Emerging Trends in Sliding Mode Control: Theory and Application (Studies in Systems, Decision and Control, 318) 9811586128, 9789811586125

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Studies in Systems, Decision and Control 318

Axaykumar Mehta Bijnan Bandyopadhyay   Editors

Emerging Trends in Sliding Mode Control Theory and Application

Studies in Systems, Decision and Control Volume 318

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. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

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Axaykumar Mehta Bijnan Bandyopadhyay •

Editors

Emerging Trends in Sliding Mode Control Theory and Application

123

Editors Axaykumar Mehta Department of Electrical Engineering Institute of Infrastructure, Technology, Research and Management Ahmedabad, Gujarat, India

Bijnan Bandyopadhyay Interdisciplinary Programme in Systems and Control Engineering Indian Institute of Technology Bombay Mumbai, Maharashtra, India

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-15-8612-5 ISBN 978-981-15-8613-2 (eBook) https://doi.org/10.1007/978-981-15-8613-2 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to My father Late Jayantilal Mehta by Axaykumar Mehta and Dedicated to My eldest brother Late Rathindranath Banerjee by Bijnan Bandyopadhyay

Preface

A variable structure system is a novel control methodology introduced for the stabilization of a dynamical system in which the system changes its structure through an appropriate switching feedback law that achieves the desired stability for the closed-loop system. In the late 50s, in the erstwhile Soviet Union, a new control strategy called the Sliding Mode Control (SMC) with switching structure was developed which has an additional feature of achieving robust stability. This has attracted many researchers to contribute to the development of SMC over the last half-century. There have been milestone developments in the field ranging from its theoretical foundation to practical applications. A new area of research, higher order SMC, was proposed in the late 80s using the higher derivatives of sliding variable to bring the sliding motion within a finite time. In the last two decades, quite significant results have been reported in both control and observation problems using higher order SMC. Particularly, the output-feedback-based control of uncertain plants has been studied using a differentiator because of its practical significance. Again, Lyapunov-based design of this algorithm via homogeneity property of the system presents yet another dimension of development in this area. There is another important area of research on SMC that investigates the implementation issues of the control algorithm. The discrete implementation of the analog-based design of SMC in a sampled-data system shows that the motion of trajectory is bounded around the sliding hyperplane rather than exactly sliding on it. While the direct discrete SMC design has focused on the problem of achieving a steady-state chattering bound via different (discrete) reaching laws, recently, a new implementation of the SMC technique was presented using the event-triggering strategy. The control law is applied when an event is generated by the sampling mechanism. The usefulness of this technique is that any given steady-state bound can be obtained for the closed-loop system. Apart from the theoretical development, the application of SMC has also witnessed promising results on many practical systems. It includes examples from the mechanical systems, robotics, multi-agent systems, electric drive system, etc. This edited collection is an attempt to present the state of the art of SMC in these areas. vii

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The objective of this book is to present a collection of the latest significant developments in the theory and engineering applications of SMC and stimulate further research in this field. It compiles the works on recent developments on SMC theory and its applications. A total of 16 chapters including both theoretical contributions and practical applications of SMC are presented. We divide the contributions into four parts: Chapters Homogeneous Sliding Modes in Noisy Environments–Sliding Mode Control based Tracking of Non-Differentiable Reference Functions contain the theoretical development of higher order sliding modes, Chapters State Boundedness in Discrete-Time Sliding Mode Control– Design of Periodic Event-Triggered Sliding Mode Control of this book focus on the discrete implementation of SMC, Chapters Sliding Modes in Consensus Control– Discrete Higher-Order Sliding Mode Leader-Following Consensus Protocols for Homogeneous Discrete Multi-Agent System discuss the application of sliding mode algorithms to multi-agent systems, and, finally, Chapters State and Disturbance Estimation Using Fast Output Sampling Approach for Robust Motion Control Systems–Industry-Grade Robust Controller Design for Constant Voltage Arc Welding Process collect the application of SMC to different practical systems which include the accurate position regulation of an electro-hydraulic actuator, ripple regulation of single-stage inverters, direct power control of grid-connected doubly fed induction generator, stabilization of underactuated mechanical systems, and control of arc welding process. The contributions of chapters that contain the theoretical development are detailed below. In chapter “Homogeneous Sliding Modes in Noisy Environments”, Avi Hanan, Adam Jbara, and Arie Levant have presented homogeneous sliding modes in noisy environments. The chapter presents controller design methods which easily produce infinitely many new homogeneous SM controllers for any relative degree. Each such controller can be equipped with a homogeneous or bihomogeneous filtering differentiator. The filtering differentiators not only preserve the exactness, robustness, and asymptotic optimality of the traditional SM-based differentiators, but also readily reject wide classes of extremely large noises. The noise-rejection properties are also inherited by the corresponding output-feedback SMC. Numerous examples demonstrate numeric differentiation, development of new SM controllers, and their output-feedback application in noisy environments. Further, the numeric comparison with Kalman filters is also performed. In chapter “A Lyapunov based Saturated Super-Twisting Algorithm”, Ismael Castillo, Martin Steinberger, Leonid Fridman, Jaime A. Moreno, and Martin Horn have proposed two different structures of saturated super-twisting algorithms combining a relay controller and a standard super-twisting algorithm. Both structures switch between a relay controller and super-twisting algorithm through a switching law that is based on Lyapunov-level curves allowing the algorithms to generate bounded control signals. The relay controller enforces the system trajectories to a neighborhood of the origin in which the super-twisting algorithm dynamics ensure finite-time convergence to the origin without saturation. In order to increase the maximal admissible bound of the perturbations, the second

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algorithm also includes a perturbation estimator setting super-twisting's integrator to the theoretically exact perturbation estimation. Finally, the experimental results are presented to validate the proposed algorithms. In chapter “Sliding Mode Control based Tracking of Non-Differentiable Reference Functions”, Shyam Kamal and Rahul Kumar Sharma have explored the class of tracking problems in nonlinear systems with non-differentiable reference signals. The classical reference tracking control techniques require the first-order derivative of the reference signals to exist and be bounded. However, the desired reference signals may not satisfy such conditions in various applications. Therefore, the conventional approach significantly restricts the allowable class of reference signals to be tracked. So, there is a need to revisit the tracking problem with a sound mathematical framework. In this context, a new approach based on fractional-order operators is discussed. The control scheme works provided the reference function satisfies the H¨older condition. Therefore, a larger class of reference signals can be addressed using the technique. A switch-controlled RL circuit is considered for illustration of the approach, and some of the possible applications have been discussed. The following chapters emphasize on the implementation aspect of SMC in discrete time. The discrete-time sliding mode control strategies are well known to provide remarkable robustness of the controlled plant with respect to disturbance and model uncertainties. However, even though such strategies confine the system state to a certain vicinity of the sliding hyperplane defined in the state space, they do not provide explicit information about the bounds of each individual state variable. Since state constraints are important in many practical applications of sliding mode control, Paweł Latosiński and Andrzej Bartoszewicz have developed explicit formulae that describe the absolute bounds of all state variables in the sliding mode in chapter “State Boundedness in Discrete-Time Sliding Mode Control”. These formulae are universal and remain valid for any discrete-time sliding mode control strategy with known quasi-sliding mode bandwidth. Additionally, they have considered the case of sliding variables with relative degree higher than one, as well as several common special cases that allow one to reduce the obtained bounds. Further, in implementation of discrete-time sliding mode controller, the states of the system are not usually accessible, or there exists unavoidable noise in the measured states. These problems may affect the performance of the systems. Therefore, utilizing functional observers can resolve such issues. In chapter “Discrete Stochastic Sliding Mode with Functional Observation”, Satnesh Singh and S. Janardhanan have presented a functional-observer-based Sliding Mode Control (SMC) for discrete-time stochastic systems, both uncertainty-free systems and systems with uncertainty. The existence conditions and stability analysis of the SMC functional observer are also given. The functional observer is designed in such a way that the effect of process and measurement noise is minimized. Furthermore, the functional observer-based SMC design with unmatched uncertainty is also considered in which a disturbance-dependent sliding function method is discussed such that the effect of unmatched uncertainty of the system is

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minimized. Finally, a simulation example is given to show the effectiveness of the method. In chapter “Design of Event-Triggered Integral Sliding Mode Controller for Systems with Matched and Unmatched Uncertainty”, Asifa Yesmin and Manas Kumar Bera have proposed the design of the aperiodic sampled-data implementation of Integral Sliding Mode Controller (ISMC) for a linear continuous-time system. In this chapter, the asymptotic stability of the system in the presence of matched and unmatched uncertainty is guaranteed with the chosen optimal value of the projection matrix and controller gain. A unique triggering strategy based on the defined measurement is proposed which also ensures the Zeno-free behavior of the system. This methodology can reduce the number of control computation in comparison to the periodic implementation of ISMC. Finally, the theoretical analyses and numerical simulations have been presented to ensure the effectiveness of the control strategy. Another contribution by Abhisek K. Behera and Bijnan Bandyopadhyay presents a Design of Periodic Event-Triggered Sliding Mode Control in chapter “Design of Periodic Event-Triggered Sliding Mode Control”. In this chapter, a new event-triggering strategy for the design of SMC is proposed. Here, the event condition is verified at regular intervals to determine the time instant for applying the control input. The main advantage of this technique is that the computational burden is reduced due to the relaxation of continuous evaluation of the event condition. It is shown that the desired stability can be achieved by the periodic event-triggered SMC like in the continuous event-triggering case. The following chapters deal with the control problem of multi-agent systems. In chapter “Sliding Modes in Consensus Control”, Massimo Zambelli and Antonella Ferrara have proposed an overview about the possibility of adopting sliding mode control in consensus control problems. In particular, after a general introduction on consensus control and its major characteristics, the theoretical concepts at the basis of consensus control are first introduced to form a common basis useful to understand the works presented throughout the entire discussion. A number of results in algebraic graph theory are reported and briefly discussed to enable an effective description of the interconnections existent between agents in multi-agent systems. Then, the problem of enforcing leaderless and leader–follower consensus is formally stated. A review of some results currently available in the literature which relies on sliding mode control to solve both leaderless and leader– follower consensus is provided, highlighting the possibilities offered by this strategy, with particular emphasis on the robustness and finite-time convergence features. In chapter “On Fixed-Time Convergent Sliding Mode Control Design and Applications”, Jyoti Mishra and Xinghuo Yu have focused on developing algorithms to provide even faster convergence speed than that of finite-time convergent algorithms. Some practical applications need strict constraints on time response due to security reasons or to ameliorate the productiveness. It is worth mentioning that the state convergence achieved in SMC during sliding can be either asymptotic or in finite time, depending on the selection of the surface. Furthermore, it primarily depends on the initial conditions of the states. This provided a motivation to focus

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on developing SMC controllers where the convergence time does not depend on initial conditions, and a well-defined theoretical analysis is provided in the chapter regarding arbitrary-order fixed-time convergent SMC design. It is evident that a higher control effort will be required by achieving so and is discussed thoroughly in the chapter, leading to a trade-off between the control effort and convergence speed. To this end, a novel distributed algorithm is developed for achieving second-order consensus in the multi-agent systems by designing a full-order fixed-time convergent sliding surface as an application to the proposed algorithm. Then, in chapter “Discrete Higher-Order Sliding Mode Leader-Following Consensus Protocols for Homogeneous Discrete Multi-Agent System”, the authors Keyurkumar Patel and Axaykumar Mehta have proposed discrete higher order sliding mode leader-following consensus protocols, namely, with reaching law approach and discrete super-twisting algorithm for the leader-following discrete homogeneous multi-agent system to achieve the global consensus in a finite time. The higher order sliding mode control facilitates to reduce the switching control chattering band, known as quasi-sliding mode band which ultimately increases the robustness property. The chapter also presents the derivation of the ultimate quasi-sliding mode band that can be achieved for both the proposed protocols. Both protocols ensure the anti-disturbance and robustness consensus performance. To compare the consensus performance and robustness properties, both the proposed protocols are implemented on a discrete homogeneous multi-agent system comprising of 2-DOF serial flexible robotic arms. From the simulation and experimental results, it is inferred that the discrete higher order protocol due to the reaching law approach outperforms the protocol using the discrete super-twisting algorithm. In the end, the last six chapters that present the different applications of sliding mode algorithms are detailed below. As mentioned earlier, the disturbance is inherent in all systems, and it affects the performance of the system. To get robust performance, compensation of disturbance is required. However, the disturbance is not known. The authors Surajkumar Sawai and Shailaja Kurode in chapter “State and Disturbance Estimation Using Fast Output Sampling Approach for Robust Motion Control Systems” deal with investigation of state and disturbance estimation using fast output samples. For that the disturbance is considered as an additional state and an extended state system is formulated. Estimation of states of an extended system is proposed using output samples taken at faster rate. These state estimates are used for devising control which is augmented with disturbance estimate to compensate its effect. Effectiveness of this method is verified in simulations while developing motion control system. Friction, backlash neglected gear characteristics, etc. act like a lumped disturbance and affect the performance of motion system. It is shown that the proposed method effectively compensates these effects and yields robust performance. The experimental validation strengthens the proposal. The authors of chapter “Accurate Position Regulation of an Electro-Hydraulic Actuator via Uncertainty Compensation-Based Controller”, Ramón I. Verdés, Alejandra Ferreira de Loza, Luis T. Aguilar, Ismael Castillo, and Leonid Freidovich, have presented robust sliding mode controller for the accurate position

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regulation of an electro-hydraulic actuator. The electro-hydraulic actuators are complex systems that face up with parameter uncertainties and neglected dynamics, which may lead to a miscalculation of the controller gains. In hydraulic systems, the overestimation of controller gains spoils the control performance, whereas the controller gain’s underestimation impacts on the error accuracy. Motivated by the previous issues, the authors have proposed a compensation-based controller to solve an electro-hydraulic actuator's position regulation problem. To this aim, a super-twisting observer identifies the uncertainties and disregarded dynamics. Later on, the identified signal is injected through the control input to counteract their effects. As a result, the position accuracy is improved. The closed-loop stability is carried out using Lyapunov theory. The advantage of the method is twofold: (1) the controller signal is continuous, i.e., without the tarnishing effect of chattering and (2) the accuracy of the position error is improved without increasing the controller gain. Simulations and experimental results in a forestry crane illustrate the effectiveness of the proposed method. In chapter “Control of Single Stage Inverters and Second-Order Ripple Regulation Using Sliding Mode Control”, Shivam Chaturvedi and Deepak Fulwani have developed sliding mode controller for single-stage Inverters and second-order ripple regulation. The Single-Stage Inverters (SSIs) are used to accomplish dc-ac conversion with buck-boost capability with increased reliability and efficiency compared to the traditional Voltage Source Inverters (VSIs). The SSIs also suffer from the Second-Order Ripple Currents (SRCs) as in the case of VSIs. These SRCs have detrimental effects such as reduced life span of the converter components, stress in energy sources, and storages. In this chapter, a sliding mode control-based SRC mitigation methodology is proposed so as to regulate the SRCs and hence improve the reliability of the system. The sliding surface consists of the voltage and current errors to generate the shoot-through control law. The inductor current is passed through a low-pass filter which has time constant less than the SRC frequency, to generate the shoot-through control law. The proposed control is robust and maintains the output ac voltage within the desired limits for both qZSIs and eqSBIs, while reducing the SRCs. Then, in chapter “Sliding Mode Direct Power Control of a Grid-Connected DFIG Using an Extended State Observer”, Ankit Shah and Axaykumar Mehta have proposed sliding mode direct power control of grid-connected DFIG using state and disturbance observer. This chapter presents Extended State Observer (ESO)-based Sliding Mode (SM)-Direct Power Control (DPC) scheme in order to get the independent control of stator active and reactive powers, considered as state variables, of a 5 kW grid-connected Doubly Fed Induction Generator (DFIG) run by variable-speed wind turbine. The stator active and reactive powers of the DFIG are controlled to optimize the extracted wind power and to follow the demand generated by transmission system operator, respectively. The ESO estimates the active power, reactive power, and lumped disturbance of the system. These estimated state variables which are free from disturbances are fed back for the design of the SM controller. This approach attenuates the amount of chattering or

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high-frequency variations in the control signal and state variables. The control algorithm is implemented in a reference frame referred to the stator of the DFIG in order to simplify the model and controller design. The dynamic performance of the proposed control scheme is compared with that of the SM-DPC scheme without ESO to evaluate the improvement in the performance by ESO. Robustness of the proposed control scheme is also shown by simulation results involving DFIG parameter uncertainties and unbalanced grid voltage. In chapter “Robust Stabilization of a Class of Underactuated Mechanical Systems of 2 DOF via Continuous Higher-Order Sliding-Modes”, the authors Diego Gutiérrez-Oribio, Ángel Mercado-Uribe, Jaime A. Moreno, and Leonid Fridman have proposed robust continuous higher order sliding mode controllers for stabilization of a class of underactuated mechanical systems of 2 DOF. In this chapter, the authors have presented a design of robust controllers for a class of underactuated mechanical systems of two DOF, using a continuous higher order sliding mode strategy. Two methods for controller design are presented. The first one generates a fifth-order sliding mode and achieves local finite-time stability. The other one is a robust controller providing global asymptotic stability. These controllers compensate matched Lipschitz disturbances and/or uncertainties, cope with an uncertain control coefficient, and generate a continuous control signal, possibly reducing the chattering effect. The authors provided evidence of the performance of the controllers using simulations for the Reaction Wheel Pendulum (RWP) and the translational oscillator with rotational actuator systems, and by means of experiments on the RWP system. Chapter “Industry-Grade Robust Controller Design for Constant Voltage Arc Welding Process” authored by Arun Kumar Paul, Manas Kumar Bera, Mangesh Waman, and Bijnan Bandyopadhyay presents industry-grade robust SOSM controller design for constant voltage arc welding process. Achieving robust arc stability of nonlinear arc is critically needed for creating good welding joints, particularly, when constant voltage arc types are used. They are popularly used in the industry. In this design, following the conventional documented procedures used in industry, the process is simplified where, using fast acting arc controllers, single input control for electric arc becomes effective. Here, the material feeding along with the movement of electrode tip is maintained at respective constant designed value, and it is independently controlled. This chapter proposes the practical design procedure of constant voltage arc welding controller using robust control concept. To ease the implementation process, a simplified second-order sliding mode control law is formulated. Moreover, to avoid any disturbance from metal feeding and tip movement, simplified SOSMC is used there as well. The ideas have been validated through wide range practical use. In conclusion, the main objective of this book is to present theoretical contributions as well as a broad range of well-worked out recent application studies in the field of SMC. The editors acknowledge the contribution of each author and reviewer to accomplish the objectives of the book. The editors would also like to thank the officials of Springer Nature Singapore, particularly to Aninda Bose, Ashok kumar, and Mannisaran Gandhi, for their cooperation and patience in

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bringing this volume. It is hoped that this book will be a reference guide for graduate students, researchers, professional engineers, and, in particular, to those working in the variable structure systems community. Ahmedabad, India Mumbai, India December 2020

Axaykumar Mehta Bijnan Bandyopadhyay

Contents

Homogeneous Sliding Modes in Noisy Environments . . . . . . . . . . . . . . . Avi Hanan, Adam Jbara, and Arie Levant

1

A Lyapunov based Saturated Super-Twisting Algorithm . . . . . . . . . . . . Ismael Castillo, Martin Steinberger, Leonid Fridman, Jaime A. Moreno, and Martin Horn

47

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shyam Kamal and Rahul Kumar Sharma State Boundedness in Discrete-Time Sliding Mode Control . . . . . . . . . . Paweł Latosiński and Andrzej Bartoszewicz

69 97

Discrete Stochastic Sliding Mode with Functional Observation . . . . . . . 119 Satnesh Singh and S. Janardhanan Design of Event-Triggered Integral Sliding Mode Controller for Systems with Matched and Unmatched Uncertainty . . . . . . . . . . . . . 145 Asifa Yesmin and Manas Kumar Bera Design of Periodic Event-Triggered Sliding Mode Control . . . . . . . . . . . 161 Abhisek K. Behera and Bijnan Bandyopadhyay Sliding Modes in Consensus Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Massimo Zambelli and Antonella Ferrara On Fixed-Time Convergent Sliding Mode Control Design and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Jyoti Mishra and Xinghuo Yu Discrete Higher-Order Sliding Mode Leader-Following Consensus Protocols for Homogeneous Discrete Multi-Agent System . . . . . . . . . . . 239 Keyurkumar Patel and Axaykumar Mehta

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State and Disturbance Estimation Using Fast Output Sampling Approach for Robust Motion Control Systems . . . . . . . . . . . . . . . . . . . . 265 Surajkumar Sawai and Shailaja Kurode Accurate Position Regulation of an Electro-Hydraulic Actuator via Uncertainty Compensation-Based Controller . . . . . . . . . . . . . . . . . . 279 Ramón I. Verdés, Alejandra Ferreira de Loza, Luis T. Aguilar, Ismael Castillo, and Leonid Freidovich Control of Single Stage Inverters and Second-Order Ripple Regulation Using Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . 305 Shivam Chaturvedi and Deepak Fulwani Sliding Mode Direct Power Control of a Grid-Connected DFIG Using an Extended State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Ankit Shah and Axaykumar Mehta Robust Stabilization of a Class of Underactuated Mechanical Systems of 2 DOF via Continuous Higher-Order Sliding-Modes . . . . . . 351 Diego Gutiérrez-Oribio, Ángel Mercado-Uribe, Jaime A. Moreno, and Leonid Fridman Industry-Grade Robust Controller Design for Constant Voltage Arc Welding Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Arun Kumar Paul, Manas Kumar Bera, Mangesh Waman, and Bijnan Bandyopadhyay

About the Editors

Axaykumar Mehta received his Bachelor of Engineering in Electrical Engineering (1996), M.Tech in Control Systems (2002) and Ph.D. (2009) degrees from Gujarat University, Ahmedabad, Indian Institute of Technology Kharagpur, and Indian Institute of Technology Bombay, respectively. He is currently associated as an Associate Professor in Electrical Engineering at the Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat. Prior to that, he was Director of Gujarat Power Engineering and Research Institute, Gujarat. He has more than 22 years of teaching experience at the undergraduate and graduate levels at various premier institutions of Gujarat. He has supervised 5 Ph.D. thesis in the domain of sliding mode control and published more than 80 research articles and book chapters in reputed journals, conference proceedings, and books. He has also authored 4 monographs on sliding mode control and edited 3 proceedings with Springer Nature Singapore. He has also published 5 patents at the Indian Patent Office Mumbai. He has successfully organized International Conference on Power Control and Communication 2019 supported by Springer India. He is also an advisor to Gujarat Council on Science and Technology, Government of Gujarat for the Design Lab project and Robotics Museum. His research interests include networked sliding mode control, control of multi-agent systems and its applications. Dr. Mehta received the Pedagogical Innovation Award from Gujarat Technological University (GTU) in 2014, and Dewang Mehta National Education Award in 2018. He is a senior member of the IEEE, member of the IEEE Industrial Electronics Society (IES) and Control System Society (CSS), life member of the Indian Society for Technical Education (ISTE), and Institute of Engineers India (IEI). Prof. Bijnan Bandyopadhyay received his B.E. degree in Electronics and Telecommunication Engineering from Indian Institute of Engineering Science and Technology, Shibpur (formerly B.E. College, Shibpur, Calcutta University) India in 1978, and Ph.D. in Electrical Engineering from IIT Delhi, India in 1986. In 1987, he joined the Systems and Control Engineering group, IIT Bombay, India, as a faculty member, where he is currently a chair professor. In 1996, he was with the Lehrstuhl fur Eleck-trische Steuerung und Regelung, RUB, Bochum, Germany, as xvii

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an lexander von Humboldt Fellow. He further visited Technical University of Ilmenau, Germany as an Alexander von Humboldt Fellow in 2016 and 2019. He has been a visiting Professor at Okayama University, Japan, Korea Advanced Institute Science and Technology (KAIST) South Korea and Chiba National University in 2007. He visited the University of Western Australia, Australia, as a Gledden Visiting SeniorFellow in 2007. Professor Bandyopadhyay is one of the recipients of UKIERI (UK India Education and Research Initiative) Major Award in 2007. He was awarded Distinguished Visiting Fellow by the Royal Academy of Engineering, London in 2009 and 2012. Professor Bandyopadhyay is a Fellow of the Indian National Academy of Engineering, National Academy of Sciences, and the Indian Academy of Sciences. He has 400 publications which include monographs, book chapters, journal articles, and conference papers. He has guided 39 Ph.D. theses at IIT Bombay. His research interests include the areas of multi rate output feedback based discrete-time sliding mode control, event-triggered sliding mode control, and nuclear reactor control. Prof. Bandyopadhyay served as Co-Chairman of the International Organization Committee and as Chairman of the Local Arrangements Committee for the IEEE ICIT, Goa, India, in 2000. He also served as one of the General Chairs of the IEEE ICIT conference, Mumbai, India in 2006. Prof. Bandyopadhyay has served as General Chair for IEEE International Workshop on VSSSMC, Mumbai, 2012. He has also served as Associate Editor of IEEE Transactions on Industrial Electronics and IEEE/ASME Transactions on Mechatronics. He is currently serving as Associate Editor of IET Control Theory and Applications. Prof. Bandyopadhyay has been awarded the IEEE Distinguished Lecturer of IEEE IE society in 2019. Prof. Bandyopadhyay is an IEEE Fellow.

Homogeneous Sliding Modes in Noisy Environments Avi Hanan, Adam Jbara, and Arie Levant

Abstract One of the main achievements of the high-order sliding mode control (HOSMC) theory is the standardized output-feedback regulation based on the robust high-order differentiation. The method employs universal HOSM controllers valid for any relative degree combined with standard HOSM differentiators. In this chapter, we present recently developed new universal controllers and filtering differentiators and demonstrate their output-feedback application in the presence of large sampling noises.

1 Introduction Sliding mode control (SMC) [27, 53, 75, 78] has been introduced to effectively control uncertain processes. The method assumes choosing a proper system output σ called sliding variable to keep it at zero. The constraint σ ≡ 0 is to provide for the desired system performance and is established in finite time by a high-frequency switching control. The control switching is inevitable due to the uncertainty of the system. Unfortunately, it produces undesired system vibrations called the chattering effect [9, 15, 34, 77]. While keeping the switching, the SM control itself can be done continuous. Its discontinuity can be shifted to the higher total derivatives of the sliding variable. The number r of the first discontinuous total time derivative σ (r ) is called the SM order [44, 46]. The conventional SMs [27, 77] feature the first SM order. Higher order A. Hanan (B) · A. Jbara · A. Levant Applied Mathematics Department, Tel-Aviv University, Tel-Aviv 6997801, Israel e-mail: [email protected] A. Jbara e-mail: [email protected] A. Levant e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_1

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SMs (HOSMs) are capable of successful chattering mitigation [10, 11, 16, 44], but are not able to completely remove it [15, 49]. Moreover, in fact, the chattering can be considered as an inherent feature of sampling-based systems [49]. The output regulation is the most straightforward application of SMC due to the simplicity of choosing the tracking error as the sliding variable. A great number of papers employ this technique, here we only cite a few: [11, 19, 23–26, 30, 33, 36, 39, 43, 46, 56, 63, 65, 67, 68, 74, 76]. SM-based differentiators [45, 46] are included in the feedback to produce finite-time (FT) exact derivatives of the sliding variable [2, 4, 7, 8, 20, 22, 29, 32, 42, 53, 66, 73]. Most above results are based on the application of the general homogeneity approach [6, 14, 41] to the SMC theory [12, 13, 35, 47, 54, 57, 65, 69, 72]. Till recently, the invention of new SM controllers has been considered a difficult task [24, 46, 48], but recently numerous new controllers have been proposed together with general construction approaches [19][18, 65, 70, 71]. Recent control template approaches [37, 51] belong to this category, and present very easy control design. Standard SM-based differentiators [46] have been recently used to construct new filtering differentiators [59, 61]. These new differentiators combine their exactness and asymptotically optimal accuracy in the presence of noises [60] with the new strong noise-filtering capabilities. In particular, they are capable of suppressing unbounded noises, provided some high-order local multiple integral of the noise is uniformly small. New hybrid differentiators [7, 58] feature the bilimit homogeneity [1]. They can be considered as hybrids of the linear filters [5] (high-gain observers) with the SM-based differentiators [46]. Such differentiators do not employ high gains, allow variable gains, and feature fast FT convergence. Recently, we have proposed equipping hybrid differentiators with the filtering capabilities [38]. In this chapter, we demonstrate the implementation of new SM control templates [37, 51] in the output-feedback HOSM control and its new filtering capabilities in the presence of very large noises due to the filtering [59] and hybrid filtering [38] differentiators. ∂ |x|m+1 = (m + Notation. Let ·m = | · |m sign(·) for any m ≥ 0. Note that ∂x ∂ xm+1 = (m + 1)|x|m . A function of a set is the set of function 1)xm and ∂x values on this set. The norm ||x|| stays for the standard Euclidean norm of x, Bε = {x| ||x|| ≤ ε} and ||x||h is a homogeneous norm. Let a  b be a binary operation for a ∈ A, b ∈ B, then A  B = {a  b|a ∈ A, b ∈ B}. ˙ ..., ξ (k) ) and Depending on the context, we use the same notation ξ k for both (ξ, ξ, (ξ0 , ξ1 , ..., ξk ). We define the finite difference operator δ j A = A(t j+1 ) − A(t j ) for any sampled function A(t j ).

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2 Preliminaries In this section, we recall some basic homogeneity and stability notions.

2.1 Stability of Differential Inclusions Let T Rn x denote the tangent space to Rn x , and Tx Rn x be the tangent space at the point x ∈ Rn x . Consider the differential inclusion (DI) x˙ ∈ F(x), x ∈ Rn x , F(x) ⊂ Tx Rn x .

(1)

Recall that a solution of (1) is any locally absolutely continuous function x(t), satisfying DI (1) for almost all t. A differential inclusion (DI) (1) is further called Filippov differential inclusion (DI), if the vector set F(x) is non-empty, compact, and convex for any x, and F is an upper semicontinuous set function [31, 47]. The latter means that the maximal distance from the vectors of F(x) to the vector set F(y) vanishes as x → y. Solutions of the Filippov DI possess most of the well-known standard properties, like the local-solution existence for the Cauchy problem, the solution extendability till the boundary of a compact, and the continuous dependence on the graph of the DI [31]. Obviously, there is no solution uniqueness. A differential equation (DE) x˙ = f (x), x ∈ Rn x with a locally essentially bounded Lebesgue-measurable right-hand side is said to be understood in the Filippov sense, if its solutions are defined as the solutions of the special Filippov DI x˙ ∈ K F [ f ](x) with   co f ((x + Bδ )\N ). (2) K F [ f ](x) = μ L N =0 δ>0

Here co denotes the convex closure operation, whereas μ L is the Lebesgue measure. Formula (2) introduces the famous Filippov procedure [31]. In the non-autonomous case, we introduce the fictitious coordinate t, t˙ = 1. Whereas there are other definitions of solutions of the DE with discontinuous right-hand side, Filippov solutions satisfy all of them, i.e., constitute the minimal set of reasonably defined solutions. A point x0 ∈ Rn x is called the equilibrium of the Filippov DI (1), if x(t) ≡ x0 is its solution. The equilibrium x0 is called (Lyapunov) stable, if all solutions starting in some of its vicinity at t = 0 are extendable till infinity in time, and for any ε > 0 there exists such δ > 0 that each solution x(t) satisfying ||x(0) − x0 || < δ satisfies ||x(t) − x0 || < ε for any t ≥ 0. A stable equilibrium x0 is called asymptotically stable (AS), if any solution x(t) starting in some of its vicinity satisfies limt→∞ ||x(t) − x0 || = 0. It is globally AS if limt→∞ ||x(t) − x0 || = 0 for any x(0) ∈ Rn x .

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An AS equilibrium x0 is called FT stable (FTS), if x0 is AS, and for each initial condition x(0) from a vicinity of x0 there exists a number T ≥ 0, such that x(t) = x0 for any t ≥ T . It is called globally FTS, if such T exists for any initial condition x(0) ∈ Rn x . The equilibrium x0 is called fixed-time (FxT) stable (FxTS) [70], if it is globally FTS and the upper transient-time bound T can be chosen uniformly for all initial conditions. A ball x0 + Bε is called FxT attractive, if all trajectories converge to it in FxT, i.e., all solutions are extendable till infinity in time, and there exists such T > 0 that for any solution x the relation x(t) ∈ x0 + Bε holds for any t ≥ T . Example 1 The origin 0 is a FxTS equilibrium of the scalar dynamic system x˙ = −x 1/3 − x 3 . Any ball Bε = {x ∈ R| |x| ≤ ε} is FxT attractive for the Filippov DI  x˙ ∈ −[1, 2]x 3 . Globally, (locally) AS Filippov DIs always have proper global (local) C ∞ -smooth Lyapunov functions [17].

2.2 Weighted Homogeneity Introduce the weights (degrees) m 1 , m 2 , . . . , m n x > 0 of the coordinates x1 , x2 , . . . , xn x in Rn x , and denote deg xi = m i . The simple linear transformation dκ (x) = (κm 1 x1 , κm 2 x2 , ..., κm n x xn x ), κ ≥ 0

(3)

is called the dilation [6]. The function f : Rn x → Rm is said to have the homogeneity degree (weight) q ∈ R, deg f = q, provided the identity f (x) = κ−q f (dκ x) holds for any x and κ > 0. We distinguish a vector function f : Rn x → Rn x , f : x → f (x) ∈ Rn x , and a vector field f : Rn x → T Rn x , f : x → f (x) ∈ Tx Rn x [75]. In its turn the vector field f (x) ∈ Tx Rn x is considered as a particular case of the vector-set field F(x) ⊂ Tx Rn x for the vector set only containing one vector, F(x) = { f (x)}. Correspondingly, a vector-set function F(x) ⊂ Rm is called homogeneous of the homogeneity degree (HD) q ∈ R, if the identity F(x) = dκ−q F(dκ x) holds for any x and κ > 0 [47]. A vector-set field F(x) ⊂ Tx Rn x (DI (1)) is called homogeneous of the homogeneity degree (HD) q ∈ R, if the identity F(x) = κ−q dκ−1 F(dκ x) holds for any x and κ > 0 [47]. It follows from the latter definition that DI (1) is invariant with respect to the combined time-coordinate transformation (t, x) → (κ−q t, dκ x), κ > 0.

(4)

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One can interpret −q as the weight of the time t, deg t = −q. In the case of a vector field (DE), the definition is reduced to the classical definition deg x˙i = deg xi − deg t [6]. Any number can be considered as deg 0, deg a = 0 for any constant a = 0. The following simple rules of the homogeneous arithmetic are easily checked: deg Aa = ∂ A = deg A − deg α, deg A˙ = deg A − a deg A, deg(AB) = deg A + deg B, deg ∂α deg t. Any continuous positive-definite function of the HD 1 is called a homogeneous norm. We denote such norms by ||x||h . They are not real norms, but any two homogeneous norms || · ||h and || · ||h∗ are still equivalent in the sense that the inequalities γ∗ ||x||h∗ ≤ ||x||h ≤ γ ∗ ||x||h∗ hold for some γ∗ , γ ∗ > 0 and any x. The following are two traditional homogeneous norms: 1

||x||h∞ = max {|xi | mi }, ||x||h = ( 1≤i≤n x





1

|x| mi ) .

i

Note that the second homogeneous norm is continuously differentiable for x = 0, provided > maxi {m i }. The weights and homogeneity degrees are defined up to proportionality. In other words, deg xi = m i , − deg t = q can always be replaced with γm i , γq for any γ > 0. Also, the HDs of all functions/fields/inclusions are multiplied by γ in that case. Obviously, such weight transformation does not preserve homogeneous norms. A function is called quasi-continuous (QC) [48], if it is continuous everywhere except the origin. In particular, any continuous function is QC. A homogeneous DI (1) is called AS (FTS, FxTS) if the origin 0 is its global AS (FTS, FxTS) equilibrium. A set D0 is called homogeneously retractable if dκ D0 ⊂ D0 for any κ ∈ [0, 1]. A Filippov DI (1) is called contractive [47], if there exist positive numbers T, ε > 0, a retractable compact D0 and a compact D1 , 0 ∈ D1 , D1 + Bε ⊂ D0 , such that for any solution x(t) the relation x(0) ∈ D0 implies x(T ) ∈ D1 . ˜ A Filippov DI x˙ ∈ F(x) is called a small homogeneous perturbation of the Filippov homogeneous DI x˙ ∈ F(x) with the same dilation and the HD, if for some ˜ (small) ε ≥ 0 the relation F(x) ⊂ F(x) + Bε holds whenever x ∈ B1 . Theorem [37, 54, 57] summarizes stability features of DIs for arbitrary homogeneous degrees. Theorem 1 Let the Filippov DI (1) be homogeneous of the HD q. Then the asymptotic stability and the contractivity features are equivalent and robust with respect to small homogeneous perturbations. • If q < 0 the asymptotic stability implies the FT stability, and the maximal (minimal) stabilization time is a well-defined upper (lower) semicontinuous function of the initial conditions [57]. Moreover, the FT stability of DI (1) implies that q < 0. • If q = 0 the asymptotic stability is exponential. • If q > 0 the asymptotic stability implies the FxT attractivity of any ball Bε , ε > 0. The convergence to 0 is slower than exponential.

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Example 2 Consider any smooth DE x˙ = f (x), f (0) = 0, f  (0) = A, x ∈ Rn x . Then x˙ = f (x) ∈ {Ax + ε||x||B1 } holds for any ε > 0 in a sufficiently small vicinity of the origin. One can consider the linear time-invariant system x˙ = Ax, x ∈ Rn x , A ∈ Rn x ×n x , as a homogeneous Filippov DI x˙ ∈ {Ax} of the HD 0 with deg xi = 1, i = 1, ..., n x . Now, due to Theorem 1, the asymptotic stability of x˙ = Ax implies the asymptotic stability of its small homogeneous perturbation x˙ ∈ {Ax + ε||x||B1 }, which, in its turn, implies the local asymptotic stability of x˙ = f (x).  Theorem [6, 12] asserts that any AS homogeneous DI admits a smooth homogeneous Lyapunov function. Theorem 2 Let (1) be an AS Filippov homogeneous DI of the HD q. Then for any natural l, k, k > max(−q, l max deg xi ), there exists a pair of continuous functions V, W : Rn x → R, V, W ∈ C ∞ (Rn x \ {0}) such that 1. V is positive definite and homogeneous, deg V = k, V ∈ C l (Rn x ); 2. W is positive definite and homogeneous of degree k + q; and 3. maxv∈F(x) V (x) · v ≤ −W (x) for all x ∈ Rn x .

2.2.1

Accuracy of Perturbed Homogeneous DIs

Consider the retarded “noisy” perturbation of the AS Filippov homogeneous DI (1) of the negative homogeneity degree q < 0 [47] x˙ ∈ F(x(t − [0, τ ]) + Bhε ), x ∈ Rn x ,

(5)

where τ , ε ≥ 0, Bhε = {x ∈ Rn x | ||x||h ≤ ε}. In principle, DI (5) requires some functional initial conditions for t ∈ [−τ , 0]. The following result [46] requires some homogeneity assumptions on these conditions [28, 57] which are always satisfied provided the solutions do not depend on the solution prehistory for t < 0. That assumption usually holds in the case when the system is a combination of a smooth dynamic system with a digital dynamic controller based on discrete output sampling starting at t = 0. So assume that the solutions of (5) do not depend on the values x(t) for t < 0. Fix any homogeneous norm || · ||h . Then the accuracy, x ∈ γ Bhρ , ρ = max[ε, τ −1/q ],

(6)

is established in FT for some γ > 0 independent of ε, τ and initial conditions. If q = 0 that accuracy is established for ρ = ε and any sufficiently small τ [28]. If q > 0 one also takes ρ = ε, but the initial value x(0) and ε are to be uniformly bounded, whereas τ is to be sufficiently small for each fixed R, x(0) ∈ B R (it is the most “fragile” case [28], since the system can escape to infinity faster than any exponent [50]). A similar result also holds for the implicit Euler integration with the step τ [28].

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3 Homogeneity Approach to Output Regulation Under Uncertainty Consider a dynamic system of the general form x˙ = a(t, x) + b(t, x)u,

σ = σ(t, x),

(7)

where x ∈ Rn x , u ∈ R , σ : Rn x +1 → R are the system control and the system output, respectively. Smooth vector fields a : Rn x +1 → T Rn x , b : Rn x +1 → T Rnx and the very dimension n x are nowhere used and can be uncertain. Solutions of (7) are assumed forward complete, i.e., infinitely extendible in time, provided the control u(t) is Lebesgue-measurable and bounded along the trajectory. The system output function σ is sampled in real time and plays the role of the tracking deviation. The control task is to keep σ as small as possible. The system (7) is assumed to possess a known relative degree r . It means [40] that the control for the first time explicitly appears in the r th total time derivative of σ, i.e., σ (r ) = h(t, x) + g(t, x)u, ∀t, x : g(t, x) = 0, (8) where both g, h : Rn x +1 → T R are unknown smooth scalar vector fields, and g does never vanish. Moreover, the functions σ r −1 = (σ, ..., σ (r −1) )T and t can always be extended to local coordinates in Rn x +1 [40]. According to the traditional SMC approach [46, 75], uncertain system dynamics (8) are extended to a quite certain controlled autonomous DI. For that end assume that σr −1 ) (9) h(t, x) ∈ H ( σr −1 ), g(t, x) ∈ G( for some convex compact upper semicontinuous scalar (vector) set functions H, G : Rr → T R. In the fixed coordinates σ r −1 , these vector-set functions are naturally treated as numeric ones. Apply some locally essentially bounded Lebesgue-measurable feedback control u( σr −1 ). The resulting Filippov DI gets the form σ (r ) ∈ H ( σr −1 ) + G( σr −1 )K F [u]( σr −1 ).

(10)

It is to become AS for a proper choice of control. Note that this approach requires the real-time estimation or availability of σ r −1 . The main idea is to make DI (10) homogeneous. Assign deg σ = 1, and let the system HD be q ∈ R, i.e., deg t = −q. Then deg σ (i) = 1 + iq holds for i = 0, 1, ..., r − 1. The required conditions deg σ (i) > 0 are ensured by the inequality deg σ (r ) = 1 + rq ≥ 0 which is in any case necessary for the feasibility of the system (Theorem 1 [57]). Thus, q ≥ −1/r is required. Also fix some homogeneous norm || · ||h .

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Assume that the set functions H, G are homogeneous, deg H = 1 + rq. Let also the control u be homogeneous, so that deg u = deg K F [u]. Without losing the generality assume that deg G = 0, i.e., deg u = 1 + rq. This implies the inclusions 1+rq

h(t, x) ∈ H ( σr −1 ) ⊆ [−C, C]|| σr −1 ||h g(t, x) ∈ G( σr −1 ) ⊆ [K m , K M ]

,

(11)

for some constants C ≥ 0, K m > 0, K M ≥ K m , and the DI [51] σr −1 ||h σ (r ) ∈ [−C, C]|| C ≥ 0, 0 < K m ≤ K M .

1+rq

+ [K m , K M ]u,

(12)

Recall that the Filippov procedure K F [·] is to be applied to u in order to produce a Filippov DI. The control u is assumed to be a Borel-measurable function of σ r −1 or of its dynamic estimation. The measurability in the sense of Borel is needed to ensure the Lebesgue measurability of the resulting control in the presence of Lebesguemeasurable noises. Note that all elementary functions are Borel-measurable. The important case q = −1/r corresponds to the standard high-order SMC (HOSMC) approach [46, 47]. In that case deg σ (r ) = 0, and (12) gets the well-known form (13) σ (r ) ∈ [−C, C] + [K m , K M ]u, deg u = 0. The corresponding assumptions |h| ≤ C and g ∈ [K m , K M ] are always at least locally true for some C, K m , K M . In the case when (13) is AS, q = −1/r , it is also FT stable (Theorem 1), and the control feedback function u( σr −1 ) is necessarily discontinuous at σ = 0 for C > 0. The motion on the set σ = 0 is said to be in the r th-order SM (r -SM), and the control is called r th-order SMC (r -SMC) [46, 47]. There are many homogeneous SM controllers solving the problem in the case q = −1/r < 0, deg u = 0, some of them appear in [11, 24, 25, 39, 67, 68, 74, 75]. The recently established powerful method [18, 19] exploits the knowledge of a concrete homogeneous control Lyapunov function or builds it for the system σ (r ) = u in order to generate an r -SM controller. Constructing a new control Lyapunov function becomes the initial non-trivial design step. The alternative approach [51] presented below removes any differentiability conditions in the control construction and, correspondingly, yields significantly more controllers for any possible r and q.

4 Homogeneous Control Templates We call two scalar functions ω, :  → R,  ⊂ Rn ω , sign-equivalent in , if sign ω(s) ≡ sign (s) whenever s ∈  and one of them is not zero. Let the (r − 1)th-order homogeneous DE,

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σ (r −1) + ϕr −1 ( σr −2 ) = 0,

(14)

be AS, and ϕr −1 be continuous, deg ϕr −1 = 1 + (r − 1)q. The following theorem extends the result [51] while exactly preserving its proof. Theorem 3 Let q ≥ −1/r . Choose any homogeneous norm || σr −1 ||h , and let σr −1 ) be any homogeneous quasi-continuous (QC) scalar function. Let φr also φr ( σr −2 ) for σ r −1 = 0. Consider the homogeneous be sign-equivalent to σ (r −1) + ϕr −1 ( controls of the form σr −1 ), (15) u = αUr ( where α > 0, and Ur is defined by one of the formulas 1+qr −deg φr

σr −1 ) = −|| σr −1 ||h Ur ( σr −1 ) = Ur (

1+qr −|| σr −1 ||h

φr ( σr −1 ),

(16)

sign φr ( σr −1 ).

(17)

Then for any sufficiently large α > 0 these controllers asymptotically stabilize DI (12). In particular, the homogeneous DE, σr −1 ) = 0, ϕr ( σr −1 ) = −αUr ( σr −1 ), σ (r ) + ϕr (

(18)

is AS for any sufficiently large α. The function ϕr is continuous for q > −1/r , if Ur is taken in the form (16). Control function (16) is QC (i.e., discontinuous only at σ r −1 = 0) for q = −1/r . It is continuous for q > −1/r , provided Ur (0) = 0 is assigned. DI (12) (in particular, (18)) is FT stable for q < 0, and exponentially stable for q = 0. If q > 0 any ball Bε attracts solutions in FxT. When applied to the general system (7) the controllers can be multiplied by any locally bounded Lebesgue-measurable function k(t, x) ≥ 1 without losing the convergence of σ to zero. The chattering of the QC r -SM controller (15), (16), obtained in the case q = −1/r , is much lower compared with (15), (17) [48]. Also, in spite of controller (15), σr −1 ) = 0, (17) looking as a classical SM controller, it does not keep the SM φr ( σr −1 ), in general, features infinite gradients. since φr (

4.1 Recursion in the Relative Degree Actually, under the condition q ≥ −1/r , Theorem 3 establishes a recursion from the (r − 1)th-order AS DE (14) to the new r th-order AS DE (18). In the sequel, we use that A + B and Aγ + Bγ are sign-equivalent for any A, B ∈ R and γ > 0.

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The initial step. Let q ≥ −1. The AS DE (14) of the order 1 can always be chosen as (19) σ˙ + β0 σ1+q = 0, β0 > 0. In order to recursively construct a homogeneous stabilizer for r = 2 one will need q ≥ −1/2. The recursive step. Let an (r − 1)th-order AS DE (14) be given, q ≥ −1/r . Choose two arbitrary homogeneous norms || · ||h , || · ||h∗ , some m > 0, and any QC function θ(s), θ : R \ {0} → R, sign-equivalent to s, e.g., θ(s) = (s + sin s)−1 . Then the σr −1 ): following are only three of the simplest choices for φr ( m  1. φr ( σr −1 ) = σ (r −1) + ϕr −1 , deg ϕr = m > 0, m 2. φr ( σr −1 ) = σ(r −1) + ϕr−1 m , deg ϕr = m > 0, σ(r −1) +ϕr −1 m , deg ϕr = 0. σr −1 ) = θ 3. φr ( m(1+(r −1)q)

(20)

|| σr −1 ||h

Alternatively one, for example, can take the function φr ( σr −1 ) =

  (r −1) m3 m 2   σ  +ϕr −1 m3 + ϕr −1 m 2  θ |σ (r −1) + ϕr −1 |m 1  σ (r −1) m 3 (1+(r −1)q) || σr −1 ||h

with deg φr = m 1 + m 2 , m 2 , m 3 > 0, m 1 + m 2 ≥ 0, etc. There are, obviously, infinitely many such constructions for each r ≥ 2. Now, according to Theorem 3, from (16), (17) obtain the new homogeneous controls (15) of the order r , 1+qr −deg φ

r σr −1 ||h∗ φr ( σr −1 ), u r = −α|| 1+qr σr −1 ||h∗ sign φr ( σr −1 ), u r = −α||

(21)

and the r th-order AS DE (18) 1+qr −deg φr

σ (r ) + βr −1 || σr −1 ||h∗

φr ( σr −1 ) = 0.

(22)

The new equation contains uncertain parameters of the auxiliary function φr as well as the uncertain parameter βr −1 . It is natural to call (21) a controller template. If q ≥ −1/(r + 1) one can now perform one more recursive step, etc. In general, one needs r − 1 recursive steps to develop a controller of the order r , provided q ≥ −1/r . But the first step (19) is trivial, since any β0 > 0 is admissible. It is reasonable to immediately assign proper values to additional design parameters which appear at each recursion step. Usually, it is done by simulation of (22).

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11

HOSMC Template Development

The most practical special case is definitely the case of SM control. Let the relative degree be r ≥ 1. Then the corresponding system HD is −1/r and deg σ (r ) = deg u = 0. In that case, it is usually convenient to proportionally change all the weights, getting q = −1, deg σ (i) = r − i for i = 0, 1, ..., r , deg t = 1 (the r -sliding homogeneity [47]). A number of r -SM controllers are readily available, and their parameters are known in advance at least till r = 5 [24, 75]. Note that the parameter α from (15) defines the control magnitude and is only assigned at the last practical control-design stage. The presented control template development is so simple that one can develop a new r -SM controller for each practical application (Sect. 9.1). In that case, the first recursion step almost always employs σ˙ + β0 σ r −1 sign σ)σ r

r −1 r

= 0, though one can, for

= 0 instead. Fix some β0 > 0. example, take “exotic” σ˙ + β0 (2 + At the next step, one has already infinitely many variants. Any equation of the ˙ = 0 is admissible, provided deg φ = deg σ, ¨ φ is QC and signform σ¨ + β1 φ(σ, σ) equivalent to σ˙ + β0 σ For example,

r −1 r .

 σ¨ + β1 tan

r −1 σ+β ˙ 0 σ r r −1 |σ|+β ˙ 0 |σ| r

 2r −4    r −1 1 1 r −1   ˙ 2 + β02 σ 2r  =0 σ  

can be taken. The equation is AS for any sufficiently large β1 > 0 to be assigned ... by simulation. The recursion process proceeds then to an AS equation for σ , etc. A complete design for r = 3, 4 is demonstrated in the simulation Sect. 9.1.

5 Filtering SM-based Differentiation The practical realization of the r -SM controllers developed in Sect. 4.1.1 requires the real-time estimation of the derivatives σ r −1 . Some popular observation methods applied in that context are based on high-gain observers [5] and SMs [75]. In the following, we present some modern methods of SM-based observation featuring fast robust exact derivatives’ estimation while keeping high accuracy in the presence of large and even unbounded noises, provided they are small in average.

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5.1 Homogeneous Differentiation 5.1.1

Standard Differentiator

The control approach to the nth-order differentiation of a noisy sampled function f 0 (t) suggests constructing an observer for the disturbed integrator chain y (n+1) = ξ(t) with the output y and the unknown disturbance/input ξ = f 0(n+1) (t). Its outputs z i , i = 0, ..., n, are to approximate the y (i) (t) in spite of y = f 0 (t) being sampled with some noise and ξ(t) being unknown. The problem is very old and is known to be ill-posed, if no restrictions are imposed on ξ. Let the input f (t) take the form f (t) = f 0 (t) + η(t) ∈ R, where η(t) is a Lebesgue-measurable bounded noise, |η(t)| < ε0 , and f 0 (t) is an n-times differentiable unknown function to be restored together with its n derivatives in spite of the unknown measurement-noise intensity ε0 . The last derivative f 0(n) is assumed to have a known Lipschitz constant L > 0, which means that f 0(n+1) (t) ∈ [−L , L] holds for almost all t. It is further denoted as f 0 ∈ Lipn L. The general differentiator [5, 46] is usually of the form z˙ i = ϕi (z 0 − f (t)) + z i+1 , i = 0, ..., n − 1, z˙ n = ϕn (z 0 − f (t)),

(23)

where ϕi is a scalar function, z i ∈ R. The system is understood in the Filippov sense [31] to allow discontinuities of ϕi . The equivalent recursive form of (23) is z˙ 0 = ϕi (z 0 − f (t)) + z 1 , z˙ i = ϕi (z i − z˙ i−1 ) + z i+1 , i = 1, ..., n − 1, z˙ n = ϕn (z 0 − f (t)).

(24)

Assuming the noise is absent (i.e., ε0 = 0), subtracting f 0(i+1) from both sides of (23), and denoting σi = z i − f 0(i) , derive σ˙ i = ϕi (σ0 ) + σi+1 , i = 0, ..., n − 1 σ˙ n ∈ ϕn (σ0 ) + [−L , L] ,

(25)

which is a DI in the error space σ n . DI (23) becomes homogeneous and FT stable for properly chosen functions ϕi . The “standard” nth-order homogeneous SM-based differentiator [46] has the form z˙ 0 z˙ 1 ... z˙ n−1 z˙ n

n 1 = −λ˜ n L n+1 z 0 − f  n+1 + z 1 , n−1 2 = −λ˜ n−1 L n+1 z 0 − f  n+1 + z 2 ,

(26) 1 n = −λ˜ 1 L n+1 z 0 − f  n+1 + z n , = −λ˜ 0 L sign(z 0 − f ),

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13

where the parameters λ˜ i > 0 of the differentiator (28) are to be chosen in advance for each n, i = 0, 1, 2, ..., n. A proper choice of the parameters λ˜ i renders the error dynamics FTS. Correspondingly, in the absence of noises, the equalities z i = f 0(i) are established in FT. For the future usage, introduce the number n d currently equal to the differentiation order n. Then in the presence of a sampling noise with the maximal magnitude ε0 , the accuracy  

 (nd −i+1)  (i)  (27) z i − f 0  ≤ γi L εL0 (nd +1) ˜ . is obtained in FT for some γi ≥ 1 only depending on the coefficients λ n ˜ Whereas γi depend on the parameters λi of (26), the asymptotic structure (27) (i.e., the powers) is fixed and cannot be improved by any differentiation algorithm exact on functions f 0 ∈ Lipn (L) [45]. Moreover, it can be shown that γi ≥ 2i/(n d +1) [60]. Therefore, an n d th-order differentiator of any nature is called asymptotically optimal, if it provides for the steady-state accuracy (27) for all signals and noises satisfying the above assumptions [60]. It is not simple to properly choose the differentiator parameters for each n. The task is facilitated by employing its recursive form [46] z˙ 0 z˙ 1 ... z˙ n−1 z˙ n

1

n

= −λn L n+1 z 0 − f (t) n+1 + z 1 , n−1 1 = −λn−1 L n z 1 − z˙ 0  n + z 2 , (28) 1

1

= −λ1 L 2 z n−1 − z˙ n−2  2 + z n , = −λ0 L sign(z n − z˙ n−1 ),

for some positive λi > 0, i = 0, 1, ..., n. Excluding z˙ i reduce (28) to the general structure (23) and the standard form (26). It is easily verified that λ˜ 0 = λ0 , λ˜ n = λn , i i+1 , i = n − 1, n − 2, ..., 1. and λ˜ i = λi λ˜ i+1 In the case of f (t) ≡ 0 systems, (26) and (28) become homogeneous of the HD −1 with deg t = 1, deg z i = deg σi = n − i + 1, i = 0, ..., n. = {λ0 , λ1 , ...} can be built [46], providing An infinite sequence of parameters λ ˜ coefficients λi of (26) for all natural n. For this end, one simply starts with any λ0 > 1 and recursively adds a sufficiently large value λn > 0 for each n = 1, 2, .... = The parameters are surprisingly easily found by simulation. In particular, λ {1.1, 1.5, 2, 3, 5, 7, 10, 12, 14, 16, 20, 26, 32, ...} are well checked for n ≤ 12. Note that a shorter sequence up to n = 7 has been published in [58, 60], while a sequence till n = 5 was the first one to appear in [46]. The corresponding parameters λ˜ i are listed in Table 1. Alternative parameters are provided in Sect. 5.2 by another sequence (37). It is always assumed in the following that the parameters λi are properly λ chosen, so that (26) is finite time stable.

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Filtering Differentiators

The following filter/observer is build on the basis of the standard differentiator (26) and, remaining exact, is capable of filtering out unbounded sampling noises. Introduce the number n f ≥ 0 which is further called the filtering order. Correspondingly, n d is further called the differentiation order. Let n d , n f ≥ 0, where n f is called the filtering order. The filtering differentiator is defined by the new form n d +n f

1

w˙ 1 = − λ˜ n d +n f L nd +n f +1 w1  nd +n f +1 + w2 , ... n f −1

n+2

w˙ n f −1 = −λ˜ n d +2 L nd +n f +1 w1  nd +n f +1 + wn f , w˙ n f = − λ˜ n d +1 L

nf n d +n f +1

w1 

n d +1 n d +n f +1

(29)

+ wn f +1 ,

wn f +1 = z 0 − f (t), n f +1

nd

z˙ 0 = − λ˜ n d L nd +n f +1 w1  nd +n f +1 + z 1 , ... n d +n f

1

z˙ n d −1 = −λ˜ 1 L nd +n f +1 w1  nd +n f +1 + z n d , z˙ n d = − λ˜ 0 L sign(w1 ), | f (n d +1) | ≤ L .

(30)

0

Parameters λ˜ i , i = 0, 1, ..., n, n = n d + n f , of (26) and (29), (30) coincide and can be taken from Table 1. For n f = 0, the fictitious variable wn f +1 turns into w1 , DEs of (29) disappear and (30) turns into the standard differentiator (26). The assumptions on the input signal are the same. It was recently shown [59] that the steady-state accuracies, n f +n d +1 |w ,  1 | ≤ γw1 Lρ    (i) z i − f 0 (t) ≤ γi Lρn d +1−i , i = 0, ..., n d

(31)

|w j | ≤ γw j Lρn d +n f +2− j , j = 2, ..., n f ,

(32)

are in FT established for

1

ρ = (ε0 /L) (nd +1) ,

(33)

and some γw1 , γw j , γi > 0 only depending on the choice of λ0 , ..., λn d +n f . It means that (29), (30) describe an alternative asymptotically optimal n d th-order differentiator. This differentiator has new significant filtering properties to be presented in Sect. 7. The accuracy estimation (32) is singled out, since it does not hold for the corresponding ρ in the presence of large noises considered there.

1.1

1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

0

1 2 3 4 5 6 7 8 9 10 11 12

1.5 2.12 3.06 4.57 6.75 9.91 14.13 19.66 26.93 36.34 48.86 65.22

2 4.16 9.30 20.26 43.65 88.78 171.73 322.31 586.78 1061.1 1890.6 3 10.03 32.24 101.96 295.74 795.63 2045.8 5025.4 12220 29064 5 23.72 110.08 455.40 1703.9 6002.3 19895 65053 206531 7 47.69 281.37 1464.2 7066.2 31601 138954 588869

Table 1 Parameters λ˜ 0 , λ˜ 1 , ..., λ˜ n of differentiator (26) for n = 0, 1, ..., 12

10 84.14 608.99 4026.3 24296 143658 812652 12 120.79 1094.1 8908 70830 534837

14 173.72 1908.5 20406 205679

17 251.99 3623.1 48747

20 386.7 6944.8

26 623.30

32

Homogeneous Sliding Modes in Noisy Environments 15

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A. Hanan et al.

5.2 Hybrid (Bi-homogeneous) Filtering Differentiators As we have seen above, the usual requirement of HOSM-based n d th-order differentiation is that the n d th derivative f 0(n d ) has a known Lipschitz constant L > 0. Presented homogeneous differentiators solve the problem both robustly and exactly. Unfortunately, a well-known drawback of these differentiators is the low convergence rate for large initial errors. After the differentiator coefficients are fixed, the Lipschitz constant L actually remains the only adjustable parameter of the standard HOSM-based differentiator [46]. Thus, it is natural to try tuning that parameter in order to accelerate the convergence, while keeping the same steady-state accuracy (31), (33). Unfortunately, filtering and standard differentiators with variable parameter L, in general, converge only locally [55, 58] (global convergence is preserved for monotonously growing differentiable L(t) [64]). These issues are settled by the so-called hybrid differentiator [58] of the general structure (24). New quasi-linear terms are for this end added to the recursive form (28) of the differentiator producing a hybrid differentiator combining the features of the homogeneous differentiator (28) and a linear filter similar to the high-gain observer (HGO) [5], but with gains which are not to be large. The following is its further modification to the hybrid filtering differentiator [38], 1

n d +n f

w˙ 1 = −λn d +n f L nd +n f +1 w1  nd +n f +1 −μn d +n f Mw1 + w2 , ...

1   nd +2 w˙ n f −1 = −λn d +2 L nd +3 wn f −1 − w˙ n f −2 nd +3

w˙ n f

(34)

−μn d +2 M(wn f −1 − w˙ n f −2 ) + wn f , 1   nd +1 = −λn d +1 L nd +2 wn f − w˙ n f −1 nd +2 −μn d +1 M(wn f − w˙ n f −1 ) + z 0 − f (t),

1   nd z˙ 0 = −λn d L nd +1 z 0 − f (t) − w˙ n f nd +1

−μn d M(z 0 − f (t) − w˙ n f ) + z 1 , 1

z˙ 1 = −λn d −1 L nd z 1 − z˙ 0 

n d −1 nd

−μn d −1 M(z 1 − z˙ 0 ) + z 2 , ...

0  z˙ n d = −λ0 L z n d − z˙ n d −1 −μ0 M(z n d − z˙ n d −1 ), n d +n f and μ n d +n f are some properly chosen positive numbers. where λ

(35)

Homogeneous Sliding Modes in Noisy Environments

17

This differentiator converges in FT and exactly, provided | f 0(n d +1) (t)| ≤ L(t) and ˙ | L/L| ≤ M hold. The convergence rate is exponential to any vicinity of the error space origin, and is easily regulated by M [58]. The accuracy is covered by Theorem 4 [38] in the sequel. The hybrid filtering differentiator (40), (38) turns into the “standard” hybrid differentiator [58] for n f = 0, into the filtering differentiator (29), (30) for M = 0, into the “standard” differentiator (23) for n f = 0, M = 0, and into the linear HGO [5] for n f = 0, L = 0, M >> 1. The coefficients of the resulting HGO are μn d , μn d μn d −1 , . . . , μn d μn d −1 · · · μ0 from the top-down and correspond to the characteristic polynomial s n d +1 + μn d s n d + μn d μn d −1 s n d −1 + · · · + μn d μn d −1 · · · μ0 .

(36)

Also here one can construct infinite double sequence of parameters valid for any n d + n f [38]. In particular, the sequence {(λ0 , μ0 ), (λ1 , μ1 ), ...} = (1.1, 2), (1.5, 3), (2, 4), (3, 7), (5, 9), (7, 13),(10, 19), (12, 23), (15, 42), (21, 43), (25, 79), (39, 98), (78, 116), . . . (37) has been experimentally validated for n ≤ 12 and can be extended up to n d + n f = n = ∞. Set (37) extends the parametric set valid till n = 7 which has been published in [7, 58]. It has been proved [58] that the sequence λi is also valid for use in the standard and filtering differentiators, but the authors prefer parameters from Table 1 in that case. Note that parameters μi produce Hurwitz polynomials (36) for each n d = 0, 1, ... [58]. Introduce the functions 1

n−i

ϕi,n (s, L) = λn−i L n−i+1 |s| n−i+1 sign s + μn−i Ms, i = 0, ..., n.

(38)

Then the proposed hybrid filtering differentiator gets the form w˙ 1 = vw1 = −ϕ0,n d +n f (w1 , L) + w2 , w˙ 2 = vw2 = −ϕ1,n d +n f (w2 − vw1 , L) + w3 , ... w˙ n f = vwn f = −ϕn f ,n d +n f (wn f − vwn f −1 , L) + z 0 − f (t).

(39)

z˙ 0 = v0 = −ϕn f +1,n d +n f (z 0 − vwn f − f (t), L) + z 1 , z˙ 1 = v1 = −ϕn f +2,n d +n f (z 1 − v0 , L) + z 2 , ... z˙ n = vn d = −ϕn d +n f ,n d +n f (z n − vn d −1 , L).

(40)

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A. Hanan et al.

The recursive form (39), (40) is identically rewritten in the standard dynamic-system form w˙ 1 = −ϕ0,n d +n f (w1 , L) + w2 , w˙ 2 = −ϕ1,n d +n f (ϕ0,n d +n f (w1 , L), L) + w3 , ... w˙ n f = −ϕn f ,n d +n f (ϕn f −1,n d +n f (...(w1 , L)..., L), L) + z 0 − f (t),

(41)

z˙ 0 = −ϕn f +1,n d +n f (ϕn f ,n d +n f (...(w1 , L)..., L), L) + z 1 , ... z˙ n = −ϕn d +n f ,n d +n f (ϕn d +n f −1,n d +n f (...(w1 , L)..., L), L),

(42)

but only the recursive form is usable in practice.

6 Discretization of Differentiators and Controllers In practice, any observer is a discrete computer-based system processing a discretely sampled noisy output of a continuous-time system. Thus, the differential equations are to be replaced with their numeric real-time integration. One also cannot apply standard numeric integration methods, since the considered observer is a discontinuous dynamic system. The simplistic Euler integration works, but significantly destroys the theoretical accuracy [7, 62]. The right discretization is to produce homogeneous discrete error dynamics analogous to that of the continuous-time sampling case. The same problems naturally appear in the implementation of output-feedback systems. In such a case also the controller is computer based.

6.1 Discretization of Differentiators Let t j be the sampling instants, 0 < t j+1 − t j = τ j ≤ τ , j = 0, 1, ..., lim j→∞ t j = ∞. Though the sampling steps are assumed bounded, their upper bound τ does not need to be available. Notation. Denote δ j A = A(t j+1 ) − A(t j ) for any function A. The proposed discrete differentiator δ j w1 = ϕ0,n d +n f (w1 (t j ), L)τ j + w2 (t j )τ j , δ j w2 = ϕ1,n d +n f (w2 (t j ) − vw1 (t j ), L)τ j + w3 (t j )τ j , ..., δ j wn f = ϕn f ,n d +n f (wn f (t j ) − vwn f −1 (t j ), L)τ j + (z 0 (t j ) − f (t j ))τ j ,

(43)

Homogeneous Sliding Modes in Noisy Environments

19

δ j z 0 = ϕn f +1,n d +n f (z 0 (t j ) − f (t j ) − vwn f (t j ), L)τ j + δ j z 1 = ϕn f +2,n d +n f (z 1 (t j ) − v0 (t j ), L)τ j +

nd  i=2

nd  i=1

zi i τ , i! j

zi τ i−1 , (i−1)! j

(44)

..., δ j z n d −1 = ϕn f +2,n d +n f (z n d −1 (t j ) − vn d −2 (t j ), L)τ j + z n d (t j )τ j , δ j z n d = ϕn f +2,n d +n f (z n d (t j ) − vn d −1 (t j ), L)τ j has additional Taylor-like terms. Functions vi , vwj , andϕi,n are defined in (39), (40), and (38). Denote the (n f , n d )th-order filtering hybrid differentiator (38), (41), (42) by (w, ˙ z˙ )T = Dn f ,n d (w, z 0 − f, z, L), where the difference z 0 − f (t) is singled out. Then the above discrete differentiator (43), (44) gets the form δ j (w, z)T = Dn f ,n d (w(t j ), z 0 (t j ) − f (t j ), z(t j ), L)τ j + Tn f ,n d (z(t j ), τ j ), ⎡

⎤ 0 T0 ⎢ ⎥ ... ⎢ ... ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ 0 ⎢ Tn f −1 ⎥ ⎢ 1 ⎥ ⎥ ⎢ 2! z 2 (t j )τ 2j + · · · + n1 ! z n d (t j )τ nj d ⎥ ⎢ d ⎢ Tn f ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ... ⎢ ... ⎥ ⎢ ⎥ nd ⎥=⎢ ⎢ ⎥.  s−i 1 ⎢ Tn f +i ⎥ ⎢ ⎥ z (t )τ (s−i)! s j j ⎥ ⎢ ⎢ ⎥ s=i+2 ⎢ ... ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ . . . ⎢Tn f +n d −2 ⎥ ⎢ ⎥ 1 2 ⎥ ⎢ ⎢ ⎥ z (t )τ ⎥ ⎣Tn f +n d −1 ⎦ ⎢ 2! n d j j ⎣ ⎦ 0 Tn f +n d 0 ⎡

⎤

(45)

(46)

Here Tn f ,n d ∈ Rn f +n d +1 . In particular, Tn f ,0 (w, z, τ ) = 0 ∈ Rn f +1 , Tn f ,1 (w, z, τ ) = 0 ∈ Rn f +2 . The following theorem easily follows from the similar result on hybrid differentiators [7, 58]. The limit case τ = 0 is formally covered in that theorem as the replacement of (45) with the continuous-time hybrid filtering differentiator (34), (35) processing the signal f 0 is corrupted by the Lebesgue-measurable noise η(t). Theorem 4 Under the assumption that | f 0(n d +1) (t)| ≤ L(t), let the absolutely con˙ tinuous function L(t) satisfy | L/L| ≤ M, and the sampling noise satisfy |η(t)/L| ≤ ε. ˆ Then differentiator (45) in FT provides for the accuracy (31), (32) with ρ = max[εˆ1/(n d +1) , τ ]. • In the case M = 0 (filtering differentiator (29), (30)), the accuracy (31), (32) holds for any ε, ˆ τ ≥ 0. • In the case M > 0, the accuracy holds for sufficiently small ε, ˆ τ ≥ 0. In the case εˆ = 0, τ = 0, the convergence is in FT and exact, and is exponential to any ball of differentiation errors.

20

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The following is some explanation. Recall that the hybrid filtering differentiator (34), (35) turns into the filtering differentiator (29), (30) for M = 0 and L = const. It is also homogeneous in bilimit. Correspondingly in a small vicinity of the manifold z 0 − f 0 = ... = z n d − f 0(n d ) = 0, w = 0 the error dynamics of the hybrid filtering differentiator (34), (35) corresponding to M > 0 and the filtering differentiator (29), (30) corresponding to M = 0 (asymptotically) coincide. The same happens to the discretization (45). This leads to the same accuracy of the both differentiators, provided ρ is small. If ρ is large enough, the linear dynamics prevail, and the hybrid filtering differentiator effectively turns into a linear low-pass filter, whose frequency response and accuracy are determined by M. It causes the corresponding change in the accuracy asymptotics for larger ε. ˆ Moreover, large τ can cause the instability of the limit linear error dynamics at infinity, correspondingly leading to the filter divergence [7, 58]. It is shown in [7] that the accuracy is not improved when additional integration steps are introduced between the actual sampling instants, or the integration makes use of the corresponding matrix exponent over each sampling interval.

