,15 = Ω15,15 = −ε3 I. Thus, the considered PPSs is asymptotically stabilized through the controller (17) with H∞ performance index γ > 0. Moreover, the periodic controller gain matrix is computed by the relation (22) Kˆ i (t) = Nˆ i (t) Mˆ i−1 (t), where the time-varying matrix function Mi and Ni are determined by

where ρi, j (t) = Ji

Mˆ i (t) = Mˆ i, j + ρi, j (t)( Mˆ i, j+1 − Mˆ i, j ),

(23)

Nˆ i (t) = Nˆ i, j + ρi, j (t)( Nˆ i, j+1 − Nˆ i, j ),

(24)

(t−(lT p +ti−1 + jσi )) . Fi

Proof In order to obtain the stabilization condition of PPS (1) through the controller (17), the similar derivations of Theorem 1 is followed by substituting K i = Kˆ i in inequalities (6) and (7), then pre and post multiplying the obtained inequality ˆ Kˆ Mˆ = Nˆ , Q˜ q = M Q q M, by diag{P −1 , . . . , P −1 , I, . . . , I } and letting P −1 = M,       7

8

q = 1, . . . , 4, we obtain the inequalities (18)–(21), which completes the proof.  Case (ii) (Reliable controller design): It should be noted that the faults in actuators are inevitable in control systems due to many practical reasons, which results in performance deterioration or even instability. In this connection, the considered PPS (1) is formulated with actuator fault in control input. In order to overcome the performance degradation of the system, the reliable control is designed in the following form u G (t) = G K˜ i (t)x(t),

(25)

where K˜ i (t) = K¯ i (t) + Δ K¯ i (t) in which Δ K¯ i (t) = U¯ K i Δi (t)V¯ K i in which U¯ K i and V¯ K i are known constant matrices with appropriate dimensions and Δi (t) is the timevarying matrix satisfying ΔiT (t)Δi (t) < I and G = diag{g1 , g2 , . . . , gk } is the actuator fault matrix with 0 ≤ gvl ≤ gv ≤ gvu ≤ 1, here gvl and gvu are the lower and upper bound of gv and assumed to be known positive constants. It is noted that if gv = 0 the vth actuator fails, gv = 1 means that vth actuator is normal. If 0 ≤ gv ≤ 1 means the vth actuator has partial failure.

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Subsequently, the fault matrix G is reformulated as G = G0 + G1 M, |M| ≤ I, gl +g u where G0 = diag{g10 , g20 , . . . , gk0 }, G1 = diag{g11 , g21 , . . . , gk1 }, gv0 = v 2 v and gv1 =

gvu −gvl . 2

Theorem 3 For given scalars αi , μ and τ , the PPS (1) is asymptotically stabilized through the controller (25) with H∞ performance index γ > 0, if there exist matrices Mi, j > 0, Ni, j , Q˜ 1 > 0, Q˜ 2 > 0, Q˜ 3 > 0, Q˜ 4 > 0, j ∈ J, i ∈ S and positive scalars ε1 > 0, ε2 > 0, ε3 > 0, ε4 > 0, ε5 > 0, such that the following LMIs hold: [Υ˜ ]19×19 < 0, ˜ 19×19 < 0, [Ω]

(26)

Mi,Ji = Mi+1,0 , M S,Ji = M1,0 , Ni,Ji = Ni+1,0 , N S,Ji = N1,0 ,

(28) (29)

(27)

where Υ1,1 = Sym(Ai Mi, j + Bi G0 Ni, j ) + αi Mi, j − + Q˜ 1 + Q˜ 2 + τ 2 Q˜ 3 +

Ji (Mi, j+1 − Mi, j ) Fi

τ4 ˜ Q4, 4

Ω˜ 1,1 = Sym(Ai Mi, j+1 + Bi G0 Ni, j+1 ) + αi Mi, j+1 −

Ji (Mi, j+1 − Mi, j ) Fi

τ4 ˜ Q4, 4 = Aτ i Mi, j+1 , Υ˜1,8 = Ci , Ω˜ 1,8 = Ci ,

+ Q˜ 1 + Q˜ 2 + τ 2 Q˜ 3 +

Υ˜1,2 = Aτ i Mi, j , Ω˜ 1,2 Υ˜1,9 = Mi, j DiT , Ω˜ 1,9 = Mi, j+1 DiT , T , Υ˜1,10 = Ω˜ 1,10 = ε1 Ui , Υ˜1,11 = Mi, j V Ai

T , Υ˜1,12 = Ω˜ 1,12 = ε2 Ui , Ω˜ 1,11 = Mi, j+1 V Ai Υ˜1,14 = Ω˜ 1,14 = ε3 Bi G0 U¯ K i , Υ˜1,15 = Mi, j V¯ KTi ,

Ω˜ 1,15 = Mi, j+1 V¯ KTi , Υ˜1,16 = Ω˜ 1,16 = ε4 Bi G1 , Υ˜1,17 = Ni,T j , Ω˜ 1,17 = Ni,T j+1 , Υ˜1,18 = Ω˜ 1,18 = ε5 Bi G1 , T Υ˜2,2 = Ω˜ 2,2 = −(1 − μ) Q˜ 1 , Υ˜2,13 = Mi, j VBi , T ˜ ˜ ˜ ˜ Ω2,13 = Mi, j+1 VBi , Υ3,3 = Ω3,3 = − Q 2 ,

36 ˜ Υ˜4,4 = Ω˜ 4,4 = −9 Q˜ 3 , Υ˜4,5 = Ω˜ 4,5 = Q3, τ −60 −192 ˜ Q 3 − 6 Q˜ 4 , Υ˜4,6 = Ω˜ 4,6 = 2 Q˜ 3 , Υ˜5,5 = Ω˜ 5,5 = τ τ2 360 30 ˜ 60 Υ˜5,6 = Ω˜ 5,6 = 3 Q˜ 3 + Q 4 , Υ˜5,7 = Ω˜ 5,7 = − 2 Q˜ 4 , τ τ τ

Fault Alarm-Based Hybrid Control Design …

213

210 ˜ 720 480 Q 4 − 4 Q˜ 3 , Υ˜6,7 = Ω˜ 6,7 = 3 Q˜ 4 , 2 τ τ τ 1200 ˜ = − 4 Q 4 , Υ˜8,8 = Ω˜ 8,8 = −γ I, τ = −γ I, Υ˜14,19 = Ω˜ 14,19 = ε3 U KT i ,

Υ˜6,6 = Ω˜ 6,6 = − Υ˜7,7 = Ω˜ 7,7

Υ˜9,9 = Ω˜ 9,9 Υ˜10,10 = Ω˜ 10,10 = Υ˜11,11 = Ω˜ 11,11 Υ˜12,12 = Ω˜ 12,12 = Υ˜13,13 = Ω˜ 13,13 Υ˜14,14 = Ω˜ 14,14 = Υ˜15,15 = Ω˜ 15,15 Υ˜16,16 = Ω˜ 16,16 = Υ˜17,17 = Ω˜ 17,17 Υ˜18,18 = Ω˜ 18,18 = Υ˜19,19 = Ω˜ 19,19

= −ε1 I, = −ε2 I, = −ε3 I, = −ε4 I, = −ε5 I.

Moreover, the corresponding gain matrices of the designed controller (25) is obtained by using the relations (22)–(24) with Kˆ i (t) = K¯ i (t), Mˆ i (t) = Mi (t) and Nˆ i (t) = Ni (t). Proof The suf cient stability conditions in Theorem 3 can be easily obtained by following the similar lines of Theorems 1–2 and Schur complement. Thus, the proof is completed. 

4 Design of Fault Alarm In this section, the alarm system is designed to timely alert the controller to switch over from robust controller to the reliable controller, which tolerates the effect of the faults occurred in system dynamics. In particular, the threshold value of the fault is estimated with the aid state observer. Thus, the robust residual observer system is considered in the following form; ⎧ ˙˜ ⎪ = (Ai + ΔAi (t))x(t) ˜ + (Aτ i + ΔAτ i (t))x(t ˜ − τ (t)) ⎨x(t) g +Bi u (t) + L i (t)(y(t) − y˜ (t)), ⎪ ⎩ ˜ y˜ (t) = Di x(t),

(30)

where x(t) ˜ ∈ Rx is the observer state; u g (t) ∈ Ru is the control input; L i (t) is the time-varying periodic observer gain and y˜ (t) ∈ R y is the observer output. Now, the error residue is de ned as x(t) ˇ = x(t) − x(t). ˜ Based on the above relation, the error system is obtained in the following form: ˙ˇ =(Ai + ΔAi (t))x(t) ˇ + (Aτ i + ΔAτ i (t))x(t ˇ − τ (t)) x(t) + Bi (u G (t) − u g (t)) + Ci v(t) − L Di x(t). ˇ

(31)

It should be noted that the relation between the control input of the state and observer is varying with respect to the presence of actuator faults.

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In this connection, the control input is selected as u G (t) = u g (t) for fault free case and u G (t) = Gu g (t) when the fault occurs. Consequently, the critical relationship between the x(t) ˇ and w(t) is de ned as follows: ˇ = Ci v. ¯ (32) (L i Di − Ai − Aτ i )x(t) where v¯ is the upper bound of v(t). Then, by the simple manipulation, we obtain ¯ ≤ ||ζi ||||v||, ¯ ||e(t)|| ≤ ||ζi v|| where ζi = Di (L i Di − Ai − Aτ i )+ Ci ,e(t) = y(t) − y˜ (t). ¯ Moreover, the Now, the threshold value of fault is de ned as T = sup ||ζi ||||v||. algorithm used to detect the fault is designed in the following form;  Sw (t) =

0, nor mal, ||e(t)|| ≤ T, 1, alar m, ||e(t)|| > T.

(33)

Finally, the hybrid control scheme based on the fault-alarm approach is given as follows; (34) u G (t) = (1 − Sw (t)) Kˆ i (t)x(t) + Sw (t)G K˜ i (t)x(t).

5 Numerical Example Consider a uncertain periodic piecewise system (2) with three subsystems and fundamental period F p = 3s and F1 = 0.9s, F2 = 0.6s, F3 = 1.5s. Subsystem 1: 

     −2.51 1.21 2.53 0.6 1 1 , Aτ 1 = , B1 = , 0.49 −0.9 1.49 0.7 1.41 0.52    T    T 0.21 1 0.51 −0.1 , D1 = , U1 = , V A1 = , C1 = 0.1 0.8 0.1 −0.51      T −0.1 0.12 0.61 V Aτ 1 = , Uˆ K 1 = U¯ K 1 = , Vˆ K 1 = V¯ K 1 = . −0.5 0.5 0.51 A1 =

Subsystem 2:  A2 =

     1 1.79 0.59 0.51 1.51 1 , Aτ 2 = , B2 = , 2.51 −1 1.5 0.61 0.95 1

Fault Alarm-Based Hybrid Control Design …

215



    T  T 0.20 1 0.19 −0.49 C2 = , U2 = , , D2 = , V A2 = 0.15 0.42 0.2 0.9  T    T 0.61 0.22 0.89 V Aτ 2 = , Uˆ K 2 = U¯ K 2 = , Vˆ K 2 = V¯ K 2 = . −0.8 0.4 0.11 Subsystem 3: 

     1.22 3 0.39 0.8 1 1 , Aτ 3 = , B3 = , A3 = 0.99 −1.1 1.33 0.6 1.82 1      T  T 0.12 0.9 0.49 0.49 C3 = , U3 = , , D3 = , V A3 = 0.31 1.1 0.1 0.15      T 0.21 0.52 0.49 ˆ ¯ ˆ ¯ V Aτ 3 = , UK 3 = UK 3 = , VK 3 = VK 3 = . 0.8 0.1 −0.1 Further, the external disturbance signal w(t) is chosen as  w(t) =

4 + 4 cos(8πt), t < 1.5, 0.01e(−0.07t) sin t, otherwise.

Moreover, the upper and lower bound of gk is considered as g1u = 0.3, g2u = 0.3, = 0.1 and g2l = 0.1 with respect to the initial condition x(0) = [0.1 − 0.1]T . Now, by solving the conditions developed in Theorems 2 and 3 with the considered parameters together with the values of α = 0.7, τ = 1.5, Ji = 1 and μ = 0.3, we

g1l

30

ˆ ||K(t)||

ˆ (t)|| ||M

ˆ (t)|| ||N

25

20

15

10

5

0

0

0.5

1

1.5

2

2.5

3

Time (seconds)

ˆ Fig. 2 Variations of || Kˆ (t)||, || M(t)|| and || Nˆ (t)|| over one period based on Theorem 2

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obtain the gain matrices of the corresponding control and observer with satisfactory H∞ performance index γ = 0.17. Further, the computed gain matrices of the corresponding robust and reliable controller with respect to the relation (22)–(24) are plotted in Figs. 2 and 3, respectively. Further, the observer gain matrix is plotted in Fig. 4. Consequently, in the presence of unknown actuator faults, the trajectories of the system (4) with respect to the robust controller (17) is plotted in Fig. 5. From Fig. 5, it can be seen that the robust

700

¯ ||K(t)||

600

||M(t)||

||N (t)||

500 400 300 200 100 0 100 0

0.5

1

1.5

2

2.5

3

Time (seconds)

Fig. 3 Variations of || K¯ (t)||, ||M(t)|| and ||N (t)|| over one period based on Theorem 3 40

||L(t)|| ¯ (t)|| ||M ¯ ||N (t)||

35 30 25 20 15 10 5

0

0.5

1

1.5

Time (seconds)

Fig. 4 Variations of observer gain matrix over one period

2

2.5

3

Fault Alarm-Based Hybrid Control Design … Fig. 5 State responses of the closed-loop system (4) with fault under robust control (17)

217

0.6

x (t) 1

0.5

x (t)

0.4

T=0.16

2

0.3 0.2 0.1 0 0.1 10

5

0

15

20

30

25

Time (seconds)

Fig. 6 State responses of the closed-loop system (4) with fault under hybrid control (34)

0.15

x1(t) x2(t)

0.1

0.05

0

0.05

0.1 0

5

10

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Time (seconds)

controller exhibits poor performance with respect to the actuator faults in system (1). Moreover, in Fig. 5, the calculated threshold value T = 0.16 is also plotted. Furthermore, the state response of the system (2) concerning the fault-alarm based switching controller (hybrid controller) is provided in Fig. 6. Moreover, the proposed hybrid controller via fault alarm-based scheme ensures the asymptotic stability of the system even in the presence of actuator faults and also the automatic switching among the controllers is achieved with the aid of timely alert of the alarm signal. Finally, the simulation results reveal the applicability and effectiveness of the fault-alarm based hybrid control scheme.

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6 Conclusion In this chapter, fault-alarm based hybrid control design for a class of uncertain periodic piecewise systems subject to time-varying delay, external disturbances, and actuator faults is discussed. Speci cally, fault-alarm is designed to automatically switch over from robust control to reliable control once if the faults occur. By constructing periodically time-varying Lyapunov–Krasovskii functional, the suf cient conditions are derived in the form of LMIs to guarantee the asymptotic stability of the addressed system together with satisfactory H∞ performance level. Finally, the effectiveness of the theoretical result is illustrated via numerical simulations.

References 1. Xie, X., Lam, J., Fan, C.: Robust time-weighted guaranteed cost control of uncertain periodic piecewise linear systems. Inf. Sci. 460, 238–253 (2018) 2. Zhu, B., Lam, J., Song, X.: Stability and L1 -gain analysis of linear periodic piecewise positive systems. Automatica 101, 232–240 (2019) 3. Li, P., Li, P., Liu, Y., Bao, H., Lu, R.: H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function. J. Frankl. Inst. 356, 6968–6988 (2019) 4. Li, P., Lam, J., Cheung, K.C.: lH∞ control of periodic piecewise vibration systems with actuator saturation. J. Vib. Control 23, 3377–3391 (2017) 5. Li, P., Lam, J., Kwok, K.W., Lu, R.: Stability and stabilization of periodic piecewise linear systems: a matrix polynomial approach. Automatica 94, 1–8 (2018) 6. Park, J.H., Lee, T.H., Liu, Y., Chen, J.: Dynamic Systems with Time Delays: Stability and Control. Springer-Nature, Singapore (2019). https://doi.org/10.1007/978-981-13-9254-2 7. Lee, T.H., Park, J.H.: A novel Lyapunov functional for stability of time-varying delay systems via matrix-re ned-function. Automatica 80, 239–242 (2017) 8. Xie, X., Lam, J., Li, P.: Finite-time H∞ control of periodic piecewise linear systems. Int. J. Syst. Sci. 48, 2333–2344 (2017) 9. Li, P., Lam, J., Cheung, K.C.: Stability, stabilization and L2 −gain analysis of periodic piecewise linear systems. Automatica 61, 218–226 (2015) 10. Xie, X., Lam, J.: Guaranteed cost control of periodic piecewise linear time-delay systems. Automatica 94, 274–282 (2018) 11. Xie, X., Lam, J., Li, P.: H∞ control problem of linear periodic piecewise time-delay systems. Int. J. Syst. Sci. 49, 997–1011 (2018) 12. Park, M.J., Kwon, O.M., Ryu, J.H.: Advanced stability criteria for linear systems with timevarying delays. J. Frankl. Inst. 355, 520–543 (2018) 13. Liu, Q., Zhou, B.: Extended observer based feedback control of linear systems with both state and input delays. J. Frankl. Inst. 354, 8232–8255 (2017) 14. Yu, J., Yang, C., Tang, X., Wang, P.: H∞ control for uncertain linear system over networks with Bernoulli data dropout and actuator saturation. ISA Trans. 74, 1–13 (2018) 15. Samidurai, R., Sriraman, R., Cao, J., Tu, Z.: Non-fragile stabilization for uncertain system with interval time-varying delays via a new double integral inequality. Math. Methods Appl. Sci. 41, 6272–6287 (2018) 16. Xiao, B., Yang, X., Karimi, H.R., Qiu, J.: Asymptotic tracking control for a more representative class of uncertain nonlinear systems with mismatched uncertainties. IEEE Trans. Ind. Electron. 66, 9417–9427 (2019) 17. Xiang, W.: Necessary and suf cient condition for stability of switched uncertain linear systems under dwell-time constraint. IEEE Trans. Autom. Control 61, 3619–3624 (2016)

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18. Sakthivel, R., Wang, C., Santra, S., Kaviarasan, B.: Non-fragile reliable sampled-data controller for nonlinear switched time-varying systems. Nonlinear Anal.: Hybrid Syst. 27, 62–76 (2018) 19. Shen, M., Lim, C.C., Shi, P.: Reliable H∞ static output control of linear time-varying delay systems against sensor failures. Int. J. Robust Nonlinear Control 27, 3109–3123 (2017) 20. Zhao, D., Karimi, H.R., Sakthivel, R., Li, Y.: Non-fragile fault-tolerant control for nonlinear Markovian jump systems with intermittent actuator fault. Nonlinear Anal.: Hybrid Syst. 32, 337–350 (2019) 21. Shen, H., Wu, Z.W., Park, J.H.: Reliable mixed passive and H∞ ltering for semi-Markov jump systems with randomly occurring uncertainties and sensor failures. Int. J. Robust Nonlinear Control 25, 3231–3251 (2015) 22. Sakthivel, R., Saravanakumar, T., Ma, Y.K., Anthoni, S.M.: Finite-time resilient reliable sampled-data control for fuzzy systems with randomly occurring uncertainties. Fuzzy Sets Syst. 329, 1–18 (2017) 23. Sakthivel, R., Divya, H., Sakthivel, R., Liu, Y.: Robust hybrid control design for stochastic Markovian jump system via fault alarm approach. IEEE Trans. Circuits Syst. II: Express Briefs (2019). https://doi.org/10.1109/TCSII.2019.2941008 24. Li, J.N., Xu, Y.F., Bao, W.D., Xu, X.B.: Passivity and fault alarm-based hybrid control for a Markovian jump delayed system with actuator failures. IEEE Access 6, 797–805 (2017) 25. Harshavarthini, S., Selvi, S., Sakthivel, R., Almakhles, D.J.: Non-fragile fault alarm-based hybrid control for the attitude quadrotor model with actuator saturation. IEEE Trans. Circuits Syst. II: Express Briefs (2020). https://doi.org/10.1109/TCSII.2020.2964906 26. Shen, Q., Jiang, B., Shi, P., Lim, C.C.: Novel neural networks-based fault tolerant control scheme with fault alarm. IEEE Trans. Cybern. 44, 2190–2201 (2014) 27. Zheng, Q., Zhang, H., Ling, Y., Guo, X.: Mixed H∞ and passive control for a class of nonlinear switched systems with average dwell time via hybrid control approach. J. Frankl. Inst. 355, 1156–1175 (2018)

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems Wenhai Qi

Abstract This chapter considers the problem of sliding mode control (SMC) design for stochastic Markov jump systems (MJSs) via a suitable event-triggered communication scheme. Event-triggered strategy, which can save computational resources, is designed to determine whether the current data should be transmitted or not in the design process. First, by the key points of stochastic Markov Lyapunov functional and observer design theory, a desired SMC law is constructed to guarantee that the system trajectories can arrive at the specified sliding surface. Then, sojourn-time-dependent sufficient conditions are established to ensure the required stochastic stability performance of the whole closed-loop system. Finally, a boost converter circuit model is given to illustrate the proposed method. Keywords Sliding mode control · Stochastic Markov jump systems · Event-triggered communication scheme · Stochastic stability

1 Introduction It is well known that Markov jump systems (MJSs) are of great value in practical applications, such as networked control systems, manufacturing systems, faultdetection systems, economic systems, power systems, and aircraft systems [1–8]. Many actual control processes subject to random abrupt variations in their structures due to random failures or repairs of components can be described by MJSs. As a special kind of hybrid systems, MJSs consist of some dynamics and mode transitions governed by a Markov process (or a Markov chain) taking values in a finite set. Over the past decades, most leading research mainly focuses on MJSs with stochastic stability [9, 10], stochastic stabilization [11, 12], filter design [13], actuator saturation [14], passivity [15, 16], finite-time control [17, 18], and positivity analysis [19]. W. Qi (B) School of Engineering, Qufu Normal University, Rizhao 276826, People’s Republic of China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_10

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On another research front, sliding mode control (SMC) has been active in a wide variety of complex systems for its insensitivity to parameter variations, its complete rejection of external disturbances, and order reduction. The purpose of SMC is to drive the state signals onto an artificially specified hyper-surface (i.e. the sliding mode surface), on which the requested stability and disturbance inhibition can be achieved [20–22]. In a limited time, the convergence of the sliding mode surface can be guaranteed when an appropriate control law is designed to suppress the adverse effects of the uncertainty and nonlinearity. Nowadays, with the help of MJSs describing complex dynamical systems, various SMC approaches have been immersed with stochastic MJSs; for details, see [23–33]. With the rapid development of computer and networked technologies, data transmission via communication networks has received considerable research attention. Compared with traditional control systems, communication network schemes have incomparable advantages in realizing remote control and detection, increasing flexibility and reliability of control system [34, 35]. However, the data transmission process inevitably suffers from limited bandwidth, network-induced delay [36], communication packet losses, and network congestion. The traditional approach is to update the control signal in a time-triggered manner, which will cause unnecessary signal transmission. For the time-triggered strategy, it will result in resource waste since some redundant sampled signals are sent even when those signals keep the same values all the time. In order to overcome this shortcoming, the event-triggered communication scheme has been put forward [37, 38], in which the sampled signals are transmitted when the updated sampled signal satisfies the predefined triggered condition. Its advantage is to avoid the transmission of redundant data and save precious communication resources. Thus, plenty of noticeable results have been devoted to the event-triggered strategy [39–43]. To name a few, a novel event-triggered scheme has been proposed for a class of fuzzy Markov jump systems with general switching policies to improve the transmission efficiency at each sampling instance [42]. By constructing a novel common sliding surface, an event-triggered asynchronous SMC law for networked Markov jump Lur’e systems has been designed to depend on the hidden mode information [43]. However, the SMC design for stochastic MJSs via a suitable event-triggered communication scheme has received little research attention, since many complicated factors, such as uncertainty, time-triggered communication scheme, and Markov process, exist in the design of SMC law. These stimulate our interests to address the event-triggered SMC design for a class of stochastic MJSs. The main contributions can be concluded as follows: (i) Considering the effects of sensor information constraints and limited network resource, the observer design theory and suitable event-triggered communication scheme is adopted to remove the constraint of unmeasurable system states and save network resource, respectively. (ii) By utilizing Lyapunov functional, sufficient conditions for stochastic stability are proposed for the resulting closed-loop system.

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

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(iii) An appropriate observer-based SMC law via the event-triggered strategy is proposed to ensure the finite-time reachability of the sliding mode surface. (iv) Representing a boost converter circuit model as MJSs shows the validity of the SMC algorithm. Notations: P T represents the transpose of P. Symbol Rn stands for the n dimensional Euclidean space, Rn×m is the set of n × m real matrices, and S = {1, 2, . . . , N } means a set of positive numbers. The superscript I denotes identity matrix with appropriate dimensions. Given a probability space (Ξ, Υ, Θ), Ξ is the sample space, Υ is the σ-algebra of subsets of the sample space and Θ is the probability measure on Υ . Symbol E{·} represents the mathematical expectation,  ∗  means the Euclidean norm of vectors. P > 0 (≥0) means P is real symmetric positive (semi-positive) definite. For simplicity, symbol ∗ is represented as an ellipsis for symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2 Preliminaries Consider a class of MJSs on the probability space (Ξ, Υ, Θ) as x(t) ˙ =(A(rt ) + ΔA(rt , t))x(t) + B(rt )u(t), y(t) =C(rt )x(t),

(1)

where x(t) ∈ Rn , u(t) ∈ Rm , and y(t) ∈ Rl are the state, control input, and output, respectively. ΔA(rt , t) is represented by ΔA(rt , t) = M(rt )H(rt , t)N (rt ), in which H(rt , t) satisfies HT (rt , t)H(rt , t) ≤ I. {rt , t ≥ 0} denotes the Markov process in S = {1, 2, . . . , N } with probability transitions:  Pr {rt+Δ¯ = β|rt = α} =

¯ α = β, παβ Δ¯ + o(Δ), ¯ α = β, 1 + παα Δ¯ + o(Δ),

(2)

N where παβ ≥ 0 denotes the transition rate from α to β for α = β, παβ = β=1,β=α −παα . For rt = α ∈ S, A(rt ), ΔA(rt ), B(rt ), C(rt ), M(rt ), H(rt , t), and N (rt ) are, respectively, denoted as Aα , ΔAα (t), Bα , Cα , Mα , Hα (t), and Nα . Consider the output signal y(t) sampled at a fixed period h. Denote the current sampled signal as y(i h) (i ∈ N) and the latest transmitted one as y(ir h) (ir ∈ N, r = 0, 1, 2, . . . , ∞, i 0 = 0, r means the number of triggering). Whether the sampled signal y(i h) is sent or not is determined by the event trigger. Then, the event-triggered condition is formulated as ir +1 h =ir h + min j∈N { j h|e T ((ir + j)h)Πα e((ir + j)h) > κα y T ((ir + j)h)Πα y((ir + j)h)},

(3)

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where e((ir + j)h) = y((ir + j)h) − y(ir h). κα and Πα , α ∈ S, mean the predefined thresholds and weighting matrices. {(ir + j)h}, j ∈ N, denotes the set of sampling instants during [ir h, ir +1 h). Remark 1 It is noted that if there is no latest signal, the zero-order holder keeps the previous input signal of observer and SMC law. Once new signals are updated by the event-triggered communication scheme, the zero-order holder transmits the data to observer and SMC law. In the network, we denote τir as the network-induced transmission delay at the sampling instants ir h. The zero-order holder successfully receives the signals at instants ir h + τir . It is assumed that τir satisfies τm ≤ τir ≤ τ M ≤ h, where dm and d M are two positive constants. It is noted that the output signal y¯ (t) keeps the same data until new one reaches the zero-order holder, that is y¯ (t) =y(ir h), t ∈ [ir h + τir , ir +1 h + τir +1 ).

(4)

Due to the existence of the network-induced transmission delay, the signal holding interval [ir h + τir , ir +1 h + τir +1 ) can be divided into the following subintervals: r Ωr, j , [ir h + τir , ir +1 h + τir +1 ) = ∪kj=0

(5)

where Ωr, j = [qr, j h + τqr, j , qr, j+1 h + τqr, j+1 ), qr, j h = ir + j h are the sampling instants during [ir h, ir +1 h), and kr = ir +1 − ir . The invisible time delays τqr, j satisfy τm ≤ τqr, j ≤ τ M ≤ h and qr, j h + τqr, j < qr, j+1 h + τqr, j+1 . Define the piecewise function η(t) = t − qr, j h, t ∈ Ωr, j . Then, we have η1 ≤ η(t) ≤ η2 , where η1 = τm and η2 = τ M + h. Furthermore, one has y¯ (t) =y(ir h) = y(qr, j h) − e(qr, j h) = y(t − η(t)) − e(qr, j h), t ∈ Ωr, j .

(6)

According to (3), it is obtained that e T (qr, j h)Πα e(qr, j h) ≤ κα y T (t − η(t))Πα y(t − η(t)), t ∈ Ωr, j .

(7)

In many practical industrial control systems, due to the limitation of sensor, special working environment, and other factors, it is difficult to measure the signal variables directly. In such case, we need to use other methods to obtain the numerical values of these physical quantities. The observer-design approach is an important method in solving these problems. Then, it is necessary and important to take the observer-design approach into account when studying the characteristics of those signal variables. Based on the event-triggered communication scheme, the observer is constructed as

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

˙ˇ =Aα x(t) ˇ + Bα u(t) + Lα ( y¯ (t) − Cα x(t ˇ − η(t))), t ∈ Ωr, j , x(t)

225

(8)

where x(t) ˇ ∈ Rn and Lα mean the estimated state and the observer gains. Then, it follows from (6) and (8) that ˙ˇ =Aα x(t) ˇ + Bα u(t) + Lα (y(t − η(t)) x(t) − Cα x(t ˇ − η(t)) − e(qr, j h)), t ∈ Ωr, j .

(9)

Define error signal e(t) = x(t) − x(t). ˇ Combining (1) and (9) leads to the error dynamics as ˇ e(t) ˙ =(Aα + ΔAα (t))e(t) + ΔAα (t)x(t) − Lα Cα e(t − η(t)) + Lα e(qr, j h).

(10)

Therefore, according to (9) and (10), one has ˙ = (A¯ α + ΔA¯ α (t))ξ(t) + A¯ dα ξ(t − η(t)) + B¯α u(t) ξ(t) + L¯ α e(qr, j h), ξ(t0 + θ) = φ(θ), θ ∈ [−η2 , 0],

(11)

where   T  Aα 0 , ξ(t) = xˇ T (t) e T (t) , A¯ α = 0 Aα   0 0 ¯ α Hα (t)N¯ α , =M ΔA¯ α (t) = ΔAα (t) ΔAα (t)   T  0 Lα Cα A¯ dα = , B¯α = BαT 0 , 0 −Lα Cα    T T T   ¯ α = 0 MαT T , N¯ α = 0 Nα . L¯ α = −Lα Lα , M Definition 1  ([9]) System (11) is said to

be stochastically stable if, for r0 ∈ S and ∞ φ(θ) ∈ Rn , E 0 ||ξ(t)||2 dt|(φ(θ), δ0 ) < T (φ(θ), r0 ), where T (φ(θ), r0 ) is a positive constant. Lemma 1 ([44]) For given matrix P > 0 and all continuously ϑ(·) in [a, b] → R2n , one has  b ˙ −(b − a) ϑ˙ T (s)P ϑ(s)ds a

≤ −(ϑ(b) − ϑ(a))T P(ϑ(b) − ϑ(a)) − 3Π T PΠ, where Π = ϑ(b) + ϑ(a) −

2 b−a

b a

ϑ(s)ds.

(12)

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Lemma 2 ([45]) Consider a continuous function as 0 < τ1 ≤ τ (t) ≤ τ2 . For ˙ [−τ2 , 0] → X 2n , and real positive-definite real matrix X ∈ R2n×2n  , a vector x: X˜ Y ≥ 0, one has matrix Y ∈ R4n×4n satisfying Y T X˜  − (τ2 − τ1 )

t−τ1

t−τ2

x˙ T (s)X x(s)ds ˙ ≤ 2ψ1T Yψ2T − ψ1T X˜ ψ1T − ψ2T X˜ ψ2T ,

(13)

where 

 x(t − τ (t)) − x(t − τ2 ) ψ1 = , x(t − τ (t)) + x(t − τ2 ) − 22 (t)   x(t − τ1 ) − x(t − τ (t)) , ψ2 = x(t − τ1 ) + x(t − τ (t)) − 21 (t) X˜ = diag{X , 3X },  t−τ1 1 1 (t) = x(s)ds, τ (t) − τ1 t−τ (t)  t−τ (t) 1 x(s)ds. 2 (t) = τ2 − τ (t) t−τ2

3 Stochastic Stability The sliding switching surface is chosen as  s(t) =Dσ x(t) ˇ −

t

Dσ [(Aσ + Bσ Fσ )x(s) ˇ − Lσ Cσ x(s ˇ − η(s))]ds,

(14)

0

where Dσ ∈ Rm×n are chosen such that Dσ Bσ are nonsingular and Fσ ∈ Rm×n will be designed later. When the system states arrive at the sliding switching surface, we get ˇ − (Dσ Bσ )−1 Dσ Lα (y(t − η(t)) − e(qr, j h)). u eq (t) =Fσ x(t)

(15)

Substituting (15) into (11) yields ˙ =(A¯¯ α + ΔA¯¯ α (t))ξ(t) + A¯¯ dα ξ(t − η(t)) + L¯¯ α e(qr, j h), ξ(t) where

(16)

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

227

  T  Aα + Bα Fα 0 , ξ(t) = xˇ T (t) e T (t) , A¯¯ α = Aα   0 0 ¯¯ H (t)N¯¯ , ΔA¯¯ α (t) = =M α α α ΔAα (t) ΔAα (t)   T  0 −Lα Cα , L¯¯ α = 0 LαT , A¯¯ dα = 0 −Lα Cα T    ¯ T ¯ = 0M M , N¯¯ = 0 N . α

α

α

α

Next, we will construct novel sufficient conditions to make sure that the closedloop system (16) can realize stochastic stability. Theorem 1 If we find symmetric matrices Πα > 0, Pα > 0, Q1α > 0, Q2α > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, matrices Xσ and Y, and scalars ε1α > 0, ε2α > 0, ∀α ∈ S, such that 

 Z˜2 Y ≥ 0, Y T Z˜2

(17)

Ω1 < 0, N N παβ Q1β ≤ R1 , παβ Q2β ≤ R2 , β=1

β=1

where N T Ω1 = a1T Pα A¯¯ α + A¯¯ α Pα + παβ Pβ + Q1α + Q2α β=1

¯¯ M ¯¯ T P + ε−1 N¯¯ T N¯¯ + η1 R1 + η2 R2 + ε1α Pα M α α α α α 1α T T

T T ¯¯ ¯¯ ¯¯ ¯¯ + ε−1 2α N α N α a1 + a1 Pα Adα a2 + a2 Adα Pα a1 T T + a1T Xα A¯¯ σ a8 + a8T A¯¯ σ Xα a1 + a1T Pα L¯¯ σ a9 + a9T L¯¯ σ Pα a1 T T T ¯¯ T ¯¯ G σ Pα a1 − a1T Pα B¯¯ σ a11 − a11 B σ Pα a1 + a1T Pα G¯¯ σ a10 + a10 + κ a T [I I]T C T Π C [I I]a + a T X A¯¯ a

+

α 2 T a8T A¯¯ dα Xα a2

α

α α

2

2

α

dα 8

− a3T Q1α a3 − a4T Q2α a4 + a8T ((η2 − η1 )2 Z2

¯¯ M ¯¯ T X )a + a T X L¯¯ a + a T L¯¯ T X a − 2Xσ + ε2α Xα M α α α 8 8 α σ 9 9 σ α 8 T T T ¯¯ T ¯¯ G σ Xα a8 − a8T Xα B¯¯ σ a11 − a11 B σ Xα a 8 + a8T Xα G¯¯ σ a10 + a10 T T − a9T Πα a9 − a10 γIa10 − a11 γIa11 − b1T Z˜1 b1 + 2b2T Yb3

− b2T Z˜2 b2 − b3T Z˜2 b3 ,

(18) (19)

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with T  b1 = a1T − a3T a1T + a3T − a7T , Z˜1 = diag{Z1 , 3Z1 },  T b2 = a2T − a4T a2T + a4T − a5T , Z˜2 = diag{Z2 , 3Z2 },   T  Y Y b3 = a3T − a2T a3T + a2T − a6T , Y = 1 2 , Y3 Y4 in which as1 ∈ R(16n+l)×2n (s1 = 1, 2, . . . , 8), and a9 ∈ R(16n+l)×l mean the block entry matrices, e.g. a2 ζ(t) = ξ T (t − η(t)), then system (16) achieves stochastic stability. Proof Choose Lyapunov functional as V(ξ(t), rt , t) = V1 (ξ(t), rt , t) + V2 (ξ(t), rt , t) + V3 (ξ(t), t) + V4 (ξ(t), t),

(20)

where V1 (ξ(t), rt , t) = ξ T (t)P(rt )ξ(t),  t  ξ T (s)Q1 (rt )ξ(s)ds + V2 (ξ(t), rt , t) =  V3 (ξ(t), t) =

t−η1  t

0

−η1



V4 (ξ(t), t) = η1

ξ T (s)R1 ξ(s)dsdθ +



−η1



+ (η2 − η1 )

ξ T (s)Q2 (rt )ξ(s)ds,

t−η2 0  t



t+θ 0

t

t

t+θ −η1 −η2

−η2

ξ T (s)R2 ξ(s)dsdθ,

t+θ

˙ ξ˙T (s)Z1 ξ(s)dsdθ 

t

˙ ξ˙T (s)Z2 ξ(s)dsdθ.

t+θ

At time t, rt = α, for α ∈ S, one has the infinitesimal operator as Γ V1 (ξ(t), rt , t) = 2ξ T (t)Pα [(A¯¯ α + ΔA¯¯ α (t))ξ(t) + A¯¯ dα ξ(t − η(t)) N + L¯¯ α e(qr, j h)] + ξ T (t) παβ Pβ ξ(t). β=1

Similarly, one has

(21)

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

229

Γ V2 (ξ(t), rt , t) ≤ ξ T (t)(Q1α + Q2α )ξ(t) − ξ T (t − η1 )Q1α ξ(t − η1 )  t N T παβ e T (s)Q1β e(s)ds − ξ (t − η2 )Q2α ξ(t − η2 ) + +



N ρ=1

παβ

β=1

t

t−η1

e T (s)Q2β e(s)ds,

(22)

t−η2

Γ V3 (ξ(t), t)



= ξ T (t)(η1 R1 + η2 R2 )ξ(t) −

t

ξ T (s)R1 ξ(s)ds

t−η1



t

−

ξ T (s)R2 ξ(s)ds,

(23)

t−η2

Γ V4 (ξ(t), t) ˙ − η1 = ξ˙T (t)(η12 Z1 + (η2 − η1 )2 Z2 )ξ(t)  − (η2 − η1 )

t−η1



t

˙ ξ˙T (s)Z1 ξ(s)ds

t−η1

˙ ξ˙T (s)Z2 ξ(s)ds.

(24)

t−η2

For the uncertain term 2ξ T (t)Pα ΔA¯¯ α (t)ξ(t), there exist scalars ε1α , such that 2ξ T (t)Pα ΔA¯¯ α (t)ξ(t) ¯¯ M ¯¯ T P ξ(t) + ε−1 ξ T (t)N¯¯ T N¯¯ ξ(t). ≤ ε1α ξ T (t)Pα M α α α α α 1α

(25)

Applying Lemma 1 gives rise to  − η1

t

˙ ξ˙T (s)Z1 ξ(s)ds

t−η1

T   ξ(t) − ξ(t − η1 ) Z1 0 ≤− t ξ(t) + ξ(t − η1 ) − η21 t−η1 ξ(s)ds 0 3Z1   ξ(t) − ξ(t − η1 ) . × t ξ(t) + ξ(t − η1 ) − η21 t−η1 ξ(s)ds 

(26)

By Lemma 2 and (17), one has  − (η2 − η1 )

t−η1

t−η2

˙ ξ˙T (s)Z2 ξ(s)ds ≤ 2κ1T Yκ2 − κ1T Z˜2 κ1 − κ2T Z˜1 κ2 ,

(27)

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where 

 ξ(t − η(t)) − ξ(t − η2 ) κ1 = , t−η(t) 2 ξ(t − η(t)) + ξ(t − η2 ) − η2 −η(t) t−η2 ξ(s)ds   ξ(t − η1 ) − ξ(t − η(t)) . κ2 = t−η1 2 ξ(t − η1 ) + ξ(t − η(t)) − η(t)−η t−η(t) ξ(s)ds 1 For any appropriately matrices Xσ , it is shown that  ˙ + (A¯¯ α + ΔA¯¯ α (t))ξ(t) + A¯¯ dα ξ(t − η(t)) 2ξ˙T (t)Xσ −ξ(t)  + L¯¯ α e(qr, j h) = 0.

(28)

For the uncertain term 2ξ˙T (t)Xσ ΔA¯¯ α (t)ξ(t) in (28), there exist scalars ε2α such that 2ξ˙T (t)Xα ΔA¯¯ α (t)ξ(t) T ¯¯ M ¯¯ T X ξ(t) ˙ + ε−1 ξ T (t)N¯¯ N¯¯ α ξ(t). ≤ ε2α ξ˙T (t)Xα M α α α α 2α

(29)

Define  ζ(t) = ξ T (t) ξ T (t − η(t)) ξ T (t − η1 ) ξ T (t − η2 )  t−η1  t−η(t) 2 2 ξ T (s)ds ξ T (s)ds η2 − η(t) t−η2 η(t) − η1 t−η(t) T  2 t T T T ˙ ξ (s)ds ξ (t) e (qr, j h) . η1 t−η1

(30)

Applying the condition (7) leads to Γ V(ξ(t), rt , t) ≤ ζ T (t)Ω1 ζ(t),

(31)

with as1 ∈ R(16n+l)×2n (s1 = 1, 2, . . . , 8) and a9 ∈ R(16n+l)×l described in Theorem 1. According to the condition (18), we have Γ V(ξ(t), rt , t) < 0,

(32)

which means that system (16) achieves stochastic stability. This completes proof.  Remark 2 Here, an appropriate mode-dependent Lyapunov function (20) is constructed, and parameter in integral term V(ξ(t), rt , t) is mode-dependent, which may reduce some conservativeness.

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

231

Next, in order to deal with some nonlinear coupling terms in (18), we will design the controller gains and observer gains for satisfying the aforementioned conditions Theorem 1 under linear matrix framework. Theorem 2 Consider Dα in (14) given as Dα = BαT P1α . If there exist symmetric matrices Πα > 0, Pα > 0, Q1α > 0, Q2α > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, ˜ and scalars ε1α > 0, ε2α > 0, α > 0, ∀α ∈ S, such that (17), matrices Y and L, (19) and ⎡

⎤ ¯¯ a T ε  P M ¯¯ α Ω2 a1T ε1α Pα M α 8 2α α α ⎣ ∗ ⎦ < 0, −ε1α I 0 ∗ ∗ −ε2α I

(33)

where Ω2 = a1T

N β=1

T παβ Pβ a1 + a1T (Pα A¯¯ α + A¯¯ α Pα + Q1α + Q2α

T ¯¯ T ¯¯ ¯¯ −1 ¯¯ + η1 R1 + η2 R2 + ε−1 1α N α N α + ε2α N α N α )a1 + a1T diag{−L˜ α Cα , −L˜ α Cα }a2 + a2T diag{−L˜ α Cα , −L˜ α Cα }T a1 T  T + α a1T Pα A¯¯ α a8 + α a8T A¯¯ α Pα a1 + a1T 0 L˜ αT a9 T   T ¯¯ G α Pα a1 − a1T Pα B¯¯ α a11 + a9T 0 L˜ αT a1 + a1T Pα G¯¯ α a10 + a10 T ¯¯ − a11 B α Pα a1 + κα a2T [I I]T CαT Πα Cα [I I]a2 + α a2T diag{−L˜ α Cα , −L˜ α Cα }a8 T

+ α a8T diag{−L˜ α Cα , −L˜ α Cα }T a2 − a3T Q1α a3 − a4T Q2α a4 + a8T ((η2 − η1 )2 Z2 − 2α Pα )a8  T   + α a8T 0 L˜ αT a9 + α a9T 0 L˜ αT a8 + α a8T Pα G¯¯ α a10 T T T ¯¯ T ¯¯ G α Pα a8 − α a8T Pα B¯¯ α a11 − α a11 B α Pα a8 + α a10 T T − a9T Πα a9 − a10 γIa10 − a11 γIa11 − b1T Z˜1 b1 + 2b2T Yb3

− b2T Z˜2 b2 − b3T Z˜2 b3 , with Pα = diag{P1α , P1α }, in which Y, Z˜1 , Z˜2 , b1 , b2 , and b3 are described in Theorem 1, then system (16) achieves stochastic stability, in which Fα are selected such that Aα + Bα Fα are −1 ˜ Lα . Hurwitz. Furthermore, the observer gains are given as Lα = P1α

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Proof First, let us define the followings Pα = diag{P1α , P1α }, L˜ α =P1α Lα , Xα = α Pα .

(34)

Then, applying Schur complement lemma to (17), we can easily get (33) without detail procedure. This completes proof. 

4 Reachability The observer-based SMC law will be constructed to ensure the reachability of sliding switching surface, which guarantees that the state trajectories can be driven onto the sliding switching surface in a finite time. Theorem 3 Consider feasible solutions in Theorem 2. Then, the finite-time attractiveness of the sliding switching surface can be realized by ˇ − ((Dα Bα )−1 Dα Lα  y¯ (t) u(t) = Fα x(t) 1 N +  παβ (Dβ Bβ )−1 s(t) + ωα )sign(s(t)), β=1,β=α 2

(35)

with the positive constant ωα . Proof Choose Lyapunov function: V(ξ(t), rt ) =

1 T s (t)(D(rt )B(rt ))−1 s(t). 2

When rt = α, we have Γ V(ξ(t), rt )

N 1 = s T (t)(Dα Bα )−1 s˙ (t) + s T (t) παβ (Dβ Bβ )−1 s(t) β=1 2 ˇ = s T (t)[u(t) + (Dα Bα )−1 Dα Lα y¯ (t) + Fα x(t)] 1 T N + s (t) παβ (Dβ Bβ )−1 s(t) β=1 2 = s T (t)[−((Dα Bα )−1 Dα Lα  y¯ (t) 1 N +  παβ (Dβ Bβ )−1 s(t) + ωα ]sign(s(t)) β=1,β=α 2 N 1 + (Dα Bα )−1 Dα Lα y¯ (t) + s T (t) παβ (Dβ Bβ )−1 s(t). β=1 2

(36)

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

233

Further, it can be led as Γ V(ξ(t), rt ) ≤ s T (t)[−(Dα Bα )−1 Dα Lα  y¯ (t) 1 N +  πσβ (Dβ Bβ )−1 s(t) + ωα )]sign(s(t)) β=1,β=α 2 + s(t)(Dα Bα )−1 Dα Lα  y¯ (t) N 1 + s(t) παβ (Dβ Bβ )−1 s(t) β=1,β=α 2  = |s(t)| −(Dα Bα )−1 Dα Lα  y¯ (t)  1 N +  παβ (Dβ Bβ )−1 s(t) + ωα ) β=1,β=α 2 + s(t)(Dα Bα )−1 Dα Lα  y¯ (t) N 1 + s(t) παβ (Dβ Bβ )−1 s(t). β=1,β=α 2 Noting s T (t) ≤ |s T (t)|, one has Γ V(ξ(t), rt ) ≤ − ωα s(t). Therefore, the finite-time reachability can be realized. This completes proof.

(37) 

Remark 3 Considering the multi-mode characteristic of MJSs, it is necessary to choose a mode-dependent Lyapunov function (36). In addition, the proposed SMC law (35) depends on the upper bound of stochastic Markov switching transition rate παβ , β = α, the feedback controller gains Fα , and the event-triggered transmit data y¯ (t). Hence, under the effects of event-triggered communication scheme, the SMC law (35) can guarantee the finite-time attractiveness and improve the system performance.

5 Case Study Consider the boost converter circuit model. As shown in Fig. 1, there exist two modes that obey the SMP {rt , t ≥ 0} taking values in S = {1, 2}. Applying Kirchoff voltage and current law gives rise to d Vc Vc di L =u(t) − Vc , C = i L − , δt = 1, dt dt R di L d Vc L =u(t), RC = −Vc , δt = 2. dt dt L

(38) (39)

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Fig. 1 Boost converter circuit

T  Define the state vector z(t) = i L Vc . Then, we can get the system matrices as    1 0 0 0 − L1 , A2 = , B1 = B2 = L . A1 = 1 1 1 0 − RC 0 − RC C 

Suppose the other system matrices given as    T Cα = 0 1 , Mα = 0 0.3 ,   Nα = 0.1 −0.1 , Hα (t) = sin(t), (α = 1, 2). The values of system parameters R, L, and C are given as R = 200 Ω, L = 2 H , and C = 200 m F, respectively. Also, the TR matrix is listed as   −0.8 0.8 . 0.6 −0.6 In this model, an appropriate SMC law is designed to ensure that the equilibrium point zero of the state space is stochastically stable and it dispels the adverse effects of parametric uncertainty.     The controller gains are chosen as F1 = −1.9789 −9.1881 and F2 = 1 0 , such that Aα + Bα Fα , α = 1, 2, are Hurwitz. For given τm = 0.01, τ M = 0.1, h = 0.02, κα = 0.1, ε1α = 0.1, ε2α = 0.1, α = 1.2, and α = 1, 2, solving Theorem 2 results in Π1 = 114.4548, Π2 = 114.9490,     0.6962 0.0101 2.8600 −0.3184 , P12 = , P11 = 0.0101 0.2976 −0.3184 2.1869     −0.2274 0.0141 L1 = , L2 = . 1.2440 0.8914 Then, BαT P1α Bα is nonsingular and the integral sliding variable is given in the form of (14), for α = 1, 2.

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems

235 rt

2.5

System mode

2

1.5

1

0.5

0 0

2

4

6

8

10

Time(s)

Fig. 2 System mode Fig. 3 Release instants

2.5

Release intervals

2

1.5

1

0.5

0

0

2

4

6

8

10

Time(s)

For given ωσ = 0.01, α = 1, 2, the SMC law based on the event-triggered communication scheme can be computed by (35).  T  T For given r0 = 2, x0 = −0.8 1.2 and xˇ0 = −0.3 0.8 , the simulation results with the event-triggered SMC law (35) are given in Figs. 2, 3, 4, 5, 6 and 7.

236 Fig. 4 System state x(t)

W. Qi 1

x1 (t) x2 (t)

0.5

0

-0.5

-1

-1.5

-2 0

2

4

6

8

10

Time(s)

Fig. 5 Observer state x(t) ˇ

1

x ˇ1 (t) x ˇ2 (t)

0.5

0

-0.5

-1

-1.5

-2

0

2

4

6

8

10

Time(s)

Figure 2 describes the system mode. Figure 3 plots the event-triggering release instants and intervals. In Figs. 4 and 5, the state responses x(t) and x(t) ˇ converge to a small region of the origin, which explain the availability of the proposed SMC approach. Figure 6 stands for the sliding variable s(t). Figure 7 represents the SMC law u(t). Hence, the dynamic performance of the closed-loop system is satisfied under the event-triggered communication scheme.

Event-Triggered Sliding Mode Control for Stochastic Markov Jump Systems Fig. 6 Sliding variable s(t)

237

0.6

s(t)

0.4 0.2 0 0.2 0.4 0.6 0.8 0

2

4

6

8

10

Time(s)

Fig. 7 SMC law u(t)

60

u(t)

40 20 0 20 40 60 80

0

2

6

4

8

10

Time(s)

6 Conclusion In this chapter, the SMC for S-MSSs has been developed by applying an observer design. In order to reduce the occupancy of network bandwidth resources, the eventtriggered communication scheme has been considered in the design of the SMC law. Novel sufficient conditions have been proposed to guarantee that the system is stochastically stable. Based on these criteria, a new event-triggered SMC law is constructed to realize the finite-time attractiveness of the sliding surface.

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Acknowledgements The author would be very grateful to thank National Natural Science Foundation of China (61703231, 61773236, and 61873331), Natural Science Foundation of Shandong (ZR2017QF001, ZR2017MF063, and ZR2019YQ29), Natural Science Foundation of Hubei Province (2017CFB501), Chinese Postdoctoral Science Foundation (2018T110670), and Excellent Experiment Project of Qufu Normal University (jp201728).

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Asynchronous Quantized Control for Markov Switching Systems with Channel Fading Jun Cheng, Yaonan Shan, and Ju H. Park

Abstract This chapter considers the asynchronous control for Markov switching systems with channel fading, where a more general class of discrete-time Markov switching system is developed by exploring the channel fading. The original systems and controller are characterized by a variety of stochastic variables. Following this, the relationship between stochastic variables is considered. In order to reduce energy consumption, a mode-dependent event-triggered communication mechanism is employed. By establishing the mode-dependent Lyapunov function, sufficient conditions are derived to guarantee the stochastic stability of closed-loop systems. Furthermore, by solving achieved results, the gains of an asynchronous controller are obtained. In the end, a simulation example is borrowed to illustrate the effectiveness of the theoretical results. Keywords Markov switching systems · Asynchronous control · Channel fading · Lyapunov function

1 Introduction Over the past few decades, as a special class of hybrid systems, Markov switching systems (MSSs) have been received increasing attention due to its powerful modeling physical applications with sudden environmental changes [1–4]. Benefit from its J. Cheng (B) College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, People’s Republic of China e-mail: [email protected] Y. Shan University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China e-mail: [email protected] J. H. Park Department of Electrical Engineering, Yeungnam University, Gyeongsan 38541, Republic of Korea e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_11

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advantage, MSSs have been applied to in many fields, such as robotics, networked systems, economic systems, et al. Accordingly, many scholars draw their attention to MSSs and many valuable results are reported, including stability and stabilization [5–8], control [9–11], filtering [12–15], sliding mode control [16, 17], et al. For instance, a class of MSS has been studied by taking unreliable communication links into consideration in [8]. The issue of dissipativity-based control for MSS with unreliable measurements is investigated in [9]. In addition, due to the scenario that limitation of bandwidth and unmeasurable state in the networked systems, the system’s performance is degrading, even instable, which may lead to the fading phenomenon. Such limited communication is not acceptable in reality. To tackle with the shortage of unreliable channels, fading channels are adopted. Note that some important results are reported on fading channels, including control [18–20], state estimation [21–23], filtering [24, 25]. Despite the successfully utilizing of fading channels in networked control systems, a few results are expressed on the topic of MSSs. On the other hand, most of signals is transmitted to others by communication networks, and the sampling data control strategy (SDCS) was the most used technology. As stated in [26–31], SDCS has been extended to many fields, including MSSs. Recently, as an improvement of conventional SDCS, an event-triggered mechanism (ETM) has been proposed. Compared with conventional SDCS, ETM based control is aperiodic control, which plays an important role in reducing useless data. Following this, many MSSs have been investigated by employing ETM [30–33]. It is remarkable that, most of the reported ETMs is not consider the systems mode information, which left room for further improvement. In most existing works, synchronization issues of MSSs are studied by assuming controllers/filters keep the same modes with plants [34, 35]. In practical, it is difficult to achieve the information of original systems, and the existing assumption is hard to be satisfied. Recently, the asynchronous controllers/filters are developed, where the controller/filter modes are different from the system modes. By introducing a hidden Markov model (HMM), the original system modes are hidden to the controllers/filters. Accordingly, many asynchronous topics of MSSs are addressed [36–40]. However, there is little literature on asynchronous control for MSSs subject to ETM and CF, which remains open and challenging for MSSs. Motivated by aforementioned discussions, in this chapter, an asynchronous controller is developed for SMMs by utilizing HMM strategy, which covers synchronous ore mode-dependent one as a special case. By a set of random variables, a modedependent CF model is expressed. An improved ETM is established, where original system information is involved in ETM. Together with mode-dependent Lyapunov functional, sufficient conditions are derived and the closed-loop MSSs are stochastically stable. Notation A > 0(A ≥ 0) means the symmetric positive (semi-positive) definite matrix; E {·} denotes the mathematical expectation; A−1 and A represent the inverse and transpose of A, respectively.  ·  stands for the Euclidean vector norm; ⊗ indicates the Kronecker product.

Asynchronous Quantized Control for Markov Switching Systems …

243

2 Preliminaries Let’s concerning on the following MSS: x(k + 1) = Aϕ(k) x(k) + B1ϕ(k) u(k) + D1ϕ(k) ω(k), z(k) = Cϕ(k) x(k) + B2ϕ(k) u(k) + D2ϕ(k) ω(k),

(1)

where x(k) ∈ Rn means the system state, u(k) ∈ Ru denotes the control input, z(k) ∈ Rz represents the controlled output, ω(k) ∈ Rw stands for the external disturbance belonging to l2 [0, ∞), Aϕ(k) , B1ϕ(k) , B2ϕ(k) , Cθϕ(k) , D1ϕ(k) , and D2ϕ(k) are the given matrices subject to proper dimensions, and ϕ(k) ∈ M  {1, 2, . . . , M} is a homogenous Markovian chain subject to the following transition probability matrix (TPM) Π = {πab }: Pr{ϕ(k + 1) = b | ϕ(k) = a} = πab (2)  where 0 ≤ πab ≤ 1 and b∈M πab = 1. As is well known that, in the point-to-point systems, the signal transition between sensor and actuator is reliable. In reality, signal transmission is always experienced with bandwidth and limitation, which may lead to the instability of conventional data transmission. Following this trend, the network induced time delays have been taken into consideration in the networked system channels, for instance, the sensor to the controller, the controller to the actuator, et al. Therefore, it is important to deal with the consumption of network resources. Event-triggered mechanism (ETM) has been introduced in [31–33], which is an efficient way to tackle with issue of limited bandwidth. With the aid of ETM, the signals are transmitted at triggered time {ks }∞ s=0 . Inspired by above discussion, in this chapter, we consider the mode-dependent ETM (MDETM) described by (x(k) − x(ks )) Ω1ϕ(k) (x(k) − x(ks )) ≤ x  (ks )Ω2ϕ(k) x(ks ),

(3)

where Ω1ϕ(k) and Ω2ϕ(k) are two positive weighting matrices to be determined. Defining e(k) as the error during current instant x(k) and latest triggered instant x(ks ), one has e(k) = x(k) − x(ks ). The MDETM (3) can be changed as follows: e (k)Ω1ϕ(k) e(k) ≤ x  (ks )Ω2ϕ(k) x(ks ).

(4)

In the networked systems the phenomenon of unreliability often inevitably occurs during data transmission. Channel fading (CF) is widely exists in the networked systems, which has been attracted increasing attention in the past years. Combing with MDETM, the improved CF is given by x(k) ¯ =

lk  t=0

αt (k)x(ks−t ),

(5)

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where lk = min{l, k} with k represents the current sampled instant and l means the given number of paths. αt (k) ∈ [0, 1], (t = 0, . . . , lk ) are the mutually independent stochastic variables, note that αt (k) satisfies the probability density functions g(αt (k)) with (6) E {αt (k)} = αt , E {(αt (k) − αt )2 } = α 2t , √ where α t = αt (1 − αt ). Recalling e(k) = x(k) − x(ks ), it is easy to achieve that e(k − t) = x(ks−t+1 ) − x(ks−t ), t = 1, 2, . . . , lk . Following this trend, one has x(ks−t ) = x(k) − e(k) − · · · − e(k − t).

(7)

It follows from (5) and (7) that x(k) ¯ =

lk 

αt (k)[x(k) − e(k) − · · · − e(k − t)]

t=0

= (α0 (k) + Dlk )x(k) − (α0 (k) + Dlk )e(k) − De(lk ), where

e(lk ) = [e T (k − 1), . . . , e T (k − lk )]T , Dlk −i+1 =

lk 

αt (k), D = [Dlk I, . . . , D1 I ].

t=i

In the networked systems, the asynchronous phenomena is encountered because of the time lags, data dropout, et al. By taking MDETM and FC into consideration, the asynchronous controller can be designed as below: u(k) = K φ(k) x(k),

(8)

where K φ(k) is the asynchronous controller to be given later. φ(k) decides the controller mode and takes finite values in N = {1, 2, . . . , N }. The discrete-time MC φ(k) satisfying the conditional probability matrix Γ = {λac } Pr{φ(k) = c | ϕ(k) = a} = rac ,

(9)

N rac = 1. where rac ∈ [0, 1], for any c ∈ N , and c=1 Under the unreliable communication environment, it is difficult to get the raw measurement signals completely and continuously. Therefore, control u(k) should be quantized before transmission: u(k) = q(u(k)),

(10)

Asynchronous Quantized Control for Markov Switching Systems …

where

245

q(u(k)) = [q1 (u 1 (k)), q2 (u 2 (k)), . . . , qt (u t (k))] ,

with u s (k), s ∈ {1, 2, . . . , t} is the sth component of u(k), and q(−u(k)) = −q(u(k)). ∀qs (u s (k)) (s ∈ {1, 2, . . . , t}), the logarithmic quantizer has the following quantization levels Ts = {±ζsl : ζsl = ρsl ζs0 , l = ±1, ±2, . . .} ∪ {0}, where ρs ∈ (0, 1) and ζs0 > 0. To be continued, the quantizer qsψ(k) (u s (k)) is expressed as below ⎧ ζsl ⎨ ζsl , < u s (k) < 1+δs qs (u s (k)) = 0, u (k) =0 s ⎩ −qs (−u s (k)), u s (k) < 0 where δs =

ζsl 1−δs

1 − ρs . 1 + ρs

Similar to [15], one has qs (u s (k)) = (I + s (k))u s (k), where | s (k) |≤ ∇s . Defining

(11)

(k) = diag{ 1 (k), 2 (k), . . . , t (k)}, ∇ = diag{∇1 , ∇2 , . . . , ∇t }.

Equation (11) indicates that q(u s (k)) = (I + (k))u(k).

(12)

The relationship between (k) and ∇ is achieved as ( (k)∇ −1 ) ( (k)∇ −1 ) ≤ I.

(13)

Letting ϕ(k) = a, φ(k) = c, and ψ(k) = d, the closed-loop MSS is derived by combining original system (1), asynchronous controller (8) and quantized controller (12): x(k + 1) = [Aa + B1a (I + Δ(k))K c (α0 (k) + Dlk )]x(k) − B1a (I + Δ(k))K c De(lk ) − B1a (I + Δ(k))K c (α0 (k) + Dlk )e(k) + D1a ω(k), z(k) = [Ca + B2a (I + Δ(k))K c (α0 (k) + Dlk )]x(k) − B2a (I + Δ(k))K c De(lk ) − B2a (I + Δ(k))K c (α0 (k) + Dlk )e(k) + D2a ω(k)

(14)

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By taking αs (k) (s = 0, 1, . . . , lk ) into consideration, one has x(k + 1) = F1 η(k) + (α0 (k) − α0 + Dlk − Dlk )F2 η(k) − (D − D)F3 η(k), z(k) = F4 η(k) + (α0 (k) − α0 + Dlk − Dlk )F5 η(k)

(15)

− (D − D)F6 η(k), where F1 = [Aa + B1a (I + Δ(k))K c (α0 + Dlk ) D1a − B1a (I + Δ(k))K c (α0 + Dlk ) B1a (I + Δ(k))K c B1a (I + Δ(k))K c · · · B1a (I + Δ(k))K c ], F2 = [B1a (I + Δ(k))K c 0 B1a (I + Δ(k))K c 0], F3 = [0 0 0 B1a (I + Δ(k))K c , B1a (I + Δ(k))K c · · · B1a (I + Δ(k))K c ], F4 = [Ca + B2a (I + Δ(k))K c (α0 + Dlk ) D2a − B2a (I + Δ(k))K c (α0 + Dlk ) B2a (I + Δ(k))K c B2a (I + Δ(k))K c · · · B2a (I + Δ(k))K c ], F5 = [B2a (I + Δ(k))K c 0 B2a (I + Δ(k))K c 0], F6 = [0 0 0 B2a (I + Δ(k))K c , B2a (I + Δ(k))K c · · · B2a (I + Δ(k))K c ],   lk lk   (αt (k) − αt ), (αt (k) − αt ), . . . , (αlk (k) − αlk ) , D − D = diag 0, 0, 0, 



t=1 

t=2 

η(k) = [x (k) ω (k) e (k) e (lk )]. Definition 1 ([12]) The MSS (15) is stochastically stable (SS) when ω(k) = 0, that is ∞   2  x(k)  < ∞. E k=0

The purpose of this chapter is designing asynchronous controller such that closedloop MSS (15) is SS and meeting the following conditions. (1) The MSS (15) is SS with ω(k) = 0. (2) ∀ω(k) = 0 and zero initial condition, the MSS (15) achieves an H∞ performance index γ , such that ∞  k=0

z  (k)z(k) < γ 2

∞  k=0

ω (k)ω(k).