6.2 Output-Feedback Discretization Consider system (7) of the relative degree r . Let it be closed by the feedback r -SMC (15) developed in Sect. 4 and exploiting the output differentiation, x˙ = a(t, x) + b(t, x)u(t j ), σ(t ˆ j ) = σ(t j , x(t j )) + η(t j ), u = αUr (z(t j )), L ≥ C + K M α sup |Ur |, L > 0, t ∈ [t j , t j+1 ), ˆ j ), z(t j ), L)τ j . δ j (w, z)T = Dn f ,r −1 (w(t j ), z 0 (t j ) − σ(t

(47)

Here σˆ = σ + η represents the sampled value of σ corrupted by the noise η. Theorem 5 Let the sampling noise satisfy |η(t)| ≤ ε0 , the sampling interval be bounded, 0 < t j+1 − t j ≥ τ , n f ≥ 0. Then the discrete output-feedback control from (47) in FT provides for the accuracy |σ i | ≤ γi ρr −i , i = 0, 1, ..., r − 1, for ρ = max[(ε0 /L)1/(n d +1) , τ ] and some γ0 , ..., γr −1 > 0. Addition of the terms Tn f ,n d (z(t j ), τ j ) in (47) is optional, but not required. Also this theorem formally covers the limit case τ = 0 corresponding to the continuous sampling of σ in the presence of the Lebesgue-measurable noise η(t), |η(t)| ≤ ε0 . The proofs of Theorems 4, 5 are based on the accuracy estimation (6) of the disturbed homogeneous systems.

Homogeneous Sliding Modes in Noisy Environments

21

7 Filtering Noises In this section, we show that the proposed differentiators and output-feedback SM controllers filter out large sampling noises, while still preserving the exactness in the absence of noises and the asymptotically optimal accuracy (27) in the presence of bounded noises.

7.1 Filtering Noises in Continuous Time Recall a few notions from [59]. A signal ν(t), ν : [0, ∞) → R, is called globally filterable [59], or a signal of the (global) filtering order k ≥ 0, if it is a locally integrable Lebesgue-measurable function, and there exists a globally bounded Caratheodory solution ξ(t), ξ : [0, ∞) → R, of the equation ξ (k) = ν. Correspondingly, ξ (k−1) (t) is a locally absolutely continuous function, if k > 0. Naturally, ν(t) is said to have the filtering order k = 0, if ν is essentially bounded. Any number exceeding sup |ξ(t)| is called a kth-order (global) integral magnitude of ν. Assumption 1 The sampled input is of the form f (t) = f 0 (t) + η(t), where f 0(n d ) is a Lispchitzian function, | f 0(n d +1) (t)| ≤ L for almost all t > 0 and known L > 0,  i.e., f 0 ∈ Lipn d L. Assumption 2 The noise η(t) admits an expansion of the form η(t) = η0 (t) + η1 (t) + ... + ηn f (t), where each ηk , k = 0, ..., n f , is a signal of the global filtering order k and the kth-order integral magnitude εk ≥ 0. Correspondingly, the noise components η1 , ..., ηn f are possibly unbounded, whereas η0 is essentially bounded,  ess supt≥0 |η0 | ≤ ε0 . Introduce parameter ρ measuring the filtered intensity of the sampling noise ρ = max



ε0  L

1 n d +1

,

ε1  L

1 n d +2

, ...,

ε  nf

L

1 n d +n f +1

 .

(48)

The following two theorems appear in [38]. Recall that for M = 0 the hybrid filtering differentiator (34), (35) turns into the filtering one (29), (30). Theorem 6 Under Assumptions 1, 2, the practical stability of the hybrid filtering differentiator (34), (35) is preserved for any ρ defined by (48). For any ρ if M = 0, and for sufficiently small ρ if M > 0, after some FT transient the hybrid filtering differentiator (34), (35) provides for the accuracy (31), i.e.,

22

A. Hanan et al. n f +n d +1 |w ,  1 | ≤ γw1 Lρ    (i) z i − f 0 (t) ≤ γi Lρn d +1−i , i = 0, ..., n d

(49)

for some γw1 , γi > 0. Example 3 The noise η = A cos(ωt) features any global filtering order k ≥ 0 and the integral magnitude A for k = 0 and 2 A/ω k for k > 0. Theorem 6 implies that the accuracy estimation (49), (48) holds for each possible expansion η = η0 + ... + ηn f . In particular, η = ηn f corresponds to ρ = (A/L)1/(n d +n f +1) ω −n f /(n d +n f +1) . Note that for sufficiently large n f the resulting noise-intensity parameter ρ of the harmonic signal approaches the number 1/ω and does not depend on A. On the other hand, the theorem provides an upper estimation valid for any possible expansion of η into a sum of filterable signals ηk . For sufficiently small A, another estimation ρ = (A/L)1/(n d +1) corresponding to η = η0 provides a better estimation and leads to the asymptotically optimal asymptotics (27). Indeed, for sufficiently small A, one gets f ∈ Lipn d L and the differentiator is to exactly differentiate the noise. k The unbounded signals η = A dtd k cos(ωt)β , β ∈ (k − 1, k), k = 1, 2, ..., feature the filtering order k and the integral magnitude A.  Consider now the output-feedback closed-loop system x˙ = a(t, x) + b(t, x)u, σˆ = σ(t, x) + η(t), u = αUr (z), L ≥ C + K M α sup |Ur |, L > 0, ˆ L). (w, ˙ z˙ )T = Dn f ,r −1 (w, z 0 − σ,

(50)

The following theorem follows [38]. Theorem 7 Under Assumption 2 after some FT transient closed system (50) converges into the region |σ i | ≤ γi ρr −i , i = 0, 1, ..., r − 1, for some constant γi > 0. The result holds for any ρ provided by (48) if M = 0, and for sufficiently small ρ if M > 0. System preserves its practical stability for any ρ. The next notion extends the corresponding definition from [59] and is employed to demonstrate that the conditions on the noise are actually of the local nature. A locally integrable Lebesgue-measurable function ν(t), ν : [0, ∞) → R, is called locally T -filterable signal of the filtering order k > 0 and the integral magnitudes a0 , a1 , ..., ak−1 ≥ 0, if there exists an infinite sequence t0 , t1 , ..., t0 ≥ 0, t j+1 − t j ≥ T > 0, j = 0, 1, ..., such that for each j there exists a Caratheodory solution ξ(t), t ∈ [t j , t j+1 ], of the equation ξ (k) (t) = ν(t) which satisfies |ξ (l) (t)| ≤ al for l = 0, 1, ..., k − 1. The number al is called the local (k − l)th-order integral magnitude of ν. Signals of local filtering order 0 are trivially defined as uniformly essentially bounded Lebesgue-measurable signals of the magnitude a0 , ess supt≥0 |ν(t)| ≤ a0 . In particular, locally filterable noises can be concatenated producing new locally filterable noises. The following lemma [59] shows that filtering differentiators can be applied when the noises are only locally filterable.

Homogeneous Sliding Modes in Noisy Environments

23

Lemma 1 Any signal ν(t) of the local T -filtering order k ≥ 0 can be represented as ν = η0 + η1 + ηk , where η0 , η1 , ηk are signals of the (global) filtering orders 0, 1, k, respectively. Their magnitudes continuously depend on a k−1 and T . In particular, in the important case k = 1 get ν = η0 + η1 , where |η0 | ≤ a0 /T , and the first-order integral magnitude of η1 is 2a0 . In the general case k > 1, fix any 1/k 1/(k−1) , ..., ak−1 ], number ρˆ0 > 0. Then, provided ρˆ ≤ ρˆ0 holds for ρˆ = max[a0 , a1 ˆ , γ1 ρ, ˆ γk ρˆk , the integral magnitudes of the signals η0 , η1 , ηk are calculated as γ0 ρ/T respectively, where the constants γ0 , γ1 , γk > 0 only depend on k and ρˆ0 .

7.2 Filtering Noises in Discrete Time Once more, let the sampling take place at the times t0 , t1 , . . . , t0 = 0, t j+1 − t j = τ j ≤ τ . Due to the Nyquist–Shannon sampling rate principle noises small in average under one sequence {t j } can be large under another. Therefore, the admissible sampling-time sequences are to exist for any τ > 0. Correspondingly, the set of such sequences is infinite. A discretely sampled signal ν : R+ → R is said to be of the global sampling filtering order k ≥ 0 and the global kth-order integral sampling magnitude a ≥ 0 if for each admissible sequence t j there exists a discrete vector signal ξ(t j ) = (ξ0 (t j ), ..., ξk (t j ))T ∈ Rk+1 , j = 0, 1, ..., satisfying the relations δ j ξi = ξi+1 (t j )τ j , i = 0, 1, ..., k − 1, ξk (t j ) = ν(t j ), |ξ0 (t j )| ≤ a. Theorems similar to Theorems 6, 7 hold also here [38, 59]. Assumption 3 The noise η(t j ) admits an expansion of the form η(t j ) = η0 (t j ) + η1 (t j ) + ... + ηn f (t j ), where each ηk , k = 0, ..., n f , is a signal of the global sampling  filtering order k and the kth-order sampling integral magnitude εk ≥ 0. Introduce parameter ρ measuring the discrete filtered sampling noise intensity,  ρ = max τ ,

ε0  L

1 n d +1

,

ε1  L

1 n d +2

, ...,

ε  nf

1 n d +n f +1

L

 .

(51)

Theorem 8 Under Assumptions 1, 3, after some FT transient, the hybrid filtering differentiator (45), (46) provides for the accuracy (49) for ρ, τ small enough if M > 0, or for any ρ if M = 0. The practical stability of the filter is preserved for any ρ if M = 0 and for sufficiently small τ if M > 0. Theorem 9 Under Assumption 3, the closed system (47) in FT stabilizes in the region |σ i | ≤ γi ρr −i , γi > 0, i = 0, 1, ..., r − 1, for ρ defined by (51). It holds for any ρ ≥ 0 if M = 0, and for sufficiently small ρ, τ if M > 0. The system practical stability is preserved for any ρ if M = 0 and for sufficiently small τ if M > 0. The following notion extends the similar one from [59].

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A discretely sampled signal ν(t j ) is said to be locally T -filterable of the local sampling filtering order k > 0 and the integral magnitudes a0 , a1 , ..., ak−1 ≥ 0, if there exists an infinite sequence tˆ0 , tˆ1 , ..., tˆ0 ≥ 0, tˆl+1 − tˆl ≥ T > 0, l = 0, 1, ..., such that for any sufficiently small τ , admissible sequence {t j }, and any l ≥ 0 there exists a discrete vector signal ξ(t j ) = (ξ0 (t j ), ..., ξk (t j ))T ∈ Rk+1 , j = j0 , j0 + 1, ..., j1 , t j0 ∈ [tˆl , tˆl + τ ), t j1 ∈ (tˆl+1 − τ , tˆl+1 ], which satisfies the relations δ j ξi = ξi+1 (t j )τ j , i = 0, 1, ..., k − 1, ξk (t j ) = ν(t j ), |ξi (t j )| ≤ ai .

(52)

Numbers ai are called the local (k − i)th-order sampling integral magnitudes of ν. Signals of local sampling filtering order 0 by definition are just bounded signals of the magnitude a0 . Similarly to the continuous-time case, one can concatenate locally filterable signals. Lemma [59] similar to Lemma 1 justifies application of Theorems 8, 9 in the case of locally filterable sampled noises. Lemma 2 Let all admissible sampling-time sequences satisfy the condition sup τ j ≤ cτ inf τ j for some cτ > 0. Then any discretely sampled signal ν(t j ) of the local sampling T -filtering order k ≥ 0 can be represented as ν = η0 + η1 + ηk , where η0 , η1 , ηk are signals of the (global) sampling filtering orders 0, 1, k. In particular, if k = 1 get ν = η0 + η1 , where |η0 | ≤ a0 /T , and the first-order integral sampling magnitude of η1 is 2a0 . If k > 1 fix any number ρ0 > 0. Then, 1/k 1/(k−1) , ..., ak−1 ] ≤ ρ0 the sampling integral magnitudes provided ρ = max[a0 , a1 of the signals η0 , η1 , ηk are calculated as γ0 ρ/T , γ1 ρ, γk ρk , respectively, where the constants γ0 , γ1 , γk > 0 only depend on k and ρ0 . It is easy to prove that any bounded continuous periodic signal featuring a local T -filtering order is transformed into a discrete signal of the same sampling filtering order, provided sup τ j ≤ cτ inf τ j holds for some cτ > 0. It follows from the convergence of the Euler approximations to the unique solutions of DEs [31]. Also the smaller τ the closer are the integral sampling magnitudes to those of the original continuous-time signal. Any bounded periodic noise of a global filtering order is trivially of the same local filtering order. Correspondingly, Lemma 2 establishes its effective suppression by a filtering or a hybrid filtering discrete differentiator. It is wrong to claim that sampling any globally filterable signal of the order k produces a discrete signal of the same global sampling filtering order k. Indeed, a multiple numeric integral of the unbounded signal from Example 3 can become very large for some concrete sampling sequence t j and even cause computer overflow. The issue is resolved by introducing a saturation of the sampled periodic unbounded signal, even if a very high saturation level is taken.

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8 Numeric Differentiation 8.1 Numeric Homogeneous Differentiation Consider the noisy input signal f (t) = f 0 (t) + η(t), f 0 (t) = 0.5 cos(t) + 0.9 sin(0.5t + log(t + 1)).

(53)

Obviously, for each k > 0, the inequality | f 0(k) (t)| ≤ 1 holds starting from some moment. Let the noise η be composed of three components η(t) = η1 (t) + η2 (t) + η3 (t), η1 (t) ∈ N (0, 0.22 ), η2 (t) = 107 cos(108 t),   − 1 1 η3 (t) = 0.1 cos(104 t) sin(104 t) 2 = 2 · 10−5 dtd sin(104 t) 2 ,

(54)

where η1 is a random Gaussian signal of the standard deviation 0.2, η2 is a large highfrequency harmonic signal, and η3 is an unbounded signal of the filtering order 1 and the integral magnitude 2 · 10−5 (Example 3). The noise components are presented in Fig. 1. Apply the discrete filtering differentiator (45) of the differentiation order n d = 2 and the filtering order n f = 8 with the parameters (37), L = 1, M = 0, and the constant sampling step τ j = τ = 10−6 . The simulation is performed over the time interval [0, 25]. Performance of the discrete filtering differentiator in the absence of noise is presented in Fig. 2. Practically exact convergence is demonstrated. The resulting accuracy is presented by the component-wise inequality

Fig. 1 Graphs of the noise components (54). η1 is a Gaussian noise, η2 is a large high-frequency harmonic noise, and η3 is an unbounded noise of the filtering order 1

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Fig. 2 Performance of the discrete filtering differentiator (45) with n d = 2, n f = 8, L = 1, M = 0, τ = 10−6 in the absence of noises, η = 0, for the input (53). Estimations of f 0 , f˙0 , f¨0 are shown

Fig. 3 Performance of the discrete filtering differentiator with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (53) corrupted by the Gaussian sampling noise η = η1 ∈ N (0, 0.22 ). Estimation of f 0 , f˙0 , f¨0 is shown

(|w1 |, |w2 |, |w3 |, |w4 |, |w5 |, |w6 |, |w7 |, |w8 |, |z 0 − f 0 |, |z 1 − f˙0 |, |z 2 − f¨0 |) ≤ (1.6 · 10−54 , 2.7 · 10−48 , 2.6 · 10−42 , 1.3 · 10−36 , 4.0 · 10−31 , 7.0 · 10−26 , 6.2 · 10−21 , 2.7 · 10−16 , 4.9 · 10−12 , 4.8 · 10−5 , 1.8 · 10−4 ). (55) Performance of the differentiator separately for each noise component is demonstrated in Figs. 3, 4, and 5. The accuracy obtained for the Gaussian noise η = η1 is (|w1 |, |w2 |, |w3 |, |w4 |, |w5 |, |w6 |, |w7 |, |w8 |, |z 0 − f 0 |, |z 1 − f˙0 |, |z 2 − f¨0 |) ≤ (1.2 · 10−23 , 3.7 · 10−20 , 6.8 · 10−17 , 6.4 · 10−14 , 3.4 · 10−11 , 1.1 · 10−8 , 1.6 · 10−6 , 1.3 · 10−4 , 2.6 · 10−3 , 2.9 · 10−2 , 1.7 · 10−1 ). (56) It has been shown in a qualitative way [52] that random non-correlated sampled noises of the same distribution feature the first filtering order, and can be practically canceled for sufficiently small sampling constant step. In other words, the integral magnitude of the noise tends to zero as the sampling rate tends to infinity, but this

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Fig. 4 The discrete filtering differentiator with n d = 2, n f = 8, L = 1, τ = 10−6 with the input (53) is almost insensitive to the noise η = η2 = 107 cos(108 t) featuring both extremely large magnitude and extremely high frequency. Estimation of f 0 , f˙0 , f¨0 is shown

Fig. 5 Performance of the discrete filtering differentiator with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (53) corrupted by the unbounded noise η = η3 of the filtering order 1. Estimation of f 0 , f˙0 , f¨0 is shown

convergence is very slow. Increasing the filtering order n f does not significantly affect the differentiator performance in that case. Contrary to the Gaussian noises harmonic noises of high frequency are very well filtered (Fig. 4). The higher the filtering order the better is the result. It is shown in Example 3 that the influence of small and large harmonic noises of the same frequency is almost the same for large n f . In that case, the noise-intensity parameter ρ i

n d +1−i

approaches 1/ω where ω is the noise frequency, and |z i − f 0(i) | ≤ γi L n d +1 ρ n d +1 . Thus, from some moment further increasing n f does not provide an accuracy improvement. The accuracy obtained for the harmonic noise η2 is (|w1 |, |w2 |, |w3 |, |w4 |, |w5 |, |w6 |, |w7 |, |w8 |, |z 0 − f 0 |, |z 1 − f˙0 |, |z 2 − f¨0 |) ≤ (1.0 · 10−25 , 4.9 · 10−22 , 1.4 · 10−18 , 2.0 · 10−15 , 1.6 · 10−12 , 8.6 · 10−10 , 3.7 · 10−5 , 19.1, 4.0 · 10−4 , 7.9 · 10−3 , 8.3 · 10−2 ). (57)

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The considered unbounded noise has the filtering order 1 (Example 3, Fig. 5). It means that increasing n f > 1 does not improve accuracy, which is determined by the noise average value. Noises of the type η1 and η3 are most difficult for the filtering differentiator. The accuracy obtained for the unbounded noise η = η3 is (|w1 |, |w2 |, |w3 |, |w4 |, |w5 |, |w6 |, |w7 |, |w8 |, |z 0 − f 0 |, |z 1 − f˙0 |, |z 2 − f¨0 |) ≤ (7.3 · 10−24 , 2.3 · 10−20 , 4.4 · 10−17 , 4.4 · 10−14 , 2.5 · 10−11 , 8.0 · 10−9 , 1.3 · 10−6 , 1.2 · 10−4 , 2.3 · 10−3 , 2.3 · 10−2 , 1.6 · 10−1 ). (58) Note that the high frequencies of η2 , η3 form a special challenge for the differentiator. In fact, the authors cannot rigorously explain, how the differentiator removes a noise of a period which is at least 16 times less than the sampling step (also see the simulation in Sects. 9.2.2 and 9.3.1). The performance of the filtering differentiator for the input (53) in the presence of the combined noise (54) is presented in Figs. 6 and 7. The resulting accuracy for t ∈ [20, 25] is provided by the component-wise inequality Fig. 6 Performance of the discrete filtering differentiator, n d = 2, n f = 8, L = 1, τ = 10−6 , for the input (53) and the combined noise (54). Estimation of f 0 , f˙0 , f¨0 is shown

Fig. 7 Performance of the discrete filtering differentiator, n d = 2, n f = 8, L = 1, τ = 10−6 , for the input (53) and the combined noise (54). A zoom of the approximation graphs z 0 , f 0 , z 1 , f˙0 and z 2 , f¨0 is shown

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Fig. 8 Performance of the discrete filtering differentiator with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (53) and the combined noise (54). The initial state is z(0) = (100, −10, 10). Estimation of f 0 , f˙0 , f¨0 is shown

Fig. 9 Performance of the discrete filtering hybrid differentiator with n d = 2, n f = 8, L = 1, M = 1, τ = 10−6 for the input (53) and the combined noise (54). The initial state is z(0) = (100, −10, 10). The convergence is significantly faster compared with the filtering differentiator (the case M = 0). Estimation of f 0 , f˙0 , f¨0 is shown

(|w1 |, |w2 |, |w3 |, |w4 |, |w5 |, |w6 |, |w7 |, |w8 |, |z 0 − f 0 |, |z 1 − f˙0 |, |z 2 − f¨0 |) ≤ (5 · 10−23 , 1.4 · 10−19 , 2.2 · 10−16 , 1.8 · 10−13 , 8.6 · 10−11 , 2.3 · 10−8 , 3.9 · 10−5 , 19, 0.003, 0.029, 0.167).

(59)

Note that w8 has seemingly absorbed the main part of the noise. Compare the accuracies (56), (57), (58) obtained separately for each noise component with the accuracy (59) obtained for the composite noise (54). One clearly sees that there is no superposition principle. The overall maximal errors are closer to the maximal errors obtained for each noise component than to their sum (Figs. 8 and 9).

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8.2 Numeric Hybrid Differentiation Filtering differentiators feature slow convergence rate from significant initial errors. Hybrid filtering differentiators provide practically the same accuracy for the considered noises, but feature much faster convergence. Consider the same noisy input but with the non-zero differentiator initial state z 0 (0) = 100, z 1 (0) = −10, z 2 (0) = 10. Then the above filtering differentiator, corresponding to M = 0, has the convergence time of about 15 time units, whereas the filtering hybrid differentiator with M = 1 demonstrates the convergence time of about 5 units. The larger the initial errors the larger the difference. For really large initial errors, implementation of the homogeneous SM-based differentiators becomes impossible (see the simulation in Sect. 9.3.1).

8.3 Comparison with the Kalman Filter Compare the performance of the standard Kalman filter (KF) and the filtering differentiator (FD). Once more consider the input signal (53) f (t) = f 0 (t) + η(t), f 0 (t) = 0.5 cos(t) + 0.9 sin(0.5t + log(t + 1)),

(60)

where η is a noise. As mentioned previously for each k from some moment, | f 0(k) | ≤ 1 holds. The filtering differentiator is once more of the differentiation order n d = 2 and the filtering order n f = 8, L = 1, τ = 10−6 . The Kalman prediction and innovation equations are xˆ j+1 =  j xˆ j , (61) y(t j ) = f (t j ) − H xˆ j , where xˆ j and y(t j ), respectively, are the estimation of ( f 0 , f˙0 , f¨0 )T and the Kalman innovation. The state transition and the measurement models are ⎡ 2 ⎤ 1 τ τ2   j = ⎣ 0 1 τ ⎦ , H = 1 0 0 , (62) 00 1 respectively. The covariance matrix of xˆ j is propagated with the noise covariance matrix ⎡ ⎤ 000 Qj = ⎣0 0 0⎦. (63) 00τ

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Fig. 10 Performance of numeric filtering differentiator with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (60) corrupted by the Gaussian noise η(t) ∈ N (0, 0.22 ). Estimation of f 0 is shown

Fig. 11 Performance of Kalman filter (KF) and the filtering differentiator (FD) with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (60) and the Gaussian noise η(t) ∈ N (0, 0.22 ). Estimation of f 0 , f˙0 , f¨0 is shown

The Kalman update is applied with an appropriate scalar measurement-noise covariance matrix R ∈ R to be specified further. First consider the Gaussian noise η(t) ∈ N (0, 0.22 ) of the standard deviation 0.2. Correspondingly, R = 0.22 is taken. Performance of both filters is presented in Figs. 10 and 11. The resulting accuracy for t ∈ [8, 10] is provided by the componentwise inequality K F : (|z 0 − f 0 |, |z 1 − F D : (|z 0 − f 0 |, |z 1 −

f˙0 |, |z 2 − f˙0 |, |z 2 −

f¨0 |) ≤ (0.003, 0.046, 0.395), f¨0 |) ≤ (0.00278, 0.02364, 0.113).

(64)

Consider a large high-frequency harmonic noise η(t) = 100 cos(108 t). In this case, two different measurement covariance matrices are considered: R = 0.22 (as previously) and R = 1002 corresponding to the noise magnitude. Performance of

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Fig. 12 Performance of the FD with n d = 2, n f = 8, L = 1, τ = 10−6 for the input (60) corrupted by the harmonic noise η(t) = 100 cos(108 t). Estimation of f 0 is shown

Fig. 13 Performance of the KF with R = 1002 and the FD with n d = 2, n f = 8, L = 1, τ = 10−6 in the presence of the harmonic noise η(t) = 100 cos(108 t). Estimation of f 0 , f˙0 , f¨0 is shown

both filters is presented in Figs. 12 and 13. The corresponding accuracies for t ∈ [8, 10] are as follows: f¨0 |) ≤ (0.0065, 0.1135, 0.99), f¨0 |) ≤ (0.03, 0.1323, 0.2754), f¨0 |) ≤ (3.9 · 10−5 , 0.003, 0.029). (65) Note that in order to provide for the good performance of the Kalman filter one needs to adjust the covariance parameter R using some knowledge on the sampling noise. Contrary to this, we do not change the parameters of the filtering differentiator, and do not need to know whether any noise is present. K F(R = 0.22 ) : (|z 0 − f 0 |, |z 1 − K F(R = 1002 ) : (|z 0 − f 0 |, |z 1 − FD : (|z 0 − f 0 |, |z 1 −

f˙0 |, |z 2 − f˙0 |, |z 2 − f˙0 |, |z 2 −

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9 Output-Feedback Control Simulation In this section, we demonstrate the efficiency, application simplicity, and robustness of the developed SM controllers and observers. Two different academic examples are presented. The first one is a disturbed integrator chain of the relative degree 3, whereas the second one is a slightly modified one-link robot inspired by the classical example [40] of the relative degree 4.

9.1 Homogeneous SM Control Development Let the relative degree be r ∈ N. Then the r -SM homogeneity weights are deg σ = r, deg σ˙ = r − 1, ..., deg σ (k) = r − k, 0 ≤ k ≤ r . Choose a homogeneous norm valid for k < r : 1

1

|| σk ||h∞ = ||(σ, ..., σ (k) )||h∞ = max[|σ| r , ..., |σ (k) | r −k ]. Any other homogeneous norm can be chosen here. Also, recall that σ r −1 constitute the r -SM homogeneous coordinates.

9.1.1

4-SMC Development

First develop a universal 4-SMC. The homogeneity weights of the sliding variables ... are deg σ = 4, deg σ˙ = 3, deg σ¨ = 2, deg σ = 1; deg t = 1. According to Sect. 4.1, start with the first-order homogeneous FTS DE σ˙ + β0 σ3/4 = 0, β0 > 0. Any value β0 > 0 is valid. Choose and substitute β0 = 1. The second-order DE has already infinitely many options (Sect. 4.1). Choose σ¨ +

1  2 σ˙ β1 || σ1 ||h∞

+

3 σ 4

1 2

= 0, β1 > 0.

According to Theorem 3, it is FTS for sufficiently large β1 > 0. Simulation shows that β1 = 1 fits. The third-order FTS DE is chosen in the form ⎢ ⎤1 ⎢ 1 2 1  3 ⎢ 2 ... 2 σ + β2 ⎣σ¨ + || σ1 ||h∞ σ˙ + σ 4 ⎥ ⎥ = 0. ⎥

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Simulation shows that β2 = 5 provides for the FT stability. At the last step, choose the 4-SM QC control ⎢ ⎤1 ⎢ ⎢ ⎤1 2 1 ⎢ ⎢ 2  1 1  3 2 ⎢ ⎥ − ⎢... 2 ⎥ ⎥ , ⎣σ¨ + || 4 σ σ σ + 5 σ ˙ + σ3 ||h∞2 ⎢ || u( σ3 ) = −α|| 1 h∞ ⎥ ⎥ ⎣ ⎥ ⎥ ⎥

(66)

where the free parameter α defines the control magnitude. Obviously, deg u = 0. When applied in the output feedback the differentiator outputs z i are to be substituted for σ (i) , i = 0, 1, 2, 3.

9.1.2

3-SMC Development

Development of a universal 3-SM controller is even simpler. The homogeneity weights of the sliding variables are deg σ = 3, deg σ˙ = 2, deg σ¨ = 1; deg t = 1. Once more start with the simplest first-order homogeneous FTS DE σ˙ + β0 σ2/3 = 0, β0 = 1. The second-order FTS DE is similarly chosen as  1 2 2 σ¨ + β1 σ˙ + σ 3 = 0, β1 > 0. Once more the simulation shows that β1 ≥ 1 suffices. Choose β1 = 2. Now the 3-SM controller is chosen as ⎢

u( σ2 ) =

1⎢ − ⎢ −α|| σ2 ||h∞2 ⎣σ¨

 + 2 σ˙ +

2 σ 3

1 2

⎤1 2

⎥ . ⎥ ⎥

(67)

It is easy to see the general form of r -SM controllers incorporating controllers (67) and (66) for r = 3 and r = 4, respectively.

9.1.3

Output-Feedback Control: Choice of Initial Observer State

In the case when only the tracking error σ is available, an observer is to provide for the estimations of σ r −1 . However, observer application requires assignment of its initial values.

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One of the ways is to choose a FxT stable observer/differentiator like [3, 21]. Such differentiator is very sensitive to sampling noises and intervals, especially to the large sampling noises we consider, and can simply diverge [54]. Another solution proposed in the past by the authors suggests approximate calculation of the initial derivative values. For this end, one uses finite differences over r sampling intervals of a reasonable length. The calculation can be repeated and some average values can be taken for larger noises. That approach leads to the immediate differentiator convergence, if the noises are small. Unfortunately, the initial error can happen to be very large in the presence of significant noises. Homogeneous differentiators [46, 59] are known to slowly converge from large initial errors (see Sect. 9.3). In the sequel, we demonstrate that the hybrid (bihomogeneous) filtering differentiators solve this problem converging fast even from large initial errors.

9.2 Output-Feedback Control of the Integrator Chain Consider the disturbed third-order integrator chain ... x = cos(x 2 + x¨ + 100t + 1) + yc (t) = cos(0.5t) + 0.6 sin t,

3+2 cos2 (1000t) u, 1+cos2 (1000t)

y = x,

(68)

where y is the output of the system and the signal yc (t) is to be tracked. The tracking error is correspondingly defined as σ y = y − yc . The relative degree of system (68) is 3. It is easy to see that the tracking error σ y satisfies ... σ y = h y (t, x, x, ˙ x) ¨ + g y (t, x, x, ˙ x)u, ¨ |h y | ≤ 2, g y ∈ [2, 3].

(69)

Therefore, each of its solution satisfies the DI ... σ y ∈ [−2, 2] + [2, 3]u.

(70)

Apply control (67) for α = 5, ⎢

u = −5 ·

1⎢ − ⎢ 2⎣ || σ y2 ||h∞ σ¨ y

⎤1 1 2  2   2 + 2 · σ˙ y + σ y 3 ⎥ ⎥ . ⎥

(71)

The Euler integration method is applied with the integration step τ = 10−6 and the initial conditions (x(0), x(0), ˙ x(0)) ¨ = (50, −50, 50).

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3-SM Control with Exact Measurements

First assume that all derivatives of σ be available in real time. The corresponding performance of the system is presented in Figs. 14, 15, and 16. The obtained tracking accuracies are |y − yc | < 4 · 10−10 , | y˙ − y˙c | < 5 · 10−7 and | y¨ − y¨c | < 5 · 10−5 for t > 25.

Fig. 14 Tracking error σ y = y − yc and its derivatives σ˙ y , σ¨ y vs time in the case of the 3-SM control (71) with full exact measurements

Fig. 15 Zoom of the tracking graphs for y(t), yc (t) and their derivatives in the 3-SM control (71) in the case of the 3-SM control (71) with full exact measurements

Fig. 16 3-SM control (71) in the case of full exact measurements

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Now consider the case when only the tracking error σ y is available, and an observer/differentiator is to provide the estimations of σ y2 . The differentiator outputs z i are substituted for σ (i) y , i = 0, 1, 2, in the controller (71). The hybrid filtering differentiator (34), (35) is chosen with L = 100, M = 0.5, n d = 2, n f = 7, z(0) = (1000, −1000, 1000). We intentionally choose large initial observer values to demonstrate its fast convergence. The parameters λi , i = 0, ..., 9, are taken from (37). The differentiator discrete version (45), (46) is employed. The performance of the output-feedback 3-SM control in the absence of noises is demonstrated in Figs. 17 and 18. The resulting accuracy is |y − yc | < 5 · 10−10 , | y˙ − y˙c | < 2 · 10−6 and | y¨ − y¨c | < 6 · 10−3 for t > 40.

9.2.2

Output-Feedback 3-SM Control in the Presence of Noises

Let now σ y = y − yc be measured with the noise η(t) = η1 (t) + η2 (t) + η3 (t), η1 (t) ∈ N (0, 0.52 ), η2 (t) = 108 sin(5 · 108 t), η3 (t) = 0.2 · sin(500000t) · | cos(500000t)|−0.5 ,

(72)

where η1 (t) is a Gaussian noise of the standard deviation 0.5, η2 (t) is a harmonic noise of extremely high magnitude and frequency, and η3 (t) is an unbounded noise (Fig. 19, Example 3). We preserve the same controller, differentiator, initial values, and the sampling interval. Note that not only the noise magnitude, but also its frequency are challenging for the differentiator. Indeed, the sampling frequency is very low compared with the frequencies of both η2 and η3 . Under the considered sampling with interval

Fig. 17 3-SM output-feedback control in the absence of noises. Control signal and convergence of the differentiator output z 0 to the tracking error σ y

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Fig. 18 3-SM output-feedback tracking performance in the absence of noises

Fig. 19 Noises (72): the Gaussian noise η1 , the harmonic noise η2 , the unbounded noise η3

τ = 10−6 both signals can be considered as discrete random signals of a not clear distribution. The numeric evaluation of the integrals is not valid, since the number of the integration points per period is less than 0.1 in the first case and less than 3 in the second. In particular, the peaks appearing in the graph of η3 (Fig. 19) are caused by some digital resonance due to the finite number of the meaningful computer number digits. Note that the differentiator indeed diverges for the harmonic-signal frequency exceeding 1010 .

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Fig. 20 Output-feedback 3-SM control (71) under the noise measurements. Control and convergence of y(t) to yc (t)

Performance of the system in the presence of the combined noise η(t) is demonstrated in Fig. 20. The tracking accuracy is (|σ y |, |σ˙ y |, |σ¨ y |) ≤ (0.15, 1, 6.5) for t > 45.

9.3 Output-Feedback Robot SMC Consider the academic example of a one-link robot with a joint elasticity, inspired by [40] (Fig. 21), J1 q¨1 = u + K (t)(q2 − q1 ) − F1 q˙1 ; J2 q¨2 = −K (t)(q2 − q1 ) − F2 q˙2 − mgn d cos(q2 ).

(73)

Here q1 and q2 are the angular positions, J1 and F1 represent inertia and viscous constants of the actuator, and K (t) is the elasticity of the spring in an uncertain way depending on the environment conditions. Control u is the torque produced at the actuator axis. Similarly, J2 and F2 are the corresponding constants of the link, m and d represent the mass and the distance to the gravity center of the link, and gn = 9.81 is the free-fall acceleration.

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Fig. 21 A one-link robot

The system would be feedback-linearizable, if K (t) were a known constant. Let J1 = 1, F1 = F2 = 1, J2 = md 2 = 1, m = 0.25, d = 2, gn = 9.81. The “unknown” function K (t) and the signal q2c (t) to be tracked are chosen as K (t) = 5 + sin t and q2c (t) = cos(0.5t) + 0.6 sin t. The tracking error is defined as σ = q2 − q2c . The system relative degree is 4, since q2(4) = ... + K /(J1 J2 )u. Correspondingly, a 4-SMC of the form (66) is applied.

9.3.1

Robot Output-Feedback 4-SM Control

Obviously, system (73) satisfies a DI of the form (13) only locally. Therefore, the developed SMC is also only locally effective. In order to start the control, one needs some initial values for the differentiator. Once more we choose large initial values of the differentiator which naturally correspond to an attempt to algebraically evaluate the initial tracking-error derivatives in the presence of large noises. Apply the hybrid filtering differentiator (45), (46) with n d = 3, n f = 7, L = 150, M = 0.5, z(0) = (10000, −12000, 20000, −10000). Note that q2(4) grows fast with the norm of the system state of (q1 , q˙1 , q2 , q˙2 ), and the value L = 150 is not exaggerated. Choose the system initial values (q1 , q˙1 , q2 , q˙2 ) = (1, −1, 1, −1). Let the sampling step be τ = 10−6 . Apply control (66) with α = 10 and z i substituted for σ (i) , i = 0, 1, 2, 3. In order to feed the control with reasonably accurate derivative estimations, the control is only applied at t = 10 providing the time for the observer convergence. Performance of the system in the absence of noises is presented in Figs. 22 and 23. The system converges to the region ... (|σ|, |σ|, ˙ |σ|, ¨ | σ |) ≤ (1.2 · 10−6 , 5.3 · 10−7 , 3.7 · 10−5 , 0.06). Let now σ be measured with the noise η = 105 cos(107 t) (Fig. 24). Once more note that the noise frequency is way too high for the sampling/integration interval

Homogeneous Sliding Modes in Noisy Environments

41

Fig. 22 Robot 4-SMC, hybrid differentiator in the feedback: tracking performance and control in the absence of noise. Control is applied from t = 10

Fig. 23 Robot 4-SMC, hybrid differentiator in the feedback in the absence of noises. Below: convergence of the tracking errors ... σ, σ, ˙ σ, ¨ σ to zero; above: convergence of the differentiator outputs z i to σ i , i = 0, 1, 2, 3. Control is applied from t = 10 in order to provide some time for the differentiator convergence

τ = 10−6 . The corresponding performance of the output-feedback controller is shown in Fig. 25. The tracking accuracy is ... (|σ|, |σ|, ˙ |σ|, ¨ | σ |) ≤ (1.9 · 10−2 , 0.026, 0.18, 4.5).

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Fig. 24 Robot 4-SMC, the noise η = 105 cos(107 t) of the sampled tracking error σ = q2 − q2c

Fig. 25 Robot 4-SMC, hybrid differentiator in the feedback. Performance in the presence of the noise η = 105 cos(107 t). Above: the tracking of q2c by the angle q2 and the graph of the angle q1 . Below: the control is applied from t = 10

Let us now check the performance of the homogeneous filtering differentiator with exactly the same parameters L = 150, n d = 3, n f = 7, but M = 0, in the absence of noises. Both differentiators are applied in the same feedback and all the parameters, initial values, etc. are the same as above. The only difference is in the parameter M. The results are presented in Fig. 26. While the filtering differentiator is very stable and converges to the exact values of σ 3 in FT, the convergence time is so long here that its application is practically impossible.

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Fig. 26 Robot 4-SMC, with a differentiator in the feedback. Comparison in the absence of noises for the same initial values z(0) = (10000, −12000, 20000, −10000). Above: convergence of the hybrid differentiator (M = 0.5). Below: practical divergence of the filtering differentiator

10 Conclusion New methodology of homogeneous SM control design and homogeneous/ bi-homogeneous SM-based observation is presented. Extensive numeric experiments demonstrate the effectiveness of the technique in the presence of very large and even unbounded sampling noises.

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A Lyapunov based Saturated Super-Twisting Algorithm Ismael Castillo, Martin Steinberger, Leonid Fridman, Jaime A. Moreno, and Martin Horn

Abstract Two different structures of saturated super-twisting algorithms are presented. Both structures switch from a relay controller to super-twisting algorithm through a switching law that is based on Lyapunov-level curves allowing the algorithms to generate bounded control signals. The relay controller provides a saturated control signal enforcing the system trajectories to reach a predefined neighborhood of the origin in which the super-twisting algorithm dynamics does not saturate, ensuring finite-time convergence to the origin. In order to increase the maximal admissible bound of the perturbations, the second algorithm also includes a perturbation estimator setting super-twisting’s integrator to the theoretically exact perturbation estimation. Experimental results are presented to validate the proposed algorithms.

I. Castillo (B) · M. Steinberger · M. Horn Institute of Automation and Control, Graz University of Technology, 8010 Graz, Austria e-mail: [email protected] M. Steinberger e-mail: [email protected] M. Horn e-mail: [email protected] L. Fridman Facultad de Ingeniería, Universidad Nacional Autónoma de México (UNAM), 04510 Mexico City, Mexico e-mail: [email protected] J. A. Moreno Instituto de Ingeniería, Universidad Nacional Autónoma de México (UNAM), 04510 Mexico City, Mexico e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_2

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1 Introduction The Super-Twisting Algorithm [8, 9] (STA) is one of the most frequently employed algorithms in sliding mode theory. It was designed to replace a First-Order Sliding Mode Controller (FOSMC) which generates a discontinuous control signal, by a continuous one to avoid high-frequency switching in the control signal. It allows the theoretically exact rejection of Lipschitz perturbations and ensures a quadratic precision of the output with respect to the sampling step due to its homogeneity properties. In addition, a second-order sliding mode is achieved in finite time, i.e., the sliding variable and its derivative are robustly driven to zero in finite time. It has been widely used in the conventional sliding mode design, where systems of higher order and nonlinear dynamics can be reduced into the desired sliding dynamics of co-dimension one, i.e., a sliding variable of relative degree one, covering a wide class of systems [5, 14, 17]. Even more, it has been used for robust exact differentiation [9] among several higher order sliding mode differentiators [3, 12]. The original version of STA, as it was introduced in Levant’s theorem 5, p. 1257 [8], is a saturated control law, i.e., the control signal is bounded. To ensure the saturation, the author proposed a switching strategy saturating the term of STA that is proportional to the square root of the state as well as the integral term separately. However, this switching logic can generate undesired oscillations along the saturation value, which is shown in the example. In contrast to the original version, the most popular form of the STA [9] is given by 1 u = −α1 |x| 2 sign (x) + z, (1) z˙ = −α2 sign (x) , where x is the state of a first-order system, and α1 , α2 , two positive constants. The first term of the control law is a nonlinear function proportional to the square root of the state while the second one is a nonlinear integral term. The STA works as a nonlinear proportional-integral controller leading to potentially unbounded control signals. However, in practical implementations, the control effort is always limited. It is well known [7] that the application of controllers with integral action in feedback loops with bounded control inputs leads to the so-called integral windup effect. This refers to the situation where a significant change in the set point causes actuator saturation and as a result the error in the integral term is accumulated significantly. This leads to undesired overshoot or even to instability. Some classic anti-windup techniques make use of disabling the integral function until the variable to be controlled has entered a region where the control signal does not saturate or use additional feedback of the difference between designed and saturated control signal. In this chapter, two different structures of Saturated Super-Twisting Algorithms (SSTA) are proposed, using an anti-windup technique in order to keep the STA’s control signal within predefined bounds. The contributions are (a) In the first SSTA, a switching law is used to combine a Relay Controller (RC) with a STA to drive the system trajectory to zero in finite time fulfilling a saturation

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condition. The switching condition is designed based on a Positively Invariant Set (PINS) formed by the level curve of the Lyapunov function lying between the saturation sets. (b) The second algorithm allows to improve the performance of the closed-loop system by means of an additional perturbation estimator which takes advantage of the time intervals where the RC controller is active. Compared to Castillo et al. [1], a detailed proof is given and the proposed approach is applied to a real-world mechanical system. Prescribed finite-time convergence gains of the estimator are obtained with the Lyapunov function from Polyakov et al. [13]. This version of SSTA allows the rejection of perturbations with higher magnitude. (c) The proposed approaches are compared to STA versions by Levant [8] and Golkani et al. [6] who also take into account saturation. Experiments on a realworld mechanical system are carried out to illustrate the performance of the proposed SSTA. The paper is organized as follows. Section 2 introduces the problem and Sect. 3 presents the first SSTA algorithm. Section 4 introduces the second algorithm using a perturbation estimator. After a detailed comparison of the different SSTA strategies using numerical simulations in Sect. 5, experimental results for a mechanical plant are presented in Sect. 6. Finally, Sect. 7 summarizes and concludes the work.

2 Problem Statement Consider the first-order perturbed system x˙ = u + φ(t),

x0 = x(0),

(2)

where x ∈ R is the state and u ∈ R the control input. Assumption The perturbation term φ(t) is a bounded and globally Lipschitz continuous function, i.e.,   φ(t) ˙  ≤ L. (3) |φ(t)| ≤ φmax ,

The goal is to robustly (with respect to the perturbation) drive the state to the origin in finite time with a saturated control signal that is continuous except at a finite number of switching instants, fulfilling |u(t)| ≤ ρ, where ρ ∈ R is a given constant.

(4)

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3 Saturated Super-Twisting Algorithm In order to guarantee boundedness of the control signal, the following dynamic switched control law, Saturated Super-Twisting Algorithm (SSTA) [2], is proposed:

u=

⎧ ⎪ ⎨u RC = −ρsign (x)

t < t1 (5a)

⎪ ⎩ u ST A = −α1 x1/2 + z  0 t < t1 z˙ = −α2 sign (x) else,

else (5b)

where the notation ab = |a|b sign (a) is used and z(t0 = 0) = 0. t1 is the time instant when the trajectory of the system reaches the neighborhood |x(t)| ≤ δ for the first time, i.e.,  (5c) t1 = inf t : |x(t)| ≤ δ , where δ is a sufficiently small positive constant to be defined later. Note that if the initial condition satisfies |x(t0 )| ≤ δ, t0 = t1 = 0, and u ST A is used from the very beginning. The RC is activated if the initial condition satisfies |x(t0 )| > δ. Then, the dynamics of the closed loop can be represented by x˙ = −ρsign (x) + φ(t) z˙ = 0, z(0) = 0.

(6)

The STA is activated if the state satisfies |x(t)| ≤ δ for the first time and its closed-loop dynamics are 1

x˙ = −α1 x 2 + z + φ(t) z˙ = −α2 x0 .

(7)

Please note that the STA remains activated for all future times even if |x(t)| > δ. Next, it is shown that the proposed algorithm forces the trajectories to zero in finite time fulfilling the saturation in the control input. Theorem 1 Suppose that the perturbation φ(t) satisfies (3) and (a) the SSTA gains in (5) satisfy α1 > 0,

α2 > 3L +

22 L 2 , α12

(8)

(b) the maximum perturbation bound satisfies φmax ≤ κρ

with

κ=

√ μ− μ 1 < μ−1 2

and

μ=

α12 + 8α2 , 2α12 + 8α2

(9)

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(c) the switching threshold δ is chosen such that

0 ≤ δ ≤ δmax ,

δmax =

√ 2 α1 φmax + δ (φmax ) , α12 + 4α2

(10)

where δ (φmax ) = ρ (ρ − 2φmax ) α12 + 8α2 . Then, all trajectories of the closed-loop system consisting of (2) and (5) converge to the origin in finite time, and u(t) fulfills (4).  A proof using Lyapunov theory is given in the appendix. Remark 1 Note that the proposed algorithm allows the trajectories of the SSTA to evolve without generating high-frequency switching between (5a) and (5b) if they start outside the set |x(t)| ≤ δ. Remark 2 Condition (9) is restrictive since κ=

1  1+ 1+


0,

α2 > 3L +

22 L 2 , α12

(14)

(b) the maximum perturbation bound satisfies φmax < ρ,

(15)

(c) the switching threshold δ is chosen such that 0 ≤ δ ≤ δmax

with

α 2 + 8α2 δmax = (φmax − ρ)2 1

2 . α12 + 4α2

(16)

(d) the perturbation estimator (12) is tuned such that   √ ˜ 6 2L , β1 ≥ max 8.8 L, with L˜ =

2

φmax + ρφmax |x0 | − δ

 β2 ≥ max 19 L˜ − 4L , 14L

(17)

|x0 | > δ.

(18)

for

Then, all trajectories of system (2) in closed loop with (5), (12), and (13), whose initial condition satisfies |x0 | > δ, will converge to the origin in finite time fulfilling (4). If the switching parameter is selected as δ = 0, the maximal perturbation bound and the best performance are obtained.  The proof is given in the appendix. Remark 3 In contrast to the first algorithm presented in this paper, the second one allows the rejection of perturbations with (a higher) maximal magnitude φmax < ρ. Remark 4 The estimator gains (17) ensure the finite-time convergence of the error dynamics to e1 = e2 = 0 in a time smaller than Te < Tcmin =

|x0 | − δ , ρ + φmax

(19)

where time Tcmin represents the minimum reaching time for the RC as shown in appendix. Remark 5 Note that if the initial value of the state is exactly at the origin, the first approach presented in Theorem 1 should be used.

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Fig. 1 Sets of different possible choices of δ in Theorem 1 (no estimator) and Theorem 2 (with estimator) for a fixed ρ. The maximum of the colored regions depend on φmax and are given by (9) and (15), respectively

5 Comparison of the Algorithms Figure 1 shows possible choices of the switching parameter δ depending on the maximum bound of the absolute value of perturbation φ(t). For perturbations whose maximum bound is less than the half of the saturation level, i.e., φmax < 21 ρ, it is possible to apply both algorithms (Theorems 1 and 2); otherwise, it is necessary to implement the estimator in Theorem 2. Two examples are presented in order to illustrate the different cases for Theorems 1 and 2 in comparison with the following algorithms: (a) STA (1) with an additional saturation function at its output (sat(STA)), i.e., 1 u = satρ (−α1 x 2 + z), where the saturation function is defined as  satρ (y) =

y ρsign (y)

for for

|y| < ρ . |y| ≥ ρ

(20)

(b) The original STA by Levant [8], Theorem 5, p. 1257, defined by u = u 1 + u 2 , 

−u |u| > ρ u˙ 1 = , −αsign (x) , |u| ≤ ρ

 −λ|xc | p sign (x) |x| > xc u2 = , (21) −λ|x| p sign (x) |x| ≤ xc

where α, λ, xc , and p are constant design parameters and ρ represents the saturation bound.

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(c) The STA by Golkani et al. [6] is defined by  1 u = −k1 sat |x| 2 sign (x) + ν ν˙ = −k2 sign (x) − k3 ν,

|ν0 | ≤

(22)

k2 , k3

where ν0 = ν(0) and k1 , k2 , k3 and are design parameters.

5.1 Example 1 Consider system (2) with perturbation φ(t) = sin(3t) − 3.5, and initial condition x0 = 20. The control input is saturated to |u| ≤ ρ = 10. The gains for the proposed SSTA are selected as in (8) with respect to φ(t), as α1 = 3 and α2 = 11.2. Choosing δ = 0, the maximum allowed perturbation (9) results in φmax = 4.89. Using (17) and (18), we get β1 = 16, and β2 = 50 for the second version of the proposed SSTA. The original STA[8] is simulated with the same gains λ = α1 , α = α2 , the extra parameters, xc = 5, p = 21 , and the integrator’s initial condition u 1 (0) = −3.3. The STA by Golkani et al. [6] is tuned to have identical gains k1 = α1 , k2 = α2 , and additional parameters k3 = 2.5, = 1.84. The initial condition for the integrator is ν0 = −4. Initial values u 1 (0) and ν0 , of course, have to comply with the conditions of the individual algorithms [6, 9] and are chosen such that the control signal is saturated in the reaching phase. Results of the simulations of all the algorithms are shown in Fig. 2. All the algorithms converge to zero in finite time. Nevertheless, the STA with saturation at its

20

sat(STA) Levant, 1993 Golkani et al., 2018 SSTA SSTA+Estimator

State

10

0

-10 10

Control

5 0 - (t) -10

-5 -10

1

0

1

2

3

1.2

4

1.4

5

6

7

Time [s]

Fig. 2 Example 1: Comparison of different versions of saturated STA

8

9

10

A Lyapunov based Saturated Super-Twisting Algorithm

55

output (sat(STA)) exhibits a significant integral windup which significantly delays the convergence. Levant’s original version[8] may show high-frequency switchings of the control signal at the saturation boundaries which can produce undesired actuator stress. Gokani et al.’s STA [6] shows a faster convergence, even its control signal does not completely reach the saturation level, i.e., the maximal admissible control is not fully exploited. The SSTA according to Theorem 1 exhibits one jump or discontinuity in control signal and a even faster convergence. The second version as stated in Theorem 2 also performs one jump in the control signal to the exact negative value of the perturbation which gives the best performance of all.

5.2 Example 2 Consider system (2) with a perturbation φ(t) = 1.4 sin(3t) − 8.5, and a saturated control input |u| ≤ ρ = 10. According to (14), the controller SSTA gains are chosen as α1 = 3, α2 = 16.6, and δ = 0. Using (17) and (18) and initial condition x0 = 20, we get β1 = 27 and β2 = 170 for the second proposed algorithm. The gains for the original STA [8] are chosen the same such that λ = α1 , α = α2 , and the extra parameters are xc = 5 and u 1 (0) = −3.3. For the STA by Golkani et al., the gains are also set to k1 = α1 and k2 = α2 with extra parameters k3 = 1.8, = 0.31, and ν0 = −8.8. The simulations results are shown in Fig. 3. The STA with saturation at its output (sat(STA)) shows an even bigger integral windup compared to example 1 and does not converge within the simulation time.

20

sat(STA) Levant, 1993 Golkani et al., 2018 SSTA+Estimator

State

10

0

-10 15

Control

10 10.05

5 0

10

- (t)

-5 9.95 4.4

-10 0

1

2

4.5

3

4.6

4

5

6

7

Time [s]

Fig. 3 Example 2: Comparison of different versions of saturated STA

8

9

10

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State

15 Estimator SSTA+Estimator

10 5 0 -5 15

1

Control

10 5 - (t)

0 -5 -10 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time [s]

Fig. 4 Example 2: Estimator convergence

Levant’s original STA[8] still shows high-frequency switching when in saturation and hitting two times the saturation levels before a slow convergence is achieved. Gokani et al.’s STA [6] is converging even slower. The algorithm as stated in Theorem 1 is not used in this example because relation (9) is not satisfied. As condition (15) is fulfilled, the proposed algorithm with perturbation estimator can be applied. It shows the best performance compared to the other algorithms and exhibits only one jump from the saturation level to the exact negative value of the perturbation. |20| = 1.0050s. Note that the estimation of the state reaching time is Tcmin = 10+9.9 The perturbation estimator converges before the state can reach |x(t)| ≤ δ = 0 (Te ≈ 0.6s), see Fig. 4. When the state reaches |x| = δ = 0 at time t1 , the SSTA’s integrator is initialized with the exact value of the perturbation z(t1 ) = xˆ2s (t1 ). Then, the trajectories are maintained in sliding x = x˙ = 0 for all future times with an equivaˆ = −1.4 sin(3t) + 8.5. Note that the maximal perturbation lent control u(t) = −φ(t) φmax = 9.9 is close to the maximal admissible control limit.

6 Experiments For testing the proposed algorithm, a Torsional Model ECP 2051 is used. It consists of inertial subsystems interconnected through springs as shown in Fig. 5. Its design allows the reconfiguration of inertia, springs, and the interconnection between subsystems. 1 http://www.ecpsystems.com

(accessed on June 25, 2020).

A Lyapunov based Saturated Super-Twisting Algorithm

57

Fig. 5 Torsional plant ECP model 205

Consider the problem of velocity tracking of a second-order mechanical system, i.e., of the bottom disk shown in Fig. 5. The inertia at the top in Fig. 5 is disconnected for the experiments. Its dynamics can be represented by ˙ = ν + ω(t), Jm q¨ + F (q, q)

(23)

where q, q˙ ∈ R are the state variables and ν ∈ R the input torque which is limited by |ν| ≤ 0.6N m. The terms in the differential equation (23) represent the moment of inertia Jm = 0.0333 kg m2 , a bounded function F representing locally Lipschitz unknown dynamics of the system and ω(t) possibly external Lipschitz disturbances. It is desired to realize exact velocity tracking of the trajectory q˙d . By defining the tracking error variable et = q˙ − q˙d , the velocity error dynamics are given by

˙ −q¨d , e˙t = γ¯ ν + γ¯ ω(t) − F (q, q)   

(24)

φ(t)

˙ which is a bounded locally Lipwith γ¯ = 1/Jm and φ(t) = γ¯ (ω(t) − F (q, q)), schitz perturbation. The perturbation is assumed to be bounded by a constant |φ(t)| ≤ φmax < γ¯ ρ. Applying the control law ν=

1 (q¨d + u) , γ¯

(25)

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State

20 0 -20 -40

Tracking Error

40 20 0 -20 -40 -60

0

10

20

30 Time [s]

40

50

60

Fig. 6 Experiment: velocity tracking of a desired polynomial trajectory q˙d including three steps at seconds 30, 35, and 45

where u ∈ R is a new control input yields e˙t = u + φ(t). Here, we can apply the SSTA as in (5) using (13) that is provided by estimator (12), where e1 = et − xˆ1 . After applying (25), and taking into account the maximum of ν, the SSTA saturation is set to ρ = 0.2. Figure 6 shows the velocity tracking of a polynomial trajectory q˙d including three steps at seconds 30, 35, and 45. A STA with a saturation function at its output without any anti-windup technique (sat(STA)) and no perturbation estimator in comparison with the proposed SSTA (with perturbation estimator, δ = 0, and a zero detection mechanism) was applied to the plant with the same gains α1 = 0.13, α2 = 0.8 and β1 = 17, β2 = 28 that were adjusted experimentally. As an alternative, one can choose a small δ > 0 instead of the zero detection mechanism. The measurements clearly show overshoots, and relative errors up to 41% produced by integrator windup because sat (STA) integrates the tracking error during all saturation intervals. In contrast, the SSTA with perturbation estimator is able to reach the desired step levels without any overshoot. The corresponding control signals of experiments are depicted in Fig. 7. In the event of step changes in the reference, the perturbation is not Lipschitz as assumed in the presented theorems. As a consequence, the algorithm is reinitialized when the super-twisting control signal u ST A reaches the saturation value ρ. The estimation error and the perturbation estimation are shown in Fig. 8. The sliding

A Lyapunov based Saturated Super-Twisting Algorithm

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0.8 0.6 0.4

Control

0.2 0 -0.2 -0.4 -0.6 -0.8

0

10

20

30 Time [s]

40

50

60

Fig. 7 Experiment: control signals of sat (STA) and of proposed SSTA with perturbation estimator

Estimation Error

40 20 0 -20 -40

Perturbation Estimation

-60 0.6 0.4 0.2 0 -0.2 -0.4

0

10

20

30 Time [s]

40

Fig. 8 Experiment: estimation error and perturbation estimation

50

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1

30

2

1,2

20

State

10 0 -10 -20

3

1,3

-30

1,1

-40 -50 28

30

32

34

36

38

40

42

44

46

48

Fig. 9 Experiment: comparison of the convergence time instances of the perturbation estimator Te,i and state t1,i (i = 1, 2, 3) after jumps in the reference

mode is lost three times during the experiment due to the jumps in the reference but is recovered afterward. Figure 9 shows the time instances Te,i (i = 1, 2, 3) when the estimator converges after the reference steps in comparison to time instances t1,i when state x converges to the reference. It can be clearly seen that also in the experiment the estimator converges in any case earlier than the state as ensured by Theorem 2. A video of the experiment can be found at https://youtu.be/-JIIfdY2-2s.

7 Conclusions Two different versions of saturated super-twisting algorithm are presented. Both versions use a dynamic switching that is based on PINS obtained from level curves of the Lyapunov function in Moreno et al. [11]. RC ensures the system trajectories to reach a PINS in finite time where the STA’s continuous control signal is able to drive system trajectory to zero in finite time fulfilling the saturation condition. In order to increment the maximum bound of the perturbation supported by the SSTA, the second version includes a perturbation estimator allowing to set the STA’s integrator to theoretically exact value of the perturbation. The proposed algorithms outperform existing algorithms as shown in numerical simulation examples. Experiments were carried out using a mechanical system that was set up as a second-order system in order to illustrate the performance of the proposed scheme with perturbation estimator.

A Lyapunov based Saturated Super-Twisting Algorithm

61

The proposed SSTA algorithm paves the ground for a wide use in real-world applications, e.g., exploiting robust optimal control [15, 16] and robust networked control [10], where saturated control is inevitable. Acknowledgements We gratefully acknowledge the financial support of (i) the Christian Doppler Research Association, the Austrian Federal Ministry for Digital and Economic Affairs and the National Foundation for Research, Technology and Development; (ii) CONACYT (Consejo Nacional de Ciencia y Tecnología) grant 282013; PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) grant IN 115419; and (iii) the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 734832.

Appendix Proof of Theorem 1 This proof is based on Castillo et al. [2] in which it was shown that the RC forces the trajectories to the neighborhood of the origin in finite time. It is shown in Moreno et al. [11] that if the parameters α1 and α2 of the STA are designed as in (8), the function     1 4α2 + α12 −α1 p11 − p12 T > 0, = Vs (x, z) = Vs (ξ1 , ξ2 ) = ξ Pξ, P = − p12 p22 −α1 2 2 (26)   1 T 2 and vector ξ = [ξ1 ξ2 ] = x z is a Lyapunov function for system (2) in closed loop with the STA (1). It ensures the finite-time convergence of the state to the origin and the exact compensation of the perturbation. Next, the admissible range for threshold δ > 0 based on Positive Invariant Sets (PINS) for the closed loop with RC and STA is derived. If the control signal u = ±ρ, one get the sets     1 1 U+ = (x, z) ∈ R2  z = ρ + α1 x 2 , U− = (x, z) ∈ R2  z = −ρ + α1 x 2 . (27) See the black dashed lines in the (x, z) plane shown in Fig. 10. The PINS contained between the two saturation sets (that only touches the saturation sets in only one point) is defined as s = {ξ ∈ R2 |Vs ≤ cs }. In order to find cs , one evaluates Lyapunov function (26) at the (upper) saturation set U+ such that Vs (ξ1 , ρ + α1 ξ1 ) − cs = 0. Saturation set U− is not considered due to symmetry. This relation can be written as quadratic equation in terms of ξ1 as a N ξ12 + b N ξ1 + c N = 0 where a N = p22 α12 − 2 p12 α1 + p11 , b N = −2ρ ( p12 − α1 p22 ), and 2 ρ − cs . To achieve a unique solution that only touches the saturation set c N = p22 in one point, the discriminant b2N − 4a N c N of this quadratic equation has to vanish. Solving for cs yields

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z

z

−

−

x

x

Fig. 10 (left) Nominal phase plane with the maximal PINS between the saturation sets. (right) phase plane with perturbation |φ(t)| ≤ φmax . Switch of control law in a neighborhood of the origin |x| ≤ δ (green). System trajectory (blue)

cs = ρ 2 μ

with

μ=

2 p11 p22 − p12 . p22 α12 − 2 p12 α1 + p11

(28)

As shown in Fig. 10(left), Vs (δmax , z = 0) = cs defines the maximum value of δ such that PINS does not exceed the saturation set in the nominal case, i.e., φmax = 0, such that Vs (x, 0) = p11 |x| = cs . The maximum admissible value for |x| to guarantee that the trajectory stays in the PINS is given by |x| =

cs , p11

resulting in

0≤δ≤

cs . p11

(29)

Subsequently, the case with perturbation is considered, where ξ2 = z + φ(t) holds. To find the maximum bound for the perturbation (9) that can be rejected, a level curve from Lyapunov function (26) in presence of the perturbation is evaluated in one of the saturation sets, i.e., Vs (ξ1 , ρ + α1 ξ1 − φmax ) = ρ , that can be represented as a quadratic equation aρ ξ12 + bρ ξ1 + cρ = 0. In contrast to the nominal case, the size and the center of the level curves now depend on the maximum perturbation as shown in Fig. 10(right). Setting the discriminant to zero, one can find the level curve that touches the boundary |u| = ρ at only one point, i.e., bρ2 − 4aρ cρ = 0, and one gets ρ = (φmax − ρ)2 μ. In order to find the relation between φmax and δ, the level curve is evaluated at x = δ and z = 0 resulting in Vs

√  δ, −φmax = ρ .

(30)

√ 1. Solving √ (30) for δ depending on φmax yields a quadratic equation in δ as 2 − aδ δ + bδ δ + cδ = 0 where aδ = p11 , bδ = 2 p12 φmax , and cδ = p22 φmax 2 μ (φmax − ρ) .

A Lyapunov based Saturated Super-Twisting Algorithm

63

Then, the parameter δ is chosen less or equal than the minimum quadratic absolute values of the roots 0 ≤ δ ≤ δmax , 

δmax = min

√

− p12 φmax + δ (φmax ) p11

2

2 

2 √ √ − p12 φmax − δ (φmax ) − p12 φmax + δ (φmax ) = , p p 11

11

(31)

2 2 + μp11 − p11 p22 φmax − 2μp11 φmax ρ + μp11 ρ 2 where δ (φmax ) = p12

2 2 2 = p12 φmax − p11 p22 φmax − μ (φmax − ρ)2 . 2. Solving (30) for the maximum perturbation bound φmax depending on the choice 2 of parameter δ, a quadratic equation in √ φmax as aφ φmax + bφ 2φmax + cφ = 0 where aφ = p22 − μ, bφ = 2μρ + 2 p12 δ, and cφ = p11 δ − μρ is considered. Then, the maximum bound φmax that the algorithm can suppress is given by the minimum absolute value of the roots     √ √  μρ+ p12 √δ−√ρ (δ)   μρ+ p12 √δ+√ρ (δ)     = μρ+ p12 δ− ρ (δ) ,  , φmax = min     μ− p22 μ− p22 μ− p22 √

2 + μp11 − p11 p22 δ + 2μp12 ρ δ + μp22 ρ 2 = (μρ+ where ρ (δ) = p12  √ 2

p12 δ − μρ 2 − p11 δ (μ − p22 ) . If δ is set to zero, φmax reduces to φmax = κρ

√ μ− μ . κ= μ−1

with

(32)

Substituting (26) and (28) into (32) results in κ=

√ √ μ− μ μ+ μ 1 √ · √ = μ (μ − 1) μ + μ 1+ μ ·

√ μ √ μ

=

1 = 1 + √1μ

1  1+ 1+

α12 α12 +8α2

(33) which is clearly less than 21 . Taking the values of the Lyapunov function (26) depending on α1 , α2 , together with (28), (32), and (31) leads to (9)–(10), respectively. This completes the proof. 

Proof of Theorem 2 The proof is performed in two steps. First, it is shown how the estimator has to be tuned to achieve convergence before the RC switches to STA. Then, conditions for the STA parameters are derived. The error dynamics of estimator (12) are e˙1 = −β1 e1 1/2 + e2 e˙2 = −β2 sign (e1 ) + φ˙

(34)

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with e2 = xˆ2 + φ(t). Therefore, there exist gains β1 and β2 depending on (3) and a time Te > 0 where e1 = e2 = 0 as shown in [11, 13]. This implies that xˆ2 = −φ(t) for all future time t > Te . For the SSTA, we design the estimator gains to make the time Te smaller than the minimum time of convergence of the state under relay control. The estimation of the minimum reaching time Tcmin of the RC is made considering the case when the perturbation helps the system trajectories to converge to a vicinity of the origin |x(t)| ≤ δ, starting from an initial condition x0 = x(0) outside of the vicinity |x0 | > δ. Then, using Lyapunov function Vc (x) = c1 |x|, c1 > 0 from Castillo et al. [2] yields the time derivative V˙c (x) = −c1 ρ + c1 sign (x) φ(t), resp.

min

|φ(t)|≤φmax

V˙c (x) = −c1 (ρ + φmax ) . (35)

If one selects c1 = 1/ (ρ + φmax ), the Lyapunov function derivative becomes V˙c ≥ −1. In order to estimate to time of convergence from the initial condition to the neighborhood δ, distance |x0 | − δ is considered in the Lyapunov function, i.e., Vc (x) ≥ Vc (x0 ) − t =

|x0 | − δ −t ρ + φmax

(36)

for t0 = 0. This shows that Vc reduces to c1 δ no earlier than Tcmin given in (19). For the second part of the proof, consider the Lyapunov function from Polyakov et al. [13] ⎧

2  k(e) ¯ 2 e2 e1 0 ⎪ m(e) ¯ ⎪ ⎪ + k0 (e)e ¯ e1 e2 = 0 s(e) ¯ ⎪ ⎪ 4 γ ⎪ ⎪ ⎨ 2 2 ¯ Ve (e1 , e2 ) = 2k e2 e1 = 0 ⎪ ⎪ β12 ⎪ ⎪ ⎪ ⎪ ⎪ |e1 | ⎩ e2 = 0, 2

(37)

¯ and k0 (e) ¯ depend on the state e1 and e2 , and k¯ where e¯ = [e1 e2 ]T . The terms k(e) is a design parameter depending on L and the gains β1 and β2 . Expressions s(e) ¯ and m(e) ¯ are also nonlinear functions of the state. Note that with the knowledge of the initial condition x0 it is possible to set xˆ1 (0) = x0 , and therefore e1 (0) = x0 − xˆ1 (0) = 0 and as a result to use the second case of (37). We choose a parametrization of the estimator gains  β1 = 2 (18L + ),

β2 = 14L + ,

(38)

with > 0, such that the conditions of Theorem 1 in Polyakov et al. [13] hold, i.e., β2 = 14L + > 5L ,

(39)

A Lyapunov based Saturated Super-Twisting Algorithm

and

65

64L < √ β12 < 8(β2 − L) 64L < (2 18L + )2 < 8((14L + ) − L) 64L < 72L + 4 < 104L + 8 .

(40)

In order to design k¯ in (37), parameter g = 8γ /β12 with γ = β2 − Lsign (e1 e2 ) as shown in Polyakov et al. [13] is considered. It may take two possible values g − = 8(β2 − L)/β12 and g + = 8(β2 + L)/β12 depending on the values of γ ∈ {β2 + L , β2 − L}. Taking into account (38), function g = 2(14L+ ±L) also varies 18L+

depending on the selection of the parameter . Note that g is monotone with respect to since its derivative with respect to is positive, i.e., dg 2(4L ± L) 2 2(14L ± L + ) = > 0. = − d

18L +

(18L + )2 (18L + )2

(41)

Therefore, the limits of g − and g + when → 0 and → ∞ are taken lim g − =

→0

lim g + =

→0

13 , 9 15 , 9

lim g − = 2,

→∞

lim g + = 2.

(42)

→∞

  , 2. The whole range of variation of g depending on γ and is g ∈ [gm , g M ] = 13 9 Parameter k¯ should belong to a intersection set of the intervals I (gm ) ∩ I (g M ) = 0, where the interval I (g) is given by  I (g) =

√

√

 √ √ exp 1/ g − 1 −π/2 − arctan 1/ g − 1 exp 1/ g − 1 π/2 − arctan 1/ g − 1 2 + , . √ √ g g g

(43)

Evaluating the endpoints of g yields I (g − ) = [1.4027, 2.01] and I (g + ) = [1.0670, 1.5509]. Parameter k¯ can be selected as k¯ = 1.4768. Using Theorem 1 in Polyakov et al. [13], we ensure that the time derivative of (37) along the trajectories of the system satisfies   V˙e ≤ −k Ve ≤ −kmin Ve

(44)

and if the bound for |e2 (0)| = |φ(0)| = φmax , the reaching time estimate can be referred to as 2  Ve (0, φmax ), (45) Te ≤ kmin with

β1 kmin = √ 8

min

g∈{g − ,g + }

∈{0,∞}

f (g, )

(46)

66

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I. Castillo et al.

   2  ⎞  ⎛ π(β1 g−8β2 )   ar ctan √−1 +   16L √ g−1  . ¯ ⎝ ⎠ f (g, ) = g k − g exp √  g−1  

(47)

Evaluating f with the two limits gm and g M and the parametrization of g, f (g, ) ∈ [ f m , f M ] = [0.1550, 2.8066], and kmin = √β18 f m . From (37), (45), and setting the reaching time of the estimator Te less than the minimum reaching time of the state Tcmin , one gets Te ≤

¯ max 8kφ < Tcmin . β12 f m

(48)

Substituting (19) in (48) and solving for β1 result in $ β1 ≥



2 + ρφmax 8k¯ φmax . fm |x0 | − δ

(49)

From β1 parametrization (38), we solve for such that

=

1 2 β − 18L . 4 1

(50)

Substituting (50) in β2 parametrization (38) yields β2 =

1 2 β − 4L . 4 1

(51)

Using the equality case of (49) results in 2

+ ρφmax 2k¯ φmax − 4L . β2 ≥ fm |x0 | − δ

(52)

Note that the value of the gains β1 and β2 tends to zero and to −4L, respectively, as x0 tends to infinity because the restriction > 0 disappeared in these expressions. Therefore, conditions (49) and (52) are expressed as the maximum of two values to get and (17). As shown before and in Castillo et al. [2], the PINS for STA (red curves in Fig. 11) move up or down in the (x, z)-plane depending on φmax . In addition, they change its size depending on δ. With the exact estimate of the perturbation, the integral control is set to z(t1 ) = −φ(t1 ) when the trajectory enters the neighborhood |x(t1 )| ≤ δ at time t1 , then equation (30) becomes

A Lyapunov based Saturated Super-Twisting Algorithm

67

Fig. 11 With the exact estimation of the perturbation the trajectories can start into a PINS of a size depending on δ

Vs

√

 δ, 0 − ρ = p11 δ − μ (φmax − ρ)2 = 0.

(53)

The set of possible choices of the parameter δ depending on φmax is then given by 0 ≤ δ ≤ δmax ;

δmax =

μ (φmax − ρ)2 . p11

(54)

The set of maximum perturbation bound φmax depending on the choice of parameter δ is $ p11 δ . (55) φmax = ρ − μ Note that the maximal rejectable perturbation depends on δ, i.e., the smaller δ, the higher maximum perturbation up to φmax < ρ, when δ = 0. The big red region in Fig. 11 equals a PINS for the choice δ = δ1 . As a result, perturbations with maximum |φmax1 | can be rejected. A choice of δ = δ2 < δ1 gives a smaller red region as shown in Fig. 11. As a consequence, perturbations with higher magnitude |φmax2 | > |φmax1 | can be eliminated. Using the values of the Lyapunov function (26) depending on α1 , α2 , together with (28), (55), and (54), we get (15) and (16), respectively. This completes the proof. 

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References 1. Castillo, I., Steinberger, M., Fridman, L., Moreno, J., Horn, M.: Saturated Super-Twisting Algorithm based on perturbation estimator. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 7325–7328. 2016 IEEE 55th Conference on Decision and Control (CDC) (2016). 10.1109/CDC.2016.7799400 2. Castillo, I., Steinberger, M., Fridman, L., Moreno, J.A., Horn, M.: Saturated Super-Twisting Algorithm: Lyapunov based approach. In: 2016 14th International Workshop on Variable Structure Systems (VSS), pp. 269–273. 2016 14th International Workshop on Variable Structure Systems (VSS) (2016). 10.1109/VSS.2016.7506928 3. Cruz-Zavala, E., Moreno, J.A.: Lyapunov functions for continuous and discontinuous differentiators. IFAC-PapersOnLine 49(18), 660–665 (2016). http://dx.doi.org/10.1016/j.ifacol.2016. 10.241. http://www.sciencedirect.com/science/article/pii/S2405896316318213 4. Davila, J., Fridman, L., Poznyak, A.: Observation and identification of mechanical systems via second order sliding modes. Int. J. Control. 79(10), 1251–1262 (2006). 10.1080/00207170600801635. http://dx.doi.org/10.1080/00207170600801635 5. Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. CRC Press (1998) 6. Golkani, M.A., Koch, S., Reichhartinger, M., Horn, M.: A novel saturated super-twisting algorithm. Syst. Control. Lett. 119, 52–56 (2018). 10.1016/j.sysconle.2018.07.001. https:// linkinghub.elsevier.com/retrieve/pii/S0167691118301178 7. Hippe, P.: Windup in Control: Its Effects and Their Prevention. Springer Science & Business Media (2006) 8. Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control. 58(6), 1247–1263 (1993) 9. Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379– 384 (1998). http://dx.doi.org/10.1016/S0005-1098(97)00209-4. http://www.sciencedirect. com/science/article/pii/S0005109897002094 10. Ludwiger, J., Steinberger, M., Horn, M.: Spatially distributed networked sliding mode control. IEEE Control. Syst. Lett. (2019) 11. Moreno, J.A., Osorio, M.: A Lyapunov approach to second-order sliding mode controllers and observers. In: 47th IEEE Conference on Decision and Control, 2008. CDC 2008, pp. 2856– 2861. IEEE 47th IEEE Conference on Decision and Control (2008) 12. Polyakov, A., Efimov, D., Perruquetti, W.: Homogeneous differentiator design using implicit Lyapunov function method. In: 2014 European Control Conference (ECC), pp. 288–293. IEEE 2014 European Control Conference (ECC) (2014) 13. Polyakov, A., Poznyak, A.: Reaching time estimation for super-twisting second order sliding mode controller via Lyapunov function designing. IEEE Trans. Autom. Control. 54(8), 1951– 1955 (2009) 14. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation. Springer (2014) 15. Steinberger, M., Castillo, I., Horn, M., Fridman, L.: Model predictive output integral sliding mode control. In: 2016 14th International Workshop on Variable Structure Systems (VSS), pp. 228–233. IEEE (2016) 16. Steinberger, M., Castillo, I., Horn, M., Fridman, L.: Robust output tracking of constrained perturbed linear systems via model predictive sliding mode control. Int. J. Robust Nonlinear Control. (2019) 17. Utkin, V.I.: Sliding Modes in Control and Optimization, vol. 116. Springer (1992)

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions Shyam Kamal and Rahul Kumar Sharma

Abstract This chapter gives an insight into the class of tracking problems in nonlinear systems for which the first-order derivative of the reference function does not exist. Using classical sliding mode control, only a restricted class of reference functions can be tracked. A solution to this problem by using fractional-order operators is proposed. The technique works provided the reference function satisfies the Hölder condition. Notably, its application to a switch-controlled R L circuit is demonstrated and some of the possible applications have been discussed.

1 Introduction Generally, the design of nonlinear control systems involves two types of problems: stabilization or regulation problem and tracking or servo problem. In stabilization problems, the states of the closed-loop system states are to be stabilized around an equilibrium point using a control law. Some of the commonly encountered stabilization problems are altitude control of aircraft, position control of the robotic arm, and temperature control of refrigerators. On the other hand, tracking problems involve control design to track a time-varying reference trajectory by the system output. The problems of making an aircraft or a robotic arm follow a specified path fall under the category of tracking problems. Without loss of generality, the tracking problem can be reduced to an equivalent stabilization problem [1]. A more generalized approach to the tracking problem is explored in this chapter. The desired reference trajectory in tracking problems may belong to any class of functions according to the specific application. Conventional sliding mode control works on a restricted class of reference functions. In this chapter, a larger class of S. Kamal (B) · R. K. Sharma Department of Electrical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, Uttar Pradesh, India e-mail: [email protected] R. K. Sharma e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_3

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reference functions has been considered by using fractional-order derivatives in the design of sliding mode control for tracking problems. A brief outline of the chapter is as follows. Section 2 provides the motivation behind the work. This is followed by the background and formulation of the tracking problem in Sect. 3. Section 4 describes sliding mode control in the context of tracking problems. Classical approaches are explored and their mathematical limitations are highlighted. A case study of switch-controlled R L circuit is given in Sect. 5. Tracking problems for constant and smooth reference functions are given in Sect. 6. Section 7 gives some details of the definitions and notions used in the study. Fractional-order sliding mode control is presented in Sect. 8. Section 9 gives the main theoretical concepts of the technique proposed in this chapter. The idea is further generalized to nonlinear systems in Sect. 10. Simulation results are used for further illustrations in Sect. 11. Finally, the chapter is concluded with Sect. 12.

2 Motivation Finite-time tracking of reference functions in nonlinear systems has been one of the major control tasks required to be performed in various applications. Numerous important contributions can be found in the literature in the direction of achieving finite-time stability [2–4]. However, the derivative of the reference function is generally required in the implementation of the control law which is a serious limitation of the well-established control techniques in this context. The tracking problem can be approached in a more generalized way by allowing the order of the derivative to take fractional values. This requires the tracking problem to be explored at the fundamental level with a sound mathematical framework using fractional calculus [5, 6]. Sliding mode control is utilized very often in achieving the desired tracking objective in finite time. It also has the capability of providing disturbance rejection and robustness to parametric uncertainties. However, the classical sliding mode scheme assumes the reference function to be differentiable with a bound for the convergence of the error dynamics. So, the conventional integer-order framework restricts the allowable class of reference functions that can be tracked. The operator based on fractional calculus can be utilized so that the reference functions which do not possess the first-order derivative and satisfy the Hölder condition [7] can also be tracked using limited control action. So, a larger class of reference functions satisfying a certain condition can be addressed using the technique proposed in this chapter. The study of fractional-order derivatives and integrals is done in a branch of mathematics known as fractional calculus [5, 7]. It has been observed that physical systems can be more accurately represented by their fractional-order models. The nature of some phenomena can inherently be described only by considering them as fractional-order systems. This makes it obvious to explore the tracking problems in a new dimension.

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3 The Tracking Problem The tracking problem in nonlinear dynamic systems is one of the most important research areas in control theory. The problem can be found in many application areas of science and engineering and can be described by considering the general nonlinear system, (1) x˙ = f (x, u) ; x ∈ D ⊂ Rn , u ∈ Rm , where f : D × Rm → Rn is locally Lipschitz in x and u. The input u(t) is a piecewise continuous, bounded function of t, ∀t ≥ 0 and domain D contains the origin x = 0. Finite-time convergence of the system states x toward the desired states xr keeping the states x bounded for all time t is required in most of the tracking problems. It is observed that the tracking problem can be reduced to an equivalent stabilization problem of the error dynamics about the origin by defining the error e = x − xr as the new state variable. The tracking problem has been explored in various contexts in the literature. One of the earlier contributions can be found in [8]. It has also been explored in the field of robotics. A sliding mode control-based approach is used in [9]. Passivity-based scheme is presented for tracking problem of flexible robot arms [10]. Attitude tracking of spacecrafts has been a major research area in the context of tracking problems. In this direction, some significant contributions are [11–13]. Another notable contribution is [14]. In these applications, there are possibilities where a sudden change in the desired reference functions may be required. The work presented in this chapter finds an important significance in such problems. An observer-based controller is designed for output tracking of fractional-order positive switched systems in [15]. The observer includes equivalent input disturbance and Smith predictor. It is able to reject the disturbance. In [8], a function space analysis is used to find the optimal input to track the system’s output in a fixed time to minimize the control energy required. Signal transformation approach to track triangular signal has been attempted by some authors. In [14], signal transformation is used to track triangular signals. This approach improves the closed-loop performance. Necessary and sufficient conditions for the stability of the control system are given for tracking triangular reference signals. In the previous approaches, the desired reference functions have been approximated by piecewise continuous functions. However, it is not possible to use this approximation to all class of desired reference functions. This situation calls for a revisit of the tracking problem with some sound mathematical framework. According to the work of [16], there exists continuous functions for which no first-order derivatives are defined but possess fractional derivatives of all orders less than one. This motivates further to have a look on the beauty of fractional calculus and how it can be utilized in tracking problems where the reference function is not differentiable.

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4 Sliding Mode Control in Tracking Problems Sliding mode control is one of the most efficient robust control techniques which performs quite effectively in the presence of disturbances and uncertainties [17–25]. In this control scheme, the system states are forced to remain on a predefined manifold called sliding surface and maintained there. Keeping the states on the sliding surface, they are driven to the origin. Therefore, the sliding mode control based technique consists of two phases. The first one is the reaching phase, where the system states are driven from there initial values to reach the sliding manifold in finite time. Then, the states undergo the sliding phase where they are further driven to the origin. On the sliding surface, the order of the system dynamics gets reduced, and robustness is obtained. Recently, the theory has been used for control and observation in several problems. In the context of tracking problems, one of the earlier contribution is [9] which discusses the design of sliding mode control law for robotic manipulators. Another work in this context is [10]. Using sliding mode techniques for attitude control of rigid body is an important class of tracking problems. Some significant works in this direction are [11–13]. Another field of application is described in [26]. However, all of the above mentioned approaches discuss the tracking problems in the integerorder framework. It is often required to calculate the derivative of the error function e(t) which itself is the difference between the state x(t) and the desired reference function xr (t). Ultimately, the information of derivative of the reference function x˙r (t) is needed for the techniques to work. So, the previous approaches restrict the class of reference functions to be tracked. It is observed that functions which do not possess the first-order derivative may possess a fractional-order derivative of order α < 1 provided the function is integrable [16]. Therefore, the above limitations of classical control strategies can be overcome by using fractional-order rate of change of the reference function D α xr (t) in the control law. The tracking problem for a reference signal xr can be considered in two different ways. In one way, a reference generator system can be taken, and it is desired to obtain some reference signal xr . In the other way, a particular reference signal is desired where the reference generator system can itself be chosen accordingly. Possible reference generator systems can be networks based on fractional elements like supercapacitor and fractional inductor or spring–dashpot fractal network. For both situations, the corresponding tracking problems can be formulated. This is illustrated by taking an example of switch-controlled RL circuit in the next section.

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5 The Switch-Controlled RL Circuit: A Case Study A switch-controlled R L circuit (DC chopper) supplied by a constant voltage source V is considered as shown in Fig. 1. The resulting dynamic behavior of the above circuit can be captured by L

di(t) = u(t)V − i(t)R, dt

(2)

where L, R, i(t), and u(t) ∈ {0, 1} are the inductance, resistance, and current through inductor and the control input, respectively. Here, the control input can be 0 or 1 corresponding to “OFF” or “ON” state of the switch, respectively. Now, suppose that it is desired to maintain the following classes of functions of current through the inductor for all time by using only limited control action u(t) ∈ {0, 1}: (a) Constant current i(t) = ir , where ir ≤ i max . (b) Smooth current i(t) = A sin(ωt), where |A| ≤ Amax and ω ≤ ωmax . (c) Non-smooth current, i.e., the desired current ir which is not differentiable (using Newton’s calculus). For example, consider a sawtooth or triangular function as the current which has a removable discontinuity point:  i(t) =

t if 0 < t ≤ 1, 2 − t, if 1 < t < 2.

(3)

Out of the above three problems, both problems (a) and (b) can be solved directly by using the concept of sliding mode control. More details can be found in [21] and the papers cited within.

u=1

L

i

u=0

V

Fig. 1 Switch-controlled R L circuit (DC chopper)

R

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6 Tracking of Constant and Smooth Reference Functions Using sliding mode control theory, the sliding variable is defined as the error between the current at time t and the desired reference current, i.e., e = i(t) − ir (t).

(4)

Further, the rate of change of error can be calculated as in both the above cases (a) and (b), the desired current ir is assumed to belong to the class of differentiable functions. Therefore, using (4), ˙ − i˙r (t). e˙ = i(t)

(5)

Using (2), (4), and (5),    L R V u(t) − s − ir (t) + i˙r (t) L R R    L R i max u(t) − s − ir (t) + i˙r (t) , = L R

e˙ =

(6)

where VR = i max . Consider that, initially, the error is negative, i.e., e < 0. Therefore, the switch must be turned “ON”, i.e., u(t) = 1 as long as e remains negative. Substituting e < 0 and u(t) = 1 in (6),    L˙ R i max + |e| − ir (t) + ir (t) . e˙ = L R

(7)

Using (7), it can be concluded that the time derivative of the error, e, ˙ is guaranteed to remain positive if the desired trajectory satisfies the condition, i max > ir (t) +

L˙ ir (t); ∀t. R

If the error is positive, i.e., e > 0, then e˙ should be negative accordingly. Therefore, the switch should go “OFF”, i.e., u(t) = 0 as long as e is positive. Substituting e > 0 and u(t) = 0 in (6),    L˙ R s + ir (t) + ir (t) . (8) e˙ = − L R Therefore, the time derivative of error, e, ˙ is guaranteed to remain negative, provided   L   ir (t) + i˙r (t) > 0; ∀t. R

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Therefore, it is clear that the desired trajectory tracking is possible using switchcontrolled R L circuit, provided the following conditions are satisfied: • the desired trajectory is differentiable and • 0 < |ir (t) + LR i˙r (t) < i max |; ∀t. Therefore, a switching strategy can be designed in the following way:  u(t) =

1 for e < 0, 0, for e > 0,

(9)

or u = 21 (1 − sgn(e)), where sgn is defined as  sgn(e) =

1 fore > 0, −1, for e < 0.

It is observed that it is not possible to discuss about the existence and uniqueness of the solution of the closed-loop system (2) in the classical sense after substitution of the required switching scheme (9), due to the associated discontinuity in the right side of the resulting differential equation. Therefore, one has to understand the solution in the sense of non-smooth theory of differential equations. There also exist several approaches in the literature which give the existence and uniqueness of the solution for differential equations with discontinuous righthand side. The most straightforward approach among them is Utkin regularization or Filippov solution method which has been followed here for the solution of the differential equation (2). In this context, one has to make sure that during sliding, the value of the equivalent control u eq should belong to the interval (0, 1), i.e., 0 < u eq < 1. The resulting expression of the equivalent control can be obtained by substituting e = e˙ = 0 into (7), u eq = −

1 i max

  L˙ ir (t) + ir (t) . R

(10)

From the above discussion, it is clear that the problem (c) is not solvable using the above methodology. This is due to the lack of the error signal’s differentiability, which further suppresses the generation and manipulation of the error dynamics based on the ordinary differential equation (5). This poses a limitation on the applicability of not only sliding mode control based design, but also that of other classes of controllers like PID, adaptive, optimal, etc. It may be argued that the desired reference function may be approximated by a piecewise continuously differentiable function, and then the same technique may be applied with some approximation. However, the above analysis is limited only to some restricted types of functions. For example, the situation may become absurd when the desired reference function is continuous everywhere but not differentiable anywhere like the Weierstrass function:

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ir (t) =



λ−μj sin(λ j t),

0 < μ < 1, λ > 1.

(11)

j≥1

The typical cases like that of problem (c) may appear if the derivative of the desired reference function ir (t): • does not exist at a finite number of points or • does not exist almost everywhere.

7 Preliminaries of Fractional Calculus The beauty of fractional calculus has made the solution of some challenging problems possible for which the classical approach of integer-order calculus fails. Fractional calculus is a generalization of integer-order calculus. It consists of generalized expressions of derivatives and integrals of non-integer order. Out of the several definitions of fractional-order derivatives, two are most commonly used which are Riemann– Liouville (R-L) and Caputo definitions [5–7]. Definition 1 The αth-order fractional-order integral of the function f : (0, ∞) → R with respect to t > 0 and terminal value t0 > 0 is given by α t0 It

1 f (t) := (α)

t t0

f (τ ) dτ, (t − τ )(1−α)

(12)

where  : (0, ∞) → R is Euler’s Gamma function: ∞ (α) :=

x α−1 e−x d x.

0

Definition 2 The R-L definition of the αth-order fractional-order derivative is given by t dm f (τ ) 1 RL α dτ, (13) t0 Dt f (t) := m (m − α) dt (t − τ )(α−m+1) t0

where m ∈ N such that m ≥ α , where α is the smallest integer greater than or equal to α where 0 < α < 1. Definition 3 The Caputo definition of the αth-order fractional-order derivative of the m times continuously differentiable function f : (0, ∞) → R or f ∈ C m ((0, ∞), R) is given by

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions

c α t0 Dt

1 f (t) := (m − α)

t t0

f (m) (τ ) dτ. (t − τ )(α−m+1)

77

(14)

A few important properties of fractional-order derivatives and integrals are as follows [27]: • For α = n, where n is an integer, the operation ct0 Dtα f (t) gives the same result as the classical differentiation of integer order n. • For α = 0, the operation ct0 Dtα f (t) is the identity operation: c α t0 Dt

f (t) = f (t).

(15)

• Fractional differentiation is a linear operation: c α t0 Dt (a f (t)

+ bg(t)) = a ct0 Dtα f (t) + ct0 Dtα g(t).

(16)

• The additive index law (semigroup property), β αc c t0 Dt t0 Dt

β

α+β

f (t) = ct0 Dt ct0 Dtα f (t) = ct0 Dt

f (t),

(17)

holds for f (t) ∈ C 1 [0, T ] for some T > 0, where α, β ∈ R+ and α + β ≤ 1 [28]. Before using the fractional-order operators, proper understanding of their operations is very essential. Visualization of a mathematical notion needs a clear interpretation of the operations involved in terms of the previously established concepts or phenomena. The search for finding proper interpretations of fractional-order integrals and derivatives has been quite long since their formulations. In this direction, a significant contribution is a work presented in [29]. Here, only the R-L definition will be considered. The αth-order fractional-order integral α t0 It

1 f (t) := (α)

t t0

f (τ ) dτ (t − τ )(1−α)

(18)

can also be expressed as α t0 It

1 f (t) := (α)

where gt (τ ) =

t f (τ )dgt (τ ),

(19)

1 {t α − (t − τ )α }. (α + 1)

(20)

t0

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Fig. 2 Geometrical interpretation of fractional-order integral [29]

The homogeneous time scale consists of equal intervals of flowing time. Apart from the concept of classical homogeneous time scale, the fractional-order derivation and integration can be considered to operate in a non-homogeneous time scale or the Cosmic time scale [29] which is composed of non-equal time intervals. The homogeneous time scale can be considered as an ideal notion for the non-homogeneous time scale. The function gt (τ ) has the scaling property gt1 (τ1 ) = gkt (kτ ) = k α gt (τ ). Consider a three-dimensional space with the axes (τ, gt , f ). The function gt (τ ) is plotted in the plane (τ, gt ) for 0 ≤ τ ≤ t. Along the obtained curve, a fence can be formed of varying height f (τ ). So, the top edge of the fence represents a line in three-dimensional space (τ, gt (τ ), f (τ )), 0 ≤ τ ≤ t. For the purpose of further illustrations, consider Fig. 2 used here from [29]. The fence can be projected onto the plane (τ, f ). The area of this projection corresponds to the integral t I1 = f (τ )dτ, (21) 0

while the same fence can be projected onto the plane (gt , f ), the area of which corresponds to the integral (19) or (18). This projection on the plane (gt , f ) is the geometric interpretation of the fractional-order integral (18) for fixed t. For gt (τ ) = τ , both the projections overlap each other showing that the classical definition of integral is a special case of the R-L definition of generalized fractional-order integral.

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8 Fractional-Order Sliding Mode Control Sliding mode control has been used for fractional-order systems [6, 30–32]. Remarkable improvements have been obtained with fractional-order sliding mode-based control law design in various applications. The technique free from reaching phase has been applied for uncertain fractional-order systems to obtain robustness throughout the state trajectories in [31]. It uses the Caputo definition (14). This work has been presented here in detail to illustrate the beauty of using fractional calculus in the context of sliding mode control. Using the theory of fractional calculus, sliding mode control law designs using two approaches are presented here which are integer reaching law approach and fractional reaching law approach. Consider a controllable commensurate fractionalorder linear time-invariant system, c α ¯ t0 Dt x(t)

 = A¯ x(t) ¯ + B¯ u(t) + d(t) ,

(22)

where x(·) ¯ ∈ Rn are pseudostates , A¯ ∈ Rn×n is the system matrix, B¯ ∈ Rn×m is the input matrix, u(·) ∈ Rm is the control input, and d(·) ∈ Rm is the disturbance which is assumed to be bounded. It is important to mention here that for fractionalorder systems, the knowledge of the initial state, x(t0 ) is not sufficient to determine the future state of the system. So, the physical variables do not strictly represent the actual states of the system. Therefore, the terminology of pseudostates is coined to represent these physical variables [33]. The same philosophy is followed throughout this presentation. There always exists an invertible matrix T ∈ Rn×n such that using the linear transformation z(t) = T x(t), (22) can be transformed into the regular form, c α t0 Dt z 1 (t) c α t0 Dt z 2 (t)

= A11 z 1 (t) + A12 z 2 (t) = A21 z 1 (t) + A22 z 2 (t) + B2 (u(t) + d(t)),

(23)

 z 1 (t) = z(t), z 1 (·) ∈ Rn−m , z 2 (·) ∈ Rm . where z 2 (t) ¯ B) ¯ is assumed controllable, the pair (A11 , A12 ) will also be conAs the pair ( A, trollable. The above system of equations can be represented as 

c α t0 Dt z(t)

= Az(t) + B(u(t) + d(t)),

(24)

   0 A11 A12 ,B= . where A = A21 A22 B2 

Assumption For a non-smooth controller, the existence and uniqueness of solutions of the system are defined in the Filippov sense [6], i.e., considering x as the pseudostates of the entire system ct0 Dtα x(t) = f (x(t), d(t)), α > 0, disturbance

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d ∈ Rm and assuming f : Rn × Rm → Rn to be locally bounded, then the solutions are defined with the differential inclusion, c α cl(co(ζ (Bδ (x) \ N ))), t0 Dt x(t) ∈ δ>0 μN =0

where cl and co denote the closure and the convex hull, respectively, Bδ (x) is the unit ball, and the sets N are all sets of zero Lebesgue measure. In [6], the sliding surface has been designed using fractional reaching law and integer reaching law. Using integer reaching law, the sliding surface for (23) is s(z, t) = t0 It1−α (c1 z 1 (t) + z 2 (t)),

(25)

where s : (t0 , ∞) × Rn → R, c1 ∈ R1×(n−1) , and for u(t) = B2−1 (v − c1 {(A11 − A12 c1 )z 1 (t)} + A12t0 It1−α s − A21 z 1 (t) − A22 z 2 (t)), where v = −k1 sign(s), it has been proved in [6] that for s to be zero in finite time, k1 > |B2 ||d|. In case of fractional reaching law, s(z, t) = c1 z 1 (t) + z 2 (t).

(26)

Using fractional-order derivative of s in (26), (23) becomes c α t0 Dt z 1 (t) c α t0 Dt s

= (A11 − A12 c1 )z 1 (t) + A12 s = c1 ct0 Dtα z 1 (t) + ct0 Dtα z 2 (t) = c1 {(A11 − A12 c1 )z 1 (t) + A12 s} + A21 z 1 (t) + A22 z 2 (t) + B2 u(t) + B2 d.

Here, the control u(t) is chosen as u(t) = B2−1 (v − c1 (A11 − A12 c1 )z 1 (t) + A12 s − A21 z 1 (t) − A22 z 2 (t)), where v = −k1 sign(s). After applying the control, the following closed-loop system results: c α t0 Dt z 1 (t) c α t0 Dt s

= (A11 − A12 c1 )z 1 (t) + A12 s, = −k1 sign(s) + B2 d.

(27)

Here, some stability concepts need to be discussed. The Lyapunov theory for general nonlinear systems has also been extended for fractional-order systems in the literature. Using Caputo definition, an n−dimensional fractional-order system can be defined as c α t0 Dt x(t)

= f (x, t); ∀t ≥ t0 ,

(28)

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81

where α ∈ (0, 1) and f (x, t) is locally bounded in x and piecewise continuous in t for all t ≥ t0 and x ∈ D, where D ⊂ Rn is a domain which contains the origin x = 0. For stability analysis of systems in (28), a fractional-order extension of the Lyapunov’s direct method was proposed in [34] which is based on the following definition: Definition 4 A continuous function γ : [0, t) → [0, ∞) is a class-K function if it is strictly increasing and γ (0) = 0. Theorem 1 Let x = 0 be an equilibrium point for the non-autonomous fractionalorder system, i.e., f (x, t) = 0, ∀t ≥ t0 . If there exists a Lyapunov function V (t, x(t)) : [t0 , ∞) × D → R and a class-K function γi (i = 1, 2, 3), such that γ1 (||x||) ≤ V (t, x(t)) ≤ γ2 (||x||) and ct0 Dtα V (t, x(t)) ≤ −γ3 (||x||), where α ∈ (0, 1), then the system (28) is asymptotically stable. Theorem 2 Let x ∈ Rn be a continuously differentiable vector-valued function. Then, for any time instant t ≥ t0 and ∀α ∈ (0, 1), 1c α  D x (t)x(t) ≤ x  (t)ct0 Dtα x(t). 2 t0 t

(29)

The above results will be used in the later sections for the Lyapunov stability analysis of fractional-order systems with the proposed control law. Since this result was derived using Caputo derivatives, the same definition will be used for stability analysis throughout the chapter unless mentioned otherwise. Here, it is important to consider the following theorem: Theorem 3 The sliding surface s in (26) becomes zer o in finite time if k1 > |B2 ||d|. Proof The Lyapunov function is selected as V = 21 s 2 . Then, Using (29), c α t0 Dt V

c α t0 Dt V

=

1c Dα s 2. 2 t0 t

≤ s ct0 Dtα s = s(−k1 sign(s) + B2 d) ≤ −k1 |s| + |s||B2 d| = −|s|(k1 − |B2 d|) 1

= −(2V ) 2 (k1 − |B2 d|) 1

≤ −η(2V ) 2 , where η = k1 − |B2 ||d| > 0. Using the above inequality, s = 0 is obtained in finite time [6] which can be derived as follows: Putting t0 = 0 in (27), c0 Dtα s = −k1 sign(s) + B2 d. Taking fractional-order integral of order α on both sides, α c α 0 I t 0 Dt s

= k1 0 Itα sign(s) + B2 0 Itα d.

(30)

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α

t t Since, 0 Itα c0 Dtα s = s(t) − c0 Dtα−1 s(0) (α) and 0 Itα c = c (α+1) , Eq. (30) becomes after finite time t = T ,

s(T ) − c0 Dtα−1 s(0)

t α−1 Tα = −k1 sgn(s(0)) + B2 0 Itα d. (α) (α + 1)

Multiplying with sign(s(0)) and using s(T ) = 0,

−c0 Dtα−1 s(0)sign(s(0))

Tα T α−1 = −k1 + B2 0 Itα (sign(s(0))d). (α) (α + 1)

Using the inequality, α 0 It (sign(s(0))d)

≤ t0 Itα |d| ≤ 0 Itα d0 = d0

Tα . (α + 1)

Equation (8) becomes −c0 Dtα−1 s(0)sign(s(0))

T α−1 Tα ≤ −(k1 − B2 d0 ) , (α) (α + 1)

which further results in T ≤

(α + 1)c0 Dtα−1 s(0)sign(s(0)) , (α)(k1 − B2 d0 )

(31)

which is always finite. Remark 1 It is clear that sliding mode has been obtained after a finite time t ≥ T where T is such that s(z, T ) = 0. Further, a modified sliding surface is proposed in which sliding starts from t ≥ t0 such that the reduced-order design methodology of the classical approach is preserved. Consider the system in the same form as in Eq. (23). The sliding surface designed for integer reaching law is s = t0 It1−α

     c1 z 1 (t) + z 2 (t) − c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 ) ,

where λ > 0 and c1 ∈ R1×n−1 are the design parameters. Note that the sliding variable, s = 0 at the initial time t = t0 . Then, the system (23) is transformed as

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions c α t0 Dt z 1 (t)

= (A11 − A12 c1 )z 1 (t) + A12

c 1−α s t0 Dt

83

   + c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 )

     s˙ = ct0 Dtα c1 z 1 (t) + z 2 (t) − c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 ) 

   = c1 (A11 − A12 c1 )z 1 (t) + A12 ct0 Dt1−α s + c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 ) .   + A21 z 1 (t) + A22 z 1 (t) + B2 u(t) + B2 f − (−λ)α c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 ) The control input is designed as 

u(t) = B2−1 v − c1 (A11 − A12 c1 )z 1 (t) + A12  

 − B2−1 A21 z 1 (t) + A22 z 1 (t) , × ct0 Dt1−α s + (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) (32) where v = −k1 sign(s). Hence, s˙ = −k1 sign(s) + B2 d + , where = B2−1 [(−λ)α ((c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) ]. Here, it is important to note that | | always remains bounded for any initial condition z(t0 ). It is proved that once the trajectories start from the sliding surface s at t = t0 , they remain on it, and then asymptotically converge to z 1 (t) = z 2 (t) = 0. Lemma 1 If k1 > |B2 d| + | |, then the trajectories are maintained on the sliding surface s = 0, ∀t ≥ t0 . Proof Consider the Lyapunov function, V = 21 s 2 . By taking the time derivative of the Lyapunov function along closed-loop subsystem s˙ = −k1 sign(s) + B2 d + , V˙ = s s˙ = s(− k1 sign(s) + B2 d + ) = − k1 |s| + s B2 d + s ≤ − k1 |s| + |s| | B2 d| + |s| | | , 1

= −(2V ) 2 (k1 − | B2 d| − | |) 1

≤ −η(2V ) 2

where η = k1 − |B2 d| − . When η = k1 − |B2 d| − > 0, Lyapunov stability theory (V = 0 and V˙ ≤ 0) ⇒ V = 0, ∀t ≥ t0 implies s = 0, ∀t ≥ t0 . This completes the proof. The expression for the finite time, T , can be derived as follows:

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√ dV ≤ −η 2V 1/2 dt T 0 dV dt ≤ − √ η 2(V )1/2 0

.

V0

0 T ≤− V0

dV

√ η 2(V )1/2

√ 2V (0) = η

Lemma 2 If the matrix (A11 − A12 c1 ) is negative definite, then the closed-loop system is asymptotically stable. Proof Take the Lyapunov function, V = 21 z 1 (t)z 1 (t). Then,

c α t0 Dt V

=

1c D α z (t)z 1 (t). 2 t0 t 1

Using (29), c α t0 Dt V

≤ z 1 (t)ct0 Dtα z 1 (t) ≤ z 1 (t)(A11 − A12 c1 )z 1 (t)

   + z 1 (t)A12 ct0 Dt1−α s + c1 z 1 (t0 ) + z 2 (t0 ) e−λ(t−t0 ) .

As s = 0 from time t = t0 , the term (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) → 0 as t → ∞. z 1 (t) and hence, the system is asymptotically stable if the matrix (A11 − A12 c1 ) is negative definite. 1

Remark 2 It is important to note that √ if we take v = −λ|s| 2 sign(s) − t ˙ ≤ , where α t0 sign(s)dτ , where α = 1.1 and λ = 1.5 such that B2 |d(t)|

is some a priori known constant, then the proposed control (32) generates continuous signal and it also results better for the chattering minimization problem, which is commonly encountered in the practical implementation of discontinuous control laws. The controller suggested above is known as Super-Twisting in the literature. Again, the trajectories that once start from the sliding surface will remain there for the subsequent time (for more detailed explanation, see [35] and the references cited therein). Now, using fractional reaching law approach, the sliding surface is defined as s = c1 z 1 (t) + z 2 (t) − (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) . Note that s = 0 when t = t0 . Using (33), (23) becomes

(33)

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions c α t0 Dt z 1 (t) c α t0 Dt s

85

  = (A11 − A12 c1 )z 1 (t) + A12 s + (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) = ct0 Dtα z 1 (t) + ct0 Dtα z 2 (t) − (c1 z 1 (t0 ) + z 2 (t0 )) ct0 Dtα (e−λ(t−t0 ) ) 

 = c1 (A11 − A12 c1 )z 1 (t) + A12 s + (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 )

.

+ A21 z 1 (t) + A22 z 2 (t) + B2 (u(t) + d(t)) − (−λ)α ((c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) (34) The control input is designed as  

 u(t) = B2−1 v − c1 (A11 − A12 c1 )z 1 (t) + A12 × s + (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 )   − B2−1 A21 z 1 (t) + A22 z 2 (t) ,

(35) where v = −k1 sign(s). From (34) and (35), ct0 Dtα s = −k1 sign(s) + B2 d + , where = B2−1 (−λ)α ((c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) . Again, the trajectories remain on the sliding surface s = 0 from the very initial time t = t0 , provided k1 > |B2 ||d| + | |. Here, the logic of the associated proof remains the same as previously in the case of Lemma 1. The related condition for the asymptotic convergence of z(t) at the equilibrium point also remains the same as in Lemma 2. This can be shown as follows. Consider the Lyapunov function, V = 21 z 1 (t)z 1 (t). Taking the fractional-order derivative, c α t0 Dt V

≤ z 1 (t)ct0 Dtα z 1 (t) ≤ z 1 (t)(A11 − A12 c1 )z 1 (t) .



−λ(t−t0 ) + z 1 (t)A12 s + (c1 z 1 (t0 ) + z 2 (t0 ))e

As s = 0 from time t = t0 , the term (c1 z 1 (t0 ) + z 2 (t0 ))e−λ(t−t0 ) → 0 as t → ∞. Further, z 1 (t) and hence the system is asymptotically stable if (A11 − A12 c1 ) is negative definite.

9 The Fractional Calculus Approach to Tracking Problem The results of tracking problems can be improved by using fractional calculus. The allowable set of reference functions in the case of classical sliding mode control can be made larger to include functions which are not first-order differentiable. Here, the tracking problem of a non-differentiable function is approached in the following two ways [30]:

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• In the first approach, a reference generator system based on fractional-order operators is selected with ON/OFF-based controller. For example, networks having resistive–inductive and resistive–capacitive fractional-order elements of order α (where α is some real number), spring–dashpot fractal elements, etc. can be used with ON/OFF feedback controller. • The second approach is based on the design of control law based on fractionalorder operators to track non-differentiable reference functions. The first approach is presented by using a switch-controlled R L (DC chopper) circuit as an example. The case is discussed in a generalized framework so that the technique can be applied for a general nonlinear system. After that, the second approach is discussed in general sense. The main idea, on which the solution to this class of problems is based, reflects from the following argument “although a class of functions do not possess firstorder derivatives at some point (for example, triangular or sawtooth function) or at any point (Weierstrass function) in their respective domains, they do have some level of smoothness which can be measured and analyzed with the help of Fractional Calculus.” For the existence of fractional-order differentiability of the given functions, the following remark is important. Remark 3 [36] tR0 L Dtα f (t) exists almost everywhere for the integrable function f (t) even if it possesses a finite number of points of discontinuity. An extensive discussion about various control issues faced in fractional-order systems can be found in [27]. By using simple calculation, it can be verified that the fractional-order derivative of the sawtooth function, i(t), as defined in (3) exists almost everywhere and can be expressed as α

D f (t) =

⎧ ⎨

t 1−α , (2−α)

⎩t

1−α

if 0 < t ≤ 1,

−2(t−1) (2−α)

1−α

, if 1 < t < 2.

(36)

Since limt→1− D α f (t) = limt→1+ D α f (t) = 1/(2 − α); therefore, D α f (t) exists at t = 1. However, its first-order derivative f˙(t) does not exist at t = 1. Similar observations can be made in case of the signal,  f (t) =

t ln t, if t > 0, 0, if t = 0.

(37)

The above signal (37) is continuous but has no derivative at the point t = 0. However, it has well-defined Riemann–Liouville fractional-order derivatives, D α f (t), of all orders up to α < 1 [7]: D α f (t) =

t 1−α [ψ(1) − ψ(2 − α) + ln t] , (1 − α)

(38)

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions

where ψ(x) =

d dt

87

ln (t) is the Psi-function. Similarly, the function f (t) =

n 

(t − tk ) ln |t − tk |, t > a = t0 ,

k=0

where a = t0 < t1 < · · · < tn is a continuous function. It lacks first-order derivative at a finite number of points. However, it possesses fractional-order derivatives, D α f (t), of orders α < 1. From the above discussion, it is clear that there are some functions which do not have first-order derivative but possess fractional-order derivatives. It is possible to capture the behavior of such class of functions using the next theorem. Lemma 3 [16] Suppose f (t) satisfies the Holder ¨ condition of order β with 0 < β ≤ 1, on [a, b], −∞ ≤ a < b ≤ ∞, or f ∈ H β ([a, b]) if | f (t + h) − f (t)| ≤ L|h|β with L not depending on h and t; t, t + h ∈ [a, b]. Then, f (t) possesses the fractional-order Riemann–Liouville derivatives of all orders α < β, and D α f (t) =

f (a) + ψ(t), (1 − α)(t − a)α

where, ψ(t) ∈ H β−α ([a, b]). Now, using fractional-order operators, the tracking problem for non-differentiable reference functions is formulated. The fractional-order derivatives and integrals can be realized practically by using fractional-order inductors or capacitors with the desired specifications. An important description about the construction and implementation of fractional-order inductors can be found in the references [37, 38] and the papers cited within. There exists a vast literature devoted to the practical applicability of other such fractional-order elements. Let the voltage across the fractional-order inductor with inductance L be v. Then, this voltage will be the α-order derivative of the flux φ(t) through it [39], i.e., v(t) =

d α φ(t) ; 0 < α < 1, dt α

α

where dtd α is the Riemann–Liouville (R-L) derivative as defined above. Using the relation φ = Li(t), d α i(t) , (39) v(t) = L dt α where i(t) is the current flowing through the fractional-order inductor. Similarly, the following relationship for the fractional-order capacitor (also known as supercapacitor or ultra-capacitor) can be obtained i(t) =

d α v(t) d α q(t) =C ; α dt dt α

0 < α < 1,

(40)

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u=1

L

i

u=0

V

R

Fig. 3 Switch-controlled fractional R L circuit

where q and C are the accumulated charge and the capacitance of the fractional-order capacitor, respectively. Consider the same DC chopper circuit again as shown in Fig. 1. Now, the simple inductor used in the circuit in Fig. 1 is replaced by a fractional-order inductor as shown in Fig. 3. For the circuit in Fig. 3, its dynamic behavior can be captured by L

d α i(t) = u(t)V − i(t)R; dt α

0 < α < 1,

(41)

where L, R, i(t), and u(t) ∈ {0, 1} are the inductance, resistance, and current through fractional-order inductor and the control input, respectively. Remark 4 For a simple illustration of the proposed technique, a circuit like Fig. 3 is considered. However, without loss of generality, the idea can be extended to derive the mathematical equation of electrical circuits composed of any finite number of resistances, fractional-order inductors, fractional-order capacitors, and voltage (current) sources. Suppose the state variables are currents flowing in the fractionalorder inductors z L ∈ Rn 2 and voltages across the fractional-order capacitors, z C ∈ Rn 1 . Using Kirchhoff’s Law, the state equations of the fractional-order linear circuit can be obtained as  dα z       L A11 A12 zL B1 dt α = + u; 0 < α, β < 1, (42) β d zC A A z B 21 22 C 2 β dt where the components of u = u(t) ∈ Rm are the current or voltage sources and Ai j ∈ Rni ×n j , Bi ∈ Rni ×m ; i, j = 1, 2. In order to solve the problem (c) in Sect. 5, one can calculate the fractional-order rate of change of the error (4),

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions

dαe d α i(t) d α ir (t) = − . α dt dt α dt α

89

(43)

Here, a sawtooth function is used as the desired voltage. It is important to recall here that the first-order or classical derivative of the desired reference voltage is not feasible. However, it is possible to calculate and further use the fractional-order derivative of the reference function, vr (t), for the manipulation of the error (4). Using Eqs. (41), (4), and (43),    R V L d α ir (t) dαe u(t) − e − i = (t) + r dt α L R R dt α    R L d α ir (t) , i max u(t) − e − ir (t) + = L R dt α

(44)

where i max = VR . Suppose that initially, the error is negative, i.e., e < 0. Therefore, one should “ON” the switch, i.e., u(t) = 1 as long as e is negative. Substituting e < 0 and u(t) = 1 in (44),    dαe R L d α ir (t) . i = + |e| − i (t) + max r dt α L R dt α

(45)

Using Eq. (45), it can be concluded that the fractional-order time derivative of the error variable e(t) is guaranteed to remain positive if the desired trajectory satisfies the condition,  L d α ir (t)   i max > ir (t) + ; ∀t. R dt α Now, it is proved that the first hitting to the error surface (sliding surface) occurs in finite time, T . For simplicity of representation, Eq. (45) is rewritten as dαe = k|e| + d, dt α    α where k = R/L > 0 and d = k i max − ir (t) + LR d dtirα(t) > 0. Applying fractional-order integral operator to both sides of (46), D −α Using the relations, D −α

dαe = D −α k|e| + D −α d. dt α

dαe t α−1 , = s(t) − D α−1 e(0) α dt (α)

and D −α d = d

tα ; (1 + α)

(46)

(47)

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(46) can be manipulated so that after finite time t = T , e(T ) − D α−1 e(0)

T α−1 Tα Tα =k +d . (α) (1 + α) (1 + α)

(48)

It is straightforward to conclude that e(t) = 0 results if time T is finite. From equation (48) and by taking initial condition of e to be negative, D α−1 e(0)

Tα Tα T α−1 =k +d (α) (1 + α) (1 + α) α−1 (1 + α)D e(0) ⇒T = . (α)(k + d)

(49)

It can be easily checked that the above calculated time, T , always comes out to α be finite. On the other hand, for positive error, i.e., e > 0, ddt αe should be negative. Therefore, one must “OFF” the switch, i.e., u(t) = 0 as long as e remains positive. Substituting e > 0 and u(t) = 0 in (44),    R L d α ir (t) dαe . e + ir (t) + =− dt α L R dt α

(50)

So, the fractional-order time derivative of the error is guaranteed to remain negative provided the desired reference function satisfies  L d α ir (t)    > 0; ∀t. ir (t) + R dt α Therefore, it is clear that the desired tracking of non-differentiable reference function is possible (using fractional-order switch-controlled R L circuit), provided  L d α ir (t)   0 < ir (t) +  < i max ; ∀t. R dt α Similarly, the desired reference function can also be tracked by utilizing fractionalorder control law design.

10 General Class of Nonlinear Systems Affine in Control Consider the following general form of an input-affine nonlinear system: x˙ = f (x) + g(x)u,

(51)

where the functions f (x) and g(x) are locally Lipschitz over a domain D ∈ Rn with f : D → Rn and g : D → Rn having bounded region D ⊂ Rn with an equilibrium

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions

91

point at x = 0. These restrictions ensure that a unique solution exists which is defined for all t ≥ t0 . It is further assumed that the function g(x) is invertible. Now, suppose that it is desired to maintain the output of a system at some reference function xr , ∀t ≥ t0 . In order to solve the problem (c) discussed in Sect. 5, the fractional rate of change of the error is calculated as D α y(t) = D α x(t) − D α xr (t), where D α =

dα . dt α

(52)

By using the property of fractional-order operator, D α y(t) = D α−1 Dx(t) − D α xr (t).

(53)

Using Eqs. (51), (53) becomes D α y(t) = D α−1 ( f (x) + g(x)u) − D α xr (t) = D α−1 f (x) + D α−1 (g(x)u) − D α xr (t).

(54)

For simplicity of representation, the effect of any type of disturbance is not considered here. It is clear that simple fractional-order P I α control is enough to stabilize the fractional-order error dynamics at the origin. Moreover, robust control strategies can be used in the presence of disturbances. The same problem formulation can befollowed  in that case. Now, suppose that the designed control law is u := g(x)−1 D 1−α ν . Then, keeping u(0) = 0, D α y(t) = D α−1 f (x) + ν − D α xr (t).

(55)

Now, defining ν := −D α−1 f (x) + D α xr (t) − kkdi D −α y(t) − kdp y(t), where k p , ki , kd > 0 are the parameters of the fractional-order P I λ D μ controller and substituting ν in (55), the expression becomes k

kd D α y(t) + ki D −α y(t) + k p y(t) = 0.

(56)

Then, the values of the controller parameters k p , ki , kd > 0 can be selected based on the work in [40] such that y(t) ≈ 0, which further implies x(t) ≈ xr (t). From the above discussion, it can be stated that if the desired reference function is not differentiable, then the classical integer-order derivative becomes unbounded which results in distorted error dynamics at the non-differentiable points. On the other hand, with fractional-order error dynamics (52), the terms remain in a bound. This also satisfies the property of Input-to-State Stability (ISS) [41]. According to this concept, the states of the fractional-order linear time-invariant system remain in a bound for all time,

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α

t

||x(t)|| ≤ ||x0 ||||E α (A(t − t0 ) )|| + ||

(t − τ )α−1 E α,α (A(t − τ )α )dτ ||.

t0

, then there Further, if the state matrix A satisfies the condition |arg(λ(A))| > απ 2 t exists a positive number M such that || t0 (t − τ )α−1 E α,α (A(t − τ )α )dτ || ≤ M. In such case, the states satisfy the condition [42] ||x(t)|| ≤ ||x0 ||||E α (A(t − t0 )α )|| + ||B|| sup ||u(τ )||M. t0 ≤τ ≤t

Suppose that the information of fractional-order derivative of the reference function is not available. Then, the fractional-order derivative term can be removed from the control input. Here, the closed-loop system (55) also contains the unknown term. With proper design of the fractional-order PID gain values, it can be shown that the closed-loop system (55) is fractional-order input-to-state stable where the input is the fractional-order derivative of the reference function. Therefore, it is clear that the proposed approach is valid for general class of nonlinear systems. This further enhances the applicability of the technique. The simulation results of the switch-controlled R L circuit have been demonstrated in the next section.

11 Simulation Results A switch-controlled R L circuit (DC chopper) is considered for the proposed approach to tracking problem [30]. A sawtooth function is taken as the reference current to be tracked. A comparison of the simulation results is done as shown in Fig. 4 for both the circuits with fractional inductor and simple inductor. The values of the parameters are taken as α = 0.5, V = 1 V, R = 10 , and L = 10 μH. A sawtooth reference current function having an amplitude of 1 V p-p and a frequency of 100 Hz is used in both the cases. At the corner points, where the reference function is not differentiable, the tracking is poor for the circuit with simple inductor as shown in the figure. For the circuit with fractional inductor, the actual variable is able to track the desired sawtooth reference function at all points accurately. This result properly demonstrates the effectiveness of the technique.

12 Conclusions From the discussion in this chapter, it is clear that the proposed technique using sliding mode control is quite effective in tracking the desired reference functions in finite time. At the same time, the sliding mode control based approach provides

Sliding Mode Control based Tracking of Non-Differentiable Reference Functions

93

Fig. 4 Current trajectories in switch-controlled R L circuit

robustness to disturbances and parametric uncertainties. It has been demonstrated that an improvement in the fundamental operation by using fractional calculus can result in a large class of reference functions to be tracked. The methodology is applicable irrespective of the non-differentiability of the reference function at some points. This enables accurate tracking in applications where the reference function changes abruptly. From the theoretical point of view, the tracking problem considered in this chapter can be explored in the direction of implementing various classes of sliding mode controllers. Continuous terminal sliding mode control based technique may be utilized to improve the results [19]. Further, arbitrary-time convergence can also be obtained by using the concept presented in [22]. Similar problems are trajectory tracking by robotic manipulators and spacecrafts [10–13, 43]. These applications require careful and precise operations of the actuators. There can be sudden obstacles in the path of motion which can dynamically alter the desired reference functions. This requires a fast control response while following the desired trajectory with high accuracy. The methodology presented in this chapter can be explored in this direction to find new results. Another evolving field is that of nanotechnology. In these applications, atomic force microscopy is frequently used which requires fast, precise, and repetitive motion of the piezoelectric actuator. Implementing the proposed technique can greatly improve their performance. Fast tracking of triangular signals with accuracy in scanning probe microscopy is a challenging problem. A triangular function contains all the odd harmonics of the fundamental frequency. When a triangular signal is fed to the piezoelectric actuator, high-frequency excitation results which further reflects as inaccurate triangular signal at the free end of the actuator causing a distorted scanned image [14, 44]. Image edge detection is another commonly encountered problem in which this technique can be used. It consists of mathematical methods which locate the points in a digital image where the image brightness changes sharply. Texture enhance-

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ment is also very important image-processing aspect in the interpretation of image data, image restoration, pattern recognition, robotics, medical image processing, and remote sensing. The fractional-order derivative operator can be used to track sharper edges of the image which can further be used for its reconstruction with higher accuracy. A similar approach is proposed in [45] using Riemann–Liouville (R-L) fractional-order derivative. Further improvements can be obtained by using the proposed technique. Therefore, it can be concluded that the approach presented in this chapter has the potential to greatly improve the control performance in various fields of diverse applications. Being a generalized approach, it covers a wide range of problems which can be handled quite effectively. The mathematical technique consists of interesting dimensions which remain to be explored.

References 1. Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice Hall (1991) 2. Haimo, V.: Finite time controllers. SIAM J. Control. Optim. 24(4), 760–770 (1986) 3. Bhatt, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control. Optim. 38(3), 751–766 (2000) 4. Moulay, E., Perruquetti, W.: Finite time stability conditions for non-autonomous continuous systems. Int. J. Control. 81(5), 797–803 (2008) 5. Podlubny, I.: Fractional Differential Equations. Academic Press (1999) 6. Bandyopadhyay, B., Kamal, S.: Stabilization and control of fractional order systems: a sliding mode approach. In: Lecture Notes in Electrical Engineering, vol. 317. Springer (2014) 7. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach Science Publishers, London, New York (1993) 8. Aoki, Y.: On a tracking problem for linear systems. J. Math. Anal. Appl. 38(2), 365–371 (1972) 9. Slotine, J.-J.E.: The robust control of robot manipulators. Int. J. Robot. Res. 4(2), 49–64 (1985) 10. Arteaga, M.A., Siciliano, B.: On tracking control of flexible robot arms. IEEE Trans. Autom. Control. 45(3), 520–527 (2000) 11. Xia, Y., Zhu, Z., Fu, M., Wang, S.: Attitude tracking of rigid spacecraft with bounded disturbances. IEEE Trans. Ind. Electron. 58(2), 647–659 (2011) 12. Su, J., Cai, K.-Y.: Globally stabilizing proportional-integral-derivative control laws for rigidbody attitude tracking. J. Guid. Control. Dyn.34(4), 1260–1264 (2011) 13. Lu, K., Xia, Y.: Adaptive attitude tracking control for rigid spacecraft with finite-time convergence. Automatica 49, 3591–3599 (2013) 14. Bazaei, A., Moheimani, S.O.R., Sebastian, A.: An analysis of signal transformation approach to triangular waveform tracking. Automatica 47, 838–847 (2011) 15. Sakthivel, R., Mohanapriya, S., Ahn, C.K., Karimi, H.R.: Output tracking control for fractionalorder positive switched systems with input time delay. IEEE Trans. Circuits Syst. II Express Briefs 66(6), 1013–1017 (2019) 16. Ross, B., Samko, S.G., Love, E.R.: Functions that have no first order derivative might have fractional derivatives of all orders less than one. R. Anal. Exch. 20(1), 140–157 (1994) 17. Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. CRC Press (1998) 18. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation. Springer, New York (2014)

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19. Kamal, S., Moreno, J.A., Chalanga, A., Bandyopadhyay, B., Fridman, L.: Continuous terminal sliding-mode controller. Automatica 69, 308–314 (2016) 20. Thomas, M., Kamal, S., Bandyopadhyay, B., Vachhani, L.: Continuous higher order sliding mode control for a class of MIMO nonlinear systems: An ISS approach. Eur. J. Control. 81, 1–7 (2018) 21. Mishra, J.P., Yu, X., Jalili, M.: Arbitrary-order continuous finite-time sliding mode controller for fixed-time convergence. IEEE Trans. Circuits Syst. II Express Briefs 65(12), 1988–1992 (2018) 22. Pal, A.K., Kamal, S., Nagar, S.K., Bandyopadhyay, B., Fridman, L.: Design of controllers with arbitrary convergence time. Automatica 112, 108710 (2020) 23. Goyal, J.K., Kamal, S., Patel, R.B., Yu, X., Mishra, J.P., Ghosh, S.: Higher order sliding mode control-based finite-time constrained stabilization. IEEE Trans. Circuits Syst. II Express Briefs 67(2), 295–299 (2020) 24. Soni, S., Kamal, S., Yu, X., Ghosh, S.: Global stabilization of uncertain SISO dynamical systems using a multiple delayed partial state feedback sliding mode control. IEEE Trans. Circuits Syst. II Express Briefs 67(7), 1259–1263 (2019) 25. Xiong, X., Kikuwue, R., Kamal, S., Jin, S.: Implicit-Euler implementation of super-twisting observer and twisting controller for second-order systems. IEEE Trans. Circuits Syst. II Express Briefs (2019). https://doi.org/10.1109/TCSII.2019.2957271 26. Cheng, J., Yi, J., Zhao, D.: Design of a sliding mode controller for trajectory tracking problem of marine vessels. IET Control. Theory Appl. 1(1), 233–237 (2007) 27. Chen, Y., Petras, I., Xue, D.: Fractional order control—a tutorial. In: American Control Conference, St. Louis, MO, USA, 1397–1411 (2009) 28. Li, C., Dang, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777–784 (2007) 29. Podlubny, I.: Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation (2001). arXiv preprint math/0110241 30. Kamal, S., Yu, X., Sharma, R.K., Mishra, J., Ghosh, S.: Non-differentiable function tracking. IEEE Trans. Circuits Syst. II Express Briefs 66(11), 1835–1839 (2019) 31. Kamal, S., Sharma, R.K., Dinh, T.N., Bandyopadhyay, B., Harikrishnan, M.S.: Sliding mode control of uncertain fractional-order systems: a reaching phase free approach. Asian J. Control. (2019). https://doi.org/10.1002/asjc.2223 32. Baleanu, D., Güven, Z.B., Machado, J.A.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2010) 33. Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: State variables and transients of fractional order differential systems. Comput. Math. Appl. 64(10), 3117–3140 (2012) 34. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-leffler stability. Comput. Math. Appl. 59(5), 1810–1821 (2010) 35. Chalanga, A., Kamal, S., Bandyopadhyay, B.: A new algorithm for continuous sliding mode control with implementation to industrial emulator setup. IEEE/ASME Trans. Mechatron. 20(5), 2194–2204 (2015) 36. Li, C.P., Zhao, Z.G.: Introduction to fractional integrability and differentiability. Eur. Phys. J. Spec. Top. 193, 5–26 (2011) 37. Krishna, M.S., Das, S., Biswas, K., Goswami, B.: Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. IEEE Trans. Electron Devices 58(11), 4067–4073 (2011) 38. Radwan, A.G., Salama, K.N.: Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901–1915 (2012) 39. Kaczorek, T., Rogowski, K.: Fractional linear systems and electrical circuits. In: Studies in Systems, Decision and Control, vol. 13. Springer (2015) 40. Podlubny, I.: Fractional-order systems and P I λ D μ -controllers. IEEE Trans. Autom. Control. 44(1), 208–214 (1999) 41. Khalil, H.K.: Nonlinear Systems. Prentice Hall (2002)

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42. Sene, N.: Fractional input stability and its application to neural network. Discret. Contin. Dyn. Syst. Ser. S 13 (2020) 43. Gennaro, D.S.: Output attitude tracking for flexible spacecraft. Automatica 38(6), 600–605 (2002) 44. Habibullah, H., Pota, H.R., Peterson, I.R., Rana, M.S.: Tracking of triangular reference signals using LQG controllers for lateral positioning of an AFM scanner stage. IEEE/ASME Trans. Mechatron. 19(4), 1105–1114 (2014) 45. Nandal, A., Rosales, H.G., Dhaka, A., Padilla, J.M.C., Tejada, J.I.G., Tejada, C.E.G., Ruiz, F.J.M., Valdivia, C.G.: Image edge detection using fractional calculus with feature and contrast enhancement. Ciircuit Syst. Signal Process. 37, 3946–3972 (2018)

State Boundedness in Discrete-Time Sliding Mode Control Paweł Latosinski ´ and Andrzej Bartoszewicz

Abstract Discrete-time sliding mode control strategies are an excellent way of ensuring robustness of the plant with respect to disturbance and model uncertainties. Such strategies mitigate the effect of perturbations on the system by confining its representative point to a specific, narrow quasi-sliding mode band defined in the state space. Sliding mode controller design is typically focused on ensuring a specific evolution of the sliding variable, which is a constructed output of the plant. On the other hand, such strategies give no explicit information about the evolution of individual state variables. Motivated by this problem, in this chapter, we have calculated ultimate bounds of each state variable during system sliding motion. In particular, we have considered conventional sliding variables with relative degree one as well as more sophisticated ones with arbitrary relative degrees. The obtained bounds of all state variables are applicable to any discrete-time sliding mode control strategy as long as its quasi-sliding mode band width is known. Moreover, we have analyzed three specific cases in which state error can be further reduced: the case of matched perturbations, dead-beat sliding hyperplane, and switching-type quasisliding motion.

1 Introduction Sliding mode control strategies have first been proposed for continuous-time systems in [8, 10], and [23]. The major advantage of these strategies is the ability to completely reject the effect of certain types of disturbance and model uncertainties on the plant by confining its representative point to a hyperplane defined in the state space. Furthermore, since virtually all modern control processes are implemented P. Latosi´nski (B) · A. Bartoszewicz Institute of Automatic Control, Lodz University of Technology, B. Stefanowskiego 18/22 St., 90924 Lodz, Poland e-mail: [email protected] A. Bartoszewicz e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_4

97

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digitally, discrete-time sliding modes have also been considered [16, 17, 24, 25]. Contrary to their continuous-time equivalent, discrete-time sliding mode controllers drive the system representative point to a specified vicinity of the sliding hyperplane referred to as a quasi-sliding mode band. By confining the system state inside that band, they guarantee a degree of robustness with respect to uncertainties. Discrete-time sliding mode controller design procedure can follow one of the two approaches established in literature. Conventionally, one first defines the control signal which depends on a specified sliding variable and then proves that the application of this signal ensures the existence of quasi-sliding motion, as done in [17] and in many other works. Alternatively, one can use the more recent reaching law approach, first proposed in [11] for continuous-time systems and in [12] for discrete-time ones. In this approach, one first specifies the desired evolution of the sliding variable and then synthesizes the control signal which ensures this particular evolution. Reaching law approach has gained significant recognition in the field of discrete-time sliding mode control and many authors have published novel strategies using this method [2, 9, 13, 15, 18]. Furthermore, authors of works such as [14] and [6] have managed to apply reaching law-based strategies without the conventional assumption of matched perturbations. Discrete-time sliding mode control strategies can aim to ensure two distinct types of system quasi-sliding motion. The first one, called switching type, follows the definition established in [12] and ensures that the system representative point crosses the sliding hyperplane in each step after crossing it for the first time. This type of sliding motion closely follows the principles of continuous-time sliding modes. The second type of quasi-sliding motion, referred to as non-switching, has been introduced in [3]. It involves confining the system representative point to a vicinity of the sliding hyperplane without the necessity to cross it and, as a result, reduces undesired oscillations during system sliding motion. A crucial part of sliding mode controller design is the proper selection of the sliding hyperplane toward which the system representative point is driven. To ensure stability of the discrete-time system during quasi-sliding motion, this hyperplane must guarantee that all closed-loop poles are placed inside the unit circle. Many methods of sliding hyperplane selection have been proposed, such as the LMI approach [7], the use of Ackermann’s formula [1], or dead-beat controller design [22]. Typically, switching hyperplane is designed so that its corresponding sliding variable has relative degree equal to one, which implies that it is affected by the control signal and matched uncertainties from the previous time instant. However, various authors have designed DSMC strategies using variables with relative degree higher than one [5, 19–21]. It has been demonstrated that the use of such variables can significantly improve sliding mode performance of the plant, particularly in switchingtype quasi-sliding motion. Design of the switching hyperplane corresponding to an arbitrary relative degree sliding variable has been discussed in [5]. However, even though proper selection of this hyperplane can ensure stable response of the system, sliding mode control strategies are still primarily concerned with the evolution of the sliding variable and provide limited information about the actual state of the system. With this in mind, in this chapter, we have calculated

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explicit bounds of all state variables in the sliding phase. In particular, we have developed formulae for conventional relative degree one sliding variables as well as arbitrary relative degree ones. The obtained formulae denoting the obtained bounds are generic and applicable to any discrete-time sliding mode control strategy with known quasi-sliding mode band width. Furthermore, we have analyzed three special cases when these bounds can be simplified. These cases concern matched perturbations, dead-beat switching hyperplane, and switching-type quasi-sliding motion.

2 Discrete-Time Sliding Mode Control In this chapter, we analyze performance of single-input discrete-time plants subject to a sliding mode control strategy. In particular, in this preliminary section, we describe the considered systems in detail and outline the basics of discrete-time sliding mode controller design. Then, we will demonstrate how the magnitude of the plant’s sliding variable is reflected in the state error of all its variables. The main contribution of our work is obtaining an explicit formula describing the error of each state variable in the sliding phase. Initially, this formula is developed for conventional relative degree one sliding variables, but the general case with arbitrary relative degree variables is also considered later in this chapter. The obtained bounds can be further simplified for many cases common in sliding mode control, which have also been analyzed in our work.

2.1 Considered Class of Systems Dynamics of the considered plants are expressed in the state space as x(k + 1) = Ax(k) + bu(k) + pd(k),

(1)

where x ∈ Rn is the state vector with known initial conditions x(0) = x 0 , u ∈ R is the control signal, state matrix A and vectors b, p are of appropriate dimensions, and d ∈ R represents perturbations affecting the plant. Since vectors b and p are not necessarily collinear, these perturbations do not satisfy matching conditions. The first step of designing a sliding mode controller for plant (1) involves selecting an appropriate sliding variable and a switching hyperplane on which the value of this variable becomes zero. A typical choice of this variable and its corresponding sliding hyperplane is σ (k) = cT x(k) = 0,

(2)

where c ∈ Rn is a vector selected to ensure cT b = 0 and to guarantee stability of the closed-loop system. Such a choice guarantees that the selected sliding variable has

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relative degree one, which implies that it is affected by the control signal from the previous step. Selection of this vector will be discussed in detail in the next section. In order to make any sliding mode control strategy applicable to the considered systems, it must be assumed that the total effect of perturbations on these systems is bounded. In particular, for all k d min ≤ d(k) ≤ d max ,

(3)

where d min and d max are constant. Furthermore, we define d avg =

1 max 1 (d + d min ) and d δ = (d max − d min ), 2 2

(4)

which represent the mean effect of perturbations on the system and its maximum admissible deviation from this mean.

2.2 Closed-Loop Stability The objective of sliding hyperplane design is to guarantee closed-loop stability of the system when its representative point remains on this hyperplane in the absence of disturbance (i.e., when σ (k + 1) = σ (k) = 0 and d(k) = 0). To that end, one first substitutes (1) into (2) and gets σ (k + 1) = cT Ax(k) + cT bu(k).

(5)

Solving (5) for u(k) when σ (k + 1) = 0, one obtains the control signal which drives the representative point of the system onto the sliding hyperplane in each step. It is expressed as u(k) = −(cT b)−1 cT Ax(k).

(6)

Substitution of (6) back into (1) yields   x(k + 1) = A − b(cT b)−1 cT A x(k).

(7)

Consequently, to ensure stability of the plant, vector c must be selected so that all eigenvalues of the closed-loop system state matrix Acl = A − b(cT b)−1 cT A are placed inside the unit circle. For such eigenvalues matrix, Acl satisfies

(8)

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 k lim Acl = 0,

(9)

k→∞

which ensures a stable response of closed-loop system (7).

2.3 Reaching Law Approach In this chapter, sliding mode control strategies designed with the use of reaching law approach will be considered. These strategies always aim to drive the system representative point to a specified vicinity of the sliding hyperplane, which is helpful for the state error analysis conducted in our work. In the reaching law approach, the target evolution of the sliding variable is first expressed with a recursive formula σ (k + 1) = f [σ (k)] + cT pd(k) − cT pd avg ,

(10)

where f is a scalar function of the sliding variable selected to ensure favorable properties of its evolution. This reaching law can then be applied to design the control signal which ensures this evolution, similar to derivations performed in (5) and (6). The control signal   u(k) = (cT b)−1 f [σ (k)] − cT pd avg − cT Ax(k) .

(11)

It should be noticed that, even though the evolution of the sliding variable (10) is affected by the unpredictable disturbance, this disturbance is not present in control signal (11). The objective of any discrete-time sliding mode control strategy is to drive the system representative point to a specific quasi-sliding mode band around the switching hyperplane. The band is expressed as 

x : |cT x| ≤ B



(12)

and its width B depends on the choice of the reaching law, though it typically cannot be smaller than d δ [4]. It is evident from relations (10) and (11) that sliding mode control strategies are designed to ensure a certain evolution of sliding variable (2). However, even though they ensure stability of the plant and its good robustness with respect to disturbance, these strategies only provide information about the evolution of constructed system output σ (k) = cT x(k) and not about the individual state variables. Even though one can suspect that achieving a narrower quasi-sliding mode band (12) results in smaller state error, relationship between the sliding variable σ and individual state variables x j for j = 1, 2, ..., n is not clear, and explicit state error bounds are difficult to obtain analytically. Therefore, in the next section of this chapter, it will be demonstrated how the quasi-sliding mode band width translates to the absolute error of all state variables.

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3 State Error in Sliding Mode Control In this section, an analytical expression describing the absolute bounds of all state variables in the sliding phase will be obtained. It will be further shown that error of each state variable is directly proportional to the quasi-sliding mode band width and the magnitude of unmatched perturbations. State vector x will be first transformed so that it depends strictly on the sliding variable rather than the control signal. Substitution of (11) into (1) yields   x(k + 1) = Ax(k) + b(cT b)−1 f [σ (k)] − cT pd avg − cT Ax(k) + pd(k) (13)   T −1 T T −1 T avg = Ax(k) − b(c b) c Ax(k) + b(c b) f [σ (k)] − c pd + pd(k).