Asynchronous Quantized Control for Markov Switching Systems …

247

3 Main Results In the following section, the H∞ performance analysis of MSS (15) will be presented and the HMM based asynchronous control strategy will be expressed. Theorem 1 Given scalars γ > 0, the MSS (15) is called SS and achieves the prescribed H∞ performance γ , if there exists matrices Pa > 0, Wac > 0, Q > 0, such that for ∀a, b ∈ M and c ∈ N , N 

rac Wac − Pa < 0,

(16)

⎤ Ξa Ψ1a Ψ2a ⎣ ∗ Ψ3a 0 ⎦ < 0, ∗ ∗ Ψ4

(17)

c=1

⎡

and

where Xa =

M 

πab Pb , b=1 Ξa = lk χ3 Qχ3 −

χ4 Qχ4 − χ3 Ω1a χ3 − χ1 Wac χ1

Ψ1a

+ (χ1 − χ3 ) Ω2a (χ1 − χ3 ) − γ 2 χ2 χ2 ,

⎡ ⎤  lk   ˆ  F7 ⎦ , = ⎣F1 1 + α 20 + α 2t F2 (F3 D)

Ψ2a

⎤  lk   ˆ  F8 ⎦ , = ⎣F4 1 + α 20 + α 2t F5 (F6 D)

t=1

⎡

t=1

  , F7 = α 1 Φ1(1) α 2 Φ2(1) · · · αlk Φl(1) k   F8 = α 1 Φ1(2) α 2 Φ2(2) · · · αlk Φl(2) , k ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ −1 −1 −1 −1 −1 Ψ3a = diag −X a , −X a , −X a , −X a , . . . , −X a , ⎪   ⎪ ⎩ ⎭ lk ⎧ ⎫ ⎨ ⎬ Ψ4 = diag −I, −I, −I, −I, . . . , −I ,   ⎭ ⎩  Φt(1)

= 0, 0, 0,

lk t  s=1



Υs ⊗ B1a (I + (k))K c ,

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 Φt(2)

= 0, 0, 0,

t 

 Υs ⊗ B2a (I + (k))K c ,

s=1

Υs = [01×(s−1),1,01×(lk −s) ], (s = 1, 2, . . . , lk ),   lk lk   2 2 2 αt , α t , . . . , αl , Dˆ = diag 0, 0, 0, k

t=1

t=2

Q = diag{Q, Q, . . . , Q}. Proof Utilizing the Schur complement to (17), such that 

F 1 X a F 1 + (1 + α 20 + + + +

χ 3

lk 

α 2t

t=1

 t 



α 2t )F 2 X a F 2







t 

Υs ⊗ K c Xa Υs ⊗ K c χ 3 t=1 s=1 s=1    ˆ  DF 3 X a F 3 + lk χ 2 Qχ 2 − χ 3 Qχ 3 − χ 2 Ω1a χ 2 (χ 1 − χ 2 ) Ω2a (χ 1 − χ 2 ) − χ  1 Wac χ 1 < 0,

and F1 X a F1 + (1 + α 20 + +

lk 

χ4

lk 

α 2t

t=1

 t 

lk  t=1

(18)

α 2t )F2 X a F2 

Υs ⊗ K c

Xa

 t 

s=1

 Υ s ⊗ K c χ4

s=1

ˆ 3 X a F3 + lk χ3 Qχ3 − χ4 Qχ4 − χ3 Ω1a χ3 + DF + (χ1 − χ3 ) Ω2a (χ1 − χ3 ) + F4 F4 + (1 + α 20 + +

χ4

lk  t=1

α 2t

 t 

lk  t=1

Υs ⊗ K c

(19)

α 2t )F5 F5   t 

s=1

 Υ s ⊗ K c χ4

s=1

ˆ 6 F6 − γ 2 χ2 χ2 − χ1 Wac χ1 < 0. + DF According to the condition that that

N

c=1 rac Wac

< Pa in (16), it is clear to observe

Asynchronous Quantized Control for Markov Switching Systems … N 

 

rac F 1 X a F 1 + (1 + α 20 +

c=1

+

χ 3

lk 

α 2t

t=1

 t 

lk 

249 

α 2t )F 2 X a F 2

t=1





Υs ⊗ K c

Xa

t 

s=1

 Υs ⊗ K c χ 3

s=1

   ˆ  + DF 3 X a F 3 + lk χ 2 Qχ 2 − χ 3 Qχ 3 − χ 2 Ω1a χ 2  +(χ 1 − χ 2 ) Ω2a (χ 1 − χ 2 ) − χ  1 Pa χ 1 < 0,

and

N 

 rac

F1 X a F1

+ (1 +

α 20

+

c=1

+

lk 

(20)

α 2t )F2 X a F2

t=1

χ4

lk 

α 2t

t=1

 t 

 Υs ⊗ K c

Xa

 t 

s=1

 Υ s ⊗ K c χ4

s=1

ˆ 3 X a F3 + lk χ3 Qχ3 − χ4 Qχ4 − χ3 Ω1a χ3 + DF + (χ1 − χ3 ) Ω2a (χ1 − χ3 ) + F4 F4 + (1 + α 20 + +

χ4

lk  t=1

α 2t

 t 

lk  t=1

Υs ⊗ K c

(21)

α 2t )F5 F5   t 

s=1

 Υ s ⊗ K c χ4

s=1

 ˆ 6 F6 − γ 2 χ2 χ2 − χ1 Pa χ1 < 0. + DF We select the following mode-dependent Lyapunov functional for MSS (15) V (k, xk , ϕk ) = V1 (k, xk , ϕk ) + V2 (k, xk , ϕk ), where

(22)

V1 (k, xk , ϕk ) = x  (k)Pϕ(k) x(k), V2 (k, xk , ϕk ) =

lk k−1  

e (q)e(q).

p=1 q=k− p

Define E {ΔV (k)} = E {V (k + 1, xk+1 , ϕk+1 ) | −V (k, xk , ϕk )}. Here, denote ϕ(k) = a, ϕ(k + 1) = b, φ(k) = c, we get

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E {ΔV1 (k)} = x  (k + 1)Pϕ(k+1) x(k + 1) − x  (k)Pϕ(k) x(k) =

N 

rac η (k) f 1 X a F1 η(k) − x  (k)Pϕ(k) x(k)

c=1

+

N 

 rac E (α0 (k) − α0 + Dlk − Dlk )2 η (k) f 2 X a F2 η(k)

c=1

+

N 

 rac He{E (Dlk − Dlk ) }η (k)F2 X a F3 η(k)

c=1

+

N 

 rac E (D − D)2 η (k)F3 X a F3 η(k)

c=1

=

N 

rac η (k)F1 X a F1 η(k) − x  (k)Pϕ(k) x(k)

(23)

c=1

+

N 

 rac (E (α0 (k) − α0 )2 η (k)F2 X a F2 η(k)

c=1

+

N 

 rac E (Dlk − Dlk )2 )η (k)F2 X a F2 η(k)

c=1

+

N 

 rac He{E (Dlk − Dlk ) }η (k)F2 X a F3 η(k)

c=1

+

N 

 rac E (D − D)2 η (k)F3 X a F3 η(k).

c=1

Then, it is easy to achieve that E {ΔV1 (k)} =

N 

rac η (k)F1 X a F1 η(k)

c=1

+

N 

rac (α 20

+

c=1

+2

c=1

+

α 2t )η (k)F2 X a F2 η(k)

t=1

N 

N 

lk 

rac

lk 

α 2t η (k)F2 X a F3 η(k)

t=1

ˆ  (k)F3 X a F3 η(k) − x  (k)Pϕ(k) x(k). rac Dη

c=1

By using Jensen’s inequality to (24), it yields that

(24)

Asynchronous Quantized Control for Markov Switching Systems …

251

E {ΔV1 (k)} = x  (k + 1)Pϕ(k+1) x(k + 1) − x  (k)Pϕ(k) x(k) ≤

N 

rac η (k)F1 X a F1 η(k) − x  (k)Pϕ(k) x(k)

c=1

+

N 

rac (1 + α 20 +

c=1

+

× Xa

α 2t )η (k)F2 X a F2 η(k)

t=1

N  c=1

lk 

lk 



rac e (lk )

 t 

α 2t

 t 

t=1

 Υs ⊗ B1a (I + (k) K c )

s=1

(25)



Υs ⊗ B1a (I + (k))K c e(lk )

s=1

+

N 

ˆ  (k)F3 X a F3 η(k). rac Dη

c=1

On the other hand, it follows from (22) that E {ΔV2 (k)} = E {lk e (k)Qe(k) −

lk 

e (k − p)Qe(k − p)}

p=1

(26)

= E {lk e (k)Qe(k) − e (lk )Qe(lk )}. Recalling (4) and using the term x(ks ) = x(k) − e(k), we have 0 ≤ −e (k)Ω1a e(k) + (x(k) − e(k)) Ω2a (x(k) − e(k)).

(27)

Firstly, when ω(k) = 0, one has E {ΔV (k)} =

N 



rac η (k)F 1 X a F 1 η(k) − η (k)χ1 Pa χ1 η(k)

c=1

+

N 

rac (1 + α 20 +

c=1

+

N 

rac η (k)χ3

+

lk  t=1



α 2t )η (k)F 2 X a F 2 η(k)

t=1

c=1 N 

lk 

α 2t

 t 

 Υs ⊗ K c

s=1



Xa

 t 

 Υs ⊗ K c χ3 η(k)

s=1

ˆ  (k)F 3 X a F 3 η(k) + lk η (k)χ1 Qχ1 η(k) rac Dη

c=1

− η χ3 Qχ3 η(k) − η (k)χ2 Ω1a χ2 η(k) + η (k)(χ1 − χ2 ) Ω2a (χ1 − χ2 )η(k) = η (k)Ξ a η(k),

(28)

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where Ξa =

N 



rac F 1 X a F 1 − χ1 Pϕ(k) χ1

c=1

+

N  c=1

+

N 

N 



α 2t )F 2 X a F 2

t=1

rac χ3

c=1

+

lk 

rac (1 + α 20 + lk 

α 2t

 t 

t=1

 Υs ⊗ K c

Xa

 t 

s=1

 Υ s ⊗ K c χ3

s=1

   ˆ  rac DF 3 X a F 3 + lk χ1 Qχ1 − χ3 Qχ3 − χ2 Ω1a χ2

c=1

+ (χ1 − χ2 ) Ω2a (χ1 − χ2 ), F 1 = [Aa + B1a (I + Δ(k))K c (α0 + Dlk ) − B1a (I + Δ(k))K c (α0 + Dlk ) B1a (I + Δ(k))K c B1a (I + Δ(k))K c · · · B1a (I + Δ(k))K c ], F 2 = [B1a (I + Δ(k))K c (α0 (k) − α0 + Dlk − Dlk ) B1a (I + Δ(k))K c (α0 (k) − α0 + Dlk − Dlk ) 0], F 3 = [0 0 B1a (I + Δ(k))K c B1a (I + Δ(k))K c · · · B1a (I + Δ(k))K c ], η(k) = [x  (k) e (k) e (lk )], χ 1 = [In×n , 0n×(1+lk )n ], χ 2 = [0n×n , In×n , 0n×(lk )n ], χ 3 = [0n×2n , In×n , . . . , In×n ]. From (22), we get E {ΔV (k)} < − inf{λmin (−Ξ a )}x  (k)x(k) < 0.

(29)

where β = inf{λmin (−Ξ a )}. From k = 0 to ∞, summing up the both sides of (29), one can derive that E

∞  k=0

 

 x (k) 

2


0, the MSS (15) is called SS and achieves the pre!a > 0, W !ac > 0, Q ! > 0, and scribed H∞ performance γ , if there exists matrices P matrices Yc , X , such that for ∀a, b ∈ M and c ∈ N , N 

!a < 0, !ac − P rac W

(36)

c=1

⎡

⎤ !a Π (1) Π (2) Ξ ⎣ ∗ Π (3) 0 ⎦ < 0, ∗ ∗ Π (3)

and

where ⎡

!a Ξ ⎣ Πa = ∗ ∗

!1a Ψ !3a Ψ ∗

⎤ !2a Ψ 0 ⎦ , Π (1) = [Λ1c ∇ Λ2c ∇ Λ3c ∇], Ψ4

Π (2) = [ε1 Θ1a ε2 Θ2a ε3 Θ1a ], Π (3) = diag{−ε1 I, −ε2 I, −ε3 I },

⎤ ⎡  lk   ! Z a (F ! Z a ⎦ , ! Z a 1 + α 2 + !3 D) ˆ  Za F !1a = ⎣F Ψ α 2t F 0 1 2 7 t=1

(37)

Asynchronous Quantized Control for Markov Switching Systems …

255

⎤  lk   ! (F ! ⎦ , ! 1 + α 2 + !6 D) ˆ  F !2a = ⎣F Ψ α 2t F 4 5 8 0 ⎡

t=1

!1 = [Aa X + B1a Yc (α0 + Dlk ) D1a − B1a Yc (α0 + Dlk ) F B1a Yc B1a Yc · · · B1a Yc ], !2 = [B1a Yc 0 B1a Yc 0], F !3 = [0 0 0 B1a Yc X, B1a Yc · · · B1a Yc ], F !4 = [Ca X + B2a Yc (α0 + Dlk ) D2a − B2a Yc (α0 + Dlk ) F B2a Yc B2a Yc · · · B2a Yc ], !5 = [B2a Yc 0 B2a Yc 0], F !6 = [0 0 0 B2a Yc , B2a Yc · · · B2a Yc ], F   !7 = α 1 Φ !1(1) α 2 Φ !2(1) · · · αlk Φ !l(1) , F k   (2) (2) (2) !8 = α 1 Φ !1 α 2 Φ !2 !l F , · · · α lk Φ k Λ1c = [0 0 0 Yc Yc · · · Yc 0 0 · · · 0] , Λ2c = [Yc 0 0 · · · 0] , Λ3c = [0 0 Yc 0 0 · · · 0] ,   Θ1a = [0 0 · · · 0 Ba 0 · · · 0 Ba 1a B2a 0 B2a 2a ],     Θ2a = [0 0 · · · 0 B1a · · · B1a 0 · · · 0 B2a B2a 0 · · · 0],       Θ3a = [0 0 · · · 0 − B1a · · · − B1a B1a · · · B1a 0 · · · 0 − B2a B2a 0 · · · 0],

!1(3) · · · α 1 Φ !1(3) α 2 Φ !2(3) · · · αlk Φ !l(3) ], 1a = [α 1 Φ k !1(4) α 2 Φ !2(4) · · · αlk Φ !l(4) ], 2a = [α 1 Φ    k  t t   (1) (2) ! ! Φt = 0, 0, 0, Υs ⊗ B1a Yc , Φt = 0, 0, 0, Υs ⊗ B2a Yc ,  !t(3) Φ

= 0, 0, 0,

s=1 t 



 Υs ⊗ B1a

!t(4) ,Φ

= 0, 0, 0,

s=1

s=1 t 



Υs ⊗ B2a .

s=1

Here, the control gain matrices are given by K c = Yc X −1 .

(38)

Proof The inequality (17) can be rewritten as follows: ⎤ Ξa Ψ 1a Ψ2a ⎣ ∗ Ψ 3a 0 ⎦ < 0, ∗ ∗ Ψ4 ⎡

(39)

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where

⎤  lk   ˆ  Z a F7 Z a ⎦ , = ⎣F1 Z a 1 + α 20 + α 2t F2 Z a (F3 D) ⎡

Ψ 1a

t=1

⎧ ⎨

⎫ ⎬

Ψ 3a = diag −P −1 , −P −1 , −P −1 , −P −1 , . . . , −P −1 ,   ⎭ ⎩ lk

√ √ √ Z a = [ πa1 I πa2 I · · · πa N I ], P = diag{P1 , P2 , . . . , PN }. For each a, b ∈ M and c ∈ N , we denote

!a X −1 , Wac = X − W !ac X −1 , Pa = X − P ! X −1 , Ωla = X − Ω !la X −1 , l = 1, 2. Q = X − Q Then, one has

!a − He(X ). −Pa−1 = P

(40)

(41)

According to (40) and (41), (39) can be rewritten as ⎤ !a Ψ 1a Ψ2a Ξ ⎣ ∗ Ψ !3a 0 ⎦ < 0, ∗ ∗ Ψ4 ⎡

where

(42)

⎧ ⎨

⎫ ⎬ " P, " P, " P, " ...,P " , !3a = diag P, Ψ   ⎭ ⎩ lk

" = diag{ P !2 − He(X ), . . . , P !N − He(X )}. !1 − He(X ), P P Pre- and post-multiple (42) by {X, I, X, X, . . . , X , I, . . . , I }, then, (42) can be    lk

rewritten as

where

⎡

!a Ξ ⎣ ∗ ∗

Ψ` 1a !3a Ψ ∗

⎤

Ψ` 2a 0 ⎦ < 0, Ψ4

! 3 − χ4 ! !ac χ1 !1a χ3 − χ1 W !a = lk χ3 Qχ Ξ Qχ4 − χ3 Ω !2a (χ1 − χ3 ) − γ 2 χ2 χ2 , + (χ1 − χ3 ) Ω

(43)

Asynchronous Quantized Control for Markov Switching Systems …

Ψ` 1a

⎤  lk   ˆ  Z a F`7 Z a ⎦ , = ⎣F`1 Z a 1 + α 20 + α 2t F`2 Z a (F`3 D)

Ψ` 2a

⎤  lk   ˆ  F`8 ⎦ , = ⎣F`4 1 + α 20 + α 2t F`5 (F`6 D)

257

⎡

t=1

⎡

t=1

F`1 = [Aa X + B1a (I + Δ(k))K c X (α0 + Dlk ) D1a − B1a (I + Δ(k))K c X (α0 + Dlk ) B1a (I + Δ(k))K c X B1a (I + Δ(k))K c X · · · B1a (I + Δ(k))K c X ], F`2 = [B1a (I + Δ(k))K c X 0 B1a (I + Δ(k))K c X 0], F`3 = [0 0 0 B1a (I + Δ(k))K c X, B1a (I + Δ(k))K c X · · · B1a (I + Δ(k))K c X ], F`4 = [Ca X + B2a (I + Δ(k))K c X (α0 + Dlk ) D2a − B2a (I + Δ(k))K c X (α0 + Dlk ) B2a (I + Δ(k))K c X B2a (I + Δ(k))K c X · · · B2a (I + Δ(k))K c X ], F`5 = [B2a (I + Δ(k))K c X 0 B2a (I + Δ(k))K c X 0], F`6 = [0 0 0 B2a (I + Δ(k))K c X, B2a (I + Δ(k))K c X · · · B2a (I + Δ(k))K c X ]. Adding (13) to (43), one has ⎡

!a Ξ ⎣ ∗ ∗

!1a Ψ !3a Ψ ∗

⎤ !2a 3 Ψ  ⎦ He(Λsc (k)Θsa ) < 0, + 0 s=1 Ψ4

Applying Schur complement to (44), one can obtain (37).

(44) 

4 Numerical Example A single link robot arm model [39], which is described by the differential equation as below x˙1 (t) = x2 (t) R gl M(ϕ(t)) R sin(x1 (t)) − − x2 (t) x˙2 (t) = − J (ϕ(t)) J (ϕ(t)) J (ϕ(t)) 1 + u(t) + ω(t) J (ϕ(t))

(45)

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Table 1 The meaning of g, R, l, M(ϕ(t)) and J (ϕ(t)) Description g R l M(ϕ(t)) J (ϕ(t))

The gravity acceleration The coefficient of the viscous friction The arm length The payload mass The inertia moment

Table 2 The values of M(ϕ(t)) and J (ϕ(t)) ϕ(t) 1 M(ϕ(t)) J (ϕ(t))

1 1

2

3

5 5

10 10

where the description of g, R, l, M(ϕ(t)) and J (ϕ(t)) are given in Table 1. Accordingly, the values of M(ϕ(t)) and J (ϕ(t)) are given in Table 2. The TPM is assumed to be ⎡ ⎤ 0.3 0.4 0.3 Π = ⎣ 0.4 0.2 0.4 ⎦ . 0.5 0.3 0.2 Applying the first-order Euler approximation method to discretize the MSS (1) with T = 0.1. Then, the parameters of MSS (1) is given as follows #

T

$

, M(ϕ(k)) TR − T glJ (ϕ(k)) 1 − J (ϕ(k)) $ # # $ 0 0 , D = = , T 1ϕ(k) T J (ϕ(k))

Aϕ(k) = B1ϕ(k)

1

Cϕ(k) = [1 0], B2ϕ(k) = 0, D2ϕ(k) = 0.1. On the other hand, the conditional probability matrix of Makov jump φ(k), is shown as ⎡ ⎤ 0.2 0.55 0.25 Π = ⎣ 0.3 0.6 0.1 ⎦ . 0.4 0.2 0.4 Additionally, one gives the fading channel lk = 1, and the probability density functions of αt (k) with the following forms: %

f (α0 (k)) = 0.0005(exp(9.89α0 (k)) − 1), 0 ≤ α0 (k) ≤ 1, f (α1 (k)) = 8.5017(exp(8.5α1 (k)) − 1), 0 ≤ α0 (k) ≤ 1.

Asynchronous Quantized Control for Markov Switching Systems … Fig. 1 System mode ϕ(k)

259

3.5

3

ϕk

2.5

2

1.5

1

0.5

20

40

60

80

100

120

140

k

Thus, it is easy to obtain that α0 = 0.8989, α1 = 0.1174, α 20 = 0.0105 and α 20 = 0.5870. For given initial condition x(0) = [0.2π − 0.2] , and the external disturbance % ω(k) =

exp(−0.1(k − 15)) sin(0.3(k − 15)), 1 ≤ k ≤ 15, 0, 16 ≤ k ≤ 80.

Fig. 2 Control mode φ(k)

3.5

3

φk

2.5

2

1.5

1

0.5

20

40

60

k

80

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Fig. 3 Triggered instants and release intervals

5 4.5

Release intervals

4 3.5 3 2.5 2 1.5 1 0.5 0

150

100

50

0

Time(k)

Fig. 4 State response

x1 (k) x2 (k)

1.5 1 0.5 0 0.5 1 1.5 20

40

60

k

80

100

120

140

According to Theorem 2, the network is SS under the designed control strategy. The modes evolution of the system and the controller are given in Figs. 1 and 2. By using ETM, the triggered instants and release intervals are presented in Fig. 3. With the designed controller gain, the state response and control input are expressed in Figs. 4 and 5, respectively. As shown in Figs. 4 and 5, the simulation result shows the feasibility of the theory.

Asynchronous Quantized Control for Markov Switching Systems … Fig. 5 Control input

261

4

3

u(k)

2

1

0

1

2 20

40

80

60

100

120

140

k

5 Conclusion In this chapter, by utilizing an HMM technique, the asynchronous control for Markov switching systems with channel fading, where a more general class of discretetime MSS is developed. The original systems and controller are characterized by two stochastic variables. Following this, the relationship between stochastic variables is considered. In order to reduce energy consumption, a mode-dependent event-triggered communication mechanism is employed. A simulation example is employed to demonstrate the correctness and effectiveness of the derived results. Acknowledgements This work of J. Cheng was supported by the National Natural Science Foundation of China (No. 61703150), and the Training Program for 1,000 Young and Middle-aged Cadre Teachers in Universities of Guangxi Province. Also, the work of J. H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. 2019R1A5A808029011).

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Fractional-Order Systems

Observer-Based Controller Design for Fractional-Order Neutral-Type Systems Yongxia Qu, Youggui Kao, and Cunchen Gao

Abstract This article is devoted to investigating how to design the observer-based controller for a type of fractional-order neutral-type systems (FONTSs) by applying the Lyapunov direct method. The feedback control system based on the observer is designed, and by solving the linear matrix inequality (LMI) feasible problems, the gains of the controller and the observer are given. Then, an observer-based guaranteed cost control of fractional-order systems is studied. At last, simulation examples are provided to demonstrate that our method is effective. Keywords Fractional-order systems · Neutral-type · Time-delay · Observer-based control · LMI approach · Guaranteed cost control

1 Introduction With the continuous improvement of modern industry and advanced technology, it is necessary to analyze and synthesize the related mathematical model, so as to achieve the higher performances of the systems to a greater extent. Recently, scholars have realized most of the systems are non-integer differential systems [1], especially for Y. Qu · C. Gao School of Mathematical Science, Ocean University of China, Qingdao 266100, People’s Republic of China e-mail: [email protected] C. Gao e-mail: [email protected] Y. Kao (B) Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, People’s Republic of China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_12

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dynamic systems with non-rigid motion, such as thermal systems, exible systems, and viscous systems [2–5]. Fractional-order differential equations (FODEs) are more accurate and appropriate to describe them. The widespread applications of FODEs in describing the nature of things have become the hot topic [6–16]. Stability is the basic character of all control systems, and, of course, includes fractional-order control systems. Li et al. [17, 18] discussed the generalized Mittag-Lef er (M-L) stability (asymptotic stability) by the characteristics of fractional order differential operators. Further, the literature [19] studied the M-L stability of FODEs with impulses. The authors of [20] gave the Lyapunov method to study the FODEs with uncertainties and time-delay. And there are more and more researches about the fractional-order discrete-time systems, such as the literature [21]. In practice, the time delay phenomenon [22] often occurs in the dynamical systems. It restricts the structure, nature, and performance of the systems to some extent. The neutral system [20, 23, 24] is a kind of important time-delay system, which exists in many engineering practices such as ship stability, population ecology and protein distribution in blood. Naturally, as a dynamical system, the rst control task for the neutral systems is to ensure the stability of the controlled system. However, until now, there have been few research results on the fractional-order neutral systems, and their control problems are of theoretical importance and practical signi cance [25–29]. Liu et al. [25] studied the Riemann–Liouville FONTSs, and discussed the asymptotical stability of it by using a new method to calculate the Lyapunov functions. In the literature [26], the Riemann–Liouville fractional-order neutral delay neural network is studied and its suf cient conditions for delay-independent stability are given. Also, the literatures [27–29] studied the stability of some kinds of FONTSs. In many practical systems, the feedback states are usually not directly available, so it is necessary to create a controller based on the observer for such a system. There are many reports about the design approach of the controller based on state observers. In [30], a neutral linear delay differential system is studied, and a method of constructing an observer-based controller is presented. Further, there are many other researches on the observer design, for instance, the guaranteed cost control based on the observer and the H∞ control based on the observer. In [31], the author considered a new control problem for uncertain neutral time-delay systems (UNTDSs), and considered how to design a guaranteed cost controller based on the observer. Then, in [32], the authors investigated how to create a H∞ controller based on the observer for UNTDSs. In [33], the authors focused on a class of stochastic systems, which is more complicated. They gave a sliding mode method to study how to design an observer-based H∞ controller, which is non-fragile at the same time. Then, in the literature [34], the author gave an integral sliding mode method to study the design of the H∞ controllers. However, to the best knowledge of ours, the similar reports on FONTSs are few. Base on the aforementioned concerns, how to design the observer-based controller for FONTSs by the knowledge of Lyapunov stability and fractional calculus is discussed in this chapter. Section 2 presents problem preliminaries and formulation. Then, in Sect. 3, an observer-based fractional order controller for FONTSs is designed to guarantee the closed-loop system asymptotically stable. By using the

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269

LMI technology, the gains of the observer and the controller are calculated respectively. And in this section, an observer-based guaranteed cost controller is designed for the fractional-order systems. In Sect. 4, the effectiveness of our ndings is veri ed by numerical examples. At last, the full chapter is summarized in Sect. 5. Notations: In this chapter, matrix M > N means that M − N is a positive de nite matrix. A unit matrix with appropriate dimension is represented as I. ∗ represents the symmetric term in a matrix.

2 Problem Formulation and Preliminaries In this section, we present some basic de nitions, lemmas, and basic calculation rules of fractional-order systems, such as Γ functions, Riemann–Liouville fractional derivative and integral, and some properties of them. Definition 1 ([3]) The Riemann–Liouville fractional-order integral with a > 0 for a function g(t) is de ned as 1 Γ (a)

−a 0 Dt g(t) =

where Γ (·) is the Γ function, Γ (s) =



t

(t − s)a−1 g(s)ds,

0

 +∞ 0

e−t t s−1 dt, s > 0.

Definition 2 ([3]) The Riemann–Liouville fractional-order derivative with a > 0 for a function g(t) is de ned as a 0 Dt g(t)

dn 1 = Γ (n − a) dt n

 0

t

g(s) ds, n − 1 ≤ a < n, t ≥ 0, (t − s)a+1−n

n is a positive integer. And when 0 < a < 1, the upper form changed to a 0 Dt g(t) =

Property 1

a 0 Dt C

=

C(t−t0 )−a Γ (1−a)

d 1 Γ (1 − a) dt

 0

t

g(s) ds, t ≥ 0. (t − s)a

holds, where C is a constant.

Property 2 When n − 1 ≤ a < n, b > 0, first calculate the integrals and then derivative, and the following equality a −b 0 Dt 0 Dt g(t)

=0 Da−b g(t) t

holds for function g(t) with good property. In particular, if g(t) is integrable, the upper relation still holds.

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Now, we consider the following FONTS: ⎧ α α ⎪ ⎨ 0 Dt x(t) = A0 x(t) + A1 x(t − ω) + B0 Dt x(t − ω) + Gu(t), x(t) = ϕ(t), t ∈ [−ω, 0], ⎪ ⎩ y(t) = Cx(t) + Du(t),

t ≥0 (1)

where 0 < α < 1, x(t) ∈ Rn is the state vector, y(t) ∈ Rq is the output vector, u(t) ∈ Rm is the control input vector, ϕ(t) ∈ C([−ω, 0], Rn ) is the initial function, and A0 , A1 , B ∈ Rn×n , C ∈ Rq×n , G ∈ Rn×m , D ∈ Rq×m are known constant real-valued matrices. Lemma 1 (Schur Complement lemma) ([35]) For a given symmetric matrix 

M11 M12 M=M = MT12 M22



T

(2)

with M11 ∈ Rr ×r , the following conditions are equivalent: (1) M < 0, (2) M11 < 0, M22 − MT12 M−1 11 M12 < 0, T (3) M22 < 0, M11 − M12 M−1 22 M12 < 0. The observer-based control of system (1) is designed by ⎧ α ˆ (t) = A0 xˆ (t) + A1 xˆ (t − ω) + B0 Dαt xˆ (t − ω) 0 Dt x ⎪ ⎪ ⎪ ⎨ + Gu(t) + L[y(t) − yˆ (t)], t ≥ 0, ⎪ ˆ xˆ (t) = ϕ(t), t ∈ [−ω, 0], ⎪ ⎪ ⎩ yˆ (t) = C xˆ (t) + Du(t),

(3)

u(t) = −K xˆ (t)

(4)

and where xˆ (t) ∈ Rn is the estimation of the state vector x(t), yˆ (t) ∈ Rq is the observer output, L ∈ Rn×q is the gain of the observer, K ∈ Rm×n is the gain of the controller, ˆ and ϕ(t) ∈ C([−ω, 0], Rn ) is the initial function of xˆ (t). Here, the vector e(t) = x(t) − xˆ (t) is the error of the estimated system. Lemma 2 ([3]) Let x(t) ∈ Rn be a derivable and continuous function, then, the following inequalities holds 1 T α T α 0 Dt (x (t)Px(t)) ≤ x (t)P0 Dt x(t) ∀α ∈ (0, 1), ∀t ≥ 0 2 where P ∈ Rn×n is a symmetric positive definite matrix. By (2)–(3), (1) and (2) can be sorted and written as

Observer-Based Controller Design for Fractional-Order Neutral-Type Systems



α ˆ (t) 0 Dt x α 0 Dt e(t)

271

= (A0 − GK)ˆx(t) + A1 xˆ (t − ω) + B0 Dαt xˆ (t − ω) + LCe(t), t ≥ 0 = (A0 − LC)e(t) + A1 e(t − ω) + B0 Dαt e(t − ω), t ≥ 0. (5)

Then, write a matrix as    ˆ (t) xˆ (t) LC A0 − GK α α x = D Z(t) = D 0 t 0 t e(t) 0 A0 − LC e(t)    α  ˆ (t − ω) A1 0 xˆ (t − ω) B0 0 0 Dt x + . + α 0 A1 0 B0 e(t − ω) 0 Dt e(t − ω)

(6)

The problem considered here is to nd the controller based on the observer (2)–(3) to ensure the system (4) above asymptotically stable, then calculate the gains of the controller and the observer.

3 Main Results In this section, we study the design of the controller based on the observer for FONTSs in this section. First, we give the main theorem as follows: Theorem 1 The system (1) with (2)–(3) is asymptotically stable if there exist some ¯ i ∈ Rn×n , J¯ i ∈ Rn×n , (i = 1, 2, 3) and matrisymmetric positive definite matrices H n×q m×n ˆ q×q , R1 ∈ R , such that ces X ∈ R , Y ∈ R ⎡

Ω11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Ω12 Ω13 Ω22 0 ¯2 ∗ −H ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 Ω15 Ω24 0 0 0 ¯ −J2 0 ¯3 ∗ −H ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 Ω26 0 0 0 −J¯ 3 ∗ ∗ ∗ ∗

Ω17 Ω27 Ω37 0 Ω57 0 ¯3 −H ∗ ∗ ∗

¯1 0 H Ω28 0 0 0 Ω48 0 0 0 Ω68 0 0 0 −J¯ 3 0 ¯2 ∗ −H ∗ ∗

ˆ 1 C, CJ¯ 1 = R where ¯ 1 AT0 + A0 H ¯ 1 − YT GT − GY, Ω11 = H ¯ 2, Ω12 = XC, Ω13 = A1 H

¯ 1 AT0 − YT GT , ¯ 3 , Ω17 = H Ω15 = B0 H Ω22 = J¯ 1 AT0 + A0 J¯ 1 − CT XT − XC,

⎤ 0 J¯ 1 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −J¯ 2

(7)

(8)

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Ω24 = A1 J¯ 2 , Ω26 = B0 J¯ 3 , Ω27 = CT XT , ¯ 2 AT1 , Ω28 = J¯ 1 AT0 − CT XT , Ω37 = H

¯ 3 BT0 , Ω68 = J¯ 3 BT0 . Ω48 = J¯ 2 AT1 , Ω57 = H

The observer gain and the control gain are given by −1

¯1 K = YH and

−1 L = XRˆ 1

¯ i−1 , Ji = J¯ i−1 , i = 1, 2, 3. respectively, where Hi = H Proof De ning the Lyapunov function    t H1 0 H2 0 T T V(Z(t)) = 0 Dα−1 Z(t) + Z(s)ds Z (t) Z (s) t 0 J1 0 J2 t−ω   t H3 0 α T α + D Z (s) 0 t 0 Dt Z(s)ds. 0 J 3 t−ω

(9)

From Property 2 and Lemma 2, the integer derivative of V(Z(t)) with respect to t along the trajectories of (4) is calculated as   H1 0 H2 0 ˙ V(Z(t)) = 0 Dαt ZT (t) Z(t) + ZT (t) Z(t) 0 J1 0 J2  H2 0 Z(t − ω) − ZT (t − ω) 0 J2  H3 0 + 0 Dαt ZT (t) Dα Z(t) 0 J3 0 t  H3 0 α T Dα Z(t − ω) − 0 Dt Z (t − ω) 0 J3 0 t T   LC H1 0 A0 − GK ≤ ZT (t) 0 A0 − LC 0 J1     LC A0 − GK H1 0 H2 0 Z(t) + + 0 J1 0 A0 − LC 0 J2   A1 0 H1 0 + 2ZT (t) Z(t − ω) 0 J1 0 A1   B0 0 H1 0 + 2ZT (t) Dα Z(t − ω) 0 J1 0 B0 0 t  H3 0 Dα Z(t) + 0 Dαt ZT (t) 0 J3 0 t

(10)

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H2 0 Z(t − ω) 0 J2  H3 0 − 0 Dαt ZT (t − ω) Dα Z(t − ω) 0 J3 0 t 

− ZT (t − ω)

= X T (t)ΦX(t) ⎡

⎤ Z(t) X(t) = ⎣ Z(t − ω) ⎦ , α 0 Dt Z(t − ω)

where

⎡

Φ11 ⎢ ∗ ⎢ ⎢ ∗ Φ=⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Φ12 Φ13 Φ22 0 ∗ −H 2 ∗ ∗ ∗ ∗ ∗ ∗

0 Φ15 Φ24 0 0 0 −J 2 0 ∗ −H 3 ∗ ∗

⎤ 0 Φ26 ⎥ ⎥  0 ⎥ ⎥ + M T H 3 0 M, 0 ⎥ 0 J3 ⎥ 0 ⎦ −J 3

LC A1 0 B0 0 A0 − GK , M= 0 A0 − LC 0 A1 0 B0 

and

with Φ11 = AT0 H 1 + H 1 A0 − K T GT H 1 − H 1 GK + H 2 , Φ12 = H 1 LC, Φ13 = H 1 A1 , Φ15 = H 1 B0 , Φ22 = AT0 J 1 + J 1 A0 − C T LT J 1 − J 1 LC + J 2 , Φ24 = J 1 A1 , Φ26 = J 1 B0 . De ne a real symmetric matrix ⎡

Σ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ Σ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Σ12 Σ13 0 Σ15 0 Σ22 0 Σ24 0 Σ26 ∗ −H¯ 2 0 0 0 ¯ ∗ ∗ −J 2 0 0 ∗ ∗ ∗ −H¯ 3 0 ∗ ∗ ∗ ∗ −J¯ 3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Σ17 Σ27 Σ37 0 Σ57 0 ¯3 −H ∗ ∗ ∗

¯1 0 H Σ28 0 0 0 Σ48 0 0 0 Σ68 0 0 0 −J¯ 3 0 ¯2 ∗ −H ∗ ∗

⎤ 0 J¯ 1 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −J¯ 2

(11)

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where ¯ 1 AT0 + A0 H ¯ 1 − Y T GT − GY, Σ11 = H ¯ 2, Σ12 = XC, Σ13 = A1 H ¯ 1 AT0 − Y T GT , ¯ 3 , Σ17 = H Σ15 = B0 H

Σ22 = J¯ 1 AT0 + A0 J¯ 1 − C T X T − XC, Σ24 = A1 J¯ 2 , Σ26 = B0 J¯ 3 , Σ27 = C T X T ,

¯ 2 AT1 , Σ28 = J¯ 1 AT0 − C T X T , Σ37 = H Σ48 = J¯ 2 AT1 , Σ57 = J¯ 3 BT0 , Σ68 = J¯ 3 BT0 . −1

−1

¯ 2 . Then, the LMI condition (6) equals to (10) ¯ 1 , LC = XC H Here, let K = Y H < 0, then it implies the following result: ⎡

¯ 1H ¯ −1 ¯ Σ11 + H Σ12 Σ13 2 H1 ⎢ −1 ⎢ ∗ Σ22 + J¯ 1 J¯ 2 J¯ 1 0 ⎢ ¯2 ⎢ ∗ ∗ −H N=⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗  −1 ¯3 0 H + N T1 N1 0 J¯ 3

0 Σ15 Σ24 0 0 0 −J¯ 2 0 ¯3 ∗ −H ∗ ∗

⎤ 0 ⎥ Σ26 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −J¯ 3

(12)

0, li represents a constant external input, h i (·) (i = 1, 2, . . . , m) and δ > 0 represent nonlinear activation functions and time delay, respectively. Memristive connection weights wi j (u j (t)) and vi j (u j (t)) have the presentations like  ∗  ∗ wi j , |u j (t)| > X j , vi j , |u j (t)| > X j , wi j (u j (t)) = (u (t)) = v wi∗∗j , |u j (t)| < X j , i j j vi∗∗j , |u j (t)| < X j , wi j (±X j ) = wi∗j or wi∗∗j , vi j (±X j ) = vi∗j or vi∗∗j , for i, j = 1, 2, . . . , m. Switching jumps X j > 0, weights wi∗j , wi∗∗j , vi∗j and vi∗∗j are all invariant scalars. The slave system is built as follows:

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D q u˜ i (t) = −ai u˜ i (t) +

m 

wi j (u˜ j (t))h j (u˜ j (t))

j=1

+

m 

vi j (u˜ j (t))h j (u˜ j (t − δ)) + li + ri (t),

(2)

j=1

where ri (t) is a controller , i = 1, 2, . . . , m, t ≥ 0. Since the discontinuity of wi j (u j (t)) and vi j (u j (t)), all solutions of master-slave systems are Filippov’s ones. With the aid of the theories of differential inclusions and set-valued maps [51], from (1) and (2), we get D q u i (t) ∈ −ai u i (t) +

m 

co[wi j (u j (t))]h j (u j (t))

j=1

+

m 

co[vi j (u j (t))]h j (u j (t − δ)) + li ,

(3)

j=1

and D q u˜ i (t) ∈ −ai u˜ i (t) +

m 

co[wi j (u˜ j (t))]h j (u˜ j (t))

j=1

+

m 

co[vi j (u˜ j (t))]h j (u˜ j (t − δ)) + li + ri (t),

j=1

for a.e. t ≥ 0, i = 1, 2, . . . , m, where 0 < q < 1, ⎧ ∗ |u j (t)| > X j , ⎨ wi j , co[wi j (u j (t))] = co{wi∗j , wi∗∗j }, |u j (t)| = X j , ⎩ ∗∗ wi j , |u j (t)| < X j . ⎧ ∗ |u j (t)| > X j , ⎨ vi j , co[vi j (u j (t))] = co{vi∗j , vi∗∗j }, |u j (t)| = X j , ⎩ ∗∗ vi j , |u j (t)| < X j . ⎧ ∗ |u˜ j (t)| > X j , ⎨ wi j , co[wi j (u˜ j (t))] = co{wi∗j , wi∗∗j }, |u˜ j (t)| = X j , ⎩ ∗∗ wi j , |u˜ j (t)| < X j . ⎧ ∗ |u˜ j (t)| > X j , ⎨ vi j , ∗ ∗∗ co{v , v }, |u˜ j (t)| = X j , co[vi j (u˜ j (t))] = ⎩ ∗∗ i j i j vi j , |u˜ j (t)| < X j ,

(4)

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with co{wi∗j , wi∗∗j } = [wi j , wi j ], co{vi∗j , vi∗∗j } = [v i j , v i j ], wi j = min{wi∗j , wi∗∗j }, wi j = max{wi∗j , wi∗∗j }, v i j = min{vi∗j , vi∗∗j }, v i j = max{vi∗j , vi∗∗j }. Or, there exist ξi j (u j (t)) ∈ co[wi j (u j (t))], ηi j (u˜ j (t)) ∈ co[wi j (u˜ j (t))], ζi j (u j (t)) ∈ co[vi j (u j (t))] and χi j (u˜ j (t)) ∈ co[vi j (u˜ j (t))] satisfying D q u i (t) = −ai u i (t) +

m 

ξi j (u j (t))h j (u j (t))

j=1

+

m 

ζi j (u j (t))h j (u j (t − δ)) + li ,

(5)

j=1

and D q u˜ i (t) = −ai u˜ i (t) +

m 

ηi j (u˜ j (t))h j (u˜ j (t))

j=1

+

m 

χi j (u˜ j (t))h j (u˜ j (t − δ)) + li + ri (t),

(6)

j=1

for a.e. t ≥ 0, i = 1, 2, . . . , m. It is assumed that the initial conditions of master-slave systems (1) and (2) are u i (s) = φi (s) and u˜ i (s) = ψi (s), respectively, for s ∈ [−δ, 0], φi (s), ψi (s) ∈ C([−δ, 0], R), i = 1, 2, . . . , m. Definition 3 ([52]) u(t) = (u 1 (t), u 2 (t), . . . , u m (t))T is called a Filippov solution of system (1) with initial condition φi (s) = (φ1 (s), φ2 (s), . . . , φm (s))T ∈ C([−δ, 0], Rm ), if u(t) is absolutely continuous on any compact interval of [0, +∞) and satis es the differential inclusions (4) or (6). The following assumption determines that the solution of system (1) or fractional differential inclusion (4), is existent and unique. Assumption 1 The functions h j are Lipschitz-continuous on R i.e., |h j (x) − h j (y)| ≤ L j |x − y| for all x, y ∈ R and j = 1, 2, . . . , m.



Lemma 1 ([32, 53]) The following inequality holds if function h(t) belongs to the nondecreasing and differential functions set, D q (h(t) − g)2 ≤ 2(h(t) − g)D q h(t), where g is any constant, t ∈ [0, ∞), 0 < q < 1.

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Lemma 2 ([50]) If Assumption 1 holds and h j (±X j ) = 0 ( j = 1, 2, . . . , m), then |co(wi j (u j (t)))h j (u j (t)) − co(wi j (u˜ j (t)))h j (u˜ j (t))| ≤ wizj L j |u j − u˜ j |, f or i, j = 1, 2, . . . , m, that is, for any ξi j (u j (t)) ∈ co[wi j (u j (t))], ηi j (u˜ j (t)) ∈ co[wi j (u˜ j (t))], |ξi j (u j (t)) − ηi j (u˜ j (t))| ≤ wizj L j |u j − u˜ j |, for i, j = 1, 2, . . . , m, wizj = max{|wi∗j |, |wi∗∗j |}.

3 Main Results In this section, adaptive synchronization schemes are taken for FMNNs to get effective and simple conditions. Let u¯ i (t) = u˜ i (t) − u i (t), for i = 1, 2, . . . , m and ⎧ ⎨ ri (t) = −αi (t)u¯ i (t) − sgn(u¯ i (t))βi (t)|u¯ i (t − δ)|, D q αi (t) = ρi |u¯ i (t)|, (7) ⎩ q D βi (t) = θi |u¯ i (t − δ)|. where ρi > 0 and θi > 0 are any scalars. u¯ i (t) → 0 as t → ∞ (i = 1, 2, . . . , m) implies that the synchronization between master system (1) and the slave system (2) is reached. The error dynamical system can be got from (5) and (6): D q u¯ i (t) = −ai u¯ i (t) +

m  [ηi j (u˜ j (t))h j (u˜ j (t)) − ξi j (u j (t))h j (u j (t))] j=1

+

m 

[χi j (u˜ j (t))h j (u˜ j (t − δ)) − ζi j (u j (t))h j (u j (t − δ)]

j=1

−αi (t)u¯ i (t) − sgn(u¯ i (t))βi (t)|u¯ i (t − δ)|,

(8)

for a.e. t ≥ 0, i = 1, 2, . . . , m. Now, we have the following main result. Theorem 1 If Assumption 1 and h j (±X j ) = 0 ( j = 1, 2, . . . , m) hold, the globally asymptotical synchronization will be derived for master-slvae systems (1) and (2) based on the controller (7). ˜ = (u˜ 1 (t), . . . , u˜ m (t))T denote two Proof Let u(t) = (u 1 (t), . . . , u m (t))T and u(t) arbitrary solutions of systems (1) and (2) with initial conditions u(t0 ) = (u 1 (t0 ), . . . , u m (t0 ))T and u(t ˜ 0 ) = (u˜ 1 (t0 ), . . . , u˜ m (t0 ))T meeting u¯ i (t0 ) = 0 for i = 1, . . . , m.

Adaptive Synchronization of Fractional-Order Delayed Memristive Neural Networks

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Apparently, u¯ i (t) = 0 is a solution of the error system (8). Since the solution of fractional order differential functions is existent and unique [24], u¯ i (t0 )u¯ i (t) > 0 for t > t0 . If u¯ i (t0 ) > 0, then u¯ i (t) > 0 and 1 D |u¯ i (t)| = Γ (1 − q)



t

u¯ i (s) ds = D q u¯ i (t), (t − s)q

t

u¯ i (s) ds = −D q u¯ i (t), (t − s)q

q

t0

and if u¯ i (t0 ) < 0, then u¯ i (t) < 0 and D q |u¯ i (t)| = −

1 Γ (1 − q)

 t0

So, D q |u¯ i (t)| = sgn(u¯ i (t))D q u¯ i (t). Building a Lyapunov function: m 

m m   1 1 2 |u¯ i (t)| + (αi (t) − αi ) + (βi (t) − βi )2 , V (t) = 2ρ 2θ i i i=1 i=1 i=1

(9)

where adaptive constants αi and βi will be decided later. Combing D q αi (t) = ρi |u¯ i (t)|, D q βi (t) = θi |u¯ i (t − δ)|, with Lemmas 1 and 2, the following inequality holds D q V (t) =

m 

D q |u¯ i (t)| +

i=1

≤

m 

m m   1 q 1 q D (αi (t) − αi )2 + D (βi (t) − βi )2 2ρ 2θ i i i=1 i=1

sgn(u¯ i (t)) −ai u¯ i (t)

i=1

+

m  [ηi j (u˜ j (t))h j (u˜ j (t)) − ξi j (u j (t))h j (u j (t))] j=1

m  + [χi j (u˜ j (t))h j (u˜ j (t − δ)) − ζi j (u j (t))h j (u j (t − δ))] − αi (t)u¯ i (t) j=1 m

 1 −sgn(u¯ i (t))βi (t)|u¯ i (t − δ)| + (αi (t) − αi )ρi |u¯ i (t)| ρ i=1 i

+

m  1 (βi (t) − βi )θi |u¯ i (t − δ)| θ i=1 i

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≤

m 

−ai |u¯ i (t)| +

i=1

−βi |u¯ i (t − δ)| =−



m 

wizj L j |u¯ j (t)| +

j=1

m 

vizj L j |u¯ j (t − δ)| − αi |u¯ i (t)|

j=1

m m m m



    ai + αi − βi − w zji L i |u¯ i (t)| − v zji L i |u¯ i (t − δ)|. i=1

j=1

i=1

j=1

  Suitably selecting αi and βi such that: ai + αi − mj=1 w zji L i > 0 and βi − mj=1 v zji L i > 0, for i = 1, 2, . . . , m. Let m

 w zji L i > 0, π1 = min ai + αi − j=1

and

m

 π2 = min βi − v uji L i > 0, j=1

then we have D q V (t) ≤ −π1

m 

|u¯ i (t)| − π2

i=1

m 

|u¯ i (t − δ)|, t ≥ t0 .

i=1

Let t > t0 , and ¯ u(t ¯ − δ)) D q V (t) = h(t, u(t), m m   |u¯ i (t)| − π2 |u¯ i (t − δ)| ≤ −π1 i=1

≤ −π1

m 

i=1

|u¯ i (t)| = −π1 Y (t) ≤ 0,

(10)

i=1

where Y (t) = We can get

m i=1

|u¯ i (t)|.

1 V (t) − V (t0 ) = Γ (q)



t

(t − s)q−1 h(s, u(s), ¯ u(s ¯ − δ))ds ≤ 0,

t0

from De nition 1, which implied that V (t) ≤ V (t0 ), t ≥ t0 . And, the boundness of u¯ i (t), αi (t) and βi (t) are direct results of (9) on t ≥ t0 . Thus, there exists a positive constant H > 0 satisfying |D q Y (t)| ≤ H, t ≥ t0 .

(11)

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299

Next, we prove limt→∞ Y (t) = 0 is true. And the method of contradiction is utilized. If (11) is not true, the inequality Y (tn ) ≥ , n = 1, 2, . . . ,

(12)

holds for some constant  > 0 and the time series {tn } meeting t0 < t1 < t2 < . . . < tn < . . .. 1 Let Q = ( Γ (q+1) ) q > 0. If tn ≤ t ≤ tn + Q, n = 1, 2, . . . , we get 4H |Y (t) − Y (tn )| =

 t 1 | (t − s)q−1 D q Y (s)ds Γ (q) t0  tn (tn − s)q−1 D q Y (s)ds| − t0

 tn 1 | [(t − s)q−1 − (tn − s)q−1 ]D q Y (s)ds = Γ (q) t0  t + (t − s)q−1 D q Y (s)ds| tn

 tn 1 [ ≤ |[(t − s)q−1 − (tn − s)q−1 ]D q Y (s)|ds Γ (q) t0  t |(t − s)q−1 D q Y (s)|ds] + tn

 tn  t H [ [(tn − s)q−1 − (t − s)q−1 ]ds + (t − s)q−1 ds] Γ (q) t0 tn H q q [(tn − t0 ) − (t − t0 ) + 2(t − tn )q ] = Γ (q + 1) 2H  ≤ (t − tn )q ≤ . Γ (q + 1) 2 ≤

From (11) and (12), this indicates Y (t) ≥ 2 , tn ≤ t ≤ tn + Q, n = 1, 2, . . .. Analogously, for tn − Q ≤ t ≤ tn and n = 1, 2 . . . , we can get  tn 1 | |Y (t) − Y (tn )| = (tn − s)q−1 D q Y (s)ds Γ (q) t0  t − (t − s)q−1 D q Y (s)ds| t0

 t 1 | [(tn − s)q−1 − (t − s)q−1 ]D q Y (s)ds = Γ (q) t0  tn (tn − s)q−1 D q Y (s)ds| + t

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 t 1 [ |[(tn − s)q−1 − (t − s)q−1 ]D q Y (s)|ds Γ (q) t0  tn |(tn − s)q−1 D q Y (s)|ds] + t  tn  t H q−1 q−1 ≤ − (tn − s) ]ds + (tn − s)q−1 ds] [ [(t − s) Γ (q) t0 t H q q [(t − t0 ) − (tn − t0 ) + 2(tn − t)q ] = Γ (q + 1) 2H  (tn − t)q ≤ . ≤ Γ (q + 1) 2 ≤

From (11) and (12) , Y (t) ≥ 2 , tn − Q ≤ t ≤ tn + Q, n = 1, 2, . . .. Hence,  Y (t) ≥ , tn − Q ≤ t ≤ tn + Q, n = 1, 2, . . . . 2

(13)

Under the assumption of these intervals are disjoint and t1 − Q > t0 , for any n = 1, 2, . . . , it holds tn−1 + Q < tn − Q < tn + Q < tn+1 − Q.

(14)

Combing with (10) and (13), when tn − Q ≤ t ≤ tn + Q,  D q V (t) ≤ − π1 2 holds. So, for any n = 1, 2, . . . , we get V (tn + Q) − V (t0 ) =

1 Γ (q)



tn +Q

(tn + Q − s)q−1 D q V (s)ds

t0

 t1 −Q  t1 +Q  t2 −Q  t2 +Q  t3 −Q 1 + + + + +··· = Γ (q) t0 t1 −Q t1 +Q t2 −Q t2 +Q  tn +Q (tn + Q − s)q−1 D q V (s)ds + ≤

tn −Q n  ti +Q 

1 Γ (q)

i=1

ti −Q

 1  ≤ − π1 2 Γ (q) i=1 n

=

 1 π1 2 Γ (q + 1)

(tn + Q − s)q−1 D q V (s)ds 

ti +Q ti −Q

n  i=1

(tn + Q − s)q−1 ds

  (tn − ti )q − (tn − ti + 2Q)q .

(15)

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301

n 1 q From (15), we get V (tn + Q) ≤ V (t0 ) + 2 π1 Γ (q+1) i=1 [(tn − ti ) − (tn − ti + q 2Q) ]. This shows that V (tn + Q) → −∞ as n → +∞, and this contradicts with V (t) ≥ 0. Hence, limt→∞ Y (t) = 0. Furthermore, globally asymptotical synchronization is acquired between the master-slave systems (1) and (2) under the controller (7). This completes proof.  Remark 1 Apparently, the adaptive control gains αi (t) and βi (t) converge to some positive constants when the master system (1) and the slave system (2) are synchronous. Remark 2 If we don’t consider the value changes of wi∗j , wi∗∗j , vi∗j , vi∗∗j for (1) and (2), i.e. we consider an FNN built by resistors, then we can obtain the adaptive synchronization conditions for the FNNs. Suppose the fractional-order delayed master-slave systems have the following form: D q u i (t) = −ai u i (t) +

m 

wi j h j (u j (t)) +

j=1

D u˜ i (t) = −ai u˜ i (t) + q

m  j=1

m 

vi j h j (u j (t − δ)) + li ,

(16)

j=1

wi j h j (u˜ j (t)) +

m 

vi j h j (u˜ j (t − δ))

j=1

+li + ri (t).

(17)

Then, the error system can be got under the controller (7): D q u¯ i (t) = −ai u¯ i (t) +

m  j=1

wi j [h j (u˜ j (t)) − h j (u j (t))] +

m 

vi j [h j (u˜ j (t − δ))

j=1

−h j (u j (t − δ))] − αi (t)u¯ i (t) − sgn(u¯ i (t))βi (t)|u¯ i (t − δ)|.

(18)

Now, we are in the position to give synchronization for FNNs. Theorem 2 Under Assumption 1, the master system (16) and the slave system (17) with the controller (7) come to globally asymptotical synchronization. Proof Making use of the Lyapunov function (9), from Assumption 1, let us carry out the fractional-order derivative computation for V (t),

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D q V (t) =

m 

D q |u¯ i (t)| +

i=1

≤

m 

m m   1 q 1 q D (αi (t) − αi )2 + D (βi (t) − βi )2 2ρ 2θ i i i=1 i=1

m  sgn(u¯ i (t)) −ai u¯ i (t) + wi j [h j (u˜ j (t)) − h j (u j (t))]

i=1

+

j=1

m 

vi j [h j (u˜ j (t − δ)) − h j (u j (t − δ))] − αi (t)u¯ i (t)

j=1 m

 −sgn(u¯ i (t))βi (t)|u¯ i (t − δ)| + (αi (t) − αi )|u¯ i (t)| i=1 m  + (βi (t) − βi )|u¯ i (t − δ)| i=1 m m m    ≤ −ai |u¯ i (t)| + |wi j |L j |u¯ j (t)| + |vi j |L j |u¯ j (t − δ)| i=1

j=1

−αi |u¯ i (t)| − βi |u¯ i (t − δ)| =−

j=1



m m

  ai + αi − |w ji |L i |u¯ i (t)| i=1

−

j=1

m  i=1

βi −

m 

|v ji |L i |u¯ i (t − δ)|.

j=1

The selection of αi and βi meets the conditions: ai + αi −

m 

|w ji |L i > 0,

j=1

and βi −

m 

|v ji |L i > 0,

j=1

for i = 1, 2, . . . , m. Let

m

 |w ji |L i > 0, π1 = min ai + αi − j=1

and

m

 π2 = min βi − |v ji |L i > 0, j=1

Adaptive Synchronization of Fractional-Order Delayed Memristive Neural Networks

303

then D V (t) ≤ −π1 q

m 

|u¯ i (t)| − π2

i=1

m 

|u¯ i (t − δ)|, t ≥ t0 .

i=1

The other proof is similar to that in the proof of Theorem 1. This completes proof.  Remark 3 As far as we know, there is no investigation about adaptive synchronization of FMNNs or FNNs with time delay. Compared with the conclusions in [28, 31, 32], the results and models in this chapter have more generality and less conservativeness for two reasons: The systems under investigation are discontinuous right-hand ones and they are FNNs with time delay. It should be emphasized that methods used in [28, 31] do not work for the systems in this chapter. Remark 4 An important thing we should emphasize is that Theorem 1 and 2 are new and still hold for the case q = 1. Moreover, results in [12, 45] are exceptional circumstances of our results. Remark 5 The control cost is obviously cut down and easy to be accomplished by designing a simple delay feedback controller (7). Clearly, the order of the fractionalorder systems affect the synchronization of FMNNs.

4 Numerical Examples We will give two examples to testify the validity of the proposed synchronization results. Example 1 Suppose the master system is described by the following FMNN: D q u(t) = −Au(t) + W (u(t))h(u(t)) + V (u(t))h(u(t − δ)) + l,

(19)

where q = 0.98, u = (u 1 , u 2 )T , h(u) = (h 1 (u 1 ), h 2 (u 2 ))T , h j (u j ) = tanh(|u j | − 1), L j = 1 ( j = 1, 2), δ = 1, l = (0, 0)T , and

w11 (u 1 ) −0.2 , A = 2I2×2 , W (u(t)) = −10 w22 (u 2 )

v (u ) −0.2 , V (u(t)) = 11 1 −0.4 v22 (u 2 )   −4, |u 1 | < 1 8.8, w11 (u 1 ) = , w22 (u 2 ) = −3.5, |u 1 | > 1 7.8,   −2.9, |u 1 | < 1 −7.9, , v22 (u 2 ) = v11 (u 1 ) = −2.8, |u 1 | > 1 −7.4,

|u 2 | < 1 , |u 2 | > 1 |u 2 | < 1 . |u 2 | > 1

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The salve system is given by: ˜ = −Au(t) ˜ + W (u(t))h( ˜ u(t)) ˜ + V (u(t))h( ˜ u(t ˜ − δ)) D q u(t) +l + r (t),

(20)

where r (t) = (r1 (t), r2 (t))T . The FMNN (19) is a chaotic system as indicated in Fig. 1. Next, The coupling strength update law (7) will be proved to work. All the initial values are (u 1 (s), u 2 (s))T = (0.16, 0.17)T , (u˜ 1 (s), u˜ 2 (s))T = (−0.28, −0.28)T , ∀s ∈ [−1, 0], βi (0) = 0.55, αi (0) = 0.51 (i = 1, 2). 13.2, α2 = 8.1, β1 = 4.1, Let ρi = 2, θi = 1, and α1 =  β2 = 8.5, by simple computing, we can get ai + αi − mj=1 w zji L i > 0, βi − mj=1 v zji L i > 0 (i = 1, 2), therefore, from Theorem 1, synchronization occurs between the master system (19) and the slave system (20). Figure 2 gives the synchronization error. Also, Fig. 3 reveals the variable evolutions for FMNNs u 1 (t), u 2 (t), u˜ 1 (t), u˜ 2 (t). We can see the master-slave systems (19) and (20) are synchronized in Figs. 2 and 3. Figures 4 and 5 demonstrate the convergence of the adaptive gains αi (t), βi (t) (i = 1, 2).

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Example 2 Suppose the master system is described by the following FMNN: D q u(t) = −Au(t) + W (u(t))h(u(t)) + V (u(t))h(u(t − δ)) + l,

(21)

where q = 0.98, u = (u 1 , u 2 )T , h(u) = (h 1 (u 1 ), h 2 (u 2 ))T , h 1 (u 1 ) = sin(|u 1 | − 1), h 2 (u 2 ) = tanh(|u 2 | − 1), L j = 1 ( j = 1, 2), δ = 1, l = (0, 0)T , and

1 w12 (u 2 ) , A = I2×2 , W (u(t)) = w21 (u 1 ) 1.8

v (u ) v (u ) V (u(t)) = 11 1 12 2 , v21 (u 1 ) v22 (u 2 )   7.1, |u 2 | < 1 0.85, |u 1 | < 1 , w21 (u 1 ) = , w12 (u 2 ) = 5.1, |u 2 | > 1 1.05, |u 1 | > 1   −1.4, |u 1 | < 1 1.1, |u 2 | < 1 , v12 (u 2 ) = , v11 (u 1 ) = −1.1, |u 1 | > 1 0.9, |u 2 | > 1   0.9, |u 1 | < 1 −1.45, |u 2 | < 1 , v22 (u 2 ) = . v21 (u 1 ) = 1.1, |u 1 | > 1 −1.65, |u 2 | > 1 The salve system is: ˜ = −Au(t) ˜ + W (u(t))h( ˜ u(t)) ˜ + V (u(t))h( ˜ u(t ˜ − δ)) D q u(t) +l + r (t),

(22)

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where r (t) = (r1 (t), r2 (t))T . System (21) is demonstrated to be chaotic in Fig. 6. The initial values are (u 1 (s), u 2 (s))T =(0.46, 0.59)T , (u˜ 1 (s), u˜ 2 (s))T =(0.2, 0.3)T , ∀s ∈ [−1, 0], β1 (0) = 0.1, β2 (0) = −0.1, α1 (0) = −0.1, α2 (0) = 0.1. Let ρi = 3.1, θi = 2.1, and α1 = 1.5, α2 =8, β1 = 3, β2 = 3.  Through calculation, we can get ai + αi − mj=1 w zji L i > 0, βi − mj=1 v zji L i > 0 (i = 1, 2). From the conditions of Theorem 1, the slave system (22) synchronizes with the master system (21). Figures 7 and 8 give the synchronization errors and the states’ trajectories of FMNN, respectively, and reveals the fact that synchronization takes place between the master-slave systems (21) and (22). Figures 9 and 10 demonstrate the convergence of the adaptive gains αi (t) and βi (t), (i = 1, 2).