Uncertainties affecting the plant can then be divided into multiple parts in the following way:   x(k + 1) = Ax(k) − b(cT b)−1 cT Ax(k) + b(cT b)−1 f [σ (k)] − cT pd avg + pd(k) − b

(14)

cT p cT p d(k) + b d(k). cT b cT b

Finally, taking reaching law (10) into account, we express the state vector as x(k + 1) = Acl x(k) + b(cT b)−1 σ (k + 1) + wd(k),

(15)

where Acl is specified by (8) and  cT p w = p−b T . c b

(16)

Before the explicit bounds of all state variables are obtained, we define vector v j = [0 . . . 0 1 0 . . . 0 ]T j−1

(17)

n− j

for j = 1, 2, ..., n. This vector allows one to express the j-th state variable as x j = v Tj x. An explicit absolute bound of each state variable in the sliding phase will now be presented. To that end, the following theorem will be formulated. Theorem 1 If the control signal for plant (1) is defined by (11), then for each j = 1, 2, ..., n the j-th state variable will at least asymptotically approach the vicinity of zero specified by the following inequality: ∞ ∞  −1   T cl i   avg  T cl i   v ( A ) b + |d | + d δ v ( A ) w  , |x j (k)| ≤B  cT b j j i=0

i=0

(18)

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103

where d avg and d δ are specified in (4) and B is the width of the quasi-sliding mode band (12). Proof Let k > 0. Since initial conditions x 0 of the system are known, repeated substitution of the left-hand side of relation (15) into its right-hand side yields x(k) = ( Acl )k x 0 +

k−1  ( Acl )i b(cT b)−1 σ (k − i) i=0

+

k−1 

(19)

( Acl ) wd(k − i − 1). i

i=0

Let k0 be the first time instant for which the system representative point has entered the quasi-sliding mode band (12). We can then rewrite (19) as x(k) = ( A ) x 0 + cl k

k−1 

( Acl )i b(cT b)−1 σ (k − i)

i=k−k0 +1

+

k−k 0

( Acl )i b(cT b)−1 σ (k − i)

(20)

i=0

+

k−1 

( Acl )i wd(k − i − 1).

i=0

For the sake of clarity, this expression will be divided into three parts that will then be analyzed separately. Taking relation (17) into account, for a given j = 1, 2, ..., n, the j-th state variable is expressed as x j (k) = v Tj x(k) = α j (k) + β j (k) + γ j (k),

(21)

where α j (k)

= v Tj ( Acl )k x 0

+

k−1 

v Tj ( Acl )i b(cT b)−1 σ (k − i),

i=k−k0 +1

β j (k) =

k−k 0

v Tj ( Acl )i b(cT b)−1 σ (k − i),

(22)

i=0

γ j (k) =

k−1 

v Tj ( Acl )i wd(k − i − 1).

i=0

It will now be shown that as k tends to infinity all state variables are bounded exactly as stated in (18). One can see from (22) that all components of the sum on the

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right-hand side of α j (k) have a common factor ( Acl )k−k0 +1 . Thus, relation (9) implies that each component converges to 0 as k tends to infinity, which further gives lim |α j (k)| = 0.

(23)

k→∞

Furthermore, (22) implies that the absolute value of β j can be expressed as 0  k−k    T cl i T −1  v j ( A ) b(c b) σ (k − i) |β j (k)| =

i=0

0    −1 k−k   v T ( Acl )i b · σ (k − i). ≤ cT b

(24)

j

i=0

Reaching law-based sliding mode controller design ensures that for all k > k0 inequality |σ (k)| = |cT x(k)| ≤ B is satisfied, where B is the width of the band (12). Consequently, relation (24) gives 0   −1 k−k  v T ( Acl )i b, |β j (k)| ≤B  cT b j

(25)

i=0

which further implies 0   T −1 k−k  v T ( Acl )i b lim sup |β j (k)| ≤ lim sup B  c b k→∞

j

k→∞

i=0

∞  −1   T cl i  v ( A ) b. =B  cT b

(26)

j

i=0

Similarly, since relation (4) implies |d(k)| ≤ |d avg | + d δ for all k, one can express the absolute value of γ j (k) as |γ j (k)| ≤

k−1    T cl i   v ( A ) w · d(k − i − 1) j i=0

k−1  T cl i    v ( A ) w. ≤ |d avg | + d δ j

(27)

i=0

This further implies ∞  T cl i    v ( A ) w . lim sup |γ j (k)| ≤ |d avg | + d δ j k→∞

i=0

(28)

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Considering relations (21), (23), (26), and (28) the ultimate bound of the j-th state variable can be expressed as   lim sup |x j (k)| ≤ lim sup |α j (k)| + |β j (k)| + |γ j (k)| k→∞

(29)

k→∞ ∞ ∞  −1   T cl i   avg  T cl i   v ( A ) b + |d | + d δ v ( A ) w , ≤0 + B  cT b j j i=0

i=0

which implies that each state variable at least asymptotically converges to the vicinity of zero identical to the one given in (18). In this section, explicit bounds of all state variables in discrete-time sliding mode control have been obtained. Furthermore, one can see from relation (18) that these bounds are directly proportional to the quasi-sliding mode band width and the magnitude of unmatched perturbations. Consequently, achieving better sliding mode performance of the system always results in smaller error of every individual state variable. In the next section, several special cases which let one further reduce the width of band (18) will be discussed.

4 Special Cases Formula (18) describing the absolute error of each state variable holds true for all systems. However, there is a number of cases commonly occurring in sliding mode control in which the obtained bound is either too conservative or not sufficiently streamlined. Three such cases will be analyzed in this section and modified forms of the bound (18) will be presented.

4.1 Matched Perturbations In sliding mode control, it is often assumed that the disturbance and model uncertainties affecting the plant satisfy matching conditions, which means they affect the system state through the same input channel as the control signal. These conditions imply that vectors b and p in relation (1) are collinear, which means they satisfy p = ϕb for a certain constant ϕ = 0. Although such conditions are quite restrictive in practical applications of sliding mode control, they offer significant benefits as far as system robustness is concerned. When perturbations affecting the plant are matched, then for any c ∈ Rn , one gets  ϕcT b cT p  = ϕb − b T = 0. w = p−b T c b c b

(30)

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As a result, the second element on the right-hand side of inequality (18) is reduced to zero and the ultimate bound of the j-th state variable can be expressed in a more compact form ∞  −1   T cl i  v ( A ) b. |x j (k)| ≤ B  cT b j

(31)

i=0

Depending on the magnitude of the disturbance, this bound can be significantly smaller than the one obtained in Theorem 1 for the case with unmatched perturbations.

4.2 Dead-Beat Control In order to ensure stability of a discrete-time plant, all eigenvalues of its closedloop system state matrix (8) must be placed inside the unit circle. One possible choice of sliding hyperplane cT x = 0 that satisfies this criterion involves placing all eigenvalues of Acl in zero, which causes this matrix to become nilpotent. In other words, ( Acl )i = 0 for all i ≥ n, where n is the dimension of matrix Acl . As a result, expression (18) specifying absolute bounds of all state variables can be greatly simplified. Indeed, if k0 is the first time instant for which the band (12) has been reached, then for any k ≥ k0 + n formula (18) is reduced to a finite sum n−1 n−1  T −1   T cl i   avg  T cl i   δ     v ( A ) w  , v ( A ) b + |d | + d |x j (k)| ≤B c b j

i=0

j

(32)

i=0

which yields significantly smaller values than the bound given in Theorem 1. Furthermore, when dead-beat sliding hyperplane is considered, the obtained bound (32) is always reached in finite time k0 + n, whereas the general case considered in Theorem 1 only ensures at least asymptotic convergence to the vicinity of zero.

4.3 Switching-Type Quasi-sliding Motion Sliding mode control strategies can aim to ensure two distinct types of discretetime quasi-sliding motion: switching type and non-switching type. As outlined in the introduction, switching-type strategies follow the classic definition introduced in [12], which states that • For any initial conditions of the system, sliding variable monotonically approaches zero and changes its sign in finite time. • After changing sign for the first time, sliding variable changes its sign again in each subsequent step. • Maximum overshoot after the sliding variable changes its sign is always nonincreasing.

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Although such strategies tend to produce undesirable oscillations due to enforced switching-type motion in the sliding phase, it can be demonstrated that they have a significant effect on the error of all state variables. Indeed, let k0 be the first time instant for which sliding variable σ has changed its sign. Then, switching-type sliding mode controller design ensures that for all k ≥ k0 sgn[σ (k + 1)] = −sgn[σ (k)] and |σ (k)| ≤ B.

(33)

Let us now consider a special case where for a given j = 1, 2, ..., n and for all k ≥ 0 elements v Tj ( Acl )k b ∈ R have the same sign. Such a case is not uncommon in various practical applications, such as plants with strictly positive parameters. Then, since σ (k) changes its sign in each step, one can conclude that only every second element of the sum in the second line of (22) will contribute toward the upper bound of |β j (k)|. Indeed, considering (26) the bound is reduced to  −1 lim sup |β j (k)| ≤B  cT b max{β even , β odd j j }, k→∞

(34)

where β even = j

∞ ∞    T cl 2i   T cl 2i+1  v ( A ) b, β odd = v ( A ) b. j j j i=0

(35)

i=0

Consequently, depending on the considered plant the absolute value of β j can be reduced up to 50% compared to the general case. Taking this into account, the absolute bound of the j-th state variable can now be expressed as ∞  T −1  T cl i    avg even odd δ   v ( A ) w  . |x j (k)| ≤B c b max{β j , β j } + |d | + d j

(36)

i=0

Remark 1 Naturally, several of the special cases discussed in this section can occur simultaneously. If all three of the presented cases (matched uncertainties, dead-beat control, and switching-type motion) are applicable to the j-th state variable of the considered plant, then the absolute value of this variable will satisfy inequality  −1 |x j (k)| ≤B  cT b max{ξ even , ξ odd j j }

(37)

in finite time, where n/2

ξ even = j

n/2−1      v T ( Acl )2i b, ξ odd = v T ( Acl )2i+1 b j j j i=0

i=0

and · , · are the floor and ceiling functions, respectively.

(38)

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5 DSMC with Arbitrary Relative Degree Sliding Variables In the previous three sections, reaching law-based controller design has been described and analyzed for conventional sliding variables with relative degree one. Now this procedure will be extended to the case of arbitrary relative degree variables. It is said that a discrete time variable σr has relative degree r with respect to input u if this input only affects the variable after r time instants, i.e.,     σr (k + r ) = f x(k), u(k) and ∀i 0 repeated substitution of the right-hand side of (52), lets us express the j-th state variable as x j (k) = v Tj x(k) = α j,r (k) + β j,r (k),

(54)

where α j,r (k) = v Tj ( Arcl )k x 0 +

k−1 

v Tj ( Arcl )i b(crT Ar −1 b)−1 σr (k − i + r − 1),

i=k−k0 +1

β j,r (k) =

k−k 0

v Tj ( Arcl )i b(crT Ar −1 b)−1 σr (k − i + r − 1),

(55)

i=0

constant x 0 represents initial conditions of the system and k0 is the first time instant for which the system representative point entered the band (50). It will now be shown that both elements in relation (55) are bounded as k tends to infinity. Since all elements of the sum on the right-hand side of α j,r (k) have a common factor of ( Arcl )k−k0 +1 , then for any initial conditions of the system relation (47) implies lim |α j,r (k)| = 0.

k→∞

(56)

Then, since for all k > k0 one gets |σr (k)| ≤ Br , relation (55) gives 0  k−k    T cl i T r −1 −1  |β j,r (k)| = v j ( Ar ) b(cr A b) σr (k − i + r − 1)

i=0

0   T r −1 −1 k−k    v T ( Acl )i b · σr (k − i + r − 1) ≤ cr A b r j i=0

0  −1 k−k   v T ( Acl )i b. ≤Br  crT Ar −1 b r j i=0

(57)

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Consequently, as k tends to infinity, one gets ∞ −1    T cl i  v ( A ) b. lim sup |β j (k)| ≤Br  crT Ar −1 b j r k→∞

(58)

i=0

Substitution of (56) and (58) into (54) yields   lim sup |x j (k)| ≤ lim sup |α j,r (k)| + |β j,r (k)| k→∞

(59)

k→∞ ∞ −1    T cl i  v ( A ) b, ≤ 0 + Br  crT Ar −1 b j r i=0

which is consistent with relation (53).



It has been demonstrated that, with the use of a reaching law-based sliding mode control strategy, all state variables are bounded regardless of the relative degree of the considered sliding variable. Furthermore, their bounds are always proportional to the quasi-sliding mode band width ensured by the particular reaching law. In the next section, additional special cases which allow one to further reduce state error during sliding mode will be discussed.

7 Special Cases in DSMC with Relative Degree r Variables Section 4 of this chapter presented three cases for which the state error can be further reduced when a conventional relative degree one sliding variable is used. One of these cases, related to matched uncertainties, has already been assumed at the beginning of Sect. 5 since it is essential for strategies using arbitrary relative degree sliding variables. The remaining two cases (dead-beat control and switching-type motion) are also applicable when such strategies are considered, as will be demonstrated shortly.

7.1 Dead-Beat Control Just like in the case of conventional sliding variables, one can design a relative degree r variable so that it ensures dead-beat performance of the closed-loop system. To that end, one must select vector cr that places all eigenvalues of matrix Arcl in zero. Then, one gets ( Arcl )i = 0 for all i ≥ n and the bound (53) is reduced to

State Boundedness in Discrete-Time Sliding Mode Control n−1 −1    T cl i  v ( A ) b. |x j (k)| ≤Br  crT Ar −1 b j r

113

(60)

i=0

Since this is a finite sum, its values will always be significantly smaller than the one given in the more general inequality (53).

7.2 Switching-Type Quasi-sliding Motion Through analogy with Sect. 4.3, one can simplify the bound (53) when system operates in switching-type quasi-sliding mode. Suppose that for a given j = 1, 2, . . . , n and for all k ≥ 0 elements v Tj ( Arcl )k b ∈ R have the same sign. Then, since variable σr (k) changes its sign in each step, only every second element of the sum (57) contributes toward the upper bound of the state variable. Indeed, this bound can now be expressed as −1  odd |x j (k)| ≤Br  cT Ar −1 b max{β even j,r , β j,r },

(61)

where β even j,r =

∞ ∞    T cl 2i   T cl 2i+1  v ( A ) b, β odd = v ( A ) b. r r j j,r j i=0

(62)

i=0

Naturally, just like for conventional relative degree one variables, multiple cases described in this section can be applicable at once to a single system, if they are relevant. In the next section, the bounds of state error developed in this chapter will be verified by means of a simulation example. Remark 2 Although the absolute bounds of state variables obtained in relations (18) and (53), and subsequently modified in Sects. 4 and 7 are always true, they represent the “worst-case scenario” for a given plant and do not always have to be attained by all state variables simultaneously. Indeed, in most practical applications, the actual values of the state variables will stay significantly below those bounds.

8 Simulation Example In this section, we consider a simple second-order discrete-time plant (1), where  A=

     1 0.5 0 10 , b= , x(0) = −0.2 1 1 5

(63)

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subject to disturbance d(k) = (−1)k/20 . To this plant, we apply two reaching lawbased sliding mode control strategies. The first strategy is the one proposed by Gao et al. [12] with the control signal   u(k) = (cT b)−1 qσ (k) − εsgn[σ (k)] − cT pd avg − cT Ax(k) .

(64)

Vector c is selected so that c = [1.5 1]T , which guarantees a stable response of the closed-loop system. The second strategy is a generalization of Gao’s reaching law to the case of relative degree two sliding variables. As described in [5], the control signal for this strategy is expressed as  u(k) = (cT2 Ab)−1 q 2 σ2 (k) − qε2 sgn[σ2 (k)] − ε2 sgn[σ2 (k + 1)]  − cT2 A pd avg − cT2 A2 x(k) ,

(65)

where c2 = [2 0]T to ensure dead-beat performance of the closed-loop system. Three cases in total are considered in this simulation: (I) Strategy (64) applied to the system with matched uncertainties ( p = b). (II) Strategy (64) applied to the system with unmatched uncertainties ( p = [0.4 0.4]T ). (III) Strategy (65) applied to the system with matched uncertainties ( p = b). Since strategies using sliding variables with relative degree higher than one greatly benefit from matching conditions, (65) will not be applied to the system subject to unmatched disturbance. For all cases, relation (4) implies d avg = 0 and d δ = 1. Then, according to [12] and [5], design parameters for strategies (64) and (65) are chosen as q = 0.5, ε = 3 and q = 0.5, ε2 = 2, respectively. Such a choice of parameters guarantees switching-type quasi-sliding motion of the system for both cases. Furthermore, since elements v Tj ( Acl )i b are non-negative for j = 1 and any i, and non-positive for j = 2 and any i, special case described in Sect. 4.3 of this chapter applies to each state variable of the plant in simulations I and II. In simulation III, i special case from Sect. 7.2 applies to the first state variable since elements v Tj ( Acl 2) b are non-positive for j = 1 and any i. Figures 1 and 2 illustrate both state variables obtained in simulations I, II, and III. Furthermore, numerical results of the simulations have been summarized in Table 1. It can be seen from the table that both variables stay within their absolute bounds calculated in this chapter in each of the considered cases. In all three cases, both state variables are well within their expected limits, considering the special conditions of matched disturbance and switching-type quasi-sliding motion, described in Sects. 4 and 7. In particular, the first state variable matches its expected bound exactly in cases I and III.

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Fig. 1 First state variable

Fig. 2 Second state variable Table 1 Maximum absolute values of the state variables Case I Case II Obtained Expected Obtained Expected

Case III Obtained

Expected

Variable x1 Variable x2

1.6667 4.0000

1.6667 6.6667

2.1333 3.2000

2.1333 5.0000

2.6667 4.0000

2.8267 7.3600

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9 Conclusions In this chapter, we have determined the bounds of each individual state variable of a discrete-time system subject to a sliding mode control strategy. In particular, we have explicitly calculated state errors in DSMC using both conventional relative degree one sliding variables and more complex relative degree r ones. This is a useful development, since sliding mode control strategies are typically not concerned with state constraints and only analyze the evolution of the selected sliding variable. Furthermore, the obtained bounds clearly illustrate that state error is always proportional to the quasi-sliding mode band width and the magnitude of unmatched disturbance. We have further shown that the obtained generic formulae, which are true for all systems, can be simplified in three relatively common special cases. Acknowledgements Paweł, Latosi´nski gratefully acknowledges financial support provided by the Foundation for Polish Science (FNP).

References 1. Ackermann, J., Utkin, V.: Sliding mode control design based on Ackermann’s formula. IEEE Trans. Autom. Control. 43(2), 234–237 (1998) 2. Bandyopadhyay, B., Janardhanan, S.: Discrete-Time Sliding Mode Control. A Multirate Output Feedback Approach. Springer, Berlin (2006) 3. Bartolini, G., Ferrara, A., Utkin, V.: Adaptive sliding mode control in discrete-time systems. Automatica 31(5), 769–773 (1995) 4. Bartoszewicz, A.: Discrete time quasi-sliding mode control strategies. IEEE Trans. Ind. Electron. 45(4), 633–637 (1998) 5. Bartoszewicz, A., Latosi´nski, P.: Generalization of Gao’s reaching law for higher relative degree sliding variables. IEEE Trans. Autom. Control. 63(10), 1–8 (2017) 6. Chakrabarty, S., Bandyopadhyay, B., Moreno, J.A., Fridman, L.: Discrete sliding mode control for systems with arbitrary relative degree output. In: International Workshop on Variable Structure Systems, pp. 160–165 (2016) 7. Da Silva, J., Tarbouriech, S.: Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. IEEE Trans. Autom. Control. 46(1), 119–125 (2001) 8. Draženovi´c, B.: The invariance conditions in variable structure systems. Automatica 5(3), 287–295 (1969) 9. Du, H., Yu, M.C., Li, S.: Chattering-free discrete-time sliding mode control. Automatica 68(3), 87–91 (2016) 10. Emelyanov, S.V.: Variable Structure Control Systems. Nauka, Moscow (1967) 11. Gao, W., Hung, J.: Variable structure control of nonlinear systems: a new approach. IEEE Trans. Ind. Electron. 40(1), 45–55 (1993) 12. Gao, W., Wang, Y., Homaifa, A.: Discrete-time variable structure control systems. IEEE Trans. Ind. Electron. 42(2), 117–122 (1995) 13. Hou, H., Yu, X., Zhang, Q., Huang, J.: Reaching law based sliding mode control for discrete time system with uncertainty. In: 27th International Symposium on Industrial Electronics, pp. 1155–1160 (2018) 14. Kurode, S., Bandyopadhyay, B., Gandhi, P.: Discrete sliding mode control for a class of underactuated systems. In: Annual Conference of the IEEE Industrial Electronics Society, pp. 3936– 3941 (2011)

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15. Ma, H., Wu, J., Xiong, Z.: Discrete-time sliding-mode control with improved quasi-slidingmode domain. IEEE Trans. Ind. Electron. 63(10), 6292–6304 (2016) 16. Mehta, A.J., Bandyopadhyay, B.: Frequency-shaped sliding mode control using output sampled measurements. IEEE Trans. Ind. Electron. 56(1), 28–35 (2008) ˇ General conditions for the existence of a quasisliding mode on the switching 17. Milosavljevi´c, C.: hyperplane in discrete variable structure systems. Autom. Remote. Control. 46(3), 307–314 (1985) ˇ Peruniˇci´c-Draenovi´c, B., Veseli´c, B., Miti´c, D.: Sampled data quasi-sliding 18. Milosavljevi´c, C., mode control strategies. In: IEEE International Conference on Industrial Technology, pp. 2640– 2645 (2006) 19. Salgado, I., Kamal, S., Bandyopadhyay, B., Chairez, I., Fridman, L.: Control of discrete time systems based on recurrent super-twisting-like algorithm. ISA Trans. 64, 47–55 (2016) 20. Sharma, N.K., Janardhanan, S.: A novel second-order recursive reaching law based discretetime sliding mode control. IFAC-PapersOnLine 49(1), 225–229 (2016) 21. Sharma, N.K., Janardhanan, S.: Discrete higher order sliding mode: concept to validation. IET Control. Theory Appl. 11(8), 1098–1103 (2017) 22. Su, W., Drakunov, S., Ozguner, U.: An O(T2 ) boundary layer in sliding mode for sampled data systems. IEEE Trans. Autom. Control. 45(3), 482–485 (2000) 23. Utkin, V.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control. 22(2), 212–222 (1977) 24. Utkin, V., Drakunov, S.V.: On discrete-time sliding mode control. In: IFAC Conference on Nonlinear Control, pp. 484–489 (1989) 25. Yu, X., Wang, B., Li, X.: Computer-controlled variable structure systems: the state of the art. IEEE Trans. Ind. Inform. 8(2), 197–205 (2012)

Discrete Stochastic Sliding Mode with Functional Observation Satnesh Singh and S. Janardhanan

Abstract This chapter presents a functional observer-based sliding mode control (SMC) for discrete-time stochastic systems, both uncertainty-free systems and systems with uncertainty are considered. Existence conditions and stability analysis of the SMC functional observer are given. The control input is calculated by a linear functional observer. The functional observer is designed in such a way that the effect of process and measurement noise is minimized. Furthermore, functional observerbased SMC design with unmatched uncertainty is discussed and a disturbancedependent sliding function method is discussed such that the effect of unmatched uncertainty of the system is minimized. An SMC is calculated using the functional observer method. Finally, a simulation example is given to show the effectiveness of the method presented.

1 Introduction In the past few decades, there has been an increasing curiosity in the control design for variable structure systems (VSS) whose structure varies under certain conditions [1, 2]. In the discrete-time implementation of the sliding mode methodology, the switching elements are replaced by a computing device which changes the structure of the system at discrete instants. Thus, a discrete-time sliding mode condition must be imposed. An essential property of a discrete-time system is that the control signal is computed and varied only at sampling instants which makes discrete-time control inherently discontinuous. Hence, unlike the case of continuous SMC law need not S. Singh (B) ASRI, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, South Korea e-mail: [email protected]; [email protected] S. Janardhanan Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_5

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necessarily be of variable structure or need to have an explicit discontinuity. Here, the system states move about the sliding manifold, but are inadequate to stay on it, hence giving the terminology quasi-sliding mode (QSM) [3]. In order words, the trajectories of discrete-time systems may not remain on a pre-designed sliding surface, because of the limitation in sampling rate. However, the state trajectories may be able to remain within a boundary layer around the sliding surface called quasi-sliding mode band (QSMB). Few studies also proved that the chattering phenomenon in DSMC has vanished if the discontinuous control part is eliminated from the feedback control law. The control law in the absence of explicit discontinuous component is then called linear control law [1, 4–6]. Various applications of SMC have been found in areas like robotic manipulators, aircrafts, chaotic systems, process control, aerospace, motor control applications, and robotics [2, 7]. The states of the system are not usually accessible, or there exists unavoidable noise in the measured states. These problems may affect the performance of the systems. Therefore, utilizing observers can resolve such issues. The issue of linear system states estimation is of great importance in many applications because the states are not available in many systems and they should be estimated. The notion of SMC for discrete-time stochastic systems has been researched in detail [8]. Most of the studies in SMC available in the literature do not consider the presence of stochastic noise in the systems. However, it has been noticed that many real-world systems and a natural process may be disturbed by different noises such as process and measurement noise. It means that stochastic system representations are more prudent to reality. Therefore, it is crucial to extend the SMC theory to stochastic systems [9]. The design of a controller for each control problem uses either state feedback controller or output feedback controller depending upon availability of measurement [10, 11]. Traditionally, SMC has been developed in an environment in which all the states of the system are available. This is not a very realistic situation for practical problems and has motivated the need for the functional observer-based SMC. The concept of the functional observer was first introduced in [10] by Luenberger. The estimation of a function of states does not necessarily requires the estimation of all the states and therefore the order of the functional observers can be relatively less than that of the full state observer. Functional observer concept has been extended to the problem of multi-linear system [12]. A necessary and sufficient condition for the design and existence of functional observer was proposed by Darouach [13]. The method of the functional observer-based SMC for an uncertain dynamical system has been designed [14, 15]. But, the problem of the functional observer-based SMC for discrete-time stochastic systems has not been investigated so far. Therefore, it is crucial to extend the functional observer-based SMC theory to stochastic systems. Functional observer-based SMC design for stochastic systems with unmatched uncertainty has not been addressed in the literature. Motivated by the preceding discussion, this chapter also explores the problem of the functional observer-based SMC design for discrete-time stochastic systems with unmatched uncertainty. The control input consists of the function of the system state and disturbance components [16, 17].

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1.1 Main Focus The focus of the chapter comes from two aspects: 1. A functional observer-based SMC is designed for linear discrete-time stochastic systems. Sliding function, stability, and convergence analysis are given for the stochastic system. Finally, the controller is calculated by a functional observer method. 2. A functional observer-based SMC is designed for discrete-time stochastic systems in the presence of unmatched uncertainty. A disturbance-dependent sliding surface method is discussed for discrete-time stochastic systems in the presence of unmatched uncertainty. Finally, SMC is calculated by a functional observer method.

1.2 Organization of Chapter The rest of the chapter is organized in the following way: Sect. 2 presents the background of stochastic SMC with and without state information after the introduction in Sect. 1. Functional observer-based SMC without uncertainty is presented in Sect. 3. In Sect. 4, functional observer-based SMC with unmatched uncertainty is presented with existence conditions and design analysis. Sections 3.4 and 4.3 validate the findings of the chapter through numerical simulations. Lastly, Sect. 5 draws conclusions on the method discussed in this chapter.

2 Background 2.1 Discrete-Time Sliding Mode Control for Stochastic Systems Consider an LTI discrete-time stochastic system with filtered probability space (Ω, F , Fk≥0 , P): x(k + 1) = Ax(k) + Bu(k) + Γ w(k).

(1)

Sliding function is constructed as s(k) = cx(k),

(2)

where state vector x(k) ∈ Rn , control action u(k) ∈ Rm , and sliding function s ∈ Rm are mentioned. The matrices A ∈ Rn×n , B ∈ Rn×m , c ∈ Rm×n , and Γ ∈ Rnטr are

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known. The plant noise w(k) ∈ Rr˜ is mean zero and Gaussian white processes with covariance Q. 

Assumption 1 cB is a nonsingular matrix.

The main target is to find an SMC u(k) such that system states (1) will be driven to and held within the band in Rn [8]: Sμc = {x ∈ Rn :| cx | μc }, μc > 0

(3)

with the probability (1 − δ), i.e., P{|s(k)|  μc } = (1 − δ) f or k > N ,

(4)

where 0 < δ  1 and N ∈ Z large enough. Sμc is named as sliding mode band (SMB). Proposition 1 ([8]) The objective (4) can be assured by the following reaching condition: (5) P{|s(k + 1)| < θ |s(k)||Fk } ≥ 1 − ε a.s, where 0 < θ < 1, 0 < ε < 0.5, and Fk ⊂ F denotes the σ -field set up by {x(k), x(k − 1), . . . , x(0); u(k), u(k − 1), . . . , u(0)}. Parameter θ is admitted in (5) for regulating the approaching speed. The notation a.s. stands for almost surely. 

Proof The detailed proof is given in [8]. Lemma 1 Let Gaussian random variable g with mean zero and variance Wg the given [8] solution of the equation P{|g| ≤ Wg } = L (Wg , −Wg , 0, σg ) = 1 − ε.

σg2 ,

and

(6)

Let g =  + g. Then the solutions of the following equation regarding  P{|g | ≤ φ} = L (φ, −φ, , σg ) = 1 − ε

(7)

have following properties: (i) if φ > Wg , then (7) has only two solutions. (ii) if φ > Wg and 0 < ε < 0.5, then the solutions of (7) are bounded by φ, i.e., || ≤ φ. Proof The proof of Lemma 1 is provided in [8].

2.1.1



Synthesis of DSMC

Now here, our main goal is to find out the control which achieves the objective as in (4) for system (1).

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123

± Theorem 1 There occur predictable values of Wc , m ± 1 (k) and m 2 (k) such that the DSMC law + 1. u(k) = u + (k) ∈ (cB)−1 (−c Ax(k) + m + 1 (k), −c Ax(k) + m 2 (k)) if s(k) > −1 θ Wc − 2. u(k) = u − (k) ∈ (cB)−1 (−c Ax(k) + m − 1 (k), −c Ax(k) + m 2 (k)) if s(k) < −1 −θ Wc 3. u(k) = −(cB)−1 c Ax(k) if |s(k)| ≤ θ −1 Wc

drives the system states (1) to and be held within SMB Sμc with a given probability (1 − δ).

Proof On considering s(k + 1) = m(k + 1) + g(k + 1). Then, m(k + 1) = c Ax(k) + cBu(k) and g(k + 1) = cΓ w(k), where g(k + 1) is Gaussian with mean zero and variance σc2 = cΓ QΓ T c T . Case 1 s(k) > θ −1 Wc In this case, (5) can be brought in the form P{−θ s(k) < s(k + 1) < θ s(k)} ≥ 1 − ε.

(8)

The condition is necessary and sufficient for the survival of solutions of (8) which is given by (9) s(k) > θ −1 Wc , where Wc is represented by  L (Wc , −Wc , 0, σc ) 

Wc −Wc

√

1 2πσc2

e−z

2

/2σc2

dz = 1 − ε,

(10)

where Gaussian random variable with mean zero and variance σc2 . After solving Eq. (10), we are getting Wc =

√

2σ (er f )−1 (1 − ε),

(11)

where erf is the error function [18]. By Lemma 1, under the condition (9) the controller is + u(k) = u + (k) ∈ (cB)−1 (−c Ax(k) + m + 1 (k), −c Ax(k) + m 2 (k)),

(12)

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where u ∈ [β1 , β2 ] means that any number in the closed interval [β1 , β2 ] can be selected as the control action value u and m i+ (k), i = 1, 2, are the two solutions of + the following equation satisfying m + 1 (k) < 0 < m 2 (k): L (θ s(k), −θ s(k), , σc ) = 1 − ε.

(13)

Case 2 s(k) < −θ −1 Wc . In this case, (5) is of the form P{θ s(k) < s(k + 1) < −θ s(k)} ≥ 1 − ε.

(14)

Similarly, the condition of necessary and sufficient for the solutions survival of (14) is (15) s(k) < −θ −1 Wc and the controller would be − u(k) = u − (k) ∈ (cB)−1 (−c Ax(k) + m − 1 (k), −c Ax(k) + m 2 (k)),

(16)

where m i− (k), i = 1, 2, are the only two solutions of the following equation con− cerning  that satisfy m − 1 (k) < 0 < m 2 (k): L (−θ s(k), θ s(k), , σc ) = 1 − ε.

(17)

Case 3 |s(k)| ≤ θ −1 Wc . In this case, the SMC should be selected as u(k) = −(cB)−1 c Ax(k).

(18)

From Lemma 1, it can be seen that control input (18) would be within the valid range of values for u(k). Hence, it can be used for the SMC implementation. It is also valid for case 1 and case 2.  Remark 1 When the SMC (18) is used, it will be s(k + 1) = cΓ w(k). In this case, the system motion (1) is termed as stochastic sliding mode (SSM).

(19)

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2.2 Observer-Based SMC for Discrete-Time Stochastic Systems Luenberger observer is very simple and effective, but has many problems in facing the random noise or certain inputs, so they do not operate properly in estimating the states. Kalman filter (KF) has solved the estimation problem in random linear systems as a linear filter by the use of minimum mean square error [19]. In the context of stochastic SMC, few works of literature are available in recent years [20]. Reaching condition would be different, in case states are not fully accessible. Output equation is described as y(k) = C x(k) + Gv(k),

(20)

where v(k) ∈ Rl is white Gaussian noise with mean zero and covariance R. Matrix G ∈ R p×l is known. Sliding function is described as sˆ (k)  c x(k), ˆ

(21)

where x(k) ˆ denotes the state estimation of original state x(k) given F˜ k , the σ -field is created by {y(k), y(k − 1), . . . , y(0); u(k), u(k − 1), . . . , u(0)}. The main task is to control the sliding variable sˆ (k) such that P{|ˆs (k)|  μ˜ c } = (1 − δ) f or k > N ,

(22)

where μ(k) ˜ > 0, and then analyze the behavior of s(k). Algorithm 1 Algorithm for State Estimation [17] 1: x(0) ˆ = x0 is initially assumed. 2: Given the noise covariance P(k − 1) of the state vector x(k − 1), and the covariance of w(.) and v(.) as Q and R, the error covariance P(k) is determined as (a) P(k|k − 1) = A P(k − 1)A T + Γ QΓ T 23 (b) P(k) = [P −1 (k|k − 1) + C T R −1 C]−1 , (c) ϒ(k) = P(k)C T R −1 . where ϒ(k) is the Kalman gain. 3: Using ϒ(k), the state vector x(k) ˆ can be determined as 24

(a) x(k|k ˆ − 1) = A x(k ˆ − 1) + Bu(k − 1), (b) x(k) ˆ = x(k|k ˆ − 1) + ϒ(k)[y(k) − C x(k|k ˆ − 1)]

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3 Functional Observer-Based SMC for Systems Without Uncertainty Functional observers take benefit of a recurring theme in state feedback control. The direct estimation will reduce the order of the observer considerably in comparison with the state observe [21]. From the practical scenario, designing the feasible order of observer will directly affect the components which lead to the cost benefits which in turn increase reliability. When this functional observer exists, its order, namely, q according to our notations, is less than the order (n − p) of a reduced-order state observer. Detailed literature of functional observer’s procedure for continuous systems has been given in [21]. The discrete-time LTI stochastic system is described as follows [22]: x(k + 1) = Ax(k) + Bu(k) + Γ w(k) y(k) = C x(k) + Gv(k),

(25)

where x(k) ∈ Rn , u(k) ∈ Rm , and y(k) ∈ R p . Matrices A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n , G ∈ R p×l , and Γ ∈ Rnטr are known constants. w(k) ∈ Rr˜ is process noise and v(k) ∈ Rl is measurement noise. Let the initial state x0 be a random vector with zero mean and given covariance matrix P0 . The covariance of plant noise is Q and covariance of measurement noise is R. Assumption 2 The initial state x(0) is a Gaussian random vector and x(0), w(k), and v(k) are mutually uncorrelated.  Assumption 3 The system (25) is controllable and observable.



The main task is to design a functional observer-based SMC law for a system (25) such that sliding mode is achieved and sliding function will lie within a specified band.

3.1 Sliding Function and Controller Design 3.1.1

Design of Sliding Function

Sliding function is similar as defined in (2) for the system (25). To get the regular form of (25), an orthonormal matrix U ∈ Rn×n can be selected such that      0(n−m)×m 0 A¯ 11 A¯ 12 ,UB = , UΓ = , U AU = ¯ ¯ B2 Γ2 A21 A22 

T

where B2 ∈ Rm×m is a nonsingular. By the transformation of state ξ(k) = U x(k), (25) has the regular form

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ξ1 (k + 1) = A¯ 11 ξ1 (k) + ξ2 (k + 1) = A¯ 21 ξ1 (k) +

(26)

A¯ 12 ξ2 (k) A¯ 22 ξ2 (k) + B2 u(k) + Γ2 w(k),

where ξ1 (k) ∈ Rn−m and ξ2 (k) ∈ Rm . The sliding function (2) can be expressed in terms of the new state ξ(k) as s(k)  cU T ξ(k)  [K Im ]ξ(k) = K ξ1 (k) + ξ2 (k).

(27)

Substituting ξ2 (k) = −K ξ1 (k) in the first equation of system (26) gives the dynamics ξ1 (k + 1) = ( A¯ 11 − A¯ 12 K )ξ1 (k),

(28)

where matrix K is chosen such that ( A¯ 11 − A¯ 12 K ) should be stable. The sliding surface (2) can be expressed in terms of the original state co-ordinates as (29) s(k) = cx(k) = cU T ξ(k) = 0.

3.2 Synthesis of Control Law Our objective is to find out the control which achieves same objective as in (4) for system (25). The analysis of controller design is similar to Theorem 1. The control can be obtained to be u(k) = −(cB)−1 c Ax(k).

(30)

The stochastic system state (25) will lie in the Sμc using (30) and σc2 = cΓ QΓ T c T .

3.3 Design of Functional Observer In this section, we design an observer for estimating the linear functional of the states. Let u(k) be a vector that is required to be estimated, where u(k) = L x(k)

(31)

and L ∈ Rm×n is a known matrix. Matrix L = −(cB)−1 c A, which represents any desired partial set of the state vector to be estimated, and can always be obtained by the SMC (30). The aim is to design an observer of the form

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ζ (k + 1) = Mζ (k) + J y(k) + H u(k) uˆ f (k) = V ζ (k) + E y(k),

(32)

where ζ (k) ∈ Rq is the observer state vector. uˆ f (k) is the estimate of u(k) functional state vector. Matrices M ∈ Rq×q , J ∈ Rq× p , H ∈ Rq×m , V ∈ Rr ×q , and E ∈ Rr × p are unknown constant. Remark 2 The observer dynamics are determined by order of matrix M, which may be chosen arbitrarily, with the restriction that its eigenvalues are distinct and different from the eigenvalues of A and all the eigenvalues of matrix M are inside the unit disk.

Theorem 2 The q-th-order observer (32) will estimate u f (k) if the following conditions hold: (i) (ii) (iii) (iv) (v)

M is a stable matrix, V T = L − EC, M T + J C − T A = 0, H = T B, and q ≥ ρ(L(In − C + C)),

where L ∈ Rr ×n is the functional gain matrix and T ∈ Rq×n is the unknown constant matrix.

Proof Let us define e(k) as the error between ζ (k) and T x(k) as e(k)  ζ (k) − T x(k).

(33)

After taking the first-order difference of (33), we get e(k + 1) = ζ (k + 1) − T x(k + 1) = Mζ (k) + J y(k) + H u(k) − T Ax(k) − T Bu(k) − T Γ w(k) = Me(k) + (M T + J C − T A)x(k) + (H − T B)u(k) + J Gv(k) − T Γ w(k).

(34)

Independence of error dynamics (34) from x(k), u(k) requires that MT + JC − T A = 0

(35)

H − T B = 0.

(36)

and

Subject to (35) and (36), the dynamics of error (34) becomes

Discrete Stochastic Sliding Mode with Functional Observation

e(k + 1) = Me(k) + J Gv(k) − T Γ w(k).

129

(37)

Hence, the error dynamics e(k) is governed by matrix M, J , and T . Output estimation error is expressed as eu (k)  uˆ f (k) − L x(k) = V ζ (k) + E y(k) − L x(k) = V e(k) + (V T + EC − L)x(k) + E Gv(k).

(38)

Independence of output error dynamics (38) from x(k) requires that V T = L − EC.

(39)

Then, the estimation error in output becomes eu (k) = V e(k) + E Gv(k).

(40)

Hence, the output error dynamics (40) is governed by matrices V and E. Matrix E can be chosen as E = LC + to satisfy the observer condition (v) of Theorem 2. Remark 3 In given system (25), if the w(k) and v(k) were zero then discrete-time stochastic system would act as a discrete-time deterministic system. In that case, if conditions (iii)–(iv) of Theorem 2 are satisfied, then (37) is reduced to e(k + 1) = Me(k). Further, if condition (ii) is satisfied, then eu (k) is reduced to V e(k). ˆ → u(k) as Since e(k) → 0 as k → ∞, it follows that eu (k) → 0 and hence u(k) k → ∞.  Further, the unknown terms J and T in observer Eq. (32) can be solved as follows. Let   J T =X, (41) where X ∈ Rq×(n+ p) is an unknown matrix. Using (41), matrices J and T can be expressed in terms of unknown matrix X as   I p× p (42) J =X 0n× p   0 p×n . T =X In×n

and

On substituting for J and T in (35), MX

      0 I 0 +X C −X A=0 I 0 I

(43)

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or MX

    0 C +X =0 I −A

VX

  0 = L − EC. I

(44)

(45)

Now augmenting the matrices (44) and (45) in a composite form would give         M 0 I C X + X =Θ V I 0 −A 

with Θ=

(46)

 0 , L − EC

where Θ ∈ R(q+r )×n is known constant matrix. By applying the Kronecker product [23], in (46), Σvec(X ) = vec(Θ) with

(47)

  T     T   M C I 0 , Σ= ⊗ + ⊗ V −A 0 I

where vec(X ) ∈ R(n+ p)q , vec(Θ) ∈ Rn(q+r ) , and Σ ∈ R(q+r )n×q(n+ p) . The error covariance of (37) propagates as P(k + 1) = M P(k)M T + J G RG T J T + T Γ QΓ T T T .

(48)

The output error covariance of (40) is Pu (k) = V P(k)V T + E G RG T E T .

(49)

Now, we can write Pu (k + 1) in terms of covariance Pu (k + 1) = V (M P(k)M T + J G RG T J T + T Γ QΓ T T T )V T + E G RG T E T . (50) To minimize the effect of process and measurement noise covariance, we choose X in (41) as       G RG T 0 T T  (51) X = argmin V X T X V  0 Γ QΓ X such that minimizing the effect of noise in functional observer. Designing a q-th-dimensional dynamical system, where

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q ≥ ρ(L(In − C + C)) to generate any required vector state function of x(k). This proves condition (v) and completes the proof of Theorem 2.  Based on the procedure explained above, the design policy can be now summarized as follows: Algorithm 2 Functional observer based SMC for stochastic systems To obtain the sliding function s(k)  cx(k). Design controller u(k), and functional to be estimated is L x(k), where L = −(cB)−1 c A. Choose the (r × q) elements of matrix V , arbitrarily. Obtain the minimum order of the functional observer, q ≥ ρ(L(In − C + C)), such that ρ(V ) = ρ(L − EC). 5: Choose arbitrarily, a stable (q × q) matrix M. 6: Solve composite form of matrix (46) by Kronecker product, For matrix X . Matrix X is chosen according to      G RG T 0  T X = argmin X X  T  X  0 Γ QΓ

1: 2: 3: 4:

subject to the equality condition (46), if satisfied, go to next step; else set q = q + 1 and go to step 4. 7: From (42) and (43), Find the values of matrix J and T . 8: Now, find the value of matrix H by H = T B. 9: As a result of steps 1 to 8, obtain a structure of functional observer as per (32).

3.4 Simulation Results Consider an unstable multi-input multi-output linear discrete-time stochastic system (25) is given, where system parameter matrices are ⎡

0 ⎢1 ⎢ ⎢0 ⎢ A=⎢ ⎢0 ⎢0 ⎢ ⎣0 0

0 0 0 0 2 0 1

0 0 0 1 1 0 0

1 0 0 0 0 1 0 0 1 0 0 0 −1 0

0 1 0 0 1 0 0

⎡ ⎤ ⎡ ⎤ ⎤ 0 00 1 ⎢0⎥ ⎢0 1⎥ 0⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢0⎥ ⎢1 0 ⎥ 0⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎥ ⎥ ⎢ ⎥ 1⎥ , B = ⎢0 1⎥ , Γ = ⎢ ⎢0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ 0⎥ ⎢1⎥ ⎢0 0⎥ ⎣ ⎦ ⎣ ⎦ 0⎦ 10 1 1 00 0

   T C = I3 03×4 , G = 1 1 1 , where I3 denotes a 3 × 3 identity matrix and 03×4 matrix consisting of 3 rows and 4 columns. The open-loop unstable eigenvalues of matrix A are obtained as

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Control input u 1

0 -10 -20 -30 -40 -50 -60

5

10

15

20

25

20

25

k (No. of samples) Fig. 1 Evolution of control input u 1 (k) 20

Control input u 2

15

10

5

0

-5

5

10

15

k (No. of samples)

Fig. 2 Evolution of control input u 2 (k)

eig(A) = 0.141, −0.316 ± 1.522i, −1.058 ± 0.722i, 1.303 ± 0.299i. The covariances of process noise and measurement noises are Q = 0.1I and R = 0.1I , respectively. The parameters θ and ε are chosen as θ = 0.9 and ε = 0.01. The initial conditions of the original system and observer are chosen arbitrarily, as x(0) = [1 3 6 4 8 5 7]T and ζ (0) = [1 2 7]T . Here, order of observer is obtained as q = 3. Figures 1 and 2 show the evaluation of control inputs u 1 (k) and u 2 (k), respectively. Figures 3 and 4 show the evolution of estimated errors eu1 (k) and eu2 (k), respectively.

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2

Sliding Function s 2

0 -2 -4 -6 s 2 (k)

-8

+W -W

-10 -12

5

10

15

c

c

20

25

20

25

k (No. of samples)

Fig. 3 Evolution of estimation error eu1 (k) 20

Estimation error eu1

15 10 5 0 -5 -10 -15 5

10

15

k (No. of samples)

Fig. 4 Evolution of estimation error eu2 (k)

It is observed that the estimation error converges to zero. Figures 5 and 6 show the sliding function s1 (k) and s2 (k) response. Sliding functions s1 (k) and s2 (k) are brought into the constant sliding mode band, which are Wc1 = 0.8162 and Wc2 = 0.5799, respectively. Order of functional observer is 3, which is less than the reducedorder observer (n − p) = 4. These simulation results demonstrate that the design procedure presented here is very simple and effective of the functional observerbased SMC design.

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Estimation error e u2

20 15 10 5 0 -5 -10 -15

5

10

15

20

25

k (No. of samples)

Fig. 5 Evolution of sliding function s1 (k) 2

Sliding Function s 1

0

-2

-4

s (k)

-6

1

+W c -W

-8

-10

5

10

15

c

20

25

k (No. of samples)

Fig. 6 Evolution of sliding function s2 (k)

4 Functional Observer-Based SMC for Stochastic Systems with Unmatched Uncertainty Consider the LTI stochastic system with unmatched uncertainty as x(k + 1) = Ax(k) + Bu(k) + Fd(k) + Γ w(k) y(k) = C x(k) + Gv(k),

(52)

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where x(k) ∈ Rn , u(k) ∈ Rm , d(k) ∈ Rrd , and y(k) ∈ R p . It is assumed that x(0) is a Gaussian random vector and x(0), d(k), w(k), and v(k) are mutually uncorrelated. The following assumptions are made for the purpose of the design of the stochastic SMC. Assumption 4 The matching condition ρ(B) = ρ[B Γ ] is satisfied but for unmatched condition ρ(B) = ρ[B F] is not necessarily satisfied (unmatched means ρ(B) < ρ[B F] and ρ(B) means the rank of matrix B).  The main target is to design functional observer-based SMC for the system (52) such that sliding mode is obtained and mismatched uncertainty on the system is minimized.

4.1 Sliding Function and Controller Design The design problem contains the following three stages. In first stage, a sliding function is designed; in the second stage, the appropriate controller inputs have been developed; and in the final stage, compensating the matched and unmatched uncertainties by the appropriated selection of the augmented system.

4.1.1

Sliding Function Design

In this section, we would like to design a sliding function in order to develop an SMC of the system (52) such that the motion of sliding is stable. Assume that U ∈ Rn×n is an orthonormal matrix so that U B = [0(n−m)×m B2T ]T , where B2 is nonsingular matrix. Let ξ(k) = [ξ1T (k) ξ2T (k)]T = U x(k), then the system (52) has the regular form ξ1 (k + 1) = A¯ 11 ξ1 (k) + ξ2 (k + 1) = A¯ 21 ξ1 (k) +

A¯ 12 ξ2 (k) + F1 dun (k) A¯ 22 ξ2 (k) + B2 u(k) + F2 dm (k) + Γ2 w(k),

(53)

where ξ1 (k) ∈ Rn−m and ξ2 (k) ∈ Rm are blocks of state vector. Matrices A¯ 11 ∈ R(n−m)×(n−m) , A¯ 12 ∈ R(n−m)×m , A¯ 21 ∈ Rm×(n−m) , and A¯ 22 ∈ Rm×m are blocks of the ¯ dun (k) and dm (k) are the unmatched and matched parts of the uncersystem matrix A. tainty. F1 ∈ R(n−m)×rd , F2 ∈ Rm×rd and Γ2 ∈ Rmטr are blocks of the uncertainty and noise matrix of system. Remark 4 The system (53) in sliding mode is independent of d(k) if and only if F1 = 0. The system in the sliding mode is free of d(k) if there exists a matrix Ξ such that F = BΞ . If m = 1 this condition is also necessary. Thus, the sliding function is expressed as [24]

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s(k)  cx(k) + k0 F1 dun (k),

(54)

where s(k) ∈ Rm is sliding function, c ∈ Rm×n is gain matrix, and k0 ∈ Rm×(n−m) is a constant matrix. Remark 5 The conventional sliding surface is described as s(k) = cx(k) but in this chapter sliding surface is constructed as s(k) = cx(k) + k0 F1 dun (k). The sliding function is dependent on state and uncertainty, whereas in the conventional sliding surface, only states are involved. By using this disturbance-aware sliding surface (54), the effect of unmatched uncertainty, which is presented in the system, can be minimized. The sliding function (54) may be rewritten using the state transformation s(k) = K ξ1 (k) + ξ2 (k) + k0 F1 dun (k).

(55)

Now, while the system is on the sliding surface (55), it can be rewritten as ξ2 (k) = −(K ξ1 (k) + k0 F1 dun (k)). Putting the values of ξ2 (k) in (53) ξ1 (k + 1) = ( A¯ 11 − A¯ 12 K )ξ1 (k) + (I(n−m) − A¯ 12 k0 )F1 dun (k),

(56)

where K is the parameter to be designed as in [24]. Remark 6 Minimization of the unmatched uncertainty effect in the sliding mode is ¯+ ¯ possible if k0 = A¯ + 12 , where A12 is the left pseudoinverse of A12 [25]. Assumption 5 The presence of disturbance in the system is slowly varying, i.e., it can be said that difference between two sampling instances, |d(k + 1) − d(k)|, is not significant. Hence, disturbance d(k) would be a good estimate of disturbance d(k + 1) [26]. 

4.1.2

Augmented System

In this section, we introduce the augmented state vector xd (k) = [x T (k) d T (k)]T ∈ R(n+rd ) and form the augmented system by combining (52) and Assumption 5. We can rewrite (52) as ¯ d (k) + Bu(k) ¯ + Γ¯ w(k) xd (k + 1) = Ax y(k) = C¯ xd (k) + Gv(k), where

(57)

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137

    B AF ∈ R(n+rd )×m A¯ = ∈ R(n+rd )×(n+rd ) , B¯ = 0 0 I     Γ ¯ Γ = ∈ R(n+rd )טr , C¯ = C 0 ∈ R p×(n+rd ) . 0

4.1.3

Controller Design

After taking the first-order difference of sliding function (54), we get s(k + 1) = cx(k + 1) + k0 F1 dun (k + 1).

(58)

Putting the values from (52) in (58) and using Assumption 5, we can write m(k + 1) = c Ax(k) + cBu(k) + cFd(k) + k0 F1 dun (k) and g(k + 1) = cΓ w(k) + k0 [I 0]F(d(k + 1) − d(k)), where variance σc2 = cΓ QΓ T c T . m(k + 1) = c Ax(k) + cBu(k) + cFd(k) + k0 F1 dun (k) = 0. The controller is obtained to be u(k) = L xd (k), where L = [−(cB)−1 c A

(59)

− (cB)−1 (c + k0 [I 0])F].

Remark 7 When the SMC u(k) = L xd (k) is used, it will be s(k + 1) = cΓ w(k) + k0 [I 0]F(d(k + 1) − d(k)).

(60)

As unmatched uncertainty is bounded, the difference of uncertainty term will also be bounded. In the next section, we discuss a functional observer-based SMC method for augmented discrete-time stochastic systems.

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4.2 Linear Functional Observer-Based SMC Design In this section, our objective is to estimate a linear function of the state vector of the form u(k) = L xd , where L and xd (k) are the state vectors of the discrete-time system (57). The control gain matrix L is calculated via the control input (59). Our aim is to design functional observer of the form ζ (k + 1) = Mζ (k) + J y(k) + H u(k) uˆ f (k) = V ζ (k) + E y(k),

(61)

where ζ (k) ∈ Rq is the state vector. uˆ f (k) ∈ Rm is desired functional estimate of u(k). Theorem 3 The qth-order observer (61) will estimate u f (k) if the following conditions hold: (i) (ii) (iii) (iv) (v)

M is a stable matrix, ¯ V T = L − E C, M T + J C¯ − T A¯ = 0, ¯ and H = T B, ¯ (n + rd − p) ≥ q ≥ ρ(L(In+rd − C¯ + C)),

where L ∈ Rr ×(n+rd ) is the known functional gain matrix and T ∈ Rq×(n+rd ) is the unknown constant matrix.

Proof Let us define e(k) as the error between ζ (k) and T xd (k) as e(k)  ζ (k) − T xd (k).

(62)

After taking the first-order difference of (62), we get e(k + 1) = ζ (k + 1) − T xd (k + 1) ¯ d (k) + (H − T B)u(k) ¯ = Me(k) + (M T + J C¯ − T A)x + J Gv(k) − T Γ¯ w(k).

(63)

If conditions (iii)–(iv) of Theorem 3 are satisfied, then (63) reduces to e(k + 1) = Me(k) + J Gv(k) − T Γ¯ w(k).

(64)

Output estimation error is described as eu (k)  uˆ f (k) − L xd (k) = V e(k) + (V T + E C¯ − L)xd (k) + E Gv(k).

(65)

Discrete Stochastic Sliding Mode with Functional Observation

139

If condition (ii) of Theorem 3 is satisfied, then above equation reduces to eu (k) = V e(k) + E Gv(k).

(66)

Hence, the output error dynamics eu (k) is governed by matrices V and E. Matrix E can be chosen as E = L C¯ + to satisfy the observer condition (v) of Theorem 3. Further, the unknown terms J and T in observer Eq. (61) can be solved as follows: 

 J T =X,

(67)

where X ∈ Rq×(n+ p+rd ) is an unknown matrix. Using (67), matrices J and T can be expressed in terms of unknown matrix X as   I p× p (68) J =X 0(n+rd )× p 

and T =X

0 p×(n+rd )



I(n+rd )×(n+rd )

.

(69)

On putting the values of J and T in condition (iii)–(iv) of Theorem 3,     0 C¯ MX +X =0 I − A¯ VX

  0 ¯ = L − E C. I

(70)

(71)

Now assembling the matrix equations (70) and (71) in the composite form would be as follows:         M 0 I C¯ X + X =Θ (72) V I 0 − A¯ with



 0 Θ= , L − E C¯

where Θ ∈ R(q+r )×(n+rd ) is a known constant matrix.

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By applying the Kronecker product[23], in (72), we can write Σvec(X ) = vec(Θ) with

(73)

      T   T 0 M C¯ I Σ= , ⊗ + ⊗ ¯ I V 0 −A

where vec(X ) ∈ R(n+ p)(q+rd ) , vec(Θ) ∈ R(n+rd )(q+r ) , (q+r )(n+rd )×(q+rd )(n+ p) . R The error covariance of (64) propagates as

and

P(k + 1) = M P(k)M T + J G RG T J T + T Γ¯ Q Γ¯ T T T .

Σ∈

(74)

The output error covariance of (66) is Pu (k) = V P(k)V T + E G RG T E T .

(75)

Now, we can write Pu (k + 1) in terms of covariance Pu (k + 1) = V (M P(k)M T + J G RG T J T + T Γ¯ Q Γ¯ T T T )V T + E G RG T E T . (76) To minimize the effect of process and measurement noise covariance, we choose X in (67) as       G RG T 0 T T V X X X = argmin  V T   ¯ ¯ 0 Γ QΓ X such that minimizing the effect of process and measurement noise in the functional observer. Designing a q-th-dimensional dynamical system, where ¯ (n + rd − p) ≥ q ≥ ρ(L(I(n+rd ) − C¯ + C)). This completes the proof.



Based on the abovementioned development, the design procedure is now summarized as follows:

Discrete Stochastic Sliding Mode with Functional Observation

141

Algorithm 3 Functional Observer based SMC with Unmatched uncertainty 1: To obtain the sliding function s(k)  cx ¯ d (k) where c¯ = [c k0 [In−m 0(n−m)×m ]F] 2: Design controller u(k), and functional to be estimated is L xd (k), where L = [−(cB)−1 c A (cB)−1 (c + k0 [I 0])F]. 3: Choose the (r × q) elements of matrix V , arbitrarily. 4: To obtain the order of the functional observer

−

¯ (n + rd − p) ≥ q ≥ ρ(L(I(n+rd ) − C¯ + C)) ¯ such that ρ(V ) = ρ(L − E C) 5: Choose arbitrarily, a stable (q × q) matrix M. 6: Solve the composite form of matrix (72) by Kronecker product, For matrix X . Matrix X is chosen according to      G RG T 0  T X = argmin X X  T ¯ ¯  X  0 Γ QΓ subject to the equality condition (73), if satisfied, go to next step; else set q = q + 1 and go to step 4. 7: From (68) and (69), Find the values of matrix J and T . ¯ 8: Now, find matrix H by H = T B. 9: As a result of steps 1 to 8, obtain a structure of functional observer as per (61).

4.3 Simulation Example and Results In this section, simulation results are given to demonstrate the validity of the approaches discussed in this chapter [2]. The covariance of process noise and measurement noises are Q = 0.1I and R = 0.1I , respectively. Let the unmatched uncertainty [−15 + 0.2 sin(0.125π k)] and matched uncertainty sin(k/2)e(−k/5) are slowly time-varying, which affect the discrete-time system. The parameters θ and ε are chosen as θ = 0.9 and ε = 0.01. The system initial conditions and observer initial conditions are chosen, arbitrarily, as xd (0) = [1 1.5 2 2.5 3 3.5 4 5 6]T and ζ (0) = [1 2 7]T . ⎤ ⎡ ⎤ 0 0 0 0 1 0 0 0 0 0 ⎢0 0⎥ ⎢ 0 −0.154 −0.004 1.54 0 −0.744 −0.032 0⎥ ⎥ ⎢ ⎥ ⎢ ⎢0 0⎥ ⎥ ⎢ 0 0.249 −1 −5.2 0 0.337 −1.12 0 ⎢ ⎥ ⎢ ⎥ ⎢0 0⎥ ⎢0.039 −0.996 0 −2.117 0 0.02 0 0⎥ ⎢ ⎥ ⎢ ⎥, ,B =⎢ A=⎢ ⎥ 0.5 0 0 −4 0 0 0⎥ ⎢0 0⎥ ⎥ ⎢ 0 ⎢20 0 ⎥ ⎥ ⎢ 0 0 0 0 0 −20 0 0 ⎢ ⎥ ⎢ ⎥ ⎣ 0 25⎦ ⎣ 0 0 0 0 0 0 −25 0⎦ 0 0 0 1 0 0 0 0 0 1 ⎡

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 T  T Γ = 0 0 0 0 0 20 0 0 , F = 1 −2 0 1 0 0 1 −0.5 .    T C = I4 : 04×3 , G = 1 1 1 0 , where I4 denotes a 4 × 4 identity matrix. Here, order of functional observer obtained is q = 3.

4.3.1

Comparative Study

For comparative study of the system performance, we can consider the two cases: (i) presence of unmatched uncertainty and (ii) absence of unmatched uncertainty effect, where F1 = 0 and k0 = 0m×(n−m) . Here, order of functional observer is obtained as q = 3. Figure 7 shows the response of control inputs u 1 (k) and u 2 (k), respectively. Figure 8 shows the evolution of estimated errors eu1 (k) and eu2 (k), respectively. Figure 9 shows the sliding function response s1 (k) and s2 (k). Sliding functions s1 (k) and s2 (k) lie between the band, which are Wc1 = 0.47 and Wc2 = 0.05, respectively. Order of functional observer is 3, which is less than the reduced-order observer (n − p) = 4. These simulations results validate the effectiveness and efficacy of the functional observer-based SMC design.

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5 Conclusions In this chapter, the problem of functional observer-based SMC for discrete-time stochastic systems with and without unmatched uncertainty has been considered. In the minimization of unmatched uncertainty effects in a sliding mode, a disturbancedependent sliding surface method is discussed. Along with the existing conditions, a strategy has been successfully used to find the functional observer-based SMC in discrete-time stochastic systems with unmatched uncertainty. Finally, a simulation example is provided to show the effectiveness of the method.

References 1. Gao, W., Wang, Y., Homaifa, A.: IEEE Trans. Industr. Electron. 42(2), 117 (1995) 2. Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory And Applications. Series in Systems and Control. Taylor & Francis (1998) 3. Ma, H., Wu, J., Xiong, Z.: IEEE Trans. Industr. Electron. 63(10), 6292 (2016). https://doi.org/ 10.1109/TIE.2016.2580531 4. Bartoszewicz, A.: IEEE Trans. Industr. Electron. 45(4), 633 (1998) 5. Bartolini, G., Pisano, A., Punta, E., Usai, E.: Int. J. Control 76(9–10), 875 (2003) 6. Yu, X., Wang, B., Li, X.: IEEE Trans. Industr. Inf. 8(2), 197 (2012) 7. Young, K.: Variable Structure Control for Robotics and Aerospace Applications. Studies in Automation and Control. Elsevier (1993) 8. Zheng, F., Cheng, M., Gao, W.B.: Syst. Control Lett. 22(3), 209 (1994) 9. Sharma, N.K., Singh, S., Janardhanan, S., Patil, D.U.: 25th Mediterranean Conference on Control and Automation, pp. 649–654 (2017) 10. Luenberger, D.: IEEE Trans. Autom. Control 11(2), 190 (1966) 11. Luenberger, D.: IEEE Trans. Autom. Control 16(6), 596 (1971) 12. Kondo, E., Takata, M.: Bull. JSME 20(142), 428 (1977). https://doi.org/10.1299/jsme1958.20. 428 13. Darouach, M.: IEEE Trans. Autom. Control 45(5), 940 (2000) 14. Ha, Q., Trinh, H., Nguyen, H., Tuan, H.: IEEE Trans. Industr. Electron. 50(5), 1030 (2003) 15. Singh, S., Janardhanan, S.: Int. J. Syst. Sci. 48(15), 3246 (2017) 16. Singh, S., Janardhanan, S.: Int. J. Syst. Sci. 50(6), 1179 (2019). https://doi.org/10.1080/ 00207721.2019.1597942 17. Singh, S., Janardhanan, S.: Discrete-Time Stochastic Sliding Mode Control Using Functional Observation. Lecture Notes in Control and Information Sciences, vol. 483. Springer, Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-030-32800-9 18. Kennedy, W.J., Gentle, J.E.: Statistical Computing. Marcel Dekker, New York (1980) 19. Kalman, R.E., Bucy, R.S.: Trans. ASME, Ser. D, J. Basic Eng. p. 109 (1961) 20. Mehta, A.J., Bandyopadhyay, B.: J. Dyn. Syst. Meas. Control, ASME 138, 124503 (2016) 21. Trinh, H., Fernando, T.: Functional Observers for Dynamical Systems, vol. 420. Springer, Berlin, Heidelberg (2012) 22. Singh, S., Janardhanan, S.: 2017 Australian and New Zealand Control Conference (ANZCC), pp. 175–178 (2017) 23. Brewer, J.: IEEE Trans. Circuits Syst. 25(9), 772 (1978) 24. Singh, S., Sharma, N.K., Janardhanan, S.: 2017 Australian and New Zealand Control Conference (ANZCC), pp. 179–183 (2017) 25. Polyakov, A., Poznyak, A.: Automatica 47(7), 1450 (2011) 26. Janardhanan, S., Kariwala, V.: IEEE Trans. Autom. Control 53(1), 367 (2008)

Design of Event-Triggered Integral Sliding Mode Controller for Systems with Matched and Unmatched Uncertainty Asifa Yesmin and Manas Kumar Bera

Abstract This book chapter explores the design of aperiodic sampled-data implementation of Integral Sliding Mode Controller (ISMC) for a linear continuous-time system with matched and unmatched uncertainty. The optimal value of the projection matrix is chosen and gain of controller is redesigned such that the proposed design strategy can ensure the asymptotic stability of the system in presence of matched and unmatched uncertainty. A triggering condition has been derived based on the defined measurement error which ensures the sufficient condition for the existence of the practical sliding mode. To guarantee the Zeno free behavior of the system, the existence of a positive lower bound of the inter-event execution time has been shown. This proposed methodology can reduce the number of control computation in comparison to the periodic implementation of ISMC. Finally, the simulation result is provided with a numerical system to illustrate the benefits of the proposed approach.

1 Introduction In modern control application, the Network Control Systems (NCSs) and CyberPhysical Systems (CPSs) offer great advantages in terms of flexibility of operation, easy installation, and cost reduction [1]. In such systems, the feedback mechanisms include communication networks through which exchange of information occurred among multiple control loops [2, 3]. The presence of the network in the control loop put several challenges. The overall performance of the control system can deteriorate due to the occurrence of packet loss, jitter, and delayed transmission. Moreover, the battery-powered smart devices connected in the network have limited energy resources that pose a requirement of efficient use of energy [4]. A. Yesmin (B) · M. K. Bera National Institute of Technology, Silchar, India e-mail: [email protected] M. K. Bera e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_6

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One potential solution to tackle these problems is the use of resource-aware design of control strategy which can reduce the usage of communication network and the network can be released for other tasks [5, 6]. Likewise, the sensors, the actuators, and the controller hardware should be used only if necessary to save energy in battery-powered equipment. To economizing the use of digital network and to manage efficiently the constrained energy budget, the appropriate choice was event-based control or event-triggered control [7–10]. The key idea of this strategy is that the control is updated only when some predefined threshold or condition is violated and that condition is known as triggering rule. This control strategy enables the effective usages of resources by increasing the control update interval. Such control scheme also lessens the communication burden in the network. Many variants and applications of event-triggered control strategy can be found in several literature [11–13] and the references therein. Any practical system which may be the part of NCSs or CPSs can be affected by the model uncertainties or external disturbances. The performance of eventtriggered strategy is affected in the presence of such perturbations. Therefore, the event-triggered implementation of robust control strategy is an active research area [14]. Sliding Mode Control (SMC) is one of the efficient robust control techniques which applies a switching control law so that the state trajectories of the system slide along a manifold called sliding surface [15–17]. SMC has gained its popularity due to its attractive features like insensitive to model uncertainties and external disturbances satisfying matching condition, ease of implementation and order reduction during the sliding motion. SMC is widely used in various engineering applications and it has been extensively discussed in literature [15–17]. The classical SMC design framework comprises two phases, namely reaching phase and sliding phase. In the reaching phase, the trajectories are compelled to slide on the sliding manifold in some definite time. After reaching the sliding manifold, in sliding phase, the plant states slide asymptotically toward the origin along the sliding surface. Many literature can be found on the event-triggered implementation of SMC as in [14], [18–24]. However, in classical SMC during the reaching phase, the robustness of the system can not be guaranteed even in the presence of matched perturbation and the resulting controller is robust against only matched perturbations. Therefore, ISMC was proposed in [25–27] to remove the reaching phase from the initial time instant. In ISMC, an integral term is introduced in the sliding variable, which enforces the sliding motion all through the system response beginning from the initial time instance so that the robustness toward the matched uncertainty can be ensured. The difference between the trajectory of the system with uncertainties and ideal trajectory of nominal system in the absence of uncertainties, both are projected by the projection matrix, is the philosophy to choose the sliding manifold. Naturally, the control law also comprises two parts–a continuous nominal control and discontinuous control. The nominal controls like state feedback, LQR, PID, and MPC, etc., are designed for nominal system and they are accountable for the desired performance of the nominal system. The discontinuous control is responsible for retaining the nominal system’s performance by rejecting the uncertainties. This can be achieved by making the sliding variable zero in finite time. To deal with the

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unmatched uncertainties, the possible solution is to combine other robust control techniques with ISMC and to choose a suitable projection matrix such that there will be no amplification of the unmatched perturbations. The ISMC design for both matched and unmatched perturbation can be found in many literatures as in [28–30]. To balance the control performance versus utilization of resources, we move from classical periodic control to aperiodic control like event-triggering, which is considered to be more effective. In this work, the key objective is to design eventtriggered ISMC to deal with both matched and unmatched uncertainty. The main contributions of this proposed method are summarized below. • Reaching law-based event-triggered ISMC has been proposed for linear continuous-time system with matched and unmatched uncertainty for fast convergence of the system trajectories. • A new event-triggering scheme is designed which involves the states as well as sliding surface. • The Zeno free behavior of the system has been ensured by showing a positive lower bound of the inter-event execution time. The remaining work is organized as follows: The preliminaries and the problem statement are described in Sect. 2. The design of the event-triggered ISMC has been presented in Sect. 3. In the same section, the closed-loop system stability and the existence of the lower bound of minimum inter-event time have been shown. The simulation results of a numerical system with event-triggered ISMC have been shown in the Sect. 4. Section 5 concludes the chapter with some remarks. Notation: The set of real numbers and integers are denoted by R and Z, respectively. The positive real numbers and integers are given as R>0 and Z>0 , respectively. Similarly, it can also be defined for negative and nonnegative cases. Rn represents the Euclidean space over real. For any a ∈ R, we write |a| for the absolute value of a. Similarly, for any vector a ∈ Rn , a denotes the 2-norms of the vector.

2 Problem statement and preliminaries 2.1 Integral sliding mode control In order to illustrate the design of ISMC based on reaching law, consider an uncertain Linear Time Invariant (LTI) system of the form x(t) ˙ = Ax(t) + (B + ΔB)u(t) + B + ΔB f m (t, x) + pun (t, x); x(0) = x0 , (1) where x(t) ∈ Rn and u(t) ∈ R represents the states and control input, respectively. The function f m (t, x) presents the matched perturbation due to the external disturbance and pun (t, x) represents the unmatched uncertainty. The matrices A ∈ Rn×n and B ∈ Rn×1 are the system matrix and input matrix, respectively. The modeling

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error ΔB is unknown and with same dimension as B. The disturbances considered here are the bounded disturbance and the uncertainty due to the modeling error can be considered as a part of the unmatched uncertainty. The unmatched uncertainties are clubbed together and the modified representation of system (1) is in the form x(t) ˙ = Ax(t) + Bu(t) + B f m (t, x) + f u (t, x),

(2)

where f u (t, x) = ΔB[u(t) + f m (t, x)] + pun (t, x). The following assumptions are made which are needed in the rest of paper. Assumption The rank of B, i.e., rank(B) = m.



Assumption The matrix pair (A,B) is controllable.



Assumption The macthed and unmatched uncertainty f m (t, x) and f u (t, x) have a known upper bound for all x and t, i.e., supt≥0 | f m (t, x)| ≤ Fm . and supt≥0 | f u (t, x)| ≤  Fu . The following lemma has been introduced which will be useful to establish the closed loop stability of the system. Lemma 1 [32] Consider a dynamical system (2). Suppose that there exists a continuous function V : D → R such that the following conditions hold 1. V is positive definite. 2. There exist positive real numbers η1 and η2 and κ ∈ (0, 1) such that V˙ (x) ≤ −η1 V κ (x) − η2 V (x). Then for any initial state x0 , the origin is globally finite time stable and the state trajectories converge to origin in some definite time.   η2 V 1−κ (x(0)) + η1 1 ln tf ≤ η2 (1 − κ) η1

(3)

and stays there forever, i.e., x = 0 for all t > t f . The design of ISMC is formulated in two steps. Firstly, we will design a stable sliding surface s, and in the second step, we will design the discontinuous control which compels the system trajectories to lie on that surface. To understand the ISMC and event-triggered ISMC, the following definitions related to ideal sliding mode and practical sliding mode are introduced as in [14]. Definition 1 Given t ∗f ≥ t0 , if ∀ x0 ∈ Rn , s = 0 ∀ t ∗f ≥ t0 , then an ideal sliding mode of a system (2) is established on the sliding manifold s = 0. Definition 2 Given t f ≥ t0 , if ∀ x0 ∈ Rn , |s| = δ ∀ t f ≥ t0 , then an practical sliding mode of a system (2) is established in the vicinity of the sliding manifold s = 0. Here, δ is known as Practical Sliding Mode Band (PSMB).

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Let the nominal control input that is designed for the system (2) be u 0 , assuming that the matched and unmatched uncertainty are zero. So the state trajectories of the ideal system can be obtained by solving the following ordinary differential equation, x(t) ˙ = Ax(t) + Bu 0 (t).

(4)

The motive is to design a control law u(t) so that the state trajectories of perturbed system (2) can yield sliding motion as well as the performance of the nominal system (4) can be hold even in the presence of perturbations. Now the control law can be framed as (5) u(t) = u 0 (t) + u 1 (t), where u 1 (t) is the discontinuous control for the compensation of the uncertainties affecting the system. Here, the nominal control u 0 (t) is considered as the state feedback controller of the form (6) u 0 (t) = −K x(t), where K ∈ R1×n is a state feedback gain to be designed so that (A − B K ) will be Hurwitz. The integral sliding manifold which can eliminate the reaching phase is designed as t s(t) = Gx(t) − Gx(0) − G

(Ax(τ ) + Bu 0 (τ ))dτ ,

(7)

0

where G is the projection matrix and can be chosen such that the det(G B) = 0. In order to optimize the effect of the unmatched uncertainties, an optimal design of projection matrix is the key. One of the optimal and simplest choices is G = B as in [16]. Differentiating equation (7) and using the relation (6), it is obtained as s˙ (t) = G x(t) ˙ − G(A − B K )x(t).

(8)

Now, substituting the system equation (2) in (8), it is obtained as s˙ (t) = G(Ax(t) + Bu(t) + B f m (t, x) + f u (t, x)) − G(A − B K )x(t) = G Ax(t) + G Bu(t) + G B f m (t, x) + G f u (t, x) − G Ax(t) + G B K x(t) = G Bu(t) + G B f m (t, x) + G f u (t, x) + G B K x(t).

(9)

In [17], Gao proposed the constant plus proportional rate reaching law for continuoustime system. The dynamic quality of the system response can be controlled by appropriate selection of the design parameter in the reaching law. The reaching law is reformulated by considering the matched perturbation as s˙ = −Q 1 s(t) − Qsign(s(t)) + G B f m (x, t) + G f u (t, x).

(10)

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where the gains Q 1 > 0 and Q > supt≥0 G BFm + supt≥0 GFu + η, η > 0 can be chosen to satisfy reachability condition [15]. The integral SMC can be designed by equating relation (9) and (10) and solving for u(t) u(t) = −(G B)−1 (G B K x(t) + Q 1 s(t) + Qsign(s(t))) .

(11)

The control law (11) ensures the finite time stability of the system (2) and it can be shown that this control law enforces sliding mode. Here, the solution of the system is understood in the sense of Filippov [31]. The practical aspect of control implementation in cyber-physical space is different from the conventional one. The conventional implementation approach is based on time-triggered protocols where the control law is updated periodically at every sampling instants. The states are sampled at certain time instants t = {t0 , t1 , · · · , ti , · · · }, i ∈ Z≥0 and based on this information, the control law is computed as τ (ti ) = κ(s(ti )). A Zero-Order Hold (ZOH) is utilized at the controller to hold the feedback information between any two sampling instants, can be expressed as u(t) = u(ti ) t ∈ [ti , ti+1 [, i ∈ Z≥0 ,

(12)

where ti , ti+1 ∈ T, T is the set of triggering time instants. In conventional timetriggering approach, the time interval ti+1 − ti is equidistant and predefined and the sequence {ti }i∈Z≥0 is periodic. This periodic update of states and control lead to wasteful use of the limited resources. To reduce the expenditure related to number of control updates and computation, the event-triggered ISMC is required to be designed without compromising the system stability and performance.

3 Design of Event-triggered ISMC For the event-triggered realization of ISMC, the control law (11) can be rewritten as u(t) = −(G B)−1 (G B K x(ti ) + Q 1 s(ti ) + Qsign(s(ti )))

(13)

for all t ∈ [ti , ti+1 ) and i ∈ Z≥0 . This control law is obtained by replacing the continuous state values x(t) by its sampled values x(ti ) at the triggering instants ti in (11). Ti is defined as the inter-event execution time where Ti := ti+1 − ti . The control is updated at every ti and is held same up to the next instant ti+1 using Zero-Order Hold (ZOH). This signifies that u(t) = u(ti ) for ∀ t ∈ [ti , ti+1 ). The error that grows in the system for all t ∈ [ti , ti+1 ) is given by e(t) = G B K x(ti ) + Q 1 s(ti ) − G B K x(t) − Q 1 s(t).

(14)

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The event-triggering strategy is proposed such that the control law (13) will be updated only when the norm of the error will be violated by some predefined threshold. Theorem 1 Consider the system (2) and the sliding surface (7). Let the positive constants α ∈ (0, ∞) and σ ∈ (0, 1] be given such that e(t) < σα

(15)

holds for all t > 0 then the control law (13) can make the system finite time stable in presence of perturbations, i.e., the practical sliding mode is achieved in the vicinity of s(t) = 0 within a band        σα   G B K   x(t) − x(ti ) Bx = x ∈ Rn : |s(t)| ≤   +  Q1 Q1 

(16)

if Q is designed as Q > supt≥0 G BFm + supt≥0 GFu + σα + η

(17)

where η is some positive scalar. Proof To show the existence of the sliding mode, assume the Lyapunov candidate as V = 21 s 2 . Differentiating V with respect to time and using the relation (9) and control law (13) it can be obtained as V˙ (s(t)) =s(t)˙s (t) =s(t)(G x(t) ˙ − G(A − B K )x(t)) =s(t)(G(Ax(t) + Bu(t) + B f m (t, x) + f u (t, x)) − G Ax(t) + G B K x(t)) =s(t)(G Bu(t) + G B f m (t, x) + G f u (t, x) + G B K x(t)) =s(t)(G B K x(t) + G B f m (t, x) − G B K x(ti ) − Q 1 s(ti ) − Qsigns(ti ) + G f u (t, x))  = − s(t) G B K x(ti ) + Q 1 s(ti ) − G B K x(t) − Q 1 s(t) + Q 1 s(t) + Qsigns(ti ) − G B f m (x, t) − G f u (t, x) . It is to be observed that the sign of sliding variable, s, does not change until it reaches the manifold. So, it can be inferred that sign(s(ti )) = sign(s(t)) as long as sign of s(t) does not change. With this above equation can be rewritten as

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V˙ (s(t)) ≤|s(t)|e(t) − Q 1 |s(t)|2 − Q|s(t)| + |s(t)|G BFm + |s(t)|GFu ≤ − |s(t)|(Q − e(t) − G BFm − GFu ) − Q 1 |s(t)|2 ≤ − |s(t)|(Q − σα − G BFm − GFu ) − Q 1 |s(t)|2 ≤ − η|s(t)| − Q 1 |s(t)|2 . Let η1 =

(18)

√ 2η and η2 = 2Q 1 and the above equation can be written as 1 V˙ (s(t)) ≤ −η1 V 2 (t) − η2 V (t).

(19)

From Lemma 1, it can be concluded that the system is finite time stable and the reaching time to the sliding manifold is estimated as

 1 2 η2 V 2 (0) + η1 t f ≤ ln . η2 η1

(20)

As the system trajectories reach the sliding manifold, it will cross the manifold if the control signal is not updated. The trajectories move away from the manifold and the error starts to grow. A time instance comes when the event condition is violated and control signal is updated. This updated control signal compels the trajectories to move toward the sliding manifold again. This process continues until the trajectories reach to the origin. The band, i.e., the PSMB can be yielded by calculating the maximum deviation of sliding trajectory in one triggering interval. From (14), it can be shown as      σα   G B K     x(t) − x(ti ).  (21) |s(ti ) − s(t)| ≤   +  Q1 Q1  Thus the sliding trajectory can move utmost by (21) from its immediate past sampled value in one triggering interval. The maximum band can be calculated by setting 

s(ti ) = 0 and is given by (16). Thus, the proof is completed. To avoid infinite number of executions in finite time, it is essential to show that the inter-event execution time is always lower bounded by some positive number which has been discussed in the following section.

3.1 Inter-Execution Event Time A proper triggering mechanism can guarantee the Zeno free behavior of the system (2), and it can be ensured if the inter-event execution time Ti = ti+1 − ti is lower bounded by some positive number. The relation (15) which is the condition for the

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event to be generated and the control signal (13) will be updated only when (15) is violated. Therefore, the triggering instant ti+1 can be given by ti+1 = inf[t > ti : e(t) ≥ σα].

(22)

Theorem 2 Consider the system (2) and the control law (13). If the event generation function is defined by (22) then the inter-event execution time Ti is lower bounded by the relation σα , (23) Ti > k1 δ + k2 + k3 + k4 where k1 = (G B K  + Q 1 G)A + (Q 1 G A + Q 1 G B K ), k2 = (G B K  + Q 1 G)B − (G B)−1 (G B K x(ti ) + Q 1 s(ti ) + Qsign(s(ti ))) , k3 = (G B K  + Q 1 G)BFm and k4 = (G B K  + Q 1 G)Fu . Proof Let Ti be the inter-event i.e., time required to grow error from execution time,

zero to σα. Let the set  = t : e(t) = 0 where t ∈ [ti , ti+1 )\. Thus for the time t and using the relation (8) and (13) , it can be obtained as 

d d e(t) = (G B K x(ti ) + Q 1 s(ti ) dt dt − G B K x(t) − Q 1 s(t)) ≤G B K x(t) ˙ − Q 1 s˙ (t) ≤G B K x(t) ˙ + Q 1 (G x(t) ˙ − G(A − B K )x) ≤(G B K  + Q 1 G)x(t) ˙ + (Q 1 G A + Q 1 G B K )x(t) ≤(G B K  + Q 1 G)Ax(t) + Bu(t) + B f m (t, x) + f u (t, x) + (Q 1 G A + Q 1 G B K )x(t)  ≤ (G B K  + Q 1 G)A + (Q 1 G A + Q 1 G B K ) x(t) + (G B K  + Q 1 G)Bu(t) + (G B K  + Q 1 G)BFm + (G B K  + Q 1 G)Fu .

(24)

While sliding motion s(t) = s˙ (t) = 0. So from the relation (8), the norm x(t) can be shown as bounded which is given by x(t) ≤ exp(−(A − B K )t)x0 (t). By construction A − B K is Hurwitz, hence x(t) ≤ δ exp(−βt) where β and δ are positive constants. Now, if the last event for states x(t) generated at time ti > 0 then for all t ∈ [ti , ti+1 ), x(t) ≤ x(ti ), i.e., x(ti ) ≤ δ exp(−βti ). Therefore, from

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the relation (24) it can be obtained as 

 d e(t) ≤ (G B K  + Q 1 G)A dt

+ (Q 1 G A + Q 1 G B K ) δ exp(−βti )

+ (G B K  + Q 1 G)B ×  − (G B)−1 (G B K x(ti ) + Q 1 s(ti ) + Qsign(s(ti ))) + (G B K  + Q 1 G)BFm + (G B K  + Q 1 G)Fu ≤k1 δ exp(−βti ) + k2 + k3 + k4 . The error can be bounded as t (k1 δ exp(−βti ) + k2 + k3 + k4 )dt.

e(t) ≤

(25)

ti

The term in the bracket can be further upper bounded by (k1 δ + k2 + k3 + k4 ). Then from the above inequality (25), it can be written as e(t) ≤(k1 δ + k2 + k3 + k4 )(t − ti ).

(26)

The next event will occur if e(t) ≥ σα. Thus, a lower bound Ti on the interexecution event time ti+1 − ti is given by (23) and holds for all the event time. This completes the proof.  .

4 Simulation Result In this section, an example is shown to validate the proposed event generation scheme and to evaluate the performance of event-triggered ISMC. The dynamics of a perturbed continuous-time linear system is described as follows ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 10 15 13 0 0.5sin(2πt) ⎦ 0 x(t) ˙ = ⎣−20 −10 17⎦ x(t) + ⎣−3⎦ (u(t) + 0.3sin(2 ∗ π ∗ t)) + ⎣ 0 15 15 5 0.5sin(2πt).

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2.5 x

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Fig. 1 Evolution of states with time-triggered ISMC 8 s

Sliding Manifold

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

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Fig. 2 Sliding manifold with time-triggered ISMC

The initial state of the system is chosen as x0 = [−1 1 1] . The projection matrix G is designed as G = B = [0 − 3 5]. The gain K is computed using MATLAB ‘place’ command, i.e., K = place(A, B, 102 × [−1.9188 + 0.0000i − 0.2296 + 0.3628i − 0.2296 − 0.3628i]). The computed value of gain is K = [165.26 52.9310 82.3188]. The other design parameters are chosen as σ = 0.4, α = 50, the reaching gain Q 1 = 4, and the switching gain Q = 42. Figs. 1 and 2 represent the evolution of states and the sliding manifold of the system with ISMC law (11). Fig. 3 represents the plot of the states of the system with event-triggered ISMC. The sliding manifold and the control input of event-triggered ISMC are presented in Figs. 4 and 5. The proposed control scheme also ensure the Zeno free behavior of the

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2

x3

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1

x2

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Fig. 3 Evolution of states with event-triggered ISMC 8

s

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Fig. 4 Sliding manifold with event-triggered ISMC

system, i.e., the lower bound of inter-execution event time is bounded away from zero which is shown in the Fig. 6. With the appropriately chosen design parameters α and σ in event condition, the numbers of event generation can be controlled. The greater value of the α leads to lesser number of event generation, i.e., less control effort. In Table 1, the comparison between time-triggered and event-triggered ISMC has been presented. From the Table 1 it can be observed that the number of control updates in event-triggered control scheme is much lesser than the time-triggered case which leads to the reduction of control computation and minimal usages of resources without compromising the stability and performance. Some of the performance indices have been determined. The performance indices as in [20] considered are

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40

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30 20 10 0 -10 -20 -30 0

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Fig. 5 Control effort with event-triggered ISMC 0.035 T

i

0.03 0.025 0.02 0.015 0.01 0.005 0 0

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Fig. 6 Inter-execution event time with event-triggered ISMC

• The RootMean Square (RMS) value of the plant state which is determined as n s 2 i=1 x (ti ) xRMS = ns  n s 2 i=1 s (ti ) • The RMS value of the sliding variable and is determined as s R M S = ns  n s 2 i=1 e (ti ) • The RMS value of error is determined as e R M S = ns where n s is the number of integration steps. The above performance indices are calculated and tabulated in Table 2 for timetriggered and event-triggered implementation of ISMC. It can be observed from Table 2 that the RMS value of the state and the sliding surface is increased slightly in the case of event-triggered implementation of ISMC. This slight performance degradation in

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Table 1 System performance for time-triggered and event-triggered control scheme Triggering scheme No. of control updates Time triggered (τ = 0.001) Event triggered

1001 314

Table 2 The simulated values of the performance indices for time-triggered and event-triggered Control mechanism Cases xRMS sR M S eR M S Time triggered (τ = 0.001) Event triggered

0.3335 0.7914

0.0527 0.6983

– 0.0383

10000 ||e(t)||

Error

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Fig. 7 Evolution of e(t) with event-triggered ISMC

case of event-triggered can be compromised because the number of control updates is significantly reduced in comparison with time-triggered implementation which is reflected in Table 1. Fig. 7 represents the evolution of e(t) which is found to be bounded.

5 Conclusion In this work, an event-triggered ISMC scheme based on constant plus proportional rate reaching law has been proposed for a linear continuous-time system with matched and unmatched uncertainty. The event-triggered-based control implementation of ISMC has considerably reduced the control computation than the periodic eventtriggered scheme. The optimal value of projection matrix is chosen to ensure the

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asymptotic stability in presence of unmatched uncertainty. The Zeno free behavior is guaranteed by ensuring a positive lower bound of the inter-execution event time. To validate the efficacy of the proposed control law, a numerical system is considered for simulation.

References 1. Gupta, R.A., Chow, M.: Networked control system: overview and research trends. IEEE Trans. Ind. Electron. 57(7), 2527–2535 (2010). July 2. Zhang, W., Branicky, M.S., Phillips, S.M.: Stability of networked control systems. IEEE Control Syst. Mag. 21(1), 84–99 (2001) 3. Hespanha, J.P., Naghshtabrizi, P., Xu, Y.: A survey of recent results in networked control systems. Proc. IEEE 95(1), 138–162 (2007) 4. Wang, F.-Y., Liu, D.: Networked control systems: theory and applications. Springer Publishing Company (2008) 5. Donkers, M.C.F., Tabuada, P., Heemels, W.P.M.H.: Minimum attention control for linear systems. Discrete Event Dyn. Syst. 24(2), 199–218 (2014) 6. Nagahara, M., Quevedo, D.E., Nešic, D.: Maximum Hands-Off Control: A Paradigm of Control Effort Minimization. In: IEEE Transactions on Automatic Control, vol. 61, no. 3, pp. 735–747. (2016) 7. Heemels, W.P.M.H., Johansson, K.H., Tabuada, P.: An introduction to event-triggered and selftriggered control. In: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 3270–3285. (2012) 8. Astrom, A.J., Bernhardsson, B.: Comparison of Riemann and Lebesgue sampling for first order stochastic systems. International Conference on Decision and Control, Las Vegas, USA (2002) 9. Tabuada, P.: Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. Autom. Control 52(9), 1680–1685 (2007) 10. Borger, D.P., Heemels, W.P.M.H.: Event-separation properties of event-triggered control systems. IEEE Trans. Autom. Control 59(10), 2644–2656 (2014) 11. Lunze, J., Lehmann, D.: A state-feedback approach to event-based control. Automatica 46(1), 211–215 (2010) 12. Fang, F., Xiong, Y.: Event-driven-based water level control for nuclear steam generators. IEEE Trans. Ind. Electron. 61(10), 5480–5489 (2014) 13. Wen, Z., Chen, M.Z.Q., Yu, X.: Event-triggered master-slave synchronization with sampleddata communication. In: IEEE Transaction Circuits Systems II, Experiment Briefs, vol. 63, no. 3, pp. 304–308 (2016) 14. Bandyopadhyay, B., Behera, A.K.: Event Triggered Sliding Mode Control: A New Approach to Control System Design. Springer (2018) 15. Edwards, C., Spurgeon, S.K.: Sliding Mode Control: Theory and Applications. CRC Press (1998) 16. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation, Birkhäuser Basel. (2014) 17. Hung, J.Y., Gao, W., Hung, J.C.: Variable structure control: a survey. IEEE Trans. Ind. Electron. 40(1), 2–22 (1993) 18. Behera, A.K., Bandyopadhyay, B.: Event-triggered sliding mode control for a class of nonlinear systems. Int. J. Control 89(9), 1916–1931 (2016) 19. Behera, A.K., Bandyopadhyay, B.: Robust sliding mode control: an event-triggering approach. IEEE Trans. Circ. Syst. II Exp. Briefs 64(2), 146–150 (2017) 20. Cucuzzella, M., Incremona, G.P., Ferrara, A.: Event-Triggered Sliding Mode Control Algorithms for a Class of Uncertain Nonlinear Systems: Experimental Assessment, pp. 6549–6554. ACC, Boston, MA (2016)

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21. Kumari, K., Behera, A.K., Bandyopadhyay, B.: Event-triggered sliding mode-based tracking control for uncertain Euler-Lagrange systems. IET Control Theory App. 12(9), 1228–1235 (2018) 22. Ferrara, A., Cucuzzella, M.: Event-triggered sliding mode control strategies for a class of nonlinear uncertain systems. In: Clempner, J., Yu, W. (eds.) New Perspectives and Applications of Modern Control Theory, pp. 397–409. Springer, Cham (2018) 23. Cucuzzella, M., Incremona, G.P., Ferrara, A.: Event-triggered variable structure control. Int. J. Control (2019) 24. Yesmin, A., Bera, M.K.: Design of event-triggered sliding mode controller based on reaching law with time varying event generation approach. Euro. J. Control 48, 30–41 (2019) 25. Utkin, V., Shi, J.: Integral sliding mode in systems operating under uncertainty conditions. In: Proceedings of 35th IEEE Conference on Decision and Control, vol. 4, pp. 4591–4596. Kobe, Japan (1996) 26. Nair, R.R., Behera, L., Kumar, S.: Event-triggered finite-time integral sliding mode controller for consensus-based formation of multirobot systems with disturbances. IEEE Trans. Control Syst. Technol. 27(1), 39–47 (2019). Jan 27. Pan, Y., Yang, C., Pan, L., Yu, H.: Integral sliding mode control: performance, modification, and improvement. IEEE Trans. Ind. Inf. 14(7), 3087–3096 (2018) 28. Castanos, F., Fridman, L.: Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans. Auto. Control 51(5), 853–858 (2006) 29. Rubagotti, M., Estrada, A., Castanos, F., Ferrara, A., Fridman, L.: Integral sliding mode control for nonlinear systems with matched and unmatched perturbations. IEEE Trans. Auto. Control 56(11), 2699–2704 (2011) 30. Xi, Z., Hesketh, T.: Brief paper: discrete time integral sliding mode control for systems with matched and unmatched uncertainties. IET Control Theory Appl. 4(5), 889–896 (2010) 31. Filippov, A.F.: Differential Equations With Discontinuous Right-Hand Sides. Kluwer, Dordrecht, The Netherlands (1988) 32. Hong, Y., Huang, J., Xu, Y.: On an output feedback finite-time stabilization problem. IEEE Trans. Auto. Control 46(2), 305–309 (2001)

Design of Periodic Event-Triggered Sliding Mode Control Abhisek K. Behera and Bijnan Bandyopadhyay

Abstract This chapter presents an event-triggering approach to the design of sliding mode control (SMC) for an uncertain linear dynamical system. In the event-triggering mechanism, only discrete measurements are used to verify the event condition for generating a sequence of triggering instants. Such a triggering mechanism is known as the periodic triggering mechanism. Due to the periodic evaluation, the event condition in the triggering mechanism can only be detected at a time instant, which is a multiple of sampling period, although it may hold before this instant. The design of SMC in the periodic event-triggering framework becomes even more complicated because of the dependency of switching gain on the triggering parameter. This chapter presents a novel method of designing the periodic triggering mechanism-based SMC that can control the plant under external perturbations. In our approach, we decouple the design of event condition and the switching gain of SMC by choosing appropriately the sampling period for the periodic evaluation. Therefore, one may design the switching gain first to obtain a suitable sampling period so that the desired bound for the state trajectories can be achieved by the event-triggered controller. Finally, the simulation results are presented to demonstrate the performance of the proposed triggering mechanism.

A. K. Behera (B) Department of Electrical Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, India e-mail: [email protected] B. Bandyopadhyay Interdisciplinary Program in Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_7

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1 Introduction The problem of designing a feedback control law for a sampled-data system is an ever-challenging task starting from its beginning formulations based on the simplified approaches. The modern technological advancement has even demanded new approaches for the control of sampled-data systems to meet the additional stringent requirements. From the technical side, one may state that the objective is to design a discrete controller to control the analog plant in a manner to satisfy the constraints which are not necessarily specified in the digital framework. It is, therefore, important for the designer to interpret these requirements in an appropriate sense for achieving the design objective with the discrete controller. The state of art in the sampled-data literature has witnessed the development of many promising techniques in the last half-century that enable us to address the issues concerning the control goals. Broadly, there are two ways of designing the controller for a sampled-data system, viz., direct and indirect approaches [1]. The direct approach analyzes the sampleddata system without any approximation that can simplify the control problem. It is, therefore, more complicated to design the sampled-data controller in this approach. The indirect approach greatly simplifies the problem by considering either a pure discrete-time design or a continuous-time design. The consequence of this approach is that the control goals for sampled-data systems may not be achievable in this design because it does not consider the aspect of open-loop behavior between two samples. Thus, the controller designed from the indirect approach may not emulate the exact behavior of a real-time sampled-data system. Recently, a new approach to analyze such systems, known as the event-triggering based implementation, was proposed in which sensing and/or actuation are/is triggered by an online verifiable condition that exclusively depends on the plant trajectory [7–12]. Though the complexities are expected to increase in this approach, the outcome in terms of reduction in traffic from sensors to actuators is very significant for the modern control applications [13–16], [37]. The main feature of the event-triggered controller is the online detection of a stabilizing event condition to initiate the control tasks, such as the transmission of state or application of control signal, etc. The implementation schemes with the continuous evaluation of event condition which is termed as continuous event-triggering are mostly considered in many works reported in the literature. This continuous triggering mechanism requires the continuous measurement of the plant state for generating a sequence of time instants. An immediate consequence can lead to the increased cost in the implementation owing to the need for sophisticated sensors to provide almost continuous measurements. This drawback can be overcome by designing the eventtriggered controller by taking into account the periodic measurements of the sensors. In this case, the continuous evaluation in the triggering mechanism is relaxed to only periodic instants where the event condition is checked for its fulfillment at periodic intervals [29, 35, 36]. This is known as the periodic event-triggering mechanism. In this chapter, our focus is to design a stabilizing periodic event-triggered controller for an uncertain plant. It is well known that sliding mode control (SMC) is a

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powerful control technique to achieve robust stabilization for the plant in the presence of a certain class of disturbances [2–6]. For stabilizing the uncertain plants with a reduced communication burden, an event-triggering approach to the design of SMC is also presented in the literature. In [17–19, 22, 30, 31], the design of continuous event-triggered SMC was presented where plant trajectory can be bounded with any arbitrary bound irrespective of bounded (matched) disturbances. The design of event-triggered SMC for nonlinear systems was discussed in [23]. A new triggering strategy was developed to achieve global stability for an uncertain linear plant [24]. Since it may not be desirable to communicate among sensors in many situations, a decentralized version of event-triggered SMC was proposed in [25]. Here, the local triggering mechanisms were used to generate the local triggering instants at which the centralized event condition is checked for its violation. The control is updated if the centralized triggering condition is satisfied. In [21, 33], quantized feedback-based event-triggered SMC was presented. Recently, a reduced-order model-based design of event-triggered SMC was presented in [38] where the time interval between two event occurrences is increased as a reduced state vector is used. As the implementation of the continuous triggering mechanism requires continuous state measurements, some variants of the continuous event-triggered SMC are also proposed in the literature. The self-triggered SMC was proposed in which the triggering instant is calculated based on the state measurements at previous triggering instants [20]. The design of a triggering mechanism based on discrete measurements is also discussed in [26–29, 32]. The works reported in [26–28] use a discrete-time model to design a discrete event-triggering strategy. In [29], the periodic triggering mechanism-based SMC was considered for a multi-input multi-output system. It is shown that the design of periodic event-triggered SMC is not straightforward because of the explicit dependency of the controller gain on the triggering parameter [29]. Nevertheless, it is possible to design the controller in a decoupled manner to guarantee the stability for the plant with periodic event-triggered implementation. The design algorithm is based on the selection of an appropriate sampling interval to evaluate the event condition periodically. The controller gain can be designed independently for a given sampling interval. In this chapter, the discussion on the design of periodic event-triggered SMC is presented for a single-input single-output system. The contributions of this chapter are listed below • We provide a design technique for the periodic event-triggering mechanism based on SMC. Our approach allows a decoupled way of designing the switching gain and the sampling period for the periodic triggering mechanism. First, we select the switching gain of SMC for some set of parameters, and then one can obtain an upper bound of the sampling period. One may design the periodic triggering mechanism with any sampling period less than the upper bound obtained for the switching gain. The inherent feature of this triggering mechanism is that the consecutive triggering instants are separated from each other by the sampling period of the triggering mechanism. Thus, the triggering sequence is devoid of the Zeno phenomenon. • The stability for the closed-loop system with the proposed periodic event-triggering scheme is also presented. The periodic triggering mechanism achieves the sim-

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ilar stability for the closed-loop system as that obtained with continuous eventtriggering strategy. It is shown that the plant trajectories can be bounded within any arbitrary bound by the event-triggered SMC. However, the switching gain in the periodic triggering mechanism is larger than the case with the continuous triggering mechanism. The rest of the content of this chapter is structured as follows: Section 2 presents the problem of periodic event-triggered controller design. The main result of this chapter is discussed in Sect. 3. It consists of formulation of periodic event-triggering strategy and the control law; and also the stability analysis for the closed-loop system. This is followed by the simulation study in Sect. 4. Finally, the concluding remarks are made in Sect. 5. Notation: The set of real numbers is denoted by R and the Euclidean space is given by Rn . The set of nonnegative integers are specified by Z≥0 . The norm of a given vector x ∈ Rn is given by x, whereas the absolute value of a scalar variable a is given by |a|. For any square matrix R, the largest and smallest eigenvalues are written as λm {R} and λ M {R}, respectively. Given a matrix R, R > 0 denotes the positive definiteness of the matrix. Finally, R represents the spectral norm of the matrix R. For any set S, ∂ S denotes its boundary.

2 Problem Statement We consider a linear time-invariant (LTI) dynamical system given by x˙ = Ax + B(u + d),

x0 := x(0)

(1)

where x ∈ Rn is the state vector, u ∈ R is the scalar control input, and d ∈ R is the external matched disturbance input; the constant matrices A and B are of appropriate dimensions. The matched disturbance can be interpreted as the uncertainties that enters into the plant through the input channel. In fact, many practical systems such as mechanical systems, electrical systems, etc., fall into the class given above. Assumption 1 The matrix pair (A, B) is controllable. Assumption 2 The disturbance, d(t), is bounded for all t ≥ 0, i.e., supt≥0 |d(t)| ≤ d0 for any known d0 > 0. Let us now discuss the implementation of the control law to be dealt with in this chapter. The feedback loop is not very conventional in the sense that the control signal is applied only at some discrete instants possibly depending on the plant state. The rationale behind such a scheme is to reduce the information traffic over the network which is an essential design constraint in modern control applications. Particularly, we consider event-triggering-based controller implementation for stabilizing the plant where a triggering rule monitors a condition to initiate the control

Design of Periodic Event-Triggered Sliding Mode Control Fig. 1 The event-triggered controller in the closed-loop system

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Periodic Triggering Mechanism Communication Network

task on its violation as shown in Fig. 1. Unlike, the classical triggering condition, an additional restriction is imposed on the triggering mechanism by using only discrete measurements of the state trajectory. As a result of this, the event condition is evaluated periodically based on the information available from the sensors. ∞ be the sequence of time instances generated by the triggering mechaLet {ti }i=0 nism at which the control is applied to the plant. Since measurements are available periodically, say h > 0 is the sampling period of these measurements, one has 0 = t0 < t1 < t2 < · · · < ti < · · · ,

lim ti = ∞,

i→∞

where Ti := ti+1 − ti ≥ h for all i ∈ Z≥0 . The control task is to design an eventtriggered feedback controller to ensure the stability for the closed-loop system satisfying the above requirements. The event-triggered controller obviously is free from the undesired Zeno phenomenon of the triggering sequence which is a remarkable outcome of the periodic event-triggering mechanism. Under the periodic event-triggering framework, we consider SMC for the control of uncertain plants against the perturbations in this chapter. The key feature in the event-triggered SMC is that the switching gain of SMC depends on the triggering parameter. So, the gain of SMC also depends on the parameters of the periodic triggering mechanism. However, this dependency introduces some obstacles in the design of the triggering mechanism independently. This obstacle is a consequence of the periodic evaluation of the triggering condition. In the periodic triggering mechanism, the triggering instants may not represent the time instants at which the event condition is satisfied because of the relaxation of continuous evaluation. In other words, it may be expected that the event condition may be satisfied before its detection by the triggering mechanism. In this case, the application of the triggering condition in the closed-loop analysis is not very straightforward. The goal of this chapter is to discuss the design of the triggering mechanism and analyze the stability of the closed-loop system with the proposed periodic event-triggered SMC. In this chapter, we discuss the impact of relaxing the continuous evaluation of the triggering mechanism, and then we analyze the design of the triggering mechanism by taking these constraints into the analysis.

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3 Main Result This section gives the main result of this chapter. The first two subsections describe the periodic event-triggering mechanism and the control law to ensure the desired stability for the system. The last part of the section presents the stability analysis for the closed-loop system.

3.1 Design of SMC A brief discussion on the design of SMC is presented below. It consists of two sequential steps: first one deals with the design of sliding manifold and another one is concerned with the control input to bring the plant trajectory to this manifold. For the design of sliding surface, we note that the input matrix can be represented as  B=

 B1 , B2

where B2 ∈ R \ {0}. This directly follows from Assumption 1. Then, applying the transformation   I −B1 B2−1 , W = 0 B2−1 the system (1) can be represented in the new coordinate z = W x as z˙ 1 = A11 z 1 + A12 z 2

(2a)

z˙ 2 = A21 z 1 + A22 z 2 + u + d,

(2b)

  where the usual notation z = z 1 z 2 ∈ Rn with some appropriate dimensions of state components. The main feature in the above dynamics is that the disturbance input only affects the subpart of dynamics directly. This suggests that the part of dynamics free from the influence of disturbance can be considered for the design of sliding manifold. For use in the subsequent discussion, we write the compact notations for system and input matrices as A = W AW −1 =



A11 A12 A21 A22

 and

B = WB =

  0 1

and as a result of that, one may represent (2) by z˙ = Az + B(u + d).

(3)

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From Assumption 1, it is guaranteed that (A, B) is a controllable pair. One can further show that this fact is equivalent to the controllability of the pair (A11 , A12 ) [5]. Let F  ∈ Rn−1 be any feedback gain such that A  11 − A12 F is Hurwitz. Then, by taking s = C z as the sliding function where C = F 1 , we define   S := z ∈ Rn : s = C z = 0 .

(4)

The set S is known as the sliding manifold. The importance of this manifold is that when the system trajectories are confined to S the dynamics become free from the external disturbance which can be seen as follows. For z ∈ S, one has z 2 = −F z 1 and subsequently,  z˙ 1 = A11 − A12 F z 1 . Since by design that A11 − A12 F is Hurwitz, the trajectory z 1 (t) → 0 as t → ∞ and eventually it guarantees the asymptotic stability of z = 0. As a result, our goal in the controller design focuses on bringing the trajectory onto S in a finite time and subsequently ensure that it does not leave thereafter. This motion is usually referred as the sliding mode. As the controller is applied in discrete manner, the sliding motion will not take place in the system like in the case of analog implementation. However, one can still characterize this motion by a relaxed notion of sliding mode as given below. Note that this notion has been widely used in the event-triggered SMC literature [31]. Definition 1 The system is said to be in practical sliding mode if for any ε > 0 there exists a τ > 0 such that the trajectories starting from anywhere ensure |s(t)| ≤ ε for all t ≥ τ . This notion is yet a powerful characterization of sliding motion because it quantifies the accuracy of state trajectory around the sliding manifold in some sense. The controller in the event-triggered framework can achieve practical sliding mode by suitable design. The proposed control law is given as  u(t) = − C Az(ti ) + K sign(s(ti )) , ∀t ∈ [ti , ti+1 ), i ∈ Z≥0 ,

(5)

where K > 0 is the switching gain to be designed later and sign(s(ti )) is the signum function which is defined as ⎧ ⎨ 1 if r > 0, 0 if r = 0, sign(r ) = ⎩ −1 if r < 0 for any r ∈ R. The control law is implemented through a zero-order hold device by which the control signal is held constant until the next triggering instant. As soon as the new measurements are received, the updated control signal is applied to the plant again.

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We will discuss now some preliminaries which are essential for our technical developments. Applying the transformation  Wr =

I 0 F1



to the dynamics (2), we obtain the transformed dynamics z˙ 1 = S11 z 1 + S12 s s˙ = S21 z 1 + S22 s + u + d,

(6a) (6b)

  where z 1 s = Wr z, S11 = A11 − A12 F, S12 = A12 , S21 = F A11 − F A12 F + A21 − A22 F, and S22 = F A12 + A22 . The above dynamics is useful in the analysis because one can characterize the motion of sliding variable and its effect on rest of the components of the state vector. To do so, we introduce following sets for the trajectories of (6). Let P1 be a symmetric and positive definite matrix such that  P1 + P1 S11 = −Q 1 S11

(7)

n−1 : z 1 P1 z 1 for any given Q 1 = Q  1 > 0. Define the following sets: 1 := {z 1 ∈ R ≤ c1 } and 2 := {s ∈ R : |s| ≤ c2 } for some positive constants c1 and c2 . Then, construct the set   

z  = Wr−1 1 ∈ Rn : z 1 ∈ 1 , s = C z ∈ 2 . s

Here, the set  represents the region in the state space of system (3). The constants c1 and c2 can be chosen appropriately to cover any arbitrary large region in the state space. One way that is relevant to our work is discussed below. Lemma 1 Let c1 > 0 and c2 > 0 be any constants such that √ P1 A12 c2  c1 ≤ . 2 2 λm {Q 1 } λ M {P1 }

(8)

Assume that s(t) ∈ 2 for all t ≥ 0. Then, z 1 (t) ∈ 1 for all t ≥ 0. Proof Consider the Lyapunov function V1 (z 1 ) = z 1 P1 z 1 for the subsystem (6a). Taking derivative along the solutions of (6a), one can see that

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V˙1 (z 1 ) = z˙ 1 P1 z 1 + z 1 P1 z˙ 1  = z 1 (S11 P1 + P1 S11 )z 1 + 2z 1 P1 S12 s

= −z 1 Q 1 z 1 + 2z 1 P1 S12 s ≤ −λm {Q 1 }z 1 2 + 2z 1 P1 S12 |s|. Note that |s(t)| ≤ c2 for all t ≥ 0 by our assumption. Then, applying Young’s inequality the second term in the above inequality can be expressed as 2z 1 P1 S12 |s| ≤

λm {Q 1 } 2P1 S12 2 2 z 1 2 + c 2 λm {Q 1 } 2

and on substitution of this in the above relation yields λm {Q 1 } P1 S12  2 z 1 2 + 2 c V˙1 (z 1 ) ≤ − 2 λm {Q 1 } 2 λm {Q 1 } P1 S12 2 2 ≤− V1 (z 1 ) + 2 c , 2λ M {P1 } λm {Q 1 } 2 2

where we use the inequality V1 (z 1 ) ≤ λ M {P1 }z 1 2 in the last step. Finally, it follows from (8) that V˙1 (z 1 ) < 0

∀z 1 ∈ ∂1

which thus proves the claim.



The above lemma actually states some more information beyond the selection of constants c1 and c2 . The importance of the result will become clear when we analyze the stability for the closed-loop system. Toward this, let us fix the constants c1 and c2 satisfying (8) for rest of the discussions.

3.2 Periodic Triggering Mechanism The main feature of the triggering mechanism is that the triggering condition is evaluated periodically as the measurements become available. Due to the periodic evaluation, the satisfaction of event condition may occur at a time instant before it is actually detected. As a result, the triggering condition cannot be used directly in the stability analysis. To get more insight into it, we first present the periodic event-triggering mechanism which is as follows: t0 = 0,

  ti+1 = inf ti + j h : CAe(ti + j h) ≥ σ α, j ∈ Z≥0 ,

(9)

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where σ > 0 and α > 0 are triggering parameters; e(t) = z(ti ) − z(t) is the sampling error for t ∈ [ti , ti+1 ). The parameter α is actually a threshold parameter which can decide the frequency of its violation, whereas the constant σ ∈ (0, 1) tightens the triggering condition to compensate some unaccounted factors in the analysis. As the triggering condition is evaluated periodically, one may expect the violation of the event condition at any time before the triggering instant. That is to say that the event condition may be satisfied at any time between two consecutive periodic time instants. In this situation, the error bound can be larger than the threshold of the triggering mechanism. The natural question then arise here is that what is the actual error bound at the triggering instant due to the periodic evaluation of triggering condition. Following lemma gives an insight into this situation and provides a discussion on the calculation of error bound. For our use in the following lemma, we define for any positive scalars r and s, the function φ(r, s) :=

 ρ(r ) + β  sA e −1 , A

(10)

where ρ(r ) = A − BC Ar and β = B(K + d0 ). Lemma 2 Consider the system (1) and the triggering rule (9). Then, e(t) < ψ(z(ti ), h),

∀t ∈ [ti + ( j − 1)h, ti + j h)

(11)

where ψ(z(ti ), h) = φ(z(ti ), h) + (σ α/CA)ehA . Proof Let 1 = {t ∈ [ti , ti + ( j − 1)h) : e(t) = 0}. Then, for all time t ∈ [ti , ti + ( j − 1)h)\ 1 , one can easily arrive at     d  d  d    e(t) ≤  e(t) =  z(t)  dt dt dt   =  Az(t) − BC Az(ti ) − B K sign(s(ti )) + Bd(t) .

(12)

Substituting the relation z(t) = z(ti ) − e(t) in the above and simplifying further, it yields d e(t) ≤ Ae(t) + ρ(z(ti )) + β. dt

(13)

The solution to the differential inequality (13) can be obtained using Comparison lemma ([34]). By taking initial condition e(ti ) = 0, it can be shown that e(t) ≤ φ(z(ti ), t − ti ),

∀t ∈ [ti , ti + ( j − 1)h).

This shows that the error bound increases exponentially in the interval [ti , ti + ( j − 1)h). Moreover, if ti+1 > ti + ( j − 1)h, then the triggering condition cannot

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be violated in the interval [ti , ti + ( j − 1)h). Therefore, the actual error bound can be obtained by calculating the increase in the error bound from σ α/(CA) in the interval [ti + ( j − 1)h, ti + j h). This considers the fact that the event can be violated immediately after the time instant ti + ( j − 1)h before it is detected at ti + j h. On solving the differential inequality (13) with initial condition σ α/(CA) over the interval [ti + ( j − 1)h, ti+1 ), one gets e(t) ≤ φ(z(ti ), t − ti − ( j − 1)h) +

σα CA

e(t−ti −( j−1)h)A

(14)

for all time t ∈ [ti + ( j − 1)h, ti + j h). Since φ(z(ti ), t − ti − ( j − 1)h) is a monotonic function of t in the interval [ti + ( j − 1)h, ti + j h), the error bound can be given as e(t) < φ(z(ti ), h) +

σα CA

ehA .

(15)

This shows that e(t) is bounded by ψ(z(ti ), h) for all time t ∈ [ti + ( j − 1)h, ti + j h). Hence, the proof is completed.  Remark 1 The error bound given by above lemma is obtained by taking into account the sampling period of the triggering mechanism. Indeed, we see that this bound is larger than the threshold of the triggering condition. The important observation here is that the error bound reduces to that of triggering condition when h → 0. That is in the continuous triggering mechanism, the error bound becomes ψ(z(ti ), 0) =

σα CA

which is same as the bound given by triggering condition. This shows that Lemma 2 only consists of additional factors due to the sampling interval. From Lemma 2, we see that the error bound increases exponentially with time which is shown in Fig. 2. The error bound at the triggering instant can be larger than the bound given by the triggering condition (9). Therefore, to ensure the stability, one may proceed with the calculation of error bound given by this lemma. However, Fig. 2 The plot of e(t) versus time, t, in periodic event-triggering mechanism

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the interdependency of the switching gain and the sampling period does not allow the designer to know error bound a priori which can be seen in our discussion. The error bound given by (11) depends on K (through β) and sampling interval. However, K must be designed from the knowledge of error bound for stability. This interdependency is main obstacle in the design of the periodic event-triggering-based SMC. Fortunately, there is a way to the design SMC independent of the sampling period without sacrificing any performance in the closed-loop system. The sampling period h > 0 is designed for the triggering mechanism (9) by the relation   δσ α 1 , (16) ln 1 + h≤ C(ρ0 + β) A  where δ > 0 is a constant and ρ0 = A − BC A Wr−1 ((c1 /λm {P1 }) + c2 ). Our objective in the next subsection is to show that for any h satisfying (16) the switching gain can be designed independently.

3.3 Stability Analysis This subsection states the main result of this chapter. First, a decoupled design of the switching gain and the sampling period is discussed. Following this design technique the analysis for the closed-loop system is discussed with the event-triggered controller. Lemma 3 Let δ > 0 be a given constant. Then, for any K > d0 + (1 + δ)α,

(17)

there exists an h  > 0 such that (16) holds for all h ∈ (0, h  ]. Proof To begin with, let K = d0 + ( p + δ)α for any p > 1. Clearly, this choice of K satisfies (17). Then, 1 A

 ln 1 +

δσ α C(ρ0 + β0 + β1 ( p + δ))


0, the following steps may be followed to verify whether (16) holds or not: (i) Choose δ and K satisfying (17). (ii) Verify whether (16) holds for the given h. (iii) If (16) does not hold, then go to (i) and repeat (i)–(iii). The above procedure always ensures the condition (16) for any h <  because δ and K can be chosen arbitrarily. From the above discussion, we now fix the sampling period h and the constant δ for a given K as per Lemma 3 in the further developments. The stability for the given switching gain can then be discussed. Theorem 1 Consider the system (6), the control law (5) and the triggering mechanism (9). Let z(0) ∈ . Then, for any ε1 > 0 there exist α > 0, K > 0 satisfying (17) and τ1 > 0 such that z(t) ∈  for all t ≥ 0 and z(t) ≤ ε1 for all t ≥ τ1 . The proof of Theorem 1 requires an intermediate result that shows the existence of practical sliding mode to prove the boundedness of the state trajectory. Below, we establish that the control law (5) exhibits the practical sliding motion in the system. Then, we use this result in the main proof. Proposition 1 Consider the system (6), the control law (5), and the triggering mechanism (9). Let s(0) ∈ 2 . Then, there exist α > 0 and K > 0 satisfying (17) such that s(t) ∈ 2 for all t ≥ 0 and the practical sliding motion is enforced in the system. Proof For any given ε > 0, assume, without loss of generality, that ε < c2 . Then, choose triggering parameter as α
0, we assume that ε < min{c2 , ε1 /κ} where  κ = Wr−1 

√ P1 A12  1+2 2 λm {Q 1 }



 λ M {P1 } . λm {P1 }

Since there is no Zeno phenomenon associated with the triggering sequence, we can analyze the behavior of the state trajectory and sliding trajectory independently over time horizon. From Proposition 1, we conclude that s(t) ∈ 2 for all t ≥ 0. Then, Lemma 1 implies that z 1 (t) ∈ 1 for all t ≥ 0. As a result of this, z(t) ∈  for all t ≥ 0 which holds by our construction of the set . This proves the first part of the conclusion of theorem. As it is shown that for s(τ ) ∈ ε it holds s(t) ∈ ε for all t ≥ τ , we have |s(t)| ≤ ε for all t ≥ τ . Now, using this fact in the derivative of V1 (z 1 ) = z  P1 z 1 and simplifying further (cf. proof of Lemma 1), we see that 2

λm {Q 1 } P1 A12  2 z 1 2 + 2 ε V˙1 (z 1 ) ≤ − 2 λm {Q 1 } ≤−

λm {Q 1 } λm {Q 1 } P1 A12 2 2 V1 (z 1 ) − V1 (z 1 ) + 2 ε 4λ M {P1 } 4λ M {P1 } λm {Q 1 }

≤−

λm {Q 1 } V1 (z 1 ) 4λ M {P1 }

∀V1 (z 1 ) ≥

8P1 A12 2 λ M {P1 } 2 ε . λ2m {Q 1 }

This shows that the trajectory z 1 (t) starting within the set 1 enters into the set {z 1 : V1 (z 1 ) ≤ 8P1 A12 2 ε2 λ M {P1 }/λ2m {Q 1 }} in some finite time, say τ1 . Now, for all t ≥ τ1 , it follows that z(t) ≤ Wr−1 (z 1 (t) + |s(t)|)    √ A  λ M {P1 } P 1 12 + 1 ε = κε ≤ Wr−1  2 2 λm {Q 1 } λm {P1 } < ε1

which thus completes the proof.



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4 Simulation Results We consider a plant with the following matrices for simulation study: 

1 2 A= −1 0



  2 and B = , 1

and the disturbance input d(t) = 0.3 sin(2t) + 0.2. By the transformation W =  1 −2 , the plant can be written as 0 1  z˙ =

   3 8 0 z+ (u + d). −1 −2 1

Let us design different constant parameters first. We see for F = 0.5 that the matrix A11 − A12 F = −1 is Hurwitz. The sliding hyperplane parameter can be given by C = 0.5 1 . We see that P1 = 0.0417 and Q 1 = 0.1 satisfy the equation (7). Then, one can easily choose c1 = 20 and c2 = 4 to hold the inequality (8). Let ε1 = 3. We take ε = 0.394 which satisfies the inequality ε < min{c2 , ε1 /κ}. Finally, from (18), choose α = 0.4 for δ = 3 and K = 2.222. We also see that h = 0.001 satisfies the upper bound of sampling period given by (16). With all these parameters, we took   the initial condition z(0) = 3 1 ∈  for the simulation study. The plot of state trajectories and sliding variable are shown in Figs. 3 and 4, respectively. The control law ensures the practical sliding mode in the system and also the ultimate boundedness of the state trajectories. The control signal is plotted in Fig. 6. Finally, the variation of inter-event time over time is shown in Fig. 5. Since the

Fig. 3 The plot of state trajectories with time

4 3 2 1 0 -1 -2 -3

0

2

4

6

8

10

Design of Periodic Event-Triggered Sliding Mode Control Fig. 4 The plot of sliding variable with time

177

3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1

Fig. 5 Inter-event time versus time

0

2

4

6

8

10

6

8

10

0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

2

4

triggering instants are multiple of sampling period, the inter-event times are lower bounded from zero and also a multiple of sampling period.

5 Conclusion This chapter presented the design of periodic event-triggered SMC for a linear dynamical system. Since the switching gain of SMC depends on the triggering parameter in the event-triggered implementation, the design of SMC in periodic event-triggering becomes more complicated. We proposed a decoupled design of

178 Fig. 6 Event-triggered SMC versus time

A. K.Behera and B. Bandyopadhyay 8 6 4 2 0 -2 -4 -6 -8 0

2

4

6

8

10

SMC and the triggering parameter such that the stability in the closed-loop system is ensured. In this case, the controller gain depends on the triggering parameter and the additional parameter that accounts for the increase in the error due to sampling period. Simulation results demonstrate the performance of closed-loop system for the periodic event-triggered SMC.

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Sliding Modes in Consensus Control Massimo Zambelli and Antonella Ferrara

Abstract In this chapter, an overview of the possibility of applying Sliding Mode Control to Consensus Control problems is proposed. The theoretical concepts at the basis of Consensus Control are first introduced to form a common basis useful to understand the results currently available in the literature, which are here adapted when needed to achieve a uniform notation. Then, some works proposing the generation of Sliding Modes for the achievement of leaderless and leader-follower consensus are reported with the aim of giving to the reader a clearer idea of the available possibilities. Effort is made to highlight the benefits of adopting such an approach, which reside generally in robustness and finite-time convergence. Keywords Sliding mode control · Nonlinear control · Multi-agent systems · Consensus control

1 Introduction The study of networked Multi-Agent Systems (MAS) has become of primary importance in the control engineering context. Effective distributed control of agents accessing only local information can efficiently solve a huge amount of problems, often not possible to tackle with centralized approaches, while enabling for the employment of less expensive systems in an highly robust and flexible way [5]. Formation control [12, 20] of ground, aerial and submarine vehicles, microgrids control M. Zambelli (B) · A. Ferrara University of Pavia, 27100 Pavia, Italy e-mail: [email protected] URL: http://www.unipv.eu A. Ferrara e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_8

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[9, 76], flocking [85] and distributed estimation [93] represent just few examples of control engineering applications which naturally fit into the category of MAS control problems. It is well understood that the networked nature of the multi-agents systems can be effectively described by means of graphs (see Sect. 2.1), in which the vertices represent agents and the edges model the information exchange. The topology of the graph associated with the MAS represents then the interconnection between the involved agents and determines the characteristics of the resulting system as a whole. Such a topology can be fixed, for instance, when still sensors are placed in an area and communicate with nearby peers to perform data fusion, or change in time. The latter case is very common and includes all of the situations in which, for instance, the agents move and have limited communication or sensing ranges (thus, the information “exchange” can be interrupted and then possibly restored), or failures in the agents or the communication channel occur. The graph representation (which allows to analyze the behaviour of the whole MAS using results from Algebraic Graph Theory) is at the basis of the analysis carried out in the majority of the results presented in this chapter, and thus reported in Sect. 2 for the readers convenience. Consensus problems [5, 36, 60, 65] play a key role in MAS control, since they appear as the natural paradigm in which many control applications can be casted. In general, reaching consensus means finding an agreement on a value of interest depending on the agents states by means of properly designed protocols exploiting only local information. A basic example is constituted by the so-called leaderless rendezvous problem [14, 54] in which agents moving in space are required to meet at a certain common position, which in the most common circumstances happen to correspond to the average value of the initial states of the agents. Problems requiring to achieve a consensus on the (unknown) average value of the initial states are referred to as average consensus problems, and they are broadly studied in the literature. As a generalization, f -consensus problems are defined [60], which require the agents to attain together a certain continuous time-varying function by means of properly designed controllers. Strategies which lead to consensus states which are not continuous functions of the agents initial state values also exist (e.g. consensus to the median), and some specific cases will be presented in this chapter. The so-called leader-follower consensus (see, e.g. [29, 30]) is somehow a further generalization of the concept consisting in making the agents track the state value of a leader (possibly with prescribed shifts), and will be discussed in a different section with respect to classical consensus problems due to the huge amount of literature results derived specifically for the case. Intuitively, it can be seen as an f -consensus problem where one agent (the leader) has a dynamic behaviour and does not cooperate with the followers, i.e. it does not implement the designed consensus protocol [60]. The leader is usually independently controlled and not affected by the behaviour of the followers. Since, additionally, it can be either a real agent or a virtual one, this paradigm guarantees a great flexibility and forms the natural framework in which the majority of the applications which involve MAS control may naturally fall. It is additionally shown in some literature results (see, e.g. [35, 73]) that leader-follower

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configurations may lead to decrease in the necessary control energy and thus the overall control cost. Independently of the specific consensus problem to be tackled, robustness must be always considered as a primary concern, especially in real-world applications (see, among many other works, [45, 52, 53]). On the one hand, the robustness of the whole network with respect to nodes (agents) failures or deliberate non-cooperative behaviours must be considered. The latter includes cases in which the misfunctioning agent is oblivious (not exploiting the available information to modify the group behaviour) or acting specifically in an harmful way on the basis of its local interactions (for instance, it may send corrupted data to the neighbours to induce particular behaviours in them). On the other hand, the robustness of single agents to local perturbations and model uncertainties is of paramount importance since without it no protocol could actually work as expected. In particular, actuators faults constitute a broad class of problems requiring robustness and are considered, e.g. in [7, 75, 84, 96]. Another valuable feature in the solution of consensus problems is the finite-time convergence [8, 32, 34, 38, 51, 82, 83, 87], which is often seen as related to an increased robustness and the possibility of decoupling the control problems in case consensus reaching is only one of different simultaneous control objectives [5]. Sliding Mode Control (SMC) [11, 18, 77] constitutes a valuable strategy to adopt in implementing consensus control algorithms, due to its intrinsic robustness to matched uncertainty and the simplicity of implementation. Although it can be used in combination with more classical consensus control laws in order to increase robustness, in the last years also algorithms inherently based on SMC (or Higher-Order SMC, i.e. HOSMC, see [1, 2, 46–48]) have been successfully developed, which additionally often exhibit finite-time convergence features. Some of the main advantages in introducing sliding modes are, among possibly others, • The possibility of inherently considering nonlinear dynamics guarantees a very broad range of applicability. The classical consensus algorithms, as will be briefly described in the following, adopt linear control strategies, which are well suited only for linear systems. The consideration of complex and nonlinear dynamics leads often to more complex algorithms both in terms of implementation and analysis (refer, for instance, to the introductory section of [70] and the references therein for a nice and broad overview); • Robustness with respect to matched uncertainty in the local agents dynamics, under the mild assumption of boundedness. This feature is of great importance for practical applicability and constitutes one of the greatest advantages in adopting SMC-based techniques. Notice that the nonlinearities in the agents dynamics may in principle be considered as uncertain dynamics to be dealt with, as is common in the context of Sliding Mode Control. Additionally, actuators malfunctionings and nonlinearities can be regarded as matched disturbance and thus rejected under suitable conditions. In fact, differences with respect to the nominal control input u i[n] (t) are modelled as [65] u i (t) = ρi (t)u i[n] (t) + ri (t)

(1)

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for t ≥ t f , where t f is the time at which the malfunctioning starts to occur. Thus, one can rewrite (1) as u i (t) = u i[n] (t) + (ρi (t) − 1)u i[n] (t) + ri (t) = u i[n] (t) + φi (t)

(2)

where φi (t) is the usually unknown term lumping the malfunctioning, bounded for bounded ρi (t) and ri (t). Obviously, in (1) the term multiplying the nominal input must always be at least such that ρi (t) > 0, otherwise no effective control can be actually exerted on the system. • Finite-time convergence, which is almost a unique feature of discontinuous local interaction laws (see, e.g. [5]); • Lightweight implementation, which is suitable for online computation on lowpower local controllers; After a brief introduction to consensus control (given in 2.2), in this chapter, an introduction to the exploitation of Sliding Modes for consensus reaching is proposed, with the aim of giving the reader an overview of the offered possibilities and the currently available results. Particular effort is made in summarizing the situations in which each algorithm can be applied and the offered features. A selection of strategies based on the generation of sliding modes are considered for both leaderless consensus and leader-follower consensus problems, trying to cover a sufficiently wide variety of addressed problems, in Sections 3.1 and 3.2, respectively.

2 Consensus Control Let us now start introducing the concepts on which the whole subsequent discussion will be conducted, and represent, therefore, the basics needed for an easier comprehension of the presented results. In particular, in 2.1, basic concepts of Algebraic Graph Theory, which allow to represent the MAS by means of graphs and matrices, are introduced. The adoption of such a description is at the basis of every analysis developed in the results presented in this chapter. Then, in 2.2, a brief introduction to the consensus problems solvable by means of the reviewed strategies is proposed. An effort is made in reporting only the useful information (in terms of the results presented later), and, therefore, the discussion is far from exhaustive. The interested reader can refer to, among many other available works, [56, 60] and the references therein for a more thorough discussion.

2.1 Algebraic Graph Theory Let us consider a Multi-Agent System (MAS) composed of a finite number N of agents, indexed by i = 1, · · · , N . Each of them is associated to a state xi (t) ∈ Rn

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Fig. 1 A generic directed graph G composed of 7 nodes (agents of the MAS). Some agents may exchange information in a bidirectional way, while some others not. Notice, for instance, that in the represented graph a directed spanning tree is present with root in 0, meaning that the information about it can somehow “flow” from agent 0 to any other node. This characteristic is of paramount importance in leader-follower consensus problems, where a minimum requirement on the topology of the graph is that a spanning tree exists with the leader (agent 0) as the root

and can exchange information only with its neighbours, which are collected into the set Ni . The information flow can be represented in a graph G = {V, E} (see Figure 1), where the vertices in V = {1, · · · , N } correspond to the agents and the edges in E = {E i j }, E i j ∈ V × V are such that E i j ∈ E ⇐⇒ i ∈ N j

(3)

Definition 1 A graph G = {V, E} is said to be undirected if j ∈ Ni ⇐⇒ i ∈ N j

(4)

which means that the information flow among nodes is bidirectional. If (4) does not hold, the graph is said directed and called a digraph. For the sake of clarity, in the following of the discussion we will always explicitly refer to graphs as directed or undirected when this characteristic has a crucial importance. When it is not the case, we will employ the term graph or simply use the variable G to denote a generic directed graph, since undirected graphs can be intuitively seen as directed graphs in which every linked pair of nodes is so in both the directions (i.e. the connections are always bidirectional). Notice that this is different from the terminology adopted, e.g. in [56], where graphs are always assumed undirected unless otherwise specified, and directed graphs are obtained “discarding” links. Definition 2 A (directed) path of length m is a sequence of distinct vertices vi0 , · · · , vim

(5)

such that E ik ik+1 ∈ E, k = 1, · · · , m − 1. The vertices vk and vk+1 are then said to be adjacent.

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Definition 3 An undirected graph is connected if, for every pair of vertices in V, there exists a path having them as end vertices. If this is not the case, the graph is disconnected. A digraph is strongly connected if for every pair of vertices in V, there exists a path having them as end vertices, and is weakly connected if the undirected graph obtained neglecting the edges orientation is connected. Definition 4 If the vertices composing a (directed) path are all distinct except for the end vertices, the path is referred to as a (directed) cycle. A graph without cycles is called a forest. Definition 5 A cut in a graph is a partition of its nodes into two sets joined by at least one edge. A minimum cut is a cut in which the two sets are joined by the minimum number of edges. Definition 6 A graph is k-connected if its minimum cuts partition it into two sets joined by at least k edges. Definition 7 A graph G  = (V  , E  ) is said to be a subgraph of G if V  ⊆ V and E  ⊆ E. In particular, if V  = V, then G  is a spanning subgraph. From Definition 7, one has that a subgraph is spanning if it comprises all the nodes of the graph, independently of the particularly considered topology. Definition 8 A component of a graph is a subset of the graph associated with a minimal partitioning of the vertex set V such that a partition is connected. Definition 9 A forest with one component is called a tree. If the tree is also a subgraph of G, then it is a (directed) spanning tree. According to Definition 9, one can intuitively argue that a (directed) graph has a (directed) spanning tree if there exists at least one agent (called the root) which has a (directed) path to all the other nodes. Suppose that the nodes of the network can be divided into two groups Vsa f e and Vunsa f e (such that Vsa f e ∪ Vunsa f e = V). The following definition can then be introduced [24]. Definition 10 A directed graph G is k-safe is the subgraph Gsa f e ⊆ G constructed considering the nodes in Vsa f e is k-connected and all the nodes in Vunsa f e are only connected to nodes in Vsa f e . We are now ready to introduce a formal representation of graphs by means of matrices. First of all, let us introduce the following definitions. Definition 11 The (in)-degree di of vertex vi is the cardinality of its neighbourhood Ni , i.e. (6) di = |Ni |, ∀i = 1, · · · , N

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The degree matrix D of a graph is then defined as the N × N diagonal matrix with elements di , namely ⎤ ⎡ d1 · · · 0 ⎥ ⎢ (7) D = ⎣ ... . . . ... ⎦ 0 · · · dN

The adjacency matrix A is instead a matrix representing the adjacency relationships between vertices, i.e.  A = [ai j ], ai j =

1 if j ∈ Ni 0 otherwise

(8)

In (8), we adopt the convention that aii = 0, meaning that no self-loops are considered. Moreover, notice that (7) and (8) hold both for undirected and directed graphs. In the latter case, they hold under the assumption that the graph is unweighted and the notion of in-degree is adopted for di (see [56] for further details), as is the case in all the consensus control cases discussed in the present chapter. Remark 1 Despite weighted graphs are widely adopted in the literature for many different purposes, the SMC-based methods considered in this chapter resort to unweighted graphs, thus described by the definition of adjacency matrix given in (8). Therefore, a thorough discussion on weighted graphs is not provided here, but the interested reader can refer to, for instance, [19, 27, 40, 79]. Proposition 1 The adjacency matrix A of an undirected graph is symmetric. The graph laplacian matrix L(G) is now ready to be introduced, it being defined as L=D−A

(9)

The first noticeable feature of (9) is that the row-sum is always null. Therefore, the following proposition holds. Proposition 2 The laplacian matrix in (9) has at least a null eigenvalue, associated to the eigenvector 1 N , which corresponds to a column vector of ones with length N . This means that (10) L1 N = 0 N hold. Moreover, L 0 (i.e. the laplacian of a graph is always positive semidefinite). Definition 12 A graph is balanced if j=i

holds for every i and j.

ai j =

j=i

a ji

(11)

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Proposition 3 A graph is balanced if and only if 1 is a left eigenvector of its laplacian L. The topology of a network can be modelled as changing over time by means of a time-varying description of its underlying graph G = G(t). In this case, also the laplacian L = L(G(t)) varies over time. It is common in the works we will present in the following to model this variation as a sequence of switchings between a discrete number p of possible topologies exhibiting certain features, i.e. G(t) ∈ {G1 , · · · , G p }

(12)

Such sequence is often considered as determined by some time-varying function ζ (t) : R → {1, · · · , p} indexing the topology in place at time t.

2.2 Consensus Control Problems In this section, a brief introduction to classical consensus control algorithms is provided, since they form the theoretical foundations on which many of the successively presented works are developed. The presentation of the concepts is anyway far from exhaustive, and the interested reader is referred to papers as [5, 6, 20, 36, 59–61, 65, 71, 72] for a deeper understanding. Besides the distinction between consensus and leader-follower consensus problems (often considered as separate categories due to the different approaches usually adopted for their solution), broad distinctions are usually done basing on the order of the considered agents. In particular, many literature results (comprising the former ones) are developed for first-order agents dynamics. Besides the simple first-order dynamics, second-order agents are also subject of a vast amount of research. Anyway, less results are available due to the intrinsic difficulties that higher-order dynamics present (one above all, it is understood that consensus values may naturally exhibit time-varying behaviours, see, e.g. [5]). Agents with orders greater than the second have also attracted attention, but less works are currently available [74, 91]. Another distinction which intuitively arise is between linear agents systems (often in form of chain of integrators) [60] and nonlinear ones [33, 62]. While the former class can be usually treated well with linear control laws, the latter requires more involved algorithms. It is valuable to underline here, as already done also in the introduction, how sliding mode control is inherently a nonlinear control approach, which is then well suited for dealing with complex dynamics by means of fairly simple laws. Notice that works on heterogeneous agents, i.e. exhibiting different dynamics, are also present [15, 95]. In this case, it is common to resort to outputconsensus strategies, which enforce then consensus on the output values instead of the states. Reaching a consensus between N agents with states xi (t) ∈ Rn means, formally, to achieve the condition

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x i (t) − x j (t) = 0

(13)

where x i (t) = ξ (t), ∀i, for ξ (t) some vector function. The concept of practical consensus is also often resorted to, which reads as

x i (t) − x j (t) ≤ δ, ∀i = j

(14)

for some small δ ≥ 0. For leader-follower consensus, the general expression for condition (13) becomes instead (15) x ik (t) − x 0k (t) = ik (t), k = 1, · · · , n where i = 0 is chosen for the (unique) leader without loss of generality and (t) is a given function representing the reference relationship (shift) between the states of each follower and those of the leader. For ik = 0 and x 0 (t) = ξ (t), it is easy to see that condition (15) degenerates into (13), corresponding to f -consensus. A simple algorithm which allows to reach average consensus among first-order integrators (thus, with n = 1), i.e. such that x˙i (t) = u i (t)

(16)

where x ∈ R, is based on the following linear law [61] u i (t) =



x j (t) − xi (t)



(17)

j∈Ni

which leads to the following collective dynamics x˙ (t) = −Lx(t)

(18)

For an undirected and connected network, it can be shown that the simple protocol (17) enforces a global exponentially stable convergence of the collective dynamics (18) to the constant equilibrium state ξ (t) = x¯ (t) =

i

xi (0) 1 N

(19)

meaning that all the agents states converge to the average value of their initial states. If the protocol (17) is applied to strongly connected undirected networks (see Def. 3), then an f -consensus problem is asymptotically solved for x¯ (t) =

γ T x(0) γT1

(20)

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where γ is a left-eigenvalue of L (i.e. γ T L = 0). Due to Proposition 3, one has that if the digraph is also balanced (see Def. 12), then (20) degenerates into (19) and thus average consensus is reached. Actually, law (17) is much more powerful than that since it can be proved to enforce consensus also in the case of time-varying topologies under the assumption that the underlying digraph has a directed spanning tree in terms of a union of timevarying topologies (see [5, 59, 71] for the details). In order to exploit the concepts just reported to model leader-follower consensus problems, let us introduce without loss of generality the leader as the agent 0. Then, since the behaviour of the leader is not affected by that of the followers, one has that the neighbourhood of the leader is always empty, i.e. N0 = ∅ and, therefore, d0 = 0 as per (6). Also, a0 j = 0, ∀ j, while a j0 = b j is not null for all the agents j which are actually somehow acquiring information from the leader. The consequent representation may follow then that in (7), (8) and (9) only adding accordingly one row and one column to the matrices A and D, which become then (N + 1) × (N + 1) square matrices. The same definitions and results stated above hold obviously also in this case, highlighting the generality of the adopted framework. Nevertheless, it is also common to adopt a different approach in modelling the presence of an uncontrolled leader, which gives more flexibility in proving some results. A pinning matrix is introduced besides the laplacian (this time corresponding exactly to that in (9), considering only the followers), i.e. ⎤ b1 · · · 0 ⎥ ⎢ B = ⎣ ... . . . ... ⎦ 0 · · · bN ⎡

(21)

where for the adopted convention bi = 1 if agent i has the leader as neighbour and bi = 0 otherwise. The following result is useful in proving many results presented in (3.2) regarding leader-follower finite-time consensus. Theorem 1 ([43]). If the (undirected) graph G  obtained considering the leader and the followers has a spanning tree, then the square matrix L + B is invertible, where L is the laplacian of the graph G generated considering only the followers and B is the network pinning matrix.