5 Conclusion In this chapter, the issue of adaptive synchronization has been concerned for FMNNs subject to time delay. Adaptive feedback control and fractional-order inequality are employed to obtain suf cient synchronization conditions. The proposed approach is

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easy to be implemented. Ultimately, we gave two examples to illustrate the correctness of the newly designed synchronization method. Acknowledgements The work of H. Bao was jointly supported by the National Natural Science Foundation of China under Grant No. 61203096, the Chinese Postdoctoral Science Foundation under Grant 2013M513924, the Fundamental Research Funds for Central Universities XDJK2013C001 and the scienti c research support project for teachers with doctor’s degree, Southwest University under Grant No. SWU112024. The work of J.H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B02002002).

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New Results on Stability of Coupled Impulsive Fractional-Order Systems on Networks Li Zhang, Youggui Kao, and Cunchen Gao

Abstract The chapter is devoted to researching the stability issue of fractionalorder impulsive coupled systems over networks (FICSNs). Five new results on stability, uniform stability, uniform asymptotic stability, finite-time stability and Mittag-Leffler stability of FICSNs are derived respectively by the graph theory and Lyapunov second technique. These stability principles have a close relation to the topology property of the network. A numerical example is put forward to verify the validity of the obtained findings. Keywords Fractional-order · Impulsive coupled systems · Lyapunov second method · Graph theory

1 Introduction Fractional calculus is an old mathematical concept and dates back to 17 century. Regards to the complexity and lack of background, not enough attention has been paid by researchers for quite a long time. Recently, fractional differential equations have been extensively applied in a variety of fields, such as neural networks, complex L. Zhang · C. Gao School of Mathematical Science, Ocean University of China, Qingdao 266100, People’s Republic of China e-mail: [email protected] C. Gao e-mail: [email protected] Y. Kao (B) Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, People’s Republic of China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_14

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switched networks, stochastic systems, hybrid systems, chaotic systems, and discrete systems, etc., see [1, 2]. On the other hand, impulsive differential equations were widely employed to simulate some real circumstances, such as population medicine, population ecology, radio engineering, neural networks, economics, dynamics, communication security, etc., see [3–10]. Simultaneously, plenty of important conclusions about the stability issue of impulsive systems have been reported [11–17]. Li [11–15] investigated the stability issue of nonlinear systems with the influence of delayed impulses on input-to-state, distributed-delayed impulses, state-independent impulses, and impulsive control. In the research works [16, 17], the finite-time stability of impulsive reaction-diffusion systems and impulsive switched delay systems with nonlinear disturbances was investigated. Particularly, there are a large number of research results about the stability issue of various impulsive fractional systems [18–23]. Hei [18] studied the finite-time stability of impulsive fractional-order systems with time-delay. Yang [19] studied the Mittag-Leffler stability for fractional nonlinear fractional-order systems with impulses. Stamova [20, 21] researched the Mittag-Leffler stability and synchronization of fractional neural networks with timevarying delays and reaction-diffusion terms using impulsive and linear controllers. Li [22] and Xi [23] investigated the impulsive synchronization and adaptive impulsive synchronization of fractional-order chaotic systems with time-delay, respectively. Besides, more and more researchers have spent more effort to study coupled nonlinear differential systems, such as social science systems, biological systems, engineering systems, and physical systems. Strenuous results about stability analysis on networks have been provided in literatures [24–31]. In particular, Chen [24] gave some criteria on uniform stability of the coupled system with constant delay over networks. Kao et al. [25–29] resolved the stability analysis of coupled reaction-diffusion systems (SCRDS). Suo et al. [30] discussed a stability analysis issue for impulsive coupled systems on networks (ICSNs). Wang et al. [31] provided networked synchronization controller design method for coupled delay dynamic networks. In addition, there are some stability analysis results of the fractional coupled systems (FCSNs) [32–39]. Li [32, 33] investigated the Mittage-Leffler stability analysis of FCSNs and global Mittag-Leffler stability for FCSNs with feedback controls. Zhang [34, 35] proposed the synchronization issues of fractional coupled delayed systems. In the chapter [36–38], hybrid synchronization, Q-S synchronization, periodically intermittent discrete observation control for synchronization of coupled fractional-order systems, respectively. Especially, the exponential stability related to the integer-order systems is generalized to the fractional-order case by means of the Mittag-Leffler stability. Li [39] studied the Mittag-Leffler stability for a new coupled system of fractional-order differential equations with impulses, which have a great significance to the investigation of FICSNs. As we all know, in order to investigate the stability analysis for integer-order nonlinear systems, the direct Lyapunov method is derived. The direct Lyapunov method that it is not necessary to find the specific solution of the equation then we can analyze the equilibrium solution stability. When we considered stability analysis for fractional order nonlinear systems, constructed systematically a global Lyapunov function for the system on networks was a significant point in the research. Li et al. [40] settled the global stability issue for coupled systems by utilizing outcomes of the

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graph theory and the direct Lyapunov method. And Li [41] utilizes the same technique for some stochastic coupled systems on networks (SCSNs). To the best knowledge of ours, there are no research findings about the finite-time stability, uniform stability, asymptotic stability as well as uniform asymptotic stability analysis of FICSNs, which inspires us to research further. Motivated by the above analysis, in this chapter, we mainly study the various stability analysis of FICSNs. Firstly, based on the results of graph theory, we provide a systematic method that constructs global Lyapunov functions for large-scale impulsive coupled systems from building blocks of individual vertex systems. Secondly, five sufficient criterions on stability, uniform stability, uniform asymptotic stability, finite-time stability, and Mittag-Leffler stability of FICSNs are derived respectively. It should be noted that the study in this aspect is meaningful and challenging. At the rest of the chapter, Sect. 2 presents a few essential definitions and lemmas. In Sect. 3, we derived five criteria on various stability for FICSNs. In Sect. 4, an example is given to verify the validity and feasibility of the outcomes. The conclusion is given in Sect. 5.

2 Preliminaries Let’s introduce a few essential lemmas and definitions. Denote Rn as the ndimensional Euclidean space, Ξ is an open set in Rn , and R+ = [0, ∞). Definition 1 ([42]) Caputo fractional-order derivative for a function h(t) ∈ C 1 ([to , +∞), R) denote as C β to Dt h(t)

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Let glk : R+ × Rnl × Rn k → Rnl reflect the effect from the vertex k to l and glk ≡ 0 represents there have no arc between l and k. The following system can be get: ⎧ n  ⎪ ⎪ C β ⎪ D x = h (t, x ) + glk (t, xl , xk ), l l ⎪ to t l ⎨ ⎪ Δxl = Iw (xl ), ⎪ ⎪ ⎪ ⎩ xl (to+ ) = xlo ,

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t = tw , w = 1, 2, . . . ,

(3)

l = 1, 2, . . . , n, assuming these conditions hold: (i) 0 < t1 < t2 < · · · < tw < · · · , and tw → ∞ as w → ∞, (ii) h l : (tw−1 , tw ]×Ξ → Rnl . glk : (tw−1 , tw ]×Rnl × Rn k → Rnl is continuous, (iii) Iw : Rnl → Rnl . When t = tw , w = 1, 2, . . ., for each xl , xk the functions h l , glk and Iw : Ξ → Rn k are right continuous and satisfy the global Lipschitz condition. Respectively, to guarantee existence, uniqueness and continuability of the solution (x1 , x2 , . . . , xn ) = 0 for t ≥ to of system (3). The solution (x1 , x2 , . . . , xn ) = 0 is, in general, piecewise continuous functions with points of discontinuity of the first type at with they are left continuous. At the moments t = tw , w = 1, 2, . . . , the following relationship can be get: x(tw− ) = x(tw ) and x(tw+ ) = x(tw ) + Iw (x(tw )). Definition 4 ([42]) (x1 , x2 , . . . , xn ) = 0 for t ≥ to is the solution of system (3): (a) Mittag-Leffler stable if x(t; to , xo ) ≤ {n[x(to )]E β (−ν(t − to )r )}s

(4)

where β ∈ (0, 1), ν > 0, s > 0, n(x) ≥ 0, n(0) = 0, n(x) contents the locally Lipschitz, and n 0 is the Lipschitz constant; E β (t) =

∞  k=0

tk Γ (βk + 1)

(5)

represents Mittag-Leffler function, and then follow two parameters β and γ: E β,γ (t) =

∞  k=0

tk Γ (βk + γ)

where β > 0 and γ > 0. When γ = 1, then obtain E β (t) = E β,1 (t). (b) Globally Mittag-Leffler stable if (a) satisfies that Ξ ∈ Rn .

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New Results on Stability of Coupled Impulsive …

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Definition 5 ([43]) (x1 , x2 , . . . , xn ) = 0, ∀t ≥ to is the solution of system (3): 







(a) stable supposing that ∀to ∈ R+ , ∀ > 0, ∃η = η(to , ) > 0, ∀xo ∈ Ξ : xo <    η, ∀t ≥ to , then x(t; to , xo ) < ;  (b) uniformly stable supposing that η in (a) is not depend on to ∈ R+ ;   (c) attractive supposing that ∀t ∈ R+ , ∃χ = χ(to ) > 0, and 







∀xo ∈ Ξ : xo < χ, lim x(t; to , xo ) = 0; to →∞

(d) equi-attractive supposing  To∗ = To∗ (to , ) > 0, and 

that







∀to ∈ R+ , ∃χ = χ(to ) > 0, 



∀ > 0, ∃



∀xo ∈ Ξ : xo < χ, ∀t ≥ to + To : x(t; to , xo < ; 

(e) uniformly attractive supposing that the χ and To∗ in (d) are not depend on t ∈ R+ ; (f) asymptotically stable supposing that (a) and (c) hold; (g) uniformly asymptotically stable supposing that (b) and (e) hold. 







Definition 6 ([44]) There are three constants ωo > 0, ωo > 0, To , ωo < ωo , the    system (3) is defined as finite-time stable, xo (to ) ≤ ωo ⇒ xo (t) ≤ ωo , ∀to ∈   (to , to + To ]. Remark 1 Assume that the equilibrium solution of system (3) is Mittag-Leffler stable, then must be asymptotic stable. There are some fundamental notations about the graph theory [19] as follows: The digraph G = (U, F) consist of a set U = {1, 2, . . . , n} vertices and a set F of arcs (l, k), which represents the vertex from l to k, B = (blk )n×n is the weight matrix. The elements blk have consistent with the weight of arcs (l, k). A weighted digraph (G, B) is called balanced assume that the W (Π ) = W (−Π ) for every directed cycle Π . The −Π represents the reverse of Π . Ψ represents a unicyclic graph. L = (qlk )n×n is the Laplacian matrix of (G, B) is defined as ⎡ ⎢ ⎢ ⎢ L=⎢ ⎢ ⎢ ⎣

k=1 q1k −q21

· · · −qn1

−q  12 k=2 q2k · · · −qn2

⎤ · · · −q1n · · · −q2n ⎥ ⎥ ⎥ · · ⎥. ⎥ · · ⎥ ⎦ ·  · · · · k=n qnk

Lemma 1 ([32]) Supporting that n ≥ 2. Denote el represent the cofactor of the l-th  diagonal element of L . n  (l,k)=1

el alk Flk (xl , xk ) =

 (Ψ ∈Θ)

W (Ψ )

 (u,v)∈E(ΠΨ )

Fuv (xu , xv ).

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The Flk (xl , xk ), l, k ∈ L represent any functions, Θ denotes the set that whole spanning unicyclic graphs of (G, B) in. W (Ψ ) represents the weight of Ψ , ΠΨ is the directional cycle of Ψ . The (G, B) is strongly connected, el > 0, l ∈ L. If (G, B) is balanced n  (l,k)=1

el alk Flk (t, xl , xk ) =

 1  W (Ψ ) {Flk (t, xl , xk ) + Fkl (t, xk , xl )}. 2 (Ψ ∈Θ) (k,l)∈E(ΠΨ ) (8)

Lemma 2 ([43]) Assume the inequality C α to Dt x(t)

≤Cto Dtα yo

(9)

is effective when t ≥ to , t = tw , w = 1, 2, . . . , then x(t; to , xo ) ≤ yo ,

t ∈ [to , ∞).

(10)

Lemma 3 (Generalized Gronwall’s inequality) ([44]) Assume that x(ζ) > 0, k(ζ) > 0 are local integrable for 0 ≤ ζ < T ∗ , T ∗ ≤ ∞, h(ζ) > 0 is the nondecreasing continuous function with 0 ≤ ζ < T ∗ ; β > 0 , content that 

ζ

x(ζ) ≤ k(ζ) + h(ζ)

(ζ − s)β−1 x(s)ds.

(11)

0

Then

 x(ζ) ≤ k(ζ) + h(ζ) 0

ζ

∞  (h(ζ)Γ (β))n i=1

Γ (nβ)

 (ζ − s)nβ−1 k(s) ds.

(12)

Let k(ζ) is a nondecreasing function for [0, T ∗ ), then x(ζ) ≤ k(ζ)E β (h(ζ)Γ (β)ζ β ).

(13)

The E β represents the Mittag-Leffler function. Lemma 4 ([45]) Let z(ζ) ∈ R is a continuous differentiable function. ∀ζ ≥ ζ0 , 1 C β β ( Dt z(ζ)2 ) ≤ z(ζ)Cζ0 Dζ z(ζ), 2 t0

∀β ∈ (0, 1).

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3 Stability Analysis for Fractional-Order Impulsive Coupled Systems on Networks Let Dw = (tw−1 , tw ) × Ξ, w = 1, 2, . . ., D =

 w=1

Dw .

We shall study the stability analysis the system (3) with A0 : fl (t, 0) = 0, t ≥ 0, A1 : Iw (0) = 0, w = 1, 2, . . . . To proof the mainly theorems of this work, the piecewise continues auxiliary Lyapunov functions [42] is used: V ∈ C[D, R+ ] : R+ × Ξ → R+ . The V is the continuous function and content the locally Lipschitz with respect to its second argument on each Dw , V (tw− , x) = V (tw , x), and V (tw− , x) = V (tw , x)(t → tw , t > tw ) exists. The following condition contents that A2 : Vl (t, 0) = 0, t ≥ 0. Theorem 1 Assume theses conditions hold: (A1) ∃ Vl (t, xl ), Flk (xl , xk ), a matrix B = (blk )m×n where blk ≥ 0 satisfy: C β to Dt Vl (t, xl )

≤

n 

blk Flk (xl , xk ), t = tw , t ≥ to , l = 1, 2, . . . , n.

(15)

k=1

(A2) Towards every directional cycle Π of the weight digraph (G, B) 

Fuv (xu , xv ) ≤ 0,

t ≥ to , x u ∈ D u , x v ∈ D v .

(u,v)∈E(Π)

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(A3) Vl (t + , xl + Iw (xl )) ≤ Vl (t, xl )

xl ∈ Ξ, t = tw , w = 1, 2, . . . ,

(17)

(A4) ∃ αl1 ∈ K , l = 1, 2, . . . , n satisfy: αl1 ( xl ) ≤ Vl (t, xl ),

(t, xl ) ∈ Dw .

(18)

l

(A5) The functions Iw ∈ [Ξ, Rn ], w = 1, 2, . . . , are nondecreasing in Ξ and (E + Iw ) : Ξ → Ξ, w = 1, 2, . . . , in which E is the identity of Ξ . n  β The V (t, x) = el Vl (t, xl ) is a Lyapunov function and contents Cto Dt l=1

V (t, x)≤0, t ≥ to , t = tw . The (x1 , x2 , . . . , xn ) = 0 for system (3) is stable. Proof Denote α1 ( x(t) ) =

n  l=1

cl αl1 ( xl (t) ) and δ = min{δ01 , δ02 , . . . , δ0n }.

Let ε > 0.T hen, ∃δ0 = δ0 (to , ε) > 0 if x ≤ δ0 < δ, later

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V (to+ , x) < α1 (ε).

(19)

x ≤δ0

Next, to proof: ∀to ∈ R+ , ∀ε > 0, ∃δ0 = δ(to , ε) > 0, ∀xo ∈ Ξ : xo < δ0 , ∀t ≥ to , then x(t; to , xo ) < ε. Suppose that is not true. There exists x(t; to , xo ) for system (3), then xo < δ0 , ∃t  > to , tw < t  ≤ tw+1 , then x(t  ) ≥ ε, and x(t; to , xo < ε, t ∈ [to , tw ]. In view of the conditions (A1) and (A2), when t ∈ Dw , we will have C β to Dt V (t, x)

β

=Cto Dt

n 

el Vl (t, xl )

l=1

≤

n 

β el (Ct0 Dt Vl (t, xl ))

l=1

≤

n 

(20) el alk Flk (xl , xk ).

l,k=1

Regards of the conditions (A2), (7), W (Ψ ) > 0, then C β to Dt V (t, x)

≤





W (Ψ )

Ψ ∈Θ

Fuv (xu , xv ) ≤ 0.

(u,v)∈E(ΠΨ )

(21)

When t = tw , make use of the condition (A3), then C β + to Dt V (t , x

≤Cto

β Dt

β

+ Iw (x)) =Cto Dt

n 

el Vl (t, xl )) ≤

l=1

n 

el Vl (t + , xl + Iw (xl ))

l=1 n 

(22)

β el (Cto Dt Vl (t, xl ))

≤ 0.

l=1

Considered condition (A5) and Iw is nondecreasing, ∃t o , satisfy tw < t o ≤ t  , then x(t o ) > ε and x(t o ; to , xo ) ∈ Ξ , when t ∈ [to , t o ], then we can obtain V (t, x(t; to , xo )) = ≤

n  l=1 n 

el Vl (t, xl (t; to , xlo )) (23) el Vl (to+ , xlo ))

=

V (to+ , xo ).

l=1

From above discussion, the inequalities hold α1 (ε) < α1 ( x(t o ) ) =

n 

el αl1 ( (xlt o ) ) ≤

l=1

= V (t ; to , xo ) ≤ o

n 

el Vl (t o ; to , xlo ))

l=1 V (to+ , xo )

< α1 (ε).

(24)

New Results on Stability of Coupled Impulsive …

321

This is contradictory. Then, we can obtain ∀ε > 0, ∃δ0 = δ0 (to , ε) > 0, ∀xo ∈  Ξ : xo < δ0 , ∀t ≥ to , ∀to ∈ R+ , therefore x(t; to , xo < ε. Corollary 1 Assume the (G, B) is balanced, later n 

el alk Flk (xl , xk ) =

l,k=1

 1  W (Ψ ) [Fuv (xu , xv ) + Fvu (xv , xu )], 2 Ψ ∈Θ (u,v)∈E(Π ) Ψ

the result for Theorem 1 is established when the condition (A2) is substitute for (A2 ) 

[Fuv (xu , xv ) + Fvu (xv , xu )] ≤ 0, t ≥ to , xu ∈ Du , xv ∈ Dv .

(u,v)∈E(ΠΨ )

(25)

Theorem 2 Assume that the requirements (A1)–(A6) set up: (A6) Existing δ0(l) > 0 and functions b1l ∈ K satisfy: Vl (t, xl ) ≤ bl ( xl ),

provided xl < δ0(l) , (t, xl ) ∈ (to , ∞) × Ξ,

(26)

the solution (x1 , x2 , . . . , xn ) = 0 for system (3) is uniformly stable. Proof For all ε > 0, there exists δ0 = δ0 (ε) < δ such that b1l (δ0 ) < αl1 (ε). Let x ∈ Ξ : xo < δ0 . From Lemma 2, we can get: α1 ( x(t; to , xo ) ) =

n 

αl1 ( xl (t; to , xo ) ) ≤

l=1

n 

el Vl (t; xl (t; to , xo ))

l=1

= V (t, x(t; to xo )) < V (to+ , xo ) =

n 

el Vl (to+ , xlo )

l=1

≤

n 

el b1l ( xlto+ )
0 and ql > 0, l = 1, 2, . . . , n. satisfy: C β t0 Dt Vl (t, xl )

≤ −ql Vl (t, xl ) +

n  l=1

blk Flk (xl , xk ) t = tw , t ≥ to .

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The solution (x1 , x2 , . . . , xn ) = 0 for system (3) is uniformly asymptotically stable. Proof Denote q = min{q1 , q2 , . . . , qn } and η = max{η1 , η2 · ··, ηn }. Let H0 > 0 be a constant satisfying {x ∈ Rn : x ≤ H0 } ⊂ Ξ , ∀t ∈ [to , ∞). We shall denote Wt,H0 = {x ∈ Ξ : V (t + , x) ≤ α1 (H0 )}. From conditions (A4) and (A6): ∃ αl1 , b1l ∈ K , (t, x) ∈ Dw , we have αl1 xl ≤ Vl (t, xl ) ≤ b1l xl ,

(28)

then we reduce Wt,H0 ⊂ {x ∈ Rn : x ≤ H0 } ⊂ Ξ. Let λ = constant > 0, which satisfies αl1 (λ) < b1l (H0 ). If xo ∈ Ξ : xo < λ, from (28) implies V (to+ , xo ) =

n 

el Vl (to+ , xlo ) ≤

l=1




α1 (η)q b1 (H0 )Γ (β + 1)

 β1

.

Assume ∀t ∈ [to , to + T ], then x(t; to , xo ) ≥ η is valid, in view of condition (A7) C β to Dt V (t, x)

=

C β t0 Dt

n  l=1

≤ ≤

n  l=1 n  l=1



el Vl (t, x) ≤

n 

β

el (Cto Dt Vl (t, xl ))

l=1 n 

el −ql Vl (t, xl ) +

 blk Flk (xl , xk )

l=1

el ql Vl (t, xl )) = −q V (t, x)

t  = tw

New Results on Stability of Coupled Impulsive …

323

and from condition (A3), we can have C β to Dt V (t, x)

≤ −q V (t, x)

t ∈ [to , ∞).

Integral of degree β on both sides of the equation gives V (t, x) −

V (to+ , x(to+ ))

−q ≤ Γ (β) =

−q Γ (β)



t

to+

V (s, x)(t − s)β−1 ds

 t n to+

el Vl (s, xl )(t − s)β−1 ds

l=1

 t n −q ≤ el αl1 xl (τ ; to , xlo ) (t − s)β−1 ds Γ (β) to+ l=1  t −q = α1 xl (τ ; to , xo ) (t − s)β−1 ds Γ (β) to+ and −q Γ (β)



t to+

α1 (η)(t − s)β−1 ds ≤

−qα1 (η) , Γ (β + 1)T β

which leads V (t, x) ≤ V (to+ , x(to+ )) −

−qα1 (η) −qα1 (η) ≤ b1 (H0 ) − < 0. β Γ (β + 1)T Γ (β + 1)T β

This is contradictory with (28). So, the assumption is not valid. This contradiction indicates ∃ t  ∈ [to , to + T ] satisfies x(t  ; to , xo ) < η. Regard to (A3), (A7), and (28), we can obtain ∀ t ≥ t  (thus for any t ≥ to + T ), the inequalities set up: α1 ( x(t; to , xo ) ) = ≤

n  l=1 n  l=1

el αl1 ( xl (t; to , xlo ) ) ≤ el Vl (t  ; x(t; t  , xlo )) ≤

n  l=1 n 

el Vl (t, x(t; to , xlo )) el b1l ( xl (t  ; to , xlo ) )

l=1

= b1 x(t  ; to , xo ) < b1 (η) < α1 (ε). Then, x(t; to , xo ) < ε, ∀t ≥ to + T . Combining with Theorem 2, the results can be obtained.  Corollary 2 Assume (G, B) is balanced, the results for Theorem 3 established if condition (A2) is substitute for (A2 ).

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Remark 2 When β = 1, the stability analysis of equilibrium solution for FICSNs shall degenerate the stability analysis of ICSNs, Theorems 1–3 in this chapter in accordance with Theorems 3.1 and 3.2 in the literature [32]. Remark 3 When

n 

glk (t, xl , xk ) = 0, the system will transform into:

k=1

⎧C β ⎪ ⎨ to Dt xl = h l (t, xl ), t = tw , Δxl = Iw (xl ), t = tw , w = 1, 2, . . . , ⎪ ⎩ xl (to+ ) = xlo .

(29)

The stability analysis of equilibrium solution for FICSNs will degenerate the stability of fractional impulsive nonlinear system, Theorem 1, 2, and 3 of this chapter consistent with Theorems 5.1, 5.2, and 5.3 in the paper [43]. Theorem 4 Assume the conditions (A2), (A3), (A5), (A7) and (A8), (A9) set up: (A8) There have r1l , r2l ∈ k satisfying: r1l xl ≤ Vl (t, xl ) ≤ r2l xl t ≥ to ,

(30)

 r2 E α (−qT α ) < , r1 

(31)

(A9) when xlto < ,

the equilibrium solution is finite-time stable. Proof In view of conditions (A7) and (A2) : C β to Dt Vl (t, xl )

≤ −ql Vl (t, xl ) +

n 

blk Flk (xl , xk ) t = tk , t ≥ t0

l=1



Fuv (xu , xv ) ≤ 0,

t ≥ to , x u ∈ D u , x v ∈ D v ,

(u,v)∈E(Π)

we can obtain:

C β to Dt Vl (t, xl )

≤ −ql Vl (t, xl ).

From condition (A3), we will get: C β to Dt Vl (t, xl )

≤ −ql Vl (t, xl ), t ∈ [to , ∞).

Integral of degree β on both sides of the equation gives Vl (t, xl ) − V (to+ , xl (to ) + Iw xl (to )) ≤

−q Γ (β)



t to+

Vl (s, xl )(t − s)β−1 ds,

New Results on Stability of Coupled Impulsive …

325

then, Vl (t, xl ) ≤ V (to+ , xl (to ) + Iw xl (to )) +

−q Γ (β)



t to+

Vl (s, xl )(t − s)β−1 ds,

which finally leads Vl (t, xl ) ≤ Vl (to , xlto ) +

−q Γ (β)



t

Vl (s, xl )(t − s)β−1 ds.

to

By using Lemma 3, we can get: Vl (t, xl ) ≤ Vl (to , xlto ) · E β (−qT β ). When considered the condition (A8), we can see: r1 x =

n 

el r1l xl

≤

l=1

n 

el Vl (t, xl )

l=1

= V (t, x) ≤

n 

el Vl (to , xlto ) · E β (−qT β )

l=1

≤

n 

el r2l xlto · E β (−qT β ) = r2 xlto · E β (−qT β ),

l=1

which leads x ≤ Let

r2 r1

r2 xlto E β (−qT β ), t ≥ to . r1

E β (−qT β ) <  , when xlto < , we can get x ≤

r2 r2 xlto E β (−qT β ) <  E β (−qT β ) < . r1 r1

Then, the proof process is ended.



Remark 4 When β = 1, the Mittag-Leffler stable of equilibrium solution for FICSNs shall translate into exponentially stable of Theorem 3.3 in [30]. Theorem 5 Let to = 0, Assume the conditions (A2), (A3), (A5), (A7) and (A10) satisfying: (A10) There have functions r1l , r2l ∈ k, such that r1l xl a ≤ Vl (t, xl ) ≤ r2l xl ab t ≥ 0.

(32)

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L. Zhang et al.

The r1l > 0, r2l > 0, a > 0 and b > 0 are arbitrary numbers. The equilibrium solution (x1 , x2 , . . . , xn ) = 0 for system (3) is Mittag-Leffler stable. Proof The proof process has the similarity with Theorem 4. When to = 0 −q Vl (t, xl ) ≤ Vl (0, xl (0)) + Γ (β)



t

Vl (s, xl )(t − s)β−1 ds.

o

By using Lemma 3, we can get: Vl (t, xl ) ≤ Vl (0, xl (0)) · E β (−qT β ). When considered the condition (A10), we can see: r1 x a =

n 

el r1l xl a ≤

l=1

= V (t, x) ≤

n 

el Vl (t, xl )

l=1 n 

el Vl (0, xl (0)) · E β (−qT β )

l=1

= V (0, x(0)) · E β (−qT β ), which leads

 x ≤

Let m =

V (0,x(0)) r1

V (0, x(0)) E β (−qT β ) r1

 a1

, t ≥ 0.

≥ 0. The we obtain x ≤ {m E β (−qT β )} a , t ≥ 0, 1

then m = 0 satisfies only if x(0) = 0. m is a Lipschitz constant, m(0) = 0, therefore the equilibrium solution (x1 , x2 , . . . , xn ) = 0 for system (3) is Mittag-Leffler stable.  Corollary 3 From Theorem 5, the equilibrium solution (x1 , x2 , . . . , xn ) = 0 for system (3) is Mittag-Leffler stable and must be asymptotically stable. Remark 5 When Iw (xl ) ≡ 0 for all l, k and the constant x  = 0 , then the MittagLeffler stable of equilibrium solution for FICSNs shall transform into the MittagLeffler stable of FCSNs in the paper [43] according with Theorem 3.1. Remark 6 Above five theorems are our main works. It is different that the results in this chapter compared with the previous in [30, 32, 43]. We consider the stability analysis for FICSNs. This is the highlight of this work.

New Results on Stability of Coupled Impulsive …

327

4 Numerical Example Let’s consider the following example: ⎧ n  ⎪ C α ⎪ ⎪ D x = α x + h (x ) − βlk (xl − xk ), t = tw , l l l l to t l ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ n ⎪  ⎪ ⎪ C α ⎪ D y = α y + h (y ) − βlk (yl − yk ), t = tw , l l l l l ⎪ t t o ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ βlk (zl − z k ), t = tw , ⎨ Cto Dtα zl = αl zl + h l (zl ) − k=1

⎪ ⎪ ⎪ ⎪ ⎪ Δxl = Iw (xl (tw )) = Cw xl (tw ) − xl (tw ), t = tw , ⎪ ⎪ ⎪ Δyl = Iw (yl (tw )) = Cw yl (tw ) − yl (tw ), t = tw , ⎪ ⎪ ⎪ ⎪ ⎪ Δzl = Iw (zl (tw )) = Cw zl (tw ) − zl (tw ), t = tw , ⎪ ⎪ ⎪ ⎪ ⎪ xl (to+ ) = xlo , ⎪ ⎪ ⎪ ⎪ ⎪ yl (to+ ) = ylo , ⎪ ⎪ ⎩ zl (to+ ) = zlo ,

(33)

where xl , yl , zl are n-dimensional column vectors, h l : Rn → Rn are continuous and satisfy that h l (xl ) − h l (yl ) ≤ L l xl − yl for any xl = yl . βlk ≤ 0, βlk = −βkl and βlk = 0 for l = k. Cw is an n × n constant matrix and λmax (Cw ) ≤ 1, k = 1, 2, . . .. Let G be a digraph with n vertices, blk = |βlk |, (l, k = 1, 2, . . . , n), B = (blk )n×n , n  βlk − L l > 0. (G, B) is strongly connected and balanced; ∃ pl , pl = −αl + k=1

Here, let X l = (xl , yl , zl ), l = 1, 2, . . . , n, then constitute the Lyapunov functions Vl (t, X l ) =

1 2 1 2 1 2 x + yl + zl . 2 l 2 2

From then, we have C D α V (t, X ) =C D α l to t l to t



 = xl αl xl + h l (xl ) −

1 2 1 2 1 2 x + yl + zl 2 l 2 2

n 





α C α C α ≤ xl (C to Dt xl ) + yl (to Dt yl ) + zl (to Dt zl )



βlk (xl − xk ) + yl αl yl + h l (yl ) −

k=1

 + zl αl zl + h l (zl ) −

n  k=1

n  k=1

 βlk (zl − z k )

 βlk (yl − yk )

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 ≤ αl + L l −

n 

 βlk (xl2 + yl2 + zl2 ) +

k=1

n 

βlk xl xk +

k=1

 ≤ αl + L l −

n 

 βlk Vl (t, X l ) +

k=1

n 

n 

βlk yl yk +

k=1

n 

βlk zl z k

k=1

βlk (xl xk + yl yk + zl z k )

k=1

= − pl Vl (t, X l ) +

n 

blk Flk (X l , X k )

k=1

where pl = −αl − L l +

n 

βlk > 0, Flk (X l , X k ) = sgn(βlk )(xl xk + yl yk + zl z k )

k=1

for l = k, and

Flk (X l , X k ) = sgn(βlk )(xl xk + yl yk + zl z k ) = −sgn(βkl )(xk xl + yk yl + z k zl ). Towards every directional cycle Π of the weighted digraph (G, B) satisfies the following relationship: 

Flk (X l , X k ) + Fkl (X k , X l ) = 0.

(l,k)∈E(ΠΨ )

Then, we can obtain: Vl (tw+ , X l tw + Iw (X l tw )) = ≤

1 (Cw xl (tw ))2 + (Cw yl (tw ))2 + (Cw zl (tw ))2 2

1 2 λ (Cw )(xl (tw ))2 + (yl (tw ))2 + (zl (tw ))2 ) = λ2 (Cw )Vl (tw , X l (tw )). 2

From the condition λmax (Cw ) ≤ 1, w = 1, 2, . . ., we can get: Vl (tw+ , X l tw + Iw (X l tw )) ≤ Vl (tw , X l (tw )). Let αl1 X l = 41 X l , b1l X l = X l , then αl1 X l ≤ Vl (t, X l ) = 21 X l ≤ b1l X l . Considering Theorem 3, the (x1 , x2 , . . . , xn ) = 0 for system (33) is uniformly asymptotically stable. Let h l (xl ) = sin(lxl /(n − 1)), αl = −2n/(n − 1), βll = 0, βlk = −1/(n − 1) if l < k and βlk = 1/(n − 1) if l > k. Let n = 3, l = 1, 2, 3, Cw = 1/2 I3 , where I3 is 3 × 3 identity matrix. Since blk = |βlk |, we can obtain that ⎡ ⎢ A=⎢ ⎣

0 1 2 1 2

1 1 2 2 0 21 1 2 0

⎤

⎡ ⎤ 1 − 21 − 21 ⎥ ⎢ 1 ⎥ ⎥, L = ⎢− 1 −1 ⎥. 2 ⎦ ⎦ ⎣ 2 − 21 − 21 1

329

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Fig. 1 Times-series of three-dimension system with initial value X 1 = [0.6, 0.2, −0.5], Y1 = [−0.4, 0.6, 0.3], Z 1 = [0.4, −0.5, 0.2]

By simple calculation, e1 = e2 = e3 = 43 , L 1o = 0.5, L 2o = 1, L 3o = 1.5. Therefore, (G, B) is balanced and strongly connected. For i = 1, 2, 3, we can compute n  1 βlk − L l = 1 > 0, λ2 (Cw ) = < 1 pl = −αl + 4 k=1 which satisfy all conditions of the system (33). Firstly, we consider the system with the parameters that not content with the conn  βlk − L l < 0, then from the Fig. 1, we can get the equilibrium dition pl = −αl + k=1

point is instability. Secondly, we adjust the parameters of the system (33) so that content all the conditions above analysis. From Fig. 2, the system is uniformly asymptotically stable. Here, the initial values are taken as X 1 = [0.6, 0.2, −0.5], Y1 = [−0.4, 0.6, 0.3], and Z 1 = [0.4, −0.5, 0.2]. Remark 7 Distinguish from the numerical example in [30], they gave an impulsive coupled system with integer derivative. In this chapter, we study the FICSNs (33) with the dimension extended to three. Next, we discuss the influence of fractional-order α of system (33).

L. Zhang et al. 0.6 x1(t)

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Fig. 2 Times-series of three-dimension system with initial value X 1 = [0.6, 0.2, −0.5], Y1 = [−0.4, 0.6, 0.3], Z 1 = [0.4, −0.5, 0.2]

x (t) 2

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0.4

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Fig. 3 α = 0.5

Keeping the other parameters fixed and altering the parameter α with α1 = 0.98, α2 = 0.75, α3 = 0.5. Figures 3, 4 and 5 show that the differences in times-series of three-dimension system with initial value X 1 = [0.6, 0.2, −0.5], Y1 = [−0.4, 0.6, 0.3], and Z 1 = [0.4, −0.5, 0.2].

valuesofx 1(t),x 2(t),x 3(t),y 1(t),y 2(t),y 3(t)andz 1(t),z 2(t),z 3(t)

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Fig. 4 α = 0.75

x (t) 2

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Fig. 5 α = 0.98

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Remark 8 From Figs. 3, 4 and 5, α is an essential element that can affect the convergence velocities. With the ascending of the fractional-order α ∈ (0, 1), accelerates the convergence velocities.

5 Conclusion In this chapter, we studied the stability, uniform stability, asymptotic stability, uniform asymptotic stability, finite-time stability and the Mittag-Leffler stability for FICSNs via utilizing the graph theory and Lyapunov second technique. We first build Lyapunov functions for individual vertex and build global Lyapunov functions for FICSNs. Then, five stability criteria of FICSNs have been derived. Moreover, our method and results can be expanded to large scale complex networks and agents. Acknowledgements The authors would be very grateful to the editor of this edited book for their helpful comments and constructive suggestions. This research is supported by the National Natural Science Foundations of China (61873071) and the Shandong Provincial Natural Science Foundations (ZR2019MF027).

References 1. Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, Amsterdam (1998) 2. Carpinteri, A., Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Berlin (1997) 3. Stamova, I.: Stability Analysis of Impulsive Functional Differential Equations. Walter de Gruyter, New York (2009) 4. Yang, Y., He, Y., Wang, Y., et al.: Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method. Commun. Nonlinear Sci. Numer. Simul. 45, 140–157 (2016) 5. Luo, R., Su, H.: The stability of impulsive incommensurate fractional order chaotic systems with Caputo derivative. Chin. J. Phys. 56(4), 1599–1608 (2018) 6. Jiao, T., Park, J.H., Zong, G., Zhao, Y., Du, Q.: On stability analysis of random impulsive and switching neural networks. Neurocomputing 350, 146–154 (2019) 7. Mathiyalagan, K., Park, J.H., Sakthivel, R.: Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities. Appl. Math. Comput. 259, 967–979 (2015) 8. Zhang, Y., Sun, J., Gang, F.: Impulsive control of discrete systems with time delay. IEEE Trans. Autom. Control 54(4), 830–834 (2009) 9. Tang, Z., Park, J.H., Wang, Y., Feng, J.: Distributed impulsive quasi-synchronization of Lur’e networks with proportional delay. IEEE Trans. Cybern. 49(8), 3105–3115 (2019) 10. Zhang, Y.: Robust exponential stability of uncertain impulsive neural networks with timevarying delays and delayed impulses. Neurocomputing 74(17), 3268–3276 (2011) 11. Li, X., Zhang, X., Song, S.: Effect of delayed impulses on input-to-state stability of nonlinear systems. Automatica 76, 378–382 (2017)

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12. Li, X., Wu, J.: Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63–69 (2016) 13. Zhang, X., Li, X.: Input-to-state stability of non-linear systems with distributed-delayed impulses. IET Control Theory Appl. 11(1), 81–89 (2017) 14. Li, X., Song, S.: Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control 62(1), 406–411 (2017) 15. Lin, D., Li, X., O’Regan, D.: Stability analysis of generalized impulsive functional differential equations. Math. Comput. Model. 55(5–6), 1682–1690 (2012) 16. Wu, K., Na, M., Wang, L., et al.: Finite-time stability of impulsive reaction-diffusion systems with and without time delay. Appl. Math. Comput. 363, 124591 (2019) 17. Tian, Y., Cai, Y., Sun, Y., et al.: Finite-time stability for impulsive switched delay systems with nonlinear disturbances. J. Frankl. Inst. 353(14), 3578–3594 (2016) 18. Hei, X., Wu, R.: Finite-time stability of impulsive fractional-order systems with time-delay. Appl. Math. Model. 40(7–8), 4285–4290 (2016) 19. Yang, X., Li, C., Huang, T., et al.: Mittag-Leffler stability analysis of nonlinear fractional-order systems with impulses. Appl. Math. Comput. 293, 416–422 (2017) 20. Stamova, I.: Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn. 77(4), 1251–1260 (2014) 21. Stamova, I., Stamov, G.: Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers. Neural Netw. 96, 22–32 (2017) 22. Li, D., Zhang, X.: Impulsive synchronization of fractional order chaotic systems with timedelay. Neurocomputing 216, 39–44 (2016) 23. Xi, H., Yu, S., Zhang, R., et al.: Adaptive impulsive synchronization for a class of fractionalorder chaotic and hyperchaotic systems. Optik-Int. J. Light Electron Opt. 125(9), 2036–2040 (2014) 24. Chen, H., Sun, J.: Stability analysis for coupled systems with time delay on networks. Phys. A: Stat. Mech. Appl. 391(3), 528–534 (2012) 25. Li, Y., Kao, K.: Stability of coupled impulsive Markovian jump reaction-diffusion systems on networks. J. Syst. Sci. Complex. 29(5), 1269–1280 (2016) 26. Kao, Y., Sun, H., Cao, H.: Stability analysis for coupled stochastic systems with time delay on networks. Math. Appl. 26, 67–75 (2013) 27. Kao, Y., Wang, C., Karimi, H.R., et al.: Global stability of coupled Markovian switching reaction-diffusion systems on networks. Nonlinear Anal.: Hybrid Syst. 13, 61–73 (2014) 28. Kao, Y., Zhu, Q., Qi, W.: Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching. Appl. Math. Comput. 271, 795–804 (2015) 29. Kao, Y., Wang, C.: Global stability analysis for stochastic coupled reaction-diffusion systems on networks. Nonlinear Anal.: Real World Appl. 14(3), 1457–1465 (2013) 30. Suo, J., Sun, J., Zhang, Y.: Stability analysis for impulsive coupled systems on networks. Neurocomputing 99, 172–177 (2013) 31. Wang, Y., Zhang, H., Wang, X., et al.: Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 40(6), 1468–1479 (2010) 32. Li, H., Jiang, Y., Wang, Z., et al.: Global Mittag-Leffler stability of coupled system of fractionalorder differential equations on network. Appl. Math. Comput. 270, 269–277 (2015) 33. Li, H., Hu, C., Jiang, Y., et al.: Global Mittag-Leffler stability for a coupled system of fractionalorder differential equations on network with feedback controls. Neurocomputing 214, 233–241 (2016) 34. Zhang, Y., Liu, S., Yang, R., et al.: Global synchronization of fractional coupled networks with discrete and distributed delays. Phys. A: Stat. Mech. Appl. 514, 830–837 (2019) 35. Zhang, H., Ye, M., Ye, R., et al.: Synchronization stability of Riemann-Liouville fractional delay-coupled complex neural networks (2018) 36. Ma, T., Zhang, J.: Hybrid synchronization of coupled fractional-order complex networks. Neurocomputing 157, 166–172 (2015)

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Dynamical Networks

Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks Le Van Hien

Abstract This chapter is concerned with the problems of positivity and stability of nonlinear time-delay systems in the area of neural networks. The concept of positivity is first developed and characterized for various models of neural networks with delays including Hopfield neural networks, bidirectional associative memory neural networks and inertial neural networks. It will be shown by utilizing the property of order-preserving of neuron activation functions that for nonnegative connection weights of neurons, state trajectories of the networks initializing from a positive cone called the admissible set of initial conditions are always nonnegative subject to nonnegative inputs. Then, the exponential stability of positive equilibriums of the underlying models is investigated by a systematic approach involving extended comparison techniques via differential and integral inequalities. Unified conditions for the existence and exponential stability of positive equilibriums are derived in the form of linear programming (LP) conditions with M-matrix, which can be effectively solved by various convex algorithms. Numerical examples and simulations are provided to demonstrate the applicability and effectiveness of the derived theoretical results. Keywords Positive neural networks · Positive equilibrium · Exponential stability · Time-varying delay · M-matrix.

1 Introduction Neural networks (NNs), including artificial neural networks (ANNs) and biological neural networks, can be found in a wide range of practical applications. An instance example can be found in the area of time series forecasting where ANNs are used for the test of ill-defined temporal problems like predictions in financial markets. In the L. V. Hien (B) Faculty of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_15

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field of pattern recognition for medical visualization aids and image processing, due to their capabilities in pattern-matching and learning, ANNs are used to solve many problems in image realization, speech recognition, parallel computation or natural language processing, which are normally difficult to solve by conventional methods [1–4]. For modelling of system dynamics and adaptivity features, ANNs can also be used in various state estimation problems as system identification and fault detection [5, 6] resorting to the approximation capability of nonlinear functions. Many other applications of neural networks (NNs) can be found in transportation, health care, industrial robotics, human-machine interface, process control, associative memory or computer security [7, 8]. In practical applications of NNs, it is important to ensure the existence and stability normally in the sense of asymptotic behavior of a unique equilibrium [9]. In addition, the implementation process of NNs in practical is frequently encountered with time delays according to many technical reasons such as the limit of switching speed of amplifiers or the signal processing transmission through layers in the network. The presence of time-delay usually makes the system behaviors more complicated and unpredictable [10–12]. Thus, in the past few decades, the problems of performance analysis and synthesis of NNs with delays have been extensively studied and, as a consequence, many results concerning the stability of NNs with delays have been reported. To mention a few, we refer the reader to [13–25] and the references therein. Positive systems are dynamical systems whose states and outputs initiating from nonnegative inputs and initial conditions are always nonnegative [26]. This type of systems can be used to model various practical systems where the associated state variables are subject to positivity constraints according to the nature of phenomena [27]. It has been well recognized in the area of systems and control that the problem of stability and/or performance analysis of dynamical time-delay systems plays a key role in the design and implement of controller/observer/filter in practical applications. Like general dynamical systems, the problem of stability analysis and control of positive systems with or without delays has received significant research attention from researchers in the past few decades [28–35]. While many important problems in the systems and control theory have been well-studied for various classes of positive time-delay systems and, in particular, linear positive time-delay systems, this area is still considerably less well-developed for positive nonlinear systems describing neural network models. It is noted that when a neural network is designed for application purposes of positive systems, for example, in identification [36], control [37], monotone-regular behavior implement [38] and various disciplines in the academia and industries including computer vision, pattern recognition, alignment and detection [39], states of the designed networks are expected to inherit positivity constraints of the practical model. Besides, for neural network systems, the nonlinearity of neuron activation functions makes the study of positive neural networks with delays more complicated and challenging. Thus, as a key and primary issue to apply in control of positive neural networks, the problem of stability analysis of positive nonlinear time-delay systems in network structures is very relevant and of great importance to study. In particular, a systematic approach and effective tools used in the analysis of positivity and stability of such models are obviously important to develop.

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Typically, the dynamics of a conventional neural network is described by firstorder differential equations. Recently, many authors have devoted attention to studies a type of network models called inertial neural networks (INNs) [40]. In the statespace model, INNs are described by second-order differential equations with inertial terms represented by the first-order derivative. The modeling of inertial terms in neural systems has strong biological and engineering backgrounds especially for those containing an inductance [41, 42]. In addition, in the presence of inertial terms, it is much more difficult and challenging to analyze the dynamic behaviors of INNs. Thus, due to both theoretical and practical reasons, the investigation of INNs has attracted considerable attention in recent years. For example, based on extended Halanay inequalities and a Lyapunov-like functional method, the problems of dissipativity and exponential convergence were studied for various types of INNs with time-varying delays in [43–46]. Lagrange stability and stabilization via sampleddata control were also investigated for Hopfield INNs in [47–49] and [50] using characteristic function and Lyapunov–Krasovskii functional methods. However, it should be mentioned here that the above papers only deal with traditional neural networks. Furthermore, the methodologies proposed in the existing literature cannot be directly applied or extended to positive nonlinear systems which describe INNs models with delays. Extending the theory of positive systems to nonlinear timedelay systems that describe INNs proves to be an interesting issue. In particular, the existence, uniqueness, and global exponential stability of the positive equilibrium of positive INNs with delays need further investigation, which will be addressed in this chapter. Another important model of neural networks which appear in artificial intelligence is referred to as bidirectional associative memory (BAM) neural networks. First introduced in the pioneering work [51] when studied stability and encoding properties of two-layer nonlinear feedback neural networks, the terminology of bidirectionality, referred to forward and backward information flows, was adapted to neural nets to produce a two-way associative search for stored paired-data associations. This model was later called BAM neural network. Typically, a BAM neural network is constructed of two neuron layers namely X-layer and Y-layer. Neurons in each layer are completely integrated into neurons in the other layer, whereas there is no interconnection among neurons in the same layer. This structure performs a twoway associative search for stored bipolar vector pairs and generalizes the single-layer autoassociative Hebbian correlation to a two-layer pattern-matched heteroassociative circuits [52]. Thus, the BAM model possesses many application prospects in the areas of pattern recognition, signal and image processing. As discussed in the preceding paragraphs, practical applications of neural networks in many areas require that the designed networks admit only one equilibrium point which is globally asymptotically stable thereby avoiding the risks of having spurious equilibrium point and being trapped in local minima [53]. Thus, it is important to study the problems of stability and performance analysis of a unique equilibrium of BAM neural networks with delays [54–57]. The problems of synchronization [58, 59], stabilization [60, 61] or impulsive control [62, 63] have also been investigated for various types of BAM neural networks with delays. However, the theory for positive BAM neural networks

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with delays has not been well investigated despite their elegant properties and potential applications. It should be noted that the methods developed for Hopfield-type neural networks cannot be simply extended to BAM neural networks with delays according to the nature of the structure themselves. Motivated from the above discussion, in this chapter, we study the stability problem of nonlinear time-delay systems in three types of neural networks. Specifically, we develop the concept of positivity for popular neural networks models with delays including Hopfield neural networks, BAM neural networks and inertial neural networks. A systematic approach involving extended comparison techniques via differential and integral inequalities is developed for studying exponential stability of positive equilibriums of considering models. By utilizing the proposed approach, unified conditions for the existence and exponential stability of positive equilibriums are derived in the form of linear programming conditions with M-matrix, which can be effectively solved by various convex algorithms. Numerical examples and simulations are provided to demonstrate the applicability and effectiveness of the derived theoretical results.

2 Preliminaries 2.1 Notation Rn the n-dimensional Euclidean space with the vector norm x = max1≤i≤n |xi |, 1n ∈ Rn is the vector with all entries equal one. Rm×n is the set of m × n-matrices. σ(A) denotes the set of eigenvalues of A ∈ Rn×n and ρ(A) = max{|λ| : λ ∈ σ(A)} is its spectral radius. vec(v1 , v2 , . . . , vk ) denotes the augmented vector formulated by stacking components of v1 , v2 , . . . , vk . For vectors x = (xi ) ∈ Rn and y = (yi ) ∈ Rn , we write x  y if xi ≤ yi and x ≺ y if xi < yi for all i ∈ [n]  {1, 2, . . . , n}. Rn+ = {x ∈ Rn : x  0} and |x| = (|xi |) ∈ Rn+ for x = (xi ) ∈ Rn . A matrix A = (ai j ) ∈ Rm×n is nonnegative, A  0, if ai j ≥ 0 for all i, j, A is a Metzler matrix if its off-diagonal entries are nonnegative and A is an M-matrix if ai j ≤ 0 for all i = j. C([a, b], Rn ) denotes the set of Rn -valued continuous functions on [a, b] endowed with the supremum norm φC = supa≤t≤b φ(t) for a φ ∈ C([a, b], Rn ).

2.2 Auxiliary Results We recall here some concepts in nonlinear analysis and the theory of monotone dynamical systems which will be used in the derivation of our results. A vector field ϕ : Rn → Rn is said to be order-preserving on Rn+ if ϕ(x)  ϕ(y) for any x, y ∈ Rn+ satisfying x  y [27]. A mapping Ψ : Rn → Rn is proper if Ψ −1 (K ) is compact for any compact subset K ⊂ Rn . It is well-known that a continuous

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mapping Ψ : Rn → Rn is proper if and only if Ψ has the property that for any sequence { pk } ⊂ Rn ,  pk  → ∞ then Ψ ( pk ) → ∞ as k → ∞. Lemma 1 ([64]) A locally invertible continuous mapping Ψ : Rn → Rn is a homeomorphism of Rn onto itself if and only if it is proper. Next, we introduce the following Brouwer’s fixed point theorem which will be used to establish the existence of equilibrium points of neural networks models. For more details, we refer the reader to Zeidler E (1986) [65, Sect. 2.3]. Let f : X → X be a mapping from a metric or topology space X into itself. An x ∈ X is said to be a fixed point of f if it is unchanged under the effect of f , that is, f (x) = x. The following result is a simple form of Brouwer fixed point theorem. Proposition 1 ([65]) Suppose that M is a nonempty, convex, compact subset of Rn (n ≥ 1) and that f : M → M is a continuous mapping. Then f possesses at least a fixed point in M. Finally, we recall some properties of nonsingular M-matrix [66]. A matrix A = (ai j ) ∈ Rn×n is an M-matrix if it can be expressed in the form A = s In − B, where B = (bi j )  0 and s ≥ ρ(B), the maximum of the moduli of eigenvalues of B (also known as the spectral radius of B). In the aforementioned expression, A is a nonsingular M-matrix if and only if s > ρ(B). The following proposition summarizes widely used properties of nonsingular M-matrix. Proposition 2 ([66]) Let A = (ai j ) ∈ Rn×n be an M-matrix. The following statements are equivalent. (i) (ii) (iii) (iv) (v)

A is a nonsingular M-matrix. All the principal minors of A are positive. A + D is nonsingular for any nonnegative diagonal matrix D. A is inverse-positive, that is, the inverse matrix A−1 exists and A−1  0. A has a convergent regular splitting, that is, A has a representation of the form A = M − N with M −1  0, N  0 and ρ(M −1 N ) < 1 (M −1 N convergent). (vi) Every regular splitting of A is convergent. (vii) There exists a vector ζ ∈ Rn , ζ 0, such that Aζ 0. It follows from Proposition 2 that if K = (ki j ) ∈ Rn×n is a nonnegative matrix whose spectral radius ρ(K ) < 1 then (In − K )−1  0 and there exists a positive vector ζ = (ζi ) such that (In − K )ζ 0. Therefore, n 

ki j ζ j < ζi , i ∈ [n].

j=1

This fact will be useful for our later derivation.

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3 Exponential Stability of Positive Hopfield Neural Networks with Time-Varying Delay This section investigates the problem of exponential stability of a unique positive equilibrium of positive nonlinear systems with multiple time-varying delays which describe Hopfield-type neural networks with nonlinear self-inhibition rates. Based on a novel comparison technique via a differential and integral inequalities, testable conditions are derived to ensure system state trajectories converge exponentially to a unique positive equilibrium.

3.1 Model Description and Preliminaries Consider the following nonlinear system with heterogeneous delays xi (t)

= − di ϕi (xi (t)) +

n 

ai j f j (x j (t))

j=1

+

n 

(1)

bi j g j (x j (t − τ j (t))) + Ji , i ∈ [n], t ≥ 0.

j=1

System (1) describes a model of Hopfield neural networks, where n is the number of neurons in the network, x(t) = (xi (t)) ∈ Rn and J = (Ji ) ∈ Rn are the state vector and the external input vector, respectively; f (x(t)) = ( f i (xi (t))) ∈ Rn , g(x(t)) = (gi (xi (t))) ∈ Rn are neuron activation functions, ϕi (xi (t)) and di > 0, i ∈ [n], are nonlinear self-excitation rates and self-inhibition coefficients; A = (ai j ) ∈ Rn×n and B = (bi j ) ∈ Rn×n are neuron connection weight matrices; τ j (t), j ∈ [n], represent + time-varying delays satisfying 0 ≤ τ j (t) ≤ τ + j for all t ≥ 0, where τ j is a known scalar. Initial condition of system (1) is specified as x(θ) = φ(θ), θ ∈ [−τ + , 0] + n where τ + = max1≤ j≤n τ + j and φ ∈ C([−τ , 0], R is a given function. Let F be the set of continuous functions ϕ : R → R satisfying ϕ(0) = 0 and there exist positive scalars lϕ− , lϕ+ such that

lϕ− ≤

ϕ(u) − ϕ(v) ≤ lϕ+ u−v

(2)

for all u, v ∈ R, u = v. It is clear that the function class F includes all linear functions ϕ(u) = γϕ u where γϕ is some positive scalar.

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Assumptions (A1) The decay rate functions ϕi , i ∈ [n], belong to the function class F. (A2) The activation functions f j (.) and g j (.) are continuous and satisfy the following conditions 0≤ f

g j (u) − g j (v) f j (u) − f j (v) f g ≤ lj , 0 ≤ ≤ l j , ∀u = v, u−v u−v

(3)

g

where l j and l j , j ∈ [n], are positive constants. Remark 1 It follows from Assumption (A2) that the functions f (x) = ( f i (xi )) and g(x) = (gi (xi )), x = (xi ) ∈ Rn , are globally Lipschitz continuous on Rn . Thus, by utilizing fundamental results in the theory of functional differential equations [67], it can be verified that for any initial function φ ∈ C([−τ + , 0], Rn ), there exists a unique solution x(t) = x(t, φ) of (1) on the interval [0, ∞), which is absolutely continuous in t. In the sequel, each solution of (1) will be denoted simply as x(t), t ∈ [0, ∞), if it does not make any confusion. Definition 1 System (1) is said to be positive if for any initial function φ ∈ C([−τ + , 0], Rn+ ) and nonnegative input vector J ∈ Rn+ , the corresponding state trajectory is nonnegative, that is, x(t) ∈ Rn+ for all t ≥ 0. Definition 2 Given an input vector J ∈ Rn+ . A vector xe ∈ Rn+ is said to be a positive equilibrium of system (1) if it satisfies the following algebraic equation − DΦ (xe ) + A f (xe ) + Bg(xe ) + J = 0,

(4)

where the vector function Φ : Rn → Rn is defined as Φ(x) = (ϕi (xi )). Definition 3 A positive equilibrium xe of (1) is said to be globally exponentially stable if there exist positive scalars β, η such that any solution x(t) of (1) satisfies the following inequality x(t) − xe ∞ ≤ βφ − xe C e−ηt , t ≥ 0.

3.2 Positivity of Model (1) In this section, sufficient conditions are derived to ensure that for any initial function φ ∈ C +  C([−τ + , 0], Rn+ ) and nonnegative input vector J ∈ Rn+ , the corresponding state trajectory of the system is nonnegative for all time. Proposition 3 Let Assumptions (A1)–(A2) hold and assume that the connection weight matrices A, B are nonnegative (i.e. A  0 and B  0). Then, system (1) is positive for any bounded delays.

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Proof Let x(t) be a solution of (1) with initial function φ ∈ C + and input vector J ∈ Rn+ . For a given > 0, let x (t) denote the solution (1) with initial condition φ (.) = φ(.) + 1n . Note that x (t) → x(t) as → 0. Thus, it suffices to show that x (t) 0 for all t ≥ 0. Suppose in contrary that there exists an index i ∈ [n] and a t f > 0 such that xi (t f ) = 0, xi (t) > 0 for all t ∈ [0, t f ) and x j (t) ≥ 0 for all j ∈ [n]. Then, ψi (t) =

n 

ai j f j (x j (t)) +

j=1

n 

bi j g j (x j (t − τ j (t))) + Ji ≥ 0

j=1

for all t ∈ [0, t f ]. On the other hand, by condition (4), we have lϕ−i ≤

ϕi (xi (t)) ≤ lϕ+i , t ∈ [0, t f ). xi (t)

This, together with (1), leads to xi  (t) ≥ −lϕ+i xi (t) + ψi (t), t ∈ [0, t f ).

(5)

By integrating both sides of inequality (5) we then obtain xi (t) ≥ e

−lϕ+i t



 φi (0) + +

t

e

lϕ+i s

 ψi (s)ds

0

≥e

−lϕ+i t

(6)

(φi (0) + ), t ∈ [0, t f ).

Let t ↑ t f , inequality (6) gives +

0 < (φi (0) + )e−lϕi t f ≤ xi (t f ) = 0 which clearly raises a contradiction. This shows that x (t) 0 for t ∈ [0, ∞). Finally, let → 0 we obtain x(t)  0 for all t ≥ 0. The proof is completed. 

3.3 Exponential Stability of a Unique Positive Equilibrium According to Eq. (4), for a given input vector J ∈ Rn , an equilibrium of (1) exists if and only if the equation Ψ (x) = 0 has a solution xe ∈ Rn , where the mapping Ψ : Rn → Rn is defined as Ψ (x) = −DΦ(x) + A f (x) + Bg(x) + J . Clearly, Ψ is continuous on Rn . Based on Lemma 1, we have the following result.

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Proposition 4 Let Assumptions (A1)–(A2) hold. Assume that the connection weight matrices A, B are nonnegative and there exists a vector v ∈ Rn , v 0, such that n    f g ai j l j + bi j l j vi < d j lϕ−j v j , j ∈ [n],

(7)

i=1

that is,

 v  −DL − ϕ + AL f + B L g ≺ 0,

where − − L− ϕ = diag{lϕ1 , . . . , l ϕn }, f

L f = diag{l1 , . . . , lnf }, g L g = diag{l1 , . . . , lng }. Then, for a given vector J ∈ Rn , system (1) has a unique equilibrium xe ∈ Rn . Proof For any vectors x, y ∈ Rn , we have Ψ (x) − Ψ (y) = −D (Φ(x) − Φ(y)) + A[ f (x) − f (y)] + B[g(x) − g(y)]. (8) According to Eqs. (4) and (3), we have sgn(xi − yi ) (ϕi (xi ) − ϕi (yi )) ≥ lϕ−i |xi − yi |, f

sgn(x j − y j )( f j (x j ) − f j (y j )) ≤ l j |x j − y j |. Thus, by multiplying both sides of (8) with S(x − y)  diag{sgn(xi − yi )}, we obtain  S(x − y) (Ψ (x) − Ψ (y))  −DL − ϕ + AL f + B L g |x − y| which leads to  |Ψ (x) − Ψ (y)|  DL − ϕ − AL f − B L g |x − y|.

(9)

For a vector v ∈ Rn , v 0, it follows from (9) that  v  |Ψ (x) − Ψ (y)|  v  DL − ϕ − AL f − B L g |x − y|.  Therefore, if Ψ (x) = Ψ (y) then, by condition (7), v  D − AL f − B L g |x − y| = 0 and |x − y| = 0 which clearly gives x = y. This shows that the mapping Ψ is an injective mapping in Rn . On the other hand, (9) also gives

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Ψ (x)∞ ≥

 1 v  DL − ϕ − AL f − B L g |x| − Ψ (0)∞ . v∞

The above estimate implies that Ψ (xk )∞ → ∞ for any sequence {xk } ⊂ Rn satisfying xk ∞ → ∞. By Lemma 1, Ψ is a homeomorphism onto Rn , and thus, the equation Ψ (x) = 0 has a unique solution xe ∈ Rn which is an equilibrium of (1). The proof is completed.  Remark 2 It should be pointed out that the approach based on homeomorphism as presented in the proof of Proposition 4 does not make sure xe is a positive equilibrium even when J is a positive input vector. The positivity of xe will be shown simultaneously with exponential stability of xe . Remark 3 Clearly, M = −D + AL f + B L g and M are Metzler matrices. In addition, (7) holds if and only if M v < 0. This condition is feasible for a vector v 0 if and only if M (and thus M) is a Metzler–Hurwitz matrix [68]. In the following, we will show that the derived conditions in Propositions 3 and 4 ensure that system (1) is positive and the unique equilibrium point xe is positive for any given input vector J ∈ Rn+ which is globally exponentially stable. Theorem 1 Let Assumptions (A1)–(A2) hold. Assume that A  0, B  0 and there exists a vector χ ∈ Rn , χ 0, such that  Mχ = −DL − ϕ + AL f + B L g χ ≺ 0.

(10)

Then, for any J ∈ Rn+ , system (1) has a unique positive equilibrium xe ∈ Rn+ which is globally exponentially stable for any delays τ j (t) ∈ [0, τ + j ]. Proof By Proposition 4, there exists a unique equilibrium xe ∈ Rn of (1). We first prove that xe is globally exponentially stable. Indeed, let x(t) = (xi (t)) be a solution of (1). Then, by (1) and (4), we have (xi (t) − x∗i ) = − di (ϕi (xi (t)) − ϕi (x∗i )) +

n 

ai j [ f j (x j (t)) − f j (x∗ j )]

j=1

+

n 

bi j [g j (x j (t − τ j (t))) − g j (x∗ j )].

j=1

We define z(t) = |x(t) − xe | and S(x(t) − xe ) = diag{sgn(xi (t) − xei )}. It follows from (11) that

(11)

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D − z i (t) = sign(xi (t) − x∗i )(xi (t) − x∗i ) ≤ −di lϕ−i |xi (t) − x∗i | +

n 

f

ai j l j |x j (t) − x∗ j |

j=1

+

n 

g

bi j l j |x j (t − τ j (t)) − x∗ j |,

(12)

j=1

where D − z i (t) denotes the upper left Dini derivative of z i (t). Therefore, it follows from (12) that D − z i (t) ≤ −di lϕ−i z i (t) +

n 

f

ai j l j z j (t) +

j=1

n 

g

bi j l j z j (t − τ j (t)).