3 Sliding Modes in Consensus Control In this section, a selection of effective results proposing the introduction of sliding mode control for consensus reaching purposes is discussed to give a broad overview of the currently available possibilities. As already stated, two of the most important features of SMC-based consensus protocols are the finite-time convergence and the complete robustness with respect to bounded matched uncertainty.

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Finite-time consensus reaching can be stated on the basis of (13) as

xi (t) − x j (t) = 0, t ≥ Tr

(22)

where Tr > 0 is the time at which convergence occurs. Respectively, a similar definition can be easily formulated for conditions (14) and (15). When Tr can be determined somehow independently of the agents initial states, usually by means of a proper selection of the controller parameters, we will refer to the resulting situation as fixed-time consensus. Due to the rejection property of SMC-based algorithms, the agents dynamics usually considered in the following works comprise an unknown bounded term. For instance, for first-order systems, the resulting description is x˙i (t) = u i (t) + φi (t), |φi (t)| ≤ 

(23)

When HOSMs are generated for chattering alleviation purposes, also the successive time derivatives of the uncertainty are usually required to be bounded. For instance, if second-order sliding modes are generated to get a continuous control for first-order ˙ ˆ is also assumed. This consideration can easily be agents, the condition |φ(t)| ≤ extended to any sliding order. Some protocols exploit the uncertainty rejection abilities of sliding mode control to robustify nominal consensus reaching laws by introducing into the control signal a corrective term. This approach may be adopted in general control applications, and can be referred to the very first purpose of Integral Sliding Mode Control (ISMC) [80], later extended to HOSM in [50]. With ISMC, bounded matched uncertainties are compensated through a suitable selection of the sliding manifold, which is a necessary requisite to perform such a robustification. The general idea may consist in generating a control signal u i (t) = u i[n] (t) + u i[d] (σi (t), t)

(24)

where u i[n] (t) is a nominal control input designed considering the ideal (nonperturbed) dynamics and u i[d] (σi (t), t) is an injected correction term dependent on a properly designed local sliding variable σi (t), with parameters usually chosen such that σi (0) = 0. The latter condition allows to achieve sliding (and thus uncertainty rejection) from the very first time instant, prescribing the corresponding dynamics to the closed-loop system. In consensus control, a discontinuous corrective term is employed in works as [64] for perturbed first-order systems (see (23)) in networks with undirected and connected topologies, with the nominal control (17). The resulting law reads as u i (t) = −α



 xi (t) − x j (t) − βsign (xi (t) + z i (t))

j∈Ni

z˙ i (t) = α



j∈Ni

 xi (t) − x j (t) ,

z i (0) = −xi (0)

(25)

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with α > 0 and β > . Notice that the chosen sliding variable, σi (t) = xi (t) + z i (t), is null at time t = 0 thus enforcing the sliding motion from the first time instant. For the subsequent time, keeping σi (t) = 0 corresponds in maintaining x˙i (t) = −z i (t), where −z i (t) corresponds to the behaviour the agent would follow if no disturbances were present. Therefore, robustness is enforced in a straightforward but powerful way. In the same work, the perturbation-estimator-based chattering alleviation method [80] is also applied and proved to be effective in enforcing average consensus reaching. An Integral Terminal Sliding Mode design is adopted in [89] for the robust finite-time consensus reaching under undirected information flow topologies, where two protocols are developed. The first one relies on a discontinuous corrective term, as most of the mentioned literature works, while in the second one the injected correction is continuous. Robustified protocols are insensitive to matched uncertainty affecting the single agents’ dynamics, but retain the convergence properties of the nominal control laws. Therefore, since usually such strategies enforce asymptotic (sometimes exponential) convergence, the same behaviour is exhibited by the robustified algorithms. In order to be able to guarantee finite-time convergence, a different approach must be adopted which relies on particular designs of the sliding manifold and the discontinuous control.

3.1 SMC for Leaderless Consensus Reaching The generation of sliding modes for the robust finite-time reaching of consensus has attracted increasing attention over the last few years (see, e.g. [63]). Works as [55] design the sliding surface in a way such that finite-time average consensus is reached in finite time. In particular, the proposal in [55] assumes that a master monitor exists which communicates with all the nodes and thus is able to compute the average consensus value. The single agents, then, are guaranteed to converge to it exponentially with speed determined by design parameters. Less strict assumptions are made in [26], where the requirement of having a directed spanning tree in the graph is made. In [67], instead, the authors propose a finite-time consensus reaching strategy for first-order agents in case of undirected and connected topologies. The results are extended in [69], where arbitrary-order agents are considered with the additional assumption that the network topology is balanced. The same authors propose an algorithm for leaderless consensus on altitude and heading in swarms of Unmanned Aerial Vehicles (UAVs) [68]. Second-order nonlinear agents dynamics are considered in [90] in undirected graphs. In this work, a full-order sliding surface design based on homogeneity theory is proposed to enforce consensus in finite time while producing a continuous control input. The problem of having a prescribed convergence time independent of the initial agents’ states in average consensus is considered, for instance, in [98] for first-order systems in undirected and connected networks.

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In [39], an algorithm to achieve consensus among agents with linear dynamics in the presence of actuators and communication faults is proposed. In particular, the proposal aims at achieving asymptotic average consensus with any given level of L2 gain attenuation. An adaptation mechanism is proposed to take into account the uncertain effects of the malfunctioning and an ISM-based design is adopted for the sliding manifold. The proposal is then adapted to track also arbitrary values different from the average of the initial states ( f -consensus). Actuator faults are addressed explicitly also in works as [75]. There, LTI agents in undirected strongly connected networks (which may switch among different strongly connected configurations) are considered in the presence of actuators loss of effectiveness and saturation faults. A weighting mechanism is proposed which enables to consider differently the contribution of healthy and malfunctioning agents. Specifically for f -consensus, many results exist reaching an agreement in a finite time through discontinuous control laws. Among them, [38] considers first-order agents in both fixed and switching undirected topologies for average consensus with respect to a monotonic function g(t), i.e.



N

1

g(xi (t)) − g(x j (0))

= 0

N j=1

(26)

The results are then extended for asymmetric networks, under the hypothesis that the detailed balance condition (see [38, Definition 3]) holds. In that case, weightedaverage consensus is reached in a finite time. In [25], finite-time average consensus is attained for directed and possibly switching networks, under the assumption that the underlying graph is connected for at least a certain fraction of a certain time interval. The length of the interval and the percentage of time in which connectedness is in place, together with the controller parameters, determine the speed of convergence. Undirected and switching topologies are considered also in [10], where under the assumption of connectedness of the network graph fixed-time convergence is reached. The latter work proved to be effective in the presence of disturbances both depending on the agents states (local quantities) and the entire network state (global quantity). One of the main drawbacks of average consensus is its sensitivity to outliers, i.e. agents having extreme state values (for various reasons, deliberate or incidental) which thus disrupt the ability of the network to compute a realistic mean. To overcome this issue, the median (which is far more robust to outlier values) is considered in [23] and [24]. In particular, the first work deals with undirected and connected networks and proposes a distributed law to achieve finite-time median consensus. The latter contribution goes far beyond, addressing specifically the cases in which agents deliberately avoid to cooperate (following thus arbitrary trajectories). It is shown that the proposed algorithm allows to achieve consensus in finite time, robustly with respect to uncooperative agents in case of k-safe networks (see Definition 10). Table 1 summarizes the discussed works dealing with leaderless consensus.

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Table 1 Results inherently based on SMC, ordered by publication year. The acronym ST is used to indicate the requirement for a spanning tree (directed if the topology is allowed to be so). The possibility of dealing with Switching topologies includes obviously also as a special case fixed ones Agents Topology Continuous Additional dyn. order input notes [38]

1

Undirected, Switching

No

[55] [67] [69]

1 1 Any

No No No

[25] [39] [26] [23] [75]

1 Any 1 1 Any

[10]

1

[98] [90]

1 2

Undirected Undirected, Connected Undirected, Connected, Balanced Directed, Switching Undirected Directed, ST Directed, Connected Directed, Connected, Switching Undirected, Switching, Connected Undirected, Connected Undirected

No No No No No

Also asymmetric topologies, under detailed balance condition assumption Needs a master monitor

Periodic connectedness LTI agents dynamics Median value consensus LTI agents dynamics

No

No Yes

3.2 SMC in Leader-Follower Consensus Sliding mode control is employed in many different literature results addressing the problem of robustly reaching a consensus with respect to the states of a dynamic leader in a finite time. Usually, the adopted approaches aim at designing a timevarying sliding manifold which can be intuitively linked to those employed for leaderless consensus in the case of agents not directly gaining information from the leader, and comprising the information about the leader otherwise. Due to this latter fact, the dynamics of the leader often determine the evolution of the sliding manifold, which must somehow be dominated by the adopted control strategy. Therefore, it is a common assumption the leader has a bounded input, i.e. |u 0 (t)| ≤ U0

(27)

for some positive known U0 , in addition to possibly being affected by bounded uncertainty. Some strategies include adaptation mechanisms which allow to drop the hypothesis of having known bounds for the leader input and the uncertainty in general, which are then only assumed to be bounded.

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A fairly broad work on leader-follower consensus reaching via sliding mode control is reported in [4], which also proposes algorithms for swarm tracking. Here, both first (with fixed-time convergence) and second-order agents (with guaranteed global exponential convergence) are considered in both fixed and switching topologies with directed and connected graphs. In particular, for first-order systems (and a fixed topology) as (16) the protocol u i (t) = −α



(xi (t) − x j (t)) − βsign(σi (t))

(28)

j∈Ni

with σi (t) =



(xi (t) − x j (t))

(29)

j∈Ni

is proposed, for some α ≥ 0 and β > 0. In (28), the considered graph comprises both the followers and the leader, and so do the sets Ni . For the second-order case, the protocol

 x i1 − αx j1 − βsign(σi (t)) u i (t) = − (30) j∈Ni

with σi (t) =



 γ x i2 − x j2

(31)

j∈Ni

and α > 0, β > 0, γ > 0. Another key contributions to the literature of leader-follower consensus reaching through SMC is [43], where second-order agents are considered in undirected topologies including a spanning tree. In this work, Terminal Sliding Mode (TSM) control [21, 81, 88] is exploited, which is suitable to enforce finite-time consensus. The design of the local sliding variables is based on the selection of the local errors ei (t) =





 x 1,i (t) + i − x 1, j (t) −  j + bi (t) x 1,i (t) + i − x 1,0 (t) (32)

j∈Ni

where k , k = 1, · · · , N is the required distance between agent k and the leader. The corresponding sliding variable for the i-th agent is chosen as σi (t) = ei (t) + (e˙i (t))α , 0 < α < 1

(33)

according to the TSM concept. A similar approach is adopted also in [41], where an Integral Terminal Sliding Mode Control is adopted for first-order agents under directed topologies including a spanning tree. The multiple surface sliding mode approach [31] is exploited in [44] to extend the previous results to higher-order agents dynamics describable by means of perturbed chains of integrators.

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Notice that the adoption of relationship (32) as well as the exploitation of Theorem 1 are central in many works presented in this section. A nonsingular TSM formulation is employed also in the strategy proposed in [94], which deals with second-order agent dynamics in directed networks containing a spanning tree. Inspired by the considerations in [43], a different approach is considered in [22], where Higher-Order Integral Sliding Modes (HOSM) are employed. In this work, both first and second-order agents affected by matched uncertainty are considered. In the former case, chattering alleviation is performed so as to generate a continuous control action. Differently from the previous work, the errors (32) are directly considered as the local sliding variables and thus made vanish in a finite time. Two features characterize the approach, namely the utilization of the Sub-Optimal SOSM algorithm [1], which only requires the knowledge of the first states in the agents dynamics to enforce second-order sliding modes, and the introduction of an Integral Higher-Order Sliding Mode [49, 50] design. In particular, the sliding variable is chosen as (34) σi (t) = ei (t) + γi (t) where γi (t) is chosen so that σi (0) = σ˙ i (0) = 0 and γi (t) = 0 t ≥ t f,i where t f,i is the time instant at which agent i is required to enforce ei (t) = 0. The choice of such time parametrizes the function γi (t), whose shape has in turn effects, for instance, on the required control magnitude. Particular behaviours of the agents can be required through the design of the transient function, for instance, to consider explicitly the agents input saturations. The leader-follower consensus is robustly attained at time T f = maxi t f,i with the desired inter-agent spaces i as in [43], thus guaranteeing prescribed-time convergence independently of the initial states. The idea of having a prescribed convergence time independent of the initial states of the agents is addressed also in [13], where agents with some nonlinear uncertain dynamics are considered in connected networks. The proposed nonlinear law comprises linear, nonlinear and discontinuous terms with gains which determine the final convergence time T f . Fixed-time consensus tracking is addressed also in [97] for second-order agents in undirected networks including a spanning tree. The authors of [70] propose two algorithms for agents with second-order Lipschitz dynamics in directed networks including a spanning tree. In particular, both an asymptotic and a finite-time convergence algorithms are proposed basing somehow on the error in (32) and the latter relies on a TSM design. Networks of heterogeneous agents with higher-order dynamics are considered in [58], which addresses the problem of finite-time consensus reaching in the presence of both matched and unmatched uncertainty. A Finite-Time Disturbance Observer (FTDO) with time-varying gain is employed to estimate the uncertainty and the control law, designed according to the super-twisting algorithm, generates a continuous input signal thus leading to chattering alleviation features. The problem of chattering alleviation is considered in [17], where second-order sliding modes are generated (through the Super-Twisting algorithm) to ensure a continuous control for first-order agents. The network topology is only required to be such that the graph laplacian (9) has a single zero eigenvalue, which is known to be

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the case for graphs containing at least a spanning tree or being connected. In the same work, conditions are given so that the convergence of the followers to the leaders state occur simultaneously. In particular, it is evidenced that a sufficient additional condition is that the subgraph including only the followers is strongly connected (see Definition 3). In [57], agents with arbitrary order dynamics are considered and consensus is attained in finite time by means of full-order HOSM generation. A mechanism for chattering alleviation is proposed, for instance, also in [66], where an adaptation algorithm for the control gain is presented. In this work, second-order nonlinear agents dynamics are considered and possible actuators faults are explicitly considered. One of the problems which appears intuitive to try to solve through leader-follower consensus is the so-called platoon control. In this particular unidimensional formation control problem, the agents are required to proceed keeping a certain distance, specified by the adopted spacing policy, one from the successive (see, e.g. [3, 16, 37, 78, 92] and references therein). Sliding modes in the reaching of such consensus are exploited, for instance, in [86], where undirected topologies are considered including a spanning tree. The design of the controller exploits multiple surface control and is specifically referred to as vehicular systems. Table 2 summarizes the discussed results in leader-follower consensus reaching.

Table 2 Results inherently based on SMC, ordered by publication year. The acronym ST is used to indicate the requirement for a spanning tree (directed if the topology is allowed to be so). The possibility of dealing with switching topologies includes obviously also as a special case fixed ones Agents Topology Continuous Additional dyn. order input notes [42] [43] [4]

1 2 1,2 1 Any 2 1

Undirected, ST Undirected, ST Undirected, Connected Undirected, ST Undirected, ST Directed, ST Connected

[17] [28] [94] [13]

No No No Yes No No No

[97]

2

Directed, ST

No

[70]

2

Undirected, ST

No

[58]

Any

Yes

[57] [66] [22]

Any 2 1,2

Undirected, Connected Undirected, ST Undirected, ST Undirected, ST

Yes No Yes, No

Underactuated agents Fixed-time convergence Fixed-time convergence 2 algorithms: asymptotic and finite-time convergence Heterogeneous dynamics, Unmatched uncertainty

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On Fixed-Time Convergent Sliding Mode Control Design and Applications Jyoti Mishra and Xinghuo Yu

Abstract This chapter discusses an algorithm to provide arbitrary-order fixed-time convergent SMC design. Through numerical simulation and comparative study, it is shown that in the proposed algorithm, the convergence speed does not depend on the initial condition. Moreover, it is also evident from the simulations that a higher control effort will be required by achieving so and is discussed thoroughly in the chapter, leading to a trade-off between the control effort and convergence speed. Subsequently, a novel distributed algorithm is developed for achieving second-order consensus in the multi-agent systems by designing a full-order fixed-time convergent sliding surface as an application to the proposed algorithm with suitable numerical simulations.

1 Introduction Physical plants are often affected by various disturbances and uncertainties. The performance of linear controllers deteriorates in the presence of such anomalies and there is always a need for designing robust controllers for such plants, where the nature of disturbance is not actually known. Sliding Mode Control (SMC) is a nonlinear robust control strategy, where the control input is actually discontinuous [1–3]. This, in fact, is a demerit in many mechanical plants , where this discontinuity leads to high-frequency switching that is undesirable [4]. Therefore, recent research in the SMC community is dealt with designing continuous robust control strategies [5]. Higher Order Sliding Mode Control (HOSMC) is one such solution. However, designing a continuous control structure for higher order systems is complicated [6, 7]. There are only a few continuous control strategies reported till date, which can J. Mishra (B) · X. Yu School of Engineering, RMIT University, Melbourne 3000, Australia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_9

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provide stability to any arbitrary-order system [8–11]. These methods often become too complex as the system order increases. Some practical applications need strict constraints on time response due to security reasons or to ameliorate the productiveness. For example, a missile or any aerial launch vehicle can be hugely affected by a strong wind gust deviating it from the desired trajectory yielding a large amount of initial tracking error [12]. It is worth mentioning that the convergence achieved in SMC during sliding can be either asymptotic or in finite time [13], depending on the selection of the surface. Furthermore, it mainly depends on the initial conditions of the states. It is definitely true that by increasing the control gain in the existing controllers, the speed of convergence can be increased. However, even if the convergence speed is increased, it is still dependent on the initial conditions. Moreover, most of the controllers have limited bandwidth. This motivated the researchers to focus on developing such controllers where the convergence time does not depend on initial conditions and a well-defined theoretical analysis is present in the literature about the so-called fixed-time convergence [14–16]. Fixed-time convergence is usually obtained by a class of polynomial state-feedback control systems and such control systems can find its application to many chemical processes, electronic circuits, mechatronics and biological systems [14]. However, there is yet no such continuous finite-time control structure providing fixed-time convergence in the open literature for higher order systems. In [17], a chattering-free full-order strategy is reported, which is quite simple to practically implement in any arbitrary-order plant. However, it provides finite-time convergence, i.e. the rate of convergence still depends on the initial conditions. The main focus of this chapter is to develop a fixed-time convergence strategy. This chapter mainly focuses on the theoretical development of a new finite-time Controller, which can definitely be able to provide fixed-time convergence by incorporating some additional non-linear terms in the control design. First, a continuous control law is developed providing fixed-time convergence for any arbitrary-order system with equivalent control cost. The proof of the fixed-time convergence for the proposed control scheme is provided in the chapter through Lyapunov analysis. Second, a theoretical convergence time is calculated for the proposed algorithm and its effectiveness over the existing state-of-the-art methods is verified through numerically simulating a practical example. Finally, a novel decoupled distributed continuous SMC providing fixed-time second-order consensus for multiagent systems. The important feature of the proposed method is that it can be generalised to arbitrary-order agent dynamics. An interesting fact about the proposed formulation is that the states of the agents are coupled on the sliding surface. However, the sliding mode states are decoupled out of this surface, which makes the design of SMC independent of the extent of coupling between the agents. The proposed surface will provide fixed-time consensus, i.e. independency to the initial condition. The robustness of the proposed controller is demonstrated in the presence of Lipschitz disturbance in agent dynamics and uncertainties in the network structure.

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2 Typical Types of Convergence When talking about convergence speed, it is obvious that the system should be dynamic. Any dynamic system can be represented by a single or a series of differential equations. It is important to understand the nature of the convergence from the solution of differential equations. Therefore, let us analyse some differential equations.

2.1 Asymptotic Convergence Consider an autonomous system of the form x(t) ˙ = −ax, a > 0

(1)

where x(t) ∈  and a ∈ . The solution for the above linear differential equation can be written as (2) x(t) = x(0)e−at where x(0) is the initial condition of x(t). It can be clearly seen from Eq. (2) that the convergence of x(t) to the origin, i.e. x(t) → 0 is obtained when t → ∞, and is popularly known as asymptotic convergence. The origin is considered to be the equilibrium point. Definition 1 A state x ∗ is the equilibrium state (or equilibrium point) of the system iff once x(t) equals to x ∗ , then it remains equals to x ∗ for all future time [18].

2.1.1

Graphical Analysis

It can be clearly noticed from Fig. 1 that the convergence rate depends on the initial condition x(0) and it is indeed asymptotic.

2.2 Finite-Time Convergence Let us now discuss a non-linear differential equation of form p

x(t) ˙ = −ax q , q > p > 0

(3)

where x(t) ∈ , a ∈  and p, q ∈ Nodd . After solving the above differential equation Eq. (3), the convergence time can be written as

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50 x(0)=0.5 x(0)=10 x(0)=20 x(0)=50

3 40 2

x(t)

30 1

20

0

10 0

0

4

1

4.5

2

3

5

4

5

t (sec.) Fig. 1 Asymptotic convergence for a = 1

p

Tconv

x(0)1− q = a(1 − qp )

(4)

It can be clearly noticed from Eq. (4) that the convergence time obtained is finite and it depends on the initial condition x(0). Definition 2 σ˙ (t) = f (σ (t)), σ (0) = σ0

(5)

where σ ∈ Rn and f : Rn → Rn is a non-linear function such that f (0) = 0, that is, the origin σ = 0 is an equilibrium point (5). The equilibrium point of system (5) is globally finite-time stable if it is globally asymptotically stable and any solution σ (t, σ0 ) of system (5) reaches the equilibrium point at some finite-time, i.e. ∀ ≥ T (σ0 ) : σ (t, σ0 ) = 0, where T : Rn → R+ ∪ {0}.

2.2.1

Graphical Analysis

It can be clearly seen from Fig. 2 that the convergence time in this case is finite with higher accuracy. Moreover, the convergence time still depends on the initial condition x(0).

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50 0

40

x(t)

30

x(0)=10 x(0)=0.5 x(0)=20 x(0)=50

-1

20 -2 4.2

10

4.4

4.6

4.8

0 -10

0

1

2

3

4

5

t (sec.) Fig. 2 Finite-time convergence for p = 1, q = 3 and a = 5

2.3 Fixed-Time Convergence Now, consider another non-linear differential equation of the form x˙ = −αx ξ − βx η , ξ > 1 > η ≥ 0

(6)

the settling time T (x0 ) is being bounded by a hypergeometric function. To prove the above claim, let us solve the above differential equation: |x0 | T (x0 ) = 0

1 d x. αx ξ + βx η

(7)

Let x = |x0 |z, then d x = |x0 |dz 1 T (x0 ) = 0

|x0 | dz α|x0 |ξ z ξ + β|x0 |η z η

|x0 | = β|x0 |η

1 0

1 1 dz η −1 ξ −η z αβ |x0 | z ξ −η + 1

|x0 | = (1 − η)β|x0 |η

1 0

1 dz 1−η . αβ −1 |x0 |ξ −η z ξ −η + 1

(8)

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Let us denote z ξ −η = t and rewrite Eq. (8) |x0 | T (x0 ) = (1 − η)β|x0 |η |x0 | = (1 − η)β|x0 |η |x0 | = (ξ − η)β|x0 |η T (x0 ) =

1

1 1+

0

1 1 0

1 0

1−η

αβ −1 |x

0

|ξ −η t

dt ξ −η

1−η ξ1−η t −η −1 ξ −η dt + αβ −1 |x0 |ξ −η t 1−η

t ξ −η −1 dt 1 + αβ −1 |x0 |ξ −η t

1−η 1−η |x0 |1−η (1, ;1 + ; −αβ −1 |x0 |ξ −η ) (ξ − η)β ξ −η ξ −η

(9)

where (.) is the hypergeometric function defined in [19]. To find an even more precise result, let us define a Lyapunov function for the above system (6) as V = x 2. (10) Taking the first derivative of Eq. (10) V˙ = 2x(−αx ξ − βx η ) = −2αx ξ +1 − 2βx η+1 = −2αV

ξ +1 2

= −2(αV 1

− 2βV

η+1 ξ +1 2 − 2

η+1 2

+ β)V

V˙ = −2(αV

η+1 2

η+1 ξ +1 2 − 2

+ β) V η+1 ξ +1 1 d 1−η V 2 = −(αV 2 − 2 + β). 1 − η dt η+1 2

Let us define V

1−η 2

= s. Hence, the above equation can be rewritten as ds αs 1+

+β

= −(1 − η)dt

(11)

ξ −1 where  = 1−η and the upper bound of the convergence time Tmax can be found out such that s = 0

On Fixed-Time Convergent Sliding Mode Control Design and Applications

Tmax

⎛ 1 ⎞  ∞ ds ds 1 ⎝ ⎠ = + 1−η αs 1+ + β αs 1+ + β 0 1 ⎞ ⎛ 1  ∞ ds ⎠ 1 ⎝ ds + ≤ 1−η β αs 1+ 0 1   1 1 1 + = 1−η β α 1 1 + = β(1 − η) α(ξ − 1)

209

(12)

Definition 3 σ˙ (t) = f (σ (t)), σ (0) = σ0

(13)

where σ ∈ Rn and f : Rn → Rn is a non-linear function such that f (0) = 0, that is, the origin σ = 0 is an equilibrium point (13). The equilibrium point of system (13) is fixed-time stable if it is globally finite-time stable and the settling time is bounded, i.e.∃Tmax > 0 : ∀σ0 ∈ Rn and T (σ0 ) ≤ Tmax .

2.3.1

Graphical Analysis

It can be clearly seen from Fig. 3 that the convergence time in this case is indeed finite with maximum accuracy. However, the convergence time does not depend on the initial condition x(0) and is always bounded. Hence, it can be concluded that all fixed-time convergent strategies are in fact finite-time convergent with the addition of some extra non-linear terms.

3 Continuous Finite-Time Sliding Mode Controller with Fixed-Time Convergence This section mainly focuses on proposing a fixed-time convergent arbitrary-order SMC. The following standard definitions are used throughout this particular section. Definition 4 The sign is the signum function and can be defined as ⎧ ⎫ f or x > 0 ⎬ ⎨ 1, sign (x) = [1, −1] , f or x = 0 ⎩ ⎭ −1, f or x < 0

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J. Mishra and X. Yu

50

0.5

40 0

x(t)

30

x(0)=0.5 x(0)=10 x(0)=20 x(0)=50

-0.5

20 10

3.5

4

4.5

5

4

5

0 -10

0

1

2

3

t (sec.) Fig. 3 Fixed-time convergence for ξ = 3, η =

1 3

and α = 1, β = 5

Definition 5 A vector field X = (x1 , ..., xn )T ∈ n is said to be homogenous iff for any λ > 0, ri > 0, i ∈ 1, ..., n , with the vector of weights r = (r1 , ..., rn )T n and the dilation matrix D(λ) = diag{λri }i=1 , where λ ∈ + , such that D(λ)x = r1 ri rn T (λ x1 , ..., λ xi , ..., λ xn ) . Definition 6 A function f : n →  is said to be r-homogeneous of degree m iff for all λ > 0 and for all x ∈ n we have g(D(λ)x) = λm g(x). The proposed control structure is smooth and free from high-frequency switching. This makes it possible to implement it practically to any physical plant. The algorithm is also generalized to any arbitrary-order system. The control structure is free from any singularity. This is the first known reported full-order finite-time continuous control for an arbitrary-order system providing fixed-time convergence. In the following, the corresponding problem formulation along with the stability proof for the proposed fixed-time converging control structure is provided.

3.1 Problem Formulation Consider an arbitrary-order non-linear system in terms of chain of integrators form:

On Fixed-Time Convergent Sliding Mode Control Design and Applications

⎫ x˙1 = x2 ⎪ ⎪ ⎪ ⎬ x˙2 = x3 ⎪ ..⎪ .⎪ ⎪ ⎪ ⎭ x˙n = f (x, t) + b(x, t)u + d(t)

211

(14)

where x1 , x2 , · · · , xn are the individual states and f, b ∈  are both smooth nonlinear known functions with x as system states with b(x, t) = 0. d(t) is an unknown ˙ Lipschitz continuous disturbance acting on the system and |d(t)| ≤ , where is a known positive constant.

3.1.1

Arbitrary-Order Fixed-Time Sliding Surface

Now, let us define the proposed sliding surface of the form described below: s = x˙n + cn |xn |αn sign(xn ) + cn−1 |xn−1 |αn−1 sign(xn−1 ) + · · · + c1 |x1 |α1 sign(x1 ) + bn |xn |βn sign(xn ) +bn−1 |xn−1 |

βn−1

(15)

β1

sign(xn−1 ) + · · · + b1 |x1 | sign(x1 ).

The above surface is fixed-time convergent, provided that the following conditions are satisfied: 1. s n + cn s n−1 + · · · + c2 s + c1 is Hurwitz 2. s n + bn s n−1 + · · · + b2 s + b1 is Hurwitz αi αi+1 βi βi+1 , βi−1 = , i = 2, .., n 3. αi−1 = 2αi+1 − αi 2βi+1 − βi where αn+1 = 1, αn = α and βn+1 = 1, βn = β, α, β ∈ . The important point to remember here is that, αi ∈ (0, 1) for i = 1, · · · , n and βi ∈ (1, 1 + ) where 0 <  < 1 is sufficiently small.

3.2 Control Design Theorem 1 For the non-linear system as described by Eq. (14), let us define the control input as u = b−1 (x, t)(u eq + u n )

(16)

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where

u eq = − f (x, t) − cn |xn |αn sign(xn ) − cn−1 |xn−1 |αn−1 sign(xn−1 ) − · · · − c1 |x1 |α1 sign(x1 ) − bn |xn |βn sign(xn ) −bn−1 |xn−1 |

βn−1

(17)

β1

sign(xn−1 ) − · · · − b1 |x1 | sign(x1 )

and u˙ n = −K sign(s)

(18)

where K > is the gain parameter (K ∈ ). The chosen control as Eq. (16) with the given sliding surface as Eq. (15) can provide convergence of the states to the origin λδ (P) + in fixed-time with the bound of the convergence time given as Tmax ≤ max c1 δ 1−αn 1 ´ m 2 , where δ = αn . c m λ ( P) 2

2 min

Proof Substituting the control given in Eq. (16) in the sliding surface dynamics given by Eq. (15), we will get s = d(t) + u n .

(19)

The first derivative of the above equation reads as ˙ + u˙ n . s˙ = d(t)

(20)

Now, let us consider a Lyapunov function as V =

1 2 s . 2

(21)

Taking the first derivative of the V , we get V˙ = s s˙ ˙ + u˙ n )s = (d(t)

˙ = −(K − d(t))|s|.

(22)

Hence, the negative definiteness of V˙ can always be maintained if the gain of the controller is chosen properly, i.e. K > , where the disturbance d(t) is sufficiently ˙ smooth and |d(t)| ≤ . The solution to the above differential equations with discontinuous right-hand side can be understood in Filippov sense [20]. Moreover, it can be clearly shown that 1 V˙ ≤ − K´ V 2

(23)

1 ˙ The above equation is a finite-time converging differential where K´ = 2 2 (K − d(t)). equation and the convergence time can be easily found by solving it. Therefore, the reaching time is obtained in finite time [21].

On Fixed-Time Convergent Sliding Mode Control Design and Applications

213

Let us analyse the dynamics of the system of Eq. (14) during sliding; we have ⎫ x˙1 = x2 ⎪ ⎪ ⎬ x˙2 = x3 ⎪ ..⎪ ⎭ .

(24)

x˙n = −cn |xn |αn sign(xn ) − · · · − c1 |x1 |α1 sign(x1 )

(25)

−bn |xn |βn sign(xn ) − · · · − b1 |x1 |β1 sign(x1 ).

In the following section, the fixed-time stability is analysed during sliding. Let us first define a Lyapunov function V1 (x) for the above system as described by Eqs. (24) and (25) with bi = 0, i = 1, 2, ..., n as V1 (x) = xT Px

(26)

where x = [x1 , x2 , ...., xn ]. The symmetric positive-definite matrix P must satisfy the Lyapunov equation as defined below: P A + A T P = −Q

(27)

where Q ∈ n is a positive-definite symmetric matrix with A defined as ⎡

0 ··· ⎢ 1 ··· ⎢ ⎢ .. A=⎢ . ··· ⎢ ⎣ 0 0 0 ··· −c1 −c2 −c3 · · · 0 0 .. .

1 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥. ⎥ 1 ⎦ −cn

(28)

The above matrix can be easily obtained by setting αi = 1, i = 1, 2, ..., n. The time derivative of V1 satisfies the following: V˙1 (x) = −xT Qx < 0.

(29)

The above equality also holds true for αi ∈ (1 − , 1), for sufficiently small  > 0 [21]. Moreover, the right-hand side of Eqs. (24) and (25) are homogenous vector < 0 and with dilation α1i , i = 1, 2, ..., n. Therefore, fields with degree m 1 = αnα−1 n the following inequality holds leading to a finite-time convergence: V˙1 (ζ ) ≤ −c1 V11+m 1 (ζ ) where c1 =

λmin (Q) , with ζ λmax (P)

(30)

= [x1α1 , x2α2 , ...., xnαn ]. The convergence time for the above V1δ ζ (t0 ) , c1 δ λδmax (P) . c1 δ

differential equation can be found easily by solving it and is given by T1 ≤ where δ =

1−αn . αn

By using Rayleigh inequality [14], we can show T1 ≤

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J. Mishra and X. Yu

Now, let us define another Lyapunov function V2 (x) for the system of Eqs. (24) and (25) with all ci = 0, i = 1, 2, ..., n as ´ V2 (x) = xT Px.

(31)

The symmetric positive-definite matrix P´ must satisfy the following Lyapunov equation as (32) P´ A´ + A´ T P´ = − Q´ where Q´ ∈ n is a positive-definite symmetric matrix with A´ defined as ⎡

0 ··· ⎢ 1 ··· ⎢ ⎢ .. ´ A=⎢ . ··· ⎢ ⎣ 0 0 0 ··· −b1 −b2 −b3 · · · 0 0 .. .

1 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥. ⎥ 1 ⎦ −bn

(33)

The above matrix can be easily obtained by setting βi = 1, i = 1, 2, ..., n. The time derivative of V2 satisfies the following: ´ < 0. V˙2 (x) = −xT Qx

(34)

The equality Eq. (34) also holds true for βi ∈ (1, 1 + 2 ), for sufficiently small 2 > 0. Moreover, the right-hand side of Eqs. (24) and (25) are again homogenous vector > 0 and with dilation β1i , i = 1, 2, ..., n. Therefore, fields with degree m 2 = βnβ−1 n the following inequality holds leading to finite-time convergence: V˙2 (ς ) ≤ −c2 V21+m 2 (ς ) where c2 =

´ λmin ( Q) ´ ,and λmax ( P)

β

β

(35)

β

ς = [x1 1 , x2 2 , ...., xn n ].

Now, considering > 0 such that V2 (ς(t0 )) > , and calculating the full time derivative along the trajectories described in Eq. (25), we have ∂ V2 (−cn |xn |αn sign(xn ) − · · · − c1 |x1 |α1 V˙2 (x) ≤ V˙2 (ς ) + ∂ xn sign(x1 )) ≤ V˙2 (ς ) ≤ −c2 V21+m 2 (ς ).

(36)

Therefore, it can be seen that V2 (x) decreases and reaches the value in a time no later than T2 = c2 m 21 m2 , which is independent of initial conditions [22, 23]. By using Rayleigh inequality, ||ς ||2 ≤

V˙2 (ς) ´ λmin ( P)

≤

´ λmin ( P)

≤ 1. Thus, the T2 will become

T2 = c m λ ( P) ´ m 2 . Therefore, the total convergence time can be achieved in Tmax ≤ 2 2 min  T1 + T2 . This completes the proof. 1

On Fixed-Time Convergent Sliding Mode Control Design and Applications

215

Remark 1 The main reason behind adding some terms in the surface with power greater than one is to make convergence faster when the initial conditions are far from the origin. Therefore, when initial conditions of the states are far away from the origin, the terms with power greater than one will dominate the convergence rate and when the states converge towards the origin, the terms with power less than one will dominate the convergence rate. It is important to mention here that almost all systems with fixed-time stable equilibrium have the property of bi-homogeneity [15]. Hence, the convergence can be achieved in fixed-time. Remark 2 If we analyse Eq. (16), all the terms in the control u are known except the term s as it contains the term x˙n , which contains the uncertainties. For calculating sign(s) in Eq. (16), let us define a function η(t) as below: t η(t) =

s dt.

(37)

0

We have t η(t) = xn +

[cn |xn |αn sign(xn ) + · · · + c1 |x1 |α1 sign(x1 )

0

(38)

+ bn |xn |βn sign(xn ) + · · · + b1 |x1 |β1 sign(x1 )]dt sign(s) can be obtained by simply equating sign(s) = sign(η(t) − η(t − τ )) where τ is a time delay. Since (η(t) − η(t − τ )) , s = lim τ →0 τ the fundamental sample time can be easily chosen as τ . The interesting fact about the above analysis is that we only need to know the value for sign(s), i.e. whether η(t) increases or decreases. This is much easier to obtain than the exact value of s.

3.3 Simulation Results Consider the system for simulation as x˙1 = x2 x˙2 = x3 x˙3 = x23 + 0.1 sin(20t) + u. The designed control for the above system using the proposed fixed-time SMC scheme is

216

J. Mishra and X. Yu

u eq = −x23 − 15|x3 |7/10 sign(x3 ) − 66|x2 |7/13 sign(x2 )− 80|x1 |7/16 sign(x1 ) − 15|x3 |21/20 sign(x3 ) − 66|x2 |21/19 sign(x2 ) − 80|x1 |21/18 sign(x1 ) and u˙ n = −10sign(s) where s = x˙3 + 15|x3 |7/10 sign(x3 ) + 66|x2 |7/13 sign(x2 ) + 80|x1 |7/16 sign(x1 ) + 15|x3 |21/20 sign(x3 ) + 66|x2 |21/19 sign(x2 ) + 80|x1 |21/18 sign(x1 ). The performance of the proposed controller is compared with the state-of-the-art methods available for arbitrary-order systems and the results are shown for the comparison (Figs. 4 and 5). For the purpose of comparison, by keeping almost similar control effort for all controllers, the convergence rate is observed. It is found that the convergence obtained in the proposed case is much faster than that of existing cases. For table space limitations, the following numberings/abbreviations reference the various methods under consideration: 1. [6] M1 , 2. [8] M2 , 3. [17] M3 , 4. Proposed  M4 . From the Table 1, it can be clearly seen that the convergence time in the proposed case is around 1.6 s (Refer Fig. 4) for the initial conditions x1 (0) = 0.5, x2 (0) = 0 and x3 (0) = 0. However, with the same initial conditions, the convergence time in both Ding et al. [6] and Shyam et al. [8] is approximately 2.3 and 3.2 s, respectively.

0

0.5

Ding et al. Shyam et al. Yong et al. Proposed

x1

0.3

−0.2

Ding et al. Shyam et al. Yong et al. Proposed

x2

0.4

0.2

−0.4

0.1 0 −0.1

−0.6

0

1

2

time (sec.)

3

4

5

0

1

2

time (sec.)

3

100

2 1

x3

0

−2 −3 0

1

2

time (sec.)

3

4

u

Ding et al. Shyam et al. Yong et al. Proposed

5

Ding et al. Shyam et al. Yong et al. Proposed

50

−1

4

0 −50

5

−100

0

1

2

time (sec.)

Fig. 4 Evolution of states and control signal for the sample dynamical system

3

4

5

On Fixed-Time Convergent Sliding Mode Control Design and Applications 5

1

Ding et al. Shyam et al. Yong et al. Proposed

4

0 −1

x2

3

x1

2

Ding et al. Shyam et al. Yong et al. Proposed

−2 −3

1

−4

0 −1 0

1

2

10

time (sec.)

3

4

−5 0

5

1

2

time (sec.)

3

4

5

400

5

200

0

0

−10

u

Ding et al. Shyam et al. Yong et al. Proposed

−5

x3

217

Ding et al. Shyam et al. Yong et al. Proposed

−200 −400

−15 −20 0

1

2

time (sec.)

3

4

5

−600

0

1

2

time (sec.)

3

4

5

Fig. 5 Evolution of states x1 − x3 and control signal with change in the initial conditions Table 1 Simulated convergence time with range of control for various initial conditions of x1 (x1 (0)) x1 (0) Convergence time (s) Control range [u min , u max ] M1 M2 M3 M4 M1 M2 M3 M4 0.5

2.3

3.2

1.8

1.6

1

3

3.5

2.2

1.8

5

5.2

6

2.8

2.5

10

6

6.5

3.2

2.6

15

7.2

7.5

3.8

3

20

8.8

8.9

4

3

25

10.1

10.5

4.5

3

50

11

11.5

8

3

100

15

15

12

3

[−89, 99] [−113, 125] [−181, 230] [−187, 335] [−147, 456] [−162, 603] [−175, 778] [−221, 2118] [−278, 7185]

[−36.9, 59.67] [−46.01, 69.82] [−61.30, 118.71] [−50, 220] [−53, 407] [−59, 696] [−65, 1090] [−82, 4858] [−104, 15000]

[−59.07, 11.05] [−80, 15.56] [−162, 88.36] [−219, 805] [−261, 1062] [−296, 2126] [−327, 3649] [−442, 8000] [−599, 21000]

[−94, 10] [−160, 18] [−584, 156] [−610, 1005] [−720, 1100] [−800, 2250] [−860, 3700] [−980, 8800] [−1800, 25000]

Moreover, in the case of Ding et al. and Shyam et al., the chattering of the control signal u is significant and is clearly visible from Fig. 4. Due to the simple structure of the proposed control and its continuous nature, it will easily find its applications in many practical fields with equivalent average control cost. The convergence of

218

J. Mishra and X. Yu

x1 is guaranteed within 1.6 s in the case of the proposed control, however it takes more time in the case of the existing controls with similar average control cost. The simulations are also performed by changing the initial conditions. For the purpose of verifying the claim made, the initial conditions are changed to x1 (0) = 5, x2 (0) = −1 and x3 (0) = 0. It can be clearly seen from the Fig. 5 that the convergence time is a function of initial conditions in the case of existing control structures, i.e. Ding et al. [6], Yong et al. [17] and Shyam et al. [8] cases. As initial condition of the states is increased by 10 times, the convergence time is also increased in previously reported methods (refer Table 1). However, in the case of the proposed control scheme, the convergence time is slightly changed to 2.5 s. The initial condition for x1 is further increased and it can be seen from Table 1 that the convergence is guaranteed within Tmax = 3 s. Hence, fixed-time convergence is verified. Remark 3 It can be clearly seen from Table 1 that by increasing the control effort to a large amount, the fixed-time convergence can be achieved. In all linear controllers, the presence of linear terms provides asymptotic convergence. In the case of finitetime controller, the finite convergence time is achieved due to the presence of some non-linear terms in the control. However, if we add further non-linear terms into the control design with some special properties, it provides fixed-time convergence with a very high control effort and behaves like a high gain controller.

4 An Application to Network Systems A complex network is defined as a group of nodes interconnected by edges, where each node represents a sub-unit and each edge represents an interaction between two nodes [24, 25]. Many real systems can be modelled as network systems, with examples including the Internet, worldwide web, online social networks, multiagent systems, power grids, transportation networks and the human brain. Over the past few decades, network science has attracted a great deal of attention from many fields of research ranging from engineering to computer science, biology, sociology and medicine [26–29]. The main contribution of this section lies in proposing a novel decoupled distributed continuous SMC providing fixed-time second-order consensus for multiagent systems. The important feature of the proposed method is that it can be generalised to arbitrary-order agent dynamics. An interesting fact about the proposed formulation is that the states of the agents are coupled on the sliding surface. However, the sliding mode states are decoupled out of this surface, which makes the design of SMC independent of the extent of coupling between the agents. The proposed surface will provide fixed-time consensus, i.e. independency to the initial condition. The robustness of the proposed controller is demonstrated in the presence of Lipschitz disturbance in agent dynamics and uncertainties in the network structure. Moreover, the efficacy of the proposed scheme is compared with one of the state-of-the-art

On Fixed-Time Convergent Sliding Mode Control Design and Applications

219

techniques and the average control effort is found to be comparatively less in the proposed scheme.

4.1 Fixed-Time Second-Order Consensus in Multiagent Systems 4.1.1

Problem Formulation

The multiagent system under consideration is 

x˙i = vi v˙i = f (xi , vi , t) + di (xi , vi , t) + bi (xi , vi , t)u i (t)

(39)

where xi ∈ n and vi ∈ n i = 1, 2, · · · , N . di (xi , vi , t) ∈ n refers to any kind of parametric uncertainty or external added disturbance with bi (xi , vi , t) ∈  = 0. Both f (xi , vi , t), bi (xi , vi , t) are two smooth known non-linear functions of xi , vi and t. The main objective is to design a control u i for each agent, which can drive the states to the consensus in fixed-time irrespective of any initial conditions. The control objectives are described as below: lim ||xi (t) − x j (t)|| = 0,

t→T

||xi (t) − x j (t)|| = 0, ∀t ≥ T and lim ||vi (t) − v j (t)|| = 0,

t→T

||vi (t) − v j (t)|| = 0, ∀t ≥ T where T is the convergence time and finite. Moreover, T is independent of initial conditions. Assumption 1 The disturbances/uncertainties acted on the above systems are considered to be Lipschitz in nature, i.e. ||di (xi , vi , t)||∞ ≤ pi ||d˙i (xi , vi , t)||∞ ≤ qi where pi , qi > 0 are known constants with || · ||∞ is the infinite norm of the any vector.

220

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Lemma 1 Let us consider an n-dimensional system: y˙ = g(y)

(40)

where y = [y1 , y2 , · · · , yn ]T and g(y) = [g1 (y), g1 (y), · · · , gn (y)]T . If the system Eq. (40) is homogenous of degree γ1 with dilation (a1 , a2 , · · · an ), and homogenous of degree γ2 with dilation (b1 , b2 , · · · bn ), and with a non-linear continuous function g, then y = 0 is the asymptotic equilibrium of the system. If homogeneity degrees γ1 < 0 in 0-limit and γ2 > 0 in ∞-limit, then the system is said to be bi-homogenous and by satisfying such property, the equilibrium point is called to be globally fixed-time stable [14, 30].

4.1.2

Full-Order Fixed-Time Sliding Surface

The first task is to design a stable fixed-time converging sliding surface as defined below: N  ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 ) σi = x¨i − (41) j=1 +ϕ3 (sig(v j − vi )α2 ) + ϕ4 sig(v j − vi )β2 )] 2β1 2α1 and β2 = 1+β . ϕ1 , ϕ2 , ϕ3 where 0 < α1 , α2 < 1 and β1 , β2 > 1 with α2 = 1+α 1 1 T and ϕ4 are four continuous odd functions, satisfying yi ϕk (yi ) > 0 if yi = 0 and ϕk (yi ) = cki yi + O(yi ) around yi = 0; where O is the complexity function. Moreover, ϕk (yi ) = [ϕk (yi1 ), ϕk (yi2 ), . . . , ϕk (yin )]T if yi = [yi1 , yi2 , . . . , yin ]T with yi ∈ Rn . sig is the signum function. A weighted undirected network G = (V, E, A) with order N consists of a set of nodes V = v1 , v2 , ..., v N , a set of undirected edges E ⊆ V × V , and a weighted adjacency matrix A = (ai j ) N ×N . An undirected edge E i j in a weighted undirected network G is denoted by the unordered pair of nodes (vi , v j ), which means that nodes vi and v j can exchange information with each other. According to the definition of adjacency matrices, weights ai j = a ji > 0 are positive if and only if there is an edge (vi , v j ) in G . A = (ai j ) N ×N is the coupling configuration matrix representing the structure of the network, which is also called the weighted adjacency matrix of the network [31, 32].

Theorem 2 If the states of the system Eq. (39) can reach to the designed sliding surface Eq. (41), then second-order consensus can be achieved in fixed time. Proof When sliding is achieved, σi = 0. Therefore, the system dynamics given in Eq. (39) can be represented as 

x˙i = vi , v˙i = uˆ i , for i = 1, ..., N

(42)

On Fixed-Time Convergent Sliding Mode Control Design and Applications

221

 where uˆ i = Nj=1 ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 ) + ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )]. The above dynamics Eq. (42) can be equivalently represented as in fixed-time convergence of error dynamics of the system as follows: 

e˙xi = evi , ˇ for i = 1, ..., N e˙vi = uˆ i − u,

(43)

N where exi = [exi1 , exi2 , . . . , exin ]T = xi − N1 j=1 x j , evi = [evi1 , evi2 , . . . ,   N N 1 1 T evin ] = vi − N ˇ= N ˆ j . The equivalent control uˆ i can be j=1 v j and u j=1 u written in terms of error dynamics as uˆ i =

N 

ai j [ϕ1 (sig(ex j − exi )α1 ) + ϕ2 (sig(ex j − exi )β1 )

j=1 α2

(44)

β2

+ϕ3 (sig(ev j − evi ) ) + ϕ4 (sig(ev j − evi ) )]. As ϕ1 , ϕ2 , ϕ3 and ϕ4 are odd functions uˇ i =

N N 1  ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 ) N i=1 j=1

(45)

+ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )] = 0. Let us define a Lyapunov function as 1  V = 2 i=1 j=1 k=1 N

N

n

exik −ex jk

ai j [ϕ1 (sig(z)α1 ) + ϕ2 (sig(z)β1 )]dz

0

(46) +

N 1

2

T evi evi .

i=1

Taking the derivative of above function, we have V˙ =

N N  

T ai j evi [ϕ1 (sig(exi − ex j )α1 ) + ϕ2 (sig(exi − ex j )β1 )]

i=1 j=1

+

N 

(47) T evi e˙vi .

i=1

The solution to the above differential equation Eq. (47) with non-differentiable continuous right-hand side can be understood in Filippov sense [20].

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J. Mishra and X. Yu N  N 

V˙ =

T ai j evi [ϕ1 (sig(exi − ex j )α1 ) + ϕ2 (sig(exi − ex j )β1 )]

i=1 j=1

+

N 

N 

T evi

i=1

ai j [ϕ1 (sig(ex j − exi )α1 ) + ϕ2 (sig(ex j − exi )β1 )

(48)

j=1

+ϕ3 (sig(ev j − evi )α2 ) + ϕ4 (sig(ev j − evi )β2 )]. Since ϕ1 , ϕ2 are odd functions, ϕ1 (sig(exi − ex j )α1 ) = −ϕ1 (sig(ex j − exi )α1 ) and ϕ2 (sig(exi − ex j )β1 ) = −ϕ2 (sig(ex j − exi )β1 ). Therefore, Eq. (48) can be rewritten as N N   T ai j evi [ϕ3 (sig(ev j − evi )α2 ) + ϕ4 (sig(ev j − evi )β2 )]. V˙ = (49) i=1 j=1

It can also be seen that since ai j = a ji , Eq. (49) can be further simplified as 1  T V˙ = (ai j + a ji )evi [ϕ3 (sig(ev j − evi )α2 ) + ϕ4 (sig(ev j − evi )β2 )] (50) 2 i=1 j=1 N

N

1  (evi − ev j )T ai j [ϕ3 (sig(ev j − evi )α2 ) + ϕ4 (sig(ev j − evi )β2 )] (51) V˙ = 2 i=1 j=1 N

N

N N n 1  V˙ = (evik − ev jk )T ai j [ϕ3 (sig(ev jk − evik )α2 ) + ϕ4 (sig(ev jk − evik )β2 )]. 2 i=1 j=1 k=1

(52) Since ϕ3 , ϕ4 are odd functions, we have • ∀evik − ev jk = 0, (evik − ev jk )ϕ3 (sig(ev jk − evik )α2 ) = −|ev jk − evik |α2 +1 < 0 and (evik − ev jk )ϕ4 (sig(ev jk − evik )β2 ) = −|ev jk − evik |β2 +1 < 0 • when evik − ev jk = 0, (evik − ev jk )ϕ3 (sig(ev jk − evik )α2 ) < 0 and (evik − ev jk ) ϕ4 (sig(ev jk − evik )β2 ) < 0. Therefore, V˙ < 0 is always satisfied. It can be seen from Eq. (52) that if V˙ = 0, then evi = ev j , ∀ j = i. When evi = ev j , the equivalent control becomes uˆ i =

N 

ai j [ϕ1 (sig(ex j − exi )α1 ) + ϕ2 (sig(ex j − exi )β1 )].

(53)

j=1

As the connectivity graph is undirected (ai j = a ji ), we have

N i=1

uˆ i = 0.

N T N exi j=1 ai j [ϕ1 (sig(ex j − exi )α1 ) + ϕ2 (sig(ex j − exi )β1 )] = That leads to i=1 N  0, which in turn leads to 21 i=1 (exi − ex j )T Nj=1 ai j [ϕ1 (sig(ex j − exi )α1 ) + ϕ2

On Fixed-Time Convergent Sliding Mode Control Design and Applications

223

(sig(ex j − exi )β1 )] = 0. As exi − ex j → 0, evi − ev j → 0 as V˙ < 0 when t → ∞; therefore xi − x j → 0, vi − v j → 0 as t → ∞ ∀i, j = 1, 2, · · · N . The system Eq. (42) with variables [(x1 )T . . . (x N )T , (v1 )T · · · (v N )T ]T is homogenous of degree γ1 = α1 − 1 < 0 with dilation (2, . . . , 2, 1 + α1 , . . . , 1 + α1 ) and is homogenous of degree γ2 = β1 − 1 > 0 with dilation (2, . . . , 2, 1 + β1 , . . . , 1 + β1 ). Since system Eq. (43) is globally asymptotically stable and locally bi-homogeneous, therefore, according to Lemma 1, the above system is globally fixed-time stable and the second-order consensus can be achieved in fixed-time on the designed sliding surface Eq. (41). This completes the proof.  Remark 4 The main reason for adding some terms in the surface Eq. (41) with a power greater than one is to make the convergence faster when the initial conditions are far from the origin. Therefore, when initial conditions of the states are far away from the origin, these terms with a power greater than one will be dominating and when the states converge towards the origin, the terms with a power less than one will be dominating. Hence, the convergence is achieved in fixed-time [15, 23].

4.1.3

Design of Control

The control for system Eq. (39) is designed to be of the form: u i (t) = bi−1 (xi , vi , t)[u ni (t) + u ei (t)]

(54)

where u ei (t) = − f (xi , vi , t) +

N 

ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 )

j=1

(55)

+ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )] and u˙ ni (t) + m i u ni (t) = −ki sgn(σi (t))

(56)

where m i > 0, ki > 0 with ki > qi + pi m i + ζi , ζi > 0. Theorem 3 Under Assumption 1, the proposed control Eq. (54) can bring the states of the multiagent system Eq. (39) to reach the sliding surface in finite time, leading them to achieve second-order consensus in fixed-time along the surface (41). Proof We analyse the following proof mainly during the reaching phase. Substituting the control as given by Eq. (54) in the surface dynamics Eq. (41) for the multiagent system as described by Eq. (39), we have

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σi = x¨i −

N 

ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 )

j=1

+ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )] σi = f (xi , vi , t) + di (xi , vi , t) + bi (xi , vi , t)u i (t) −

N 

ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 )

j=1

+ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )] σi = di (xi , vi , t) − m i u ni (t) − ki sgn(σi (t)).

(57)

Now, taking the first derivative of Eq. (57): σ˙ i = d˙i (xi , vi , t) + u˙ ni (t).

(58)

Substituting u ni (t) in Eq. (58): σ˙ i = d˙i (xi , vi , t) − m i u ni (t) − ki sgn(σi (t)).

(59)

It can be clearly seen that u˙ ni (t) + m i u ni (t) = −ki sgn(di (xi , vi , t) + u ni (t))

(60)

which can be further written as d (u ni + di ) + m i (u ni + di ) = −ki sgn(σi ) + d˙i + m i di dt

(61)

which can be written as  d  mi t e (u ni + di ) = em i t [−ki sgn(u ni + di ) + d˙i + m i di ]. (62) dt   When u ni + di > 0, it can be seen from Eq. (62) that dtd em i t (u ni + di ) < 0 as ki > qi + pi m i + ζi and sgn(u ni + di ) = 1. Therefore, σi will start to decay towards Similarly, σi = 0, i.e. towards the sliding surface.   when u ni + di < 0, it can again be verified from Eq. (62) that dtd em i t (u ni + di ) > 0 as ki > qi + pi m i + ζi and sgn(u ni + di ) = −1. Therefore, σi will again start to decay towards σi = 0. The most important observation here is that whenever σi , i.e. u ni + di is non-zero, the control dynamics Eq. (54) forces σi dynamics to converge towards zero. Now, it is important to prove that the convergence is achieved in finite time. Let us define a Lyapunov function as described below:

On Fixed-Time Convergent Sliding Mode Control Design and Applications

1 T σ (t)σi (t). 2 i=1 i

225

n

V =

(63)

Taking the first derivative of Eq. (63) V˙ =

n 

σiT (t)σ˙ i (t).

(64)

i=1

From Eq. (59), it can be verified that σiT (t)σ˙ i (t) = σiT (t)d˙i (t) − σiT (t)m i u ni (t) − ki |σi (t)|

(65)

which can be further simplified as σiT (t)σ˙ i (t) ≤ [σiT (t)d˙i (t) − qi |σi (t)|] + [−σiT (t)m i u ni (t) +m i pi |σi (t)|] − (ki − qi − pi m i )|σi (t)|.

(66)

Substituting Eq. (66) in Eq. (64), we have V˙ ≤ −

n 

ζi |σi (t)|, as ki > qi + pi m i + ζi

i=1

V˙ ≤ −ζmin

n  n 

|σi j (t)|,

(67) (68)

i=1 j=1

√ ≤ −ζmin V .

(69)

Therefore, for the system as described by Eq. (39), convergence to the sliding surface σi can be achieved in finite time. It can be clearly seen that, if we choose u˙ ni (t) = −ki sgn(σi (t))

(70)

where ki > qi + ζi , ζi > 0, then Theorem 3 can easily be proven in usual way. Adding an extra term m i u ni (t) makes the control to be a low-pass filter, which helps to smoothen out the effects of disturbances and makes the convergence faster. However, from Eq. (59) it can be seen that a negative feedback of control u ni , is not a good idea in actual practise. Therefore, if we just modify dynamics of u ni as u˙ ni (t) = −m i σi (t) − ki sgn(σi (t))

(71)

where m i > 0, ki > 0 with ki > qi + ζi , ζi > 0, then with above control the analysis will become simpler as described below.

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Let us analyse the Lyapunov function: V˙ =

n 

σiT (t)σ˙ i (t)

(72)

σiT (t)[d˙i (xi , vi , t) + u˙ ni (t)].

(73)

i=1

using Eq. (58), we can further write V˙ =

n  i=1

Now substituting Eq. (71) in above V˙ ≤ −

n 

ζi |σi (t)| − m i σiT (t)σi (t)

(74)

i=1

√ ≤ −ζmin V .

(75)

After the trajectories reach to the surface σi , the consensus can be achieved in fixed time as described in Theorem 1. This completes the proof.  Remark 5 If we analyse Eq. (54), all the terms in the control u i (t) are known except the term σi (t) as it contains the term x¨i , which contains the uncertainties. For calculating sgn(σi (t)) in Eq. (54), let us define a function δi (t) as below: t δi (t) =

σi (t) dυ

(76)

0

δ(t) = x˙i −

t  N 0

ai j [ϕ1 (sig(x j − xi )α1 ) + ϕ2 (sig(x j − xi )β1 )

j=1

(77)

+ ϕ3 (sig(v j − vi )α2 ) + ϕ4 (sig(v j − vi )β2 )]dυ sgn(σi (t)) can be obtained by simply equating sgn(σi ) = sgn(δ(t) − δ(t − τ )) where τ is a time delay. Since (δ(t) − δ(t − τ )) , τ →0 τ

σi = lim

the fundamental sample time can be easily chosen as τ . The interesting fact about the above analysis is that we only need to know the value of sgn(σi ), i.e. whether δ(t) increases or decreases, which is much easier to obtain than the exact value of σi .

On Fixed-Time Convergent Sliding Mode Control Design and Applications

1

227

4

5

3

2 Fig. 6 Undirected graph with five agents

4.1.4

Simulation Results

Consider each agent with system dynamics as given below: 

x˙i = vi , v˙i = vi3 + u i + di , i = 1, 2...5

(78)

where di = sin 30t as a Lipschitz continuous and bounded disturbance. The control input u i is as described in Eq. (54) with α1 = 0.7, β1 = 1.2 and α2 = 0.8235, β2 = 1.0909. Each agent is connected to each other by following graph Fig. 6. The adjacency matrix for the graph shown in Fig. 6 can be written as ⎡

0 ⎢1 ⎢ A=⎢ ⎢0 ⎣0 0

1 0 1 0 0

0 1 0 1 1

0 0 1 0 0

⎤ 0 0⎥ ⎥ 1⎥ ⎥ 0⎦ 0

The control gain is ki > qi + pi m i + ζi . The definitions for qi , pi are given  in Assumption 1. In this example, qi = 30, pi = 1 and ζi = 1. Moreover, x¯ = Nj=1 xj , N = 5 in our simulations. The simulations are carried out for various values of N m i to verify the influence of this filtering coefficient (m i ) on the control performance. Figure 7 represents the evolution of position for various agents with respect to time. It can be clearly seen from Fig. 7 that the consensus is achieved within 11.12 s. even in the presence of a Lipschitz disturbance. The corresponding position error trajectories for various agents are shown in Fig. 8. It is interesting to note that, irrespective of various initial conditions for five agents, the convergence can be achieved approximately in 11.12 s. It can be clearly seen that the change in the filtering coefficient (m i ) has almost no effect in terms

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200

1 2 3 4 5

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180 160

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Fig. 7 Trajectory of position xi w.r.t t for various m i under the influence of Lipschitz disturbance 8

8

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6 mi=0

4

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1 2 3 4 5

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4

-4

1 2 3 4 5

-6 -8 -10

0

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time (sec.)

Fig. 8 Trajectory of position error xi − x¯ w.r.t t for various m i under the influence of Lipschitz disturbance

On Fixed-Time Convergent Sliding Mode Control Design and Applications 20

20

20 1 2 3 4 5

18 16 14

1 2 3 4 5

18 16 14

mi=0

10

16 14

mi=10

10 8

8

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2

0

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1 2 3 4 5

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12

12

12

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0

0

10

20

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Fig. 9 Trajectory of velocity vi w.r.t t for various m i under the influence of Lipschitz disturbance

Fig. 10 Trajectory of control u i w.r.t t when m i = 40, using Eq. (54) under the influence of Lipschitz disturbance

of achieving consensus. The most important aspect of the proposed control is that it will provide a second-order consensus, i.e. the velocity v for each agent will also achieve consensus (refer Fig. 9). The corresponding control (Fig. 10) to achieve fixed-time consensus is smooth and continuous.

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0.5 0 -0.5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.14

0.16

0.18

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0.14

0.16

0.18

0.2

time (sec.) 1 0.5

mi=10

0 -0.5

0

0.02

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0.12

time (sec.) 1 mi=40

0.5 0 -0.5

0

0.02

0.04

0.06

0.08

0.1

0.12

time (sec.)

Fig. 11 Trajectory of sliding surface σ w.r.t t for various m i , using Eq. (41) under the influence of Lipschitz disturbance

The proposed control can bring all the state trajectories to the sliding surface σ in finite-time and can be clearly seen in Fig. 11 irrespective of any filtering coefficient m i . Robustness Testing 1: Uncertainties in weight of the edges The robustness of the proposed controller is tested by varying the weight assignment to various links. The adjacency matrix for the corresponding analysis can be written as ⎡ ⎤ 05000 ⎢5 0 5 0 0 ⎥ ⎢ ⎥ ⎥ A=⎢ ⎢0 5 0 5 5⎥ ⎣0 0 5 0 0⎦ 00500 As we can see, the weights are increased to five times greater than that of the previous case. By keeping all the simulation parameters similar, it can be seen that the consensus is still achieved in 11.12 s. (refer Figs. 12, 13 and 14). 2: Uncertainties in the orientation of the edges In this analysis, we have changed the graph structure as simulated before Fig. 6. The link between agent 3 and agent 5 is broken. A new link is established between agent 4 and agent 5 during the simulation. The corresponding adjacency matrix for the graph is given as

On Fixed-Time Convergent Sliding Mode Control Design and Applications 200

200

180

200

180 mi=0

160

180 mi=10

160 140

140

120

120

120

100

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60 1 2 3 4 5

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160

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231

0

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40 20 0

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Fig. 12 Trajectory of position xi w.r.t t for various m i under the influence of Lipschitz disturbance with weighted links 20

20

20

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18

18 mi=0

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mi=10

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2

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0

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1 2 3 4 5

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8

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0

0

10

20

time (sec.)

Fig. 13 Trajectory of velocity vi w.r.t t for various m i under the influence of Lipschitz disturbance with weighted links

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Fig. 14 Trajectory of control u i w.r.t t when m i = 40, using Eq. (54) under the influence of Lipschitz disturbance with weighted links

⎡

0 ⎢1 ⎢ A=⎢ ⎢0 ⎣0 0

1 0 1 0 0

0 1 0 1 0

0 0 1 0 1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0

It can be seen from Figs. 15, 16 and 17 that the proposed control scheme is robust. Comparison with Existing Finite-time Control: In this section, we have carried out a comparison study with a state-of-the-art technique recently published. A detailed comparison is performed with the proposed method. Figure 18 shows the evolution of position trajectory for various agents. It can be clearly seen that the consensus is achieved approx. in 22 s which is almost twice than that of the proposed case. The corresponding velocity and control trajectories are shown in Figs. 19 and 20, respectively. A thorough comparison is made with the method [32] and is tabulated in Table 3. It can be seen that in the proposed case, the margin of error is much less. Moreover, the convergence time (Ts ) is approx. half (11.12 s) than that which was previously reported. The steady-state accuracy (ess ) is also improved. In order to achieve this, the maximum control action is not too much when compared with the previously reported case [32]. Furthermore, the average control effort in our proposed case is lower than previously reported and can be seen from Table 2. The corresponding convergence time with respect to various initial conditions for five agents is tabulated in Table 3.

On Fixed-Time Convergent Sliding Mode Control Design and Applications 200

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200

180

180

180 mi=0

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mi=10

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mi=40

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233

0

10

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1 2 3 4 5

40 20 0

time (sec.)

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10

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time (sec.)

Fig. 15 Trajectory of position xi w.r.t t for various m i under the influence of Lipschitz disturbance with links reassignment 20

20

20 1 2 3 4 5

18 16 14 12

1 2 3 4 5

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mi=0

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20

time (sec.)

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time (sec.)

Fig. 16 Trajectory of velocity vi w.r.t t for various m i under the influence of Lipschitz disturbance with links reassignment

5 Summary In this chapter, various convergence algorithms have been discussed. First, a fullorder sliding mode strategy has been proposed guaranteeing fixed-time convergence for arbitrary-order systems. The effectiveness of the proposed scheme has been shown by comparing it with a number of well-known methods. A better performance has

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Fig. 17 Trajectory of control u i w.r.t t when m i = 40, using Eq. (54) under the influence of Lipschitz disturbance with links reassignment 8

8

8 1 2 3 4 5

6 4

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-8 -10

-4

-4

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6

0

10

20

time (sec.)

-10

0

10

20

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Fig. 18 Trajectory of position xi w.r.t t for various m i under the influence of Lipschitz disturbance as proposed in [32]

been obtained with the proposed scheme in terms of convergence time. Later on, a new decoupled fixed-time convergent scheme has been proposed with continuous control for second-order consensus for multiagent systems. A distributed full-order fixed-time convergent sliding surface has been designed based on the bi-homogeneity property, under which the sliding mode states are decoupled. The designed control has been applied on the decoupled sliding mode states to ensure the trajectories reach the surface in finite time. Once the trajectories reach the surface, the control

On Fixed-Time Convergent Sliding Mode Control Design and Applications 20

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mi=0

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8

8

8

6

6

6

4

4

4

2

2

2

0

10

20

time (sec.)

0

0

10

1 2 3 4 5

18

10

0

235

mi=40

10

20

time (sec.)

0

0

10

20

time (sec.)

Fig. 19 Trajectory of velocity vi w.r.t t for various m i under the influence of Lipschitz disturbance as proposed in [32]

Fig. 20 Trajectory of control u i w.r.t t when m i = 40 under the influence of Lipschitz disturbance as proposed in [32]

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Table 2 Average control effort ||u||2 comparison Agents Reported [32] ||u||2

Proposed ||u||2

7.7318 × 104

1 2 3 4 5

7.5398 × 104 5.9939 × 104 5.5663 × 104 5.7671 × 104 5.2648 × 104

× 104

6.0500 5.6387 × 104 6.3670 × 104 5.2970 × 104

Table 3 Simulated convergence time with control range for various agents i

Finite-time Convergence Scheme [32]

Fixed-time Convergence Scheme (Proposed)

emin xi

emax xi

Ts

ess

[u min , u max ]

emax xi

Ts

ess

1

−9.5887

2.3458

22.5

10−4

emin xi

-1140.6,6.0

−7.4492

1.2104

11.1

0

−1395.6,17

2

−4.3242

1.1785

22.5

10−4

−638.59,0.5

−3.4533

0.5975

11.1

0

−722.7,0.5

3

−0.7919

1.7984

22.5

10−4

−666.87,0.5

−0.3842

1.0922

11.1

0

−750.6,0.5

4

−1.4810

6.3548

22.5

10−4

−2206.9,0.5

−0.7031

4.4336

11.1

0

−2221.6,0.5

5

−1.2516

6.7802

22.5

10−4

−1008.6,0.5

−0.7230

6.2537

11.1

0

−1024.3,0.5

[u min , u max ]

has ensured that they will never leave the surface and the second-order consensus along the surface has been obtained in fixed time. The designed control in this chapter has shown to be smooth and has exhibited less chattering, which is in fact a major challenge in the SMC design. The robustness of the proposed scheme has been verified in the presence of Lipschitz disturbance and link uncertainties.

References 1. Utkin, V.: Discussion aspects of high-order sliding mode control. IEEE Trans. Autom. Control 61(3), 829–833 (2016) 2. Yu, S., Long, X.: Finite-time consensus tracking of perturbed high-order agents with relative information by integral sliding mode. IEEE Trans. Circuits Syst. II Express Briefs 63(6), 563– 567 (2016) 3. Tan, S.-C., Lai, Y.-M., Tse, C.K.: A unified approach to the design of PWM-based sliding-mode voltage controllers for basic DC-DC converters in continuous conduction mode. IEEE Trans. Circuits Syst. I Regul. Pap. 53(8), 1816–1827 (2006) 4. Levant, A.: Chattering analysis. IEEE Trans. Autom. Control 55(6), 1380–1389 (2010) 5. Moreno, J.A., Osorio, M.: Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans. Autom. Control 57(4), 1035–1040 (2012) 6. Ding, S., Levant, A., Li, S.: Simple homogeneous sliding-mode controller. Automatica 67, 22–32 (2016) 7. Cruz-Zavala, E., Moreno, J.A.: Homogeneous high order sliding mode design: a Lyapunov approach. Automatica 80, 232–238 (2017) 8. Kamal, S., Chalanga, A., Moreno, J., Fridman, L., Bandyopadhyay, B.: Higher order supertwisting algorithm. In: 13th International Workshop on Variable Structure Systems (VSS). IEEE, pp 1–5 (2014)

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Discrete Higher-Order Sliding Mode Leader-Following Consensus Protocols for Homogeneous Discrete Multi-Agent System Keyurkumar Patel and Axaykumar Mehta

Abstract This chapter presents higher-order discrete sliding mode protocols namely with reaching law approach and discrete supertwisting algorithm are proposed for the leader-following discrete homogeneous multi-agent system to achieve the global consensus in finite time. The higher-order sliding mode control facilitates to reduce the switching control chattering band, known as quasi-sliding mode band which ultimately increases the robustness property. The chapter also presents the derivation of ultimate quasi-sliding mode band that can be achieved for both the proposed protocols. Both the protocols ensure the anti-disturbance and robustness consensus performance. To compare the consensus performance and robustness properties, both the proposed protocols are implemented on a discrete homogeneous multi-agent system comprise of 2-DOF serial flexible robotic arms. From the simulation and experimental results, it is inferred that the discrete higher-order protocol due to the reaching law approach outperforms the protocol using the discrete supertwisting algorithm. Keywords Discrete-time Sliding Mode Control (DSMC) · Discrete Multi-Agent System (DMAS) · Discrete Super-Twisting algorithm (DSTA) · Discrete First-Order Sliding Mode Control (DFSM) · Reaching Law (RL) · Multi-Agent System (MAS)

1 Introduction In the last decade, the distributed cooperative control has been the focused topic among research fraternity due to its salient applications in various domains such K. Patel (B) · A. Mehta Institute of Infrastructure Technology Research and Management, Ahmedabad 380026, Gujarat, India e-mail: [email protected] A. Mehta e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_10

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as industrial automation, intra-vehicle control for collision avoidance, autonomous underwater vehicle control, smart-grids, robotic formation, and many uncountable applications [1–6]. The main ideology in cooperative control of MAS is to design distributed local controllers on each agent by gathering information from the neighboring agent for reaching a certain global goal. To perform this task, different cooperative control algorithm such as rendezvous, flocking, consensus, formation, etc., are proposed in the literature [7–9]. Among all these cooperative control techniques, the consensus algorithm is widely researched in recent years [10–12]. Consensus of MAS can be categorized into two types: The first one is the average consensus in which each agent updates its information with neighboring agent and agree upon a certain degree of average value among them [13, 14]. The second one is the leader-following consensus in which all the follower agents continuously observed the leader information and try to set with each other [15]. Further, the leader-following consensus is divided into two groups (i) first-order consensus in which simple one state or position state information is available for consensus [16] and (ii) the second-order consensus in which position and velocity state information is available for consensus [17]. Many academician and researchers have proposed different protocols for the consensus of MAS. Liu and Jia [18] investigated H∞ consensus control of MAS for switching topology with subject to external uncertainties. Shariati and Tavakoli [19] investigated consensus protocol for a descriptor base approach to robust leaderfollowing consensus of ambiguous MAS with delay constraint using the PID technique. Huang and Pan [20] proposed a distributed observer type algorithm for the leader-following consensus using LMI technique and the separation principle. Hua et al. [21] proposed an adaptive leader-following consensus protocol which is selfreliant system parameters and only requires the relative neighborhood state information for the second-order switching nonlinear MAS. Chen et al. [22] designed a leader-following consensus protocol for nonlinear strick -feedback MAS with state time-delays under directed topology. All these protocols achieve the consensus asymptotically and also not robust against disturbance and uncertainties. Hence, many researchers proposed the consensus protocol using sliding mode control approach due to its inherent properties like insensitivity to parameter variation and finite time convergence. Ren and Chen [23] investigated the finite time leaderfollowing consensus protocol using terminal sliding mode control for second-order nonlinear MAS. Yu et al. [24] designed a distributed decoupled sliding mode (SM) protocol for second-order consensus, in which, they applied the proposed decoupled SM protocol to solve the coupled network. Mu et al. [25] proposed an integral sliding mode control for the leader-following consensus of multi-agent system with the quad-copter application. All these protocols are in the continuous-time domain for continuous MAS. Simultaneously, the researchers and academicians across the world have also proposed the protocol in a discrete domain for the consensus of homogeneous DMAS, which is convenient to implement using the digital controllers. Mahmoud and Khan [26] investigated dynamic output feedback based discrete consensus control of DMAS for leader-following topology with fixed and switching topologies using LMI algorithm. Wang et al. [27] designed a leader-following consensus of discrete linear MAS having communication constraint. Zhang et al. [28]

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proposed observer based global consensus of DMAS using state-feedback. Recently, the authors in the paper [29], have proposed a global consensus for DMAS using reaching law approach. In the paper, the authors defined a global sliding surface with enhanced reaching law with relative degree-1 system to achieve the global leaderfollowing consensus. However, the proposed consensus protocol was influenced by high chattering which deteriorates the consensus performance. To reduce the effect of chattering, the academicians and scientists opted for higher-order discrete sliding mode control [30]. Recently, Chakrabarty et al. [31], proposed switching type control effort using switching type reaching law with relative degree-1 system and relative degree-2 system. It is observed that switching type reaching law with the relative degree-2 system has exhibited less chattering compared to the switching type reaching law with the relative degree-1 system. Inspired by the aforementioned results, the authors got motivated to design the discrete higher-order sliding mode protocols namely with reaching law approach and discrete super- twisting algorithm for the leader-following consensus of homogeneous multi-agent system in finite time. The significant contributions of the chapter are narrated below: – Proposed two types of Discrete Higher-order Sliding Mode (DHSM) consensus protocols namely with Reaching Law (RL) approach and Discrete Super-Twisting Algorithm (DSTA) for leader- following Discrete Homogeneous Multi-Agent System (homogeneous DMAS) with disturbance. – Derived the ultimate quasi-sliding mode using RL approach and also derived the sufficient condition for the global stability of the homogeneous DMAS. – Simulation and experimental implementation of both the protocols on a homogeneous DMAS setup comprise of 2-DOF flexible joint robotic arms. – Extensive comparative analysis for the consensus performance and robustness properties for each consensus protocol. The chapter is structured as follows: Briefing of graph theory and problem construction is discussed in Sect. 2. The DHSM consensus protocols using RL approach and DSTA along with the stability analysis are proposed in Sect. 3. The system description, the mathematical model of 2-DOF flexible joint robotic arm, and the experimental setup is discussed is in Sect. 4. The simulation and experimental results along with comparative analysis are discussed in Sect. 5, followed by conclusion in Sect. 6.

2 Graph Theory and Problem Construction 2.1 Graph Theory [32] The weighted digraph G = (V, E, A) consists of a set of N nodes. Where a separate set of edges are defined as V = {v1 , v2 , . . . , v N }, E ⊂ V × V and the adjacency matrix A = [b˜i j ] ∈ R N ×N . A line which is linked between vertex j and vertex i is defined as

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(v j , vi ), which means that the information passes from vertex j to vertex i. The weight b˜i j of connecting lines (v j , vi ) is nonnegative, if b˜i j > 0 and (v j , vi ) ∈ E, otherwise, b˜i j = 0. One assumption is that there is no self-loops present in this graph topology, i.e., b˜ii = 0, ∀i ∈ N˜ , where N˜ = {1, 2, . . . , N }. If (v j , vi ) ∈ E, then vertex j is said to be a neighbor vertex of i. Degree matrix has the information of the number of in or out edges of particular vertex and this matrix is present in a diagonal form which ˜ = diag{di } ∈ R N ×N with di =  ˜ is further defined as D j∈Ni bi j and the Laplacian ˜ matrix as L = D − A. Hence, L1 N = 0. The agent i can obtain the information from the agent j then this graph is called directed graph. While in the case of an undirected graph, the informations are exchanged bidirectionally. Let us take index 0 for leader agent and the follower agents indexes as 1, 2, . . . , N . The positive numbers b˜i0 , i = 1, . . . , N is used to represent the interaction of follower agents with leader agent. The interaction weight matrix known as pinning gain matrix is defined as B˜ = diag(b˜10 , . . . , b˜ N 0 ). If the follower agent i obtains the information from the leader agent, then the value of b˜i0 > 0, otherwise b˜i0 = 0. Lemma 1 ([23]) If the digraph G has a rooted spanning tree, then the matrix (L + B) is invertible.

2.2 Problem Construction of Leader-Following Homogeneous DMAS Consider the discrete linear system for ith agent xi (k + 1) = Gxi (k) + H (u i (k) + Di (k)) ∀i ∈ N,

(1)

where i = 1, . . . , N , G ∈ Rn×n and H ∈ Rn×m is the system matrix and input matrix of ith agent, respectively. State vector xi (k) ∈ Rn and the input vector u i (k) ∈ Rm , Di ∈ Rm is matched disturbance imposed on ith agent. Assumption 1 The interaction graph topology of the homogeneous DMAS is not changing with time. From (1), we may write global homogeneous DMAS as X (k + 1) = (I N ⊗ G)X (k) + (I N ⊗ H )(u(k) + D(k)),

(2)

where state vector X (k) = [x1 (k), x2 (k), . . . , x N (k)]T ∈ Rn N and input vector u(k) = [u 1 (k), u 2 (k), . . . , u N (k)]T ∈ Rm N , disturbance vector D(k) = [D1 (k), D2 (k), D3 (k), . . . , D N (k)]T ∈ Rm N , where by ||D|| ≤ Λ, Λ > 0, ⊗ denotes the Kronecker product. Further, Eq. (2) is written in simplified form as X (k + 1) = G˜ X (k) + H˜ (u(k) + D(k)), where G˜ = (I N ⊗ G), H˜ = (I N ⊗ H ). Lets define the leader agent dynamical system as

(3)

Discrete Higher-Order Sliding Mode Leader-Following Consensus Protocols …

x0 (k + 1) = Gx0 (k) + H u 0 (k).

243

(4)

where x0 (k) ∈ Rn is the state vector of the leader agent. Problem Statement: To design and analyze a DSMC consensus protocols using reaching law and DSTA for global homogeneous DMAS such that all the follower agents (3), perfectly synchronize with the leader agent (4), in finite time by exchanging their information with the neighborhood agent. The local error function for the leader-follower network is defined as [29] ei (k) =



bi j [xi (k) − x j (k)] + bi0 [xi (k) − x0 (k)].

(5)

j∈N

Using Eq. (5) and according to Lemma 1 the global error for consensus is defined as [29] ˜ + B) ˜ −1 )(L + B)) ˜ ⊗ In )x, ξ(k) = ((I N + D ˜ (6) Further, Eq. (6), can be rewritten as ˜ ξ(k) = (ζ ⊗ In )x.

(7)

where x˜ = X (k) − 1 N ⊗ x0 (k) ∈ Rn N . The eigenvalues of the weighted matrix ˜ + B) ˜ −1 )(L + B)) ˜ obtained using Gersgorin circle criteria [33], which ζ = ((I N + D reside in the unit circle. ˜ Eq. (7), can be rewrittene as Consider (ζ ⊗ In ) = β and substituting x, ξ(k) = β(X (k) − 1 N ⊗ x0 (k)).

(8)

3 DSM Protocol for the Consensus of Homogeneous DMAS In this section, inspired by [31, 34, 35], we proposed DHSM protocol using the reaching law approach. To understand the discrete higher-order sliding mode control, first, let us understand the concept of relative degree in discrete-time domain.

3.1 Review of Higher-Order Discrete Sliding Mode Control Having Relative Degree More Than One Relative degree of discrete-time systems can be easily understood from the continuoustime concept [30]. In discrete-time domain the derivative operator has been changed and becomes difference operator. Definition 1 ([31]) For a general discrete-time system defined as

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x˙ = gd (k, x(k), u(k)),

(9)

the output y(k) has relative degree r if y(k + r ) = jr (k, x(k), u(k)) and y(k + z) = jz (k, x(k)), ∀ 0 ≤ z < r , where u(k) is the control input and y(k + P) illustrates the P unit delay of output y. From the above definition, first time control input appears in the r th delay of the output y(k) but not prior to that. Simply for LTI system (G, H, E) example, it can be defined as C T G z−1 H = 0 ∀z = 1 to (r − 1) and C T G r H = 0. Let us consider a discrete-time Linear time invariant (LTI) system in the canonical form as x1 (k + 1) = G 11 x1 (k) + G 12 x2 (k) x2 (k + 1) = G 21 x1 (k) + G 22 x2 (k) + H2 u(k) + H2 D(k)

(10)

where x1 (k) ∈ Rn−m , x2 (k) ∈ Rm are the states and u(k) ∈ Rm is the control input. The disturbance D with known bound and assumed as ||D(k)|| ≤ Λ. System matrix parameters G 11 ∈ R(n−m)×(n−m) , G 12 ∈ Rm×(n−m) , G 21 ∈ Rm×(n−m) , G 22 ∈ Rm×m and H2 ∈ Rm×m and assumed det(H2 ) = 0. Written in standard canonicalform G 11 G 12 x(k + 1) = Gx(k) + H (u(k) + D(k)) for LTI system, where G = , and G 21 G 22   0 . For the system defined in Eq. (10), a relative degree-2 output can be H= H2 written as σ2 (k) = C2T x(k) (11) where C ∈ Rm×(n−m) . Further, advancing Eq. (11) σ2 (k + 1) = C2T (Gx(k) + H (u(k) + D(k)).

(12)

In order to obtain the second-order sliding mode control u must appear in k + 2 instant. Hence, C2T designed such that C2T H = 0 and C2T G H = 0. So Eq. (12) can be rewritten as (13) σ2 (k + 1) = C2T (Gx(k)). Now, control input u(k) disappears from the system dynamics Eq. (13). Further advancing Eq. (13), we may write σ2 (k + 2) = C2T G 2 x(k) + C2T G H (u 2 (k) + D(k)). Now control input u(k) can be derived from Eq. (14).

(14)

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245

3.2 Discrete Higher-Order Sliding Mode (DHSM) Protocol Using Reaching Law Approach for Consensus of DMAS The DHSM protocol using reaching law approach for consensus of homogeneous DMAS can be written in the form of Theorem 1 as under. Theorem 1 For the global homogeneous DMAS, (3) is said to achieve the leaderfollowing consensus in a finite time using the protocol given as u(k) = [−(csT β G˜ H˜ )−1 [csT β G˜ 2 X (k) + csT β(−1 N ⊗ (Gx0 (k + 1)))− Γ 2 s˜2 (k) + Γ ηsgn(˜s2 (k)) + ηsgn(˜s2 (k + 1))]], where (csT β G˜ H˜ )−1 is non-singular. Proof Sliding surface of ith agent for relative degree-2 is defined as s˜i2 (k) = csTi ei (k),

(15)

where csi is the sliding gain to be designed using LQ technique. Using Eq. (15), the global sliding surface is defined as s˜2 (k) = csT ξ(k),

(16)

where csT = I N ⊗ csTi . Using Eq. (8), we may write Eq. (16), as

Further,

s˜2 (k) = csT β(X (k) − 1 N ⊗ x0 (k)).

(17)

s˜2 (k + 1) = csT β(X (k + 1) − 1 N ⊗ x0 (k + 1)).

(18)

Substituting Eqs. (3) and (4), into Eq. (18), we may get s˜2 (k + 1) =csT β(G˜ X (k) + H˜ (u(k) + D(k)) − 1 N ⊗ x0 (k + 1)).

(19)

Now selecting csT such that csT β H˜ = 0 and csT β H = 0, so Eq. (19) rewritten as s˜2 (k + 1) = csT β(G˜ X (k) − 1 N ⊗ Gx0 (k)). Further, advancing Eq. (20) s˜2 (k + 2) = csT β(G˜ 2 X (k) + G˜ H˜ (u(k) + D(k)) − 1 N ⊗ Gx0 (k + 1)).

(20)

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Now inspired from [36], for linear system, the reaching law with relative degree-1 of ith agent can be written as s˜i1 (k + 1) = γi s˜i (k) − ηi sgn(˜si (k)) + f 1i (k),

(21)

From Eq. (21), reaching law with relative degree-2 for the consensus is defined as s˜i2 (k + 2) = γi2 s˜i2 (k) − γi ηi sgn(˜si2 (k)) −ηi sgn(˜si2 (k + 1)) + f 2i (k),

(22)

From Eq. (22), we may write the global reaching law for the global consensus of homogeneous DMAS as s˜2 (k + 2) = Γ 2 s˜2 (k) − Γ ηsgn(s˜2 (k)) − ηsgn(s˜2 (k + 1)) + f 2 (k),

(23)

where Γ = [γ1 , γ2 , γ3 , . . . , γ N ] ∈ Rn N , η = [η1 , η2 , η3 , . . . , η N ] ∈ Rn N > 0 are controller gain tuning parameter. f 2 (k) = csT β G˜ H˜ D(k) and | f 2 (k)| ≤ D. Comparing Eq. (21) with Eq. (23), we may write csT β(G˜ 2 X (k) + G˜ H˜ (u(k) + D(k))1 N ⊗ Gx0 (k + 1)) = Γ 2 s˜2 (k) − Γ ηsgn(s˜2 (k)) − ηsgn(s˜2 (k + 1)) + f 2 (k)

(24)

Further Eq. (24), can be derived in terms of the global leader-following consensus protocol as u(k) = [−(csT β G˜ H˜ )−1 [csT β G˜ 2 X (k) + csT β(−1 N ⊗ (Gx0 (k + 1))) − Γ 2 s˜2 (k) + Γ ηsgn(˜s2 (k))+ ηsgn(˜s2 (k + 1))]],

(25)

This completes the proof. The ultimate band δ2i of the sliding surface s˜i2 (k) for ith agent is obtained using Eq. (22), by considering s˜i2 (k + 2) = s˜i2 (k) and maximum disturbance f 2i (k) = D2i (k) s˜i2 (k) = γi2 s˜i2 (k) − γi ηi sgn(˜si2 (k))− ηi sgn(˜si2 (k + 1)) + D2i (k).

(26)

Considering δ2i = s˜i2 (k) in above Eq. (26), ultimate band is derived as δ2i =

(1 − γi )ηi + D2i (k) 1 − γi2

(27)

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247

3.3 DHSM Protocol for the Consensus of Homogeneous DMAS In this section, we propose DHSM protocol using DSTA for the consensus of homogeneous DMAS in the form of Theorem 2 as under. Theorem 2 For the global homogeneous DMAS (3), is said to achieve the leaderfollowing consensus in a finite time using the protocol given as u(k) = [−(csT β H˜ )−1 [csT β G˜ X (k) + csT β(−1 N ⊗ Gx0 (k)  + H u 0 (k))]] − D(k) − h | s˜ (k) |sgn(˜s (k)) + w(k),

(28)

where (csT β H˜ )−1 is non-singular. Proof Sliding surface of ith agent is defined as s˜i (k) = csTi ei (k),

(29)

where csTi is the sliding mode gain to be identified using LQ technique. Using (29), the global sliding surface can be written as s˜ (k) = csT ξ(k).

(30)

s˜ (k) = csT β(X (k) − 1 N ⊗ x0 (k)).

(31)

s˜ (k + 1) = csT β(X (k + 1) − 1 N ⊗ x0 (k + 1)).

(32)

Using Eq. (8), we may write

Further,

Inserting (3) and (4), into Eq. (32) s˜ (k + 1) = csT β(G˜ X (k) + H˜ (u(k) + D(k)) − 1 N ⊗ Gx0 (k) + H u 0 (k)).

(33)

For finding equivalent control (u eq ), substituting s˜ (k + 1) = 0 in the Eq. (33), we may write 0 = csT β(G˜ X (k) + H˜ (u eq (k) + D(k)) − 1 N ⊗ Gx0 (k) + H u 0 (k)). Further

u eq (k) = [−(csT β H˜ )−1 [csT β G˜ X (k) + csT β(−1 N ⊗ Gx0 (k) + H u 0 (k))]] − D(k).

(34)

(35)

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Now motivated from the discrete supertwisting algorithm [34], let us define discrete supertwisting algorithm based discrete sliding mode control for global homogeneous DMAS as  u s (k) = −h | s˜ (k) |sgn(˜s (k)) + w(k) w(k + 1) = w(k) − F T sgn(˜s (k)) (36) where, h = [h 1 , h 2 , h 3 , . . . , h N ] ∈ Rn N , F = [F1 , F2 , F3 , . . . , F N ] ∈ Rn N are gain parameters, T = [T1 , T2 , T3 , . . . , TN ] is the sampling time. Total consensus effort is the addition of u eq (k) + u s (k), therefore, using Eqs. (35) and (36), we may get u(k) = [−(csT β H˜ )−1 [csT β G˜ X (k) + csT β(−1 N ⊗ Gx0 (k)  + H u 0 (k))]] − D(k) − h | s˜ (k) |sgn(˜s (k)) + w(k).

(37)

This completes the proof. Next, the condition for global stability of the homogeneous DMAS with the proposed DSMC protocol defined in Eq. (37), is derived in the form of Theorem 3 as under. Theorem 3 The Lyapunov based global stability of leader-following consensus of homogeneous DMAS using protocol defined in Eq. (37), is guaranteed for the global error dynamics defined in Eq. (8), drives towards the global sliding surface (30) and remain on it for gain h, F > 0 and the condition is given as 0 ≤ α < s˜ T (k)˜s (k).

(38)

Proof Let us consider the global sliding surface defined in Eq. (31)

Further

s˜ (k) = csT β(X (k) − 1 N ⊗ x0 (k)).

(39)

s˜ (k + 1) = csT β(X (k + 1) − 1 N ⊗ x0 (k + 1)).

(40)

Lyapunov function is defined as V(k) = s˜ T (k)˜s (k).

(41)

Forward derivative function of Eq. (41), can be written as ΔVs (k) = V(k + 1) − V(k),

(42)

ΔVs (k) = s˜ T (k + 1)˜s (k + 1) − s˜ T (k)˜s (k),

(43)

It is required to ensure ΔVs (k) < 0 for the stability. Now substituting s˜ (k + 1) value (40) into (43), we get

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ΔVs (k) = [csT β(X (k + 1) − 1 N ⊗ x0 (k + 1))]T [csT β(X (k + 1) − 1 N ⊗ x0 (k + 1))] − s˜ T (k)˜s (k).

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(44)

Inserting the value of X (k + 1) and x0 (k + 1) from Eqs. (3) and (4), we get ΔVs (k) = [csT β[(G˜ X (k) + H˜ (u(k) + D(k)) − 1 N ⊗ Gx0 (k)) + H u 0 (k)]]T [csT β[(G˜ X (k) + H˜ (u(k) + D(k)) − 1 N ⊗ Gx0 (k)) + H u 0 (k)]] −˜s T (k)˜s (k). (45) Now substituting the protocol defined in Eq. (37) into Eq. (45), we may get   ΔVs (k) = [−h | s˜ (k) |sgn(˜s (k)) + w(k)]T ∗ [−h | s˜ (k) | sgn(˜s (k)) + w(k)] − s˜ T (k)˜s (k),

(46)

and denoting,   α = [−h | s˜ (k) |sgn(˜s (k)) + w(k)]T ∗ [−h | s˜ (k) | sgn(˜s (k)) + w(k)]. ΔVs (k) = α − s˜ T (k)˜s (k).

(47)

From Eq. (47), the term α can be set close to zero by selecting the proper value of h and F such that ΔVs (k) < 0. Hence the global stability of homogeneous DMAS is guaranteed. This completes the proof.

4 Experimental Setup 4.1 2-DOF Serial Flexible Joint Robotic Arm (2DOFSFJ) The 2-Degree-of-Freedom (DOF) Serial Flexible Joint (2DOFSFJ) mechanism is portrayed in Fig. 1. The system consists of 2 DC motors driving harmonic gearboxes (zero backlashes) and a two-bar serial linkage. Both the links are rigid. The first link is coupled to the primary drive by means of a versatile joint. At its end, the second harmonic drive is coupled to the second rigid drive by means of another flexible joint. Each motor and each versatile joint is equipped with optical encoders. Each flexible joint uses two springs which can be modified. A thumbscrew mechanism is accessible to maneuver every spring end to completely different anchor points on its support bars, as desired. The system parameters of each joint are given in Table 1.

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Fig. 1 2-DOF serial flexible joint robotic arm. Courtesy: Quanser Inc.

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4.2 Mathematical Model The dynamical equations of the flexible joints are solved using Euler-Lagrange method. The entire robotic arm system is divided into two stages (i) drive stage1 and (ii) drive stage-2. θ12 , θ21 represent the angular position of stage-1 with respect to drive stage-2 and stage-2 angular position with respect to stage-1. Similarly, θ˙12 , θ˙21 represent the relative angular velocity of two drive stages. Using linear EulerLagrange method, first, let us write the dynamic equation for driving stage-1 of 2-DOFSFL robotic arm system as θ¨11 (t) =

−K s1 θ11 (t) K s θ12 (t) B11 θ˙11 (t) kt21 Im 1 + 1 − + J11 J11 J11 J11

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−K s1 θ11 (t) K s θ12 (t) B12 θ˙12 (t) − 1 − . J12 J12 J12

From (51), the state space model for the driving stage-1 is obtained as

(48)

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Table 1 System parameters of 2DOFSFJ robotic arm System parameters First drive stage Im Kt τ θ11 θ˙11

Motor armature current (A) Torque constant (N m/A) Torque produced at the load shaft (N m) Angular position (rad) Angular velocity (rad/s) 1st rigid link absolute angular position link to driving stage-2 1st rigid link absolute angular velocity link to driving stage-2 Moment of inertia (kg m2 ) Moment of inertia (compounded with stage-2) (kg m2 ) Viscous damping coefficient (N m s/rad) Viscous damping coefficient (compounded with stage-2) (N m s/rad) Torsional stiffness constant (N m/rad)

θ12 θ˙12 J11 J12 B11 B12 Ks Second drive stage θ21 2nd rigid link absolute angular position link to driving stage-1 (rad) θ˙21 2nd rigid link absolute angular velocity link to driving stage-1 (rad/s) θ22 Angular position (rad) θ˙22 Angular velocity (rad/s) J21 Moment of inertia (compounded with stage-1) (kg m2 ) J22 Moment of inertia (kg m2 ) B21 Viscous damping coefficient (compounded with stage-1) (N m s/rad) B22 Viscous damping coefficient (N m s/rad)

⎡

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(49)

where G 1 and H1 are the system matrix and input matrix of 2DOFSFJ robotic arm stage-1, respectively. T represents the transpose of the matrix. Similarly, for the driving stage-2, the dynamics can be written using EulerLagrange equations of motion as θ¨21 (t) =

−K s2 θ21 (t) K s θ22 (t) B21 θ˙21 (t) kt2 Im 2 + 2 − + , J21 J21 J21 J21

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θ¨22 (t) =

K s2 θ21 (t) K s θ22 (t) B22 θ˙22 (t) − 2 − . J22 J22 J22

(50)

And the state space model for the driving stage-2 is obtained as ⎡

0 ⎢ 0 ⎢ −K G 2 = ⎢ s2 ⎣ J21 K s2 J22

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22 − −B J22 T K t2 ∈ Rn×m , 0 J21

(51)

where G 2 and H2 are the system matrix and input matrix of 2DOFSFJ robotic arm stage-2, respectively. To obtain the discrete-time model, the dynamic model (49) and (51), are discretized at sampling rate T = 0.002 and is obtained as ⎤ 0.9997 0.0002696 0.001865 1.818 × 10−7 ⎢7.81 × 10−5 0.9999 5.028 × 10−8 0.001999 ⎥ ⎥ G1 = ⎢ ⎣ −0.2634 0.2634 0.868 0.0002695 ⎦ 0.9993 0.07809 −0.07809 7.455 × 10−5 ⎡

  H1 = 0.0002673 3.545 × 10−9 0.2612 7.042e × 10−6

(52)

Similarly, for stage-2 ⎡

⎤ 0.9979 0.002078 0.001739 1.416 × 10−6 ⎢0.0007443 0.9993 4.63 × 10−7 0.001994 ⎥ ⎥ G2 = ⎢ ⎣ −1.983 1.983 0.7499 0.002075 ⎦ 0.7433 −0.7433 0.0006783 0.994   −8 H2 = 0.0004521 5.827 × 10 0.4315 0.0001149

(53)

4.3 Protocol Implementation The experimental setup shown in Fig. 2, comprises of four 2-DOF serial flexible joint robotic arms having the communication topology as shown in Fig. 3. Among four robotics arms systems, 3 systems including leader are virtual and 1 system is actual one. The 2DOFSFJ robotic arm indexed as 0 acts as leader and 1, 2, and 3 indexed act as followers. In this study, position (θ11 ) of stage-1 and position (θ21 ) of stage-2 are to be considered as a single agent parameter for consensus. The simulation and experimental analysis are carried out using Matlab R15 interfacing with QUARC software [37]. QUARC is the most suitable software to design, develop, and validate applications in the real-time domain on hardware using Matlab-Simulink.

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Fig. 2 Experimental setup for leader-follower consensus Virtual Leader Agent

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A square signal with amplitude 30 (deg) with frequency of 0.1 Hz for stage-1 of 2DOFSFJ and an amplitude 20 (deg) with frequency of 0.1 Hz for stage-2 are given to the leader agent as a reference. The laplacian matrix L, adjacency matrix A, pinning ˜ are given as gain matrix B˜ and degree matrix D ⎡

⎤ ⎡ ⎤ 2 00 000 L = ⎣−1 0 0 ⎦ , A = ⎣1 0 0 ⎦, −1 0 0 100 ⎡

⎤ ⎡ ⎤ 100 200 ˜ = ⎣0 0 0 ⎦ . B˜ = ⎣0 0 0 ⎦ , D 000 000

(54)

The robustness property of the derived protocol for the consensus of homogeneous DMAS is checked by applying a slowly varying matched disturbance with magnitude Di (k) = 0.0002 ∗ cos(0.01k) to ith agent.

5 Result Discussion In this section, the simulation and experimental results are discussed for DHSM protocols using RL approach and DSTA. Both the DHSM protocols are compared with DFSM protocol for global consensus of homogeneous DMAS.

5.1 Result Discussion Using Reaching Law The protocol defined in (25), is applied to the homogeneous DMAS comprises of 2-DOF flexible joint robotic arm. The controller gains for the follower agent of homogeneous DMAS are chosen as Γ = [0.03 0.03 0.03], η = [0.05 0.05 0.05], respectively. The sliding gain for surface of ith follower agent for the stage-1 and stage-2 are calculated using LQ technique as csTi 1 = [23.8619 − 2.1913 1.7969 3.2838] and csTi 2 = [22.06 − 6.62 1.66 0.4056], respectively. Figures 4 and 5, show the simulation and experimental results of position consensus for 2DOFSFJ robotic arm of stage-1 and stage-2, respectively. It is inferred from the results that the follower agents perfectly synchronize with the leader. Figures 6 and 7 show the simulation and experimental results of sliding surface of the individual ith agent of homogeneous DMAS comprises 2DOFSFJ robotic arm stage-1 of stage-2, respectively. It is inferred from the results that the sliding variable is remaining in the band. Figures 8 and 9 show the simulation and experimental results of protocol (u) of the individual ith agent of homogeneous DMAS for stage-1 and stage-2, respectively.

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Fig. 6 Surface (S) of individual agent of 2DOFSFJ robotic arm stage-1

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5.2 Result Discussion Using DSTA The protocol defined in (37) is applied to the homogeneous DMAS comprises of 2DOF flexible joint robotic arm. The controller gains for the follower agent of homogeneous DMAS are chosen as h = [0.03 0.03 0.03], F = [0.015 0.015 0.015], respectively. The sliding gain for surface of ith follower agent for the stage-1 and stage-2 are calculated using LQ method as csTi 1 = [31.5313 − 5.2904 2.9705 3.8968] and csTi 2 = [19.9708 − 6.1516 1.6612 0.3431] respectively. Figures 10 and Fig. 11 show the simulation and experimental results of position consensus for stage-1 and stage-2 of homogeneous DMAS. It is observed from the results that all the follower agents attained the consensus with the leader in finite time steps. Figures 12 and 13 show the simulation and experimental results of sliding surface of ith follower agent for homogeneous DMAS of stage-1 and stage-2, respectively. It

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Fig. 12 Surface of individual agent of 2DOFSFJ robotic arm stage-1

Fig. 13 Surface of individual agent of 2DOFSFJ robotic arm stage-2

is inferred from the results that we are getting less chattering using DSTA compared to DFSM but higher than the reaching law approach. Figures 14 and 15 show the protocol (u) of the individual follower agent for stage-1 and stage-2, respectively. The proposed protocols are compared with DFSM protocol given in [29], for the consensus performance and it is defined as ˜ −1 [csT γ AX ˜ (k) + csT γ (−1 N ⊗ Ax0 (k)) u(k) = [−(csT γ B) −(1 − QT )˜s (k) + E T sgn(˜s (k))]] − D(k),

(55)

˜ −1 is non-singular and E = [1 , 2 , 3 , . . . ,  N ] ∈ Rn N > 0, Q = where (csT γ B) [q1 , q2 , q3 , . . . , q N ] ∈ Rn N > 0. The controller gains for follower agents are chosen as E = [0.5 0.5 0.5], Q = [0.03 0.03 0.03], respectively. The sliding gain for surface of ith follower agent for the stage-1 and stage-2 are calcu-

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Fig. 14 Consensus protocol (u) of 2DOFSFJ robotic arm stage-1

Fig. 15 Consensus protocol (u) of 2DOFSFJ robotic arm stage-2

lated using LQ technique as csTi 1 = [29.8664 − 5.9043 2.9685 − 0.0043] and csTi 2 = [21.4614 − 17.7421 1.1829 0.4598] respectively. Figures 16, 17, 18, and 19 show the simulation and experimental results of position consensus and sliding surface for stage-1 and stage-2, respectively. It is inferred from the results that follower agents attained the consensus with the leader agent in finite time step and the chattering is more in DFSM as compared to proposed consensus protocols. Figures 20 and 21 show the consensus protocol (u) follower agent for stage-1 and stage-2, respectively, which is further applied to individual agent into the same network of homogeneous DMAS. From the results, it is observed that the proposed consensus protocols are far better compared to DFSM. Tables 2 and 3 show the simulation and experimental error performance indices for the leader-following consensus of homogeneous DMAS using the proposed two consensus protocols and DFSM consensus protocol. It is inferred from the tables that consensus protocol hav-

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Fig. 20 Consensus protocol (u) of 2DOFSFJ robotic arm stage-1

Fig. 21 Consensus protocol (u) of 2DOFSFJ robotic arm stage-2

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Table 2 Error performance index (ISE) for simulation study Stage-1 Stage-2 ISE DHSM with RL DSTA DFSM

e11 2.04 × 10−16 8.80 × 10−16 2.84 × 10−12

e21 2.04 × 10−16 2.58 × 10−16 2.84 × 10−12

e31 2.04 × 10−16 2.58 × 10−16 2.84 × 10−12

e12 6.17 × 10−16 7.80 × 10−12 1.25 × 10−13

e22 6.17 × 10−16 7.80 × 10−12 1.25 × 10−13

e32 6.17 × 10−16 7.80 × 10−12 1.11 × 10−3

e22 2.03 × 10−3 4.14 × 10−3 4.48 × 10−2

e32 2.03 × 10−3 4.14 × 10−3 4.48 × 10−2

Table 3 Error performance index (ISE) for experimental study Stage-1 Stage-2 ISE DHSM with RL DSTA DFSM

e11 2.60 × 10−4 3.77 × 10−4 8.76 × 10−4

e21 2.60 × 10−4 8.34 × 10−4 4.46 × 10−4

e31 2.60 × 10−4 8.34 × 10−4 4.46 × 10−4

e12 2.03 × 10−3 2.43 × 10−3 4.25 × 10−2

ing reaching law approach is more efficient compared to other consensus protocols. The results also reveal that the DSTA consensus protocol performs better compared to DFSM.

6 Conclusion In this chapter, two discrete higher-order sliding modes (DHSM) protocols are proposed for leader-following consensus of the discrete homogeneous multi-agent system (homogeneous DMAS) configured with a fixed, directed interaction graph topology. The protocols are designed using (i) reaching law approach and (ii) discrete supertwisting algorithm for the global consensus of homogeneous DMAS. The proposed protocols synchronize the follower agents with the leader agent in finite time steps. The protocols are validated in simulation, as well as experimentally, on homogeneous DMAS comprising of 2-DOF serial flexible joint robotic arms. The comparative results reveal that the proposed DHSM protocol using reaching law approach outperforms the DHSM protocol with supertwisting algorithm. Further, the consensus performance due to DHSM protocols is better than consensus due to DFSM protocol. Finally, the robustness property of each protocol is checked by applying slowly varying matched disturbance to each follower agents.

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22. Chen, K., Wang, J., Zhang, Y., Liu, Z.: Leader-following consensus for a class of nonlinear strick-feedback multiagent systems with state time-delays. IEEE Trans. Syst. Man Cybern. Syst. 1–11 (2018) 23. Ren, C.-E., Chen, C.P.: Sliding mode leader-following consensus controllers for second-order non-linear multi-agent systems. IET Control Theory Appl. 9(10), 1544–1552 (2015) 24. Yu, W., Wang, H., Cheng, F., Yu, X., Wen, G.: Second-order consensus in multiagent systems via distributed sliding mode control. IEEE Trans. Cybern. 47(8), 1872–1881 (2017) 25. Mu, B., Zhang, K., Shi, Y.: Integral sliding mode flight controller design for a quadrotor and the application in a heterogeneous multi-agent system. IEEE Trans. Ind. Electron. 64(12), 9389–9398 (2017) 26. Mahmoud, M.S., Khan, G.D.: Leader-following discrete consensus control of multi-agent systems with fixed and switching topologies. J. Frankl. Inst. 352(6), 2504–2525 (2015) 27. Wang, Y., Cheng, L., Wang, H., Hou, Z.-G., Tan, M., Yu, H.: Leader-following consensus of discrete-time linear multi-agent systems with communication noises. In: 2015 34th Chinese Control Conference (CCC). IEEE (2015) 28. Zhang, L., Chen, M.Z., Su, H.: Observer-based semi-global consensus of discrete-time multiagent systems with input saturation. Trans. Inst. Meas. Control 38(6), 665–674 (2015) 29. Patel, K., Mehta, A.: Discrete-time sliding mode control for leader following discrete-time multi-agent system. In: IECON 2018—44th Annual Conference of the IEEE Industrial Electronics Society. IEEE (2018). https://doi.org/10.1109/iecon.2018.8591273 30. Levant, A.: Principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007) 31. Chakrabarty, S., Bandyopadhyay, B., Bartoszewicz, A.: Discrete-time sliding mode control with outputs of relative degree more than one. In: Recent Developments in Sliding Mode Control Theory and Applications. InTech (2017) 32. Patel, K., Mehta, A.: Discrete-time sliding mode protocols for leader-following consensus of discrete multi-agent system with switching graph topology. Eur. J. Control. 51, 65–75 (2020). https://doi.org/10.1016/j.ejcon.2019.06.011 33. Varga, R.S.: Geršgorin and His Circles. Springer, Berlin, Heidelberg (2004) 34. Salgado, I., Kamal, S., Bandyopadhyay, B., Chairez, I., Fridman, L.: Control of discrete time systems based on recurrent super-twisting-like algorithm. ISA Trans. 64, 47–55 (2016) 35. Moreno, J.A., Osorio, M.: A Lyapunov approach to second-order sliding mode controllers and observers. In: 2008 47th IEEE Conference on Decision and Control, pp. 2856–2861 (2008) 36. Gao, W., Wang, Y., Homaifa, A.: Discrete-time variable structure control systems. IEEE Trans. Ind. Electron. 42(2), 117–122 (1995) 37. Sharma, S., Srivastava, D., Patel, K., Mehta, A.: Design and implementation of second order sliding mode controller for 2-DOF flexible robotic link. In: 2018 2nd International Conference on Power, Energy and Environment: Towards Smart Technology (ICEPE), pp. 1–6 (2018)

State and Disturbance Estimation Using Fast Output Sampling Approach for Robust Motion Control Systems Surajkumar Sawai and Shailaja Kurode

Abstract In this paper, Multirate output feedback (MROF) is used for simultaneous estimation of states and disturbance. Augmenting the state-space model by considering the disturbance as an extended state. MROF is designed for extended system to get the estimates of states and disturbance. Effectiveness of the method is shown by simulation and experimentation.

1 Introduction Precise motion control is important in many industrial applications such as manipulator arms, CNC machines, XY positioner, Permanent-magnet linear motors, wafer scanner system, inspection machine, and many more [1–5]. Here part of the system is moved in controlled and accurate manner in linear or rotary fashion, depending upon the need. Motion control consists of control of motion directions and expected position of the system. If load varies, the performance gets affected. Similarly variation of supply voltages, neglected gears nonlinearity, friction affect the performance. The design of robust controller for motion control is one of the important concerns for achieving robust performance. It must ensure accuracy against disturbance, while operating at different speeds. This calls for robust controller. Robust controller not only guarantees stability of system but also robust performance [6]. Proportional-integral-derivative (PID) controllers are commonly used for controller design [7, 8]. However, these controller are not robust. Statefeedback controller is easy to design but needs information of all states to implement it. Also it does not always guarantee robust performance. It is essentials to operate motion control system (MCS) to ensure stability and robustness. One of the factors that affects the performance of system is disturbance. S. Sawai (B) · S. Kurode College of Engineering, Pune, India e-mail: [email protected] S. Kurode e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_11

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Disturbance includes load variation, supply variation, neglected gear and friction nonlinearities, uncertain parameters, etc. The disturbance may be vanishing or nonvanishing. To compensate its effect it is required to have its measure and its compensation in control. Appropriate controller, i.e., Feedforward [9] and feedback controller [10] along with disturbance compensation are used, if the disturbance affecting the system is known or measurable. However in most of cases disturbance is not measurable hence it is needed to estimate it. Robust controller reported in literature are adaptive control [11], sliding mode control [12], fuzzy-based method [13]. In order to implement above controller information of all states is required which is not always possible. Hence observers are used to get information on states. For implementing control digitally, discrete controller are designed. Discrete-time representation of system is used for this purpose [14]. To implement discrete state feedback control, information of states of discrete system is required. Many methods are available in literature, these yield information of states [15–18]. These are basically dynamic estimation methods. Multirate output feedback (MROF) however is algebraic approximation. It considers stack of output which is sampled at faster rate and control input at previous sampling instant to construct states algebrically [19, 20]. It provides accurate state estimation for the linear nominal system [21, 22]. In [23, 24] states are estimated using the MROF but estimation of the disturbance is not attempted. Input and Output are sampled at same rate in most of the practical systems. Systems where in control signal and output of system are sampled at two different sampling rate is known as multirate systems [25]. Figure 1 shows the pictorial representation of multirate system [26]. Here output is sampled at faster rate, i.e., Δ s and control input is sampled at slow rate, i.e., τ s. Stacks of fast sampled output and input can be used to construct information of states, which is further used to synthesis control. This method is named as multirate output feedback. MROF has gain popularity as it required less processing time due to its algebraic nature, moreover it is simple to implement. MROF can be used to synthesize the controller for the linear time invariant system [27], In [28] it is shown that when function observer is combined with MROF then controller is designed with less number of output sample. Application specific contributions can be found in [29–33]. In this work, MROF is examined to estimate the states and disturbance of uncertain system.

2 Motivation Last two decades witnessed advancement in microcontrollers and digital processors, because of which implementation of controller digitally is feasible and economical option. Implementation of controller, digitally provides many advantages such as low cost, flexibility, and ease of implementation. This fact has motivated to design discrete control for motion control purpose. Discrete state feedback controller is easy to design but needs information of all states for implementation. Moreover system

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uncertainties and external disturbance affect the performance. Investigation of robust performance in such scenario has been driving motivation for investigation of state and disturbance estimation using MROF and devising control for compensation of disturbance.

3 Problem Statement Typical motion control system is represented by Single Input Single Output (SISO) second-order system with inherent uncertainties as below  x˙ = Ax + bu + d y = cx

(1)

where x is state vector and x ∈ 2×1 , u is input and u ∈ 1×1 , and y is output and y ∈ 1×2 , d is lumped matched uncertainties. The system matrix A and input matrix b are     0 0 1 ,b = (2) A= a3 a1 a2

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where a1 , a2 and a3 are system parameters. The discrete representation of (1), sampled at τ rate is x(k + 1) = Φτ x(k) + Γτ u(k) + Dτ d(k) y(k) = cx(k)

 (3)

where x(k) is state vector and x(k) ∈ 2×1 , u(k) is input and u(k) ∈ 1×1 and y(k) is output and y(k) ∈ 1×2 . Φτ , Γτ , Dτ and c are matrices of appropriate dimension, d(k) is lumped disturbance. The pair (Φτ , Γτ ) is controllable and the pair (Φτ , c) is observable. It is required to estimate x(k) and d(k) and design the controller to get robust performance.

4 MROF for State and Disturbance Estimation State-space representation of (1) in discrete-time system using Euler’s first-order approximation is 

        x1 (k + 1) 1 τ x1 (k) 0 0 = + u(k) + d(k). a1 τ (1 + a2 τ ) x2 (k) a3 τ τ x2 (k + 1)

(4)

In MROF scheme, output is sampled at faster rate as compare to control input. The idea here is that disturbance in the system is considered as an additional state of system. MROF is then examined for estimation of states of extended system. Defining x3 (k) = d(k) as extended state, system in (4) can be written as ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎤⎡ ⎡ ⎤ x1 (k + 1) 1 τ 0 0 x1 (k) 0 ⎣x2 (k + 1)⎦ = ⎣a1 τ (1 + a2 τ ) τ ⎦ ⎣x2 (k)⎦ + ⎣a3 τ ⎦ u(k) + ⎣0⎦ ed (k) x3 (k + 1) 0 0 1 x3 (k) 0 1 ⎡ ⎤ (5)

x1 (k) y(k) = 1 0 0 ⎣x2 (k)⎦ . x3 (k) This is called τ system. Here, ed (k) = d(k + 1) − d(k). The above system can be written as xe (k + 1) = Φτ xe (k) + Γτ u(k) + Dτ ed (k) y(k) = cxe (k)

 (6)

where xe (k) is state vector of extended system and xe (k) ∈ 3×1 . A system represented in (6) is obtained by sampling (4) at τ rate. If the same system is sampled at Δ rate.

State and Disturbance Estimation Using Fast Output Sampling …

xe (k + 1) = ΦΔ xe (k) + ΓΔ u(k) + DΔ ed (k) y(k) = cxe (k)

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 .

(7)

This is called Δ system. ΦΔ , ΓΔ and DΔ are corresponding matrices of Δ system. Consider the control input of system is sampled at τ sec and output of system is sampled at Δ = Nτ sec, where N is greater than or equal to observability index of system [34]. At (k − 1)th instant (8) y(k − 1) = cxe (k − 1). At ((k − 1) + Δ)th instant, xe ((k − 1) + Δ) = ΦΔ xe (k − 1) + ΓΔ u(k − 1) + DΔ ed (k − 1)

(9)

and the corresponding output is y((k − 1) + Δ) = cxe ((k − 1) + Δ) = cΦΔ xe (k − 1) + cΓΔ u(k − 1) + cDΔ ed (k − 1).

(10)

At ((k − 1) + 2Δ)th instant, xe ((k − 1) + 2Δ) = ΦΔ xe ((k − 1) + Δ) + ΓΔ u(k − 1) + DΔ ed ((k − 1) + Δ).

(11)

It is assumed that d(k − 1)  d(k − 1 + Δ), Substituting (9), to get xe (k − 1 + 2Δ) =ΦΔ (ΦΔ xe (k − 1) + ΓΔ u(k − 1) + DΔ ed (k − 1)) + ΓΔ u(k − 1) + DΔ ed (k − 1) and

(12)

y((k − 1) + 2Δ) = cxe ((k − 1 + 2Δ)) = cΦΔ2 xe (k − 1) + cΦΔ ΓΔ u(k − 1) + cΦΔ DΔ ed (k − 1) + cΓΔ u(k − 1) + cDΔ ed (k − 1).

(13)

As we know that control input to the system between two sampling instant is constant, this fact can be utilized to construct the states of τ system from output sampled at Δ rate system [27, 35]. Now defining the stack of three outputs ⎡

⎤ y(k − 1) y(k) = ⎣ y((k − 1) + Δ) ⎦ . y((k − 1) + 2Δ)

(14)

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From Eqs. (8), (10) and (12) the above becomes y(k) = C0 xe (k − 1) + D0 u(k − 1) + E0 ed (k − 1) where C0 , D0 and E0 are ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ c 0 0 ⎦ , E0 = ⎣ ⎦. cΓΔ cDΔ C0 = ⎣cΦΔ ⎦ , D0 = ⎣ 2 cΦΔ c(ΦΔ + I )ΓΔ c(ΦΔ + I )DΔ

(15)

(16)

For slow varying disturbance d(k) − d(k − 1) is very small, hence E0 ed (k − 1)  0, therefore (15) becomes y(k) = C0 xe (k − 1) + D0 u(k − 1).

(17)

From Eq. (17) we can obtain the state x(k − 1) in term of output y(k) and control input u(k − 1) as below xe (k − 1) = (C0T C0 )−1 C0T [y(k) − D0 u(k − 1)].

(18)

Substituting the Eq. (18) in (6) to get the estimation of x(k) algebraically xˆ e (k) = Φτ [(C0T C0 )−1 C0T [y(k) − D0 u(k − 1)]] + Γτ u(k − 1) + Dτ ed (k). (19) For slow varying disturbance Dτ ed (k)  0 therefore xˆ e (k) = Φτ [(C0T C0 )−1 C0T [y(k) − D0 u(k − 1)]] + Γτ u(k − 1)

(20)

= L y yk + Lu u(k − 1) where

L y = Φτ (C0T C0 )−1 C0T Lu = Γτ − Φτ (C0T C0 )−1 C0T D0 .

(21)

Equation (20) gives the estimation of x1 (k), x2 (k) which are states and x3 (k) which is disturbance. The system represented in (3) is controlled using state feedback controller that uses estimated state. This is called controller without disturbance compensation. u 1 (k) = [−k1 ex1 (k) − k2 ex2 (k)]. The error states are defined as

(22)

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ex1 (k) = x1d (k) − xˆ1 (k) ex2 (k) = x2d (k) − xˆ2 (k) where x1d (k) and x2d (k) are desired values of x1 (k) and x2 (k). Effect of disturbance can be compensated by augmenting the control as below u 2 (k) = [−k1 ex1 (k) − k2 ex2 (k)] − τ d(k).

(23)

Using disturbance estimate above equation takes the form u 2 (k) = [−k1 ex1 (k) − k2 ex2 (k)] − τ xˆ3 (k).

(24)

This is the proposed controller with disturbance compensation.

5 Simulation Results To validate method, motion control system configured on Industrial mechatronic drive unit (IMDU) is considered. The mathematical model of it is, x˙1 = x2 x˙2 = 201.1u + d,

(25)

where x1 is angular position θ and x2 is angular velocity. The discrete-time representation of the above sampled at τ = 0.06 s is 

        1 0.06 x1 (k) 0 0 x1 (k + 1) = + u(k) + d(k). 0 1 12.066 τ x2 (k + 1) x2 (k)

(26)

Performance of estimator and controller were tested in simulation. Multirate chosen were τ = 0.06 s and Δ = 0.02 s. Φτ and Γτ of extended system are as below ⎡

⎤ ⎡ ⎤ 1 0.06 0 0 Φτ = ⎣0 1 0.06⎦ , Γτ = ⎣12.066⎦ . 0 0 1 0

(27)

The L y and Lu from (20) are ⎡

⎤ ⎡ ⎤ −0.002 0.003 0 0 L y = 1 × 103 ⎣ 0.1 −0.25 0.15⎦ , Lu = ⎣ 0 ⎦ . 2.5 −5 2.5 −201.1

(28)

272 Fig. 2 Estimation of state x1 (k) using MROF in simulation

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The system in (25) was simulated in MATLAB. To test the performance of estimator, control applied was u(k) = 0.5. Figures 2, 3, and 4 show performance of the estimator. It is evident that estimated states and disturbance follows actual states and disturbance respectively. The controller in (22) and (23) were tested using estimated information from (20). Control gain k1 and k2 were chosen as 0.5870 and 0.0943 to get pole at 0.6 and 0.75. External disturbance d(k) = 10 sin(0.15k) was deliberately added. While testing performance of controller the desired states xd (k) = [sin(0.15k) 0.94 cos(0.15k)]T were considered, Figs. 5, 6 and 7 show simulated performance of controller. It can be seen from Figs. 5 and 6 that the proposed controller u 2 (k) provides more robust performance. Table 1 show quantitative analysis. It is very obvious that proposed controller is more accurate and consumes less control energy.

State and Disturbance Estimation Using Fast Output Sampling … Fig. 5 Evolution of state x1 (k) with controller u 1 (k) and u 2 (k) in simulation

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Fig. 7 Comparison of the evolution control input u(k) without and with disturbance compensation

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Table 1 Qualitative analysis in simulation Controller ||u||2 State feedback control 0.4989 u 1 (k) State feedback control 0.4853 u 2 (k)

||u||∞

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Fig. 8 Experimental setup Fig. 9 Evolution of x1 (k) in experiment

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Table 2 Qualitative analysis in experiment Controller ||u||2 State feedback control 88.27 u 1 (k) State feedback control 86.72 u 2 (k)

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16,772

6 Experimental Results The Industrial Mechatronic Drive Unite (IMDU) in laboratory is used to test and validate the proposed method. Figure 8 shows experimental setup. It consists of two motor shaft and two free-spinning shaft with optical encoder. It is equipped with Data acquisition card and processor to establish real-time interface. The controller in (22) and (23) were implemented experimentally on the set up. The gains and disturbance were same as used in simulation with xd (k) = [100 sin(0.15k) 94.3 cos(0.15k)]T . Figures 9, 10 and 11 show experimental results. Figure 9 show evolution of state x1 (k). Figure 10 shows estimated disturbance and Fig. 11 depict the control effort (Table 2).

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7 Conclusions MROF has been investigated for state and disturbance estimation for uncertain system in simulation and experimentation. MROF gives fairly accurate estimates of states and disturbance of uncertain systems. Proposed controller gives better results as compared to conventional state feedback controller. The proposed method yields 76.42% improvement in accuracy and 1.75% reduction in control effort as compared to the controller without compensation.

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Accurate Position Regulation of an Electro-Hydraulic Actuator via Uncertainty Compensation-Based Controller Ramón I. Verdés, Alejandra Ferreira de Loza, Luis T. Aguilar, Ismael Castillo, and Leonid Freidovich Abstract Electro-hydraulic actuators are complex systems with uncertainties in their parameters and disregarded dynamics due to its complexity. This paper presents a disturbance observer-based controller method for the accurate position regulation of an electro-hydraulic actuator. To this aim, a super-twisting algorithm-based observer identifies the plant uncertainties and neglected dynamics, theoretically, in finite-time. Thus, a compensation based controller is designed to counteract the uncertainty and neglected dynamics effects through feedback, improving the position regulation accuracy. The closed-loop analysis is carried out using Lyapunov theory. The feasibility of the controller is validated through high-fidelity simulations and experiments in a forestry crane.

1 Introduction Electro-hydraulic actuators are widely used in heavy-duty machinery dedicated to agricultural, construction, industry, fishing, digging, and forestry tasks, to mention a few. Electro-hydraulic actuators possess attractive features: they may exert high actuation forces during long operating periods in rough conditions, and they offer a good trade-off between the delivered force and the actuator size [1]. Aviation, lifts, pick-and-place tasks, e.g., ships and forestry cranes) are examples of heavyduty tasks where precision, repeatability, stability, and robustness are also required. However, electro-hydraulic actuators are complex systems with uncertainties and nonlinearities, which change along the time due to parameter variations, pressure

R. I. Verdés · A. Ferreira de Loza (B) · L. T. Aguilar Instituto Politécnico Nacional—CITEDI, Avenida Instituto Politécnico Nacional No. 1310, Colonia Nueva Tijuana, 22435 Tijuana, Mexico e-mail: [email protected] I. Castillo · L. Freidovich Department of Applied Physics and Electronics, Umeä University, SE-901 87 Umeä, Sweden © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_12

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drops, temperature fluctuations, etc. Therefore, the control of such systems is a challenging topic. Several robust control strategies have been proposed to undertake the position tracking problem in electro-hydraulic systems. For example, Mili´c et al. [2] propose a robust H∞ controller for an electro-hydraulic system. A linearized model of the system was considered for the control synthesis. Similarly, Fallahi et al. [3] proposed an H∞ control strategy for precise position control on an electro-hydraulic actuator based on a linearization scheme. However, linearization assumption may not be affordable in practice due to the parameter variations, which cannot be safely disregarded. Alternatively, Puglisi et al. [4] verified the robustness of a PI controller by an extensive simulation campaign under several working points and parameter variations. In this respect, sliding-mode control approaches are an attractive alternative to cope with uncertainties and parameter variations. Sliding mode control offers insensitivity and finite-time convergence characteristics [5–7] and copes with matched uncertainties using a discontinuous control action; therefore, the chattering effect may occur. An alternative to diminishing chattering was proposed by Tahoumi et al. [5] proposing a hybrid control scheme that evolves from a discontinuous twisting controller to a continuous linear PD controller. The control scheme switches depending on the accuracy of the error. A different approach is to combine a nominal controller with a robust compensation term to add robustness while ensuring a precise control task, such as in Komnsta et al. [6] and Yung et al. [7]. In [6], a nominal control action and an integral sliding-mode compensator were brought together. Whereas in [7] proposed the combination of a proportional controller with a relay controller and a disturbance compensator. These approaches, as mentioned earlier, still present chattering. To overcome the detrimental effect of chattering, high-order sliding-mode techniques are considered [8]. Schmidt and Andersen [9] proposed twisting and super-twisting second-order sliding-mode controllers to deal with the position tracking of a hydraulic-valve cylinder drive. Nevertheless, the authors in [9] considered the continuous approximation of the sign-function leading to the loss of robustness. Shtessel et al. [10] proposed an adaptive gain super-twisting to control an electro-pneumatic actuator. In practice, however, only a real second-order sliding-mode was achieved. High-order sliding-mode (HOSM) observers gave a significant boost to control uncertain systems. HOSM observers are exploited to estimate the unknown parameters and disregarded dynamics in uncertain systems, theoretically in finite-time, [11–13]. In [13] a second-order observer estimates the unknown load forces in a hydraulic actuator. In this regard, HOSM compensation based methods are an alternative approach to deal with neglected dynamics, uncertainties, and parameter variations. Compensation-based controllers exploit the robustness of HOSM observers to estimate the uncertainties, later on, the reconstructed signals are feedback through the controller to compensate for their effects. Due to its appealing characteristics, compensation-based controllers using HOSM techniques have been tested experimentally in diverse applications, see, for instance, [14–17]. Particularly, in [18] dealt with the position regulation of a hydraulic actuator. In [19] dealt with disturbance identification for an electro-hydraulic actuator. To this aim, the authors count on a detailed model of the system, position, velocity,

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and pressure measurements. Still, many practical problems only count on position measurements. The precise position control of an electro-hydraulic actuator is considered in [20, 21]. In the same spirit that [19], both [20, 21] consider a detailed model that contemplates the pressure dynamics and their measurements. Therefore, the authors combine a disturbance/parameter identification procedure with a backstepping controller. In this way, the backstepping strategy can reach, and therefore compensate, the disturbances affecting the piston dynamics. Yao et al. [20] uses an adaptation scheme to identify the parameters, whereas [21] uses a HOSM differentiator. The main shortcoming, however, is the complexity in control law computation due to backstepping. Alternatively, in this work, we propose a simplified approach which only relies on position measurements. The contribution of this paper resides on the control of electro-hydraulic actuators facing up with neglected dynamics, which may lead to a miscalculation of the controller gains. In hydraulic systems, the overestimation of controller gains spoils the control performance; whereas a controller gain underestimation impacts on the error accuracy. Motivated by the previous issues, the goal of this paper is to propose a robust compensation-based controller to solve the position regulation problem of an electro-hydraulic actuator. To this aim, a super-twisting observer identifies the uncertainties and disregarded dynamics. Later on, the identified signal is injected through the control input to counteract their effects. As a result, the position accuracy is improved. The closed-loop stability is carried out using Lyapunov theory. The advantage of the proposed method is twofold: (1) the controller signal is continuous avoiding this way the tarnishing effect of chattering, (2) the accuracy of the position error is improved without increasing the controller gain. Simulations and experimental results for a hydraulic crane illustrate the effectiveness of the proposed method. The paper is organized as follows. Section 2 introduces the model description and problem statement. Section 3 proposes the control methodology; there, the stability analysis is carried out. Section 4 presents simulation and experimental results. Finally, Sect. 5 summarizes the conclusions.

2 Dynamic Model and Problem Statement Double acting hydraulic cylinders drive most of the heavy-duty mechanical systems due to high-force requirements in both directions of travel. In such systems, it is crucial to take into consideration the dynamics of the actuator for control and automation purposes. In this section, we provide a dynamic model and essential properties of a double-acting hydraulic cylinder (Fig. 1) with symmetrical spool and constant supply pressure. The supply pressure (Ps ) and the return or exit pressure (Pt ) are constant values settled by the hydraulic service unit. The spool valve position xv (t) inside its sleeve is governed by an input voltage u(t). The spool position determines the flows Q A (t) and Q B (t) to chamber A, and chamber B, respectively. The flow Q A (t) is

282 Table 1 Physical parameters Nomenclature x x˙ xv PA , PB Ps , Pt Q A, Q B V A (x), VB (x) V A (0), VB (0) A A, AB β cv xo Fh u t ∈ R≥0

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Definition Piston position Piston velocity Spool position Pressure in chambers A, B Supply pressure, tank pressure Flux to (from) chamber A, B Volume of chamber A, B Initial volume of chamber A, B Area of side A, B of piston Oil bulk modulus Discharge coefficient Overlap value Hydraulic force Control signal to the valve Time

positive when oil goes into chamber A and the flow Q B (t) is positive when the oil goes out of chamber B. For protection purposes, the manufacturer constraints the flows Q A (t) and Q B (t) up to a maximum flow through each valve. Thus the following property results. Property 1 There exists a positive constant q¯ such that [Q A Q B ] ≤ q. ¯ Since the length of the rod physically restricts the position of the piston x(t), the following property holds. Property 2 There exists an a priori known upper bound x¯ such that 0 ≤ x(t), |x| ˙ ≤ x. ¯ The expressions

V A (x) = V A (0) + x A A VB (x) = VB (0) − x A B

(1)

specify the volumes of the chambers A and B at a given position x(t) where V A (0) and VB (0) are the volumes of the chambers A and B when the piston is at x = 0. The constants A A and A B are defined in Table 1. Due to a set of safety valves, the pressures PA (t) and PB (t) in the chambers A and B cannot exceed the supplied and return pressures (Pt , Ps ). Therefore, the pressures at the cylinder have the following property. Property 3 The pressures PA (t) and PB (t) in the chambers A and B are such that PA (t), PB (t) ∈ [Pt , Ps ] for all t ≥ 0.

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In summary, the domain of operation of the hydraulic cylinder is defined as the set

W = {x, x, ˙ Q A , Q B , PA , PB ∈ R : 0 ≤ x(t), |x| ˙ ≤ x, ¯ [Q A Q B ] ≤ q¯ and PA (t), PB (t) ∈ [Pt , Ps ]} ,

(2)

therefore, the electro-hydraulic cylinder dynamics presented in the sequel is valid only inside W. The model of an electro-hydraulic cylinder consists of the pressure dynamics equations for both cylinder chambers, the valve flow equation, and the piston motion equation [6]. The valve spool dynamics is considered faster than the piston dynamics, and therefore, it is neglected [22, 23].

2.1 Pressure Dynamics Equations The pressure dynamics in cylinder chambers are β (−x˙ A A + Q A ), V A (x) β P˙B = (x˙ A B + Q B ). VB (x) P˙ A =

(3) (4)

The parameter β is oil bulk modulus, which typically depends on the pressure. However, for mineral oils and for common pressures lower than 45 × 106 Pa and temperatures between −40 and 120 ◦ C, one may assume a mean value for the bulk modulus [23]. It worth pointing out that Eqs. (3)–(4) are valid only inside W.

2.2 Equations of Fluid Motion The dynamics of the spool valve behave faster than the piston dynamics; thus, the relation xv (t) = u(t) is usually considered in the literature [6, 7], see Fig. 1. With this in mind, the equations of fluid motion through the valve orifices are  Q A = cv (−xo + u) sg(−xo + u) Ps − PA  − cv (−xo − u) sg(−xo − u) PA − Pt ,

(5)

 Q B = cv (−xo − u) sg(−xo − u) Ps − PB  − cv (−xo + u) sg(−xo + u) PB − Pt ,

(6)

where the function sg(), for a scalar variable  ∈ R, is defined by

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Control Spool Valve

: Fluid volume : Pressure

: Pressure

: Fluid volume

: Piston position

Fig. 1 Electro-hydraulic cylinder controlled by a critically centered control valve with symmetrical spool

 sg() =

1, for  ≥ 0, 0, for  < 0.

(7)

Without loss of generality we consider the same discharge coefficient cv in (5)–(6), in other words, we assume that spool valve orifices are identical. The strictly positive constant value xo > 0 represents the overlap in the valve orifices. As can be seen in (5)–(6) cross-chamber leakage and external leakage have been neglected. Due to the overlap of the valve orifices in the spool valve (see Fig. 2), the relation between the control input u(t) and the flows Q A,B present a dead-zone nonlinearity, which is compensated as in [24], i.e., ⎧ ⎪ ⎨u 0 + xo , if u 0 > 0, u = D(u 0 ) = 0, if |u 0 | = 0, ⎪ ⎩ u 0 − xo , if u 0 < 0,

(8)

where u 0 (t) is the control signal to be designed in the next section, whereas u(t) is the actual control signal to be applied to the system, which is indeed normalized, i.e., u(t) ∈ [−1, 1].

2.3 Piston Motion Equation The equation of piston motion arises by applying Newton’s second law [23]. The resulting equation is (9) m x¨ = Fh − Fext ,

Accurate Position Regulation of an Electro-Hydraulic Actuator … Fig. 2 Flux characteristic versus control input

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Q A,B

-x 0

x0

u

Overlap region

where m considers the piston and the load masses, Fext agglutinates the effects of the friction and gravity forces; Fh is the generated hydraulic force with the following expression (10) Fh = PA A A − PB A B . To sum up the entire system dynamics, first, we define the state space vector z = [z 1 z 2 z 3 z 4 ]T = [x − x d x˙ PA PB ]T , where x d is the desired constant piston position. Then, combining (3)–(6) together with (8) and (9) yields z˙ 1 = z 2 , Fext Fh − , z˙ 2 = m m β (−z 2 A A + cv φ A (z 3 )u 0 ), z˙ 3 = V A (z 1 ) β (z 2 A B + cv φ B (z 4 )u 0 ), z˙ 4 = VB (z 1 ) y = z1

(11) (12) (13) (14) (15)

with   φ A (z 3 ) = sg(u 0 ) Ps − z 3 + sg(−u 0 ) z 3 − Pt ,   φ B (z 4 ) = sg(−u 0 ) Ps − z 4 + sg(u 0 ) z 4 − Pt ,

(16) (17)

where the functions φ A (z 3 ) and φ B (z 4 ) satisfy φ A (z 3 ), φ B (z 4 ) ≤



Ps , ∀z 3 (t), z 4 (t) ∈ W.

(18)

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For the sake of convenience the nonlinear functions φ A and φ B are expressed as φ A (z 3 ) = φ¯ A + φ A , φ B (z 4 ) = φ¯ B + φ B ,

(19) (20)

where φ¯ A and φ¯ B correspond to mean values; whereas φ A and φ B represent unknown and bounded functions given that the pressures z 3 ≡ PA , z 4 ≡ PB are not measured. The only available measurement for the control law design is the piston error position z 1 (t). To simplify the control law synthesis of the system (11)–(15), a plausible approach presented in [7], is to consider the hydraulic force dynamics. Thus, taking the time derivative of (10) it yields F˙h = z˙ 3 A A − z˙ 4 A B ,

(21)

which assumes that the valve is open (either to the left or to the right position, as shown in Fig. 1). Hence, substituting (13)–(14) into (21), we obtain 1 ˙ Fh = −ϕ0 (z 1 )˙z 1 + φ(z 1 , z 3 , z 4 )u 0 , β

(22)

with A2B A2A + , V A (z 1 ) VB (z 1 )  A A φ A (z 3 ) A B φ B (z 4 ) φ(z 1 , z 3 , z 4 ) = cv + . V A (z 1 ) VB (z 1 ) ϕ0 (z 1 ) =

(23) (24)

The bulk modulus β in (22) is a large constant, β = 17 × 108 , therefore dynamics in (22) can be seen as a singular perturbation [22], i.e., β1 F˙h = 0. Thus, clearing z˙ 1 in (22) yields (25) z˙ 1 = γ (z)u 0 , where γ (z) =

φ(z 1 , z 3 , z 4 ) ϕ0 (z 1 )

(26)

is an unknown function. Now, considering (12), when the piston is in motion (u 0 = 0), the following holds ¯ 0 < |Fext | < |Fh | ≤ F,

(27)

where F¯ > 0. Moreover, inside W, the control input u 0 is bounded, as a consequence, the right-hand side of (12) is bounded. In [7] is shown that the velocity z 2 cannot grow faster than a linear function. Then, without loss of generality, the velocity dynamics

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in (12) is expressed as follows ¯ z˙ 2 ≤ a,

(28)

where a¯ is the maximum acceleration. Under the singular perturbation premise, the dynamics in (13)–(14) are disregarded. Thus, taking into account (25) and (28), the system dynamics (11)–(15) is rewritten as follows z˙ 1 = γ (z)u 0 , z˙ 2 ≤ a, ¯ y = z1.