(13)

j=1

We now establish an exponential estimate for z(t). From (10) we have − di lϕ−i χi +

n  f g (ai j l j + bi j l j )χ j < 0, i ∈ [n].

(14)

j=1

Consider the following scalar functions Hi (η) = (η − di lϕ−i )χi +

n  j=1

f

ai j l j χ j + (

n 

g

+

bi j l j χ j )eητ , η ≥ 0, i ∈ [n].

j=1

Clearly, the function Hi (η) is continuous and strictly increasing on [0, ∞), Hi (0) < 0 and Hi (η) → ∞ as η → ∞. Thus, there exists a unique positive scalar ηi such that Hi (ηi ) = 0. Let η0 = min1≤i≤n ηi and define the following functions for i ∈ [n] ρi (t) =

χi φ − xe C e−η0 t , t ≥ 0, χ+

and ρi (t) = ρi (0), t ∈ [−τ + , 0], where χ+ = min1≤i≤n χi . Note also that + ρ j (t − τ j (t)) = eη0 τ j (t) ρ j (t) ≤ eητ ρ j (t) for any t ≥ 0. Therefore,

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−di lϕ−i ρi (t) +

n 

n 

f

ai j l j ρ j (t) +

j=1

g

bi j l j ρ j (t − τ j (t))

j=1



≤ − di lϕ−i χi +

n 

f

ai j l j χ j +

n 

j=1

 + g bi j l j χ j eη0 τ

j=1

1 × φ − xe C e−η0 t χ+ Hi (η0 ) − η0 χi ≤ φ − xe C e−η0 t . χ+

(15)

Since Hi (η) is increasing in η, Hi (η0 ) ≤ 0 for all i ∈ [n]. Thus, (15) gives ρi (t) ≥ −di lϕ−i ρi (t) +

n 

f

ai j l j ρ j (t) +

j=1

n 

g

bi j l j ρ j (t − τ j (t))

(16)

j=1

for all t ≥ 0 and i ∈ [n]. Combining (13) with (16) we obtain D − ζi (t) ≤ −di lϕ−i ζi (t) +

n 

f

ai j l j ζ j (t) +

j=1

n 

g

bi j l j ζ j (t − τ j (t)),

(17)

j=1

where ζi (t) = z i (t) − ρi (t). It follows from (17) that ζi (t) ≤ e

−di lϕ−i t

+

n  j=1

ζi (0) + g

bi j l j

n 

 f ai j l j

j=1



t

−

t

−

edi lϕi (s−t) ζ j (s)ds

0

edi lϕi (s−t) ζ j (s − τ j (s))ds, t ≥ 0.

(18)

0

Obviously, ζ(0)  0. For any t f > 0, if ζ(t)  0 for all t ∈ [0, t f ) then from (18), ζi (t f ) ≤ 0 for all i ∈ [n], that is, ζ(t f )  0. This shows that ζ(t)  0 for all t ≥ 0. Consequently, x(t) − xe ∞ ≤ ( max χi /χ+ )φ − xe C e−η0 t 1≤i≤n

by which we can conclude the exponential stability of the equilibrium xe . Finally, for any initial function φ ∈ C + , by Proposition 3, the corresponding trajectory x(t)  0 for all t ≥ 0. Thus, xe = limt→∞ x(t)  0. This shows that xe is the unique positive equilibrium of system (1). The proof is completed.  Remark 4 The result of Theorem 1 can be extended to a general class of neural networks with delays. More precisely, if the neuron activation functions f j (.) and

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g j (.) satisfy Lipschitz condition l −j f ≤

f j (u) − f j (v) g j (u) − g j (v) ≤ l +j f , l −jg ≤ ≤ l +jg , u = v, u−v u−v

for real constants l −j f , l +j f , l −jg and l +jg and assume that there exists a vector χ ∈ Rn , χ 0, such that  −DL − ϕ + |A|L f + |B|L g χ ≺ 0 where |A| = (|ai j |) and |B| = (|bi j |), then system (1) has a unique equilibrium xe which is globally exponentially stable.

3.4 Simulations Consider a nonlinear system in the form of (1), which describes a Hopfield neural networks with linear decay rates (i.e. ϕi (xi ) = xi ) and Bolzmann sigmoid activation functions [69] x

f j (x j ) = g j (x j ) =

1 − exp(− θ jj ) x

1 + exp(− θ jj )

, θ j > 0, j = 1, 2, 3.

(19)

The connection weight matrices are given by ⎡

⎤ 0.5 0.6 0.25 A = ⎣0.7 0.4 0 ⎦ , 0.2 0.15 0.1

⎡

⎤ 0.15 0 0.35 B = ⎣ 0.2 0.3 0.3 ⎦ , 0.1 0.4 0.25 f

D = diag{1.0, 1.2, 1.5}.

g

Clearly, condition (3) is satisfied with l j = l j = 2θ1 j . For illustrative purpose, let θ1 = 2.0, θ2 = 1.0 and θ3 = 1.0, we have L f = L g = diag{0.25, 0.5, 0.5} and hence ⎡ ⎤ ⎡ ⎤ 1 −0.2375 (−D + AL f + B L g ) ⎣1⎦ = ⎣ −0.475 ⎦ ≺ 0. 1 −0.975 By Theorem 1, for a given input vector J ∈ R3+ , system (1) has a unique positive equilibrium xe ∈ R3+ which is globally exponentially stable. Let J = (1.0, 1.5, 0.5) ∈ R3+ . By solving Eq. (4) using Matlab Toolbox, the equilibrium xe is obtained as xe = (2.0059, 2.1552, 0.8057) .

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2.5

3

2

2.5

2

x 1 (t)

x 2 (t)

1.5 1.5

1 1 0.5

0.5

0

0 0

5

10

t

20

25

0

30

5

t

10

(a) x1 (t)

20

25

30

(b) x2 (t)

2.5

x 3 (t)

2

1.5

1

0.5 0

5

10

t

20

25

30

(c) x3 (t) Fig. 1 State trajectories of the system with τ (t) = 5| sin(0.1t)|

Figure 1 presents 20 state trajectories of the system with a common delay τ j (t) = 5| sin(0.1t)|. Moreover, a phase diagram of 50 state trajectories is also given in Fig. 2. It can be seen that all the conducted state trajectories converge to the positive equilibrium xe . This validates the obtained theoretical results.

4 Exponential Stability of Positive Inertial Neural Networks with Heterogeneous Time-Varying Delays This section considers the problem of exponential stability of positive equilibrium of positive nonlinear systems which describe inertial neural networks model with heterogeneous time-varying delays. By utilizing comparison techniques via differential inequalities and a method of using homeomorphisms in nonlinear analysis, sufficient conditions for the existence, uniqueness and global exponential stability of a positive equilibrium are derived in terms of linear programming via M-matrices.

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Fig. 2 Convergence of the system to the EP xe = (2.0059, 2.1552, 0.8057)

4.1 Model Description Consider a model of inertial neural networks (INNs) with time-varying delays which is described by the following second-order differential system  d xi (t) d 2 xi (t) − b = − a x (t) + ci j f j (x j (t)) i i i dt 2 dt j=1 n

+

n 

di j f j (x j (t − τi j (t))) + Ii , t ≥ 0, i ∈ [n],

(20)

j=1

where n is the number of neurons, xi (t) is the state of ith neuron at time t and Ii is external input. ai > 0 is the damping coefficient, bi > 0 is the rate at which ith neuron resets its potential to the resting state in isolation when disconnected from the network and external inputs, ci j and di j are constants representing the neuron connection weights. f j (.), j ∈ [n], are neuro activation functions and τi j (t), i, j ∈ [n], are heterogeneous time-varying delays satisfying 0 ≤ τi j (t) ≤ τ + , where τ + > 0 is a known scalar. We denote x(t) = (xi (t)) ∈ Rn as the state vector, A = diag{a1 , a2 , . . . , an }, B = diag{b1 , b2 , . . . , bn }, C = (ci j )n×n , D = (di j )n×n and I = (Ii ) ∈ Rn . Then, system (20) can be written in the following vector form x  (t) = −Ax  (t) − Bx(t) + C f (x(t)) + D f (xτ (t)) + I, where f (x(t)) = ( f j (x j (t))) and f (xτ (t)) = ( f j (x j (t − τi j (t)))).

(21)

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In regard to (21), each initial condition of (20) is defined by compatible vectorvalued functions φ = (φi ) ∈ C([−τ + , 0], Rn ) and φˆ = (φˆ i ) ∈ C([−τ + , 0], Rn ) : xi (s) = φi (s), xi (s) = φˆ i (s), s ∈ [−τ + , 0], i ∈ [n].

(22)

By using the following state transformation ηi yi (t) =

d xi (t) + ξi xi (t), i ∈ [n], dt

(23)

where ηi = 0 and ξi , i ∈ [n], are constants, system (20) can be represented in the following form 

x  (t) = −Dξ x(t) + Dη y(t), y  (t) = −Dα y(t) + Dβ x(t) + Dη−1 [C f (x(t)) + D f (xτ (t)) + I ] ,

(24)

where y(t) = (y1 (t), y2 (t), . . . , yn (t)) ∈ Rn and Dξ = diag{ξ1 , ξ2 , . . . , ξn }, Dα = diag{α1 , α2 , . . . , αn }, αi = ai − ξi , βi =

ηi−1 (αi ξi

Dη = diag{η1 , η2 , . . . , ηn }, Dβ = diag{β1 , β2 , . . . , βn }, − bi ), i ∈ [n].

Proposition 5 Assume that the neuron activation functions f j (.), j ∈ [n], of model (20) satisfy Assumption (A2). Then, for any initial condition (22), there exists a unique solution of system (20) defined on the interval [0, ∞), which is absolutely continuous in t. Proof Let x(t) be a solution of system (20) with initial condition (22). Then, z(t) = (x  (t), y  (t)) ∈ R2n is a solution of system (24) with initial condition xi (s) = φi (s), yi (s) = ηi−1 (ξi φi (s) + φˆ i (s)), s ∈ [−τ + , 0].

(25)

In addition, by (A2), the function f (x) = ( f i (xi )), x = (xi ), is Lipschitz continuous on Rn . Thus, by utilizing fundamental results in the theory of functional differential equations [67], we can conclude that system (24) possesses a unique solution z(t) = ˆ on the interval [0, ∞), which is absolutely continuous in t. The proof is z(t, φ, φ) completed. 

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4.2 Positivity of INNs with Heterogeneous Delays A solution x(t) of system (20) is said to be positive if its trajectory is confined within the first orthant, that is, x(t) ∈ Rn+ for all t ≥ 0. For a fixed transformation (23), where ηi > 0 and ξi > 0, i ∈ [n], we define the following set of admissible initial functions for system (20)  AT = (φi ), (φˆ i ) ∈ C([−τ + , 0], Rn ) : φi (s) ≥ 0,

 ψi = ηi−1 (ξi φi (s) + φˆ i (s)) ≥ 0, s ∈ [−τ + , 0], i ∈ [n] .

(26)

Note that AT includes all nonnegative nondecreasing functions φi , where φˆ i (s) = φi (s). Definition 4 System (20) is said to be positive if any solution x(t) of (20) initiating from AT with input vector I = (Ii ) ∈ Rn+ is positive. The following lemma will be use in establishing conditions for the existence of positive state transformation in (23). Lemma 2 For given coefficients ai > 0 and bi > 0, i ∈ [n], there exists a transformation (23) with ηi > 0 and ξi > 0, i ∈ [n], such that Dα 0 and Dβ 0 if and only if the following condition holds ai2 − 4bi > 0, i ∈ [n].

(27)

Proof (Necessity) If there exist ηi > 0 and ξi > 0, i ∈ [n], satisfying Dα 0 and Dβ 0 then, we have ξi (ai − ξi ) − bi > 0, i ∈ [n]. Observe that ξi (ai − ξi ) − bi =

1 4

 2  2 ai − 4bi − ξi − 21 ai . Thus,

ai2 − 4bi > (2ξi − ai )2 ≥ 0, i ∈ [n]. (Sufficiency) Let condition (27) hold. Then, the constants

ξil =

ai −

 ai2 − 4bi 2

, ξiu =

ai +

 ai2 − 4bi 2

are well-defined, ξil > 0, and ξiu < a. In addition, since ξi (ai − ξi ) − bi = (ξiu − ξi )(ξi − ξil ) > 0 for all ξi ∈ (ξil , ξiu ), we have

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0 < ξi < ai , −ξi2 + ai ξi − bi > 0, i ∈ [n],

(28)

for any ξi ∈ (ξil , ξiu ). It follows from (28) that Dα 0 and Dβ 0 for any ηi > 0 and ξi ∈ (ξil , ξiu ). The proof is completed.  In the following, we assume that the coefficients ai and bi , i ∈ [n], of (20) satisfy condition (27). Then, by Lemma 2, there exists a transformation (23), where the coefficients ξi , ηi satisfy 

ξil

√

ai −

ai2 −4bi 2

ξiu

0 < ηi < ξi , ξil < ξi < ξiu , i ∈ [n],

(29)

√

ai +

a 2 −4b

i i where = and = . Hereafter, we fix a transformation sat2 isfying the above condition. The positivity of system (20) will be characterized as in the following result.

Theorem 2 Assume that the neuron activation functions f j (.), j ∈ [n], of model (20) satisfy Assumption (A2). If condition (27) is fulfilled and the connection weight matrices C = (ci j ), D = (di j ) are nonnegative (C  0, D  0), then system (20) is positive for any bounded delays. Proof Let x(t) be a solution of system (20) initiating from AT and nonnegative input vector. We will show that the corresponding solution z(t) = (x  (t), y  (t)) of system (24) is positive. Note at first that if yi (t) ≥ 0, t ∈ [0, t f ), for some t f > 0, then it follows from the first equation of (24) that  t   eξi s yi (s)ds xi (t) = e−ξi t φi (0) + ηi 0

≥ φi (0)e−ξi t ≥ 0, t ∈ [0, t f ). Thus, it is only necessary to show that y(t) = (yi (t))  0 for t ≥ 0. Let z (t) = (x  (t), y  (t)) be the solution (24) with initial functions φi and ψi = ψi + , where > 0 is a given scalar. Since φ(s) = (φi (s))  0 and ψ (s) = (ψi (s))  1n for all s ∈ [−τ + , 0], it follows from (24) that y (t) = (yi (t)) 0, t ∈ [0, t f ), for some small t f > 0. Suppose in contrary that there exist a t1 > 0 and an index i ∈ [n] such that yi (t1 ) = 0, yi (t) > 0, t ∈ [0, t1 ),

(30)

and y (t)  0 for all t ∈ [0, t1 ]. Then, by multiplying with eαi t , from (24) we have

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n  d  yi (t)eαi t = βi xi (t)eαi t + ηi−1 ci j eαi t f j (x j (t)) dt j=1

+ ηi−1

n



 di j eαi t f j (x j (t − τi j (t))) + Ii eαi t .

(31)

j=1

Therefore, it follows from (31) and the fact Dβ e Dα t x (t)  0, t ∈ [0, t1 ], that  t e Dα (s−t) f (x (s))ds Dη y (t)  e−Dα t Dη ψ (0) + C 0  t  Dα (s−t) ˆ e f (xˆ (s))ds + Dα−1 E n − e−Dα t I, +D

(32)

0

where fˆ(xˆ (s)) = ( f j (x j (s − τi j (s))) and E n is the identity matrix in Rn×n . Since the vector fields F1 (x) = C f (x) and F2 (x) = D f (x) are order-preserving, x (t)  0 and xˆ (t)  0 for t ∈ [0, t1 ], from (32), we have  −1 E n − e−Dα t I, y (t)  e−Dα t ψ (0) + Dαη

(33)

where Dαη = Dα Dη . Let t ↑ t1 in (33) and note also that E n − e−Dα t1 0, we readily obtain y (t1 )  e−Dα t1 1n 0, which yields a contradiction with (30). Therefore, x (t)  0 and y (t) 0 for all t ≥ 0. Let ↓ 0 we then obtain z(t) = lim →0 z (t)  0. The proof is completed.  Let A˜ T be the subset of AT defined by A˜ T = {φ = (φi ) ∈ C 1 ([−τ + , 0], Rn+ ) : φi (s) ≥ 0}. By Theorem 2, we have the following result. Corollary 1 Under the assumptions of Theorem 2, system (20) is positive with respect to initial conditions defined by (22), where (φi ) ∈ A˜ T and φˆ i = φi .

4.3 Positive Equilibrium In this section, we derive conditions for the existence of positive equilibrium points for INNs in the form of (20). Similar to Definition 1, we give the following definition of positive equilibrium of (1).

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Definition 5 Given an input vector I ∈ Rn+ . A vector x∗ ∈ Rn+ is said to be a positive equilibrium point (EP) of system (20) if the following algebraic equation holds − Bx∗ + C f (x∗ ) + D f (x∗ ) + I = 0.

(34)

The following auxiliary result will be used to issue the existence of positive EPs for system (20). Lemma 3 For given matrices Dη 0 and Dξ 0, a vector x∗ ∈ Rn+ is a positive EP of system (20) if and only if the augmented vector (x∗ , y∗ ), where y∗ = Dη−1 Dξ x∗ , is a positive EP of system (24), that is, 

−Dξ x∗ + Dη y∗ = 0, Dη (−Dα y∗ + Dβ x∗ ) + C f (x∗ ) + D f (x∗ ) + I = 0.

(35)

Proof Note from (34) that Dη (−Dα y∗ + Dβ x∗ ) = −Dα Dξ x∗ + (Dα Dξ − B)x∗ = −Bx∗ . 

Thus, the proof is then straightforward from (34) and (35).

By Lemma 3, system (20) possesses an EP if and only if system (24) does. As revealed from (35), for a given input vector I ∈ Rn , an EP of (20)  exists if and x only if the equation H(χ) = 0 has a solution χ∗ ∈ R2n , where χ = , x, y ∈ Rn , y and the mapping H : R2n → R2n is defined by  H(χ) =

(Dαξ

 −Dξ x + Dη y , − B)x − Dαη y + C f (x) + D f (x) + I

(36)

where Dαξ = Dα Dξ and Dαη = Dα Dη . Clearly, the mapping H(.) defined in (36) is continuous on R2n . Based on Lemmas 2 and 3, we have the following result. Theorem 3 Assume that the assumptions of Theorem 2 are satisfied and there exists a vector χ0 ∈ R2n , χ0 0, such that χ 0

 Dαξ

Dη −Dξ − B + (C + D)L f −Dαη

 ≺ 0.

(37)

  x Then, for each input vector I ∈ R , there exists a unique EP χ∗ = ∗ of system y∗ (24). n

Positivity and Stability of Nonlinear Time-Delay Systems …

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Proof We define the mappings H1 , H2 : R2n → R2n by H1 (χ) = −Dξ x + Dη y, H2 (χ) = (Dαξ − B)x − Dαη y + (C + D) f (x) + I.     x1 x and χ2 = 2 in R2n , we have Then, for any two vectors χ1 = y1 y2 H1 (χ1 ) − H1 (χ2 ) = −Dξ (x1 − x2 ) + Dη (y1 − y2 ). Therefore, S(x1 − x2 )[H1 (χ1 ) − H1 (χ2 )] = −Dξ |x1 − x2 | + S(x1 − x2 )Dη (y1 − y2 )  −Dξ |x1 − x2 | + Dη |y1 − y2 |,

(38)

where S(x1 − x2 ) = diag{sgn(x1i − x2i )}. By Assumption (A2) and Dαξ − B = Dβ Dη 0, similar to (38), we have S(y1 − y2 )[H2 (χ1 ) − H2 (χ2 )]  −Dαη |y1 − y2 | + (Dαξ − B)|x1 − x2 | + (C + D)| f (x1 ) − f (x2 )|  −Dαη |y1 − y2 | + (Dαξ − B)|x1 − x2 | + (C + D)L f |x1 − x2 |, (39) f

f

f

where L f = diag{l1 , l2 , . . . , ln }. From (38) and (39), we have diag{S(x1 − x2 ), S(y1 − y2 )}[H(χ1 ) − H(χ2 )]  −M|χ1 − χ2 |,

(40)

 −Dη Dξ . B − Dαξ − (C + D)L f Dαη On the other hand, it is clear that 

where M =

−diag{S(x1 − x2 ), S(y1 − y2 )}[H(χ1 ) − H(χ2 )]  |H(χ1 ) − H(χ2 )|.

(41)

Thus, combining (40) and (41) gives |H(χ1 ) − H(χ2 )|  M|χ1 − χ2 |.

(42)

Let χ0 be a vector satisfying (37). From (42), we have  χ 0 |H(χ1 ) − H(χ2 )|  χ0 M|χ1 − χ2 |.

(43)

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If H(χ1 ) − H(χ2 ) = 0 then it is clear from (37) and (43) that χ1 = χ2 . Thus, the mapping H(.) is injective in R2n . In addition, inequality (43) also gives H(χ)∞ ≥

1 χ M|χ| − H(0)∞ , χ0 ∞ 0

which ensures that H(χk )∞ → ∞ for any sequence {χk } in R2n satisfying χk ∞ → ∞. Thus, the continuous mapping H(.) is proper. By Lemma 2, H(.) is a homeomorphism onto R2n . Consequently, the equation H(χ) = 0 has a unique solution χ∗ ∈ R2n , which is a unique EP of (24). The proof is completed.    −Dη Dξ is an M-matrix, condition Remark 5 Since M = B − Dαξ − (C + D)L f Dαη (37) holds if and only if M is a nonsingular M-matrix. Thus, condition (37) can be verified by various equivalent conditions for nonsingular M-matrices as given in Proposition 3. Remark 6 Let condition (26) hold. Then, as shown in Lemma 2, there exist scalars ξi , ηi , i ∈ [n], satisfying (29). Furthermore, condition (37) is feasible for a vector χ0 0 if and only if the inequality  v  −B + C L f + DL f ≺ 0

(44)

is feasible for a vector v ∈ Rn , v 0. Indeed, if (37) holds then, by specifying u 0 = [E n 0n ]χ0 and v0 = [0n E n ]χ0 , we have  u 0 − Dα v0 ≺ 0,   Dξ (Dα v0 − u 0 ) + −B + L f (C  + D  ) v0 ≺ 0, which implies that (44) is feasible with v = v0 . Conversely, if (44) is feasible for a vector 0 ≺ v∈Rn then, for an ∈(0, mini∈[n] ai ), let ξi = ξiu − , Dξl = diag{ξil }, Dξu = diag{ξiu } and u = Dξl v 0. We have Dα = Dξl + E n and u − Dα v = − v ≺ 0. In addition, Dξ (Dα v − u) = (Dξu − E n )v → 0 as ↓ 0. Thus, there exists an > 0 such that ξi , i ∈ [n], satisfy (29) and Dξ (Dα v − u) + [−B + L f (C  + D  )]v ≺0. By selecting 0 < ηi < ξi , i ∈ [n], it can be verified that the vector u χ0 = satisfies (37). v

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359

4.4 Exponential Stability of Positive Equilibrium We now prove under the Assumptions of Theorem 2 that the unique equilibrium χ∗ of system (24) is positive and globally exponentially stable for any delays τ j (t) ∈ [0, τ + ]. Definition 6 An equilibrium point x∗ of (20) is said to be globally exponentially stable (GES) if there exist positive constants κ, λ such that the following exponential estimate holds for any solution x(t) of (20)   ˆ C e−λt , t ≥ 0. x(t) − x∗ ∞ ≤ κ max φ − xC , φ

(45)

Theorem 4 Assume that the neuron activation functions f j (.), j ∈ [n], satisfy Assumption (A2). If condition (27) is satisfied, C = (ci j )  0, D = (di j )  0 and there exists a vector χˆ 0 ∈ R2n , χˆ 0 0, such that  Dαξ

 Dη −Dξ χˆ 0 ≺ 0. − B + (C + D)L f −Dαη

Then, for any input vector I ∈ Rn+ , there exists a unique positive EP χ∗ = of system (24) which is GES for any delays τi j (t) ∈ [0, τ + ].

(46)   x∗ ∈ R2n + y∗

Proof The proof will be divided into four steps. Step 1. Since M is an M-matrix, conditions (37) and (44) are equivalent. Thus, by Theorem 3, for any input vector I ∈ Rn+ , there exists a unique EP χ∗ of system (24), which satisfies 

−Dξ x∗ + Dη y∗ = 0, (Dαξ − B)x∗ − Dαη y∗ + (C + D) f (x∗ ) + I = 0.

(47)

Step 2. We define the following transformations u i (t) = |xi (t) − xi∗ |, vi (t) = |yi (t) − yi∗ |, uˆ i (t) = |xi (t − τi (t)) − xi∗ |, i ∈ [n], u(t) = (u i (t)), v(t) = (vi (t)), u(t) ˆ = (uˆ i (t)). It follows from (24) and (47) that D + u i (t) = sgn(xi (t) − xi∗ )(xi (t) − xi∗ ) = −ξi u i (t) + ηi sgn(xi (t) − xi∗ )(yi (t) − yi∗ ) ≤ −ξi u i (t) + ηi vi (t).

(48)

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L. V. Hien

Similarly, we have D + (ηi vi (t)) ≤ −αi ηi vi (t) + (αi ξi − bi )u i (t) +

n 

ci j | f j (x j (t)) − f j (x j∗ )|

j=1

+

n 

di j | f j (x j (t − τi j (t))) − f j (x j∗ )|

j=1

≤ −αi ηi vi (t) + (αi ξi − bi )u i (t) +

n    f f ci j l j u j (t) + di j l j uˆ j (t) . j=1

(49) Combining (48) and (49) gives D

+



     u(t) u(t) u(t) ˆ  −Ξ1 + Ξ2 . Dη v(t) v(t) 0

   Dξ 0 0 −Dη . where Ξ1 = and Ξ2 = DL f 0 B − Dαξ − C L f Dαη We will show that there exists a scalar λ > 0 such that   1 + ξ∞ u(t)  φ∗ e−λt χˆ 0 , ∀t ≥ 0, Dη v(t) min1≤i≤n χˆ 0i

(50)



(51)

  ˆ C . where φ∗  = max φ − x∗ C , φ Step 3. By condition (46), (−Ξ1 + Ξ2 ) χˆ 0 ≺ 0. Thus, there exists a λ∗ > 0 such that   + (52) λE 2n − Ξ1 + eλτ Ξ2 χˆ 0  0 for any λ ∈ (0, λ∗ ]. ˆ = κφ∗ e−λt As revealed in (50)–(52), we consider the scaling function ζ(t) 1+ξ∞ χˆ 0 , t ≥ 0, where κ = min1≤i≤n χˆ 0i . It is clear that ζ(t)  κφ∗ χˆ 0 for all t ∈ [−τ + , 0], where ζ(t) = [u  (t) Dη v  (t)] . ˆ Thus, for a fixed θ > 1, ζ(0) ≺ θζ(0). ˆ does not hold for all Assume in contrary that the estimation ζ(t) ≺ θζ(t) t > 0, then there exists a t f > 0 and an index i ∈ [2n] such that ˆ ζi (t f ) = θζˆi (t f ), ζ(t) ≺ θζ(t), t ∈ [0, t f ).

(53)

+ Note also that ζˆ j (t − τi j (t)) = eλτi j (t) ζˆ j (t) ≤ eλτ ζˆ j (t) for any j ∈ [n]. Therefore, by (51), we have

Positivity and Stability of Nonlinear Time-Delay Systems …

ˆ  −Ξ1 ζ(t) ˆ + Ξ2 ζˆd (t), D + ζ(t)

361

(54)

where ζˆd (t) = col{ζˆ j (t − τi j (t)), ζˆn+ j (t)}, j ∈ [n]. If i ∈ [n] then, from (48) and (54), we have     D + ζi (t) − θζˆi (t) ≤ −ξi ζi (t) − θζˆi (t) , and hence, ζi (t) − θζˆi (t) ≤ (ζi (0) − θζˆi (0))e−ξi t , t ∈ [0, t f ). Let t ↑ t f we obtain ζi (t f ) − θζˆi (t f ) ≤ (ζi (0) − θζˆi (0))e−ξi t f < 0, which clearly contradicts with (53). Similarly, if i = n + k, k ∈ [n], then from (49) and (54), we have     D + ζi (t) − θζˆi (t) ≤ −αk ηk ζi (t) − θζˆi (t) , and hence, ζi (t) − θζˆi (t) ≤ (ζi (0) − θζˆi (0))e−αk ηk t , t ∈ [0, t f ), which also ˆ gives a contradiction with (53) by letting t ↑ t f . This shows that ζ(t) ≺ θζ(t) for all t ≥ 0. Let θ ↓ 1 we obtain (51), which ensures that the EP χ∗ of system (24) is GES. Step 4. Positivity of χ∗ : For a given vector u 0 ∈ Rn+ , let z 0 (t) = [x  (t) y  (t)] be the solution of system (24) with constant initial condition determined as φi = u 0i and φˆ i = 0. By Theorem 2, z 0 (t)  0 for all t ≥ 0. On the other hand, according to the result of Step 3, we have ˆ diag{In , Dη }|z 0 (t) − χ∗ | ≤ ζ(t), which converges exponentially to zero. Therefore, χ∗ = lim z 0 (t)  0. t→∞

This shows that the unique EP χ∗ of system (24) is positive and GES. The proof is completed.  We now summarize the results of this section in the following theorem. Theorem 5 Consider a model of INNs with heterogeneous delays in the form of (20). Assume that the neuron activation functions f j (.) of (20) satisfy Assumption (A2) and the following conditions hold (i) ai2 − 4bi > 0, i ∈ [n]; (ii) C = (ci j )  0, D = (di j )  0;

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L. V. Hien

(iii) There exists a vector v ∈ Rn , v 0, such that  −B + C L f + DL f v ≺ 0.

(55)

Then, the following assertions are true (a) System (20) is positive subject to initial conditions in AT . (b) For a given input vector I ∈ Rn+ , there exists a unique positive EP x∗ of (20) which is GES for any delays τi j (t) ∈ [0, τ + ]. Moreover, there exists a positive constant λ∗ such that any solution x(t) of (20)–(21) satisfies the following exponential estimation x(t) − x∗ ∞ ≤

  (1 + σ0 )σ1 ˆ C e−λ∗ t , t ≥ 0, max φ − xC , φ σ2

(56)

        where σ0 = maxi 21 ai + ai2 − 4bi , σ1 = maxi vi , 21 ai − ai2 − 4bi vi and      σ2 = mini vi , 21 ai − ai2 − 4bi vi . Remark 7 The mapping F : Rn → Rn defined by F(x) = (C + D) f (x) is an order-preserving vector field by condition (ii) of Theorem 4. Thus, for non-zero input vector I ∈ Rn+ , the positive EP x∗ mentioned in statement (b) is different from the origin. Moreover, x∗ is strictly positive (i.e. x∗ 0) if the input vector is positive (i.e. I 0). Remark 8 Conditions (55) and (i) and (ii) ensure that any solution of system (20) initiating from AT is eventually positive for positive input vector I 0. Specifically, let x(t) be a solution of (20) with initial condition in AT . Since x(t) converges exponentially to the unique EP x∗ 0, x(t) 0 for all t in a sufficiently large interval [t f , ∞).

4.5 Numerical Examples In this section, two numerical examples with simulations are given to illustrate the effectiveness of the results presented in the preceding section. Example 1 Consider a class of nonlinear systems which describe cooperative-type INNs in the form of (20) with Boltzmann sigmoid activation functions [69]  1 − exp −  f j (x j ) = 1 + exp −

xj θj xj θj

, θ j > 0, j = 1, 2, 3.

(57) f

Clearly, the activation functions (57) satisfy Assumption (A2) with l j =

1 2θ j

.

Positivity and Stability of Nonlinear Time-Delay Systems … Fig. 3 Convergence of positive solutions to x∗

363

5

4

Responex(t)

x 2 (t)

x 1 (t)

3

2

x 3 (t) 1

0 0

10

20

t

30

40

50

Let the coefficients ai , bi , and connection weights ci j , di j be given by a1 = 2.0, a2 = 1.8, a3 = 2.5, b1 = 1.25, b2 = 0.75, b3 = 1.5, ⎡ ⎤ ⎡ ⎤ 0.3 0.6 0.7 0.15 0.5 0.4 C = ⎣1.0 0.8 0.2⎦ , D = ⎣ 0.3 0.5 0.6⎦ . 0.1 0.25 0.4 0.15 0.35 0.6 For θ1 = 2.0, θ2 = 2.0 and θ3 = 1.0, we have L f = diag{0.25, 0.25, 0.5}. Thus, ⎡ ⎤ ⎡ ⎤ 1 −0.0375 (−B + C L f + DL f ) ⎣2⎦ = ⎣ −0.125 ⎦ ≺ 0. 1 −0.6375 Therefore, conditions (i) and (ii) of Theorem 4 and condition (55) are fulfilled. By Theorem 4, for an input vector I ∈ R3+ , the system has a unique positive EP x∗ ∈ R3+ , which is GES for any bounded delays τ j (t), j = 1, 2, 3. To simulate the result, we take I = (1.5, 1.0, 2.0) ∈ R3+ then, by solving Eq. (27) we obtain x∗ = (2.8703, 4.7071, 2.313) . A simulation result of 20 sample state trajectories with a common delay τi j (t) = 1 + 4| sin(10πt)| is represented in Fig. 3. It can be seen that all conducted state trajectories are positive and converge to the unique positive EP x∗ , which support the obtained theoretical results. Example 2 Consider the following INN model with a time-varying delay τ (t)    0.28 0.33 ln(1 + exp(x1 (t))) x (t) + 2x (t) = − 0.9x(t) + tanh(x2 (t)) 0.36 0.15    0.12 0.18 ln(1 + exp(x1 (t − τ (t)))) + I. + tanh(x2 (t − τ (t))) 0.21 0.16 



(58)

364 Fig. 4 Convergence of positive solutions of (58) to a unique positive EP

L. V. Hien 6

5

x2

4

3

2

1

1

2

3

x1

4

5

6

It is easy to verify that the activation functions of model (58) satisfy Assumption (A2) and  conditions (i), (ii)  of Theorem 4 are satisfied. In addition, since B − (C + 0.5 −0.51 is a nonsingular M-matrix, condition (55) is also satisfied. D)L f = −0.57 0.59 By Theorem 4, system (58) is positive and, for a given input vector I ∈ R2+ , there exists a unique positive EP x∗ which is GES. In the simulation result given in Fig. 4, we simulate with τ (t) = 5| sin(20πt)| and input vector I = (1, 1) . It can be seen that the corresponding state trajectories converge to the unique point x∗ , which strongly supports the obtained theoretical results.

5 Exponential Stability of Positive BAM Neural Networks with Multiple Delays This section is concerned with the exponential stability of a unique equilibrium point of a class of nonlinear time-delay systems which describe positive neural networks in bidirectional associative memory (BAM) with multiple time-varying delays and nonlinear self-excitation rates. A systematic approach involving comparison techniques via differential inequalities in combination with the use of Brouwer’s fixed point theorem and M-matrix theory is presented.

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5.1 Model Description and Preliminaries Consider the following nonlinear time-delay system concerning BAM neural networks with multiple delays ⎧ m m   ⎪ ⎪  ⎪ x (t) = −α ϕ (x (t)) + a f (y (t)) + bi j f j (y j (t − σ j (t))) + Ii , i i i ij j j ⎪ i ⎨ j=1

j=1

n n   ⎪ ⎪  ⎪ y (t) = −β ψ (y (t)) + c g (x (t)) + d ji gi (xi (t − τi (t))) + J j , ⎪ j j j ji i i ⎩ j i=1

i=1

(59) where n and m are the number of neurons in X -layer and Y -layer, respectively; xi (t) and y j (t) are the state variables; ϕi , ψ j are decay rate functions and αi > 0, β j > 0 are self-inhibition coefficients. For linear decay rates (that is, ϕi (xi ) = xi and ψ j (y j ) = y j ), αi and β j are respectively the rates at which ith and jth neurons will reset their potential to the resting state in isolation when disconnected from the network and external inputs. In system (59), f j (.) and gi (.) are neuron activation functions, ai j , bi j , c ji and d ji are neuron connection weights. Ii and J j are external inputs to the ith neuron and jth neuron, respectively. τi (t) and σ j (t) denote the communication delays between neurons which satisfy 0 ≤ τi (t) ≤ τ , i ∈ [n], 0 ≤ σ j (t) ≤ σ, j ∈ [m],

(60)

where τ and σ are known positive constants. Initial conditions of system (59) are specified as follows x(t0 + ξ) = x 0 (ξ), ξ ∈ [−τ , 0],

y(t0 + θ) = y 0 (θ), θ ∈ [−σ, 0],

(61)

where x(t) = (xi (t)) ∈ Rn and y(t) = (y j (t)) ∈ Rm denote the state vectors in neuron fields FX and FY , respectively, and x 0 ∈ C([−τ , 0], Rn ), y 0 ∈ C([−σ, 0], Rm ) are initial functions specifying initial states of the system. We denote the matrices A = (ai j ) ∈ Rn×m , B = (bi j ) ∈ Rn×m , C = (c ji ) ∈ Rm×n , D = (d ji ) ∈ Rm×n , Dα = diag{α1 , α2 , . . . , αn }, Dβ = diag{β1 , β2 , . . . , βm }, and vectors I = (Ii ) ∈ Rn , J = (J j ) ∈ Rm . System (59) can be written as 

x  (t) = −Dα Φ(x(t)) + A f (y(t)) + B f (y(t − σ(t))) + I, y  (t) = −Dβ Ψ (y(t)) + Cg(x(t)) + Dg(x(t − τ (t))) + J,

where the associated vector-valued functions are given by

(62)

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 Φ(x(t)) = (ϕi (xi (t))) , Ψ (y(t)) = ψ j (y j (t)) ,   f (y(t)) = f j (y j (t)) , f (y(t − σ(t))) = f j (y j (t − σ j (t))) , g(x(t)) = (gi (xi (t))) , g(x(t − τ (t))) = (gi (xi (t − τi (t)))) . We recall here the set F of continuous functions ϕ : R → R satisfying ϕ(0) = 0 and there exist positive scalars l − (ϕ), l˜+ (ϕ) such that the following condition lϕ− ≤

ϕ(u) − ϕ(v) ≤ lϕ+ , u = v. u−v

(63)

Assumption (A3) The decay rate functions ϕi and ψ j , i ∈ [n], j ∈ [m], belong to the function class F. Remark 9 If the functions ϕi , ψ j are Lipschitz continuous which have lower bounded derivatives in the sense that γi = inf ϕi (u) > 0, μ j = inf ψ j (u) > 0 u∈R

u∈R

then Assumption (A3) is satisfied, where lϕ−i = γi and lψ−j = μ j . On the other hand, it follows from (63) that any function ϕ in F is also a continuous and strictly increasing function. Thus, there exists continuous inverse function ϕ−1 of ϕ. Moreover, ϕ−1 also belongs to F with lϕ−−1 = 1/lϕ+ and lϕ+−1 = 1/lϕ− . Assumption (A4) The neuron activation functions f j (.), gi (.), i ∈ [n], j ∈ [m], are f g continuous, f j (0) = 0, gi (0) = 0, and there exist positive constants L j , L i such that 0≤

f j (u) − f j (v) gi (u) − gi (v) f g ≤ Lj, 0 ≤ ≤ L i , u = v. u−v u−v

(64)

The following result will be useful for our later derivations. Proposition 6 Let Assumptions (A3) and (A4) hold. For any initial functions x 0 ∈ C([−τ , 0], Rn ), y 0 ∈ C([−σ, 0], Rm ) specified in (61), there exists a unique solution (x(t), y(t)) of system (62) defined on [t0 , ∞), which is absolutely continuous in t.   x(t) Proof We denote augmented state vector χ(t) = and the function space y(t)   0  x Cd = φ = 0 : x 0 ∈ C([−τ , 0], Rn ), y 0 ∈ C([−σ, 0], Rm ) . y Define a function F : [t0 , ∞) × Cd → Rn+m by       −Dα Φ(x 0 (0)) + A f j (y 0j (0)) + B f j (y 0j (−σ j (t))) + I F (t, φ) = ,   −Dβ Ψ (y 0 (0)) + C gi (xi0 (0)) + D gi (xi0 (−τi (t))) + J

Positivity and Stability of Nonlinear Time-Delay Systems …

367

then system (62) can be written in the following abstract functional differential equation [67]  0 x  (65) χ (t) = F (t, χt ) , t ≥ t0 , χt0 = 0 , y   x where χt = t ∈ Cd and xt ∈ C([−τ , 0], Rn ), yt ∈ C([−σ, 0], Rm ) are defined as yt xt (ξ) = x(t + ξ), ξ ∈ [−τ , 0], yt (θ) = y(t + θ), θ ∈ [−σ, 0]. By Assumptions (A3) and (A4), the function F(t, φ) is continuous and satisfies Lipschitz condition with respect to φ. Thus, the problem described by Eq. (65) possesses a unique solution χ(t) which is absolutely continuous in t on a maximal interval [t0 , t f ). We now show that t f = ∞. In contrast, if t f < ∞ then lim supt→t f χ(t) = ∞. On the other hand, it follows from (65) that 

t

χ(t) = χ(t0 ) +

F(s, χs )ds, t ∈ [t0 , t f ).

(66)

t0

In addition, we can deduce by a direct computation from the expression of function F(t, φ) that there exist positive scalars 1 , 2 such that  ˆ + yˆ (t) , F(t, χt ) ≤ 1 + 2 x(t) where x(t) ˆ = sup−τ ≤ξ≤0 x(t + ξ) and yˆ (t) = sup−σ≤θ≤0 y(t + θ). Therefore, according to (66), we have 

t

χ(t) ≤ χ(t0 ) + 1 (t − t0 ) + 2

(s)ds,

t0

where (t) = x(t) ˆ + yˆ (t), which leads to  (t) ≤ 2χ(t0 ) + 2 1 (t − t0 ) + 2 2

t

(s)ds.

(67)

t0

By the Gronwall–Bellman inequality [70], from (67), we readily obtain (t) ≤



1  2 2 (t−t0 ) e − 1 + 2χ(t0 )e2 2 (t−t0 ) , t ∈ [t0 , t f ),

2

which yields lim sup χ(t) ≤ t→t f



1  2 2 (t f −t0 ) e − 1 + 2χ(t0 )e2 2 (t f −t0 ) < ∞.

2

This contradiction shows that t f = ∞. The proof is completed.



368

L. V. Hien

A solution χ(t) = vec(x(t), y(t)) of system (62) is said to be a positive if its for all t ≥ t0 . We trajectory is confined within the first orthant, that is, χ(t) ∈ Rn+m + define the following admissible set of initial conditions for system (62)   0 x A = φ = 0 ∈ C([−τ , 0], Rn ) × C([−σ, 0], Rm ) : y x (ξ)  0, ∀ξ ∈ [−τ , 0], 0

 y (θ)  0, ∀θ ∈ [−σ, 0] . 0

(68)

Definition 7 System (62) is said to be positive if any solution χ(t) of (62) initiating is positive. from A with nonnegative input vector vec(I, J ) ∈ Rn+m + Definition 8 For a given vector J = vec(I, J ) ∈ Rn+m , an augmented vector χ∗ = vec(x ∗ , y ∗ ), x ∗ ∈ Rn , y ∗ ∈ Rm , is said to be an equilibrium point (EP) of system (62) if it satisfies the following algebraic system 

−Dα Φ(x ∗ ) + (A + B) f (y ∗ ) + I = 0 −Dβ Ψ (y ∗ ) + (C + D) g(x ∗ ) + J = 0.

(69)

χ∗ is a positive EP if it is an EP of system (62) and χ∗  0. Definition 9 A positive EP χ∗ = vec(x ∗ , y ∗ ) of system (62) is said to be globally exponentially stable (GES) if there exist positive scalars κ, γ satisfy the following exponential estimation  x(t) − x ∗  + y(t) − y ∗  ≤ κ x 0 − x ∗ ∞ + y 0 − y ∗ ∞ e−γ(t−t0 ) , t ≥ t0 .

5.2 Positivity of BAM Neural Networks with Delays The positivity of BAM neural networks model (62) is presented in the following theorem. Theorem 6 Let Assumptions (A3) and (A4). Assume that the connection  weight A B matrices A, B, C, and D are nonnegative (equivalently, M =  0). C  D Then, system (62) is positive subject to bounded delays (60) and admissible initial conditions in A. The following auxiliary result will be employed to prove Theorem 6. Lemma 4 Given a function ϕ in class F and q(t) is a nonnegative continuous function on [0, ∞). Then, any solution of the initial value problem x  (t) = −ϕ(x(t)) + q(t), t ≥ t0 , x(t0 ) = x0 > 0,

(70)

Positivity and Stability of Nonlinear Time-Delay Systems …

369

satisfies x(t) > 0 for t ≥ t0 . Proof Suppose in contrary that there exists a t¯ > t0 such that x(t¯) = 0 and x(t) > 0 for t ∈ [t0 , t¯). By condition (63), we have lϕ− ≤

ϕ(x(t)) ≤ lϕ+ , t ∈ [t0 , t¯). x(t)

This, in regard to (70), leads to x  (t) ≥ −lϕ+ x(t) + q(t), t ∈ [t0 , t¯).

(71)

By taking integral both sides of (71) we then obtain    t + + elϕ (s−t0 ) q(s)ds x(t) ≥ e−lϕ (t−t0 ) x0 + t0

≥e

−lϕ+ (t−t0 )

x0 , t ∈ [t0 , t¯).

(72)

Let t ↑ t¯, inequality (72) gives 0 < x0 e−lϕ (t¯−t0 ) ≤ x(t¯) = 0 +

which yields a contradiction. This shows that x(t) > 0 for t ∈ [t0 , ∞). The proof of Lemma 4 is completed.  Remark 10 It should be mentioned here that for any function ϕ ∈ F and positive scalar λ, the function ϕλ (x) = λϕ(x) is also belonging to F. Thus, the conclusion of Lemma 4 is still valid for function ϕλ (x), where λ is a positive scalar. Proof (Proof of Theorem 6) For an initial function φ ∈ A and input vector J = (I, J ) ∈ Rn+m + , by Proposition 6, there exists a unique solution χ(t) = (x(t), y(t)) of system (62) on interval [t0 , ∞). We will show that χ(t) is a positive solution. Note at first that, by Assumption (A4), the vector fields A f (y) and B f (y) are orderpreserving on Rm + [27]. Thus, if y(t) 0 for t ∈ [−σ, t1 ) then qi (t) 

m  j=1

ai j f j (y j (t)) +

m 

bi j f j (y j (t − σ j (t))) + Ii ≥ 0, t ∈ [t0 , t1 ), i ∈ [n].

j=1

By Lemma 4, xi (t) ≥ 0 for all t ∈ [t0 , t1 ), i ∈ [n], and therefore χ(t)  0, t ∈ [t0 , t1 ). Let χ (t) = vec(x (t), y (t)) be the solution of system (62) with initial condition φ = φ + 1n+m , where > 0 is fixed. Since χ (t0 )  1n+m 0, there exists a t1 > t0 such that χ (t) 0 for t ∈ [t0 , t1 ).

370

L. V. Hien

Suppose in contrary that there exist a t˜ > t0 and an index j ∈ [m] such that y j (t˜) = 0,

y j (t) > 0, t ∈ [t0 , t˜),

(73)

and y (t)  0 for t ∈ [t0 , t˜]. By Lemma 4 and Assumption (A4), x (t)  0 for all t ∈ [−τ , t˜]. Therefore, q j (t) 

n 

c ji gi (xi (t)) +

i=1

n 

d ji gi (xi (t − τi (t))) + J j ≥ 0, t ∈ [t0 , t˜].

i=1

Similar to the proof of Lemma 4, we have y j (t) ≥ e

−β j lψ+ (t−t0 )

≥ e



j

−β j lψ+ (t−t0 ) j

 y 0j (0)

t

+ +

e

β j lψ+ (s−t0 ) j

 q j (s)ds

t0

, t ∈ [t0 , t˜).

(74)

Let t ↑ t˜ in (74), we obtain y j (t˜) ≥ e

−β j lψ+ (t˜−t0 ) j

>0

which contradicts with (73). Therefore, y (t) 0 and x (t)  0 for all t ≥ t0 . Let ↓ 0 we obtain χ(t) = lim ↓0 χ (t)  0 for t ∈ [t0 , ∞). The proof is completed. 

5.3 Positive Equilibriums In this section, we utilize Brouwer’s fixed point theorem, to establish the existence of a positive EP for model (62). Noticing that a vector χ∗ = vec(x ∗ , y ∗ ) ∈ Rn+m is an EP of model (62) if and only if it satisfies the following algebraic system ⎧   ⎨ D −1 (A + B) f (y ∗ ) + I = Φ(x ∗ ) α   ⎩ D −1 (C + D)g(x ∗ ) + J = Ψ (y ∗ ), β

(75)

where J = (I, J ) ∈ Rn+m is a given input vector. Since the functions ϕi , ψ j are continuous and strictly increasing, there exist inverse functions ϕi−1 and ψ −1 j , i ∈ [n], j ∈ [m], which also belong to the function class F. Let Φ −1 and Ψ −1 be the vectorvalued functions defined by Φ −1 (x) = (ϕi−1 (xi )) and Ψ −1 (x) = (ψ −1 j (x j )). Then, system (75) can be written as

Positivity and Stability of Nonlinear Time-Delay Systems …

371

 −1  −1 Φ Dα ((A + B) f (y ∗ ) + I ) = x ∗

Ψ −1 Dβ−1 ((C + D)g(x ∗ ) + J ) = y ∗ .

The above system reveals that each EP χ∗ = vec(x∗ , y∗ ) of system (62) is a fixed point of the mapping H : Rn+m → Rn+m defined by  H(χ) =

  Φ −1  Dα−1 ((A + B) f (y) + I ) 

Ψ −1 Dβ−1 ((C + D)g(x) + J )

,

(76)

where χ = vec(x, y), x ∈ Rn and y ∈ Rm . Specifically, we rewrite the mapping H(χ) defined in (76) as   H(χ) = h 1 (y) · · · h n (y) h˜ 1 (x) · · · h˜ m (x) , where ⎛

⎞   m  1 h i (y) = ϕi−1 ⎝ ai j + bi j f j (y j ) + Ii ⎠ , i ∈ [n], αi j=1 %  n & 1  ˜h j (x) = ψ −1 c ji + d ji gi (xi ) + J j , j ∈ [m]. j β j i=1 The, according to Eqs. (75) and (76), a vector χ∗ ∈ Rn+m is an EP of model (62) if and only if it is a fixed point of the mapping H(χ), that is, H(χ∗ ) = χ∗ . Based on the Brouwer’s fixed point theorem, we have the following result. Theorem 7 Let Assumptions (A3) and (A4) hold. Assume that  ρ

0n×n K1 K2 0m×m

 < 1,

where K1 = (ki1j ) ∈ Rn×m , K2 = (k 2ji ) ∈ Rm×n and 1 αi lϕ−i 1 k 2ji = β j lψ−j ki1j =

 f |ai j | + |bi j | L j ,  g |c ji | + |d ji | L i .

Then, system (62) possesses at least an equilibrium point.

(77)

372

L. V. Hien

Proof By Proposition 2 and condition (77), In+m − K is a nonsingular M-matrix,  0n×n K1 . where K = K2 0m×m Thus, there exists nonnegative inverse matrix (In+m − K)−1  0. In addition, there exists a vector η ∈ Rn+m , η 0, such that (In+m − K)η 0. We now define the following vector       δ η −1 θ1 + 1 , = (In+m − K)  θ2 η2 ' () *

(78)

η

where θ1 = (θi1 ) ∈ Rn , θ2 = (θ2j ) ∈ Rm and θi1 =

1 1 |Ii |, i ∈ [n], θ2j = |J j |, j ∈ [m]. − αi lϕi β j lψ−j

It is obvious that δ 0 and  0. Furthermore, by (78), we have       δ δ θ =K + 1 + (In+m − K) η.   θ2

(79)

Based on Eq. (79), we define the following convex compact subset    + δ n+m + . B= χ∈R + |χ|   For any χ = vec(x, y) ∈ B, x ∈ Rn , y ∈ Rm , we have |x|  δ and |y|  . Thus, |xi | ≤ δi , i ∈ [n], |y j | ≤  j , j ∈ [m]. On the other hand, since ϕi ∈ F, we have + −1 + +ϕ (u)+ ≤ (1/l − )|u|. ϕi i This, in combination with Assumption (A4), gives

Positivity and Stability of Nonlinear Time-Delay Systems …

373

+ ⎛ ⎞+ +  m  + + −1 1  +  ai j + bi j f j (y j ) + Ii ⎠++ |h i (y)| = ++ϕi ⎝ αi j=1 + + + + + m + + 1 ++  ≤ ai j + bi j f j (y j ) + Ii ++ + − αi lϕi + j=1 + ⎞ ⎛ m 1 ⎝  ≤ |ai j | + |bi j | | f j (y j )| + |Ii |⎠ αi lϕ−i j=1 ⎞ ⎛ m f 1 ⎝  ≤ |ai j | + |bi j | L j |y j | + |Ii |⎠ αi lϕ−i j=1 ≤

m 

ki1j  j + θi1 , i ∈ [n].

(80)

j=1

Similarly, we also have + %  n &++ +  1 + + −1 |h˜ j (x)| = +ψ j c ji + d ji gi (xi ) + J j + + + β j i=1 & % n  g 1 ≤ |c ji | + |d ji | L i |xi | + |J j | − β j lψ j i=1 ≤

n 

k 2ji δi + θ2j ,

j ∈ [m].

(81)

i=1

It follows from (76) and (80)–(81) that    K1  θ + 1 K2 δ θ2     δ θ =K + 1 .  θ2 

|H(χ)| 

(82)

From Eqs. (79) and (82) we readily obtain   δ |H(χ)| ≺  for any χ ∈ B meaning that H(B) ⊂ B. Therefore, the continuous mapping H(χ) defined by (76) maps the convex compact subset B ⊂ Rn+m into its self. By Brouwer’s fixed point theorem [65], the mapping H has at least one fixed point χ∗ ∈ B (i.e.  H(χ∗ ) = χ∗ ), which is also an EP of system (62). The proof is completed.

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L. V. Hien

Remark 11 In the proof of Theorem 7, if the connection weight matrices A, B, C, D, and input vectors I, J are nonnegative, then ui 

1 αi

and vj 

1 βj

 m

 ai j + bi j f j (y j ) + Ii

 ≥ 0, i ∈ [n],

j=1

 n

 c ji + d ji gi (xi ) + J j

 ≥ 0, j ∈ [m],

i=1

for any x  0 and y  0. Thus, h i (y) = ϕ−1 (u i ) ≥ 0 and h˜ j (x) = ψ −1 j (v j ) ≥ 0 for  n+m χ = vec(x, y)  0. Therefore, H Rn+m ⊂ R . This shows that H : B+ → B+ , + + . Since where B+ = B ∩ Rn+m +    + δ + B+ = χ ∈ Rn+m + 0  χ   is also a convex compact subset of Rn+m , by Brouwer’s fixed point theorem, the continuous mapping H has at least one fixed point χ∗+ ∈ B+ , which is a positive EP of system (62). We summarize this result in the following corollary. Corollary 2 Under the assumptions of Theorem 7, if the connection weight matrices A, B, C and D are nonnegative, then for a given input vector J = vec(I, J ) ∈ Rn+m + , system (62) has at least one positive EP χ∗ ∈ Rn+m + . Remark 12 Let Ω be an (n + m) × (n + m) block matrix of the form  Ω=

U V W Z



where U ∈ Rn×n , V ∈ Rn×m , W ∈ Rm×n and Z ∈ Rm×m . If Z is nonsingular, then we have 

Therefore,

U − V Z −1 W V 0m×n Z



 =

U V W Z



 0n×m In . −Z −1 W Im

det(Ω) = det(U − V Z −1 W ) det(Z ).

By utilizing equality (83), for any λ = 0, we have

(83)

Positivity and Stability of Nonlinear Time-Delay Systems …

375

 λIn −K1 −K2 λIm   1 = λm det λIn − K1 K2 λ 

det(λIn+m − K) = det

= λm−n det(λ2 In − K1 K2 ). The above identity shows that λ ∈ σ(K) \ {0} if and only if μ = λ2 ∈ σ(K1 K2 ) \ {0}. Consequently, ρ(K) < 1 ⇐⇒ ρ(K1 K2 ) < 1. Based on the above observation, we have the following results. Proposition 7 Condition (77) holds if and only if one of the two following conditions does ⎛ ⎞   m    1 1 −1 f g ⎠ < 1, (84) ρ⎝ l (ψ j ) |ai j | + |bi j | |c jk | + |d jk | L j L k αi lϕ−i j=1 β j n×n % &  n  1  1 −1 f g ρ l (ϕi ) |c ji | + |d ji | (|aik | + |bik |) L k L i < 1. (85) β j lψ−j i=1 αi m×m  A B 0 Corollary 3 Let Assumptions (A3) and (A4) hold. Assume that C  D and either condition (84) or condition (85) is satisfied. Then, for a given input vector J = vec(I, J ) ∈ Rn+m + , system (62) has at least a positive equilibrium. 

5.4 Exponential Stability of Positive Equilibrium In this section, we will prove that under the assumptions of Theorems 6 and 7 system (62) has a unique positive EP χ∗ , which is globally exponentially stable. Theorem 8 Let Assumptions (A3) and (A4) hold. Assume that the connection weight matrices are nonnegative and one of the three conditions (77), (84), or (85) is satisfied. Then, for any input vector J = vec(I, J ) ∈ Rn+m + , system (62) has a unique , which is GES for any delays τ (t) ∈ [0, τ ] and σ j (t) ∈ [0, σ]. positive EP χ∗ ∈ Rn+m i +

376

L. V. Hien

Proof By Theorem 7, for a given input vector J = vec(I, J ) ∈ Rn+m + , there exists a positive EP χ∗ = vec(x ∗ , y ∗ ) of system (62) which satisfies the following algebraic system  −Dα Φ(x ∗ ) + (A + B) f (y ∗ ) + I = 0 (86) −Dβ Ψ (y ∗ ) + (C + D)g(x ∗ ) + J = 0. We now prove exponential attractivity of the EP χ∗ . Since conditions (77), (84) and (85) are equivalent to the one that In+m − K is a nonsingular M-matrix, by Proposition 2, there exists a positive vector ξ = vec(ξ 1 , ξ 2 ) 0 such that (In+m − K) ξ 0, where ξ 1 = (ξi1 ) ∈ Rn and ξ 2 = (ξ 2j ) ∈ Rm . Thus, for sufficiently small > 0, we have − αi lϕ−i ξi1 +

m 

f

(ai j + bi j )L j ξ 2j < − , i ∈ [n],

(87)

j=1

−β j lψ−j ξ 2j

n  g + (c ji + d ji )L i ξi1 < − ,

j ∈ [m].

(88)

i=1

Let χ(t) = vec(x(t), y(t)) be a solution of system (62). It follows from Eqs. (62) and (86) that ⎧   x(t) − x ∗ = − Dα Φ(x(t)) − Φ(x ∗ ) + A( f (y(t)) − f (y ∗ )) ⎪ ⎪ ⎪ ⎨ + B( f (y(t − σ(t))) − f (y ∗ ))   ∗  ⎪ y(t) − y = − Dβ Ψ (y(t)) − Ψ (y ∗ ) + C(g(x(t)) − g(x ∗ )) ⎪ ⎪ ⎩ + D(g(x(t − τ (t))) − g(x ∗ )).

(89)

System (89) can be written in detail as m    ai j f j (y j (t)) − f j (y ∗j ) (xi (t) − xi∗ ) = −αi ϕi (xi (t)) − ϕi (xi∗ ) + j=1

+

m 

 bi j f j (y j (t − σ j (t))) − f j (y ∗j ) , i ∈ [n],

(90)

j=1

and n    (y j (t) − y ∗j ) = −β j ψ j (y j (t)) − ψ j (y ∗j ) + c ji gi (xi (t)) − gi (xi∗ ) i=1

+

n  i=1

 d ji gi (xi (t − τi (t))) − gi (xi∗ ) , j ∈ [m].

(91)

Positivity and Stability of Nonlinear Time-Delay Systems …

377

Next, we define the following functions ζi (t) = |xi (t) − xi∗ | and η j (t) = |y j (t) − y ∗j | for t ≥ t0 , i ∈ [n], j ∈ [m]. By Assumption (A1), +  + sgn(xi (t) − xi∗ ) ϕi (xi (t)) − ϕi (xi∗ ) = +ϕi (xi (t)) − ϕi (xi∗ )+ ≥ lϕ−i |xi (t) − xi∗ |. Thus, from Eq. (90), we have D + ζi (t) = sgn(xi (t) − xi∗ )(xi (t) − xi∗ )  = −αi sgn(xi (t) − xi∗ ) ϕi (xi (t)) − ϕi (xi∗ ) +

m 

 ai j sgn(xi (t) − xi∗ ) f j (y j (t)) − f j (y ∗j )

j=1

+

m 

 bi j sgn(xi (t) − xi∗ ) f j (y j (t − σ j (t))) − f j (y ∗j )

j=1

≤ −αi lϕ−i ζi (t) +

m 

f  L j ai j η j (t) + bi j η j (t − σ j (t)) , i ∈ [n],

(92)

j=1

where D + denotes the upper-right Dini derivative. Similarly, we have D + η j (t) ≤ −β j lψ−j η j (t) +

n 

g L i c ji ζi (t) + d ji ζi (t − τi (t)) , j ∈ [m].

(93)

i=1

Based on the above differential inequalities, we now derive an exponential domination of positive solutions of system (62). Let us consider the following function for each i ∈ [n] i (λ) = (λ − αi lϕ−i )ξi1 +

m 

f  L j ai j + bi j eλσ ξ 2j , λ ∈ [0, ∞).

j=1

The function i (λ) is continuous and strictly increasing on [0, ∞), i (0) < − and i (λ) → ∞ as λ → ∞. Thus, there exists a unique positive scalar λi such that i (λi ) = 0 and (λ) < 0 for all λ ∈ [0, λi ). By the same arguments we can conclude that, for each j ∈ [m], there exists a unique positive scalar λˆ j such that ˆ j (λˆ j ) = 0, where

378

L. V. Hien

ˆ j (λ) = (λ − β j lψ−j )ξ 2j +

n 

g L i c ji + d ji eλτ ξi1 , λ ∈ [0, ∞).

i=1

  Let λ0 = mini, j λi , λˆ j , then λ0 > 0 and i (λ0 ) ≤ 0, ˆ j (λ0 ) ≤ 0 for all i ∈ [n], j ∈ [m]. Therefore, − αi lϕ−i ξi1 +

m 

f  L j ai j + bi j eλσ ξ 2j ≤ −λξi1 , i ∈ [n],

(94)

g L i c ji + d ji eλτ ξi1 ≤ −λξ 2j , j ∈ [m],

(95)

j=1

−β j lψ−j ξ 2j

+

n  i=1

for any λ ∈ (0, λ0 ].   Suggested by inequalities (94) and (95), we denote C(ξ) = 1/ mini, j ξi1 , ξ 2j and consider the following domination functions ˆ = C(ξ)φ∗ ∞ e−λ(t−t0 ) ξ 1 , η(t) ˆ = C(ξ)φ∗ ∞ e−λ(t−t0 ) ξ 2 , t ≥ t0 , ζ(t)

(96)

where φ∗ ∞ = x 0 − x ∗ C + y 0 − y ∗ C and a constant λ ∈ (0, λ0 ]. For convenience, we also denote 

   ˆ ζ(t) ζ(t) υ(t) = , υ(t) ˆ = = C(ξ)φ∗ ∞ e−λ(t−t0 ) ξ. η(t) η(t) ˆ Then, for any t ≥ t0 , we have D + υ(t) ˆ = −λυ(t) ˆ and   λτ  e ζˆi (t) ζˆi (t − τi (t))  λσ . υˆ d (t)  ηˆ j (t − σ j (t)) e ηˆ j (t) 

(97)

We now show that υ(t)  υ(t) ˆ for all t ≥ t0 . Note that, for t ≤ t0 , υ(t)  C(ξ)φ∗ ∞ ξ ˆ 0 ). as C(ξ)ξ  1n+m . Thus, for a fixed θ > 1, υ(t0 ) ≺ θυ(t Suppose that υ(t) ≺ θυ(t) ˆ does not hold for all t > t0 . Then, there exist a t¯ > t0 and an index k ∈ [n + m] such that ˆ t ∈ [t0 , t¯]. υk (t¯) = θυˆ k (t¯), υk (t) < θυˆ k (t), t ∈ [t0 , t¯), υ(t)  θυ(t),

(98)

Without loss of generality, we can assume k ∈ [n]. Then, from equations (92), (94) and (97), we have

Positivity and Stability of Nonlinear Time-Delay Systems …

D + υk (t) ≤ −αk lϕ−k υk (t) +

m 

379

f  L j ak j η j (t) + bk j η j (t − σ j (t))

j=1

≤ −αk lϕ−k υk (t) +

m 

f  θL j ak j ηˆ j (t) + bk j ηˆ j (t − σ j (t))

j=1

≤

−αk lϕ−k υk (t)

∗

+ θC(ξ)φ ∞

 m

f Lj

  2 −λ(t−t0 ) λσ ak j + bk j e ξ j e

j=1

 ≤ −αk lϕ−k υk (t) + θC(ξ)φ∗ ∞ −λ + αk lϕ−k ξk1 e−λ(t−t0 )  = −αk lϕ−k υk (t) − θυˆ k (t) − λθυˆ k (t), t ∈ [t0 , t¯). (99) It can be deduced from inequality (99) that   D + υk (t) − θυˆ k (t) ≤ −αk lϕ−k υk (t) − θυˆ k (t) , t ∈ [t0 , t¯), and hence  − υk (t) − θυˆ k (t) ≤ υk (t0 ) − θυˆ k (t0 ) e−αk lϕk (t−t0 ) , t ∈ [t0 , t¯).

(100)

Let t ↑ t¯ in inequality (100) we obtain  0 = υk (t¯) − θυˆ k (t¯) ≤ υk (t0 ) − θυˆ k (t0 ) e−αk l(ϕk )(t¯−t0 ) < 0. This contradiction shows that υ(t) ≺ θυ(t) ˆ for all t ≥ t0 . Let θ ↓ 1 we get υ(t)  υ(t). ˆ Since , C(ξ)ξ  C(ξ) max ξi1 , ξ 2j 1n+m , i, j

we have

, υ(t) ≤ C(ξ) max ξi1 , ξ 2j φ∗ ∞ e−λ(t−t0 ) i, j

and, therefore, x(t) − x ∗  + y(t) − y ∗  ≤ 2υ(t) ≤ κφ∗ ∞ e−λ(t−t0 ) , t ≥ t0 ,   where κ = 2C(ξ) maxi, j ξi1 , ξ 2j . This shows that the EP χ∗ of (62) is exponentially attractive. Finally, we will prove the uniqueness of positive EP. Assume that χ¯ ∗ = vec(x¯ ∗ , y¯ ∗ ) is also a positive EP of system (62). Since χ¯ ∗ can be regarded as a stationary solution of (62) with constant initial condition φ = χ¯ ∗ , by the aforementioned exponential attractivity, we have   ∗ x¯ − x ∗  +  y¯ ∗ − y ∗  1 − κe−λ(t−t0 ) ≤ 0.

380

L. V. Hien

Let t → ∞ we obtain χ¯ ∗ = χ∗ , which shows the uniqueness of χ∗ . The proof is completed.  Remark 13 As the mappings F(y) = (A + B) f (y) and G(x) = (C + D)g(x) are order-preserving vector fields, it can be seen from the proof of Theorem 8 that, for any input vectors I ∈ Rn+ , J ∈ Rm + such that I and J are not simultaneously zero, the unique positive EP χ∗ of system (62) satisfies  ∗

χ 

  Φ −1 Dα−1 I 

Ψ −1 Dβ−1 J

and, therefore, χ∗ ∈ Rn+m \ {0}. Moreover, if I 0 and J 0 then χ∗ is strictly + ∗ positive (i.e. χ 0). Remark 14 The result of Theorem 8 also guarantees that all state trajectories of system (62) are eventually positive for positive input vectors I 0, J 0. More precisely, let χ(t) = vec(x(t), y(t)) be a general solution of system (62) (the initial condition φ is not necessary to be positive). Since χ(t) converges exponentially to the unique EP χ∗ which is strictly positive, there exists a t f > t0 such that χ(t) 0 for all t ∈ [t f , ∞).

6 Simulations Consider a class of nonlinear time-delay systems which describe cooperative-type BAM neural networks with the following Boltzmann sigmoid activation functions  1 − exp −  f j (y j ) = 1 + exp − f

yj f θj yj f θj



 1 − exp − , gi (xi ) =  1 + exp −



xi g θi xi , g θi

i, j = 1, 2,

g

where θ j > 0 and θi > 0 are weighted coefficients, and decay rate functions ϕi (x) = 2x + sin2 (0.25x), ψ j (y) = 2y − sin2 (0.25y), i, j = 1, 2.

(101)

Clearly, the decay rate functions (101) satisfy Assumption (A3) with lϕ−i = lψ−j = 1.75 and lϕ+i = lψ+j = 2.25 (i, j = 1, 2). In addition, it is easy to verify that Assumpf

tion (A4) is satisfied with L j =

1 f 2θ j

g

and L i =

For illustrative purpose, the parameters are given as

1 g. 2θi f g θ j , θi and

connection weight matrices

Positivity and Stability of Nonlinear Time-Delay Systems … Fig. 5 Convergence of state trajectories to the unique positive EP χ∗ for the input vector J = (1.0, 2.0, 2.0, 1.0) and delays τi (t) = 1 + 3| sin(10πt)|, σ j (t) = 4| cos(5πt)|

381

3

y 1 (t)

Response(x(t),y(t))

2.5

x 2 (t) x 1 (t)

2

y 2 (t)

1.5

1 0

f

10

20

t

30

40

50

g

θ j = 1.6, θi = 1.2, i, j = 1, 2, Dα = Dβ = diag{1, 1},     2.25 1.18 0.54 1.16 A= , B= , 0.95 1.26 1.46 0.94     0.64 1.48 1.25 0.56 C= , D= . 1.16 0.37 0.54 1.42     0.4982 0.4179 0.45 0.4857 , K2 = , and hence Then, we have K1 = 0.4304 0.3929 0.4048 0.4262 ⎤ 1.0 0 −0.4982 −0.4179 ⎢ 0 1.0 −0.4304 −0.3929 ⎥ ⎥ I4 − K = ⎢ ⎦ ⎣ −0.45 −0.4857 1.0 0 −0.4048 −0.4262 0 1.0 ⎡

is a nonsingular M-matrix. By Theorem 8, for a given input vector J = vec(I, J ) ∈ R4+ , the system has a unique positive EP χ∗ which is GES for any bounded time-varying delays. For J = (1.0, 2.0, 2.0, 1.0) , by solving system (69) with Matlab Toolbox, we obtain χ∗ = (1.9164, 2.222, 2.5467, 1.8116) . A simulation result of 20 sample state trajectories for the delays τi (t) = 1 + 3| sin(10πt)| and σ j (t) = 4| cos(5πt)| is portrayed in Fig. 5. The corresponding phase diagram is also presented in Fig. 6. Clearly, the conducted state trajectories of the system are positive and converge to the unique positive EP χ∗ , which validates the obtained theoretical results.

382

L. V. Hien 3

2.5

2

y2

x2

2.5

1.5

1

2

1.75

0

1.5 x

1

1

2

(a) x1 and x2

2.5

3

1.5

0

1

y1

2

2.5

3

(b) y1 and y2

Fig. 6 Convergence of solutions to the positive EP χ∗ in phase space

7 Conclusion This chapter has dealt with the positivity and exponential stability of positive nonlinear systems which describe dynamics of various models of neural networks with multiple time-varying delays. A systematic approach involving comparison techniques via differential and integral inequalities in combination with nonlinear analysis has been presented. By utilizing the proposed method, LP-based conditions for the existence and exponential stability of positive equilibriums have been derived in LP setting for M-matrix, which can be effectively solved by various convex algorithms. Numerical simulations have been presented to demonstrate the applicability and effectiveness of the derived theoretical results.

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Fault Detection and Isolation Methodology in Finite Frequency Domain for Constrained Networked Systems Yue Long and Ju H. Park

Abstract This chapter is concerned with transmission-dependent fault detection and isolation strategy in a finite frequency domain for networked systems with constrained communication capacities. By considering the possible network-induced phenomena, a switched system with multiple stochastic parameters is firstly derived to model the considered features. Then, drawing support from the geometric method and the finite frequency stochastic performance indices, a transmission-dependent fault detection and isolation scheme is developed in a finite frequency domain. Sufficient conditions are established to guarantee the desired performances and characterize the filter gains. The proposed filters can complete the mission of detection and isolation well only by partially available measurements, which is shown by an application to the VTOL aircraft. Keywords Fault detection and isolation · Finite frequency · Transmission dependent · Constrained capacity

1 Introduction The increasing scale together with the corresponding requirement on high security of the modern control systems has promoted the researches on networked systems [1–6] and fault detection (FD) technologies. One the one hand, the development of the networked systems introduced some advantages such as decreased system wiring, easy reconfigurability and so on [8], it also brings some challenges [3, 9–11]. An Y. Long (B) School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China e-mail: [email protected] J. H. Park Department of Electrical Engineering, Yeungnam University, 38541 Gyeongsan, Republic of Korea e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_16

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inevitably one is communication constraint [1], which is a restriction on the quantity of the transmitted data [7]. Consequently, there have been many researches concerned about such kind of problems, one can see [8, 9, 12, 13] for more information. On the other hand, great efforts have been made to the FD technologies and the related fault-tolerant control techniques, for such issue, one can see [14–18] and the references therein for more information. We can see from the above statements, it is natural to proceed with the researches on FD methods for a networked system. As a result, there are fruitful results concerned with such issuees. For example, in [19, 20], the FD schemes against to network-induced time-delays have been proposed, and with consideration of the packet dropout, FD methods have been developed in [21–24]. Nevertheless, the referred FD methods above have been proposed in the timedomain. In practical systems, the faults always occurred in low-frequency range [25]. With aid of the GKYP (generalized Kalman-Yakubovich-Popov) lemma, many FD methods have been proposed in a finite frequency domain [25–27]. Even though there have been some results concerned on FD techniques for networked systems, such as [28]. However, the fault dectetion isolation (FDI) method for networked systems with respect to frequency features has been in its infancy. Motivated by the above discussions, in this chapter, an FDI strategy is developed for networked systems under constraint channels. A switched system with multiple parameters is firstly derived to model the whole networked system. Then, with the help of the geometric method, a novel lemma is developed to ensure the decoupling of the faults to fault detection and isolation filters and ensure the required finite frequency performances of the derived systems. Subsequently, with aid of the derived lemma, a set of filters can be constructed corresponding to the faulty modes and transmission intervals. Notations: The following notations are used throughout this chapter. l2 denotes the Hilbert space of square-integrable functions; E(·) denotes the expectation operator; I represents the identity matrix with compatible dimensions.  is utilized to represent  a term that is induced by symmetry; A represents the generalized inverse of A; A∗ means the transpose of A. Denoting He(A) := A + A T be the sum of a square matrix A and its transpose A T .

2 System Modeling and Problem Preliminaries The considered networked system is with the structure as Fig. 1, wherein the plant is as the form of

Fault Detection and Isolation Methodology in Finite Frequency Domain …

389

Fig. 1 Structure of the Networked Systems

x˙t = Ao xt + Bc wt +

L 

Foι f ιt

ι=1

yt1 = c1 xt .. .

(1)

ytm = cm xt where xt ∈ X with dimension n and yti ∈ Y i , i = 1, 2, . . . , m, with dimension 1 are systems states and the measured outputs, wt ∈ Rw is the external disturbances, which belongs to l2 [0, ∞), f ιt ∈ Fι with dimension p represents the possible failures, and Foι denotes the fault signature. Ao , Bo , Foι and ci are known matrices. Here, a general assumption, that is Ac is stable, is made to ensure the stability of the filtering error dynamics. Further, during the transmission, it is assumed ¯ (m¯ ≤ m), nodes for that m measurements y i , i = 1, 2, . . . , m are packaged into m, transmission in a finite capacity networks, wherein only one node can get the accession to the channel at each transmission instant tk . And then this transmitted node can be received by the FDI center at the arrival instant rk = tk + τk . As in [29], in this chapter, it is also assumed a new transmission will begin after the last one is completed, i.e. tk ≤ r (k) ≤ tk+1 , in which t0 = 0. In addition, as shown in Fig. 2, the transmission-induced delay τk = rk − tk and transmission interval h k = tk+1 − tk are time-varying. Without loss of generality, it is assumed that τk and h k are i.i.d. and conformed to h k ⊆ {h σk }, 0 ≤ τk < h k , σk ∈ H = {1, 2, . . . , N } for k ∈ N+ . Remark 1 By adding a buffer at the front end of the channel, the assumption h k ⊆ ¯ the packet is set to be {h σk } can be realized. Under such assumption, if t − tk > h, ¯ lost, where h = max{h σk }. It is supposed the transmission of every data node is scheduled by a stochastic protocol, which is directed by m¯ + 1 (i.e. 0, 1, . . . , m) ¯ Bernoulli distributed white sequences with P(ki = 1) = E(ki ) = φi , and P(ki = 0) = 1 − E(ki ) = 1 − φi , where ki = 1 implies the ith node is accessed to the networked channel at the kth transmission instant and k0 = 1 means packet loss. It is not difficult to derive

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Fig. 2 Illustration of the transmission

m¯

i=0 φi = 1 from the practical point of view. Thus, the received data by the FDI center can be renewed by

yct =

m¯ 

m¯    ki Ti ytk + I − ki Ti yctk + vtk

i=0

i=0

(2)

∀t ∈ [rk , rk+1 ), where vtk denotes the channel noise, and Ti is defined as Ti = diag{Dki (1), Dki (2), . . . , Dki (ı), . . . Dki (m)} where Dki (ı) = 1 if the corresponded output ykı is transmitted in node i, otherwise Dki (·) = 0. Remark 2 During recent decades, there appeared many researches, for example [30, 31], concerning the data processing protocols. In which how to choose the accessing probability has been deeply discussed. Among them, the stochastic one modelled by Bernoulli distribution is an easier protocol to be realized. Thus it is natural to assume that the probability φi is designed in advance, i.e. they are known beforehand. Then, defining y = [y1T , . . . , ymT ]T and C = [c1T , . . . , cmT ]T , the discretely dynamics between tk and tk+1 = tk + h k of (1) can be derived as xk+1 = Aσk xk + Bσk wk +

L  ι=1

yk = C xk

Fσιk f ιk

(3)

Fault Detection and Isolation Methodology in Finite Frequency Domain …

391

h h for h k = h σk , where xk =xtk , Aσk = e Ac h k , Bσk = 0 k e Ac s Bc ds and Fσιk = 0 k e Ac s Fcι ds. It is clearly that (3) is switching in H due to its reliance on h k . For convenience of understanding, some definitions are firstly given for the following switched system. xk+1 = Aς xk , (4) yk = Cς xk . Definition 1 ([32]) For system (4) with ς ∈ N and the subspace Q ⊆ X , if Aς Q ⊆ Q, the subspace Q is defined to be Aς -invariant. Denoting Q|Aς ς∈N as the largest Aς −invariant subspace involved in Q, the following definition is given. Definition 2 ([33]) A subspace S ⊆ X is called to be finite unobservable subspace N (f.u.o.s.) for system (4) if S = ∩ς=1 ker Hς Cσk |Aς + Gς Cς , ς ∈ H holds for maps Gς : Y → X and Hς : Y → Y. Defining a class of f.u.o.s. of X for system (3) as S(X ). It is easy to see S(X ) is a nonempty close set. Thus the set of f.u.o.s. containing F can be signified by S(F) for arbitrary F ⊆ X , which has the smallest unit denoting by S = inf S(F) and could be computed by following the procedure of the Algorithm in [34, 35]. For calculated f.u.o.s. S ⊂ X , we subsequently introduce the factor system of (4), which could be expressed by x˘k+1 = A¯ ς x˘k , (5) y˘ (k) = Mς x˘k , where x˘ ∈ S ⊂ X is the state with dimension n − dim(S ), A¯ ς = (Aς + Gς Cς : X /S ) and Mς is the solution of Mς T = Hς Cς , in which T : X → X /S is the canonical projection satisfying T (Aς + Gς Cς ) = A¯ ς T . In this chapter, the set of the residuals rιk with dimension 1 for fault detection and isolation are generated by x˜ιk+1 = Aιf σk x˜ιk + B ιf σk yck , rιk = C ιf σk x˜ιk ,

(6)

where x˜ι ∈ X is the state of the filter, its dimension is same with the factor system of (3) calculated below, Aιf σk , B ιf σk and C ιf σk are filter gains to be determined, and the value of yck is yck = limt→rk+ yct . ι ι ι For a certain f.u.o.s. Sι σk , we can find two maps Gσk and Hσk such that Sσk = ∩σNk =1 ker Hσι k C|Aσk + Gσι k C , where Hσι k can be derived by ker Hσι k C = Sι σk + ker C. Further, denoting Tσιk : X → X /Sι σk , the factor system of (3) corresponded to ι ι ι ι ι ι ι ι Sι σk can be obtained by Mσk Tσk = Hσk C and Tσk (Aσk + Gσk C) = Aσk Tσk with Aσk = ι ι ) and B = T B . Subsequently, defining the transmission (Aσk + Gσι k C : X /Sι σk σk σk σk error as et = yct − yt , the dynamics of the whole FDI system can be obtained by

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ξιk+1 = Aισk ξιk + Bισk dk + Z3 Tσιk

L 

Fσιk f ιk ,

ι=1

(7)

rιk = Cισk ξιk , where ξιk =

Aισk =



T    T T Tσιk xk x˜ιkT ekT , dk = wkT vkT , Cισk = Z2T Cσι k 0 , and



 ι ι ι Aισk 12 A Bσι k Z1 Eσι k Z1 ι σk + Bσk Cσk  , B = , σk −Z1T Cσι k Eσι k Z1 I Z1T Cσι k I − Aισk − Bσι k Cσι k Aισk 22

with T  Z3 = I 0 −(Mισk )T , Aισk 12

=

Bσι k Z1 Hσι k Hσιk

−

Aισk 22 = Hσι k Hσιk − Hσι k

Bσι k Z1 Hσι k

 m¯ 

 m¯ 



 ki Ti Hσιk ,

i=0

ki Ti Hσιk ,

i=0

and

Aισk Bσι k Z1 Cσι k Eσι k Z2



⎡

0 Aισk 0 ⎢ 0 Aιf σ B ιf σ k k =⎢ ⎣ Mισ 0 Bσι k k 0 C ιf σk 0

0 0 0 0

⎤ I 0⎥ ⎥. 0⎦ I

After the above manipulations, the phenomena induced by networked transmission were transformed into the switching of the system (7) and its stochastic parameter ki . Next, we are going to make the following general assumption. Assumption The sequences {ki , i ∈ {0, 1, . . . , m}} ¯ are independent from the initial  state x0 and the noises wk and vk . Now, with aid of the Definition 3 in [36], the FDI methodology addressed in this chapter can be concluded as: designing a set of FDI filters corresponding to h k and the faults as the form of (6), such that the system (7) is stochastic stable. And the residual rιk , ι ∈ L is sensitive to the fault f ιk in certain frequency range ensuring by G r f ιkE− > β, and decoupled from the others f κk , κ ∈ {1, 2, . . . , L} and κ = ι. In addition, rιk is also designed to the robust to dk by achieving rιk E2 ≤ γ 2 dk E2 .

(8)

Fault Detection and Isolation Methodology in Finite Frequency Domain …

393

3 Fault Detection and Isolation Filter Synthesis To make convenience of the analysis, the system (7) is transformed into the following form firstly  ηιk+1 =

Aσιk

+

m¯ 

 (φi −

ki )A˜σιk i

ηιk +

Bσι k dk

+

Z3 Tσιk

Fσιk f ιk ,

ι=1

i=0

rιk =

L 

(9)

Cσιk ηιk ,



0 Bσι k Z1 Hσι k Ti Hσιk ˜ , Bσι k = Bισk , Cσιk = Cισk , and where Aσk i = 0 Hσι k Ti Hσιk 

Aσιk =

ι ι ι ι A  σk + Bι σk Cσk ι ι Aσιk 12 T ι Z1 Cσk I − Aσk − Bσk Cσk Aσk 22



 m¯ with Aσιk 12 = Bσι k Z1 Hσι k Hσιk − Bσι k Z1 Hσι k φi Ti Hσιk and Aσιk 22 = Hσι k Hσιk − i=0  m¯ ι Hσι k i=0 φi Ti Hσk . In the following, the subscripts corresponded to σk and σk+1 are respectively replaced by ı and j . Lemma 1 For stochastic stabled system (9), if there exist matrices Pıι = (Pıι )T >  L κ 0, Pjι = (Pjι )T > 0, Qıι = (Qıι )T > 0 and f.u.o.s. Sıι = inf S κ=1,κ=ι Fı , ι = 1, 2, . . . , L, such that Sıι ∩ Fıι = 0, and

ι = 1, 2, . . . , L ,

⎤T ⎤ ⎡ ι ⎤T ⎡ ι ⎤ ⎡ ι Aı Fıι Aı ι Fıι Cı 0 Cı 0 ⎣ ΨA˜ι 0 ⎦ ıι ⎣ ΨA˜ι 0 ⎦ + ⎣ 0 0 ⎦  ⎣ 0 0 ⎦ < 0 ı ı 0 I 0 I I 0 I 0

(10)

⎡

(11)

holds ∀ı, j ∈ H, the residual rι will be with a finite frequency stochastic H− index β (β > 0) to f ιk when dk = 0, and decoupled from f κ , κ ∈ {1, . . . , L}, κ = ι, where Fıκ is the subspace containing Fıκ , κ ∈ {1, 2, . . . , L}, Fıι = Z3 Tı ι Fıι ,  2 ˜ιT T , ΨA˜ıι = α02 A˜0ıιT α12 A˜1ıιT · · · αm ¯ Amı ¯ ⎡

⎤ −I 0 0   = ⎣ 0 −Ψ 0 ⎦ 0 0 β2 I and (i) In LF range

394

Y. Long and J. H. Park

⎡

⎤ −Pjι 0 Qıι ⎦. ıι = ⎣ 0 −Ψıι 0 ι ι ι 0 Pı − 2 cos ϑl Qı Qı

(12)

(ii) In MF range ⎡

⎤ −Pjι 0 e jϑc Qıι ⎦ ıι = ⎣ 0 −Ψıι 0 − jϑc ι ι ι e Qı 0 Pı − 2 cos ϑw Qı

(13)

where ϑc = (ϑ2 + ϑ1 )/2, ϑw = (ϑ2 − ϑ1 )/2. (iii) In HF range ⎡

⎤ −Pjι 0 −Qıι ⎦ ıι = ⎣ 0 −Ψıι 0 −Qıι 0 Pıι + 2 cos ϑh Qıι

(14)

    2 ι 2 ι  2 ι 2 ι where Ψ = diag α02 I α12 I · · · αm with ¯ I , Ψı = diag α0 Pj α1 Pj · · · αm¯ Pj αi2 = E{(ki − φi )2 } = φi (1 − φi ). Proof Since Fıκ ∈ Sıι for κ = ι, we can see Tı ι Fıκ = 0, κ = ι, from (10). Then, we can rewrite the system (9) into 

 m¯  Aı ι + (φi − ki )A˜ıiι ηιk + Bıι dk + Z3 Tı ι Fıι f ιk

ηιk+1 =

i=0

rιk =

(15)

Cıι ηιk .

From the above equations, it is easy to see the f κ , κ ∈ {1, 2, . . . , L}, κ = ι are decoupled from the ιth FDIF. Next, we are going to prove the finite frequency stochastic H− index to f ιk . We consider the LF case firstly for system (15), deriving from Euler’s formula and by some calculations, the restriction of the system states in LF case can be written into ∞    T T T ηιk ηιk+1 ≥ 0. + ηιk+1 ηιk − 2cosϑιw ηιk ηιk

(16)

k=0

Recalling the property μT Qıι ν = tr(νμT Qıι ) for arbitrary vectors μ and ν, where tr(·) denotes the trace of the matrix. Then we have  ι T T T Qı Jc (k) = tr ηιk ηιk+1 + ηιk+1 ηιk − 2cosϑιw ηιk ηιk T T ι T ι = ηιk+1 Qıι ηιk + ηιk Qı ηιk+1 − 2 cos ϑl ηιk Qı ηιk

= ζιkT 1 ζιk ≥ 0

(17)

Fault Detection and Isolation Methodology in Finite Frequency Domain …

395

 T T T f ιk in which ζιk = ηιk and

1 =



11 Qıι Z3 Tı ι Fıι 1 ι ι T ι (Z3 Tı Fı ) Qı 0

 ι m¯   ˜ι T Qι + Qι A ι + m¯ (φi − ki )A˜ι − where 11 ı ı ı 1 = Aı + i=0 (φi − ki )Aıi i=0 ıi 2cosϑl Qıι . To achieve our purpose, considering the index as follows,  Jr, f = E

∞  

− rιkT rιk

+β

2

f ιkT f ιk



.

(18)

k=0

 T We define the functional Vs = ηιk (k)Pıι ηιk and Vs = E Vs − Vs . From the process to prove the stability of the system, i.e. Theorem 2, one can obtain Vs > 0 and Vs < 0 for Pıι > 0. Then, we have  Jr, f ≤ E ≤E =E

∞  

k=0 ∞ 



− rιkT rιk − rιkT rιk

+β

2

f ιkT f ιk

+β

2

f ιkT f ιk

− Vs



− Vs + Vs



(19)

k=0 ∞ 

   ζ T (k) 1 + 2 ζ(k)

k=0

in which

ι ι∗ ι 12 11 2 + Pı − Cı Cı 2 2 = 21 2 22 2



  ι m¯  ˜ι T P ι A ι + m¯ (φi − ki )A˜ι , 12 = where 11 j ı 2 = − Aı + i=0 (φi − ki )Aıi i=0 ıi  ι m¯  ι m¯ 2 ι T ι 21 ι ι ι ι T ι ˜ − Aı + i=0 (φi − ki )Aıi Pj Z3 Tı Fı , 2 = −(Z3 Tı Fı ) Pj Aı + i=0 (φi − ki )A˜ι and 22 = −(Z3 T ι F ι )T P ι (Z3 T ι F ι ) + β 2 I . ıi

ı

2

ı

j

ı

ı

  Inaddition, recalling the properties of E i=0 (φi − ki ) = 0 and E (φi − ki )2 = φi (1 − φi ). It is easy to get (11) is equivalent to 2 + 1 < 0 with E (φi − ki )2 = αi2 , which yields to  m¯

 Jr, f = E

∞   k=0

which means G r f ιkE− > β.

− rιkT rιk

+β

2

f ιkT f ιk



≤0

(20)

396

Y. Long and J. H. Park

Furthermore, the conclusions for MF and HF cases can be developed similarly. This proof is completed.  Even though sufficient conditions make sure the desired performance and decoupling of the faults was developed in the above lemma. However, it is obvious that the conditions contain nonlinear items and cannot be solved directly. In the following, we will deal with this problem. Theorem 1 Assume the system (9) is stochastic stable, the residual rιk is decoupled stochastic H− index β to f ιk when dk = from f κk , κ = ι, and is with a finite frequency  L κ matrices Xı ι , 0 if there exist f.u.o.s. Sıι = inf Q κ=1,κ=ι Fı ⎡, ι = 1, 2, . . . , L, ⎤ ι Pı11 ∗ ∗ ι ι Pı22 ∗ ⎦ > 0, PjιT = Gıι , Yı ι , Nı ι , Rıι , A fιı , B ιf ı , C fι ı and PıιT = Pıι = ⎣ Pı21 ι P ι P ι Pı33 ⎡ ι ⎡ ι ı31 ı32 ⎤ ⎤ Pj 11 ∗ ∗ Qı11 ∗ ∗ ι ι Qı22 ∗ ⎦ > 0, such that Pjι = ⎣ Pjι 21 Pjι 22 ∗ ⎦ > 0, QıιT = Qıι = ⎣ Qı21 ι ι ι Qı32 Qı33 Qı31 Pjι 31 Pjι 32 Pjι 33 (10) and ⎤ ⎡ −Pjι ∗ ∗ ∗ ∗ ⎢ 0 −Ψ ι ∗ ∗ ∗ ⎥ ı ⎥ ⎢ ι ι ⎢ ϒ (21) 0 ϒ ∗ ∗ ⎥ ı4 ⎥ 0 and i , i = 1, . . . , m, ¯ where ⎡

ι ϒı1

ι ϒı4 ι ϒı2 ι ϒı3 ι ϒı6 ι ϒı5

with

⎤ ι ιT ιT Qı11 − Xı ι Qı21 − Yı ι Qı31 ι ι ιT ⎦, − Gıι Qı22 − Yı ι Qı32 = ⎣ Qı21 ι ι ι ι Q Q Q − N ı ı33 ⎡ ι11 ı31 ⎤ ı32 ϒı4 ∗ ∗ ι21 ι22 ϒı4 ∗ ⎦, = ⎣ ϒı4 ι31 ι32 ι33 ϒ ϒ ϒ  ı4T ιı4 Tı4 ι =  −Iv Xı −Iv Yı 0 , ι 2 ι ι 1 ϒı2 · · · 2m¯ ϒı2 =  20 ϒı2 , ιT T =  0 Rı Iv 0 , ι ι FıιT Tı ιT GıιT + IvT A fιı ϒı5 = ϒı5

Fault Detection and Isolation Methodology in Finite Frequency Domain …

397

ι11 ι ι ϒı4 = Pı11 − 2 cos ϑl Qı11 + He(Xı ι Aıι + B ιf ı Mıι ), ι21 ι ι ϒı4 = Pı21 − 2 cos ϑl Qı21 + Gıι Aıι + B ιf ı Mıι + A fιTı + C fιTı IvT, ι22 ι ι ϒı4 = Pı22 − 2 cos ϑl Qı22 + He(A fιı ) ι31 ι ι ϒı4 = Pı31 − 2 cos ϑl Qı31 + Nı ι Mıι − Nı ι Mıι Aıι + HσιTk HσιTk B ιT fı

− HσιTk

m¯ 

φi Ti HσιTk B ιT fı,

i=0 ι32 ι ι ιT ϒı4 = Pı32 − 2 cos ϑl Qı32 + HσιTk HσιTk B ιT f ı − Hσk

m¯ 

φi Ti HσιTk B ιT fı,

i=0 ι33 ι ι = Pı33 − 2 cos ϑl Qı33 + He(Nı ι Hσι k Hσιk − Nı ι Hσι k ϒı4

m¯ 

φi Ti Hσιk )

i=0 

ι ϒı5 = FıιT Tı ιT Xı ιT + IvT Xı ι Aıι + IvT B ιf ı Mıι , 

ι ϒı5 = − FıιT Tı ιT MıιT Nı ιT + IvT B ιf ı Hσι k Hσιk − IvT B ιf ı

m¯ 

Hσι k (φi Ti

i=0

−

i2 αi2 Ti )Hσιk .

Proof It is seen that the decoupling of the fault f κk form the ιth FDI filter, κ = ι, has been proved in Lemma 1. Subsequently, we mainly focus on deriving the linear conditions to ensure the finite frequency sensitivity performance. As shown in Lemma 1, this performance in LF range can be captured by (11), which can be transformed into   T  Aı ιT ΨAT˜ι I 0 Aı ιT ΨAT˜ι I 0 ι ı ı 0, Pdj = Pdj = ⎣ Pdj 21 Pdj 22 ι ι ι Pdj 31 Pdj 32 Pdj 33 such that

⎡

ıι11 ⎢ 0 ⎢ ι31 ⎢

⎢ ıι41 ⎣ ı 0

⎤ ∗ ∗ ∗ ∗ ⎥ ∗ ∗

ıι22 ∗ ⎥ ι ⎥ 0. where Pdı Then, one has

  ΔVι (ηιk , k) = E Vι (ηιk+1 , k + 1)|ηιk , k − Vι (ηιk , k) T ι T ι = E{ηιk+1 Pdj ηιk+1 } − ηιk Pdı ηιk .

(30)

Applying the mathematical expectation on it, one can derive ΔVι (ηιk , k)  =

T ηιk

+E

Aı

ιT

ι Pdj Aı ι

 m¯ 



m¯  ι A˜ı ι + 2E (φi − ki )A˜ı ιT Pdj i=0







(31)

ι ι A˜ı ι − Pdı (φi − ki )2 A˜ı ιT Pdj ηιk .

i=0

 m¯   On the other hand, it is not difficult to get E i=0 (φi − ki ) = 0 and E (φi − ki )2 = φi (1 − φi ). Together with φi (1 − φi ) = αi2 , (31) is equivalent to T S ηιk ΔVι (ηιk , k) = ηιk   m¯  T ιT ι ι 2 ˜ιT ι ι ι ˜ = ηιk Aı Pdj Aı + (αi Aı Pdj Aı ) − Pdı ηιk

(32)

i=0

m¯ ι ι ι where S = Aı ιT Pdj Aı ι + i=0 (αi2 A˜ı ιT Pdj . A˜ı ι ) − Pdı From (28), one can see S < 0 by utilizing Schur’s complement lemma. Therefore, the system (9) is stochastic stable when dk = 0 and f ιk = 0. Next, we will consider the index as follows for dk = 0 when η(0) = 0

Fault Detection and Isolation Methodology in Finite Frequency Domain …

J(rιk , dk ) =

401

∞     E rιkT rιk − γ 2 dkT dk k=0

∞     E rιkT rιk − γ 2 dkT dk + δV (ηιk , k) = k=0









(33)

+ E V (ηι0 , 0) − E V (ηι∞ , ∞) ∞     E rιkT rιk − γ 2 dkT dk + δV (ηιk , k) , ≤ k=0

one has

 E rιkT rιk − γ 2 dkT dk + δV (ηιk , k) m¯    2 ιT ι T ι Aı ιT Pdj = ηιk Aı ι + αi A˜ı Pdj A˜ı ι + CıιT Cıι i=0

−

ι Pdı

   ι ηιk + dkT BıιT Pdj Bıι − γ 2 I dk

(34)

= ξιkT Hıι ξιk 

ι  T ∗ Hı11 T T ι ι where ξιk = ηιk dk with Hı11 and Hı = = ι ι Aı ι BıιT Pdj Bıι − γ 2 I BıιT Pdj   m¯ ι ι ι Aı ιT Pdj αi2 A˜ı ιT Pdj Aı ι + i=0 + CıιT Cıι . Then by performing the simA˜ı ι − Pdı  ilar procedure of the proof for Theorem 1, we have E rιkT rιk − γ 2 dkT dk + δV (ηιk , k) < 0 from (28). Summing the above equality with respect to k from k = 0 to ∞, one  has J(rιk , dk ) < 0, which is equal to (8). This proof is completed. Remark 3 The conditions in the above theorem are investigated without consideration of the frequency due to the fact that the disturbances may be appeared with uncertain frequency. If its range can be known in advance, it is not complex to develop the conditions in a finite frequency domain by following the line of the Lemma 1 and Theorem 1. Till now, according to Theorems 1–2, the gains of the FDI filter can be calculated out through (26) with the feasible solutions of the following optimization problem max β s.t. (16.10), (16.21), (16.28), ı, j ∈ N

(35)

for acceptable robustness level γ to the disturbance dk . Remark 4 For the sake of getting linear conditions, some scalars i and νi need to be given beforehand in Theorems 1–2, which are utilized to offset the effectiveness of the parameters ki on the system. They could be decided by the heuristic method. Without loss of generality, they can be chosen as i2 = νi2 = φi (1 − φi ).

402

Y. Long and J. H. Park

After getting rιk , ι = 1, . . . , L, successfully, the next step is determining the evaluation function and making the judgement. In this chapter, the detection logic is chosen as Jrι (τ ) > Jthι ⇒ f ιk = 0, ι = 1, . . . , L . where the evaluation functions Jrι and the thresholds Jthι are determined by % & kτ &1  r T rιk Jrι (τ ) = ' τ k=k ιk

(36)

0

and

Jthι =

sup

dk ∈l2 , f ιk =0

Jrι (τ ),

(37)

where [k0 , kτ ] implies the evaluation time window.

4 Simulation Example In this section, a linearized longitudinal model of a VTOL aircraft will be applied to show the effectiveness of the obtained results. The applied system borrowed from [38] is with the following form: ⎡

−9.9477 −0.7476 0.2632 ⎢ 52.1659 2.7452 5.5532 x˙t = ⎢ ⎣ 26.0922 2.6361 −4.1975 0 0 1 ⎡ ⎡ ⎤ ⎤ −2.035 0.1761 ⎢ 0.4847 ⎥ ⎢ 7.5922 ⎥ ⎢ ⎥ ⎥ +⎢ ⎣ 3.3051 ⎦ f 1t + ⎣ 4.4900 ⎦ 0.8451 0  1 yt = 0 1 0 1 xt ,  yt2 = 1 0 1 1 xt ,

⎡ ⎤ ⎤ 5.0337 00 ⎢ ⎥ −24.4221 ⎥ ⎥ x t + ⎢ 0 1 ⎥ wt ⎣ ⎦ −19.2774 1 0⎦ 0 00 f 2t ,

wherein each element of the states denotes the horizontal velocity, vertical velocity,  T pitch rate and pitch angle, respectively. Also, Fc1 = −2.035 0.4847 3.3051 0.8451  T and Fc2 = 0.1761 7.5922 4.4900 0 selecting as the first and third column of the input matrix denotes the fault signatures, therefore they can expresses the faults occurring on the first and third actuator, respectively. Further, the two outputs are assumed to be transmitted into the networked channel by two nodes. It means, in company with the situation of packet dropout, the matrices Ti , i = 0, 1, 2 can be obtained as

Fault Detection and Isolation Methodology in Finite Frequency Domain … Table 1 Comparison of β for different H and φi Rate-1 Rate-2 β = 2.3885 β = 0.4789

TR-1 TR-2

β = 2.3657 β = 0.4682

Table 2 β corresponded to different γ C-1 C-2 γ = 0.25 β = 0.6729

γ = 0.68 β = 1.1508

 00 T0 = , 00

403

Rate-3

Rate-4

β = 2.3029 β = 0.4091

Infeasible Infeasible

C-3

C-4

γ=1 β = 2.3885

γ=3 β = 8.0936

10 T1 = 00



 00 T2 = . 01

Also, σk is assumed to be varying in the set H = {1, 2, 3, 4}. In addition, the f.u.o.s. in Theorems 1–2 can be obtained as Sı1 = F2 and Sı2 = F1 . Subsequently, the corresponded factor systems as form of (5) along with the derived subspaces can be calculated. Then, aiming at indicating the relationships between networked phenomena and the system performances, the optimization problem (35) is firstly solved for ϑl = 0 and an admissible robust level γ = 1, and different H and φi , the optimal solutions of β are given in the following Table 1. Where ‘TR’ represents ‘transmission intervals’ h σk varying in H, and

TR-1 : h 1 = 1s; h 2 = 1.2s; h 3 = 2.5s; h 4 = 3s. TR-2 : h 1 = 0.8s; h 2 = 2s; h 3 = 2.5s; h 4 = 3.5s.



Also, ‘Rate’ means ‘accessing rate’ φi , and ⎡

⎤ Rate-1 : φ0 = 0.02; φ1 = 0.49; φ2 = 0.49; ⎢ Rate-2 : φ0 = 0.29; φ1 = 0.36; φ2 = 0.35; ⎥ ⎢ ⎥ ⎣ Rate-3 : φ0 = 0.8; φ1 = 0.1; φ2 = 0.1; ⎦ . Rate-4 : φ0 = 1; φ1 = 0; φ2 = 0. From the calculated results, it is easy to see that the performance index β is related to h k and φi closely, i.e., for fixed h k , lower φ0 results in larger β, and for certain φ0 , longer h k results in smaller β. It is to say, higher rate of packet dropout or longer transmission delays will lead to poor performance of fault detection. Next, it is the step to validate the relationship between the robustness level γ and sensitiveness level β. Specifically, choosing Rate-1 and TR-1 to be a general case, various β can be gotten for different γ, which are presented in Table 2.

404

Y. Long and J. H. Park

accessed node

3 2.5 2 1.5 1

0

10

20

30 transmission instant k

40

50

60

0

10

20

30 transmission instant k

40

50

60

3

hk

2 1 0

Fig. 3 The accessed node and h k at each transmission instant in CASE-1

It is obviously from Table 2, that β is strongly influenced by γ, i.e., larger γ results in larger β, that implies better detection ability can be reached by some expense of robustness. Without loss of generality, the simulation results will be performed in the following for the situation of Rate-1, TR-1 and C-3. Additionally, the comparison with the existed FD method proposed in [39] will be done to illustrate the advantages of the developed FDI methodology. In [39], the authors utilized a weighting matrix to restrict the signal frequency, an H∞ based FD scheme without consideration of fault isolation has been further addressed. For the purpose of fairy, the simulation results of [39] would be issued in the fault isolation framework of this chapter. Then, the simulation will be done under the two Cases as follows with the zero-initial condition. In addition, the disturbance w1k and the channel noise vk are chosen as bound-limited white noise with a power of 0.8 and 0.2, respectively. The disturbance w2k is selected as w2k = 0.1 sin(0.2k) when 0s < t < 100 s. CASE-1: The transmitted node i and h k are varying as shown in Fig. 3. In this CASE, the following two faulty modes will be simulated. CASE-1-Fmode1: Stuck failure with value 1.5 acting upon the 1st actuator when 30s < t < 70s, and the 3rd actuator works normally. CASE-1-Fmode2: The 1st actuator is in failure-free mode, and the stuck failure with value 1.5 occurred on the 3rd actuator when 30s < t < 70s. Figures 4 and 5 has shown the generated residuals rιk and the related evaluation functions Jrι (τ ) for each fault case in CASE-1, respectively. In Fig. 4, we can see

Fault Detection and Isolation Methodology in Finite Frequency Domain …

0.1

r1k

r2k

0.1 0.05 0

0.05

0

50 time (s)

0

100

50 time (s)

100

0.02

0

50 time (s)

100

0.02

2

1 r

0

0.03 Jr (τ)

0.03 J (τ)

405

0.01 0

0.01 0

50 time (s)

0

100

Fig. 4 The residuals and the corresponding evaluation function for CASE-1-Fmode1 (red line—by our method, blue line—by existing method)

0.1 r2k

r

1k

0.1 0.05 0

0

50 time (s)

0

100

0

50 time (s)

100

0

50 time (s)

100

0.03 J (τ)

0.03

0.02

2 r

0.02

r

J1(τ)

0.05

0.01 0

0.01 0

50 time (s)

100

0

Fig. 5 The residuals and the corresponding evaluation function for CASE-1-Fmode2 (red line—by our method, blue line—by existing method)

Jr1 (τ ) > Jth1 when t = 34s by our scheme and when t = 35s by existing one, and Jr2 (τ ) < Jth2 over the running time. In addition, in Fig. 5, it is seen Jr1 (τ ) < Jth1 over all the running time, and Jr2 (τ ) > Jth2 when t = 31s by our methodology and when t = 34s by existing one. That is to say, by the developed geometric FDI scheme, the occurred failures could be isolated effectively and detected earlier. CASE-2: The transmitted node i and h k are varying as shown in Fig. 6. In this CASE, the following two faulty modes will be considered.

406

Y. Long and J. H. Park

accessed node

2 1.5 1 0.5 0

0

10

20

30 transmission instant k

40

50

60

0

10

20

30 transmission instant k

40

50

60

3

hk

2 1 0

Fig. 6 The accessed node and h k at each transmission instant in CASE-2

r2k

0.05 0

0

50 time (s)

0.05 0

100

0.04

0.04

0.03

0.03

0.02

2

J1r (τ)

0.1

Jr (τ)

r1k

0.1

0.01 0

0

50 time (s)

100

0

50 time (s)

100

0.02 0.01

0

50 time (s)

100

0

Fig. 7 The residuals and the corresponding evaluation function for CASE-2-Fmode1 (red line—by our method, blue line—by existing method)

CASE-2-Fmode1: Stuck failure with value 1.5 acting upon the 1st actuator when 30s < t < 70s, and the 3rd actuator works normally. CASE-2-Fmode2: The 1st actuator is in failure-free mode, and the stuck failure with value 1.5 occurred on the 3rd actuator when 30s < t < 70s.

Fault Detection and Isolation Methodology in Finite Frequency Domain …

1

r2k

0.05 0

Jr (τ)

0.1

0

50 time (s)

0.05 0

100

0.04

0.04

0.03

0.03 J 2r (τ)

r1k

0.1

0.02 0.01 0

407

0

50 time (s)

100

0

50 time (s)

100

0.02 0.01

0

50 time (s)

100

0

Fig. 8 The residuals and the corresponding evaluation function for CASE-2-Fmode2 (red line—by our method, blue line—by existing method)

Similarly, Figs. 7 and 8 showed the generated residual and its evaluation functions for the two fault cases in CASE-2. In Fig. 7, it is obvious Jr1 (τ ) > Jth1 when t = 31 s by the proposed method and when t = 31 s by existing one, also Jr2 (τ ) < Jth2 over all the running time. Further, from Fig. 8, we can seen Jr1 (τ ) < Jth1 all the time and Jr2 (τ ) > Jth2 when t = 31 s by our scheme and t = 32 s by existing one. The results in this CASE also showed the effectiveness of the FDI methodology investigated in this chapter.

5 Conclusions This chapter has investigated a transmission-dependent fault detection and isolation methodology for constrained networked systems in a finite frequency domain. A switched system with multiple stochastic parameters was firstly derived to model the whole system. Then, with aid of the geometric approaches and the definitions of finite frequency performances, a novel FDI scheme has been proposed. Some sufficient conditions have been derived further to determine the FDI gains. Finally, the effectiveness and the advantages of the proposed methodology has been illustrated by an application to a VTOL aircraft. Acknowledgements This work of J.H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. 2019R1A5A808029011).

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Mean-Square Stochastic Stability of Delayed Hybrid Stochastic Inertial Neural Networks R. Krishnasamy, A. Manivannan, and Raju K. George

Abstract This work considers the problem of mean-square stochastic stability analysis of hybrid stochastic inertial neural networks (INNs) with the effect of timedelays. Here, the hybrid stochastic INNs are represented as the combination of a two-level system in which the first level is directed by the system of second-order differential equations and the second level is directed by the discrete set representing the switch (jump) nodes. Initially, sufficient delay-dependent mean-square stability conditions for the hybrid INNs with time-delays are established using variable transformation techniques and some integral inequalities. This process is carried out mainly based on Lyapunov theory and linear matrix inequality technique. Some general cases on time-delay such as time-varying delay, interval time-varying delay, and mode-dependent delay are considered and corresponding results are established. Finally, numerical examples pertaining to the validation and effectiveness of the derived theoretical results are demonstrated. Keywords Inertial neural networks · Lyapunov stability · Markovian jump parameters · Brownian motion · Linear matrix inequality

R. Krishnasamy (B) Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore, Tamil Nadu, India e-mail: [email protected] A. Manivannan School of Advanced Sciences - Division of Mathematics, Vellore Institute of Technology-Chennai Campus, Chennai 600127, India e-mail: [email protected] R. K. George Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_17

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1 Introduction One of the most important characteristics of any dynamical system is stability which enables us to analyze or predict how the system behaves as time increases. More insight can be gathered if the system is stable since it gives us the information about how the dynamics of the system depends on the initial data, that is continuous dependence on initial conditions. Also, in some cases, some other characteristics such as stabilizability, synchronization, state estimation, etc. can be applied to understand their behavior. In dynamical systems, there may exist delays or time-delays between the applied input and its response. These types of systems can be modeled with an idea of time-delay and they are called dynamical systems with time-delay or simply time-delay systems [1–5]. There are many factors that affects the stability of time-delay systems. Some of them are switching effects, impulses, parameter/system uncertainties, disturbances, etc. Above factors inspired many researchers to focus their investigation on the qualitative aspects of the dynamical systems, for example see [6–13] and therein. The human brain consists of one billion neurons that are interconnected to themselves in a complex manner. They constitute a network called neural network (NN). In the same manner analogy to NN, one can design a network called artificial NN (ANN) in order to solve many complex problems occurring in real-life situations. ANNs can be viewed as a dynamical system and also on the application point of view, the dynamical behavior of ANNs is carried out. There are different types of ANNs which include Cohen-Grossberg NNs, bidirectional associative NNs, cellular NNs, etc. Dynamical analysis for the above types of NNs are carried out extensively by numerous researchers due to its relevance in the real-life situations, for details see, [3, 8, 14–17] and references therein. Switched or jump dynamical system is a class of dynamical system which consists of a finite number of subsystems and a finite number of switches or modes. Here, the switch from one mode to other is coordinated by a switching law and there are two types of switching namely arbitrary switching and constrained switching. Markovian switching is a type of arbitrary switching where the switching is taken to be random among the subsystems. During switching, there will be associated a time-delay when the dynamics of the system changes from one subsystem to another subsystem and those type of time-delay is called mode-dependent time-delay, for details, see [5, 15, 18–23] and references therein. During recent years, a special class of ANNs namely INNs got much attention due to its physical and biological relevance. The dynamics of INNs model is somewhat complicated related to the ordinary ANNs model due to the presence of secondorder derivative term. Hence, there is an increasing interest among the research community to analyze the behavior of INNs and also the factors that affect the various dynamics. As mentioned before, the stability of INNs is also affected by numerous factors namely time-delays, impulses, uncertainties (which occur in the

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system parameters), external noise, internal disturbances (which occurs in system parameters), switching effects and so on, see [24–26]. Also, there are different kinds of time-delay namely discrete time-delay, continuous time-delay, neutral time-delay, mode-dependent time-delay, and these time-delays can be constant, time-varying (which may be continuous and either differentiable or non-differentiable), interval time-varying, proportional and some times random also, for more results, see [14, 24, 27–30]. Stability of INNs with these effects can be analyzed by transforming the INNs into a system of differential equations using a variable transformation technique. Here, INNs will be converted into a set of differential equations of first-order type. On the other hand, it can be handled without transforming the original system also, for details see Refs. [31, 32]. In both of these techniques, stability INNs can be carried out with the concepts of Lyapunov stability theory and linear matrix inequality (LMI) [6]. Here, the idea is to investigate the stability of the concerned without actually solving it. The problem of stability is converted into a convex optimization problem which can be solved by using the LMI technique, refer [6]. Besides Lyapunov technique, there are also other techniques to investigate stability namely Lagrangian method, M-matrix approach, etc. Authors in [33] analyzed the stability for the problem of inertial Cohen-Grossberg NNs subject to the effects of Markovian jump parameters (MJPs) that are described by a continuous-time finite-state Markov chain. While the authors in [34] studied the synchronization of INNs subject to MJPs and implemented their results in image processing as an application. Authors in [35] obtained the reachable set estimation for inertial Markov jump BAM NN with bounded input disturbances in which transition rates are assumed to be partially unknown. Authors in [36] established stability results for INNs in which time-delay is assumed to be mode-dependent. Authors in [37] discussed the exponential stability of time-delay NNs in which both MJPs and Brownian motion are taken into account. Also, construction of LKF and estimation of integral terms play a vital role in obtaining stability conditions of the system under consideration, for details see [38] and [39]. Hence, from the above discussions, it can be seen that the results are available for the stability of INNs pertaining to MJPs but the literature related to the problem of stability of INNs with both MJPs and Brownian motion is to be investigated which motivates this problem. Through the above discussions, we can conclude that the problem of hybrid stochastic INNs with time-delay is not fully investigated in the literature. Thus, it is essential to analyze the stability behavior of INNs with the effects of disturbances and time-delay. Here, the special case of time-delay, that is mode-dependent time-delay is also considered to investigate the stability of INNs with time-delay and MJPs in the absence of disturbances. Following are key factors taken into consideration while obtaining the stability results: • Activation function is taken to be bounded and monotonic;

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• Effect of intrinsic noise is considered with the help of Markovian jump parameters, extrinsic noise is analyzed with the help of stochastic calculus; • Time delay and its derivatives are assumed to be bounded; • Integral inequalities such as Jensen’s and Wirtinger’s inequality are applied to evaluate some integral terms. Moreover, sufficient stability results are derived based on constructing a piecewise positive functional which incorporates the information about the bounds of time-delay. Manipulated results are stated in terms of LMIs that depend both on the time-delay, derivatives and also jump modes. Further, formulated conditions are examined through sufficient numerical examples. The contents of this chapter are organized in the following manner. Section 2 gives the model description, and the lemmas, definitions, and assumptions which are used in the sequel are presented. Section 3 presents the corresponding stability conditions for the problem under study. Section 4 demonstrates the effectiveness of the derived results through numerical examples. Section 5 draws the conclusion. Notations Rn and Rn×m denote n-dimensional Euclidean space and the set of all n × m matrices with real entries respectively; M ≥ N (M > N ) represent that M − N is positive-semidefinite (positive-definite) matrix, where M and N are symmetric matrices. A T indicates the transpose of the matrix A and the symbol ∗ denotes the transposed elements in the symmetric matrix.; I is the identity matrix;  ·  denotes Euclidean norm in Rn . Matrices that are not defined explicitly are considered to have appropriate dimensions. (, F, {Ft }t≥0 , P) denotes the probability space. {Ft }t≥0 fulfilling the usual conditions. L2F0 ((−∞, 0]; Rn ) represents the collection of all F0 -measurable. E(·) denotes the mathematical expectation with respect to the given probability measure P.

2 Preliminaries INNs model incorporated with the effects of time-delay is described by a system of differential equations as m¨ i (t) = − ai m˙ i (t) − bi m i (t) +

n 

ci j f j (m j (t))

j=1

+

n 

di j f j (m j (t − τ (t)) + Ii , i = 1, 2, . . . , n,

(1)

j=1

where m¨ i (t) denotes the inertial term of the ith neuron at time t; m i (t) is the ith neuron’s state at time t. f j (·) denotes the activation function of jth neuron at time t and f j (0) = 0, j ∈ {1, 2, . . . , n}; ci j and di j are the connection weights associated to the neurons with out and with time-delays respectively. ai and bi are positive

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constants. bi is the rate of passive decay of the ith neuron. Ii represents the extraneous input on the ith neuron at time t. Time-varying delay τ (t) satisfies 0 ≤ τ1 ≤ τ (t) ≤ τ2 , τ˙ (t) ≤ μ < 1,

(2)

where τ1 , τ2 and μ are positive constants. Initial condition of (1) is given by m i (s) = φi (s) and dtd (m i (s)) = ψi (s), for −τ2 ≤ s ≤ 0 where φi (s) and ψi (s) are continuous and bounded. Now, let pi (t) = m˙ i (t) + γi m i (t), i = 1, 2, . . . , n.

(3)

Next, using the transformation (3), we transform system (1) into a set of first order differential equations and the transformed system is given as ⎧ ⎪ m˙ i (t) = −γi m i (t) + pi (t), ⎪ ⎪ ⎪ n ⎪ ⎨ p˙ (t) = −(a − γ ) p (t) − (b − (a − γ )γ )m (t) +  ci j f j (m j (t)) i i i i i i i i i j=1 ⎪ n ⎪  ⎪ ⎪ ⎪ + di j f j (m j (t − τ (t))) + Ii . ⎩

(4)

j=1

Initial conditions of (4) is given by m i (s) = φi (s) and pi (s) = ψi (s) + φi (s), for −τ2 ≤ s ≤ 0. Let us shift the equilibrium point of (4) to the origin using the transformation x(t) = m(t) − m¯ and y(t) = p(t) − p, ¯ where (m, ¯ p) ¯ is the equilibrium point of (4). Hence, the resulting system in vector form is given as 

x(t) ˙ = −Γ x(t) + y(t) y˙ (t) = −Ay(t) − Bx(t) + Cg(x(t)) + Dg(x(t − τ (t))),

(5)

where x(s) = φ(s) − m ∗ and y(s) = ψ(s) + φ(s) − p ∗ , s ∈ [−τ2 , 0] denote the initial condition. Here, x(t) ∈ Rn×1 and y(t) ∈ Rn×1 are the state vectors of (5), g(x(t)) = [g1 (x1 (t)), . . . , gn (xn (t))]T with g(x(t)) = f (x(t) + m ∗ ) − f (m ∗ ), A = diag{(a1 − γ1 ), . . . , (an − γn )}, B=diag{b1 − (a1 − γ1 )γ1 , . . . , bn −(an − γn )γn }, C = (ci j )n×n and D = (di j )n×n . Dynamics of INNs (5) can be affected by the effects of noises which are classified into two types namely intrinsic and extrinsic noises. Extrinsic noise which occurs as an external disturbance is taken as the white noise since it gives constant power spectral density at different frequencies. Here, the white noise is specifically assumed to be Gaussian white noise. Basically, the INN model represents the electronic circuit model with elements inductor, capacitor, and resistor. Further any electric circuit system may subject to

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disturbances which will spoil the nature of the system, refer [17]. Hence, the existence of the model considered in this work is guaranteed. Thus, taking these effects into account, system (5) can be written as ⎧ ⎪ ⎨d x(t) = [−Γ x(t) + y(t)]dt, dy(t) = [−Ar (t) y(t) − Br (t) x(t) + Cr (t) g(x(t)) + Dr (t) g(x(t − τ (t)))]dt ⎪ ⎩ +[Er (t) x(t) + Fr (t) y(t)]η(t)dt.

(6)

Here, {r (t), t ≥ 0} is a Markov chain which is right-continuous on the probability space and takes value in a finite space S = {1, 2, . . . , N } with generator  = (πi¯ j¯ ) N ×N given by 

¯ ¯ ¯ (t) = i} ¯ = πi¯ j¯ δt + o(δt), i f i = j, Pr {r (t + δt) = j|r ¯ 1 + πi¯i¯ δt + o(δt), i f i¯ = j, = 0, πi¯ j¯ ≥ 0 represents the transition rate from mode  i¯ at time t to mode j¯ at time t + δt if i¯ = j¯ and πi¯i¯ = − πi¯ j¯ . η(t) denotes the where δt > 0 and lim

δt→0

o(δt) δt

i¯ = j¯

1-dimensional white noise process. Using the fact that the white noise process is the time derivative of a Brownian motion, we have η(t)dt = dω(t), where ω(t) is an one-dimensional Brownian motion defined on (, F, {Ft }t≥0 , P) which satisfies E[dω(t)] = 0 and E[dω(t)]2 = dt. Here, it is assumed that Markov chain r (t) and Brownian motion ω(t) are independent of each other. For simplicity, denote r (t) = k, then system (6) is written as  d x(t) = [−Γ x(t) + y(t)]dt, (7) dy(t) = g1k (t)dt + g2k dω(t), where g1k (t) = −Ak y(t) − Bk x(t) + Ck g(x(t)) + Dk g(x(t − τ (t))) and g2k = E k x(t) + Fk y(t). In order to prove our main results, we state an assumption, some lemmas and a definition. Assumption 1 Activation function of the neuron g j (·) is assumed to satisfy the following condition with g j (0) = 0 l −j ≤

g j (u) − g j (v) ≤ l +j , ∀ u, v ∈ R and u = v, u−v

where j = 1, 2, . . . , n, l −j and l +j are some real scalars.



Lemma 1 ([3]) For any Z 1 , Z 2 ∈ Rn and a diagonal matrix Q > 0 with compatible dimension, the following condition satisfies

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±2Z 1T Z 2 ≤ Z 1T Q −1 Z 1 + Z 2T Q Z 2 . Lemma 2 ([3]) For any positive-definite matrix M, any scalars a and b with b > a, and a vector-valued function y˙ (t) : [a, b] → Rn such that the integrals in the following inequality are well defined, then b

b

y˙ (s)M y˙ (s)ds ≥ (b − a) T

a

y˙ (s)ds

b

T M

a

 y˙ (s)ds .

a

Lemma 3 ([38]) For any positive matrix M, and vector function x(t) ˙ : [a, b] → Rn , the following inequality holds b a

⎡

⎤ 4M 2M −6M 1 ϒ(·) ⎣ ∗ 4M −6M ⎦ ϒ T (·), x˙ T (s)M x(s)ds ˙ ≥ (b − a) ∗ ∗ 12M 

where ϒ(·) =

x (b) x (a) T

T

1 (b−a)

 b

T  x(s)ds

.

a

Definition 1 The zero solution of (7) is said to be mean-square asymptotically stable if for every system mode and all finite states φ(t) and ψ(t) then it satisfy lim E(x(t)2 + y(t)2 ) = 0.

t→∞

3 Stability Results In this section, mean-square asymptotic stability of INNs with time-delay, disturbances and MJPs is analyzed. If, in Assumption 1, l −j = 0 and l +j > 0, then g j (·) is said to be of Lurie type activation function. Further, if l +j + l −j = 0 for all j = 1, 2, . . . , n, then Assumption 1 becomes |g j (u) − g j (v)| ≤ l j |u − v|, ∀u, v ∈ R and u = v,

(8)

which is a Lipschitz type activation function. This section presents stability results of system (7) corresponding to the above two cases. Now, we present the first main result.

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Theorem 1 If the the time-delay τ (t) and activation function g(·) satisfy conditions (2) and (8) respectively, then system (7) is said to be mean-square asymptotically stable if there exist symmetric positive-definite matrices P1k , P2k , Q 1 , Q 2 , Q 3 , R1 , positive diagonal matrices S1 , S2 such that the following symmetric LMI holds for all k ∈ S: ⎤ ⎡ k T T T M3k Φ M1k M2k ⎢ ∗ −S1 0 0 ⎥ ⎥ ⎢ (9) ⎣ ∗ ∗ −S2 0 ⎦ < 0, ∗ ∗ ∗ −P2k where Φ k = (φkp,q )5n×5n , with φk1,1

= −P2k Ak −

AkT

P2k + τ21 R1 +

N 

πk i¯ P2k ,

¯ i=1

φk1,2 = −P2k Bk + P1k Γ + τ21 R1 , φk2,2 = −2P1k Γ + Q 1 + Q 2 + Q 3 + τ21 R1 + L T S1 L +

N 

πk i¯ P1k ,

¯ i=1

1 1 1 R1 , φk3,4 = R1 , φk4,4 = −Q 2 − R1 , τ21 τ21 τ21   = −(1 − μ)Q 3 + L T S2 L , M1k = CkT P2k 0 0 0 0 ,   = DkT P2k 0 0 0 0 , M3k = [Fk E k 0 0 0] , = τ2 − τ1 ,

φk3,3 = −Q 1 − φk5,5 M2k τ21

and remaining entries in the symmetric block are zero. Proof Consider the following LKF t V (k, x(t), y(t)) =x (t)P1k x(t) + y (t)P2k y(t) + T

x T (s)Q 1 x(s)ds

T

t−τ1

t +

t x T (s)Q 2 x(s)ds +

t−τ2

x T (s)Q 3 x(s)ds

t−τ (t)

−τ1 t +

x˙ T (s)R1 x(s)dsdθ, ˙

(10)

−τ2 t+θ

where P1k , P2k , Q 1 , Q 2 , Q 3 and R1 are unknown symmetric positive-definite matrices.

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Now, stochastic derivative of (10) along the state trajectory of (7) using Itô formula can be obtained as follows: d V (k, x(t), y(t)) =LV (k, x(t), y(t)) + 2y T (t)P2k g2k (t)dω(t),

(11)

where L represents the diffusion operator and ˙ + 2y T (t)P2k g1k (t) LV (k, x(t), y(t)) ≤2x T (t)P1k x(t) + x T (t)(Q 1 + Q 2 + Q 3 )x(t) − x T (t − τ1 )Q 1 x(t − τ1 ) − x T (t − τ2 )Q 2 x(t − τ2 ) − (1 − μ)x T (t − τ (t))Q 3 x(t − τ (t)) + τ21 x˙ T (t)R1 x(t) ˙ t−τ1 N  T T x˙ (s)R1 x(s)ds ˙ + x (t) πk i¯ P1k x(t) − ¯ i=1

t−τ2

+ y T (t)

N 

T πk i¯ P2k y(t) + trace(g2k (t)P2k g2k (t)).

(12)

¯ i=1

From Lemma 2 and condition (8), we have 2y T (t)P2k Ck g(x(t)) ≤ y T (t)P2k Ck S1−1 CkT P2k y(t) +x T (t)L T S1 L x(t), 2y T (t)P2k Dk g(x(t − τ (t))) ≤ y T (t)P2k C Dk S2−1 DkT P2k y(t) +x T (t − τ (t))L T S2 L x(t − τ (t)),

(13) (14)

where S1 and S2 are positive-definite diagonal matrices and L = diag{l1 , l2 , . . . , ln }. Then, Lemma 3 yields, t−τ1

x˙ T (s)R1 x(s)ds ˙ ≥ t−τ2

    R1 −R1 1  T x(t − τ1 ) T . x (t − τ1 ) x (t − τ2 ) ∗ R1 x(t − τ2 ) τ21

(15)

Last term of inequality (12) can be written as       FkT y(t) T g2k P (t)P2k g2k (t) = y T (t) x T (t) E . [F ] 2k k k E kT x(t) Combining (12)-(16), we have

(16)

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LV (k, x(t), y(t)) ≤   T −1 T −1 T −1 S1 M1k + M2k S2 M2k + M3k P2k M3k ξ(t), ξ T (t) Φ k + M1k

(17)

  where ξ T (t) = y T (t) x T (t) x T (t − τ1 ) x T (t − τ2 ) x T (t − τ (t)) . Next, taking mathematical expectation on both sides of inequality (11) gives Ed V (k, x(t), y(t)) ≤ LV (k, x(t), y(t)).

(18)

Then, according to Lyapunov stability theorem, for the system (7) to be asymptotically stable in the mean-square, we must have Ed V (k, x(t), y(t)) < 0 for all non-zero x(t) and y(t). Thus, Ed V (k, x(t), y(t)) < 0 implies LV (k, x(t), y(t)) < 0 which is true only if T −1 T −1 T −1 S1 M1k + M2k S2 M2k + M3k P2k M3k < 0. Φ k + M1k

(19)

Since inequality (19) is non-linear, apply Schur complement lemma [3] which gives inequality (9). Following the similar proof of Theorem 1 in [37], we can obtain lim E(x(t)2 + t→∞

y(t)2 ) = 0. Thus, according to Definition 1, system (7) is mean-square asymptotically stable which completes the proof of this theorem.  Remark 1 Construction of LKF and the inequalities which we use to estimate the bound of integral terms play a significant role in obtaining conservative results on time-delay. In general, Wirtinger’s inequality is preferable over Jensen’s inequality to estimate the bound of an integral term in the problem of stability analysis of dynamical time-delay systems so as to obtain less conservative results, for reference see [39]. This is obvious from the existing literature related to the qualitative behavior of dynamical systems. Taking this factor into account, Wirtinger’s inequality is used t−τ 1 T x˙ (s)R1 x(s)ds. ˙ Thus, according to Lemma to estimate the bound of integral term t−τ2

3, we have t−τ1

− t−τ2

1 x˙ (s)R1 x(s)ds ˙ ≤− τ21 T



 x (t − τ2 ) x (t − τ1 ) T

T

1 τ21

t−τ 1

T  x(s)ds

t−τ2

⎡

⎤ x(t − τ2 ) 4M 2M −6M ⎢ x(t − τ ) ⎥ 1 ⎥. × ⎣ ∗ 4M −6M ⎦ ⎢ 1 ⎣ 1 t−τ ⎦ x(s)ds ∗ ∗ 12M τ21 ⎡

⎤

t−τ2

This effects to the following changes:

(20)

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 T ¯ ξ (t) = y T (t) x T (t) x T (t − τ1 ) x T (t − τ2 ) ⎛ t−τ ⎞T  1 T ⎝ ⎠ , x(s)ds x (t − τ (t)) t−τ2

and   M¯ 1k = CkT P2k 0 0 0 0 0 ,

  M¯ 2k = DkT P2k 0 0 0 0 0 ,

M¯ 3k = [Fk E k 0 0 0 0] .

It is noted that the corresponding results are summarized in the following corollary. These results can be directly obtained from Theorem 1 and hence it is omitted. Corollary 1 Suppose conditions (2) and (8) hold, system (7) is mean-square asymptotically stable if there exist symmetric positive-definite matrices P1k , P2k , Q 1 , Q 2 , Q 3 , R1 , diagonal matrices S1 , S2 such that symmetric LMI holds for all k ∈ S: ⎡

Φ¯ k ⎢ ∗ ⎢ ⎣ ∗ ∗

T M¯ 1k −S1 ∗ ∗

⎤ T T M¯ 2k M¯ 3k 0 0 ⎥ ⎥ < 0, −S2 0 ⎦ ∗ −P2k

where Φ¯ k = (φ¯ kp,q )6n×6n , 4 φ¯ k1,1 = φk1,1 , φ¯ k1,2 = φk1,2 , φ¯ k2,2 = φk2,2 , φ¯ k3,3 = −Q 1 − R1 , τ21 2 6 4 φ¯ k3,4 = − R1 , φ¯ k3,6 = φ¯ k4,6 = 2 R1 , φ¯ k4,4 = −Q 2 − R1 , τ21 τ21 τ21 12 φ¯ k5,5 = φk5,5 , φ¯ k6,6 = − 3 R1 , τ21 and all other entries are zero. Remark 2 If the activation function is assumed to satisfy Assumption 1, then the corresponding asymptotic stability condition of system (7) can be obtained with construction of the following LKF Vˆ (k, x(t), y(t)) =V (k, x(t), y(t)) + 2

+2

n  i=1

⎡ ⎣λ2i

n  i=1

⎡ ⎣λ1i

⎤ xi (s)  gi (s) − li− s ds ⎦ 0

⎤ xi (s) + li s − gi (s) ds ⎦ . 0

(21)

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Now, according to Assumption, there exist positive diagonal matrices T1 and T2 such that − 2{g T (x(t))T1 g(x(t)) − x T (t)(L 1 + L 2 )g(x(t)) + x T (t)L 1 T1 L 2 x(t)} ≥ 0, (22) and − 2{g T (x(t − τ (t)))T2 g(x(t − τ (t))) − x T (t − τ (t))(L 1 + L 2 )g(x(t − τ (t)) + x T (t − τ (t))L 1 T2 L 2 x(t − τ (t))} ≥ 0.

(23)

Following theorem provides asymptotic mean-square stability conditions under above situations. Theorem 2 Suppose condition (2) and Assumption 1 hold, system (7) is said to be mean-square asymptotically stable if there exist symmetric positive-definite matrices P1k , P2k , Q 1 Q 2 , Q 3 , R1 , diagonal matrices Λ1 , Λ2 then the following symmetric LMI satisfies for all k ∈ S: Φˆ k < 0, where Φˆ k = (φˆ kp,q )7n×7n , φˆ k1,1 = φk1,1 , φˆ k1,2 = φk1,2 + Λ2 L 2 , φˆ k1,5 = P2k Ck , φˆ k1,6 = P2k Dk + Λ1 − Λ2 , φˆ k2,2 = −2P1k Γ + Q 1 + Q 2 + Q 3 + τ21 R1 + 2L 1T Λ1 − 2L 2T Λ2 − 2L 1 T1 L 2 , φˆ k2,6 = −Λ1 + Λ2 + L 1 + L 2 , φˆ k3,3 = φk3,3 , φˆ k3,4 = φk3,4 , φˆ k4,4 = φk4,4 , φˆ k5,5 = −(1 − μ)Q 3 + L 1 + L 2 − L 1 T2 L 2 , φˆ k6,6 = −2T1 , φˆ k7,7 = −2T2 and all other entries are zero. Proof This theorem follows directly from Theorem 1 with the help of (21)-(23). Hence, the detail is omitted.  Here, we have another result that follows directly from Theorem 2. Corollary 2 Suppose conditions (2) and (20), and Assumption 1 hold, system (7) is said to be mean-square asymptotically stable if there exist positive-definite symmetric matrices P1k , P2k , Q 1 Q 2 , Q 3 , R1 , diagonal matrices Λ1 , Λ2 such that the following symmetric LMI holds for all k ∈ S: Φ˜ k < 0, where Φ˜ k = (φ˜ kp,q )8n×8n ,

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φ˜ k1,1 =φˆ k1,1 , φ˜ k1,2 = φˆ k1,2 , φ˜ k1,5 = φˆ k1,5 , φ˜ k1,6 = φˆ k1,6 , φ˜ k2,2 =φˆ k2,2 , φ˜ k2,6 = φˆ k2,6 , φ˜ k3,3 = φ¯ k3,3 , φ˜ k3,4 = φ¯ k3,4 , 6 6 φ˜ k3,8 = 2 R1 , φ˜ k4,4 = φ¯ k4,4 , φ˜ k4,8 = 2 R1 , φ˜ k5,5 = φˆ k5,5 , τ21 τ21 12 φ˜ k6,6 =φˆ k6,6 , φ˜ k7,7 = φˆ k7,7 , φ˜ k8,8 = − 3 R1 τ21 and all other entries are zero. Remark 3 In this work, the activation function is assumed to be bounded and monotone, and a special case namely Lipschitz continuity is also assumed while deriving the sufficient conditions of the concerned system. Required delay-dependent meansquare asymptotic conditions are given for each of these cases separately. Remark 4 The problem considered in this work is distinct from those in the existing literature in the following aspects: (i) here both types of disturbances namely internal and external types are taken into account whereas in the existing literature either external or internal type, i.e., INNs with Brownian motion or MJPs are considered; (ii) two types of assumptions are made on the activation functions and the conditions for each case are summarized separately. Remark 5 Letting g2k = 0, Γ = I and τ (t) = τk (t), the system (7) changes to the following form  d x(t) = [−x(t) + y(t)]dt, dy(t) = −Ak y(t) − Bk x(t) + Ck g(x(t)) + Dk g(x(t − τk (t)))dt.

(24)

τk (t) is the mode-dependent time-varying delay which satisfies 0 ≤ τk (t) ≤ τ2k , τ˙k (t) ≤ μk < 1, ∀ k ∈ S, where τ2k and μk are constants for any k ∈ S. Corresponding stability results for the above system is summarized in [36]. Remark 6 In general, the presence of more cross-terms and appropriate manipulation of integral inequalities will reduce the conservatism on time-delay. Cross terms can be increased through the usage of free-weighting matrix technique. This involves zero equations with some free weighting matrices. Delay-dependent stability conditions summarized in Sect. 3 can be still made less conservative with the above approach. Even though this method will give less conservative results, on the other hand, it will automatically increase the computation complexity also. Hence, in this work, we focus our attention towards Wiritnger’s inequality technique to obtain less conservative results.

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4 Numerical Example In this section, numerical examples of stochastic INNs with MJPs is demonstrated to show the effectiveness and reliability of the obtained theoretical results. Sufficient LMI conditions obtained in Sect. 3 are verified through MATLAB LMI solvers. Example 1 Consider system (7) with two jump modes whose parameters are described as follows Mode 1

⎡

⎤ ⎡ ⎤ 5 0 0 0.2 0 0 A1 = ⎣ 0 5.5 0 ⎦ , B1 = ⎣ 0 0.9 0 ⎦ , 0 0 7 0 0 0.4 ⎡ ⎤ ⎡ ⎤ −2.2 0.4 0 1 1.2 0 C1 = ⎣ 1 −1.6 0 ⎦ , D1 = ⎣ 0.9 −1.5 0 ⎦ , −0.5 0 −0.8 0.3 0.5 −1.0

Mode 2

⎡

⎤ ⎡ ⎤ 2.5 0 0 0.2 0 0 A2 = ⎣ 0 4 0 ⎦ , B2 = ⎣ 0 0.9 0 ⎦ , 0 04 0 0 0.1 ⎡ ⎤ ⎡ ⎤ −1.5 0.4 0 2.1 1.6 0 C2 = ⎣ 1 −1.1 0 ⎦ , D2 = ⎣ 1.6 −1.1 0 ⎦ , −0.7 0 −2.2 0.4 0.8 −1.2

with  =

 −5 5 , E 1 = 0.1I, F1 = 0.2I, 4 −4

E 2 = diag{0.1, 0.3, 0.2}, F2 = diag{0.2, 0.5, 0.2}. If we choose τ2 = 0.3, τ1 = 0, μ = 0.5 and L = diag{0.1, 0.1, 0.1}, then the meansquare asymptotic stability conditions for the system (7) with the above parameters can be verified based on the LMI condition specified in Theorem 1. Solving the LMI condition (9) results in the following feasible matrices

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⎡

P11

P21

P12

P22

⎤ 160.8966 9.9924 −30.6903 = ⎣ 9.9924 132.7698 1.9028 ⎦ , −30.6903 1.9028 148.6468 ⎡ ⎤ 163.7432 10.4369 −29.8908 = ⎣ 10.4369 134.9727 1.8454 ⎦ , −29.8908 1.8454 150.7557 ⎡ ⎤ 44.3752 0.9817 −6.2472 = ⎣ 0.9817 46.9122 0.4032 ⎦ , −6.2472 0.4032 38.6418 ⎡ ⎤ 65.7336 3.6342 −8.8578 = ⎣ 3.6342 66.2666 0.3200 ⎦ , −8.8578 0.3200 57.4530

and ⎡

Q1

Q2

Q3

R1

⎤ 34.5425 3.0359 −9.1952 = ⎣ 3.0359 43.7191 0.7996 ⎦ , −9.1952 0.7996 45.1519 ⎡ ⎤ 34.5425 3.0359 −9.1952 = ⎣ 3.0359 43.7191 0.7996 ⎦ , −9.1952 0.7996 45.1519 ⎡ ⎤ 57.0554 5.0476 −15.4378 = ⎣ 5.0476 73.3584 1.2818 ⎦ , −15.4378 1.2818 75.9318 ⎡ ⎤ 14.2899 0.3920 −1.1768 = ⎣ 0.3920 19.4208 0.1662 ⎦ . −1.1768 0.1662 18.7159

These feasible matrices obtained using MATLAB LMI solvers ensures that the LMI condition (9) given in Theorem 1 is negative-definite which corresponds to the LKF (10). Hence, according to Definition 1 and LMI condition (9), the considered system (7) corresponding to the considered parameters mean-square asymptotically stable. In order to compare the results obtained in Theorem 1, Corollary 1 and Theorem 2 of Sect. 3, we found the maximum allowable upper bound of time-delay for the above-mentioned parameters. Maximum bound of time-delay is found to be 2.73, 2.98 and 2.87 for Theorem 1, Corollary 1 and Theorem 2, respectively. Also, from the LMI conditions, one can see that the size of the LMI obtained in Theorem 2 is less compared to Theorem 1. Further, numerical simulation is carried out for the considered system with the same set of parameters expect the time-delay which is taken to be 0.9 + sin2 (t). Figures 1 and 2 shows the state trajectories of subsystem 1 and 2 respectively whereas Fig. 3 shows the behavior of the Markov chain r (t) for the state transition probability matrix .

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0.3 x1

x(t)

0.2

x2 x3

0.1

0 0

5

10

15

20

25

Time (sec) 0.2 y

y(t)

0.15

1

y2 y

0.1

3

0.05 0 0

5

10

15

20

25

Time (sec) Fig. 1 State trajectories of subsystem 1

The numerical simulation clearly depicts that both the subsystems are asymptotically stable in the mean-square. Example 2 Consider the INNs (24) with two modes described by the following parameters Mode 1:     0.7 0 −0.2 0 A1 = , B1 = , 0 2.1 0 −0.4     −1.43 0.1 0 0.2 , D1 = , C1 = 0.4 −1.1 −0.02 −0.1 Mode 2: 

   0.9 0 −0.4 0 , B2 = , A2 = 0 0.8 0 −0.4     −2.1 0.5 0.4 0.6 C2 = , D2 = ; 0.5 −1 −0.1 −0.6

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0.3 x1 x2

x(t)

0.2

x

3

0.1

0

0

5

10

15

20

25

Time (sec) 0.25 y

y(t)

0.2

1

y2

0.15

y3

0.1 0.05 0 0

5

10

15

20

25

Time (sec) Fig. 2 State trajectories of subsystem 2 3

2.5

Mode r(t)

2

1.5

1

0.5

0 0

10

20

30

40

50

Time (t)

Fig. 3 Markovian jump modes

60

70

80

90

100

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with h 2 = 11, μ1 = 0.4, μ2 = 0.3, L 1 = 0, L 2 = I .



 −0.3 0.3 Also, the transition probability matrix is assumed as  = . 0.7 −0.7 Stochastic stability condition given in for the system with the above parameters is validated through MATLAB LMI solvers and the corresponding feasible matrices are listed as follows     0.1032 0.0015 0.1644 −0.0031 , P12 = , P11 = 0.0015 0.1181 −0.0031 0.1605     0.0231 −0.0006 0.0242 0.0141 , P22 = , P21 = −0.0006 0.0087 0.0141 0.0342     0.0524 −0.0002 0.0237 −0.0000 , Q2 = , Q1 = −0.0002 0.0510 −0.0000 0.0229     0.0084 −0.0001 0.0085 −0.0000 Q3 = , Q4 = , −0.0001 0.0079 −0.0000 0.0083     0.0107 −0.0001 0.0129 0.0001 , R1 = , Q5 = −0.0001 0.0098 0.0001 0.0159   1.2212 −0.0017 R2 = . −0.0017 1.2866 For the purpose of simulation, the mode-dependent time-varying delay is assumed to be τ1 (t) = 0.5 + 0. sin2 (t) when the system is in Mode 1 and τ2 (t) = 0.75 + 0.2 sin2 (t) when the system is in Mode 2 and the activation function is assumed to be tanh(x(t)) for both the modes. Other parameters are taken as given above in the problem. Corresponding state trajectories for two different time-varying delay is simulated and are given below remaining parameters are chosen as above. Figures 4 and 5 confirms that both subsystems converge to the equilibrium point. Hence, the considered system with the given parameters is stable in the sense of Lyapunov. Jump modes of (24) for the above parameters is simulated and it is given in Fig. 6. Now, when there is no jump in the system dynamics, that is, there are no subsystems, stability of the system (24) is investigated by taking A1 = A2 , B1 = B2 , C1 = C2 , D1 = D2 , μ1 = μ2 and  = 0. Corresponding feasible matrices obtained are given below  P11 =  Q1 =  Q3 =

 0.1291 0.0000 , 0.0000 0.1290 0.0727 −0.0006 0.0850 −0.0009

  0.0362 −0.0005 P21 = , −0.0005 0.0091    −0.0006 0.0300 −0.0001 , Q2 = , 0.0660 −0.0001 0.0281    −0.0009 0.0725 −0.0001 , Q4 = , 0.0762 −0.0001 0.0718

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0.2 x1 (t)

x(t)

0.15

x2 (t)

0.1 0.05 0 0

1

2

3

4

5

6

7

8

Time (sec) 0.04 y1 (t)

y(t)

0.03

y2 (t)

0.02 0.01 0 0

1

2

3

4

5

6

7

8

Time (sec) Fig. 4 State trajectories of subsystem 1



 0.1032 −0.0010 Q5 = , −0.0010 0.0914   1.4532 0.0020 . R2 = 0.0020 1.4487



 0.0157 0.0002 R1 = , 0.0002 0.0173

Remark 7 The work [40] analyzed the synchronization behavior of the switched NNs with the effects of mode-dependent time-delay and impulses. Also, [36] investigated the problem of stability analysis for INNs with mode-dependent time-delays in which the disturbance is taken to be intrinsic noise. Literature review reveals that there exists no work in the area of stability of INNs with both intrinsic and extrinsic noises. Hence, the effect of these noises for the stability problem of INNs is taken into an investigation in this work which is the generalization of [36].

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0.2 x1 (t)

x(t)

0.15

x2 (t)

0.1 0.05 0 0

1

2

3

5

4

6

8

7

Time (sec) 0.04 y1 (t)

y(t)

0.03

y2 (t)

0.02 0.01 0 0

1

2

3

5

4

6

7

8

Time (sec) Fig. 5 State trajectories of subsystem 2 3

2.5

r(t)

2

1.5

1

0.5

0 0

10

20

30

40

50

Time(sec) Fig. 6 Markovian jump modes

60

70

80

90

100

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5 Conclusion The problem of stochastic stability of INNs along with time-delay, disturbances, and MJPs is investigated in this chapter. Sufficient delay-dependent stochastic stability conditions are obtained in terms of LMIs which can be solved using LMI solvers through MATLAB software. Here, both types of noises namely intrinsic and extrinsic noises are taken in to account which is different from the existing literature concerning INNs. Also, stability results for mode-dependent Markovian jump INNs are investigated in the absence of external disturbances. INNs Numerical example illustrating the merits of the proposed method are given. Problem considered in this work generalizes some of the results of existing literature in the domain of INNs. Also, the problem discussed in this work can be extended with the assumption that some of the transition rates in the transition matrix are unknown. Further, it can be generalized with the incorporation of uncertainties in both system matrices and transition rates. Acknowledgements Part of this work was done when the first author was a Post-Doctoral fellow in the Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala and it was supported by DST-SERB through NPDF scheme (file No. PDF/2016/002992 dated 01/04/2017).

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39. Liu, K., Fridman, E.: Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica 48, 102–108 (2012) 40. Xie, X., Liu, X., Xu, H.: Synchronization of delayed coupled switched neural networks: Modedependent average impulsive interval. Neurocomputing 365, 261–272 (2019)

Exponential Stability and Stabilization of Stochastic Neural Network Systems via Switching and Impulsive Control Ticao Jiao and Ju H. Park

Abstract The exponential stability and stabilization problem of stochastic neural network systems via a switching and impulsive control are dealt with in the chapter. To achieve the desired performance, a new type of switching and impulsive controllers are designed. With comparison to the previous results, the main characteristics of our proposed controller lies in the asynchronous behavior between switchings and impulses. That is, the occurrence instants of switchings and impulses are not the same. To describe the generation mechanisms of switchings and impulses, the concepts of dwell time and impulsive interval are employed. Then, based on multiple Lyapunov functions approach, sufficient conditions for mean square exponential stability are first established in terms of linear matrix inequalities (LMIs), based on which gain matrices of switching and impulsive controllers are presented. We, at last, provide two numerical examples to verify the applicability of the proposed theoretical results. Keywords Neural network systems · Switching · Impulsive · Dwell time · Impulsive interval

1 Introduction During past decades, neural network systems have been widely applied in practical fields such as signal processing, image reconstruction, artificial intelligence and associative memory, etc. These applications strongly rely on the stability, asymptotic stability, and synchronization of neural network systems. Therefore, the great T. Jiao (B) School of Electrical and Electronic Engineering, Shandong University of Technology, Shandong 255000, People’s Republic of China e-mail: [email protected] J. H. Park Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_18

435

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interest of researchers have been attracted to the dynamical behaviors of neural network systems and a large number of articles and monographs concerning Hopfield, stochastic, switching, uncertain, and reaction-diffusion network models have been published (see, e.g. [1–7] and references therein). As is well known, many physical and industrial models can be well described by switched systems, for instance, the continuous stirred tank reactor with two modes feed stream and mass-spring-damper systems in [8, 9]. Switched (neural network) systems have gone through extensive studies in the literature. For instance, in [10], standard H∞ control of switched systems was investigates, in which any internal stability on switched subsystems is not required. By constructing an impulse-timedependent discretized copositive Lyapunov function, [11] addressed the stability analysis and L∞ -gain characterization for positive Takagi-Sugeno fuzzy systems subject to impulse effects. Via the method of mode-dependent average dwell time, exponential stability criteria of delayed switched neural networks were obtained in [12]. In [13], a novel method was introduced to describe the switching law, based on which the state estimation problem for a class of discrete-time switched neural networks was considered. On the other hand, some real-world systems usually undergo instantaneous state changes at discrete times. The natural and reasonable way to model such a class of dynamical systems is impulsive differential equations, which have shown their enormous power in the fields of theory and applications during the past decades. Especially, the increasing popularity has been achieved in neural networks [2, 14, 15]. In [15], from the viewpoint of destabilizing impulses, new criteria on stochastic stability with the impulse effects were derived if the feasible bound of impulses is imposed. Then, a general class of neural networks with heterogeneous impulsive effects was introduced in [16] and the synchronization problem was dealt with. Combining neural network systems, impulsive systems with switched systems, it is natural to develop a more all-around system, i.e., switching and impulsive neural networks. Despite the apparent abundance of applications, switching and impulsive neural networks have not received the attention they deserve in the past decade [2]. In [17], the asymptotic synchronization problem for a class of uncertain hybrid switching and impulsive complex networks was investigated by partitioning the dwell time. When time delay is considered, by means of Halanay inequality and LyapunovKrasovskii functionals, stability issue of impulsive and switching systems was studied in [18, 19]. It is worth noting that in the works [2, 17–21], the switching signal and the impulsive law have been integrated as an impulsive and switching law, which is described by using the concept of (average) dwell time. In other words, these results assume that the switchings and impulses occur at the same time instants. Note that such an assumption is much too conservative and out of step with reality. For instance, when switchings and state-dependent impulses are simultaneously considered, the resulting equivalent system by using B-equivalence method, in fact, owns different impulsive and switching instants [22, 23]. Another example is the switched networks with mode-dependent impulsive effects. In each switching interval, impulses have happened multiple times. In this case, it means that switchings and impulses are

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asynchronous [16]. Therefore, compared with [2, 17–19], it is more reasonable and practical to make an investigation of dynamic systems with asynchronous switchings and impulses [25]. To the authors’ knowledge, the research of such systems has not received sufficient attention and many open problems have not been successfully solved. Motivated by the above analysis, we in this chapter address the issue of exponential stability and stabilization of stochastic neural network systems via switching and impulsive control. Here, the switching instants differ from the impulsive ones. Our main contributions in the chapter lie in the following aspects: 1. In contrast with the neural networks investigated in the previous works (e.g. [2, 17–19]), the neural networks studied in the chapter take a general form, which allows the asynchronism of switchings and impulses. 2. The approach of average dwell time and (inverse) average impulsive interval is employed to control the occurrence frequency of switchings and impulses. Then, combining it with multiple Lyapunov functions approach, sufficient criteria ensuring mean square exponential stability are developed in terms of LMIs, based on which gain matrices of switching and impulsive controllers are proposed. The rest of this chapter is organized as follows. Section 2 presents the preliminary results. The main results are given in Sect. 3. Through two examples the obtained results are studied in Sect. 4. Finally, the concluding remarks are provided in Sect. 5.

2 Preliminary Results Consider the following class of stochastic neural networks with asynchronous impulses and switchings (SNNAIWs) described by   dx(t) = − Cr (t) x + Dr (t) Fr (t) (x) + u 1 (t) dt + G r (t) (x, t) dω(t), t = tk , x(tk ) = Ax(tk− ) + u 2 (tk− ), t = tk , x(t0 ) = x0 , r (t0 ) = r0 ,

(1)

where x(t) ∈ Rn denotes the neuron systems state vector; r (t) is a right-continuous function, which controls the switching sequences among m networks; the discontinuous point set {tk }k∈N+ is the impulsive instants; x(tk− ) = limt→tk− x(t) and u 1 (t), u 2 (t) are control input signals. For i ∈ M = {1, . . . , m}, Ci = diag{ci1 , . . . , cin } with ci j > 0 ( j = 1, . . . , n) and G i ∈ Rn×n , A ∈ Rn×n are constant matrices; Fi (·), G i (·) are continuous functions with proper dimensions. ω(t) is a standard Wiener process vector defined on the complete probability space (, F, Ft , P) with a filtration Ft satisfying the usual conditions.

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The following assuming conditions are imposed on the architecture defining system (1). Assumption 1 ([18]) For i ∈ M, the neuron activation function Fi (x) = [Fi1 (x1 ), . . . , Fin (xn )]T satisfying: Fi j (0) = 0, bi j ≤

Fi j (y) ≤ b¯i j , j = 1, . . . , n, y

(2)

where y ∈ R\{0}, bi j and b¯i j are known constants. Assumption 2 For the noise density function G i (x, t), there exist positive definite matrices Q i such that trace{G iT (x, t)G i (x, t)} ≤ x T Q i x.

(3)

Remark 1 It is worth stressing that the introduced SNNAIW model (1) is more general than those in some existing works. For example, a special case of model (1) without stochastic noises and asynchronous impulses and switchings is investigated in [2, 17–19]. For the investigation of exponential stabilization of system (1), the following stochastic stability and stabilization definitions are recalled. Definition 1 ([26]) The trivial solution x = 0 of system (1) with u 1 (t) = 0, u 2 (t) = 0 is said to be exponentially stable in mean (ES-M) if there exist a constant γ > 0 and a class K∞ function α, such that for all x0 ∈ Rn \{0}, E|x(t)| ≤ α(|x(t0 )|)e−γ(t−t0 ) .

(4)

Definition 2 The trivial solution x = 0 of system (1) is said to be exponentially stabilized in mean (E-S-M) if there exist control inputs u 1 (t) and u 2 (t) such that (4) holds.

3 Main Results In this section, our attention is first focused on the stability analysis of system (1) with u 1 (t) = 0 and u 2 (t) = 0, then, on the basis of the developed stability criteria, the asynchronous switching and impulsive controller is to be designed.

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3.1 Stability Analysis 3.1.1

Every Subsystem Is ES-M

In this subsection, we consider the case where every subsystem is ES-M but the impulse is destabilizing. To establish the main results, let us recall the definitions of average dwell time and average impulsive interval, respectively. Definition 3 (Average Dwell Time [27])Let Nr (t  , t  ) denote the number of switchings on an interval [t  , t  ). If there exists two positive numbers Nr 0 and ηs , such that Nr (t  , t  ) ≤ Nr 0 +

t  − t  , ηs

(5)

then we say that the switching signal r (t) has an average dwell time ηs . Definition 4 (Average Impulsive Interval [28]) For the impulsive sequences {tk }k∈N+ , let Ni (t  , t  ) be the number of impulses on an interval [t  , t  ). If there exist two numbers Ni0 > 0, ηi > 0 such that Ni (t  , t  ) ≤ Ni0 +

t  − t  , ηi

(6)

then the impulsive sequence {tk }k∈N+ is said to have an average impulsive interval ηi . Now, we have the following main result. Theorem 1 Under Assumptions 1 and 2, if there exist positive definite matrices Pi , real parameters μ ≥ 1, δ > 1, γ¯ > 0, ci > 0, ρi j > 0 (i, j ∈ M) satisfying   −Pi Ci − Ci Pi − B i1 Λi + ci Q i + γ¯ Pi Pi Di + B¯ i2 Λi < 0, ∗ −Λi

(7)

Pi ≤ ci In , Pi ≤ μP j , i = j,

(8) (9)

A T Pi A ≤ δ Pi , ln μ ln δ − > 0, γ¯ − ηs ηi

(10) (11)

where Λi = diag{ρi1 , . . . , ρin }, B i1 = diag{bi1 b¯i1 , . . . , bin b¯in }, B i2 = diag{

bi1 + b¯i1 b + b¯in , . . . , in }, 2 2

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then system (1) is ES-M when the switching signal r (t) has an average dwell time ηs and the impulsive sequence {tk }k∈N+ an average impulsive interval ηi . Proof Fix any t > t0 . On the interval [t0 , t), assume that the switching sequence is {t1s , t2s , . . . , tls1 } and the impulsive sequence {t1 , t2 , . . . , tl2 }, which satisfy t0 < t1s < t2s < · · · < t sj1 < t1 < t sj1 +1 < · · · < t sj2 < t2 < t sj2 +1 < · · · < t sjk < tl2 < t sjk +1 < · · · < tls1 < t. Therefore, Ni (t0 , t) = l2 , Ns (t0 , t) = l1 . Moreover, the candidate Lyapunov function is defined as Vr (t) (x(t), t) = x T (t)Pr (t) x(t). From system (1), we have ¯ r (t) (x(t), t) LVr (t) (x(t), t) + γV   T = 2x (t)Pr (t) − Cr (t) x + Dr (t) Fr (t) (x) + trace{G rT(t) (x, t)Pr (t) G r (t) (x, t)} + γx ¯ T (t)Pr (t) x(t).

(12)

It follows from Assumption 2 and condition (8) that trace{G rT(t) (x, t)Pr (t) G r (t) (x, t)} ≤ ci trace{G rT(t) (x, t)G r (t) (x, t)} ≤ ci x T (t)Q i x(t).

(13)

On the other hand, Assumption 1 implies that 0 ≤ (Fi j (x j ) − bi j x j )(b¯i j x j − Fi j (x j )). Thus, for any ρi j > 0, it holds 0≤

n 

ρi j (Fi j (x j ) − bi j x j )(b¯i j x j − Fi j (x j )),

(14)

j=1

which can be rewritten as the following compact form 

0≤ x

T

FiT (x)

    −B i1 Λi B¯ i2 Λi x . Fi (x) ∗ −Λi

Substituting (13) and (15) into (12) leads to

(15)

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LVr (t) (x(t), t) + γV ¯ r (t) (x(t), t)   T = 2x (t)Pr (t) − Cr (t) x + Dr (t) Fr (t) (x) + ci x T (t)Q i x(t) + γx ¯ T (t)Pr (t) x(t)     T T  −B i1 Λi B¯ i2 Λi x + x Fi (x) , Fi (x) ∗ −Λi

(16)

by which as well as condition (7), we have ¯ r (t) (x(t), t) ≤ 0, LVr (t) (x(t), t) + γV that is, ¯ r (t) (x(t), t). LVr (t) (x(t), t) ≤ −γV

(17)

We next introduce a new function ¯ Vr (t) (x(t), t). W¯ (x(t)) = eγt

By Itô formula and (17), one has ¯ ¯ Vr (t) (x(t), t) + eγt LVr (t) (x(t), t) LW¯ (x(t)) = γe ¯ γt ¯ ¯ ≤ γe ¯ γt Vr (t) (x(t), t) − γe ¯ γt Vr (t) (x(t), t)

= 0,

(18)

from which it follows that W¯ (x(t)) is a continuous non-negative supermartingale. Thus, for ∀t0 < s1 < s2 , E[W¯ (x(s2 ))|Fs1 ] ≤ W¯ (x(s1 )).

(19)

Then, by taking expectation on both sides of (19), we get EW¯ (x(s2 )) ≤ EW¯ (x(s1 )).

(20)

¯ 2 −s1 ) EVr (s1 ) (x(s1 ), s1 ). EVr (s2 ) (x(s2 ), s2 ) ≤ e−γ(s

(21)

Therefore, we have

With this in mind, we first focus on the interval [t0 , t1s ). It is obvious that ¯ 1 −t0 ) EVr (t0 ) (x(t0 ), t0 ). EVr (t0 ) (x(t1 ), t1s ) ≤ e−γ(t s

At the switching instant t1s from (8), it yields

(22)

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EVr (t1s ) (x(t1s ), t1s ) ≤ μEVr (t0 ) (x(t1s ), t1s ).

(23)

Combining (22) with (23), one obtains ¯ 1 −t0 ) EVr (t0 ) (x(t0 ), t0 ). EVr (t1 ) (x(t1s ), t1s ) ≤ μe−γ(t s

(24)

By repeating the above derivation procedure until the switching instant t sj1 , we arrive at ¯ j −t0 ) 1 EVr (t0 ) (x(t0 ), t0 ). EVr (t sj ) (x(t sj1 ), t sj1 ) ≤ μ j1 e−γ(t s

(25)

1

Note that at the impulsive point t1 , by (10), it can be deduced that EVr (t sj ) (x(t1 ), t1 ) = Ex T (t1 )A T Pr (t sj ) Ax(t1 ) 1

1

≤ δEx T (t1− )Pr (t sj ) x(t1− ).

(26)

1

Thus, we further get ¯ j +1 −t0 ) 1 EVr (t0 ) (x(t0 ), t0 ). EVr (t sj ) (x(t sj1 +1 ), t sj1 +1 ) ≤ δμ j1 e−γ(t s

(27)

1

By recurrence method, we finally get ¯ 0) EVr (t0 ) (x(t0 ), t0 ) EVr (t) (x(t), t) ≤ δ Ni (t0 ,t) μ Ns (t0 ,t) e−γ(t−t ¯ 0) = eln δ Ni (t0 ,t)+ln μNs (t0 ,t)−γ(t−t EVr (t0 ) (x(t0 ), t0 )

(28)

Using Definitions 3 and 4, inequality (28) can be computed as EVr (t) (x(t), t) ≤ e

ln δ Ni0 + lnη δ (t−t0 )+ln μNs0 + lnη μ (t−t0 )−γ(t−t ¯ 0) i

i

× EVr (t0 ) (x(t0 ), t0 ) = eln δ Ni0 +ln μNs0 EVr (t0 ) (x(t0 ), t0 )e

−(γ− ¯ lnη δ − lnη μ )(t−t0 ) i

i

.

(29)

From the definition Vr (t) (x(t), t), it derives from (29) that E|x(t)|2 ≤

λmax (Pi ) ln δ Ni0 +ln μNs0 −(γ− ¯ lnη δ − lnη μ )(t−t0 ) i i e E|x(t0 )|2 e . λmin (P j )

(30)

Via moment inequality (30) can be further calculated that E|x(t)| ≤

ln μ

δ −(γ− ¯ ln ηi − ηi λmax (Pi ) ln δ N +ln μN i0 s0 E|x(t )|2 e − 2 e 0 λmin (P j )

)(t−t0 )

.

(31)

Therefore, system (1) is ES-M based on Definition 1. The proof is complete. 

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Remark 2 From (7) and (17), we can derive that continuous subsystems are stable. However, the impulse is destabilizing according to (10). To ensure ES-M of the system (1), both a switching signal and an impulsive signal need to be skillfully designed satisfying (11).

3.1.2

No Subsystem Is ES-M

We now turn our attention to the case where any subsystem is not ES-M but the impulse is stabilizing. To achieve this goal, the approach of an inverse average impulsive interval is applied. Definition 5 (Inverse Average Impulsive Interval [28]) For the impulsive sequences {tk }k∈N+ , let Ni (t  , t  ) be the number of impulses on an interval [t  , t  ). If there exist two numbers Ni0 > 0, ηi > 0 such that Ni (t  , t  ) ≥ −Ni0 +

t  − t  , ηi

(32)

then the impulsive sequence {tk }k∈N+ is said to have an inverse average impulsive interval ηi . Theorem 2 Under Assumptions 1 and 2, if there exist positive definite matrices Pi , real parameters μ ≥ 1, 0 < δ < 1, ci > 0, ρi j > 0 (i, j ∈ M) satisfying   −Pi Ci − Ci Pi − B i1 Λi + ci Q i − γ¯ Pi Pi Di + B¯ i2 Λi < 0, ∗ −Λi

(33)

Pi ≤ ci In , Pi ≤ μP j , i = j,

(34) (35)

A T Pi A ≤ δ Pi , ln μ ln δ + < 0, γ¯ + ηs ηi

(36) (37)

stop where Λi , B i1 , B i2 have been defined in Theorem 1, then system (1) is ESM when the switching signal r (t) has an average dwell time ηs and the impulsive sequence {tk }k∈N+ an inverse average impulsive interval ηi . Proof As in Theorem 1, for any t > t0 , consider the interval [t0 , t) and the switching sequence and the impulsive sequence are assumed to be {t1s , t2s , . . . , tls1 } and {t1 , t2 , . . . , tl2 }, which satisfy t0 < t1s < t2s < · · · < t sj1 < t1 < t sj1 +1 < · · · < t sj2 < t2 < t sj2 +1 < · · · < t sjk < tl2 < t sjk +1 < · · · < tls1 < t. Then, the candidate Lyapunov function is defined as Vr (t) (x(t), t) = x T (t)Pr (t) x(t).

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From system (1), we have ¯ r (t) (x(t), t) LVr (t) (x(t), t) − γV   T = 2x (t)Pr (t) − Cr (t) x + Dr (t) Fr (t) (x) + trace{G rT(t) (x, t)Pr (t) G r (t) (x, t)} − γx ¯ T (t)Pr (t) x(t).

(38)

Substituting (13) and (15) into (38) leads to ¯ r (t) (x(t), t) LVr (t) (x(t), t) − γV   T = 2x (t)Pr (t) − Cr (t) x + Dr (t) Fr (t) (x) + ci x T (t)Q i x(t) − γx ¯ T (t)Pr (t) x(t)      −B i1 Λi B¯ i2 Λi x + x T FiT (x) , Fi (x) ∗ −Λi

(39)

by which as well as condition (33), we have LVr (t) (x(t), t) − γV ¯ r (t) (x(t), t) ≤ 0, that is, ¯ r (t) (x(t), t). LVr (t) (x(t), t) ≤ γV

(40)

¯ Vr (t) (x(t), t). With the means of Itô formula and (40), one Let W¯ (x(t)) = e−γt has ¯ ¯ Vr (t) (x(t), t) + e−γt LVr (t) (x(t), t) LW¯ (x(t)) = − γe ¯ −γt ¯ ¯ ≤ − γe ¯ γt Vr (t) (x(t), t) + γe ¯ γt Vr (t) (x(t), t)

= 0,

(41)

from which it follows that W¯ (x(t)) is a continuous non-negative supermartingale. Thus, for ∀t0 < s1 < s2 , E[W¯ (x(s2 ))|Fs1 ] ≤ W¯ (x(s1 )).

(42)

Then, taking expectation on both sides of (19) gives EW¯ (x(s2 )) ≤ EW¯ (x(s1 )).

(43)

¯ 2 −s1 ) EVr (s1 ) (x(s1 ), s1 ). EVr (s2 ) (x(s2 ), s2 ) ≤ eγ(s

(44)

Therefore,

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With this in mind, we first focus on the interval [t0 , t1s ). It is obvious that ¯ 1 −t0 ) EVr (t0 ) (x(t0 ), t0 ). EVr (t0 ) (x(t1 ), t1s ) ≤ eγ(t s

(45)

At the switching instant t1s from (8), it yields EVr (t1s ) (x(t1s ), t1s ) ≤ μEVr (t0 ) (x(t1s ), t1s ).

(46)

Combining (45) with (46), one obtains ¯ 1 −t0 ) EVr (t0 ) (x(t0 ), t0 ). EVr (t1 ) (x(t1s ), t1s ) ≤ μeγ(t s

(47)

By repeating the above derivation procedure until the switching instant t sj1 , we arrive at ¯ j −t0 ) 1 EVr (t0 ) (x(t0 ), t0 ). EVr (t sj ) (x(t sj1 ), t sj1 ) ≤ μ j1 e−γ(t s

(48)

1

Considering the relation proposed in (26), then we get ¯ j +1 −t0 ) 1 EVr (t0 ) (x(t0 ), t0 ). EVr (t sj ) (x(t sj1 +1 ), t sj1 +1 ) ≤ δμ j1 e−γ(t s

(49)

1

After a simple recurrence, we finally have ¯ 0) EVr (t0 ) (x(t0 ), t0 ) EVr (t) (x(t), t) ≤ δ Ni (t0 ,t) μ Ns (t0 ,t) eγ(t−t ¯ 0) EVr (t0 ) (x(t0 ), t0 ). = eln δ Ni (t0 ,t)+ln μNs (t0 ,t)+γ(t−t

(50)

Using Definitions 3 and 5, the inequality (28) can be computed as − ln δ N + ln δ (t−t )+ln μNs0 + lnη μ (t−t0 )+γ(t−t ¯ 0)

i0 0 ηi EVr (t) (x(t), t) ≤ e × EVr (t0 ) (x(t0 ), t0 )

i

= e− ln δ Ni0 +ln μNs0 EVr (t0 ) (x(t0 ), t0 )e

(γ+ ¯ lnηsμ + lnη δ )(t−t0 ) i

.

(51)

Based on the definition Vr (t) (x(t), t), it derives from (29) that E|x(t)|2 ≤

λmax (Pi ) − ln δ Ni0 +ln μNs0 (γ+ ¯ ln μ + ln δ )(t−t0 ) e E|x(t0 )|2 e ηs ηi , λmin (P j )

(52)

which together with moment inequality results in E|x(t)| ≤

ln μ

(γ+ ¯ η s λmax (Pi ) − ln δ N +ln μN i0 s0 E|x(t )|2 e e 0 λmin (P j )

δ + ln ηi )(t−t0 ) 2

.

Therefore, it is concluded that system (1) is ES-M. The proof is complete.

(53)

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Remark 3 It is obvious that continuous subsystems are unstable based on (33) and (40). However, condition (36) implies the stabilizing effect of impulses. By designing both a switching signal and an impulsive signal satisfying (37), ES-M of the system (1) can still be guaranteed. Inspired by the above two proposed cases, we next consider a special case, i.e., stable subsystems with stabilizing impulses. Theorem 3 Under Assumptions 1 and 2, if there exist positive definite matrices Pi , real parameters μ ≥ 1, 0 < δ < 1, ci > 0, ρi j > 0 (i, j ∈ M) satisfying   −Pi Ci − Ci Pi − B i1 Λi + ci Q i + γ¯ Pi Pi Di + B¯ i2 Λi < 0, ∗ −Λi

(54)

Pi ≤ ci In , Pi ≤ μP j , i = j,

(55) (56)

A T Pi A ≤ δ Pi , ln μ ln δ − > 0, γ¯ − ηs ηi

(57) (58)

where Λi , B i1 , B i2 take the same form defined in Theorem 1, then system (1) is ESM when the switching signal r (t) has an average dwell time ηs and the impulsive sequence {tk }k∈N+ an inverse average impulsive interval ηi . Proof It can be proven by referring to the proof processes of Theorems 1 and 2 and therefore the details are omitted here.  Remark 4 Although conditions (11) and (58) are expressed exactly the same, they are fundamentally different caused by the parameter δ. To see it clearly, from (11) or (58) we have γ¯ − lnηiδ > lnηsμ . Note that γ¯ − lnηiδ with 0 < δ < 1 is bigger than γ¯ − lnηiδ with δ > 1, which illustrates that condition (58) provides more freedom with the implementation of the switching signal r (t).

3.2 Controller Design The issue of controller design for the system (1) is to be dealt with by feat of the results developed in Sect. 3.1. The asynchronous switching and impulsive controllers u 1 (t) and u 2 (t) are designed as u 1 (t) =K r (t) x(t), u 2 (tk− ) =K 2 x(tk− ).

(59)

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By substituting (59) into (1), we immediately have the following closed-loop system   dx(t) = − (Cr (t) − K r (t) )x + Dr (t) Fr (t) (x) dt + G r (t) (x, t) dω(t), t = tk , x(tk ) = (A + K 2 )x(tk− ), t = tk , x(t0 ) = x0 , r (t0 ) = r0 .

(60)

The controller gain matrices can be obtained by the following result. Theorem 4 Under Assumptions 1 and 2, if there exist positive definite matrices Pi , matrices Hi , H2 , real parameters μ ≥ 1, 0 < δ < 1, ci > 0, ρi j > 0 (i, j ∈ M) satisfying ⎡

−Pi (Ci − Hi ) − (Ci − Hi )Pi ⎣ −B i1 Λi + ci Q i + γ¯ Pi ∗

⎤ Pi Di + B¯ i2 Λi ⎦ −Λi

< 0,

(61)

Pi ≤ ci In ,

(62)

Pi ≤ μP j , i = j,   −δ Pi A T Pi + H2 < 0, ∗ −Pi ln μ ln δ γ¯ − − > 0, ηs ηi

(63) (64) (65)

where Λi , B i1 , B i2 take the same form defined in Theorem 1, then system (1) is ES-M when the switching signal r (t) has an average dwell time ηs and the impulsive sequence {tk }k∈N+ an inverse average impulsive interval ηi . Moreover, the controller gain matrices are K i = Hi , K 2 = Pi−1 H2T . Proof We can prove it by simply using Theorem 3 and hence omit the proof details.

4 Numerical Examples In this section, two simulation examples are provided to illustrate the effectiveness of our results. Example 1 Consider system (1) with the following parameters

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⎡

⎤ ⎡ ⎤ ⎡ ⎤ 0 −1 0 0.3 −1 0.8 0 0⎦ , D1 = ⎣ 0 −1 0.6⎦ , G 1 = ⎣−3 −0.5 0 ⎦ , 6 1.4 0 2 0.1 0.2 −0.1 ⎤ ⎡ ⎤ ⎡ ⎤ 00 0.75 −0.5 0 1 −0.7 0.1 3 0⎦ , D2 = ⎣ 1.1 −0.1 0⎦ , G 2 = ⎣0.8 1 −0.5⎦ , 03 1.4 0 2 0.1 0.2 −0.1 ⎡ ⎤ 100 A = 1.4 ∗ ⎣0 1 0⎦ , 001

6 C1 = ⎣0 0 ⎡ 3 C2 = ⎣0 0

0 6 0

and the neuron activation functions Fi j (x j ) are defined as Fi j (x j ) = f i j (|x j + 1| − |x j − 1|) where f i j ∈ R. By choosing f 11 = 1.1, f 12 = f 13 = 1.2, f 21 = f 22 = f 23 = 0.7, it is easy to verify the stability of the continuous subsystems. Moreover, from the definition of matrix A, we have the destabilizing impulses. In order to examine the stability of system (1) defined by the above parameters, we next turn our attention to Theorem 1. To obtain a set of feasible solutions satisfying conditions (7)–(10), let μ = 2, δ = 2, c1 = 1.3, c2 = 2, γ¯ = 1.4, and then we have the following results: ⎡

P1

P2

Λ1

Λ2

⎤ 1.2858 −0.0451 −0.0057 = ⎣−0.0451 1.1125 −0.0409⎦ , −0.0057 −0.0409 1.1053 ⎡ ⎤ 1.8788 0.0430 0.0894 = ⎣0.0430 1.5101 −0.3957⎦ , 0.0894 −0.3957 1.2994 ⎡ ⎤ 2.5694 0 0 0 ⎦, = ⎣ 0 5.4837 0 0 22.2668 ⎡ ⎤ 11.7520 0 0 10.1126 0 ⎦ , =⎣ 0 0 0 5.4046

which implies the feasibility of Theorem 1. To perform the simulation results, choose the initial conditions x(0) = [3; −3.8; −3], r (0) = 1, and ηs = 2, ηi = 1.3, which satisfy condition (11). The system states are plotted in Fig. 1, from which we have that the stability of system (1) can still be guaranteed if the destabilizing impulses do not happen frequently. The corresponding switching signal r (t) and impulsive instants are shown in Figs. 2 and 3, respectively.

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4 x (t) 1

x (t)

3

2

x (t) 3

System state

2 1 0 −1 −2 −3 −4

15

10

5

0

Time(s)

Fig. 1 Time variation of system states 3 Switching signal r(t) 2.5

2

1.5

1

0.5

0

0

5

10

Time(s)

Fig. 2 The evolutionary process of switching signal r (t)

15

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T. Jiao and J. H. Park 2 Impulsive time instants 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

10

5

0

15

Time(s)

Fig. 3 Times-series of impulsive instants

Example 2 We next consider system (1) defined by the following parameters ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 1.2 0 0 0.3 0.1 0 1 0 0 1 −0.1⎦ , C1 = ⎣ 0 2 0⎦ , D1 = ⎣ 0 0.4 0⎦ , G 1 = ⎣ 1 0 02 1 2 1 −0.7 −0.4 1 ⎡ ⎤ ⎡ ⎤ 2.3 0 0 0.3 0.1 0 C2 = ⎣ 0 2.3 0 ⎦ , D2 = ⎣ 0 0.4 0⎦ , 0 0 2.3 1 2 1 ⎡ ⎤ ⎡ ⎤ 1 0 0 100 G 2 = ⎣ 0.1 1 −0.1⎦ , A = 0.1 ∗ ⎣0 1 0⎦ , −0.7 −0.4 1 001 and the neuron activation functions Fi j (x j ) are the same with that defined Example 1. By choosing f 11 = f 12 = f 13 = 1.2, f 21 = f 22 = f 23 = 2, from the simulation results plotted in Figs. 4 and 5, we can see that the continuous subsystems are unstable. Note that the definition of matrix A implies the stabilizing effectiveness of impulses. Therefore, we next apply Theorem 2 to examine the stability of system (1) defined by the above parameters. Letting μ = 3, δ = 0.2, c1 = 0.3, c2 = 0.2, γ¯ = 0.4, we have the following results satisfying conditions (33)–(36),

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4 x1(t) x2(t)

3

x (t) 3

System state

2 1 0 −1 −2 −3 −4

0

50

100

150

Time(s)

Fig. 4 Time variation of the first subsystem states

⎡

P1

P2

Λ1

Λ2

⎤ 0.2631 0.0142 −0.0311 = ⎣ 0.0142 0.2595 0.0172 ⎦ , −0.0311 0.0172 0.2151 ⎡ ⎤ 0.1716 0.0060 −0.0122 = ⎣ 0.0060 0.1806 0.0092 ⎦ , −0.0122 0.0092 0.1531 ⎡ ⎤ 0.6676 0 0 = ⎣ 0 1.7864 0 ⎦ , 0 0 0.5230 ⎡ ⎤ 0.9148 0 0 1.2015 0⎦ , =⎣ 0 0 00.3511

which implies the ES of system (1). To perform the simulation results, choose ηs = 2, ηi = 0.3, and obviously condition (37) holds. Choosing the initial conditions x(0) = [3; −3.8; −3] and r (0) = 1, we plot the system states in Fig. 6, the corresponding switching signal r (t) in Fig. 7 and impulsive instants in Fig. 8, from which it is shown that the stability of system (1) with the unstable continuous subsystems can be achieved if the stabilizing impulses do happen frequently.

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3

x3(t)

System state

2 1 0 −1 −2 −3 −4

150

100

50

0

Time(s)

Fig. 5 Time variation of the second subsystem states 4 x (t) 1

x (t)

3

2

x3(t)

System state

2 1 0 −1 −2 −3 −4

0

0.5

1

Time(s)

Fig. 6 Times-trajectories of system states

1.5

2

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3 Switching signal r(t) 2.5

2

1.5

1

0.5

0

0

5

10

15

Time(s)

Fig. 7 The evolutionary process of switching signal r (t) 2 Impulsive time instants 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

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Time(s)

Fig. 8 Times-series of impulsive instants

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5 Conclusion In this chapter, the exponential stability and stabilization problem for a class of stochastic neural network systems has been addressed. Comparing with the previous results, the main distinction here is the asynchronization of switchings and impulses. By employing the concepts of dwell time and impulsive interval as well as multiple Lyapunov functions approach, sufficient conditions for mean square exponential stability and stabilization have been proposed in terms of LMIs. Acknowledgements This work of T. Jiao was supported by National Natural Science Foundation of China (61703249) and China Postdoctoral Science Foundation (2019M652351). Also, the work of J.H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. 2020R1A2B5B02002002).

References 1. Wang, J., Huang, T., Wu, H., Ren, S.: Analysis and Control of Coupled Neural Networks with Reaction-Diffusion Terms. Springer, Berlin (2018) 2. Li C, Feng G, Huang T (2008) On hybrid impulsive and switching neural networks. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 38(6), 1549–1560 3. Zhang, B., Xu, S., Zong, G., Zou, Y.: Delay-dependent exponential stability for uncertain stochastic Hopfield neural networks with time-varying delays. IEEE Trans. Circuits Syst. I: Regul. Pap. 56(6), 1241–1247 (2008) 4. Park, J.H., Shen, H., Chang, X.H., Lee, T.H.: Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals. Springer, Cham, Switzerland (2018). https://doi. org/10.1007/978-3-319-96202-3 5. Shen, H., Zhu, Y., Zhang, L., Park, J.H.: Extended dissipative state estimation for Markov jump neural networks with unreliable links. IEEE Trans. Neural Netw. Learn. Syst. 28(2), 346–358 (2017) 6. Park, J.H., Kwon, O.M., Lee, S.M.: LMI optimization approach on stability for delayed neural networks of neutral-type. Appl. Math. Comput. 196(1), 236–244 (2008) 7. Sun, H., Zong, G., Ahn, C.K.: Quantized decentralized adaptive neural network PI tracking control for uncertain interconnected nonlinear systems with dynamic uncertainties. IEEE Trans. Syst. Man. Cybern.: Syst. (2019). https://doi.org/10.1109/TSMC.2019.2918142 8. Niu, B., Zhao, X., Fan, X., Cheng, Y.: A new control method for state-constrained nonlinear switched systems with application to chemical process. Int. J. Control 88(9), 1693–1701 (2015) 9. Gordillo G.L.O. (2015) Observers design for uncertain descriptor systems: application to control and diagnosis. Ph.D. thesis 10. Fu, J., Ma, R., Chai, T., Hu, Z.: Dwell-time-based standard H∞ control of switched systems without requiring internal stability of subsystems. IEEE Trans. Autom. Control 64(7), 3019– 3025 (2019) 11. Zhu, B., Suo, M., Chen, L., Li, S.: Stability and L 1 -gain analysis for positive Takagi-Sugeno fuzzy systems with impulse. IEEE Trans. Fuzzy Syst. 26(6), 3893–3901 (2018) 12. Liu, C., Yang, Z., Sun, D., Liu, X., Liu, W.: Stability of switched neural networks with timevarying delays. Neural Comput. Appl. 30(7), 2229–2244 (2018) 13. Zhang, L., Zhu, Y., Zheng, W.X.: State estimation of discrete-time switched neural networks with multiple communication channels. IEEE Trans. Cybern. 47(4), 1028–1040 (2016) 14. Xia, J., Chen, G., Sun, W.: Extended dissipative analysis of generalized Markovian switching neural networks with two delay components. Neurocomputing 260, 275–283 (2017)

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15. Huang, T., Li, C., Duan, S., Starzyk, J.A.: Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans. Neural Netw. Learn. Syst. 23(6), 866–875 (2012) 16. Zhang, W., Tang, Y., Wu, X., Fang, J.: Synchronization of nonlinear dynamical networks with heterogeneous impulses. IEEE Trans. Circuits Syst. I: Regul. Pap. 61(4), 1220–1228 (2013) 17. Yang, X., Lu, J., Ho, D.W., Song, Q.: Synchronization of uncertain hybrid switching and impulsive complex networks. Appl. Math. Modell. 59, 379–392 (2018) 18. Li, C., Ma, F., Feng, G.: Hybrid impulsive and switching time-delay systems. IET Control Theory Appl. 3(11), 1487–1498 (2009) 19. Liu, J., Liu, X., Xie, W.C.: Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica 47(5), 899–908 (2011) 20. Zong, G., Yang, D.: H∞ synchronization of switched complex networks: a switching impulsive control method. Commun. Nonlinear Sci. Numer. Simul. 77, 338–348 (2019) 21. Jiao, T., Park, J.H., Zong, G., Zhao, Y., Du, Q.: On stability analysis of random impulsive and switching neural networks. Neurocomputing 350, 146–154 (2019) 22. Zhang, X., Li, C., Huang, T.: Hybrid impulsive and switching Hopfield neural networks with state-dependent impulses. Neural Netw. 93, 176–184 (2017) 23. Ren, W., Xiong, J.: Stability analysis of impulsive switched time-delay systems with statedependent impulses. IEEE Trans. Autom. Control 64(9), 3928–3935 (2019) 24. Zhang, W., Tang, Y., Miao, Q., Du, W.: Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. 24(8), 1316–1326 (2013) 25. Jiao, T., Park, J.H., Zong, G.: Stability criteria of stochastic nonlinear systems with asynchronous impulses and switchings. Nonlinear Dyn. 97(1), 135–149 (2019) 26. Mao X, Yuan C (2006) Stochastic Differential Equations with Markovian Switching. Imperial college press 27. Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2011) 28. Lu, J., Ho, D.W., Cao, J.: A unified synchronization criterion for impulsive dynamical networks. Automatica 46(7), 1215–1221 (2010)

Hybrid-Triggered Synchronization of Delayed Complex Dynamical Networks Subject to Stochastic Cyber-Attacks Xiaojian Yi and Yajuan Liu

Abstract This chapter is concerned with the synchronization problem for a class of delayed complex networks with random occurring cyber-attacks. First, a hybrid event-triggered communication scheme is proposed to reduce the transmission load of networks, which includes the event-triggered scheme and the time-triggered scheme in one framework. Then, a synchronization error dynamic system is obtained for the considered networks, and a new sufficient condition is derived such that the synchronization error system is asymptotically stable in the mean-square sense. Furthermore, it is also shown that the controller gain matrices can be determined by solving some matrix inequalities. Finally, two simulation examples are introduced to validate the main results. Keywords Synchronization · Complex dynamical networks · Hybrid-triggered scheme · Stochastic cyber-attacks

1 Introduction Many natural and man-made systems can be expressed as networks, where many nodes interact through the network link [1–4]. Complex dynamical networks have attracted increasing attention in various fields such as social networks, the World Wide Web, food webs, computer networks, and so on [5]. A complex network refers to a set of coupled interconnected nodes, where each node can be considered as a dynamical system. One of the important research topics of complex dynamical

X. Yi School of Mechanical Engineering, Beijing Institute of Technology, Beijing, People’s Republic of China e-mail: [email protected] Y. Liu (B) School of Control and Computer Engineering, North China Electric Power University, Beijing, People’s Republic of China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_19

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networks is synchronization. Synchronization is regarded as a kind of collective behavior which is exhibited in many practical systems, see, e.g., chemical reactions, biological systems, and secure communication [6–8]. Thus, the problem of synchronization of complex dynamical networks has been studied by many researchers and many important works have been reported in recent years [9–11]. In general, not all nodes in complex dynamical networks can synchronize up to each other because of the complexities of node dynamics and also the topological structure of the network. Therefore, the synchronization control problem of complex networks deserves more attention. In the last a few years, various control schemes have been successfully applied to achieve synchronization of complex dynamical networks, see, e.g., adaptive control [10], impulsive control [9] and non-fragile control [11]. However, it should be noted that all results mentioned above are only focused on continuous-time controllers, which may not be easily implemented in real-world sometimes [12]. In order to solve this situation, sampled-data [13, 14] and event-triggered control [15] schemes are proposed to reduce the transmission load of network more effectively. Furthermore, the hybrid event-trigger scheme was established for stabilization of networked control systems [16]. Recently, the synchronization for complex dynamical networks with sampled-data was investigated in [17, 18]. In addition, the event-triggered synchronization for complex dynamical networks has also been studied in [19, 20]. However, the hybrid-driven communication scheme for synchronization for the complex dynamical network has not been investigated yet, which is one aim of this study. On the other hand, nodes in the network usually communicate with each via communication networks. For example, the vehicle in transportation system communicates with other vehicles via wireless networks. However, the network not only brings convenience and but also introduces security threats [21–23]]. Recently, the security issue has attracted increasing attention. For example, in [24], the deception attacks were taken into account in dealing with the problem of consensus for multiagent systems. The consensus of heterogeneous multi-agent systems with DoS attach was addressed in [25]. The problem of H∞ control of networked cascade control systems with stochastic cyber attacks and hybrid-driven-based transmission scheme was investigated in [26]. The problem of quantized stabilization for Takagi–Sugeno fuzzy systems with stochastic cyber-attack were studied in [27]. However, for the synchronization of complex dynamical networks with cyber-attacks, few results have been published, especially when the hybrid-triggered scheme is used, which is another motivation of this study. Motivated by the above discussions, this paper is concerned with the study of hybrid-triggered synchronization for delayed complex dynamical networks stochastic cyber-attacks. A hybrid-triggered scheme can be represented by Bernoulli variable and a type of nonlinear function is used to describe the features of cyber-attacks. The main contributions of this chapter are listed: (1) The hybrid triggered scheme is first considered in the study of synchronization control of complex dynamical networks while taking the effect of stochastic cyber attacks into account;

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(2) A closed-loop synchronization error system model is first established by considering the limited network resources and stochastic cyber-attacks; (3) According to the constructed model, the synchronization criterion for the considered complex dynamical networks is obtained by using the Lyapunov stability theory. Moreover, the controller gain matrices are calculated by solving a set of liner matrix inequality (LMIs). Notations. Throughout this paper, I denotes the identity matrix with appropriate dimensions, T is the matrix transposition. Rn denotes the n-dimensional Euclidean space, and Rm×n is the set of all m × n real matrices, ∗ denotes the elements below the main diagonal of a symmetric block matrix. For A, A > 0 means that the matrix A is a real symmetric positive definite.

2 Problem Statement Suppose we have a network with N nodes, and the it is described by

x˙i (t) = φ(xi (t)) + c

N 

gi j Ax j (t − τ (t)) + u i (t),

(1)

j=1

where xi (t) denotes the state vector and u i (t) denotes the control input; φ : Rn → Rn represents a continuous vector-valued function; the positive scalar constant c is the coupling strength; τ (t) describes a time-varying delay, which satisfies 0 ≤ τ (t) ≤ τ M , τ˙ (t) ≤ μ, where τ M and μ are known real constants; A = (ai j )n×n is the innercoupling matrix; G = (gi j ) N ×N represents an outer-coupling configuration matrix where G i j is defined as: if there exists a connection between node i and j, then gi j > 0; gi j = 0, otherwise, the diagonal elements of matrix G are given as gii = −

N 

gi j (i = 1, 2, . . . , N ).

(2)

j=1, j=i

For constant matrices U1 and U2 with appropriate dimensions. The nonlinear function φ : Rn → Rn meet the following sector-bound condition [12]: [φ(x1 ) − φ(x2 ) − U1 (x1 − x2 )]T × [φ(x1 ) − φ(x2 ) − U2 (x1 − x2 )] ≤ 0, ∀ x1 , x2 ∈ Rn ,

(3)

We now define the synchronization error signal as ρe(t) = xi (t) − s(t), where s(t) ∈ Rn the state of the target node with s˙ (t) = φ(s(t)). Thus, the error dynamics can be described by as

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ρ˙i (t) = f (ρi (t)) + c

N 

gi j Aρ j (t − τ (t)) + u i (t),

(4)

j=1

where i = 1, 2, . . . , N and f (ρi (t)) = φ(xi (t)) − φ(s). In this chapter, the hybrid-driven-based scheme including event-triggered scheme and time-triggered scheme is established, and it is employed to reduce the communication load. In the following part, the two cases are discussed in detail.

2.1 Time-Triggered Scheme When the network resource is not limited, we choose to use the ‘time-triggered’ mechanism to transmit the data, then the control input is calculated based on the following data: ρi1 (t) = ρi (tk h), t ∈ [tk h + dtk , tk+1 h + dtk+1 )

(5)

where h denotes the sampling period, tk denotes some integers satisfying {t1 , t2 , . . .} ∈ {0, 1, 2, . . .}, dtk is the communication delay [28]. Similar to [18], Eq. (5) can be rewritten as ρi1 (t) = ρi (t − d(t))

(6)

where d(t) ∈ [0, d M ], d M is the upper bound of d(t).

2.2 Event-Triggered Scheme When the network resource is not limited, we choose to use the ‘event-triggered’ mechanism. The release time under the event-triggered scheme can be expressed as follows [15] tk+1 = tk h + in f j≥1 { j h|rikT (t)Ωrik (t) ≤ σρiT (tk h)Ωρi (tk h)},

(7)

where rik (t) = ρi (tk h) − ρi (tk h + j h) denotes the threshold error, Ω is positive, σ ∈ [0, 1), and j = 1, 2, . . . . In order to analyze more easily, similar to the method in [27], dividing the interval [tk h + dtk , tk+1 h + dtk+1 ) into several intervals, and assuming that there exists a con   stant N satisfying [tk h + dtk , tk+1 h + dtk+1 ) = Nj=1 j , where j = [tk h + j h + dtk + j , tk h + j h + h + dtk + j+1 ], j = {1, 2, . . . , N }, N = tk+1 − tk − 1, and define

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η(t) = t − tk h − j h, 0 ≤ dtk ≤ η(t) ≤ h + dtk + j+1 = η M , then the error data can be rewritten as ρi2 (t) = ρi (t − η(t)) + rik (t).

(8)

Remark 1 In the event-triggered scheme (7), the parameter σ is a positive scalar, which is related to the transmission frequency. Furthermore, when σ = 0, the eventtriggered scheme reduces to the time-triggered scheme. Inspired by the work [26], the time-triggered scheme combined with the eventtriggered scheme are adopted to reduce the communication burden, and a stochastic variable α(t) is introduced to trigger one of the above transmission scheme: ρ¯i (t) = α(t)ρi1 (t) + (1 − α(t))ρi2 (t) = α(t)ρi (t − d(t)) + (1 − α(t))(ρi (t − η(t)) + rik (t)),

(9)

where α(t) take value on [0, 1] with E{α(t)} = α. ¯ In practical systems, the transmitted date over communication networked is susceptible to be attacked, which may cause degradation of the system performance. As in [27], the cyber-attack is modeled as nonlinear function g(ρi (t)) and the corresponding time delay is supposed as θ(t) ∈ [0, θ M ]. Inspired by the work in [26], a Bernoulli random variable β(t), which is independent to α(t), is used to express the stochastic cyber attacks. Therefore, the signal transmitted to the controller, can be expressed as follows ρˆi (t) = β(t)g(ρi (t − θ(t))) + (1 − β(t)){α(t)ρi (t − d(t)) +(1 − α(t))[ρi (t − η(t)) + rik (t)]}.

(10)

where the cyber-attacks g(u) are supposed to satisfy ||g(x)||2 ≤ ||V x||2

(11)

with V is a constant matrix describing the upper bound of nonlinearity. Remark 2 Inspired by the work [27], a Bernoulli variable is used to represent the probability of stochastic cyber-attack. When β(t) = 1, which means that the cyberattacks are implemented successfully. Otherwise, when β(t) = 0, it means that the data is transmitted normally. Then, the controller can be written as u i (t) = β(t)K i g(ρi (t − θ(t))) + (1 − β(t))K i {α(t)ρi (t − d(t)) +(1 − α(t))[ρi (t − η(t)) + rik (t)]}.

(12)

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Subsisting (12) into (4), the closed-loop system (4) can be described as ρ˙i (t) = f (ρi (t)) + c

N 

gi j Aρ j (t − τ (t)) + β(t)K i g(ρi (t − θ(t)))

j=1

+(1 − β(t))K i {α(t)ρi (t − d(t)) +(1 − α(t))[ρi (t − η(t)) + rik (t)]}

(13)

where i = 1, 2, . . . , N . Based on Kronecker product, the system (14) can be rewritten as ρ(t) ˙ = f¯(ρ(t)) + c(G ⊗ A)ρ(t − τ (t)) + β(t)K g(ρ(t − θ(t))) +(1 − β(t))K {α(t)ρ(t − d(t)) +(1 − α(t))[ρ(t − η(t)) + rk (t)]}

(14)

where ρ(t) = [ρ1T (t), ρ2T (t), . . . , ρTN (t)]T , f¯(ρ(t)) = [ f T (ρ1 (t)), f T (ρ2 (t)), . . . , f T (ρ N (t))]T , g(ρ(t)) ¯ = [g T (ρ1 (t)), g T (ρ2 (t)), . . . , g T (ρ N (t))]T , K = diag[K 1 , K 2 , . . . , K N ], T T rk (t) = [r1k (t), r2k (t), . . . , r NT k (t)]T .

(15)

The following lemmas will be given to obtain the main results of this work. Lemma 1 ([29]) For M > 0 ∈ Rn×n and a ≤ s ≤ b, the following inequality holds: − (b − a)

b a

˙ ≤ − [δ (b) − δ (a)]T M [δ (b) − δ (a)] . δ˙ T (s)M δ(s)ds

Lemma 2 ([30]) Let D ∈ Rm is an open subset, and f1 , f2 , · · ·, f N : Rm → R have positive values D. Then, one has min 

{αi |αi >0,

i

 1   fi (t) = fi (t) + max mi j (t) mi j (t) αi =1} αi i i i= j

subject to 

 fi (t) mi, j (t) m mi j : R → R, m j,i (t)  mi, j (t), ≥0 . mi, j (t) f j (t)

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Lemma 3 ([31]) For R > 0, there exists a differentiable function {y(v)|v ∈ [a, b]},  −(b − a)

b a

y˙ T (s)R y˙ (s)ds ≤ −Ω1T RΩ1 − 3Ω2T RΩ2 ,

where Ω1 = y(b) − y(a), Ω2 = y(b) + y(a) −

2 b−a



b

y(s)ds. a

3 Main Results In this section, a synchronization criterion for complex dynamical networks with hybrid event-triggered scheme and stochastic cyber-attack will be discussed in Theorem 1. For simplify, eˆi ∈ R(19n)×n denotes block entry matrix. For example, e2 = [0 I 0 .  . . 0, 0]T . And define some notations as follows: 16

 ¯ αH ¯ Ψ = 0 cS(G ⊗ A) 0 − S (1 − β) ¯ 0 0 0 (1 − β)(1 − α)H ¯  ¯ 0 .  . . 0 S β¯ H (1 − β)(1 − α)H ¯ , 7

(I ⊗ U1 )T (I ⊗ U2 ) (I ⊗ U2 )T (I ⊗ U1 ) + , U¯ 1 = 2 2 (I ⊗ U1 )T + (I ⊗ U2 )T , U¯ 2 = − 2  ξ T (t) = ρT (t) ρT (t − τ (t)) ρT (t − τ M ) ρ(t) ˙ ρT (t − d(t))  t−d(t) 1 ρ (s)ds ρT (s)ds ρ (t − d M ) d M − d(t) t−d M t−d(t)  t 1 ρT (t − η(t)) ρT (t − η M ) ρT (s)ds η(t) t−η(t)  t−η(t) 1 ρT (s)ds ρT (t − θ(t)) ρT (t − θ M ) η M − η(t) t−η M  t  t−θ(t) 1 1 ρT (s)ds ρT (s)ds θ(t) t−θ(t) θ M − θ(t) t−θ M T

1 d(t)



t

T

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 ¯ f (ρ(t)) g(ρ(t ¯ − θ(t))) rk (t) , Π1 = [e1 − e5 e1 + e5 − 2e7 ], Π2 = [e5 − e6 e5 + e6 − 2e8 ], Π3 = [e1 − e9 e1 + e9 − 2e11 ], Π4 = [e9 − e10 e9 + e10 − 2e12 ], Π5 = [e1 − e13 e1 + e13 − 2e15 ], Π6 = [e13 − e14 e13 + e14 − 2e16 ], ⎡ ⎤ −R1 R1 − T1 T1 R¯ 1 = ⎣ ∗ −2R1 + T1 + T1T R1 − T1 ⎦ , ∗ ∗ −R1

Σ = e1 Pe4T + e4 Pe1T + e1 Q 1 e1T − (1 − μ)e2 Q 1 e2T + e1 (Q 2 + Q 3 )e1T T −e3 Q 2 e3T − e6 Q 3 e6T + e1 (Q 4 + Q 5 )T − e10 Q 4 e10 T 2 T −e14 Q 5 e14 + τ e4 R1 e4 − [e1 e2 e3 ] R¯ 1 [e1 e2 e3 ]T   R¯2 T2 2 T [Π1 Π2 ]T +d M e4 R2 e4 − [Π1 Π2 ] ∗ R¯ 2   R¯3 T3 [Π3 Π4 ]T +η 2M e4 R3 e4T − [Π3 Π4 ] ∗ R¯ 3   R¯4 T4 [Π5 Π6 ]T +θ2M e4 R4 e4T − [Π5 Π6 ] ∗ R¯ 4   U¯1 U¯ 2 [e1 e17 ]T +[e1 + e4 ]Ψ + Ψ T [e1 + e4 ]T − 1 [e1 e17 ] ∗ I T T + 2 e13 (I ⊗ V )T (I ⊗ V )e13 − 2 e18 e18 T +σe9 (I ⊗ Ω)e9T − e19 (I ⊗ Ω)e19 .

¯ > 0, 1 > Theorem 1 For given scalars τ > 0, μ, d M > 0, η M > 0, θ M > 0, α, ¯ β, 0, 2 > 0, σ > 0, the closed loop system (14) is asymptotically stable, if there exist symmetric positive definite matrices P > 0,Q 1 , Q 2 ,Q 3 , Q 4 , Q 5 ,, R1 , R2 ,R3 ,R4 , any matrices T1 , T2 , T3 , T4 , S = diag{S1 , S2 , . . . , S N }, H = diag{H1 , H2 , . . . , HN }, with appropriate dimensions such that the following LMIs hold Σ < 0,   R¯ i Ti ≥ 0, i = 1, 2, 3, 4. ∗ R¯ i

(16) (17)

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Moreover, the designed controller gain could be calculated as K j = S −1 j H j , j = 1, 2, . . . , N .

(18)

Proof Construct the following Lyapunov function candidate V (t) =

8 

Vi (t),

(19)

i=1

where V1 (t) = ρT (t)Pρ(t),  t ρT (s)Q 1 ρ(s)ds, V2 (t) =  V3 (t) = V4 (t) =

t−τ (t) t



ρT (s)Q 2 ρ(s)ds +

t−τ M  t

ρT (s)Q 4 ρ(s)ds +

t−η M



t

V5 (t) = τ M

t−τ M t

 V6 (t) = d M V7 (t) = η M V8 (t) = θ M

t−d M  t t−η M  t t−θ M

t

ρT (s)Q 3 ρ(s)ds,

t−d M  t

ρT (s)Q 5 ρ(s)ds,

t−θ M



t

ρ˙T (u)R1 ρ(u)duds, ˙

s

  

t

ρ˙T (u)R2 ρ(u)duds, ˙

s t

ρ˙T (u)R3 ρ(u)duds, ˙

s t

ρ˙T (u)R4 ρ(u)duds. ˙

s

By calculating the time derivative V (t) along the trajectory of (14), one has ˙ E{V1 (t)} = 2ρT (t)P ρ(t),

(20)

E{V2 (t)} ≤ ρT (t − τ (t))Q 1 ρ(t − τ (t)) − (1 − μ)ρT (t)Q 1 ρ(t), E{V3 (t)} = ρT (t)(Q 2 + Q 3 )ρ(t)

(21)

−ρT (t − τ M )Q 2 ρ(t − τ M ) − ρT (t − d M )Q 3 ρ(t − d M ), E{V4 (t)} = ρT (t)(Q 4 + Q 5 )ρ(t) −ρT (t − η M )Q 4 ρ(t − η M ) − ρT (t − θ M )Q 5 ρ(t − θ M ),  t 2 T ˙ − τM ρ˙T (s)R1 ρ(s)ds, ˙ E{V5 (t)} = τ M ρ˙ (t)R1 ρ(t)

(22) (23) (24)

t−τ M t

(25)

 2 T ρ˙ (t)R2 ρ(t) ˙ − dM E{V6 (t)} = d M

t−d M

ρ˙T (s)R2 ρ(s)ds, ˙

(26)

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 E{V7 (t)} = η 2M ρ˙T (t)R3 ρ(t) ˙ − ηM E{V8 (t)} = θ2M ρ˙T (t)R4 ρ(t) ˙ − θM

t

t−η M  t

ρ˙T (s)R3 ρ(s)ds, ˙

(27)

ρ˙T (s)R4 ρ(s)ds. ˙

(28)

t−θ M

In view of Lemmas 1 and 2, for any matrix T1 , one has  −τ M

t

ρ˙T (s)R1 ρ(s)ds ˙

t−τ M



= −τ M

t

t−τ (t)



t−τ (t)

ρ˙T (s)R1 ρ(s)ds ˙ − τM

ρ˙T (s)R1 ρ(s)ds ˙

t−τ M

≤ X T (t) R¯ 1 X (t)

(29)

where X (t) = [ρT (t) ρT (t − τ (t)) ρT (t − τ M )]T and ⎤ −R1 R1 − T1 T1 R¯ 1 = ⎣ ∗ −2R1 + T1 + T1T R1 − T1 ⎦ . ∗ ∗ −R1 ⎡

Applying Lemmas 1 and 2 gives  −d M

t

ρ˙T (s)R2 ρ(s)ds ˙

t−d M



= −d M

t

t−d(t)

 ρ˙T (s)R2 ρ(s)ds ˙ − dM

t−d(t)

ρ˙T (s)R2 ρ(s)ds ˙

t−d M

dM T ˆ dM γ¯ 1 R2 γ¯ 1 − γ¯ T R¯ 2 γ¯ 2 d(t) d M − d(t) 2    T  R¯ 2 T2 γ¯ 1 γ¯ , ≤− 1 γ¯ 2 γ¯ 2 ∗ R¯ 2 ≤−

where γ¯1 = [γ1T γ1T ]T , γ¯2 = [γ3T γ4T ]T , γ1 = ρ(t) − ρ(t − d(t)),  t 2 ρ(s)ds, γ2 = ρ(t) + ρ(t − d(t)) − d(t) t−d(t) γ3 = ρ(t − d(t)) − ρ(t − d M ),  t−d(t) 2 ρ(s)ds. γ4 = ρ(t − d(t)) + ρ(t − d M ) − d M − d(t) t−d M

(30)

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Similar to (30), we can obtain  − ηM

t



γ¯ e˙ (s)R3 e(s)ds ˙ ≤− 3 γ¯ 4

T 

T

t−η M

R¯ 3 T3 ∗ R¯ 3



 γ¯ 3 , γ¯ 4

(31)

where γ¯3 = [γ5T γ6T ]T , γ¯4 = [γ7T γ8T ]T , γ5 = ρ(t) − ρ(t − η(t)), 2 η(t) γ7 = ρ(t − η(t)) − ρ(t − η M ),



t

γ6 = ρ(t) + ρ(t − η(t)) −

ρ(s)ds,

t−η(t)

γ8 = ρ(t − η(t)) + ρ(t − η M ) −

2 η M − η(t)



t−η(t)

ρ(s)ds.

t−η M

And,  − θM

t

 e˙ T (s)R4 e(s)ds ˙ ≤−

t−θ M

γ¯ 5 γ¯ 6

T 

R¯ 4 T4 ∗ R¯ 4



 γ¯ 5 , γ¯ 6

(32)

where T T T T T γ¯5 = [γ9T γ10 ] , γ¯6 = [γ11 γ12 ] ,

γ9 = ρ(t) − ρ(t − θ(t)), 2 θ(t) = ρ(t − θ(t)) − ρ(t − θ M ),



γ10 = ρ(t) + ρ(t − θ(t)) − γ11

γ12 = ρ(t − θ(t)) + ρ(t − θ M ) −

t

ρ(s)ds,

t−θ(t)

2 θ M − θ(t)



t−η(t)

ρ(s)ds.

t−θ M

Furthermore, based on system (14), for a matrix S with any appropriate dimension and a scalar , it can be concluded that:  ˙ + f¯(ρ(t)) + c(G ⊗ A)ρ(t − τ (t)) 2[ρT (t)S + ρ˙T (t)S] −ρ(t) +β(t)K g(ρ(t − θ(t))) + (1 − β(t))K {α(t)ρ(t − d(t))  +(1 − α(t))[ρ(t − η(t)) + rk (t)]} = 0.

(33)

On the other hand, one can obtain ε1 [ f (ρi (t)) − U1 ρi (t)]T [ f (ρi (t)) − U2 ρi (t)] ≤ 0,

(34)

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which is equivalent to  − ε1

ρ(t) f¯(ρ(t))

T 

U¯1 U¯2 ∗ I



 ρ(t) ≥ 0. f¯(ρ(t))

(35)

From (11), one has ε2 ρi (t − θ(t))ViT Vi ρi (t − θ(t)) −ε2 g(ρi (t − θ(t)))g(ρi (t − θ(t))) ≥ 0.

(36)

Combining (20) to (36) and taking the event-triggered condition (7) into account yield E{V˙ (t)} ≤ ξ T (t)Σξ(t).

(37)

If LMIs (16)–(17) holds. the closed loop system (14) is asymptotically stable.

4 Numerical Example In this section, two numerical examples are provided to illustrate the effectiveness and feasibility of the designed approach. Example 1 A CDN with three nodes (1) is considered in this example. The outercoupling matrix is assumed to be G = (G i j ) N ×N with ⎡

⎤ −1 0 1 G = ⎣ 0 −1 1 ⎦ . 1 1 −2 The inner-coupling matrix A is given as 

 10 A= . 01 The nonlinear function f (·) is taken as  f (xi (t)) =

 −0.5xi1 + tanh(0.2xi1 ) + 0.2xi2 , 0.95xi2 − tanh(0.75xi2 )

which implies that f (·) satisfies (3) with  U1 =

   −0.5 0.2 −0.3 0.2 , U2 = . 0 0.95 0 0.2

Hybrid-Triggered Synchronization of Delayed Complex Dynamical …

469

Letting the nonlinear signals of cyber attacks is g(xi (t)) = [− tanh T (xi1 (t)) − tanh T (xi2 (t))]T , the parameter of event-triggered scheme is σ = 0.02, the parameters of delay is given as τ = 0.25, μ = 0.5, d M = 0.1, η M = 0.2, θ M = 0.1. Our aim is to demonstrate the effectiveness of the derived controller design approach. Let α¯ = 0.2, β¯ = 0.4, which means that the control system is subject to hybrid-triggered scheme with probability of 20% and the switch probability of cyber-attack is 40%. By solving the LMIs (16) and (17), the controllers gains and the event-triggered gain matrices are obtained as 

   −0.3049 −0.4032 −0.3049 −0.4032 , K2 = , −0.3799 −2.5177 −0.3799 −2.5177     −0.2912 −0.3574 0.1713 −0.0110 , Ω1 = , K3 = −0.3347 −2.2603 −0.0110 0.0681     0.1718 −0.0115 0.1789 −0.0118 , Ω3 = . Ω2 = −0.0115 0.0685 −0.0118 0.0691 K1 =

For the initial condition x1 (0) = [8 − 3]T , x2 (0) = [4 − 8]T , x3 (0) = [−5 6]T and s(0) = [1 − 0]T , the state response of error systems with event-triggered and cyber-attacks is given in Fig. 1. Figure 2 presents the switching probability between the time-triggered and event-triggered scheme under α¯ = 0.2. Figure 3 shows the switching probabilities between the cyber-attacks under β¯ = 0.4. Figure 4 describes the graph of event-triggered instants and released intervals.

8 6 4

e(t)

2 0 −2 −4 −6 −8

0

1

2

3

4

5

t

Fig. 1 State response of e(t)

6

7

8

9

10

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Fig. 2 The signal α(t) in Example 1

Fig. 3 The signal β(t) in Example 1

Event−triggered instants and released intervals

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 4 Event-triggered instants and released intervals

Obviously, the above figures effectively demonstrate the usefulness of the designed synchronization method for complex dynamical networks with stochastic cyberattacks and hybrid-triggered scheme. Example 2 Consider the Chua’s circuit as the unforced isolated node of CDN (1), which is described by the following model

Hybrid-Triggered Synchronization of Delayed Complex Dynamical …

471

s˙1 = σ1 (−s1 + s2 − v(s1 )) s˙2 = s1 − s2 + s3 s˙3 = −σ2 s2 where σ1 = 10, σ2 = 14.87, and v(s1 ) = bs1 + 0.5(a − b)r (s1 ), in which a = −1.27, b = −0.68, and r (s1 ) = (|s1 + 1| − |s1 − 1|). Denote s = [s1 s2 s3 ]T and ⎡

⎤ ⎡ ⎤ −σ1 − σ1 b σ1 0 −0.5σ1 (a − b)r (s1 ) 0 0 1 −1 1 ⎦ + ⎣ 0 0 0⎦. f (s) = ⎣ 0 −σ2 0 0 00 It is easy to know that f (s) satisfies with ⎡

⎤ ⎡ 2.7 10 0 −3.2 10 −1 1 ⎦ , V = ⎣ 1 −1 U =⎣ 1 0 −14.87 0 0 −14.87

⎤ 0 1⎦. 0

And, the inner-coupling matrix A and out-coupling matrix G are given as ⎡

⎤ 100 A = ⎣0 1 0⎦ 001 and ⎡

−3 ⎢ 1 G=⎢ ⎣ 1 1

1 −2 1 0

1 1 −2 0

⎤ 1 0 ⎥ ⎥. 0 ⎦ −1

Let the nonlinear signals of cyber attacks be g(xi (t)) = [− tanh T (xi1 (t)) − tanh T (xi2 (t)) − tanh T (xi3 (t))]T . Also, the parameter of event-triggered scheme is σ = 0.03. The parameters of delay are given as τ = 0.1, μ = 0.5, d M = 0.1, η M = 0.3, and θ M = 0.2. Here, the objective of this example is to demonstrate the effectiveness of the derived controller design approach. Let α¯ = 0.3, β¯ = 0.6, which means that the control system is subject to hybrid-triggered scheme with probability of 30% and the switch probability of cyber-attack is 60%. By solving the LMIs (16) and (17), the controllers gains and the event-triggered gain matrices are obtained as

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⎡

⎤ −8.2740 −8.3071 0.9043 K 1 = ⎣ −1.3559 −6.8659 −0.5217 ⎦ , 1.0260 12.3906 −8.5663 ⎡ ⎤ −8.1898 −8.4044 0.9417 K 2 = ⎣ −1.4055 −6.7465 −0.4822 ⎦ , 1.0474 12.5701 −8.5101 ⎡ ⎤ −8.1919 −8.4040 0.9413 K 3 = ⎣ −1.4059 −6.7445 −0.4812 ⎦ , 1.0472 12.5700 −8.5112 ⎡ ⎤ −8.2183 −8.4863 0.9651 K 4 = ⎣ −1.4434 −6.7287 −0.4545 ⎦ , 1.0557 12.7148 −8.5572 ⎡ ⎤ 1.5623 0.1428 0.5632 Ω1 = ⎣ 0.1507 2.3389 −0.1202 ⎦ , 0.5652 −0.1202 2.2105 ⎡ ⎤ 1.1679 0.1604 0.5728 Ω2 = ⎣ 0.1608 2.3427 −0.1208 ⎦ , 0.5629 −0.1208 2.2402 ⎡ ⎤ 1.1625 0.1543 0.5728 Ω3 = ⎣ 0.1617 2.3478 −0.1304 ⎦ . 0.5783 −0.1304 2.3482 ⎡ ⎤ 1.1723 0.1658 0.5826 Ω4 = ⎣ 0.1658 2.3562 −0.1317 ⎦ . 0.5713 −0.1317 2.3614 For the initial condition x1 (0) = [0 − 3 0]T , x2 (0) = [5 − 6 1]T , x3 (0) = [1 − 2 − 4]T , x4 (0) = [−1 − 1 6]T and s(0) = [1 − 1 0]T the state response of error systems with event-triggered and cyber-attacks is shown in Fig. 5. Figure 6 describes the switching probability between the time-triggered and the event-triggered scheme under α¯ = 0.3. Figure 7 represents the switching probabilities between the cyber-attacks under β¯ = 0.6. Figure 8 gives the graph of eventtriggered instants and released intervals. It can be seen that the above figures effectively demonstrate the usefulness of the designed synchronization method for complex dynamical networks with stochastic cyber-attacks and hybrid-triggered scheme.

Hybrid-Triggered Synchronization of Delayed Complex Dynamical …

Fig. 5 State response of e(t)

Fig. 6 The signal α(t) in Example 2

Fig. 7 The signal β(t) in Example 2

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event−triggered instants and released instants

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

2

4

6

8

10

12

14

16

18

20

t

Fig. 8 Event-triggered instants and released intervals

5 Conclusions We have studied the synchronization control of delayed complex networks with stochastic cyber-attacks. In order to alleviate the communication burden, we have adopted the hybrid-triggered transmission scheme. Based on the Lyapunov stability theory and matrix inequality method, sufficient conditions were obtained such that the synchronization error system is asymptotically stable in the mean-square sense and the controller gain matrices were calculated by solving some matrix inequalities. Finally, two simulation studies have been presented to validate the proposed method. Acknowledgements This work was supported by the National Natural Science Foundation of China under the grant (71801196, 61803153).

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Cluster Synchronization on Derivative Coupled Lur’e Networks: Impulsive Pinning Strategy Ze Tang, Dong Ding, and Ju H. Park

Abstract This chapter is concerned with the global and exponential synchronization issue for a class of nonidentically delay coupled Lur’e networks with stochastic disturbance and derivative couplings. Considering that the topological structure of the coupled Lur’e networks consists of several groups, cluster synchronization instead of complete synchronization is discussed. In order to present the synchronization criteria, a novel controller combining with the impulsive control and pinning control strategies is designed, which will be imposed on those Lur’e systems in the cluster but have directed connections with the Lur’e systems in the other clusters. By applying the extended comparison principle of the impulsive differential equations, the definition of the average impulsive interval, the mathematical taxonomy on impulsive parameters simultaneously, the achievement of the cluster synchronization on nonidentically derivative coupled Lur’e networks are guaranteed. Furthermore, the exponential convergence velocity of the derivative coupled Lur’e networks with stochastic disturbance is precisely estimated. Eventually, several numerical simulations demonstrate the effectiveness and applicability of the established synchronization technique. Keywords Cluster synchronization · Nonidentical Lur’e systems · Stochastic disturbance · Derivative couplings · Impulsive control · Complex networks.

Z. Tang (B) · D. Ding Engineering Research Center of Internet of Things Technology Applications (Ministry of Education), Jiangnan University, Wuxi 214122, People’s Republic of China e-mail: [email protected] J. H. Park Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Gyeongsan 38541, Republic of Korea e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_20

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1 Introduction Since the last two decades of the last century, with the quick development of information technology, the Internet and human society have stride forward into the era of networks. In daily lives, typical examples of complex networks could be found in many situations, such as the Internet, WWW, giant electrical power grids, global transportation network, brains in the living body, various economic and political social networks [1, 2]. As we all know that human beings and the environment they living are closely connected not only in natural science but also in some artificial societies. Dynamical systems are always used to describe this kind of connections and evolutions. For discussing a system, it should either explore the evolution rules of the individuals, or the influences affected by the other individuals. To smoothly investigate this kind of relationships, algebraic graph theory has been presented in [3], and therefore, the relationships among different individuals could be described as a complex dynamical network. In the graph, every node stands for an individual, and each path denotes the connections between different nodes. From the viewpoint of mathematics, complex networks can be written as an array of coupled ordinary differential equations. During the past three decades, the study on the complex networks has attracted lots of scientists’ attention from different research fields. In fact, synchronization [3–6] could be understood as the regulation on an array of coupled dynamical systems in order to force them to behave uniformly by regulating the coupling weights, coupling strengths or other system parameters of the complex networks. The phenomena of synchronization are common in daily life. Until now, different synchronization patterns of complex networks, such as (complete) synchronization, lag synchronization, phase synchronization, anti-synchronization, cluster synchronization have been deeply and widely discussed by the scholars from all over world, and as a result, many outstanding achievements have been presented [7, 8]. For different purposes in practical application, different synchronization patterns should be flexibly introduced. For instance, in some special situations, sometimes, the systems in the complex dynamical networks are required to be separated into some subgroups, namely, clusters, according to the topological structures of the whole complex networks. In this situation, only those systems in the same clusters are requested to behave uniformly while it doesn’t matter for the systems in different clusters [9–13]. To study this issue, the cluster synchronization was discussed in [9] for a kind of complex networks with time-varying delayed couplings and stochastic disturbance by designing the randomly occurring controllers. It should be noted that the networked dynamic systems naturally have time delays through various channels due to several reasons in hardware structure and signal processing [14]. Moreover, Liu and Chen discussed the finite-time and fixed-time problems for cluster synchronization in [10] for the complex networks by introducing the distributed control protocol in order to guarantee the final synchronization. As it is known to us that the impulse signal input to dynamical systems represent a kind of outer discontinuous forces at the impulsive instants, which makes the system to be perturbed or promoted suddenly and discontinuously. From the working

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mechanism of the impulses, it could be found that impulsive control is a kind of representative discrete-time control schemes, which provides the instantaneous motion only at the impulsive instants. From the viewpoint of control time, comparing with general continuous control methods, impulsive control could save the control costs [16, 22]. However, when an impulse signal is introduced to a dynamical system, it could be either the control motion or some kinds of stochastic disturbances [15], [17–20]. To study the impulsive control problems, the key factor is the impulsive effect, which determines the strength of the generated impulse signal. In this regard, to utilize the impulsive control method, not only the advantageous impulse but also the disadvantageous impulse should be considered [28]. Although authors in [21] extended our previous work [15] from complete synchronization to projective synchronization, the impulsive effects did not well discussed, and the results were not applicable to general coupled neural networks. However, to the best of our knowledge, the cluster synchronization problem of nonidentically coupled Lur’e networks with derivative coupling and multiple timevarying delays, until now, has not drawn much attention. Both the significance in theoretical research and the importance in practical applications stimulate us to proceed with this problem in this chapter. This chapter is mainly concerned with the global and exponential synchronization on a kind of derivative coupled stochastic complex networks consisting of nonidentical Lur’e systems by jointly inducting the impulsive control and pinning feedback control strategies. The main findings of this chapter can be outlined as follows: (1) In order to describe more general complex network models, this chapter considers a nonidentically coupled Lur’e networks with three different kinds of coupling methods, namely, general state coupling, delay state coupling, and derivative state coupling; (2) Considering the cluster-tree topological structure of the coupled Lur’e networks, a novel pinning feedback controller is designed, and it will be imposed only on these Lur’e systems in current cluster which have direct connections with the Lur’e systems in the other clusters; (3) By considering the derivative coupling in networks, a novel Lyapunov functional is constructed, which is closely related to the derivative coupling strength and derivative coupling matrix; (4) Based on the general comparison principle and parameter variation formulas for impulsive differential equations, sufficient criteria for global and exponential cluster synchronization are derived by discussing different functions of the impulsive effects. The rest parts of this chapter are arranged as follows. In the second section, the nonidentically coupled Lur’e networks model with derivative coupling and stochastic disturbances will be introduced and meanwhile, some definitions, lemmas, and assumptions will be presented as well. Then, the global exponential synchronization of the derivative coupled Lur’e networks will be studied in the third section. In Sect. 4, we give some numerical simulations to verify the and theoretical analysis and the control strategy proposed in this chapter. We will draw the conclusion of this chapter in the final section.

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Notations. H T denotes the transpose of the matrix H . Rn denotes the n-dimensional Euclidean space. Rn×n is the set of n × n real matrices. diag{· · · } stands for a diagonal matrix. The symbol  ·  stands for the Euclid norm of the matrix or the vector. λmax (H ) stands for the largest eigenvalue of matrix H , and max1≤i≤N {·} denotes to take the maximum value. A = [ai j ] ∈ Rn×n denotes the matrix with elements ai j . If a matrix H is positive definite (semi-definite), then we denote it as H > 0(H ≥ 0). In stands for the identity matrix with n dimension. H = (h i j ) denotes a R N ×N matrix H with elements h i j for i, j = 1, 2, . . . , N . T race(H ) denotes the trace of matrix ¯ t−>0+ u(t+h)−u(t) . N + = H . The set N + denotes the positive integers. D + u(t) = lim h n {1, 2, . . .} Let C([−h, 0], R ) be the continuous functions class, where the function maps from the interval [−h, 0] to the filed Rn . The dimension of these vectors and matrices will be cleared in the context.

2 Model Description and Preliminaries 2.1 Network Structure Statement Firstly, to proceed with this section, some statements on the nonidentically derivative coupled Lur’e networks with stochastic disturbance and time-varying delay are presented in this subsection. Consider that the nonidentically derivative coupled Lur’e networks consist of N nonidentical Lur’e systems, and meanwhile, it could be split into l clusters, which satisfies 2 ≤ l < N . If the Lur’e system i included int the cluster k, then define a function ϕi = k. Define the set that all Lur’e systems in cluster k is Wk and the set that all Lur’e systems in cluster j which have directed connections with the Lur’e systems  in the ¯ j . In summary, there satisfies the conclusion that (a) Wk Wq = other clusters is W  ∅, for k = q and k, q = 1, 2, . . . , l; (b) lk=1 Wk = {1, 2, . . . , N } based on the discussions above.

2.2 Problem Formulation Let discuss the following nonidentically coupled Lur’e networks with derivative couplings and stochastic disturbance dz i (t) = [ϕAi z i (t) + ϕBi f ϕi (C z i (t)) + 1

N  j=1

+ 3

N  j=1

gi j Γ z j (t) + 2

N  υi j Γ z j (t − h(t)) j=1

ρi j Γ z˙ j (t) + u i (t)]dt + p(t, ˜ z i (t), z i (t − h(t)))dω(t),

(1)

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where the state vector of the i-th Lur’e system is z i (t) = [z i1 (t), z i2 (t), . . . , z in (t)]T ∈ Rn for i = 1, 2, . . . , N , which is an n dimension vector; Three matrices ϕAi ∈ Rn×n , ϕBi ∈ Rn×m and C ∈ Rm×n are constant parameter matrices; The nonlinear Lur’e function f ϕi : Rm → Rm is a memoryless vector-valued function, which is a continuous differentiable function on all real number fields, and it satisfies f ϕi (0) = 0; The three positive constants 1 , 2 , 3 are the coupling strengths for general coupling, time-varying delay coupling and derivative coupling terms, respectively; Positive definite diagonal matrix Γ = diag{r1 , . . . , rn } ∈ Rn×n denotes the inner linking connections within the Lur’e systems; In this chapter, without loss of generality, simply let Γ = In ; Constant function h(t) is the communication transmission timevarying delay satisfying 0 ≤ h(t) ≤ h; Two square matrices G = [gi j ] ∈ R N ×N and Υ = [υi j ] ∈ R N ×N stand for two different outer communication methods among different Lur’e systems, and they are called outer-coupling matrices, which are totally determined by the topological structures of the coupled Lur’e networks. Furthermore, two outer-connectedcoupling matrices are assumed to satisfy with the fol lowing condition gii = − Nj=i j=1 gi j and υii = − Nj=i j=1 υi j , where gi j > 0 and υi j > 0 mean that if there is a directed connection from the Lur’e system j to the Lur’e system i (i = j), otherwise for gi j = 0 and υi j = 0; In addition, the matrix  = [ρi j ] ∈ R N ×N describes the derivative coupling method among different Lur’e systems, and for simplicity in this chapter, we consider that  is a symmetrical matrix satisfying ρi j = ρ ji > 0 if there a directed connection between the Lur’e system j and the Lur’e system i (i = j), otherwise for ρi j = ρ ji = 0; The noise intensity function p˜ : R+ × Rn × Rn → Rn×m is a matrix-valued function satisfying the initial values p(t, ˜ 0, 0) = 0n×m ; Constant m-dimensional vector-valued function ω(t) ∈ Rm describes the Brownian motion; And u i (t) means a kind of control inputs, it will be discussed in detail later. For the nonidentically derivative coupled Lur’e networks (1), the initial conditions could be given as follows z i (t) = ζi (t) ∈ C([−h, 0], Rn ). Remark 1 In view of the concept on the derivative outer-coupling matrix , the maximum eigenvalue of matrix  is zero, while other eigenvalues of matrix are less than zero. Therefore, all of the eigenvalues of matrix  can be arrayed in the following sequence 0 = λ1 ≥ λ2 ≥ · · · ≥ λ N . Then, the minimum eigenvalues of the positive definite matrix I N − 3  is λmin (I N − 3 ) = 1, while the maximum eigenvalue λmax (I N − 3 ) ≥ 1. The feature of this structured matrix I N − 3  will play an essential role in construction of the Lyapunov functional. Consider the isolated Lur’e systems with time-varying delay and stochastic disturbance in different clusters ˜ sϕi (t), sϕi (t − h(t)))dω(t), dsϕi (t) = [ϕAi sϕi (t) + ϕBi f ϕi (Csϕi (t))]dt + p(t, (2) where sϕi (t) = [sϕ1 i (t), sϕ2 i (t), . . . , sϕn i (t)]T ∈ Rn is the state vector of the target Lur’e system in the ϕi -th cluster for i = 1, 2, . . . , N . In this chapter, our objective is to study the cluster synchronization between the nonidentically derivative coupled Lur’e networks (1) and the isolated Lur’e systems

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(2). In fact, the problem studied in this chapter could be viewed as special leaders and followers problems in each cluster. In particular, the trajectory sϕi (t) of the above Lur’e system (2) could be viewed as the leader for the ϕi -th cluster while the Lur’e systems in the ϕi -th cluster could be regarded as its followers. Let δz i (t) = z i (t) − sϕi (t) be the error vector, where δz i (t) = [δz i1 (t), δz i2 (t), . . . , n δz i (t)]T ∈ Rn for i = 1, 2, . . . , N . To realize the cluster synchronization of the nonidentically derivative coupled Lur’e networks, we propose the following impulsive pinning controllers u i (t) = u¯ i (t) + u˜ i (t),

(3)

where u¯ i (t) =

+∞  (ηδz i (t) + μδz i (t − τ (t)))∂(t − ts ), s=1

is the impulsive control input with another time-varying delay and furthermore, if ¯ φi , let the Lur’e system i satisfies that i ∈ W u˜ i (t) = − ϑi δz i (t) − 1

N N N    gi j sϕ j (t) − 3 ρi j s˙ϕ j (t) − 2 υi j sϕ j (t − τ (t)), j=1

j=1

j=1

¯ φi . Constant otherwise, just let u˜ i (t) = 0 for the Lur’e system satisfies i ∈ Wφi − W function τ (t) is the time-varying delay in the controlled state satisfying 0 ≤ τ (t) ≤ τ¯ . Furthermore, for the two different time-varying delays, let τ = max{h, τ¯ }. Two different scalars η and μ are called impulsive effects for the general error state δz i (t) and the time-varying delayed error state δz i (t − τ (t)) of the i-th Lur’e system; Function ∂(·) is the Dirac impulsive function; Let’s define a impulse sequence ζ = {t1 , t2 , . . .} with the impulsive instants tk , which is a strictly increasing sequence of time series, that is, it satisfies the condition tk−1 < tk and limk→+∞ tk = +∞ for k ∈ N + . The constant ϑi ≥ 0 is called the negative feedback control gain for i = 1, 2, . . . , N . Denote Dϑ = diag{ϑ1 , ϑ2 , . . . , ϑn } for later usage. Based on the design of the controller and the definition of the error vector, the nonidentically derivative coupled error Lur’e networks with stochastic disturbance and multiple time-varying delays could be represented as ⎧  δ z˙ i (t) = [ϕAi δz i (t) + ϕBi Φϕi (Cδz i (t)) + 1 Nj=1 gi j δz j (t) ⎪ ⎪   ⎪ ⎪ ˙ j (t) − ϑi δz i (t)]dt ⎨ +2 Nj=1 υi j δz j (t − h(t)) + 3 Nj=1 ρi j δz + p(t, δz i (t), δz i (t − h(t)))dω(t) ⎪ ⎪ ⎪ (t ) = ηδz i (tk− ) + μδz i (tk− − τ (tk− )), Δδz i k ⎪ ⎩ −τ ≤ t ≤ 0, δz i (t) = i (t),

(4)

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where the nonlinear structured function Φϕi (Cδz i (t)) = f ϕi (C z i (t)) − f ϕi (Csϕi (t)), ˜ z i (t), z i (t − h(t))) and the matrix-valued function p(t, δz i (t), δz i (t − h(t))) = p(t, − p(t, ˜ sϕi (t), sϕi (t − h(t))), i = 1, 2, . . . , N . Additionally, function i (t) is the initial condition of the above error networks for −τ ≤ t ≤ 0.

2.3 Preliminaries To derive the main results in this chapter for the cluster synchronization of the nonidentically derivative coupled Lur’e networks, we are now presenting some preliminaries including definitions, lemmas and assumptions. Definition 1 (Global Exponential Synchronization) Consider the nonidentically ˜ δ, derivative coupled error Lur’e networks (4). If there are two positive scalars F, ¯ ¯ and a time instant T > 0, such that for any time t > T and any initial values χi (·) satisfying the following inequality E[δz i (t)2 ] ≤ F˜

sup E[χi (s)2 ]e−δt ,

k∈[t−τ ,t]

(5)

then it is said that global exponential synchronization the between the nonidentically derivative coupled Lur’e networks (1) and the target isolated Lur’e systems (2) is eventually realized in a mean-square. Furthermore, the positive scalar δ is named as the convergence velocity of the exponential synchronization. Definition 2 (Cluster Synchronization [27]) Consider the nonidentically derivative coupled Lur’e networks (1) and the isolated target Lur’e systems (2). For the Lur’e systems in the same cluster, i.e., ϕi = ϕ j there holds the condition limt→+∞ δz i (t) = 0, while for Lur’e systems in different clusters, namely, ϕi = ϕ j , there holds that limt→+∞ δz i (t) = 0 with i, j = 1, 2, . . . , N , then it is said that the cluster synchronization between the nonidentically derivative coupled Lur’e networks (1) and the isolated target Lur’e systems (2) is eventually realized. Definition 3 (Average Impulsive Interval [28]) Define the impulsive sequence ζ = {t1 , t2 , . . .} during the time interval on (t1 , t2 ). If one could find two constants T0 > 0 and Ta > 0 satisfying t2 − t1 t 2 − t1 − T0 ≤ Tζ (t2 , t1 ) ≤ + T0 , ∀ T ≥ t ≥ 0, Ta Ta

(6)

where Tζ (t2 , t1 ) means how many times that the impulses occur on impulsive sequence ζ, then it could be estimated that the average impulsive interval is not larger than Ta . Lemma 1 ([29]) Suppose that if there exists a nonlinear function W (t) > 0, such that the impulsive differential inequalities satisfied

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D + W (t) ≤ −ψW (t) + ς supk∈[t−τ ,t] {W (k)}, W (t) = ϕ(t), t ∈ [t0 − τ , t0 ],

(7)

then one could drive the conclusion that W (t) ≤ W˜ (t)e−δ(t−t0 ) , t ≥ t0 ,

(8)

where ψ and ς are two scalars satisfying ψ > ς ≥ 0, the function ϕ(t) is piecewisely continuous, the notation W˜ (t) = supk∈[t−τ ,t] {W (k)} and the constant δ > 0 is the unique solution to the system parameter equation δ − ψ + ςeδτ = 0. Lemma 2 ([30]) Suppose that there are two scalars ψ and ς satisfying ψ > ς ≥ 0. Similar to Lemma 1, for the impulsive differential inequalities (7), if there exists a nonlinear function W (t) > 0 such that for any t > t0 ≥ 0, there holds E[W (t)] ≤ E[W˜ (t0 )]e−λ(t−t0 ) ,

(9)

where the constant δ > 0 is the unique solution to the system parameter equation δ − ψ + ςeδτ = 0 and W˜ (t0 ) = supk∈[t0 −τ ,t0 ] {W (k)}. Lemma 3 ([31]) Let’s discuss a nonlinear impulsive system with stochastic disturbance

dY (t) = F(t, C(t))dt + Θ(t, Y (t))db(t), t = tk , (10) Y (tk+ ) − Y (tk− ) = u k (Y (tk− )), k ∈ N + , where Y (t) is the state vector, F(·, ·), Θ(·, ·), C(·), u k (·) are constant functions. If there exist a positive function V (t, Y ) and two different nonlinear functions , χk with (t, 0) = χk (0) = 0 satisfying the conditions (a) There exist two positive scalars l1 , l2 satisfying l1 · Y (t) ≤ V (t, Y ) ≤ l2 ·  Y (t)  with t ≥ t0 ; (b) Define a continuous concave function as  : R+ × R+ → R. Denote VY (t, Y ) 2 (t,Y ) ∂V (t,Y ) (t,Y ) ) , ∂Y2 , . . . , ∂V∂Y ] and VY (t, Y ) = ( ∂∂YVi(t,Y ) . For the above con= [ ∂V∂Y ∂Y j n×n 1 n ditions, there satisfies LV (t, Y ) = Vt (t, Y ) + VY (t, Y )(t, Y ) + 21 trace[Θ T (t, Y ) VY Y Θ(t, Y )]; (c) If there exist continuous concave functions χk : R+ → R+ which satisfies the inequality V (tk+ , Y (tk+ )) ≤ χk (V (tk− , Y (tk− ))), then the solution of the following comparison system ⎧ ⎪ ˙ = φ(t, w(t)), t = tk , ⎨w(t) w(tk+ ) = χk (w(tk− )), k ∈ N + , ⎪ ⎩ w(t0 ) = E[V (t0 , x0 )]

(11)

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is exponentially stable, which directly implies that the exponential stability of the trivial solution of the nonlinear impulsive system with stochastic disturbance (10) is obtained, where E(·) means the mathematical expectation. Assumption 1 Consider the nonlinear Lur’e system function f ϕi (·). Suppose that the function f ϕi (·) satisfies with the Lipschitz condition with the Lipschitz constant h ϕi > 0(i = 1, 2, . . . , N ). That is, there exists a positive constant h ϕi for any two n-dimensional vectors x, y ∈ Rn satisfying  f ϕi (x) − f ϕi (y) ≤ h ϕi x − y, i = 1, 2, . . . , N .  Assumption 2 Consider the matrix-valued function p(t, ·, ·) ∈ Rn×m . Suppose that for two constant matrices Q, Q h ∈ Rn×n and any vectors x(t), x(t − h(t)) ∈ Rn , the function p(t, ·, ·) could be calculated as trace(( p(t, x(t), x(t − h(t))T p(t, x(t), x(t − h(t))) ≤ Qx(t)2 + Q h x(t − h(t))2 ,

where trace denotes the trace on the matrices. That is, the function p(t, ·, ·) is a uniformly Lipschitz function in the form of matrix norm. 

3 Main Results In this section, we will discuss the cluster synchronization of a kind of nonidentically derivative coupled Lur’e networks (1) with multiple time-varying delays and stochastic disturbance by applying an impulsive pinning controller. According to the extended comparison principle of the impulsive differential equations, the definition of the average impulsive interval and the mathematical taxonomy on impulsive parameters, the achievement of the cluster synchronization will be eventually guaranteed. In addition, the exponential convergence velocity of the derivative coupled Lur’e networks will be precisely estimated. Theorem 1 Consider the nonidentically derivative coupled error Lur’e networks (4) under the control strategy (3) with Assumptions 1 and 2. Suppose that for the impulse time series ζ = {t1 , t2 , . . .}, the average impulsive interval is less than Ta , which has been defined in Definition 3. If there exist a negative feedback control gain matrix Dϑ = diag{ϑ1 , ϑ2 , . . . , ϑ N } with at least one ϑi > 0(i = 1, 2, . . . , N ) and three positive scalars α, β, h ϕi , such that for the two different impulsive effects η and μ defined in controller (3) satisfying (i) The matrix-valued inequality  Π=

1 G s + αI N − Dϑ 21 2 Υ 1  ΥT −β I N 2 2

≤ 0;

(12)

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(ii) The following inequality if there exists a positive constant ι ¯ τ λ , eτ λ } ≤ ι, max{a¯ + be

(13)

where τ = max{h, τ¯ }, a¯ = λmax (I N − 3 )(1 + η)(1 + η + μ), b¯ = λmax (I N − 3 )(1 + η + μ)μ, and the parameter λ > 0 is the unique solution to the following constructed parameter equation λ − ψ + ςeλτ = 0 satisfying the requirement ψ > ς ≥ 0, and parameters are defined as follows T

ψ=−

min1≤i≤N {λmin (2ϕA1 + ϕBi ϕBi + h ϕi CC T + Q Q T − 2αIn )}

λmax (I N − 3 ) T ς = max {λmax (Q h Q h + 2β In )};

,

1≤i≤N

(iii) The exponentially convergence velocity in mean-square ln ι − λ < 0, Ta

(14)

then the trajectory of the nonidentically derivative coupled error Lur’e networks (4) is exponentially stable. That is, the global and exponential synchronization between the nonidentically derivative coupled Lur’e networks (1) and the target isolated Lur’e systems (2) is finally realized by imposing the impulsive pinning controller (3). Proof Based on the discussion in Remark 1, we construct the Lyapunov functional by thinking the coupling strength 3 and the derivative coupling matrix  1 V (t, δz(t)) = δz T (t)((I N − 3 ) ⊗ In )δz(t), 2

(15)

and define the vector as δz(t) = [δz 1T (t), δz 2T (t), . . . , δz TN (t)]T ∈ Rn N . To study the impulsively controlled Lur’e networks, the V (t, δz(t)) of the impulse instants tk and non-impulsive time interval [tk−1 , tk ) should be discussed, respectively. At first, let’s consider the situation that V (t, δz(t)) on impulsive time instant t = tk , k ∈ N + . Considering the impulsive relationship Δδz i (tk ) = ηδz i (tk− ) + μδz i (tk− − τ (tk− )) in the impulsively controlled error Lur’e networks (4) gives δz i (tk+ ) = (1 + η)δz i (tk− ) + μδz i (tk− − τ (tk− )). Recalling the truth in Remark 1 λmax (I N − 3 ) ≥ λmin (I N − 3 ) = 1, it yields the following conclusion

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V (tk+ ,δz(tk+ )) =

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1 T + δz (tk )((I N − 3 ) ⊗ In )δz(tk+ ) 2

1 λmax (I N − 3 )δz T (tk+ )δz(tk+ ) 2 1 = λmax (I N − 3 )((1 + η)δz T (tk− ) 2 + μδz T (tk− − τ (tk− )))((1 + η)δz(tk− ) + μδz(tk− − τ (tk− ))) 1 = λmax (I N − 3 )((1 + η)2 δz T (tk− )δz(tk− ) 2 + 2(1 + η)μδz T (tk− )δz(tk− − τ (tk− )) ≤

+ μ2 δz T (tk− − τ (tk− ))δz(tk− − τ (tk− ))) 1 ≤ λmax (I N − 3 )((1 + η)2 δz T (tk− )δz(tk− ) 2 + (1 + η)μδz T (tk− )δz(tk− ) + (1 + η)μδz T (tk− − τ (tk− ))δz(tk− − τ (tk− )) + μ2 δz T (tk− − τ (tk− ))δz(tk− − τ (tk− ))) λmax (I N − 3 ) ≤ ((1 + η)(1 + η + μ)δz T (tk− ) 2λmin (I N − 3 ) × ((I N − 3 ) ⊗ In )δz(tk− ) + (1 + η + μ)μδz T (tk− − τ (tk− )) × ((I N − 3 ) ⊗ In )δz(tk− − τ (tk− ))) = λmax (I N − 3 )((1 + η)(1 + η + μ)V (tk− ) + (1 + η + μ)μV (tk− − τ (tk− ))) ¯ (tk− − τ (tk− )), = aV ¯ (tk− ) + bV

(16)

where we define the parameters a¯ = λmax (I N − 3 )(1 + η)(1 + η + μ), b¯ = λmax (I N − 3 )μ(1 + η + μ). For the index k ∈ N + , calculating the mathematical expectation of the inequality (16) gives ¯ ¯ (tk− , δz(tk− ))] + bE[V (tk− − τ (tk− ), δz(tk− ))]. E[V (tk+ , δz(tk+ ))] ≤ aE[V

(17)

After finishing the analysis of the impulsive instants tk , let’s consider the derivative on V (t) during the non-impulsive time interval t ∈ [tk−1 , tk ) for k ∈ N + . Based on the definition of the Itoˆ differential equation on the nonlinear impulsive system with stochastic disturbance, calculating the derivative values by applying LV (t, δz(t)) in Lemma 3 with considering the impulsively pinning controlled error Lur’e networks (4) gives the following results

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LV (t,δz(t)) = δz T (t)((I N − 3 ) ⊗ In )δ z˙ (t) +

N 1 trace( p(t, δz i (t), δz i (t − h(t)))T p(t, δz i (t), δz i (t − h(t)))) 2 i=1

≤ δz T (t)δ z˙ (t) − 3 δz T (t)( ⊗ In )δ z˙ (t) +

N 1 T δz i (t)Q T Qδz i (t) 2 i=1

N 1 T + δz i (t − h(t))Q hT Q h δz i (t − h(t)) 2 i=1

=

N 

δz iT (t)(ϕA1 δz i (t) + ϕBi Φϕi (Cδz i (t)) + 1

i=1

+ 2

N  gi j δz j (t) j=1

N 

υi j δz j (t − h(t)) + 3

j=1

− 3 δz T (t)( ⊗ In )δ z˙ (t) +

N 

˙ j (t) − ϑi δz i (t)) ρi j δz

j=1 N 1 T δz i (t)Q T Qδz i (t) 2 i=1

N 1 T + δz i (t − h(t))Q hT Q h δz i (t − h(t)) 2 i=1

N 1 T T ≤ δz i (t)(2ϕA1 + ϕBi ϕBi + h ϕi CC T + Q Q T − 2αIn )δz i (t) 2 i=1

+

N 1 T δz i (t − h(t))(Q hT Q h + 2β In )δz i (t − h(t))) 2 i=1

n n   ˜ k (t) + 2 δz ˜ k (t − h(t)) ˜ T (t)(1 G − Dϑ + αI N )δz ˜ T (t)Υ δz + δz k k k=1 n 

−β

k=1

˜ k (t − h(t)) ˜ T (t − h(t))δz δz k

k=1

 T ≤ min λmin (2ϕA1 + ϕBi ϕBi + h ϕi CC T + Q Q T − 2αIn 1≤i≤N

×

N 1 T δz i (t)δz i (t) 2 i=1

+ max {λmax (Q hT Q h + 2β In )} 1≤i≤N

1 T δz (t − h(t))((I N − 3 ) ⊗ In )δz(t − h(t)) 2 n  Y T (t)Π Y (t) +

×

k=1

≤ −ψV (t) + ς V (t − h(t)),

(18)

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˜ T (t), δz ˜ T (t − h(t)))T , δz ˜ k (t) = (δz k (t), δz k (t), where define the vector as Y(t) = (δz 1 2 k k σ k T . . . , δz N (t)) and denote ψ = − λmax (I N −3 ) with σ = min1≤i≤N {λmin (2ϕA1 + T

ϕBi ϕBi + h ϕi CC T + Q Q T − 2αIn )}, and ς = max1≤i≤N {λmax (Q hT Q h + 2β In )}. Based on the above discussion, for the non-impulsive time interval t ∈ [tk−1 , tk ), one could get the following conclusion by applying Lemmas 1 and 2 E[V (t, δz(t))] ≤ E[V˜ (tk−1 , δz(tk− ))]e−λ(t−tk−1 ) ,

(19)

where denote the initial conditions as V˜ (tk−1 , δz(tk−1 )) =

sup

s∈[tk−1 −τ ,tk−1 ]

V (s, δz(s)).

In order to derive the criteria on global exponential synchronization of the nonidentically derivative coupled Lur’e networks, in the next procedure, based on the mathematical induction method, the following conclusion will be verified for a given positive constant ι and for any time satisfying t > t0 > 0 E[V (t, δz(t))] ≤ ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) ,

(20)

where the initial conditions for V (t, δz(t)) on time instant t0 is defined as V˜ (t0 , δz(t0 )) = sups∈[t0 −τ ,t0 ] V (s, δz(s)). In the first step, let’s discuss the time during the interval [tk−1 , tk ) with k = 1. According to the conclusion in inequality (19), there exists a scalar ι > 0 such that E[V (t, δz(t))] ≤E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) =ι0 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) =ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) . In the next step, suppose that the conclusion (20) is satisfied for k = s(s > 1). Then, in the following, by applying the mathematical induction method, we will prove that the conclusion (20) is still satisfied for k = s + 1. That is, for the situation k = s(s > 1) and the time t ∈ [ts−1 , ts ), one obtains the following results by considering the condition (13) in Theorem 1: E[V (ts ,δz(ts ))] ≤ aE[V (ts− , δz(tk− ))] + bE[V (ts− − τ (ts ), δz(ts− ))] ≤a · ιs−1 E[V˜ (t0 , δz(t0 ))] · e−λ(ts −t0 ) + b · ιs−1 E[V˜ (t0 , δz(t0 ))]e−λ(ts −τ (ts )−t0 ) ¯ λτ )ιs−1 E[V˜ (t0 , δz(t0 ))]e−λ(ts −t0 ) ≤(a¯ + be ≤ιs E[V˜ (t0 , δz(t0 ))]e−λ(ts −t0 ) .

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Therefore, we will derive the conclusion for k = s + 1 on the basis of k = s. Namely, for the time t ∈ [ts , ts+1 ), one has E[V (t,δz(t))] ≤ E[V˜ (ts , δz(ts ))]e−λ(t−ts ) =

sup

σ∈[ts −τ ,ts ]

= max{

E[V (σ, δz(σ))]e−λ(t−ts )

sup

σ∈[ts −τ ,ts )

E[V (σ, δz(σ))], E[V (ts , δz(ts ))]}e−λ(t−ts )

≤ max ιs−1 E[V˜ (t0 , δz(t0 ))]e−λ(ts −τ −t0 ) ,  ιs E[V˜ (t0 , δz(t0 ))]e−λ(ts −t0 ) e−λ(t−ts ) = max{eλτ , ι}ιs−1 E(V˜ (t0 ))e−λ(t−t0 ) ≤ιs E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) . Above all, based on the Mathematical Induction Method, we have proved that there satisfies E[V (t, δz(t))] ≤ ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) for any time t ∈ [tk−1 , tk ) with k ∈ N + . It should be noted that a positive scalar ι is introduced in the conclusion (20) to show the effectiveness of the inequality. However, an important problem is how to select the value ι such that (20) is satisfied for different impulsive effects. Therefore, by introducing the concept of an average impulsive interval, the global and exponential cluster synchronization issue on the nonidentically derivative coupled Lur’e networks will be studied by clarifying different values of the scalar ι > 0. (a) Considering the scalar satisfies ι ∈ (0, 1), then it gives the following results for the time t ∈ [tk−1 , tk ), k ∈ N + E[V (t, δz(t))] ≤ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) ≤ι

t−t0 Ta

−T0

E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 )

=ι−T0 E[V˜ (t0 , δz(t0 ))]ι

t−t0 Ta

e−λ(t−t0 )

ln ι

=ι−T0 E[V˜ (t0 , δz(t0 ))]e( Ta −λ)(t−t0 ) .

(21)

(b) For the positive scalar satisfying ι ∈ (1, +∞), one could obtain the following results for the time t ∈ [tk−1 , tk ), k ∈ N + E[V (t, δz(t))] ≤ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) ≤ι

t−t0 Ta

+T0

E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 )

=ιT0 E[V˜ (t0 , δz(t0 ))]ι

t−t0 Ta ln ι

e−λ(t−t0 )

=ιT0 E[V˜ (t0 , δz(t0 ))]e( Ta −λ)(t−t0 ) .

(22)

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(c) For the special situation that the scalar ι = 1, we could get the following results for the time t ∈ [tk−1 , tk ), k ∈ N + E[V (t, δz(t))] ≤ιk−1 E[V˜ (t0 , δz(t0 ))]e−λ(t−t0 ) ln ι

=E[V˜ (t0 , δz(t0 ))]e( Ta −λ)(t−t0 ) .

(23)

Based on the discussions above, it could be found that, for different values of the scalar ι, similar conclusions (21), (22) and (23) have been derived. According to Definition 1, the global exponential synchronization could be verified, that is, one could find two constants δ > 0 and F˜ > 0 such that the inequality (5) satisfied for any initial condition χi (t) ∈ C([−τ , 0], Rn ). Furthermore, considering the construction of the Lyapunov function (15), it gives the following results δz i (t)2 ≤

2V (t, δz(t)) = 2V (t, δz(t)). λmin (I N − 3 )

(24)

By taking the mathematical expectation on inequality (24) gives ln ι

E[δz i (t)2 ] ≤ F˜ E[V˜ (t0 , δz(t0 ))]e( Ta −λ)(t−t0 ) ,

(25)

where V˜ (t0 , δz(t0 )) = sups∈[t0 −τ ,t0 ] V (s, δz(s)). It further implies that the trajectories of the nonidentically coupled error Lur’e networks (4) are globally and exponentially stable with the convergence velocity 21 ( lnTaι − λ). That is, the global exponential cluster synchronization between the nonidentically derivative coupled Lur’e networks (1) and the target isolated Lur’e systems (2) is finally realized in mean-square by imposing the impulsive pinning controller (3). By now, we completely finish the proof.  Remark 2 Actually, the impulsive interval is one of the most important factors in impulsive control issues. In some related references like [22, 23], the impulsive interval Ta was roughly selected as Ta = mink {tk − tk−1 } or Ta = maxk {tk − tk−1 }. With similar steps in these references, if we deliberately let Ta = mink {tk − tk−1 } briefly in this chapter, then it gives a smaller exponential convergence velocity from the discussion in Theorem 1. Correspondingly, a bigger impulsive effect μ and pinning control gains di are required for the achievement of the quasi-synchronization, which tremendously increase the requirements on the controller designing and the control costs. Similarly, if take Ta = maxk {tk − tk−1 } in this chapter, similar problems would occur. From this point of view, both Ta = mink {tk − tk−1 } and Ta = maxk {tk − tk−1 } result in imprecise synchronization errors and convergence velocities. To avoid this situation, in this chapter, the concept of average impulsive interval has been introduced [22, 28]. From the definition in Definition 2, it gives that how many times that the impulses occur during the time interval (t, T ) could be evaluated by jointly using the time

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interval (t, T ), the freedom indicator T0 and the positive constant Ta , which generally satisfies mink {tk − tk−1 } ≤ Ta ≤ maxk {tk − tk−1 }. Based on the discussion above, different from previous works as we mentioned before, the introduction of average impulsive interval in this chapter would efficaciously reduce the conservatism in synchronization conditions and the requirements on impulsive effect and control gains. Remark 3 For simplicity of the calculations, the constant impulsive effects η and μ have been considered in this chapter. In fact, due to the complicated working environment and uncertainties of the control input, in many practical applications, the impulsive effects on both general state and delayed state are time-varying as the impulses signal generates. As a result, in order to make the main theorem is applicable to this situation, the impulsive pinning feedback controller designed in this chapter should be substituted by the time-varying impulsive effects ηk and μk rather than constant ones. Correspondingly, the condition (ii) in Theorem 1 should be replaced as: The following inequality is satisfied if there exists a positive constant ι max{a¯ k + b¯k eλτ , eλτ } ≤ ι, k = 1, 2, . . .

(26)

where τ = max{h, τ¯ } a¯ k = λmax (I N − ε3 ρ)(1 + ηk )(1 + ηk + μk ), b¯k = λmax (I N − ε3 ρ)(1 + ηk + μk )μk , k ∈ N + , and the parameter λ > 0 is the unique solution to the following constructed parameter equation λ − ψ + ςeλτ = 0 satisfying the requirement ψ > ς ≥ 0, and parameters are defined as follows: T

ψ=−

min1≤i≤N {λmin (2ϕA1 + ϕBi ϕBi + h ϕi CC T + Q Q T − 2αIn )}

λmax (I N − 3 ) T ς = max {λmax (Q h Q h + 2β In )}.

,

1≤i≤N

4 Numerical Simulation In this section, a numerical simulation will be presented to verify the effectiveness of the theoretical analysis and the control strategy. In the simulation, assume that there are totally 6 Lur’e systems in the coupled complex dynamical networks, which could be separated into two different clusters, i.e., N = 6 and let ϕ1 = ϕ2 = ϕ3 = 1, ϕ4 = ϕ5 = ϕ6 = 2. For different clusters in the Lur’e complex, let’s discuss the following isolated Chua’s circuits with time-varying delay and stochastic disturbance

Cluster Synchronization on Derivative Coupled Lur’e Networks: …

493

dz ϕi (t) = [ϕAi z ϕi (t) + ϕBi f ϕi (Cϕi z ϕi (t))]dt + p(t, ˜ z ϕi (t), z ϕi (t − h))dω(t),

(27)

where ϕi = 1 for i = 1, 2, 3 while ϕi = 2 for i = 4, 5, 6, and z ϕi (t) = (z ϕi (t)1 , z ϕi (t)2 , z ϕi (t)3 )T ∈ R3 denotes the state vector of the i-th Lur’e system in the ϕi cluster, the time-varying delay h(t) = 1 + 0.1 sin(0.1t), the system matrices are given as follows ⎡

⎡ 45 ⎤ ⎤ − 30 15 0 7 7   1A = ⎣ 1 −1 0 ⎦ , 1B = ⎣ 0 ⎦ , C1 = 1 0 0 , 0 −25.5 0 0 ⎡

⎤ ⎡ ⎤ −3.2 10 0 5.9   ⎦ , 2B = ⎣ 0 ⎦ , C2 = 1 0 0 , 0 2A = ⎣ 1 −1 0 −15 −0.0385 0 and the nonlinear vector-valued function f 1 (C1 z(t)) = f 2 (C2 z(t)) =

1 1 (z (t) + 1) − (z 1 (t) − 1), 2

the stochastic matrix p(t, ˜ z(t), z(t − h(t))) = 0.5 · z(t)I3 , and let ω(t) be the 3-dimensional Brownian motion. The phase graphs of the Lur’e systems in two different clusters are given in Fig. 1. As stated in this chapter previously, two different clusters will be considered in the coupled Lur’e networks with three kinds of couplings, namely, general state coupling, derivative coupling, time-varying delayed state coupling. The topological structure of the time-varying delay coupled Lur’e networks is presented in Fig. 1b, where the 3-th Lur’e system in the ϕi = 1-th cluster and the 4-th Lur’e system in the ϕi = 2-th cluster are directed linked with each other. That is, the pinning feedback control term u˜ i (t) in the impulsive pinning controller (3) will be directly imposed on these two Lur’e systems with u˜ 3 (t) and u˜ 4 (t). To describe three kinds of couplings for the coupled Lur’e networks (1), let the coupling matrices as follows ⎛

−2 ⎜ 0 ⎜ ⎜ 2 G=⎜ ⎜ 0 ⎜ ⎝ 0 0

0 −3 3 0 0 0

2 3 −8 3 0 0

0 0 3 −5 2 0

0 0 0 2 −5 4

⎛ ⎞ −5 0 ⎜ 1 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎟, Υ = ⎜ 0 ⎜ 2 0 ⎟ ⎜ ⎟ ⎝ 2 3 ⎠ 2 −4

1 −5 3 0 1 2

0 3 −5 0 1 1

2 0 1 −5 0 0

0 1 0 1 −5 2

⎞ 2 0 ⎟ ⎟ 2 ⎟ ⎟, 2 ⎟ ⎟ 1 ⎠ −5

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4

2

0

-2

-4 0.4 0.2

4 2

0

0

-0.2

-2 -0.4

-4

3 2 1 0 -1 -2 -3 0.4 0.2

4 2

0

0

-0.2

-2 -0.4

-4

Fig. 1 The phase graph of two different Lur’e systems in (27)

Cluster Synchronization on Derivative Coupled Lur’e Networks: … 0.5

495

0.08

0.45

0.07

0.4 0.06 0.35 0.05 2

E1(t)

1

E1(t)

0.3 0.25 0.2

0.04 0.03

0.15 0.02 0.1 0.01

0.05 0

0

2

4

6

8

0

10

0

2

4

t

6

8

10

6

8

10

6

8

10

t 0.25

0.5 0.45

0.2

0.4 0.35

0.15 1

E2(t)

1

E3(t)

0.3 0.25

0.1

0.2 0.15

0.05

0.1 0.05 0

0

2

6

4

8

0

10

0

2

4

t

t 0.5

0.7

0.45 0.6 0.4 0.5

0.35

0.4

2

E3(t)

2

E2(t)

0.3 0.25

0.3

0.2 0.15

0.2

0.1 0.1 0.05 0

0

2

4

6

t

8

10

0

0

2

4

t

Fig. 2 The synchronization error curves of three states in two different clusters

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⎛

−2 ⎜ 1 ⎜ ⎜ 0 ρ=⎜ ⎜ 1 ⎜ ⎝ 0 0

1 −5 2 1 1 0

0 2 −4 2 0 0

1 1 2 −8 2 2

0 1 0 2 −4 1

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟, 2 ⎟ ⎟ 1 ⎠ −3

let coupling strengths are set with some positive constants ε1 = 0.5, ε2 = 0.2, ε3 = 0.2, feedback control strength d1 = 0, d2 = 0, d3 = 0.8, d4 = 0.5, d5 = 0, d6 = 0, and the time-varying coupling delay as τ (t) = 0.2 sin(0.2t). In view of the construction of the Lyapunov function, it could be obtained that λmax (I6 − ε3 ρ) = 2.9792. In order to design the impulsive control input terms u¯ i (t) in the impulsive pinning feedback controller (3), take the nonuniform distribution of impulses into account [28]. Let the impulsive sequence as ζ = {c, 2c, . . . , (T0 − 1)c, T0 Ta , T0 Ta + c, T0 Ta + 2c, . . . , T0 Ta + (T0 − 1)c, 2T0 Ta , . . .}. It further denotes that tk − tk−1 = c if mod(k, T0 ) = 0 and T0 (Ta − z) + c, if mod(k, T0 ) = 0, where 0 ≤ c ≤ Ta . Set the average impulsive interval is no larger than Ta = 0.02, and the free adjust constants T0 = 2, c = 0.01. By the parameters, it could be verified that ψ > ς > 0. It also could be calculated that the solution to the nonlinear system parameter equation λ − ψ + ςeλτ = 0 is λ = 3.8912. To reflect the synchronization procedure of the nonidentically derivative coupled Lur’e networks (1), the synchronization error for each state in different clusters should  j

be defined. Let E 1 (t) =

j 1 (δz 1 (t)2 3

j

j

+ δz 2 (t)2 + δz 3 (t)2 ), j = 1, 2, 3 be the syn-

chronization error of each state in the  j j j 1 (δz 4 (t)2 + δz 5 (t)2 + δz 6 (t)2 ), j = 1, 2, 3. 3

first

cluster

and

j

E 2 (t) = j

In Fig. 2, the synchronization error curves of three states in the first cluster E 1 (t) j and in the second cluster E 2 (t) are presented, where the error approach to zero within about two seconds. From the figures, it can be found that the cluster synchronization between the nonidentically derivative coupled Lur’e networks and the target isolated Lur’e system could be achieved within about two seconds.

5 Conclusion In this chapter, the global exponential synchronization issue of a class of nonidentically derivative coupled Lur’e dynamical networks with time-varying delays and stochastic disturbance has been investigated. To be more applicable to general complex network models, general state coupling, delayed state coupling and derivative state coupling have been considered. By thinking that the cluster topological structure of the Lur’e networks, a novel impulsive pinning controller has been proposed, which has been imposed on the Lur’e systems in the clusters which have directed connections with the Lur’e systems in the other clusters. In view of the extended

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comparison principle of the impulsive differential equations, the concept of average impulsive interval and the mathematical induction method, sufficient criteria for the cluster synchronization of the coupled Lur’e networks have been obtained. Meanwhile, the exponential convergence velocity of the cluster synchronization has been precisely estimated. Furthermore, a numerical simulation has been given to demonstrate the availability of the theoretical analysis. Acknowledgements This work was supported in part by the National Natural Science Foundation of China with Grant No. 61803180, in part by the Natural Science Foundation of Jiangsu Province with Grant No. BK20180599, in part by the National Key Research and Development Program of China with Grant No. 2018YFB1701903, and in part by the 111 Project with Grant No. B12018. The work of J. H. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. 2019R1A5A808029011).

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16. He, W.L., Qian, F., Cao, J.D.: Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control. Neural Netw. 85, 1–9 (2017) 17. Tang, Z., Park, J.H., Zheng, W.X.: Distributed impulsive synchronization of Lur’e dynamical networks via parameter variation methods. Int. J. Robust Nonliear Control 28(3), 1001–1015 (2018) 18. Zhang, W.B., Tang, Y., Wu, X.T., Fang, J.A.: Synchronization of nonlinear dynamical networks with heterogeneous impulses. IEEE Trans. Circuits Syst. I: Reg. Pap. 61(4), 1220–1228 (2014) 19. Tang, Y., Gao, H.J., Zhang, W.B., Kurths, J.: Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses. Automatica 53, 346–354 (2015) 20. Lu, J.Q., Ho, D.W.C., Cao, J.D., Kurths, J.: Exponential synchronization of linearly coupled neural networks with impulsive disturbances. IEEE Trans. Neural Netw. 22(2), 329–336 (2011) 21. Kumar, R., Sarkar, S., Das, S., Cao, J.D.: Projective synchronization of delayed neural networks with mismatched parameters and impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. To be published (2019). https://doi.org/10.1109/TNNLS.2019.2919560 22. Lu, J.Q., Wang, Z.D., Cao, J.D., Ho, D.W.C., Kurths, J.: Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay. Int. J. Bifurc. Chaos 22(7), 1250176 (12 pages) (2012) 23. Zhang, W.B., Tang, Y., Miao, Q.Y., Fang, J.A.: Synchronization of stochastic dynamical networks under impulsive control with time delays. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1758–1768 (2014) 24. Xu, Y.H., Zhou, W.N., Fang, J.A., Xie, C.R., Tong, D.B.: Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling. Neurocomputing 173(3), 1356–1361 (2016) 25. Zheng, S.: Pinning and impulsive synchronization control of complex dynamical networks with non-derivative and derivative coupling. J. Frank. Inst. 354(14), 6341–6363 (2017) 26. Deng, L.P., Wu, Z.Y., Wu, Q.C.: Pinning synchronization of complex network with nonderivative and derivative coupling. Nonlinear Dyn. 73(1–2), 775–782 (2013) 27. Wu, W., Zhou, W.J., Chen, T.P.: Clsuter synchronization of linearly copuled complex networks under pinning control. IEEE Trans. Circuits Syst. I: Reg. Pap. 56(4), 829–839 (2009) 28. Lu, J.Q., Ho, D.W.C., Cao, J.D.: A unified synchronization criterion for impulsive dynamical networks. Automatica 46(7), 1215–1221 (2010) 29. Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic, New York (1996) 30. Tang, Z., Park, J.H., Lee, T.H., Feng, J.W.: Mean square exponential synchronization for impulsive coupled neural networks with time-varying delays and stochastic disturbances. Complexity 21(5), 190–202 (2016) 31. Liu, B.: Stability of solutions for stochastic impulsive systems via comparison approach. IEEE Trans. Autom. Control 53(9), 2128–2133 (2008)

Finite-Time Synchronization Control for Markovian Jump Memristive Neural Networks with Reaction-Diffusion Terms Xiaona Song, Jingtao Man, and Shuai Song

Abstract This chapter investigates the issue of finite-time synchronization for a class of memristive neural networks, where the reaction-diffusion items and Markovian jump parameters are both considered. Specifically, the drive and response systems proposed in this chapter follow the inconsistent Markov chains, which might be in accordance with realistic applications better. Meanwhile, a novel discontinuous controller is designed, so that the difficulty in deriving the synchronization criterion of time-space related second-order differential systems (SODSs) can be solved. In the process of dealing with the derivative of Lyapunov–Krasovskii functional (LKF), a canonical Bessel–Legendre inequality (BLI), which converts the limited interval required in traditional BLI to a general one, is employed. The main work is reflected as follows: first, the original SODSs is converted into a first-order differential system by employing a suitable variable substitution and then the error system in a compact first-order differential form is established; second, a less conservative criterion of finite-time synchronization is obtained by constructing a new LKF and utilizing the canonical BLI and free-weighting matrix methods. Ultimately, a numerical example and an application study are exploited to illustrate the feasibility and practicability of this chapter, thus the acquired theoretical results can be well supported. Keywords Finite-time synchronization · Reaction-diffusion memristive neural networks · Inconsistent Markov chains · Discontinuous controller · Canonical Bessel–Legendre inequality X. Song (B) · J. Man School of Information Engineering, Henan University of Science and Technology, Luoyang 471023, People’s Republic of China e-mail: [email protected] J. Man e-mail: [email protected] S. Song School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9_21

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1 Introduction The past few decades have witnessed the rapid development of neural networks (NNs), which have been applied to various areas such as image processing, pattern recognition, combinatorial optimization, associate memory design, etc [1–3]. With growing interest, an extraordinary volume of results on NNs have emerged (see e.g. [4–6] and references therein). However, in the hardware implementation of NNs, there is a common challenge that if we use traditional silicon-based semiconductor devices to simulate the synaptic electrical characteristics in biological neurons, only one synapse in a neuron will need dozens of transistors, resistors, capacitors, etc., which means that the implementation of complex NNs is impractical [7]. Fortunately, the invention of memristor solves this problem well. Because memristors are variable resistors and exhibit memory characteristics and their information storage and processing are very similar to human brain synapses. These characteristics show that memristive devices have many application prospects, such as device modeling and signal processing. Among them, the function of imitating synaptic behavior is not available in three types of basic circuit elements: resistance, capacitance and inductance [8]. Based on what has been discussed above, an increasing number of scholars have been focusing on memristive NNs (MNNs) and many relative results have been published. Based on the excitatory and inhibitory of memristive synaptic weights and by employing a new convex combination technique, [9] obtained a stabilization criterion for a class of T–S fuzzy MNNs via the switched fuzzy sampled-data control scheme. To reduce the complexity of available methods in dealing with the statedependent memristive connection weights and conservatism of existing event-trigger mechanisms, [10] designed an exponential-attenuation-based switching event-trigger scheme to realize the global stabilization of delayed MNNs. Moreover, a class of MNNs with discontinuous activation functions and mixed time-varying delays were proposed in [11], where an adaptive state-feedback controller and a state-feedback controller were proposed, respectively, to synchronize the drive and response systems within a finite time. As we know, reaction-diffusion NNs have become a research hotspot. The one reason is that diffusion phenomena will be obvious when electrons of NNs move in an electromagnetic field that is nonuniform. Another reason is they have been widely applied to biology, neuroscience, chemistry and some other fields. Therefore, considering reaction-diffusion terms in the model of MNNs is reasonable [12–14]. On the other hand, Markovian jumping phenomena often occur in real systems due to some factors such as sudden environmental changes and component failures [15, 16]. As a result, it is necessary to set the parameters as Markovian jumping ones during modeling and some results on Markovian reaction-diffusion MNNs have been appeared [17–19]. Nevertheless, all of the aforementioned papers about the issue of synchronization for Markovian systems suppose that the drive and response systems share the same Markov chain. While strictly speaking, in real systems, the drive and response

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systems are more likely to follow different Markovian jumping laws. Thus, in this chapter, we investigate the Markovian systems subject to inconsistent Markov chains. Then, the challenge of dealing with the inconsistent parameters in the error system must be faced. To solve this problem, a novel discontinuous controller is designed in this chapter, so that the influence of the inconsistent parameters can be avoided. This chapter’s main target is to propose an effective finite-time synchronization criterion with the designed control scheme to guarantee the response system can be synchronized with the established drive system. To sum up, this chapter mainly contributes to the following three aspects: (1) This chapter proposes a new class of drive and response MNNs, where the Markovian jumping parameters and reaction-diffusion terms are both contained. Note that the biggest innovation of this system model is that the drive and response systems subject to two different Markov chains, which is more practical than existing results on the synchronization of Markovian systems that suppose the drive and response systems share the same Markovian jumping law such as [20–22]. Moreover, to cope with the challenge caused by inconsistent parameters in the error system, inspired by [23], a novel discontinuous controller is proposed, such that the aforementioned problem can be solved. (2) A novel LKF is constructed, where the triple and quadruple integral items are involved to capture more information of the time-varying delay, so that the conservatism of obtained results can be reduced. Additionally, by combining canonical BLI and free weighted matrix (FWM) methods, the conservatism of synchronization criterion proposed in this chapter can be further reduced. (3) A numerical example and an application example are provided to illustrate the effectiveness and practicability of the main results. What should be emphasized is that the application is realized with a cryptosystem established by the Markovian reaction-diffusion MNNs, which is the first attempt in secret communication. Notations: Throughout this chapter, E denotes the mathematical expectation operator; Rn represents the n-dimensional Euclidean space; col{· · · } means expressing the elements in {· · · } as column vectors; †x = [0n,(x−1)n , In , 0n,(13−x)n ], (x = 1, 2, . . . , 13) is a block entry matrix; For an arbitrary matrix A, Sym{A} = A + A T ; I represents the identity matrix with appropriate dimensions; Additionally, for convenience, this chapter adopts some abbreviations as follows: §δ = §δ (υ, t) (§ = m, s, p, u), ℵlh = ℵl (υ, t − h(t)) (ℵ = m, s, p), where υ = (υ1 , υ2 , . . . , υq )T ∈ Ω ⊂ Rq with Ω = {υ ||υk | ≤ k } and k > 0, (k = 1, 2, . . . , q) are constants, and the meaning of m, s, p and u will be explained hereinafter.

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2 Preliminaries Suppose {rt } is a right-continuous Markov chain on the probability space that takes values in a finite-state space S = {1, 2, . . . , N }, and  Pr {rt+Δt = μ |rt = σ } =

σ = μ πσμ Δt + o(Δt), 1 + πσσ Δt + o(Δt), σ = μ,

where Pr is the transition probability from mode σ at time t to mode μ at time  t + Δt; πσμ . Δt > 0, lim [o(Δt)/Δt] = 0 , and πσμ > 0 for σ = μ, and πσσ = − Δt→0

μ∈S,σ=μ

Consider the following Markovian reaction-diffusion MNNs as the drive system ∂m δ  ∂ = ∂t ∂υk k=1 q

+

n 

  n  ∂m δ xδk − aδ m δ + cδlσ (m δ )gl (m l ) ∂υk l=1

dδlσ (m δ )gl (m lh ),

(1)

l=1

where m δ is the δth neuron’s state, δ = 1, 2, . . . , n; xδk > 0 represents the transmission diffusion operator; aδ > 0 is a given constant; and cδlσ (m δ ) and dδlσ (m δ ) correspond to the feedback connection strength and delayed feedback connection strength, respectively, and  cδlσ (m δ ) =



cδlσ ,  cδlσ ,

|m δ | ≤ Yδ , dδlσ (m δ ) = |m δ | > Yδ ,





dδlσ ,  dδlσ ,

|m δ | ≤ Yδ , |m δ | > Yδ .

Additionally, gl (·) is the activation function and h(t) is time-varying delay, which satisfy the following assumptions: gl (τ1 ) − gl (τ2 ) ≤ εl+ , τ1 − τ2 dh(t) ≤ h < ∞, 0 < h 1 ≤ h(t) ≤ h 2 , dt

|gl (τ )| ≤ gl , εl− ≤

where τ , τ1 , τ2 , εl− , εl+ , h 1 , h 2 and h are constants, τ1 = τ2 , and gl (m lh ) = gl (m l (υ, t − h(t))). For a bounded and continuous function φδ (υ, o), the boundary and initial conditions of (1) are defined as m δ (υ, t) = 0, (υ, t) ∈ ∂Ω × [−h 2 , +∞), m δ (υ, o) = φδ (υ, o), (υ, o) ∈ Ω × [−h 2 , 0].

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Now, we introduce another Markov chain {κt } that takes values in T = {1, 2, . . . , M} and depends on rt with the following conditional probability: Pr {κt = |rt = σ} = σ , where σ ≥ 0 for all σ ∈ S and  ∈ T , and

 ∈T

σ = 1.

With this Markovian jumping law, the response system can be modeled as: ∂sδ  ∂ = ∂t ∂υk k=1 q

+

n 

  n  ∂sδ xδk − a δ sδ + cδl (sδ )gl (sl ) ∂υk l=1

dδl (sδ )gl (slh ) + u δ ,

(2)

l=1

where u δ is the control input, and  cδl (sδ ) =

 , cδl  cδl ,

|sδ | ≤ Yδ , dδl (sδ ) = |sδ | > Yδ ,



 , dδl  dδl ,

|sδ | ≤ Yδ , |sδ | > Yδ .

Moreover, sδ (υ, t) = 0, (υ, t) ∈ ∂Ω × [−h 2 , +∞), sδ (υ, o) = ψδ (υ, o), (υ, o) ∈ Ω × [−h 2 , 0], where ψδ (υ, o) has the same characteristic as φδ (υ, o). Define pδ = sδ − m δ , then, the error system can be described as ∂ pδ  ∂ = ∂t ∂υk k=1 q

+ +

n  l=1 n 

  n  ∂ pδ xδk − a δ pδ + cδl (sδ ) fl ( pl ) ∂υk l=1

dδl (sδ ) fl ( plh ) +

n 

(cδl (sδ ) − cδlσ (m δ )) gl (m l )

l=1

(dδl (sδ ) − dδlσ (m δ )) gl (m lh ) + u δ ,

l=1

where fl ( pl ) = gl (sl ) − gl (m l ) and fl ( plh ) = gl (slh ) − gl (m lh ). Obviously, εl− ≤ fl τ(τ ) ≤ εl+ , τ ∈ R, and pδ (υ, t) = 0, (υ, t) ∈ ∂Ω × [−h 2 , +∞), pδ (υ, o) = ψδ (υ, o) − φδ (υ, o), (υ, o) ∈ Ω × [−h 2 , 0]. For simplicity, we give the following compact form of error system (3):

(3)

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∂p  ∂ = ∂t ∂υk k=1 q

  ∂p Λk − A p + C (s) f ( p) ∂υk

+ D (s) f ( ph ) +  + u,

(4)

where p = ( p1 , p2 , . . . , pn )T , u = (u 1 , u 2 , . . . , u n )T , Λk = diag{x1k , x2k , . . . , xnk }, A = diag{a1 , a2 , . . . , an }, C (s) = (cδl (s))n×n , D (s)=(dδl (s))n×n , ph = p(υ, t − h(t)), f (·)=( f 1 (·), f 2 (·), . . . , f n (·))T ;  is short for (υ, t) and = (C (s) − Cσ (m)) g(m) + (D (s) − Dσ (m)) g(m h ), Cσ (m) = (cδlσ (m))n×n , Dσ (m) = (dδlσ (m))n×n , and g(·) = (g1 (·), g2 (·), . . . , gn (·))T . From aforementioned switching rules, the number of combination of C (s) and D (s) is 2n at mode . Then, the 2n cases at mode  can be ordered as (C1 , D1 ), (C2 , D2 ), . . . , (C2n , D2n ). At any υ and t, the combination of C (s) and D (s) must be one of the above 2n cases. Namely, there exist some ϕ such that C (s)=Cϕ , and D (s)=Dϕ . We define  1, C (s)=Cϕ , D (s)=Dϕ , ξϕ (υ, t) = 0, other wise. Obviously,

2n  ϕ=1

ξϕ (υ, t) ≡ 1.

As a result, the error system (4) can be rewritten as: ∂p  ∂ = ∂t ∂υk k=1 q

where C˜ = C˜ (υ, t) =

  ∂p Λk − A p + C˜ f ( p) + D˜  f ( ph ) +  + u, ∂υk 2n  ϕ=1

˜  = D˜  (υ, t) = ξϕ (υ, t)Cϕ and D

2n  ϕ=1

(5)

ξϕ (υ, t)Dϕ .

To achieve the target of synchronization, we propose a discontinuous controller u as: ˜ u = −K  p − H (υ, t)sign( p)

(6)

where K  is the controller gains to be designed. sign(·) is a standard symbolic func T tion; sign( p) ˜ = sign( p˜ 1 ), sign( p˜ 2 ), . . . , sign( p˜ n ) , p˜ δ = pδ + p˙ δ ; p˙ δ = ∂ pδ /∂t, H (υ, t) = diag{H1 (υ, t), H2 (υ, t), . . . , Hn (υ, t)}, and ⎧ h1δ , ⎪ ⎪ ⎨ h2δ , Hδ (υ, t) = h3δ , ⎪ ⎪ ⎩ h4δ ,

|m δ | ≤ Yδ |m δ | ≤ Yδ |m δ | > Yδ |m δ | > Yδ

and and and and

|sδ | ≤ Yδ , |sδ | > Yδ , |sδ | ≤ Yδ , |sδ | > Yδ ,

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     n          where hiδ = c˜iδl + d˜iδl g¯ l , (i = 1, 2, 3, 4), c˜1δl = cδl − cδlσ , c˜2δl = cδlσ − cδl , l=1                         c˜3δl = cδlσ − cδl , c˜4δl = cδlσ − cδl , d˜1δl = dδl − dδlσ , d˜2δl = dδlσ − dδl ,             d˜3δl = dδlσ − dδl , and d˜4δl = dδlσ − dδl . Remark 1 As we all know, the invention of memristor greatly reduced the cost in the implementation of NNs, making the application of complex NNs possible. Therefore, MNNs are a class of system that is closely integrated with reality and it is necessary to consider various possible problems in the actual systems during the modeling of MNNs. For one thing, when studying the multilayer cellular NNs or electrons shift in a non-uniform electromagnetic field, the diffusion phenomena will be unavoidable. For another, due to the random failures, repairs of the network components, and/or sudden environment switching, the random mode switching would occur in NNs. Above discussions imply that it is significant to consider reaction-diffusion terms and Markovian jump parameters on the basis of the existing MNNs models. What should be emphasized is that in existing results on the synchronization of Markovian jump systems, drive and response systems’ jumping rules are usually set to be the same such as [24, 25]. However, it seems more reasonable for drive and response systems to jump by following different Markov chains. From where we stand, the reason that existing papers don’t consider the situation of inconsistent Markovian jumping laws is that it is a big challenge to deal with the inconsistent parameters in error systems. To solve this problem, an innovative part −H (υ, t)sign( p) ˜ is tactfully employed in the controller designed in this chapter. Then, the aforementioned challenge is overcome. More details will be shown in subsequent sections. Definition 1 ([26]) If the Markovian reaction-diffusion MNNs (5) satisfies E( p(υ, t1 ) 2 ) ≤ 1 =⇒ E( p(υ, t2 ) 2 ) ≤ 2 , t1 ∈ [−h 2 , 0], t2 ∈ (0, T ∗ ], where 0 ≤ 1 < 2 , then, the system (5) is bounded within an FTI with respect to (1 , 2 , T ∗ ). Lemma 1 ([27]) Let ω(υ) be a real-valued function belonging to C 1 (Ω) and satisfies ω(υ) |∂Ω = 0. Then  ω (υ)dυ ≤ 2

Ω

2k

    ∂ω(υ) 2    ∂υ  dυ. Ω

k

where Ω is a cube |υk | < k (k = 1, 2, . . . , q). Lemma 2 ([28]) For given scalars c and d, d > c, if there exist an integer N ≥ 0, an n × n positive constant matrix P, a matrix B with the appropriate dimensions, and a differentiable function ω : [c, d] → Rn that is vector-valued, the following inequality holds:

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d

− c

  ω˙ T (x)P ω(x)d ˙ x ≤ς NT X˜ TN Y˜ NT B + B T Y˜ N X˜ N + (d − c)B T P N−1 B ς N ,

where ς N = col{ω(d), ω(c), γ1 , . . . , γ N }, P = diag{P, 3P, . . . , (2N + 1)P}, ⎡

I ⎢0 ⎢ ⎢ X˜ N = ⎢ 0 ⎢ .. ⎣.

−I −I −I .. .

0 I 0 .. .

0 0 2I .. .

··· ··· ··· .. .

0 0 0 .. .

0 −I 0 0 0 N I

⎤

⎡ ⎤ I 0 ··· 0 ⎥ ⎥ ⎢ I YN 1 · · · 0 ⎥ ⎥ ˜ ⎢ ⎥ ⎥ , Y N = ⎢ .. .. . . .. ⎥ , ⎥ ⎣. . . . ⎦ ⎦ I YN 2 · · · YN 3

with 

(d − x)i−1 ω(x)d x, (i = 1, 2, . . . , N ), (d − c)i c    1 1+1 I, = (−1)1 1 1    N N +1 I, = (−1)1 1 1    N N+N I. = (−1) N N N

γi = YN 1 YN 2 YN 3

d

Lemma 3 ([29]) For any constant matrix B > 0 and scalars h 2 > h 1 > 0, the following estimation holds:  −

−h 1



−h 2

t t+α

2 ≤− 2 h 2 − h 21

ω T (s)Bω(s)dsdα 

−h 1

−h 2



t

 ω (s)dsdαB T

t+α

−h 1 −h 2



t

ω(s)dsdα.

t+α

3 Main Results This section proposes a finite-time synchronization criterion of the drive and response Markovian reaction-diffusion MNNs. Before deriving our main results, we give some necessary donations below:

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      χT = p T , f T ( p) , χhT = phT , f T ( ph ) , χhT1 = phT1 , f T ( ph 1 ) ,   χhT2 = phT2 , f T ( ph 2 ) , h 21 = h 2 − h 1 , ph 1 = p(υ, t − h 1 ),  q  q q  x1k   x2k xnk ph 2 = p(υ, t − h 2 ), Λ˜ = diag , ,..., , 2 2 2 k=1 k k=1 k k=1 k  t (t − α)i−1 ϑi = p(υ, α)dα, h i1 t−h 1  t−h 1 (t − h 1 − α)i−1 ϑ˜ i = p(υ, α)dα, (i = 1, 2, . . . , N ), h i21 t−h 2 ℘ = col{ p, 1 , 2 , . . . ,  N }, i = col{ϑi , ϑ˜ i },  ∇ = col p, p, ˙ ph 1 , ph , ph 2 , f ( p), f ( ph 1 ), f ( ph ), f ( ph 2 ),  t−h 1  t−h 1  t  p(υ, α)dα, p(υ, α)dα, p(υ, ˙ α)dα, 1 , 2 , . . . ,  N . t−h 1

t−h 2

t−h 2

Theorem 1 For given scalars 1 > 0, ι > 0, and T ∗ > 0, and an integer N ≥ 0, the synchronization error system (5) is bounded over an FTI if there exist a posiN N , P2σ ,..., PN } > 0, Ψ > 0, tive scalar 2 ; symmetric matrices PσN = diag{P1σ   



 



 (N +1)σ T T T T T T 11 12 > 0, T

= 11 12 > 0, T

= 11 12 T =





> 0, Q > 0, R > 0, ∗ T22 ∗ T22 ∗ T22 ˜ and diagonal F1 > 0, F2 > 0; arbitrary matrices Υ j ( j = 1, 2, 3, 4, 5), M, and M; matrices Θm (m = 1, 2, 3), such that the following LMIs hold: ⎤ Σ h 1λ¯ 1T M h 21λ¯ 2T M˜ ⎦ < 0, ⎣ ∗ −Q N 0 ∗ ∗ −R N

(7)

  N ∗ 2 , 1 ςeιT < min λmin P1σ

(8)

⎡

σ∈S

where Σ = Σ1 + Σ2 + Σ3 + Σ4 + Σ5 , λ¯ 1 = col{†1 , †3 , E¯ 1 †13 , . . . , E¯ 1 † N +12 }, λ¯ 2 = col{†3 , †5 , E¯ 2 †13 , . . . , E¯ 2 † N +12 }, Q N = diag{Q, 3Q, . . . , (2N + 1)Q}, R N = diag{R, 3R, . . . , (2N + 1)R},   ς = λmax ((2N + 1)I ) max λmax PσN σ∈S



+ λmax

" Λk Ψ

k=1



+ h 2 λmax I + (E + ) E + λmax T + T

! q 

h 321 2

λmax (R) +

+

h 31 λmax (Q) 2

h 31 λmax (F1 ) 6

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  h3 T + h 1 λmax I + (E + ) E + λmax T

+ 21 λmax (F2 ) 6   + h 21 λmax I + (E + )T E + λmax T

, with Σ1 =Sym{XT PσN Y} + XT



πσμ PμN X − 2†2T Ψ †2

μ∈S

 ˜ + 2Ψ A + 2Ψ K  †1 + Sym{†1T Ψ C˜ †6 } − †1T 2ΛΨ − Sym{†1T (Ψ + AT Ψ T + K T Ψ T )†2 } + Sym{†1T Ψ D˜  †8 } + Sym{†2T Ψ C˜ †6 } + Sym{†2T Ψ D˜  †8 },







Σ2 =†1T (T11 + T11 )†1 + Sym{†1T (T12 + T12 )†6 }  T

 T

− (1 − h) †4 T11 †4 + †8 T22 †8 + Sym{†4 T T12 †8 }











+ †6T (T22 + T22 )†6 − †5T T11 †5 − †3T (T11 − T11 )†3









− Sym{†3T (T12 − T12 )†7 } − †7T (T22 − T22 )†7





− †5T T12 †9 − †9T T22 †9 ,   T T T Σ3 =h 1λ¯ 1 X˜ N Y˜ N M + M T Y˜ N X˜ N λ¯ 1 + †2T (h 21 Q + h 221 R)†2   + h 21λ¯ 2T X˜ TN Y˜ NT M˜ + M˜ T Y˜ N X˜ N λ¯ 2 ,  2  h1 h 2 − h 21 2 T Σ4 =†2T F1 + 2 F2 †2 − 2†1T F1 †1 − 2 †10 F1 †10 2 2 h1

2 2h 2 Sym{†1T F1 †10 } − 2 21 2 †1T F2 †1 h1 h2 − h1 2h 21 T 2 + 2 † F2 †11 − 2 †T F2 †11 , h 2 − h 21 1 h 2 − h 21 11   Σ5 =Sym{ †6T − E − †1T Θ1 E + †1 − †6 }   + Sym{ †8T − E − †4T Θ2 E + †4 − †8 }      + Sym{ †6T − †8T − E − †1T − †4T Θ3 E + (†1 − †4 ) − †6 + †8 }  T + Sym{ †1T Υ1 + †2T Υ2 + †6T Υ3 + †8T Υ4 + †12 Υ5 (†3 − †5 − †12 )}, +

and X = col{†1 , †13 , . . . , † N +12 }, Y = col{†2 , † N }, † N = col{‡1 , ‡2 , . . . , ‡ N }, ‡i = col{− h11 †3 , − h121 †5 }, E¯ 1 = [I, 0], E¯ 2 = [0, I ]. Proof Consider the following LKF candidate for error system (5): V ( p, t) =

4  i=1

Vi ( p, t),

(9)

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where   V1 ( p, t) =

℘

Ω

T

Ω

t



t−h 2



+

0



t

−h 1 t+θ −h 1  t



  Ω

dυ,

# χT (υ, θ)T

χ(υ, θ)dθ dυ,



h1

+h 21 V4 ( p, t) =



χT (υ, θ)T

χ(υ, θ)dθ

t−h 1  t−h 1

Ω

Λk Ψ

∂p ∂υk

χT (υ, θ)T χ(υ, θ)dθ

t

  V3 ( p, t) =

∂υk



t−h(t)

+ +

+

 q   ∂p T k=1

  V2 ( p, t) =

PσN ℘

−h 2 t+θ 0  0 t

p˙ T (υ, α)Q p(υ, ˙ α)dαdθ # p˙ T (υ, α)R p(υ, ˙ α)dαdθ dυ,

−h 1 α t+β −h 1  0  t −h 2

p˙ T (υ, θ)F1 p(υ, ˙ θ)dθdβdα #

p˙ (υ, θ)F2 p(υ, ˙ θ)dθdβdα dυ.

α

T

t+β

Then, the time derivative of V ( p, t) can be obtained as LV ( p, t) = LV1 ( p, t) + LV2 ( p, t) + LV3 ( p, t) + LV4 ( p, t), where   LV1 ( p, t) =

Ω

2℘ T PσN ℘˙ + ℘ T  q

+2

k=1

∂p ∂υk



πσμ PμN ℘

μ∈S

T



Λk Ψ

∂ p˙ ∂υk

# dυ.

Based on the error system (5), the following equation is obviously true: 

 ∂p Λk − A p + C˜ f ( p) + D˜  f ( ph ) ∂υk  +  − K  p − H (υ, t)sign( p) ˜ .

∂p  ∂ 0 =2 pΨ ˜ − + ∂t ∂υk k=1 q



  One can conclude from the expression of H (υ, t) that 2 pΨ ˜  − H (υ, t)sign( p) ˜ ≤ 0 for all cases.

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Moreover, by using Lemma 1 and boundary conditions, we have  Ω

     q q   ∂ ∂p ∂ ∂p Λk dυ = Λk dυ 2 ( p + p) ˙ Ψ ∂υk ∂υk ∂υk ∂υk Ω k=1 k=1      q   ∂p T ∂ p˙ T ˜ ≤ −2 Λk Ψ p ΛΨ p + dυ. ∂υk ∂υk Ω k=1

2 pΨ ˜

As a result, LV1 ( p, t) ≤

⎧  ⎨



˜ p + 2( p + p)Ψ ˙ πσμ PμN ℘ − 2 p T ΛΨ ⎩ μ∈S  # ∂p × − − A p + C˜ f ( p) + D˜  f ( ph ) − K  p dυ. ∂t Ω

2℘ T PσN ℘˙ + ℘ T

One can readily obtain the following inequality based on some simple estimation:  LV2 ( p, t) ≤

$

 χT T + T

χ − (1 − h)χhT T χh Ω 

% T T −χh1 T

χh2 dυ. T − T

χh1 − χh2

Additionally, we have   LV3 ( p, t) =

 h 21 p˙ T

Ω



Q p˙ +

t−h 1

−h 21

h 221 p˙ T

R p˙ − h 1

t

p˙ T (υ, α)Q p(υ, ˙ α)dα

t−h 1

#

p˙ (υ, α)R p(υ, ˙ α)dα dυ. T

t−h 2

On the other hand, the B–L inequality provided in Lemma 2 can yield that 

t

p˙ T (υ, α)Q p(υ, ˙ α)dα   ≤ h 1 η NT X˜ TN Y˜ NT M + M T Y˜ N X˜ N + h 1 M T Q−1 M ηN , N

− h1

t−h 1

where η N = col{ p, ph 1 , ϑ1 , . . . , ϑ N }. Similarly,  − h 21

t−h 1

t−h 2

≤

h 21 η˜ NT



p˙ T (υ, α)R p(υ, ˙ α)dα

 ˜ X˜ TN Y˜ NT M˜ + M˜ T Y˜ N X˜ N + h 21 M˜ T R−1 N M η˜ N ,

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where η˜ N = col{ ph 1 , ph 2 , ϑ˜ 1 , . . . , ϑ˜ N }. Furthermore, it is straightforward that  

h 21 T h 2 − h 21 T p˙ F1 p˙ + 2 p˙ F2 p˙ 2 2 Ω  0  t p˙ T (υ, θ)F1 p(υ, ˙ θ)dθdα −

LV4 ( p, t) =

−h 1 t+α −h 1  t

# p˙ T (υ, θ)F2 p(υ, ˙ θ)dθdα dυ.

 −

−h 2

t+α

By recalling Lemma 3, the following inequalities hold:  −



0 −h 1

2 ≤− 2 h1

t

p˙ T (υ, θ)F1 p(υ, ˙ θ)dθdα

t+α 0







t

p˙ (υ, θ)dθdαF1

0



t

p˙ T (υ, θ)dθdα

T

−h 1

t+α

−h 1

t+α

 T   2 & h1 p h1 p & =− 2 F1 , t t h 1 t−h 1 p(υ, α)dα t−h 1 p(υ, α)dα

and  −

−h 1

−h 2

≤−



t

p˙ T (υ, θ)F2 p(υ, ˙ θ)dθdα

t+α

 T   h p h p 2 & t−h 1 21 & t−h 1 21 , F 2 h 22 − h 21 t−h 2 p(υ, α)dα t−h 2 p(υ, α)dα

where  F1 =

   F1 −F1 F2 −F2 , F2 = . ∗ F1 ∗ F2

The FWM method is applied to the following equation to further reduce the conservatism:    t−h 1 T 2 p Υ1 + pΥ ˙ 2 + f ( p)Υ3 + f ( ph )Υ4 + p(υ, ˙ θ)dθΥ5 t−h 2

 ×

 ph 1 − ph 2 −

t−h 1



p(υ, ˙ θ)dθ = 0.

t−h 2

By considering the activation function of this chapter, we have

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 T   0 ≤ 2 f ( p) − E − p Θ1 E + p − f ( p) , T    0 ≤ 2 f ( ph ) − E − ph Θ2 E + ph − f ( ph ) , $ % %T $ 0 ≤ 2 f ( p) − f ( ph ) − E − [ p − ph ] Θ3 E + [ p − ph ] − f ( p) + f ( ph ) . It is obvious that if (16) holds, LV ( p, t) < 0. That is, there exists a constant ι > 0, such that LV ( p, t) < −ιV ( p, t).

(10)

Pre- and post-multiplying inequity (10) by e−ιt , and using Dynkin’s formula, it follows that E {V ( p, t)} < eιt E {V ( p, 0)}    ≤ eιt λmax ((2N + 1)I ) max λmax PσN σ∈S

+ λmax

! q 

" Λk Ψ

k=1

  T + h 2 λmax I + (E + ) E + λmax T

  T + h 1 λmax I + (E + ) E + λmax T

  h 31 λmax (Q) + h 21 λmax I + (E + )T E + λmax T

2 # h3 h3 h3 + 21 λmax (R) + 1 λmax (F1 ) + 21 λmax (F2 ) 2 6 6       ##   T  T ∂p T ∂p dυ × sup p p dυ, p˙ p˙ dυ, sup ∂υk ∂υk −h 2 ≤t≤0 1≤k≤q Ω Ω Ω  ∗ ≤ eιT ς1 dυ, +

Ω









∂p ∂υk

T

where 1 = sup p p , p˙ p˙ , sup −h ≤t≤0 1≤k≤q  2 N $ % $ % N p , then Since min λmin P1σ E p 2 ≤ E p T P1σ T

T

∂p ∂υk

 # .

σ∈S

$ % E p 2
Yδ |m δ (t)| > Yδ

and and and and

|sδ (t)| ≤ Yδ , |sδ (t)| > Yδ , |sδ (t)| ≤ Yδ , |sδ (t)| > Yδ .

For the above systems, the following corollary can be obtained: Corollary 1 For given scalars ˜ 1 > 0, ι˜ > 0, and T˜ ∗ > 0, and an integer N˜ ≥ 0, the synchronization error system (14) is bounded over an FTI if there exist a posiN ˜N N , P2σ , . . . , P˜(N } > 0, Ψ˜ > 0, tive scalar ˜ 2 ; symmetric matrices P˜σN = diag{ P˜1σ   



 



 +1)σ T˜11 T˜12 T˜11 T˜12 T˜11 T˜12 ˜ ˜ > 0, T˜

= > 0, T˜

= T˜ =





> 0, Q > 0, R > 0, ˜ ˜ ∗ T22 ∗ T22 ∗ T˜22 ˜ˆ and diagonal ˆ and M; F˜1 > 0, F˜2 > 0; arbitrary matrices Υ˜ j ( j = 1, 2, 3, 4, 5), M, matrices Θ˜ m (m = 1, 2, 3), such that the following LMIs hold: ⎤ Σ˜ h 1λ¯ 1T Mˆ h 21λ¯ 2T M˜ˆ ⎥ ⎢ ⎦ < 0, ⎣ ∗ −Q˜ N˜ 0 ˜ ∗ ∗ −R N˜ ⎡

 ˜∗ N ˜ ι˜T < min λmin P˜1σ ˜ 1 ςe ˜ 2 , σ∈S

where Σ˜ = Σ˜ 1 + Σ˜ 2 + Σ˜ 3 + Σ˜ 4 + Σ˜ 5 , ˜ 3 Q, ˜ . . . , (2 N˜ + 1) Q}, ˜ Q˜ N˜ = diag{ Q, ˜ ˜ = diag{ R, ˜ 3 R, ˜ . . . , (2 N˜ + 1) R}, ˜ R N



 h 3

 ς˜ = λmax (2 N˜ + 1)I max λmax P˜σN + 1 λmax Q˜ σ∈S 2



 h3

 T + h 2 λmax I + (E + ) E + λmax T˜ + 21 λmax R˜ 2



 h3

 T + h 1 λmax I + (E + ) E + λmax T˜

+ 21 λmax F˜2 6

 h3

  + h 21 λmax I + (E + )T E + λmax T˜

+ 1 λmax F˜1 , 6

(16)

(17)

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with Σ˜ 1 =Sym{XT P˜σN Y} + XT



 πσμ P˜μN X − 2†2T Ψ †2 − †1T 2Ψ˜ A + 2Ψ˜ K  †1

μ∈S

+ +

Sym{†1T Ψ˜ C˜ †6 } − Sym{†1T (Ψ˜ + AT Ψ˜ T + K T Ψ˜ T )†2 } Sym{†1T Ψ˜ D˜  †8 } + Sym{†2T Ψ˜ C˜ †6 } + Sym{†2T Ψ˜ D˜  †8 },







+ T˜11 )†1 + Sym{†1T (T˜12 + T˜12 )†6 } Σ˜ 2 =†1T (T˜11   T ˜

T ˜

− (1 − h) †4 T11 †4 + †8 T22 †8 + Sym{†4 T T˜12 †8 }











+ †6T (T˜22 + T˜22 )†6 − †5T T˜11 †5 − †3T (T˜11 − T˜11 )†3 T ˜



T





− Sym{†3 (T12 − T˜12 )†7 } − †7 (T˜22 − T˜22 )†7





− †5T T˜12 †9 − †9T T˜22 †9 ,   T T T Σ˜ 3 =h 1λ¯ 1 X˜ N˜ Y˜ N˜ Mˆ + Mˆ T Y˜ N˜ X˜ N˜ λ¯ 1 + †2T (h 21 Q˜   T ˜ T 2 ˜ T ˜ T ˜ˆ ˜ ˜ ˆ ˜ + h 21 R)†2 + h 21λ¯ 2 X N˜ Y N˜ M + M Y N˜ X N˜ λ¯ 2 ,

Σ4 =†2T (

h 21 ˜ h 2 − h 21 ˜ 2 T ˜ 2 F1 †10 + Sym{†1T F˜1 †10 } F1 + 2 F2 )†2 − 2†1T F˜1 †1 − 2 †10 2 2 h1 h1

2h 221 T ˜ 2h 21 T ˜ 2 † F† + 2 † F† − 2 †T F˜2 †11 , 2 1 2 1 2 1 2 11 − h1 h2 − h1 h 2 − h 21 11   Σ5 =Sym{ †6T − E − †1T Θ˜ 1 E + †1 − †6 }   + Sym{ †8T − E − †4T Θ˜ 2 E + †4 − †8 }      + Sym{ †6T − †8T − E − †1T − †4T Θ˜ 3 E + (†1 − †4 ) − †6 + †8 }

 T ˜ + Sym{ †1T Υ˜1 + †2T Υ˜2 + †6T Υ˜3 + †8T Υ˜4 + †12 Υ5 (†3 − †5 − †12 )}. −

h 22

4 Numerical Examples In order to prove the validity and practicability of the main results obtained in this chapter, this section provides two simulation examples. First, for simplicity, we simplified the original systems’ models, that is, we set n = 1, 2, σ = 1, 2,  = 1, 2 and q = 1, then the drive and response systems can be described as

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0

1

2

3

4

5

6

4

5

6

t 3 2 1 0

0

1

2

3

t Fig. 1 The Markov stochastic processes of the drive and response systems

∂m δ  ∂ = ∂t ∂υk k=1 +

2 

  2  ∂m δ xδk − aδ m δ + cδlσ (m δ )gl (m l ) ∂υk l=1

dδlσ (m δ )gl (m lh ),

(18)

l=1

and ∂sδ  ∂ = ∂t ∂υk k=1 +

2 

  2  ∂sδ xδk − a δ sδ + cδl (sδ )gl (sl ) ∂υk l=1

dδl (sδ )gl (slh ) + u δ .

(19)

l=1

For drive system (18) and response system (19), by choosing π11 = −0.5, π12 = 0.5, π21 = 0.8 and π22 = −0.8, 11 = 0.8, 12 = 0.2, 21 = 0.4, 22 = 0.6, the Markovian jumping laws of system (18) and system (19) can be shown as in Fig. 1. Example 1 We set x11 = x21 = 0.01, a1 = 10.9, a2 = 11.2, Y1 = 0.2 and Y2 = 0.15, and the other parameters depend on the systems’ modes:

Finite-Time Synchronization Control for Markovian Jump Memristive … 







517 



Mode 1: c111 = 0.2, d111 = −0.4, c111 = 0.3, d111 = −0.5, c121 = 0.15, d121 =       −0.2, c121 = 0.1, d121 = −0.16, c211 = −0.4, d211 = 0.6, c211 = −0.2, d211 = 0.1,     c221 = 0.5, d221 = −0.3, c221 = 0.4, d221 = −0.2. 









Mode 2: c112 = −0.06, d112 = −0.5, c112 = −0.05, d112 = −0.6, c122 = 0.05,       = −0.1, c122 = 0.1, d122 = −0.15, c212 = −0.4, d212 = 0.5, c212 = −0.5, d212     = 0.6, c222 = 0.2, d222 = −0.5, c222 = 0.15, d222 = −0.4. Moreover,  d122

m 1 (υ, o) = −1.2sin(υ), (υ, o) ∈ Ω × [−h 2 , 0], m 2 (υ, o) = 1.4sin(υ), (υ, o) ∈ Ω × [−h 2 , 0], s1 (υ, o) = −2.1sin(υ), (υ, o) ∈ Ω × [−h 2 , 0], s2 (υ, o) = 1.3sin(υ), (υ, o) ∈ Ω × [−h 2 , 0]. t

2e With considering h(t) = 15+5e t and u = 0, the trajectories of the error system can be described with Fig. 2. It is obvious that the error of drive system (18) and response system (19) is unbounded, that is, the drive and response systems cannot achieve synchronization without controller.

Fig. 2 The dynamic behaviors of the error system without controller

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Fig. 3 The dynamic behaviors of the error system with the controller (6)

Then, we design the controller in (19) as (6), by solving the inequalities proposed in Theorem 1, we can obtain the following controller gains: 

   6.1195 −7.5517 6.0916 −6.9876 K1= , K2= . 7.5164 6.4396 7.6044 7.0124 Furthermore, with these controller gains, the dynamic behaviors of the error system with the controller (6) are shown in Fig. 3, from which we can conclude that the error system of (18) and (19) is bounded, that is, the drive and response systems can achieve synchronization with the controller (6). Additionally, we can obtain that ς = 16.71 and 1 = 0.82 by some calculations, then, set ι = 0.1, T ∗ = 4, from the inequation (17), there exists 2 ≥ 3.1485, so that the systems (18) and (19) are synchronous over an FTI. Example 2 Inspired by the applications of chaotic systems in Pseudo Random Number Generators (PRNG) proposed in [31–33], this example aims to apply our system to the field of PRNG. Note that Markovian reaction-diffusion MNNs are ideal tools for generating chaos because of their internal complexity. Therefore, applying Markovian reaction-diffusion MNNs to Pseudo Random Number Generators (PRNG) is reasonable.

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Fig. 4 The chaotic behaviors of the drive system (18)

This example realizes the encryption and decryption functions by complex dynamics of Markovian reaction-diffusion MNNs proposed in this chapter. The following parameters are chosen for system (18): a1 = 6.6, a2 = 6.5, Y1 = 1.6, Y2 = 0.5. 









When M O D E = 1: c111 = 2, d111 = −1.4, c121 = 20.1, d121 = 0.7, c211 = 0.1,        d211 = 0.1, c221 = 2.05, d221 = −1.3, c111 = 2.1, d111 = −1.6, c121 = 20.5, d121 =     0.9, c211 = 0.1, d211 = 0.1, c221 = 2.2, d221 = −1.1. 









When M O D E = 2: c112 = 2.1, d112 = −1.5, c122 = 21.1, d122 = 0.6, c212 =       0.2, d212 = 0.14, c222 = 2.15, d222 = −1.2, c112 = 2.0, d112 = −1.5, c122 = 20.5,      d122 = 1, c212 = 0.1, d212 = 0.11, c222 = 2.21, d222 = −1.3. And the other parameters are set the same as those in Example 1. With these parameters, the state trajectories of the system (18) is shown in Fig. 4, where a typical spatiotemporal chaotic attractor can be found. Now, we define a pseudo random number sequence in t ∈ [tstar t , tend ]:  x(t) = ω(p(t), q(t)) =

1, p(t) ≤ q(t), 0, p(t) > q(t),

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x(t)

0.6 0.4 0.2 0

0

10

20

30

40

50

60

70

80

t

Fig. 5 PRNG produced by reaction-diffusion IMNNs (18)

with mˆ 1 (t) , q(t) = max {mˆ 1 (t)}

p(t) =

t∈[tstar t ,tend ]

mˆ 2 (t) , max {mˆ 2 (t)}

t∈[tstar t ,tend ]

& & and mˆ 1 (t) = Ω m 1 (υ, t)dυ, mˆ 2 (t) = Ω m 2 (υ, t)dυ. The PRNG can be produced by above function, which is described with Fig. 5. It can be seen from Fig. 5 that the sequence generated by Markovian reaction-diffusion MNNs (18) is random. 1 0.8

o(t)

0.6 0.4 0.2 0

0

10

20

30

40

t

Fig. 6 Original signals o(t)

50

60

70

80

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1 0.8

e(t)

0.6 0.4 0.2 0

0

10

20

30

40

50

60

70

80

50

60

70

80

t

Fig. 7 Encrypted signal generated by o(t) and x(t) 1 0.8

o'(t)

0.6 0.4 0.2 0

0

10

20

30

40

t

Fig. 8 Decrypted signal generated by e(t) and x(t)

Choose o(t) as the original signal that is represented by Fig. 6, and we define e(t) = o(t) ⊕ x(t) as the encrypted signal. Then, the trajectory of e(t) can be described with Fig. 7. It is obvious that the encrypted signal and original signal are quite diverse by comparing Figs. 6 and 7, namely, the encryption algorithm we designed is very efficient. The decryption algorithm is the inverse of encryption algorithm, thus, o (t) = e(t) ⊕ x(t). The combination of PRNG in Fig. 5 and the encrypted signal in Fig. 7 can generates the decrypted signal, which can be represented by Fig. 8. By the comparison of Figs. 6 and 8, it can be seen that the decrypted signal and original signals are identical, which means that the signal is successfully decrypted.

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5 Conclusion This chapter aims to solve the problem of synchronization for Markovian reactiondiffusion MNNs over an FTI. In the process of modeling, the drive and response systems are set to obey inconsistent Markovian jumping laws, which is more practical. However, the challenge of handling the inconsistent parameters in the error system comes with it. To deal with this problem, a discontinuous controller is designed, so that the influence of the inconsistent parameters can be avoided. Moreover, by combining the novel LKF with the canonical BLI and FWM methods, a less conservative result on synchronization for reaction-diffusion MNNs over an FTI is obtained. Finally, a numerical example and an application study are provided to support the main results. Acknowledgements This work supported by the National Natural Science Foundation of China (Nos. 61976081, U1604146), and Foundation for the University Technological Innovative Talents of Henan Province (No. 18HASTIT019).

References 1. Park, J.H., Kwon, O.M., Lee, S.M.: LMI optimization approach on stability for delayed neural networks of neutral-type. Appl. Math. Comput. 196(1), 236–244 (2008) 2. Park, J.H.: Robust stability of bidirectional associative memory neural networks with time delays. Phys. Lett. A 349(6), 494–499 (2006) 3. Bao, H., Park, J.H., Cao, J.: Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw. 81, 16–28 (2016) 4. Li, D., Liu, Y., Tong, S., et al.: Neural networks-based adaptive control for nonlinear state constrained systems with input delay. IEEE Trans. Cybern. 49(4), 1249–1258 (2018) 5. Ding, S., Wang, Z., Zhang, H.: Event-triggered stabilization of neural networks with timevarying switching gains and input saturation. IEEE Trans. Neural Netw. Learn. Syst. 29(10), 5045–5056 (2018) 6. Xiao, Q., Huang, T., Zeng, Z.: Stabilization of nonautonomous recurrent neural networks with bounded and unbounded delays on time scales. IEEE Trans. Cybern. (2019). https://doi.org/ 10.1109/TCYB.2019.2922207 7. Bao, H., Park, J.H., Cao, J.: Non-fragile state estimation for fractional-order delayed memristive BAM neural networks. Neural Netw. 119, 190–199 (2019) 8. Mathiyalagan, K., Anbuvithya, R., Sakthivel, R., et al.: Non-fragile H∞ synchronization of memristor-based neural networks using passivity theory. Neural Netw. 74, 85–100 (2016) 9. Wang, X., Park, J.H., Zhong, S., et al.: A switched operation approach to sampled-data control ttabilization of fuzzy memristive neural networks with time-varying delay. IEEE Trans. Neural Netw. Learn. Syst. (2019). https://doi.org/10.1109/TNNLS.2019.2910574 10. Fan, Y., Huang, X., Shen, H., et al.: Switching event-triggered control for global stabilization of delayed memristive neural networks: an exponential attenuation scheme. Neural Netw. 117, 216–224 (2019) 11. Li, X., Zhang, W., Fang, J., et al.: Finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays. Neurocomputing 340, 99–109 (2019) 12. Wang, S., Guo, Z., Wen, S., et al.: Finite/fixed-time synchronization of delayed memristive reaction-diffusion neural networks. Neurocomputing 375, 1–8 (2020)

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Index

A Activation function, 343, 423 Actuator fault, 88, 209 Adjacency matrix, 63 Artificial neural networks, 412 Asymptotic stability, 17 Asynchronization, 173 Asynchronous impulses and switchings, 437 Asynchronous switching, 194 Average dwell time, 174, 194, 439 Average impulsive interval, 439 B BAM neural networks, 340 Bernoulli distribution, 87 Bernoulli variable, 461 Betweenness-centrality, 63 Bidirectional associative memory, 364 B–L inequality, 510 Boundary condition, 147 Brouwer’s fixed point theorem, 341, 370, 371 Brownian motion, 416, 481 C Caputo fractional-order derivative, 315 Caputo’s derivative, 277, 293 Chaos, 518 Chaotic system, 304 Chua’s circuit, 470 Cluster synchronization, 490 Communication network, 222 Complex networks, 457, 458, 478 Copositive Lyapunov function, 41, 436

Coupled reaction-diffusion systems, 314 Cyber attacks, 458

D Decay rate function, 343 Decentralized controller, 104 Descriptor systems, 124 Detectability, 13 Detectable, 4, 136 Dini derivative, 377 Dirac impulsive function, 482 Distributed parameter system, 158 Dwell time, 174 Dynkin’s formula, 512

E Eigenvalue, 9 Eigenvector-centrality, 62 Equilibrium point, 368 Event-triggered communication, 222 Event-triggered scheme, 224, 460 Event-triggered transmission, 175 Exponential stability, 149

F Fault alarm, 202 Fault-alarm approach, 214 Fault detection, 148, 387 Fault dectetion isolation, 388 Fault estimation, 148 Fault-tolerant control, 84, 124, 134 Filippov solution, 295 Finite-time controller, 173

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. H. Park (ed.), Recent Advances in Control Problems of Dynamical Systems and Networks, Studies in Systems, Decision and Control 301, https://doi.org/10.1007/978-3-030-49123-9

525

526 Finite-time reachability, 223, 233 Finite-time stability, 38, 314 Finite-time stable, 324 Finsler’s lemma, 70, 397 Fractional calculus, 268 Fractional-order derivative, 301 Fractional-order differential equations, 268 Fractional-order impulsive coupled systems, 313 Fractional-order neural networks, 292 Fractional-order neutral-type systems, 267 Functional detectable, 14 Functional observability/detectability, 7 Functional observers, 4, 7, 11

G Galerkin method, 149  functions, 269 Generalized Gronwall’s inequality, 318 Genetic algorithm, 23 Gronwall–Bellman inequality, 367 Guaranteed cost, 279

H Halanay inequality, 436 H∞ attenuation, 68 H∞ attenuation performance, 111 H∞ control, 268 H∞ fault estimation, 149 Homeomorphism, 346, 350 Hopfield neural networks, 340 H∞ performance, 112, 154 H∞ performance index, 206 Hurwitz, 6, 234 Hybrid control, 214 Hybrid systems, 221 Hybrid triggered scheme, 458 Hyperchaotic Rossler system, 96

I Impulse signal, 478 Impulsive effects, 38 Impulsive pinning controllers, 482 Impulsive positive systems, 38 Inertial neural networks, 340, 351, 411 Infinite observability, 124 Input delay, 10 Integral sliding mode control, 104 Interconnected system, 104 Invariant zero, 8, 15 Inverse average impulsive interval, 443

Index Itô formula, 419, 441 J Jensen inequality, 16, 28, 69, 111, 250, 414 K Kalman-Yakubovich-Popov lemma, 388 Kamke function, 315 Kronecker product, 4, 242, 462 L Laplacian matrix, 63, 79, 317 Linear matrix inequality, 22, 124, 435 Lipschitz condition, 86, 316, 367, 485 Lipschitz continuity, 423 Low pass filters, 150 Luenberger observer, 3, 33 Lur’e systems, 479 Lyapunov function, 129, 272, 301, 319 Lyapunov functional, 486 Lyapunov–Krasovskii functional, 19, 91 Lyapunov-Krasovskii functional, 206 Lyapunov stability, 38 M Markov chain, 221, 413, 500 Markov jump systems, 83, 221, 223 Markov process, 83, 221 Markovian jump, 413 Markovian systems, 501 Master-slave systems, 294 Matching condition, 30, 127 Mean square exponential stability, 437 Mean square stable, 93 Memristive neural networks, 500 Memristor, 292, 500 Memristor neural networks, 291 Metzler, 40, 48 Metzler matrix, 340, 346 Metzler–Hurwitz matrix, 346 Mittag-Leffler function, 316, 318 Mittag-Leffler stability, 268, 314 M-matrix, 341 Mode-dependent Lyapunov function, 233 Multi-agent systems, 62 N Networked systems, 387, 388 Neural networks, 337, 500 Neutral system, 268

Index O Observability, 124 Observability/detectability, 7 Observer, 221, 268

P Packet loss, 172, 183, 389 Partial differential equations, 147 Periodic piecewise control, 204 Periodic piecewise systems, 202, 203 Periodic systems, 201 Perron–Frobenius theorem, 38, 72 Piece-wise Lyapunov function, 177 Poincare-Wirtinger inequality, 153 Popov-Hautus-Rosenbrock test, 136 Positive equilibrium, 343 Positive systems, 38, 338 Positivity, 343 Probability density function, 84 Pseudo random number generators, 518 Python, 8

Q QR decomposition, 135

R Random occurring uncertainty, 84 Random time-varying delays, 85 Reachability, 115 Reaction-diffusion, 500 Reliable control, 202 Riemann–Liouville, 268 Riemann–Liouville fractional derivative and integral, 269 Robust controller, 206 Rosenbrock matrix, 8

S Sampled-data control, 87, 172

527 Schur complement, 95, 178, 194, 209, 420 Semi-Markov jump chaotic systems, 84 Sliding mode control, 114, 222 Sliding mode observer, 124 Sliding mode scheme, 24 Sliding surface, 107, 221 Sliding switching surface, 226 Sojourn time, 86 Spatiotemporal chaotic attractor, 519 Spectral radius, 340 S-procedure, 209 Stochastic jumping systems, 83 Stochastic neural networks, 437 Stochastic stability, 227 Substitute consensus protocol, 61 Switched system , 388 Sylvester equation, 6, 12, 17 Synchronization, 86, 291, 314, 463, 478

T Takagi-Sugeno fuzzy systems, 436 Time delay, 10, 62, 268, 292, 461 Time-triggered scheme, 460 Transmission delays, 185 T-S fuzzy-based model, 104 T-S fuzzy systems, 114

U Uniformly stable, 317

W Weak infinitesimal operator, 91 Weibull distribution, 96 Wiener process, 437 Wirtinger-based inequalities, 111 Wirtinger’s inequality, 414

Z Zero-order holder, 175