(29) (30) (31)

The controller u 0 can be designed from (29). Concerning (29), notice that the control coefficient γ (z) is unknown. In the present setup, as well as in all industrial hydraulics systems, γ (z) is positive for all z due to a set of safety valves, which maintain a non-zero pressure difference between the chambers A and B [7]. Hence, inside W, there exist a positive constants γ¯ such that 0 < γ (z) < γ¯ . Such constant is introduced below

 √ 1 1 . (32) + γ¯  cv ϕ Ps A A V A (0) VB (0) + x¯ A B Now, since V A (0), VB (0), A A , and A B are known, and z 1 (t) is measured, V A (z 1 ), and VB (z 1 ) are known. In contrast, φ A (z 3 ) and φ B (z 4 ) are unknown. Therefore, the unknown function γ (z) > 0 in (26) can be rewritten as γ (z) = γ0 (z) + γ (z),

(33)

where γ0 (z) > 0 is a known nominal function and γ (z) represents the uncertain term arising from disregarded nonlinear dynamics. Thus, tacking into account, (19)– (20) and (23)–(24) the following expressions arises

 cv φ¯ A VB (z 1 ) + α φ¯ B V A (z 1 ) AA α 2 V A (z 1 ) + VB (z 1 )

 cv φ A VB (z 1 ) + αφ B V A (z 1 ) γ (z) = AA α 2 V A (z 1 ) + VB (z 1 ) γ0 (z) =

(34) (35)

where α = A B /A A . As a result, γ (z) can be considered as an unknown but bounded function. Taking into account (33), the dynamics (29) can be rewritten as z˙ 1 = γ0 (z)u 0 + d1 ,

(36)

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where d1 = γ (z)u 0 represents the total uncertainties and disregarded dynamics in (36). For the sake of brevity d1 is considered a perturbation whose dynamics is unknown but bounded, i.e., |d1 | ≤ δ¯1 , d˙1 ≤ L ,

(37) (38)

where δ¯1 , L > 0 are strictly positive constants. The later constant L can be found and adjusted experimentally as long as the variables remain inside W [7], see also [13]. For the uncertain system (36), let us define the control objective as: design a control law, u 0 (t), such that, the piston position error z 1 (t) → ¯ in spite of the total uncertainties and disregarded dynamics d1 , achieving a small enough steady-state error ¯ > 0, i.e., x(t) − x d ≤ ¯ . (39) In the next section, we design the control input u 0 (t) as a proportional controller plus a compensation term. To this aim, a super-twisting observer (STO) is designed to identify the unknown input d1 in (36).

3 Controller Design This section presents the design of a controller for the system (36). A super-twisting observer (STO) recovers information of the unknown input d1 . Later on, this information is used in the control law design to compensate for the effects of d1 .

3.1 Unknown Input Observer Let us start with the unknown input identification. The following STO for (36) is proposed z˙ˆ 1 = −λ1 ˆz 1 − y1/2 + γ0 (z)u 0 + dˆ1 , (40) d˙ˆ = −λ ˆz − y0 , 1

2

1

where zˆ 1 , dˆ1 are the estimated values of y and d1 , respectively; ·ζ = |·|ζ sign(·). Finally, λ1 , λ2 are the observer gains selected according to [25] as √ λ1 = 10.3280 L,

λ2 = 1.6667L ,

(41)

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where the gain L is the Lipschitz constant presented in (38), and may be adjusted experimentally, see [7]. Throughout, the precise meaning of the differential equations with discontinuous right-hand side, is defined in the sense of Filippov [26].

3.2 Design of u0 For the system described in (36), consider the control input u 0 (t) as u 0 (t) =

 1 −k p z 1 − dˆ1 , γ0 (z)

(42)

where k p > 0 is a proportional gain and dˆ1 is the compensation term provided by the observer (40). The main result is thus summarized in the following Proposition. Proposition 1 Let the conditions (37)–(38) be satisfied. Then, the trajectories of the electro-hydraulic system (29) driven by the controller (42) and the unknown input observer (40) are ultimately uniformly bounded, provided k p > 0 and λ1 , λ2 given in (41). Proof We begin the analysis by writing down the closed-loop dynamics. Then, we split the proof into two steps: first, we expose the observer convergence and second, we tackle the position error convergence using Lyapunov theory. Before we sum up the closed-loop dynamics, let us start defining the observation errors η1 (t)  zˆ 1 (t) − y(t) and η2  dˆ1 − d1 . Taking their time derivatives yield to the observer error dynamics η˙ 1 = −λ1 η1 1/2 + η2 , η˙ 2 = −λ2 η1 0 − d˙1 .

(43)

The overall closed-loop system conforms by the piston position error dynamics (29)– (30), together with the control input (42) and the observation error dynamics (43) as z˙ 1 = −k p z 1 − η2

(44)

z˙ 2 ≤ a¯

(45)

η˙ 1 = −λ1 η1 1/2 + η2 η˙ 2 = −λ2 η1 0 − d˙1 .

(46)

Let us start with the observation error dynamics given by the subsystem (46). According to Moreno and Osorio [25, Theorem 3], selecting the gains λ1 and λ2 as in (41)

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allow the establishment of a sliding-mode in finite-time, i.e., the trajectories of (46) reach the origin for all t ≥ t f , that is η(t) ≡ 0, ∀t ≥ t f ,

(47)

in consequence, zˆ 1 ≡ z 1 , dˆ1 ≡ d1 holds for all t ≥ t f . Besides, [25] also guarantees that η(t) ≤ η(0) ∀t ≥ 0. (48) Now, we analyze the position error convergence whose dynamics corresponds to (44). To this end, let us consider the Lyapunov candidate function V (z 1 ) = z 12 /2. Taking its time derivative along the trajectories of (44) yields to V˙ (z 1 ) = −k p z 12 − z 1 η2 ,

(49)

considering (48), the above equation produces V˙ (z 1 ) ≤ −k p z 12 + |z 1 | η.

(50)

Taking into account the bound in (48), Eq. (50) results in   V˙ (z 1 ) ≤ − k p − θ z 12 , η(0) , ∀ |z 1 | ≥ θ

(51)

where 0 < θ < k p . Inequality (51) shows that trajectories of system (44) remain uniformly ultimately bounded [22, Theorem 4.18]. Moreover, once η ≡ 0 for all t ≥ t f , (51) results in V˙ (z 1 ) ≤ −z 12 , ∀t ≥ t f .

(52)

Inequality (52) shows that trajectories of system (44) tends to the origin as t → ∞. As a result, the control goal in (39) is attained. The proof is complete. 

3.3 Position Accuracy This section discusses the position accuracy. To this aim, we contrast the position error achieved using a nominal proportional controller against the results with the compensation based controller.

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Previously, it was stated that the disturbance is exactly compensated, i.e., |d1 − dˆ1 | ≡ 0. However, in [27] was stated that |d1 − dˆ1 | ≤ 1 τs , where 1 > 0 is a positive constant and τs is the sampling time. Therefore, we conduct a position accuracy analysis considering the results as shown in [27]. A. Nominal proportional control (NPC), i.e., u 0 = −k p z 1 /γ0 (z). Using a proportional control law in (36), and considering V = 21 z 12 as a candidate Lyapunov function, its time derivative along the trajectories of (36) results in V˙ ≤ −(k p − θ ) |z 1 |2 − |z 1 |(θ |z 1 | − δ¯1 ) δ¯1 ≤ −(k p − θ ) |z 1 |2 , ∀|z 1 | > , θ as a consequence, the proportional controller attains a position accuracy given by ¯ := δ¯1 /θ with 0 < θ < k p . B. Compensation-based control (CBC), i.e., u 0 as in (42). In this case, the position accuracy ¯ is directly related to disturbance identification precision. Previously, it was stated that the disturbance is exactly compensated, i.e., |d1 − dˆ1 | ≡ 0. However, in [27] was stated that the identification accuracy yields to |d1 − dˆ1 | ≤ O(τs ), where τs is the sampling time. Taking into account the latter, and considering the control law in (42), the closed-loop analysis in (52) yields to V˙ ≤ −(k p − θ ) |z 1 |2 − |z 1 |(θ |z 1 | − O(τs )) O(τs ) . ≤ −(k p − θ ) |z 1 |2 , ∀|z 1 | > θ Thus, the position accuracy is in the order of the sampling time τs , i.e., ε¯ := O(τs )/θ with 0 < θ < k p . Table 2 summarizes the position accuracy analysis. In both cases, the position error reaches an ultimate bound. The CBC approach, however, shrinks such a bound, resulting in an improvement of the closed-loop position accuracy. One may presume that increasing the proportional gain k p will, theoretically, decrease ¯ for the NPC case. However, in practice, increasing k p saturates the control action in the presence of small position error, provoking the occurrence of undesired oscillations [28]. Opposite to this, the CBC improves the position accuracy without the necessity of increase the gain k p . Next section presents simulations and experimental results to illustrate the feasibility of our approach.

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Table 2 Position accuracy comparison Controller A. B.

NPC CBC

Table 3 Physical parameters A A (m2 ) A B (m2 ) −3 1.26 × 10 0.76 × 10−3 Pt (Pa) 5 × 105

Ps (Pa) 180 × 105

Position error accuracy x(t) − x d ≤ ¯

¯ =

¯ =

δ¯1 θ , 00 dt dt

Case II: When s > 0 then, dst = 0 and s˙ < 0 Using (9), dvc1 di L1 = E + vc1 , C = i L1 − i dc L dt dt s˙ =

dvc1 di L1 +κ 0 dt dt

(29) (30)

Case II: When s > 0 then, dst = 0 and s˙ < 0, using (15), and (16), L

dvc di L = E + −vc , C = i L − i dc dt dt s˙ =

di L1 dvc1 +κ 0, (1 − 2dst )ir e f < 0 and when ev < 0, (1 − 2dst )ir e f > 0. Hence, the function V˙ is negative definite and the voltage and current error asymptotically converge to zero [18]. Hence, the system with proposed controller is stable.

8.1 Derivation of Low Pass Filter Time Constant for Current Reference Generation The low pass filter time constant τ has a significant role for system stability and impedance shaping. The bounds on the value is derived by deriving the small signal model of the converter. Here, the small signal modeling, and τ of a eqSBI is derived. Similar derivation can be carried out for q-ZSIs. The state space model with small perturbations in i L , vc and ir e f as iˆL , vˆc and iˆr e f , and duty cycle perturbation is dˆ is x˙ˆ = A xˆ + B dˆ + C,

(37)

⎡ E−Vc ⎤ ⎤ ⎡ 2VC ⎤ 0 L L L ⎦ , C = ⎣ ICL ⎦, and where xˆ = [iˆL vˆc irˆe f ]T , A = ⎣ 1−2D 0 0 ⎦, B = ⎣ −2I C C 1 0 − τ1 0 0 τ Vc , I L , and D are steady state values of capacitor voltage and inductor current and duty cycle, respectively. When the system is in sliding regime, i.e., s = (i L − ir e f ) + κ(vc − vr e f ) = 0, s˙ = (i˙L − ir˙e f ) + κ v˙c = 0 and then the third order system can be reduced to a second order system. Also, the duty cycle perturbation can be written in terms of the small signal voltage and current perturbations. ⎡

0

2D−1 L

  1 iˆL − iˆr e f ) κ(I L + (1 − 2D)iˆL ) (E − V ) + (2D − 1)vˆc ˆ − − d= , β τ C L

(38)

where β = 2V − 2κCI L . L Finally, substituting all values and deriving condition of τ such that the coefficients of characteristic equation is always positive gives τ
0 mean that the doubly-fed induction machine works as a generator and motor, respectively, whereas Q s < 0 and Q s > 0 indicate that the DFIG supplies reactive power to the grid and draws reactive power from the grid, respectively. Electromagnetic torque of the generator can be expressed using the following equation:

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  s   s 3 P L m λsβ λr α − λsα λrβ 3 P L m  λs × λr = Tg = 4 4 (σ L s L r ) (σ L s L r )

(14)

where the number of poles of the DFIG is denoted by P. Differentiating (12) and (13) gives instantaneous variations in stator active and reactive powers as  3 ˙ d Ps = vsα i sα + i sα v˙sα + vsβ i˙sβ + i sβ v˙sβ dt 2  d Qs 3 ˙ = vsβ i sα + i sα v˙sβ − vsα i˙sβ − i sβ v˙sα dt 2

(15a) (15b)

As shown in (15a), the variation of network voltage is required. Considering an ideal network,     2π 2π vas = Vs cos (ωs t) ; vbs = Vs cos ωs t − ; vcs = Vs cos ωs t + (16) 3 3 Transforming vas , vbs and vcs into stationary reference frame (α − β components): vsα = Vs cos (ωs t) and vsβ = Vs sin (ωs t)

(17)

Thus, the variation in instantaneous network voltage can be obtained as v˙sα = −ωs Vs sin (ωs t) = −ωs vsβ v˙sβ = ωs Vs cos (ωs t) = ωs vsα

(18)

Based on (10), the α, β components of variations in instantaneous stator current can be written as  1 Lr di sα (19) vr α − Rr ir α − = (vsα − Rs i sα ) dt σ Lm Lm   ωg Lr σ L m i sβ + − λsβ σ Lm Lm   1 di sβ Lr  vrβ − Rr irβ − = vsβ − Rs i sβ (20) dt σ Lm Lm   ωg Lr σ L m i sα + (21) λsα + σ Lm Lm Substituting (18) and (19) into (15a) and arranging them in matrix form yields  d Ps = G + BVrs dt Q s

(22)

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where  G=

G1 G2



⎡

⎤ Rs L r    −ω sli p ⎢ σ L2 ⎥ Ps 3 ωg L r −vsβ vsα λsα m ⎥ =⎢ − ⎣ Rs L r ⎦ Q s 2 σ L 2m vsα vsβ λsβ ωsli p 2 σ Lm −

B=

   2 2 3 L r vsα 3 Rr vsα vsβ + vsβ ir α − 0 2 σ L m vsβ −vsα irβ 2 σ L 2m

  3 1 vsα vsβ v and Vrs = r α vrβ 2 σ L m vsβ −vsα

(23)

where slip angular frequency ωsli p = ωs − ωg . As mathematical model of DFIG is derived in stationary reference frame, superscript (s) is avoided above the components of stator and rotor vectors in (12)–(23) for simplicity.

3 Control of DFIG In the partial load zone or zone-I of the wind turbine operation, the DFIG rotor speed (g ) and thus the wind turbine rotor speed (t ) are changed at different wind speeds (ν) in order to get the wind turbine to operate at the optimal tip-speed ratio. This results in the extraction of optimum power from the wind and delivers it into the network or grid. Figure 5 shows a typical 5 kW wind turbine power-rotor speed characteristics for different wind speeds. The characteristics have been obtained using the C p -μ characteristics available in Fig. 2a.

Fig. 5 Wind turbine power-speed characteristics and maximum power point operation

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For a given wind speed, each power curve has a Maximum Power Point (MPP) at which the optimal tip-speed ratio, μopt , is achieved. In order to get the wind turbine to operate on the optimal tip-speed ratio (MPP line shown in Fig. 2) in the partial load zone, DFIG torque (Tg ) is regulated to vary its rotor speed. Therefore, in order   to control the rotor speed of the DFIG, the reference for the stator active power Ps∗ [22] is expressed in terms of the optimum value of the DFIG torque (Tg )opt to be tracked as

  where Tg opt

2   Ps∗ ∼ Tg opt ωs = P     5 K opt 1 ρπ C p opt R 2 = and K =  opt g n 3g 2 μ3opt

(24)

(25)

  C p opt is the optimum value of wind turbine power coefficient or conversion efficiency corresponding to optimal value of tip-speed ratio μopt for the blade pitch angle β = 0.

4 Proposed ESO-based SM-DPC Scheme ESO is a disturbance estimation technique which can estimate states and lumped disturbances of the system. It considers the system’s internal and external dynamics as lumped disturbance of the system which is taken as an additional state of the system [23]. The main advantage of the ESO-based SM-DPC control is that the tracking and stabilization of nominal dynamics of the plant are taken care by the feedback control or SM-DPC, whereas ESO improves the dynamic performance of the controller during mismatched disturbances.

4.1 Extended State Observer Considering Ps and Q s as state variables x1 and x2 , respectively, as well as vr α and vrβ as control inputs, (22) can be rewritten in terms of state variables and inputs as ˙ = G (x1 , x2 ) + BU X    u v x1 s and U = Vr = r α = 1 where X = x2 vrβ u2 For the system in (26), the following ESO is designed.

(26)

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Z˙ =  − 2P (Z − X) + BU ˙ = −P2 (Z − X) 

(27)

     x1 G z1   =X= and  = G = 1 where Z = z2  x2 G2 1 and G 2 are estimated values of x1 , x2 , G 1 and G 2 , respectively. Matrix P  x1 ,  x2 , G is the desired double-pole of ESO and its value is −P =

 −6000 0 0 −6000

4.2 Sliding Manifold In order to fulfil control objectives of tracking or sliding along the predefined active and reactive power trajectories, sliding manifolds or surfaces are chosen as T  X S = X∗ − Z; where S = S1 S2 and Z = 

(28)

T  In (28), the matrix X∗ = x1∗ x2∗ . x1∗ and x2∗ are reference values of stator active power (Ps ) and stator reactive power (Q s ), respectively. The sliding manifolds S1 = 0 and S2 = 0 represent the precise regulation of Ps and Q s , respectively. S1 = S2 = S˙1 = S˙2 = 0 when the system states hit the sliding manifolds and move or slide along the surface. Differentiation of (28) yields S˙ = −Z˙ = − + 2P (Z − X) − BU

(29)

4.3 Control Law An SMC law which will steer the state orbits to equilibrium points is derived using the Lyapunov stability criterion. In order to derive the closed-loop stability of a system, the following quadratic Lyapunov function is used. W=

1 T S S≥0 2

(30)

Time derivative of W is given by     ˙ = ST S˙ = ST −Z˙ = ST −  + 2P (Z − X) − BU W

(31)

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˙ is definitely negative with S = 0. The The control law is chosen such that W following control law is chosen:   USMESO = Vrs SMESO     = B−1 Ksign X∗ − Z −  + 2P (Z − X) (32) 

K1 0 where K = 0 K2

(33)

K 1 and K 2 are positive controller gains in (33).

4.4 Stability Substituting (32) in (31), it can be seen that       K1 0 sign (S1 ) sign (S1 ) ˙ = −ST K 1 0 = − S1 S2 W 0 K 2 sign (S2 ) 0 K 2 sign (S2 ) 0, so that the controller stabilizes the system asymptotically.

5 SM-DPC Scheme For the SM-DPC scheme without ESO, the following sliding surfaces are chosen:   ∗ x1 − x1 S1 = ∗ S = X −X= S2 x2 − x2 ∗

(35)

Time derivative of (35) yields ˙ = −G (x1 , x2 ) − BU S˙ = −X

(36)

For deriving the SMC law, a similar procedure is followed as mentioned in Subsections (4.3) and (4.4). Based on that, the following control law is chosen which maintains the stability of state variables on sliding manifolds.

Sliding Mode Direct Power Control of a Grid-Connected …

    USMC = Vrs SMC = −B−1 Ksign (S) + G (x1 , x2 )

339

(37)

The matrix K is as per (33) and K 1 and K 2 are positive controller gains. The chosen value of matrix K for both the control schemes (i.e. ESO-based SM-DPC and SM-DPC) is as follows: K=

 0 4.8 × 106 0 4.8 × 106

(38)

6 System Implementation Figure 6 shows the schematic diagram of the proposed ESO-based SM-DPC strategy. Three-phase stator voltages, stator currents and rotor currents are measured. Threephase stator voltages, currents and rotor currents are converted into α-β components in the stationary and rotor reference frames, respectively. Stator flux in the stationary reference frame as well as active and reactive powers are computed from stator voltages and currents. α-β components of the rotor current in the rotor reference   frame are converted into stationary reference frame using rotor position angle θg measured using a shaft encoder. The proposed ESO-based SM-DPC scheme mentioned in Sect. 4 and the SM-DPC scheme without ESO mentioned in Sect. 5 directly generate the voltage references as per (32) and (37), respectively, in the stationary reference frame for RSC as per the instantaneous errors of stator active and reactive powers, α-β components of stator voltage, stator current, rotor current and stator flux vectors in the stationary reference frame. Afterwards, it is transformed into the rotor reference frame as Vrr = Vrs e− jθg

(39)

(39) shows the rotor voltage vector which is given to the Sinusoidal Pulse-Width Modulation (SPWM) unit. The SPWM unit generates required gate signals for the IGBT-based RSC. It is to be noted that synchronous coordinate transformation similar to VC schemes is not required to implement the proposed control scheme.

7 Simulation Results WECS having a 5 kW grid-connected DFIG run by a variable-speed wind turbine is modelled in a MATLAB/Simulink environment. The parameters of the DFIG are listed in Table 4 of Appendix A. The performance of the proposed control scheme based on ESO and its comparison with the SM-DPC scheme without ESO is shown by simulation results. The value of the DC-link capacitor (C) is 5000 μF. The  task of the GSC is to control a DC-link voltage and grid-side reactive power Q g . The

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Fig. 6 Schematic diagram of ESO-based SM-DPC scheme

VC scheme based on the alignment of the grid voltage vector along the d-axis of a synchronously rotating reference frame mentioned in [20, 22] is implemented to control GSC. This VC scheme based on grid voltage orientation  ∗  is not included here. The references for the nominal converter DC-link voltage Vdc and grid-side reactive   power Q ∗g are set at 720 V and 0 kVAR, respectively. The switching frequencies of GSC and RSC are set at 2.45 kHz  and5 kHz, respectively. The values of resistor and inductor of grid-side filter Rg , L g shown in Fig. 1 are 10 m and 8 mH, respectively. During the simulation, DFIG starts as a motor and reaches synchronous speed in 0.4 sec. The DC-link voltage is regulated as GSC is enabled first. The DFIG is then excited by the RSC. As this starting process is not shown, representative results are shown only after 0.4 sec. The system is set to work in the partial load zone of the wind turbine in which energy maximization is sought. In order to investigate the dynamic performance of the ESO-based SM-DPC scheme, step variation of wind speed shown in Fig. 7 is applied to the DFIG.

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Fig. 7 A step variation of wind speed signal

7.1 Comparative Studies 7.1.1

Dynamic Performance in the Absence of Uncertainties and Disturbances

Figure 8a, b shows the control of stator active power (in kW) corresponding to the wind profile displayed in Fig. 7 by the SM-DPC scheme and the ESO-based SMDPC scheme, respectively, from 0.4 to 3 sec in the absence of uncertainties and disturbances in the plant model. The central aim to regulate the stator active power is to vary the DFIG’s rotor speed in order to make the wind turbine operate at MPPs as shown in Fig. 5 in its partial load zone. Stator active power reference (Ps∗ ) is shown by (24). Negative stator active power indicates that the DFIG is in the generating mode. The red curve shows the reference value of stator active power. Zoomed boxes inside Fig. 8a, b show the performance of control schemes from 0.6 to 1.2 sec and from 0.9 to 0.91 sec. It can be seen that the regulation of stator active power by the SM-DPC scheme is poor as compared to that by the ESO-based SM-DPC scheme because there are large pulsations in the stator active power even in the absence of uncertainties and disturbances whereas, ESO-based control enhances the tracking performance of the SM-DPC scheme as it estimates the stator active power in which high-frequency switching is attenuated. It is computed from the zoomed boxes inside the figures that the amount of chattering or pulsations in stator active power is reduced by 26.4% by the ESO-based SM-DPC scheme. Similarly, the control of stator reactive power (in kVAR) by the SM-DPC scheme and the ESO-based SM-DPC scheme are shown in Fig. 9a, b, respectively, from 0.4 to 3 sec in the absence of uncertainties and disturbances in the plant model. The reference value of stator reactive power is shown in the red curve.

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(a) SM-DPC scheme

(b)ESO based SM-DPC scheme

Fig. 8 DFIG’s stator active power (kW) in the absence of parametric uncertainties and external disturbances corresponding to step variation of wind speed

(a) SM-DPC scheme

(b)ESO based SM-DPC scheme

Fig. 9 DFIG’s stator reactive power (kVAR) in the absence of parametric uncertainties and external disturbance

Negative and positive reactive powers indicate that the DFIG supplies reactive power to the grid and draws reactive power from the grid, respectively. It can be seen that the tracking of stator reactive power by both the control schemes is satisfactory. Zoomed boxes inside Fig. 9a, b which shows the performance of control schemes from 1.5 to 2 sec show that there is not much improvement in the attenuation of pulsations in stator reactive power by the ESO-based SM-DPC scheme as compared to the SM-DPC scheme.

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7.1.2

343

Dynamic Performance in the Presence of Uncertainties and Disturbances

Further to investigate, the performance of the ESO-based SM-DPC scheme is compared with that of the SM-DPC scheme in the presence of uncertainties in DFIG parameters and disturbances. Variations in the grid voltage are considered as an external disturbance. The amount of external disturbance and variations in DFIG parameters considered in the simulation are mentioned in Table 1 and Table 2, respectively. Figure 10 shows the control of stator active power from 0.4 to 3 sec by the SM-DPC and the ESO-based SM-DPC schemes in the presence of uncertainties and disturbances. In the presence of mentioned disturbances and parametric uncertainties, control by the SM-DPC scheme is further degraded which can be seen from Fig. 10a. Zoomed boxes inside the figures show the tracking behaviour from 0.5 to 1.3 sec and from 0.8 to 1 sec when the wind speed is 7.1 m/sec. Figure 10b shows that the tracking behaviour by the ESO- based SM-DPC scheme is improved as compared to that by the SM-DPC scheme even in the presence of such parametric uncertainties and external disturbances. It can be noted that the dynamic performance by the ESObased SM-DPC scheme is satisfactory and reduces the chattering by 73.6%. Figure 11a, b shows the variation of DFIG electromagnetic torque by the SM-DPC scheme and the ESO-based SM-DPC scheme, respectively, from 0.4 to 3 sec in the presence of parametric uncertainties and external disturbances due to step variation in wind speed  as shown in Fig. 7. The reference value of DFIG electromagnetic  torque Tg∗ is given by the optimum value of torque to be tracked Tg opt depicted in (25). It is shown by the red curve. Similar to stator active power regulation shown in Fig. 10a, substantial torque variation can be seen in Fig. 11a which shows the performance of the SM-DPC scheme. These torque variations produce severe stresses in the mechanical drive train reducing the life of the wind energy generation system. Zoomed boxes inside the figures show a variation of torque during 0.6 to 1.2 sec.

Table 1 Unbalanced grid voltages Grid phase voltage Nominal value (V) Vas Vbs Vcs

219.39 219.39 219.39

Table 2 DFIG parametric uncertainties Parameters Nominal value Rs Rr Lm

720 mΩ 748.246 mΩ 85.8 mH

Actual value (V)

% Change

213.62 230.94 239.60

−2.63 +5.26 +9.21

Actual value

% Change

828 mΩ 860.48 mΩ 72.932 mH

+15 +15 −15

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(a) SM-DPC scheme

(b)ESO based SM-DPC scheme

Fig. 10 DFIG’s stator active power (kW) corresponding to step variation of wind speed in the presence of parametric uncertainties in the plant model and unbalanced grid voltages

(a) SM-DPC scheme

(b)ESO based SM-DPC scheme

Fig. 11 Electromagnetic torque of DFIG in the presence of DFIG’s parametric uncertainties in the plant model and unbalanced grid voltages

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(a) SM-DPC scheme

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(b)ESO based SM-DPC scheme

Fig. 12 Variation of DFIG rotor speed (g ) and DC-link voltage (Vdc ) in response to step change of wind speed by the SM-DPC and the ESO-based SM-DPC schemes

Figure 11b shows that the ESO-based SM-DPC control substantially attenuates these torque variations which further improves the health of the drive train. Figure 12a shows the variations of DFIG rotor speed (g ) by the SM-DPC and the ESO-based SM-DPC schemes in response to step change in wind speed. The reference rotor speed is shown by the red curve according to power maximization. It can be seen from Fig. 12a that there is no substantial change in tracking the reference rotor speed by the two control schemes. The main reason is that due to the inertia of the wind turbine and generator rotors, mechanical dynamics works as a low-pass filter rejecting any sudden and sharp variation in the rotor speed. It is to be clearly noted that while deriving optimum torque reference, stator resistance and rotor resistance losses are neglected, while in deriving the DFIG model, the stator resistance and rotor resistance are considered. This is the reason why speed tracking by the two control schemes is not exact to the reference values. Figure 12b shows the variations in DC-link voltage (Vdc ) in response to step variation of wind speed by the SM-DPC and the ESO-based SM-DPC schemes. The DC-link voltage is regulated by the GVOVC scheme in both the cases, i.e. when RSC is controlled by the SM-DPC scheme and the ESO-based SM-DPC scheme. It is observed from Fig. 12b that DC-link voltage is effectively regulated having maximum steady-state variation of ±1 volt from the set value of 720 volt, ensuring proper exchange of active power between GSC and grid.

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Fig. 13 Reference, estimated and measured values of stator active power

7.2 Estimation of Stator Active and Reactive Powers Figures 13 and 14 show the actual (magenta curve), estimated (blue curve) and reference (black curve) values of stator active and reactive powers, respectively, by the ESO-based SM-DPC scheme from 0.4 to 3 sec during the disturbances and parametric uncertainties mentioned in Tables 1 and 2. Zoomed boxes inside Fig. 13 show simulation results from 0.6 to 1 sec and from 0.9 to 0.901 sec. It is computed from the zoomed plots of stator active power that high- frequency chattering is attenuated by 90% in the estimated value of stator active power as compared to its measured value and estimation error by the proposed ESO-based SM-DPC remains less than 1.2%. Similarly, zoomed boxes inside Fig. 14 show the simulation results from 1.5 to 2 sec and from 1.7 to 1.701 sec. It is computed from the zoomed plots of stator reactive power that high-frequency chattering is attenuated by 88% in the estimated value of stator reactive power as compared to its measured value and estimation error by the proposed ESO remains less than 0.8%.

8 Conclusion This chapter has presented he ESO-based SM-DPC scheme for the control of stator active and reactive powers of a grid-connected DFIG in the stationary reference frame. In the partial load zone operation of the wind turbine, the DFIG’s electromagnetic torque is varied by controlling its stator active power in order to operate the wind turbine on the MPP line to get maximum power from the wind when wind speed changes. The main objective to control the stator reactive power is to follow the grid requirement. Specifically, control by the SM-DPC scheme without an observer and the ESO-based SM-DPC scheme are simulated on a grid-connected DFIG-based

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Fig. 14 Reference, estimated and measured values of stator reactive power

wind turbine system having a capacity of 5 kW, and results are compared for stated objectives. Decoupled control of stator active and reactive powers are attained by both the control schemes. But the ESO improves the tracking performance of the SM-DPC scheme and reduces the chattering or pulsations in DFIG’s stator active power and torque in the absence of uncertainties and external disturbances. In the presence of DFIG parameter variations and unbalanced grid voltages, the performance of the SM-DPC scheme without ESO further degrades, whereas in the presence of similar disturbances, the ESO-based SM-DPC scheme attenuates the effect of these disturbances and provides robustness against them. ESO works like a patch to the controller scheme enhancing its disturbance attenuation capability. Its implementation is not complicated. The chattering attenuation in the active power reduces the fluctuation of the generated electric power which may cause flickering in weak grids. With estimation error of less than 1.2% and 0.8%, respectively, ESO estimates stator active and reactive powers in this highly disturbed environment. The ESO-based SM-DPC scheme is implemented in the DFIG’s stationary reference frame, resulting in simple implementation which does not require complex synchronous frame transformation and rotor current decoupling. It is shown that the ESO-based control scheme exhibits robust performance despite parametric uncertainties and external disturbances in fulfilling control objectives.

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Appendix A See Tables 3 and 4. Table 3 Wind turbine and drive train parameters (Pt )rated = 5 kW μopt = 8.1 νrated = 11 m/sec   C p opt = 0.48

Table 4 DFIG parameters   Rated power Pg rated   Rated torque Tg rated Rated stator line voltage Rated stator current Stator frequency Stator and rotor connection No. of poles Stator to rotor turn ratio Stator resistance (Rs ) Stator leakage inductance (L σ s ) Rotor resistance (referred to stator) (Rr ) Rotor leakage inductance (referred to stator) (L σ r ) Mutual inductance (L m )

R = 2.0326 m β = 0◦ ρ = 1.225 kg/m3 n g = 4.3

5 kW 31.8 Nm 380 Volt 8.36 Amp 50 Hz Star 4 0.54 720 mΩ 5.8 mH 750 mΩ 6.0 mH 85.8 mH

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Robust Stabilization of a Class of Underactuated Mechanical Systems of 2 DOF via Continuous Higher-Order Sliding-Modes Diego Gutiérrez-Oribio, Ángel Mercado-Uribe, Jaime A. Moreno, and Leonid Fridman Abstract In this chapter we design robust controllers for a Class of underactuated mechanical systems of two DOF, using a continuous Higher Order Sliding-Mode strategy. Two kinds of controller designs are presented: One generates a fifth-order sliding-mode and achieves Local Finite-Time Stability (LFTS). The other one is a robust controller that provides Global Asymptotic Stability (GAS). These controllers compensate matched Lipschitz disturbances and/or uncertainties, cope with an uncertain control coefficient and generate a continuous control signal, possibly reducing the chattering effect. We provide evidence of the performance of the controllers using simulations for the Reaction Wheel Pendulum (RWP) and the Translational Oscillator with Rotational Actuator (TORA) systems, and by means of experiments on the RWP system.

1 Introduction Underactuated mechanical systems have less control inputs than Degrees Of Freedom (DOF). Control design is a challenging task for this class of systems, which are numerous and very useful in practice. Some examples are given by aircraft and spacecraft vehicles, helicopters, flexible-link and mobile robots, locomotive systems, D. Gutiérrez-Oribio (B) · L. Fridman Departamento de Control y Robótica, División de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad Nacional Autónoma de México (UNAM), 04510 Ciudad de México, Mexico e-mail: [email protected] L. Fridman e-mail: [email protected] Á. Mercado-Uribe · J. A. Moreno Eléctrica y Computación, Instituto de Ingeniería, Universidad Nacional Autónoma de México (UNAM), 04510 Ciudad de México, Mexico e-mail: [email protected] J. A. Moreno e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_15

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snake-type and swimming robots, acrobatic robots, satellites, surface vessels, and underwater vehicles. Many aspects are discussed in the Survey paper [27], that also contains a vast literature. Underactuation may appear due to several reasons: 1. The natural dynamics of the system, as, e.g., in aircraft, spacecraft and underwater vehicles, helicopters, and locomotive systems without wheels. 2. Due to the failure of an actuator in an otherwise full actuated system. 3. By design with the purpose of illustrating some features of this class of systems: the Acrobot, the Pendubot, the Cart-Pendulum system, the Furuta Pendulum, the TORA system, the Reaction Wheel Pendulum (RWP). Since underactuated systems are not globally feedback linearizable (see, e.g., [37]), the control design for pendulum-like systems is usually separated in two tasks: First swinging-up from the downward position to (a neighborhood of) the upright position of the pendulum, and then switching to a (local) stabilizing controller. This includes swing up control using energy-based methods for the RWP [3, 47], the Acrobot [44], the Pendubot [46], the cart-pendulum system [8], and the Furuta pendulum [4]. With respect to the stabilizing controller, some examples are partial linearization methods [49, 50], passivity-based methods [18], and EnergyShaping Method [1, 57], using small nested saturations [52], stabilization by output feedback [53], discontinuous stabilizing feedback [40], adaptive control [14], and sliding mode control techniques [3, 15, 17, 51]. The local stabilizing controller in the previous approaches can be designed using a local feedback linearization design. However, these controllers can be applied only locally, since they posses singularities. A way to overcome this difficulty and to arrive at a Global controller for a class of underactuated mechanical systems of two DOF is to find a partially linearizing output that renders the zero dynamics asymptotically stable (see [13, 26] and the references therein). In this case, stabilizing the linearized subsystem results in a global asymptotic stabilization. However, this approach is not useful when the zero dynamics is not asymptotically stable. Olfati-Saber [35, 37, 38] has gone one step further and has classified the underactuated mechanical systems (having some symmetries) in eight classes, according to the properties of the normal form to which they can be globally transformed. In particular, systems with two DOF belonging to the Class-I in Olfati-Saber’s classification, can be globally transformed to a partially linearized system in strict feedback form. In this case, the zero dynamics can be globally stabilized using a virtual control variable and a back-stepping design leads to global stabilization of the equilibrium point [37, 38]. Moreover, in [37] the non-linearities are supposed to be fully compensated so that the stabilization can be performed through a linear controller. For other classes in Olfati-Saber’s taxonomy a global control design method has been also proposed [37]. A major drawback of these global results [35, 37, 38] is that they do not consider the action of external disturbances and/or the effect of model or parameter uncertainties. They also rely on full compensation of the system’s non-linearities, what is clearly hardly achieved in a real setup. Dealing with non constant uncertainties and/or disturbances requires a Robust Control strategy, such as, e.g., Sliding-Mode Control.

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Paper [58] proposes a First-Order Sliding-Mode strategy for underactuated systems, that aim at dealing with the uncertainties/perturbations. This approach has two main disadvantages: (i) Since the control law is discontinuous, the chattering effect can be prohibitive for a real application. (ii) The whole design is based on a form which is valid only locally, and so no global results are obtained. The effect of chattering can be eventually attenuated by using Higher-Order Sliding-Mode (HOSM) controllers. This has been studied in several works as, e.g., [10, 11, 20, 28, 41]. However, these works only achieve local stabilization, and they are based on the modification of a First-Order Sliding-Mode control strategy. Continuous Higher-Order Sliding-Mode Algorithms (CHOSMA, see [7, 12, 21, 33, 34, 54]) are a class of homogeneous sliding mode controllers capable of compensating Lipschitz uncertainties and/or perturbations theoretically exactly, but using a continuous control signal. When the actuator is fast (see [39]), the chattering effect caused by the discontinuity and discretization is strongly attenuated. These algorithms consist in a static homogeneous finite-time controller for the nominal model of the system and a discontinuous integral action, aimed at estimating and compensating the uncertainties and perturbations. They are an extension of the (classical) Super-Twisting [22, 23, 42, 43], and are related to the Continuous Twisting Algorithm (CTA) [30, 31, 56] and Discontinuous Integral Algorithm (DIA) [33, 34]. The objective of this work is to design robust controllers for the Class-I of underactuated mechanical systems of two DOF, presented in [37], using a continuous HOSM strategy. Two kinds of controller designs are presented: (i) One based on a model of the systems valid only locally. The controller generates a fifth-order sliding-mode and achieves Local Finite-Time Stability (LFTS). (ii) The other uses the globally valid cascade normal form proposed by Olfati-Saber [35, 37, 38] to design a robust controller that provides Global Asymptotic Stability (GAS). A third-order sliding manifold is generated, which is reached in finite-time, but the convergence to the equilibrium is asymptotic. All these controllers are able to compensate matched Lipschitz disturbances and/or uncertainties, cope with an uncertain control coefficient and generate a continuous control signal, possibly reducing the chattering effect. The proposed strategy is valid for the whole Class-I of systems, but we use as study cases in this chapter the RWP and the TORA systems, both as full fourth order systems. Finally, we provide evidence of the performance of the controllers, in simulations for both the RWP and the TORA system, and through experiments carried on in laboratory setup of the real RWP system. Some related results have been reported in our previous work [15]. However, in [15] only the RWP system was considered, and for it a Third-Order Discontinuous Integral Algorithm (3-DIA) was used for the local finite-time stabilization of the third-order system’s origin, i.e., neglecting one of the states (the wheel position). The outline of this work is as follows. The presentation of the Class-I and the systems belonging to such class considered in this work as the study cases (the RWP and the TORA systems), are given in Sect. 2. In Sect. 3 is presented the controller design to reach LFTS of the Class-I origin and in Sect. 4 the control design to reach GAS of the Class-I origin. Each section presents simulations of both kinds of stability

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in the RWP and TORA systems and experimental validation on the RWP system. Conclusions of this paper are presented in Sect. 5. Some preliminary results and the proof of the Theorem 1 are presented in the Appendices. Notation: Define the function ·γ := | · |γ sign(·), for any γ ∈ R≥0 .

2 Class-I of Mechanical Systems with Two DOF Although underactuated (mechanical) systems are not globally state feedback linearizable, it is possible to obtain a (global) partial feedback linearization [13, 26, 29, 35, 38]. Olfati-Saber [37, 38] has classified the underactuated mechanical systems in eight classes, according to the properties of the normal form to which they can be globally transformed. We will consider in this work the simplest case of systems with two DOF that can be transformed to a partial linearized system in strict feedback form, which has been named by Olfati-Saber [37] as Class-I systems. For this class, he [37, 38] proposes a back-stepping approach to globally stabilize an equilibrium point. Class-I of underactuated mechanical system of two Degrees-of-Freedom (DOF), having as configuration vector q = [q1 , q2 ]T , have an inertia matrix depending only on q2 , i.e., M = M(q2 ), and the variable q2 is actuated. The Lagrangian is   1 T m 11 (q2 ) m 12 (q2 ) L(q, q) ˙ = q˙ q˙ − V (q), m 21 (q2 ) m 22 (q2 ) 2

(1)

where V (q) is the potential energy of the system, and the Euler–Lagrange equations of motion are m 11 (q2 )q¨1 + m 12 (q2 )q¨2 + d11 (q2 )q˙1 q˙2 + d12 (q2 )q˙22 − g1 (q1 , q2 ) = 0, 1 1 m 21 (q2 )q¨1 + m 22 (q2 )q¨2 − d21 (q2 )q˙12 + d22 (q2 )q˙22 − g2 (q1 , q2 ) = τ, 2 2 , i = 1, 2 and di (q2 ) = where gi (q1 , q2 ) = − ∂ V∂q(q) i Using the global change of variables

d m (q ). dq2 i 2

x1 := q1 , x2 := q˙1 , x3 := q2 , x4 := q˙2 , the system achieves the state space representation

(2)

(3)

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x˙1 = x2 ,

  m 22 (x3 ) d11 (x3 )x2 x4 + d12 (x3 )x42 − g1 (x1 , x3 ) x˙2 = − η(x3 ) 1  2 m 12 (x3 ) 2 d21 (x3 )x2 − 21 d22 (x3 )x42 + g2 (x1 , x3 ) + τ , − η(x3 ) x˙3 = x4 ,   m 21 (x3 ) d11 (x3 )x2 x4 + d12 (x3 )x42 − g1 (x1 , x3 ) + x˙4 = η(x3 )   m 11 (x3 ) 21 d21 (x3 )x22 − 21 d22 (x3 )x42 + g2 (x1 , x3 ) + τ , η(x3 )

(4)

where η(x3 ) = m 11 (x3 )m 22 (x3 ) − m 12 (x3 )m 21 (x3 ) > 0. System (4) can be transformed globally to a cascade normal form. Proposition 1 ([37, Proposition 3.9.1], [38, Theorem 1]) Consider the Class-I underactuated system with two DOF (4). Then the following global change of coordinates, obtained from the Lagrangian of the system, z 1 = x1 + γ (x3 ), z 2 = m 11 (x3 )x2 + m 12 (x3 )x4 , z 3 = x3 , z 4 = x4 ,

(5)

transforms the dynamics of the system into a cascade nonlinear system in strict feedback form z2 , m 11 (z 3 ) z˙ 2 = g1 (z 1 − γ (z 3 ), z 3 ) ,

z˙ 3 = z 4 ,

z˙ 1 =

z˙ 4 = u ,

(6)

where  γ (x3 ) = 0

x3

m 12 (θ ) dθ , m 11 (θ )

g1 (q1 , q2 ) = −

∂ V (q) , ∂q1

(7)

and u is the new control variable obtained from a collocated partial feedback linearization (see [45, 49])

τ=

    2 12 (z 3 )z 4 η(z 3 )u − m 21 (z 3 ) d11 (z 3 ) z2 −m z + d (z )z − g (z − γ (z ), z ) 4 12 3 1 1 3 3 4 m 11 (z 3 ) 

1 z 2 − m 12 (z 3 )z 4 − d21 (z 3 ) 2 m 11 (z 3 )

2

m 11 (z 3 ) 1 + d22 (z 3 )z 42 − g2 (z 1 − γ (z 3 ), z 3 ). 2 (8)

System (6) is in the Byrnes–Isidori Normal Form, with the zero dynamics subsystem (z 1 , z 2 ). Moreover, it is also in the strict feedback form, since the zero dynamics

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is driven only by z 3 , the output of the master system (z 3 , z 4 ). These properties characterize the Class-I systems. For them, even if the zero dynamics is not asymptotically stable, i.e., the system is not assumed to be strictly minimum phase, the global stabilization of the system is achieved by first globally stabilizing the zero dynamics subsystem, and then globally stabilizing the master subsystem.

2.1 Uncertain Model In presence of parameter or dynamics uncertainties as well as external perturbations, the nominal model (6) is no longer valid. A more appropriate model can be given by the equations z2 + δ1 (z) , m 11 (z 3 ) z˙ 2 = g1 (z 1 − γ (z 3 ), z 3 ) + δ2 (z) , z˙ 1 =

z˙ 3 = z 4 + δ3 (z) , z˙ 4 = β(t, z) [u + ϕ(t)] + δ4 (z),

where the terms δi (z) represent vanishing perturbations, i.e., δi (0) = 0, while the uncertain control coefficient β(t, z) and the matching perturbation term ϕ(t) are assumed to be bounded (without changing sign) and Lipschitz continuous as a function of time, respectively, i.e., 0 < bm ≤ |β(t, z)| ≤ b M ,

dϕ(t) dt ≤ L .

(9)

In what follows we will only consider the non-vanishing perturbations β(t, z) and ϕ(t), that is, we use the uncertain model z2 , m 11 (z 3 ) z˙ 2 = g1 (z 1 − γ (z 3 ), z 3 ) , z˙ 1 =

z˙ 3 = z 4 , z˙ 4 = β(t, z) [u + ϕ(t)] ,

(10)

to design the controllers. It is well known (see, e.g., [19, Chap. 9] for smooth nonlinear systems, or [6, Corollary 5.5] for homogeneous systems) that for sufficiently small and vanishing perturbations with a triangular structure, i.e., δi (z) ≤ (z 1 , . . . , z i ), if the origin of the system (10) is Asymptotically Stable (GAS) so is also the origin of the perturbed system. We do not provide further details of these results here. The presence of the non-vanishing perturbation ϕ(t) in (10) prevent a continuous static state-feedback controller to achieve convergence to the origin z = 0. Our objective in this chapter is to design dynamic and homogeneous state-feedback controllers that are able to stabilize the origin z = 0 despite of the presence of the Lipschitz perturbation term ϕ(t), and the uncertain control coefficient β(t, z), and using a continuous control signal u(t). We use a local homogeneous approximation of system (10) to design a locally finite-time stabilizing dynamic controller in Sect. 3.

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For global (asymptotic) stabilization of the origin z = 0 we use the globally valid model (10) to design a dynamic and homogeneous feedback controller in Sect. 4. This extends the results presented in [35, 37, 38]. In the next paragraphs we present two examples of Class-I systems, which will be used to illustrate the results in simulations and experiments.

2.2 Application Systems: RWP and TORA The Acrobot, the Reaction Wheel Pendulum (RWP), and the Translational Oscillator with Rotational Actuator (TORA) systems are all Class-I underactuated systems and can be globally transformed into the form (6) using an explicit global change of coordinates (5) [37, Corollary 3.9.1]. In this work the RWP and the TORA will be studied and controlled, so their dynamics will be presented in the following sections.

2.2.1

Reaction Wheel Pendulum (RWP)

Consider the RWP shown in Fig. 1. This figure describes a pendulum rotating in a vertical plane and its pivot pin is mounted in a stationary base. The pivot pin of the wheel is attached to the pendulum. The axes of rotation of the pendulum and the wheel are parallel to each other. The wheel is actuated by the torque τ (Nm). The states vector is x = [x1 , x2 , x3 , x4 ]T , where x1 (rad) is the angle between the upward direction and the pendulum, which is measured counter-clockwise (x1 = 0 for the upright position of the pendulum), x2 (rad/s) is the pendulum angular velocity, x3 (rad) is the wheel angular position, and x4 (rad/s) is the wheel angular velocity. The system dynamics was originally presented in [48] and can be described by the following equations:

Fig. 1 Reaction wheel pendulum system

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d22 W sin (x1 ) + d12 b2 x4 − d12 τ , D −d21 W sin (x1 ) − d11 b2 x4 + d11 τ , x˙4 = D

x˙1 = x2 ,

x˙2 =

x˙3 = x4 ,

(11)

where d21 = d12 = d22 = J2 ,

2 d11 = m 1lc1 + m 2 l12 + J1 + J2

D = d11 d22 − d12 d21 > 0 ,

m¯ = m 1lc1 + m 2 l1 ,

W = mg. ¯

According to Proposition 1 the following global change of coordinates (similar to the one used in [36]) z 1 = d11 x1 + d12 x3 ,

z 2 = d11 x2 + d12 x4 ,

z 3 = x1 ,

z 4 = x2 ,

(12)

brings the RWP system to the cascade form z˙ 1 = z 2 , z˙ 2 = W sin (z 3 ) ,

z˙ 3 = z 4 , d22 W sin (z 3 ) + b2 (z 2 − d11 z 4 ) − d12 τ . z˙ 4 = D

(13)

The feedback control transformation τ=

d22 W sin (z 3 ) + b2 (z 2 − d11 z 4 ) − Du , d12

(14)

partially linearizes the system, and we obtain the normal form (6) z˙ 1 = z 2 , z˙ 2 = W sin (z 3 ) ,

z˙ 3 = z 4 , z˙ 4 = u .

(15)

The used system parameters were obtained with an off-line identification algorithm and they are given in Table 1.

Table 1 Parameters of the reaction wheel pendulum system Name Description m¯ d11 J2 b2 g

Equivalent mass of the system Equivalent moment of inertia of the system Moment of inertia of the wheel Friction coefficient of the wheel Acceleration of the gravity

Value 0.191 (kgm) 0.0543 (kgm2 ) 0.0027 (kgm2 ) 0.01 (Ns/m2 ) 9.81 (m/s2 )

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2.2.2

359

Translational Oscillator with Rotational Actuator (TORA)

Consider the TORA system shown in Fig. 2. This figure describes a pendulum rotating in a vertical plane and its pivot pin is mounted in a mass m 1 . The actuation on the pendulum by the torque τ (Nm) makes the mass to move in a translational manner. The states vector is x = [x1 , x2 , x3 , x4 ]T , where x1 (m) is the position of the mass m 1 , x2 (m/s) is the velocity, x3 (rad) is the angle between the downward direction and the pendulum, which is measured counter-clockwise (x3 = 0 for the downright position of the pendulum), and x4 (rad/s) is the pendulum angular velocity. The system’s dynamics can be described by the following equations: η1 (kx1 − η3 x42 ) + η2 (τ − gη3 ) , η4 η2 (kx1 − η3 x42 ) + (m 1 + m 2 )(τ − gη3 ) x˙4 = , η4

x˙1 = x2 ,

x˙2 = −

x˙3 = x4 ,

(16)

where η1 = m 2 r 2 + I , η2 = m 2 r cos(x3 ) , η3 = m 2 r sin(x3 ) , η4 = (m 1 + m 2 )η1 − η22 > 0 .

The global change of coordinates z 1 = x1 +

η3 , m1 + m2

z 2 = (m 1 + m 2 )x2 + η2 x4 ,

z 3 = x3 ,

z 4 = x4 , (17)

together with the feedback transformation τ=

η3 u − η2 (kx1 − m 2 r z 42 sin(z 3 )) + m 2 gr sin(z 3 ), m1 + m2

bring the system to the normal form (6)

Fig. 2 TORA system

(18)

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Table 2 Parameters of the TORA system Name Description m1 m2 k r I g

Value

Mass of the platform Mass of the pendulum Coefficient of the spring Length of the pendulum Moment of inertia of the pendulum Acceleration of the gravity

10 (kg) 1 (kg) 5 (kg/s2 ) 1 (Ns/m2 ) 1 (kgm2 ) 9.81 (m/s2 )

z2 , m1 + m2 km 2 r sin(z 3 ) z˙ 2 = −kz 1 + , m1 + m2 z˙ 1 =

z˙ 3 = z 4 , z˙ 4 = u .

(19)

The used system parameters are the same as in [37], and are given in Table 2.

3 Design of a Local Finite-Time Stabilizing (LFTS) Controller 3.1 Problem Statement In this section, we aim to design a controller to attain local finite-time stability of the origin z = 0 of the uncertain system (10). Since we will use for u homogeneous controllers, we consider the following local homogeneous approximation of system (10) for this purpose z˙ 1 = α1 z 2 ,

z˙ 3 = z 4 ,

z˙ 2 = α2 z 3 ,

z˙ 4 = β(t, z) [u + ϕ(t)] ,

(20)

where α1 , α2 ∈ R are given by 1 , α1 = m 11 (0)

∂g1 (z 1 − γ (z 3 ), z 3 ) α2 = . ∂z 3 z=0

Local controllability of the system (10) assures that α1 = 0, α2 = 0. Since the nominal system (20), i.e., with β(t, z) constant and ϕ(t) ≡ 0, is homogeneous of negative degree, system (10) can be seen as a “perturbation” of (20)

Robust Stabilization of a Class of Underactuated Mechanical Systems …

z˙ 1 = α1 z 2 + δ1 (z) ,

z˙ 3 = z 4

z˙ 2 = α2 z 3 + δ2 (z) ,

z˙ 4 = β(t, z) [u + ϕ(t)] ,

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where the perturbation terms δ1 (z) and δ2 (z) are of higher homogeneity degree. Approximation models similar to (20) have been used in the literature to attain a local exact feedback linearization for local stabilization of several underactuated systems, as, e.g., [3, 4, 8, 14, 15, 17, 18, 40, 44, 46, 47, 51]. In this section, we design a local finite-time stabilizing dynamic controller for the equilibrium point z = 0 of system (20), despite of the presence of β(t, z) and ϕ(t), that uses a continuous control signal. Due to Hermes’ Theorem for homogeneous systems [6, Corollary 5.5], this controller will also locally stabilize in finite-time the origin of system (10).

3.2 Controller Design Since the approximation system (20) is an uncertain chain of integrators of order 4, we can use a Discontinuous Integral Algorithm (DIA) [15, 32–34] to stabilize the origin. These algorithms consist in a static homogeneous finite-time controller for the nominal model of the system and a discontinuous integral action, aimed at estimating and compensating the uncertainties and perturbations. For system (20) we propose here the Fourth Order Discontinuous Integral Algorithm (4-DIA), given by

 15 5 5 5 5 5 5 5 5 5 u = −k4 z 4  2 + k32 z 3  3 + k32 k23 z 2  4 + k32 k23 k14 z 1 + ζ,

 5 5 5 0 ζ˙ = −k I 1 z 1 + k I 2 z 2  4 + k I 3 z 3  3 + k I 4 z 4  2 .

(21)

The first term in (21) is a continuous and homogeneous static feedback controller, providing finite-time convergence for the nominal system (without perturbations), and is obtained using a back-stepping-like approach as in [9]. The second term in (21) is a discontinuous integral term that ensures theoretically exact compensation of Lipschitz disturbances ϕ(t). This is in contrast to the classical integral controllers (see, e.g., [19, Chap. 14]), which are only able to fully compensate constant perturbations ϕ(t). Note that due to the fact that the discontinuous function sign is integrated, the control signal generated by (21) is continuous, so that the chattering effect can be strongly attenuated (see [39]). Introducing a new state variable, z 5 = ζ + ϕ(t), which corresponds to the sum of the integral variable ζ and the non-vanishing perturbation term ϕ(t), the dynamics of the closed-loop system, formed by system (20) with the controller (21), is

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z˙ 1 = α1 z 2 ,

z˙ 2 = α2 z 3 , z˙ 3 = z 4 ,

 15 5 5 5 5 5 5 5 5 5 2 2 3 2 3 4 z˙ 4 = β(t, z) −k4 z 4  2 + k3 z 3  3 + k3 k2 z 2  4 + k3 k2 k1 z 1 + z 5 z˙ 5 ∈ −k I 1





5

 5 0

5

z 1 + k I 2 z 2  4 + k I 3 z 3  3 + k I 4 z 4  2

 ¯ L¯ + − L,

 

(22)

,

where L¯ = kLI 1 . The following theorem presents the stability analysis of the 4-DIA applied to the uncertain Class-I system (10). Theorem 1 The origin of the uncertain system (10) is finite-time locally stable, even in presence of uncertainties fulfilling conditions (9), when the control τ is given by (8) and (21), and the set of gains ki and k I i , for i = 1, 2, 3, 4, are properly chosen. Proof The proof is presented in the Appendix 2. We basically show in the proof, using a homogeneous Lyapunov function, that the origin of system (22) is Globally Finite-Time stable for all uncertainties satisfying (9). Since (20) is a local approximation of the plant (10), local finite-time stability follows for the latter system, using the same Lyapunov function. Remark 1 Since the closed-loop system with the controller of Theorem 1 is homogeneous with homogeneity degree d = −1 and weights (r1 , r2 , r3 , r4 , r5 ) = (5, 4, 3, 2, 1), the point z = 0 is a fifth-order sliding-mode. Due to homogeneity properties [24], the theoretical precision of the states after the transient are |z 1 | < Δ1 τ¯ 5 , |z 2 | < Δ2 τ¯ 4 , |z 3 | < Δ3 τ¯ 3 , |z 4 | < Δ4 τ¯ 2 and |z 5 | < Δ5 τ¯ , where Δi > 0 with i = 1, . . . , 5 and τ¯ as the sample time.

3.2.1

Homogeneous Approximation of RWP and TORA Systems

The nominal local approximation of the RWP system is z˙ 1 = z 2 ,

z˙ 2 = W z 3 ,

z˙ 3 = z 4 ,

z˙ 4 = u ,

and for the TORA system is z˙ 1 =

1 z2 , m1 + m2

z˙ 2 =

km 2 r z3 , m1 + m2

z˙ 3 = z 4 ,

z˙ 4 = u .

3.3 Gain Design and Scaling The calculation of appropriate values for the gains of the 4-DIA are provided in Appendix 3, and they are given by the expressions (38)–(42). The gains k I 2 , k I 3 and

Robust Stabilization of a Class of Underactuated Mechanical Systems … Table 3 Set of gains for the 4-DIA Set k1 1 2 3

2 2 2

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k2

k3

k4

2 2 3

5 20 7

45 55 45

k I 4 can take any value. For the maximization of the functions in the design of k2 , k3 , k4 , and k I 1 , it is necessary to choose the value of k1 , γ1 , γ2 , γ3 > 0, and m ≥ 9. Some set of gains obtained with this procedure are shown in Table 3. Only the gains for the state feedback controller are shown, since the gain k I 1 depends on the size of the disturbance and k I 2 , k I 3 and k I 4 are arbitrary. Gain Scaling As a consequence of homogeneity it is possible to perform a useful gain scaling: k = (k1 , k2 , k3 , k4 , k I 1 , k I 2 , k I 3 , k I 4 ) →   1 1 1 1 1 2 3 kλ = λ 5 k1 , λ 4 k2 , λ 3 k3 , λ 2 k4 , λk I 1 , λ− 4 k I 2 , λ− 3 k I 3 , λ− 2 k I 4 .

(23)

If the set of gains k stabilizes the origin for a perturbation of size L, then for every λ > 0 the set of gains kλ also stabilizes the origin for a perturbation of size λL. This can be easily proved: Applying the linear transformation zˆ i = λz i , i = 1, . . . , 5, 0 < λ ∈ R, to system (22) with gains k leads to an equivalent system with gains kλ .

3.4 Simulation Results 3.4.1

RWP System

The Local Finite-Time Stabilizing (LFTS) controller given by the law (14), the transformation (12) and the 4-DIA described by (21) with gains k1 = k2 = 2, k3 = 5, k4 = 45, k I 1 = 8, k I 2 = k I 3 = k I 4 = 0, and scaling λ = 0.2, was implemented in Matlab Simulink with the Runge–Kutta’s integration method with fixed step and a sampling time equal to 1 × 10−4 (s). As perturbation the unbounded signal ϕ(t) = 0.1 sin(t) + 0.1t + 0.5 was used, and β = 1 was taken as control coefficient. Two initial conditions are presented: one “near” the upward equilibrium x0 = [−1, 0, 0, 0]T , and one “far” from it x0 = [π, 0, 0, 0]T , which corresponds to the downward possition. Figure 3 shows that for both initial conditions the controller is able to stabilize the origin, although theoretically only local stability is guaran-

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10-10 2

2

0 -2

1

14

14.5

15

0 -1 0

2

4

6

8

10

2

4

6

8

10

2 0 -2 -4 -6 0 300

5

200

0

100

10-9

-5 14

14.5

15

0 -100 0

2

4

6

8

10

2

4

6

8

10

400 200 0 -200 -400 0

Fig. 3 Simulated states for the closed loop of the Local Finite-Time Stabilizing (LFTS) controller for the RWP

teed. This simulation illustrates that the controller solves both, the swing-up and the stabilization problems, but this does not follow from the stability analysis. Figure 4 presents the behavior of the output signal ζ of the integrator. It identifies exactly and in finite time the (unbounded) perturbation. Figure 5 shows that the control signal is continuous.

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2 0 -2 -4 0

5

10

15

Fig. 4 Simulation results illustrating how the output of the integral signal identifies the perturbation in the LFTS of the RWP 5

0

-5 0

2

4

6

8

10

12

Fig. 5 Behavior of the continuous control signal for the LFTS of the RWP

3.4.2

TORA

For the LFTS, the control (18), the transformation (17) and the 4-DIA described by (21) with k1 = k2 = 2, k3 = 5, k4 = 45, k I 1 = 8, k I 2 = k I 3 = k I 4 = 0, and scaling λ = 0.2, were implemented in Matlab Simulink with the Runge–Kutta’s integration method with fixed step and a sampling time equal to 1 × 10−4 (s). The simulations were made with an unbounded perturbation ϕ(t) = 0.1 sin(t) + 0.1t + 0.5, and β = 1. In Figs. 6, 7, and 8 the simulations results are shown. As in the LFTS simulations of the RWP system, two initial conditions were tested to check the performance of the algorithm: close to the origin x0 = [0, 0, −1, 0]T and far from the origin x0 = [0.5, 0, π, 0]T . The plots show that for both initial conditions the four states reach the origin in finite-time with a continuous control signal shown in Fig. 8. In Fig. 7 the integrator ζ is shown and it is observed how it identifies the (negative value) of the unbounded perturbation.

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3.5 Experimental Validation: RWP System The experimental setup for the RWP, illustrated in Fig. 1, was developed at the Institut für Regelungs-und Automatisierungstechnik of TU Graz, Austria. The experiments are performed in Matlab Simulink over a data-acquisition board connected to the RWP system. Since the actual RWP system only measures positions, a second order robust exact differentiator [25] was implemented in order to estimate the velocities x2 , x4 required by the controller. The control signal τ from the algorithm is converted to a voltage signal by the expression V = −0.2778τ + 3.6 × 10−5 x4 . This signal is saturated between [−0.9, 0.9] (V) and, after an amplification stage, is connected to a 12 (V) DC motor. The control (14) and the 4-DIA described by (21) with k1 = k2 = 2, k3 = 20, k4 = 55, k I 1 = 0.3 k I 2 = k I 3 = k I 4 = 0 and scaling gain λ = 0.09, were implemented in the real RWP system. This was done using the Runge–Kutta’s integration method with fixed-step and a sampling time equal to 1 × 10−4 (s). The experiments were made with two initial conditions, x0 = [−0.5, 0, 0, 0]T and x0 = [π, 0, 0, 0]T , to see if the same behavior as in simulations is obtained. In the experiment starting in the downward position the wheel position encoder was reset

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once the pendulum reached a position near to the origin. This was made to avoid that the wheel has to perform several rotations. Figure 9 shows how the four states reach the origin, even in presence of uncertainties in the parameters and non-modeled dynamics. Fig. 10 shows the continuous control signal τ generated by the 4-DIA, while Fig. 11 presents the voltage signal controlling the DC motor. Note that during the swinging-up phase the control reaches the saturation, and this explains why the 4-DIA controller produces some oscillations of the pendulum until the upward position is reached. This phenomenon does not appear in the simulations, where there is no saturation, and the pendulum goes to the upward position in just one movement. These results show that the 4-DIA is capable of stabilizing the four states even with saturation in the real control.

4 Design of a Globally Asymptotically Stabilizing (GAS) Controller 4.1 Problem Statement In this section, the objective is to design a controller to (robustly) attain global asymptotic stability of the origin z = 0 of the uncertain system (10), in spite of the uncertainties and/or perturbations satisfying (9). To reach our objective we follow the main strategy used in [35, 37, 38] for the Class-I of underactuated systems without uncertainties/perturbations. Since system (10) is a cascade system in strict feedback form, the controller design can be divided in two steps: Step 1: For the virtual control variable z 3 of the Slave subsystem (z 1 , z 2 ) in (10) find a smooth control law z 3 = k(z 1 , z 2 ) that renders the origin (z 1 , z 2 ) = 0 Globally Asymptotically Stable (GAS). Step 2: For the (nominal linear) Master subsystem (z 3 , z 4 ) a Back-Stepping design leads to the global stability of the state z = 0. We keep Step 1 unmodified but for Step 2 we design a discontinuous and homogeneous integral term, which is able to cope with the uncertainties and/or perturbations satisfying (9).

4.2 Controller Design 4.2.1

Virtual Controller for the Slave Subsystem

An interesting feature of the Slave subsystem for the Class-I systems is that the function g1 (z 1 − γ (z 3 ), z 3 ) is usually periodic in z 3 . For example, for the RWP

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it is given by g1 (z 1 − γ (z 3 ), z 3 ) = W sin(z 3 ), while for the TORA system it is 2r sin(z 3 ). This issue leads to a challenging global g1 (z 1 − γ (z 3 ), z 3 ) = −k1 z 1 + mkm 1 +m 2 stabilization problem for the Slave subsystem. A nice solution proposed in [35, 37, 38] is to use a saturated virtual control law, z 3 = σ (κ(z 1 , z 2 )), where σ (w) is a sigmoidal function, i.e., a smooth, saturated, and monotonic increasing function, as, e.g., tanh(w) or arctan(w), and κ(z 1 , z 2 ) is a smooth scalar function of the states. Below it will be shown for the two application examples, the RWP and the TORA systems, how GAS of the Slave subsystem can be attained. Moreover, Input-to-State Stability (ISS) of the system with respect to z 3 is also achieved. This is important for the stability of the cascade system.

4.2.2

Robust Controller for the Master Subsystem

Once the virtual GAS controller for the slave is obtained, the global stabilization of the origin of the whole system is reduced to the global asymptotic stabilization of the transformed system, which in the nominal case is given by

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z2 , m 11 (μ1 + σ (κ(z 1 , z 2 ))) z˙ 2 = g1 (z 1 − γ (μ1 + σ (κ(z 1 , z 2 ))), μ1 + σ (κ(z 1 , z 2 ))), μ˙ 1 = μ2 ,

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(25)

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(26)

The origin of system (24) is globally asymptotically stable if the new control variable ν is designed properly to globally stabilize the origin of the μ1 , μ2 subsystem. This task, in principle, is rather simple, since it is a second-order linear system. However, in presence of uncertainties and perturbations, a more realistic model is z2 , m 11 (μ1 + σ (κ(z 1 , z 2 ))) z˙ 2 = g1 (z 1 − γ (μ1 + σ (κ(z 1 , z 2 ))), μ1 + σ (κ(z 1 , z 2 ))), μ˙ 1 = μ2 , z˙ 1 =

(27)

μ˙ 2 = β(t, z)[ν + ϕ(t)], where β(t, z) and ϕ(t) are assumed to satisfy the conditions (9). In this case, the design of a robust control ν is vital to achieve the task of driving the states to the origin globally. Our objective in this section is the design of a robust and global stabilizing controller for system (27), despite the presence of Lipschitz uncertainties/disturbances, i.e., |ϕ(t)| ˙ ≤ L, and uncertain control coefficient β(t, z). In order to attenuate the chattering effect we ask the control signal to be continuous. We propose two Continuous Higher Order Sliding Modes Algorithms (CHOSMA): the second-order Continuous Twisting Algorithm (2-CTA) (introduced in [55, 56]): 1

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(28)

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 13 3 3 ν = −k2 μ2  2 + k12 μ1 + ζ

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(29)

which is an integral extension of the static controller presented in [9]. In both algorithms, the static part ν drives the states μ1 , μ2 to the origin in finite-time, while the integral part ζ rejects the disturbance ϕ(t). Note that since the discontinuous function sign is in the integrator, the generated control signal in both algorithms is continuous. The following theorem is the main result of this section. Theorem 2 The origin of μ1 , μ2 -system in (27) is globally finite-time stable, despite the presence of the Lipschitz disturbances/uncertainties ϕ(t) and the uncertain coefficient β satisfying (9). Moreover, z 1 , z 2 will be driven to the origin globally and asymptotically, when the control τ takes the form of (8) and (26), with the state transformations (5) and (25), and ν takes the form of (28) or (29). Remark 2 As a consequence of the Theorem 2, the states of the original system (2), are globally and asymptotically driven to the origin. Proof Since the Slave system is ISS with respect to μ1 (t) [35, 37], it suffices to analyze the stability of the Master subsystem μ˙ 1 = μ2 , μ˙ 2 = β(t, z)[ν + ϕ(t)].

(30)

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(31)

μ˙ 3 = −k3 μ1 0 − k4 μ2 0 + ϕ(t) ˙ . For the case with known coefficient β, the origin is proven to be global and finite-time stable in [56], when the gains k1 , k2 , k3 , k4 are properly designed, using the following homogeneous Lyapunov function: 5

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(32)

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2 −3 1 3 5 5 γ1 |ξ1 | 3 + ξ1 ξ2 + k1 2 |ξ2 | 2 + |ξ3 |5 , 5 5 5 −1

where ξ1 = μ1 − ξ3 3 , ξ2 = μ2 , ξ3 = k1 2 k2−1 μ3 . The derivative of V is given by   3   3  13 3 1 3 3 −3 V˙ (μ) ∈ −β (t, z) k2 k1 2 k12 ξ1 + ξ2  2 k12 ξ1 + ξ2  2 + k12 ξ3 3 − k12 ξ3 +       2 2 −1 γ1 ξ1  3 + ξ2 ξ2 − k I 1 k1 2 k2−1 −3 γ1 ξ1  3 + ξ2 |ξ3 |2 + ξ3 4 × 

   3 0 ˜ L˜ , ξ1 + ξ3 3 + k I 2 ξ2  2 − − L, where L˜ =

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Using Lemma 2 we conclude that the first term is negative semi 3  3 definite and it vanishes only on the set S1 = k12 ξ1 + ξ2  2 = 0 . Evaluating V˙ on this set we obtain       −1 V˙ S1 ∈ − γ1 k1−1 − 1 ξ22 − k I 1 k1 2 k2−1 |ξ3 |2 3 γ1 k1−1 − 1 ξ2 + ξ3 2 × 

 0    3 −3 ˜ L˜ . ξ3 3 + k I 2 − k1 2 ξ2  2 − − L, If γ1 > k1 the first term is negative semi-definite, and it is zero only on the set S2 = {ξ2 = 0}. Evaluating V˙ S1 on this set we get    −1 ˜ L˜ . V˙ S1 ∩S2 ∈ −k I 1 k1 2 k2−1 ξ3 4 ξ3 0 − − L, This is negative if L˜ = kLI 1 < 1. By Lemma 1 we can render V˙ S1 < 0 selecting k I 1 > 0 small. Using Lemma 1 again we conclude that it is possible to make V˙ < 0 selecting k2 > 0 sufficiently large. Therefore, the controllers (28) or (29) globally stabilize in finite-time μ1 = μ2 = 0 for the μ-subsystem of (27). Then, the origin of the system (27) is globally asymp-

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totically stable. Furthermore, by the change of coordinates (5) and (25), the original states of the system (2) will be driven to the origin asymptotically and globally.  Remark 3 The systems (31) and (32) are homogeneous of degree d = −1 and weights (r1 , r2 , r3 ) = (3, 2, 1). Due to homogeneity properties [24], the theoretical precision of the states after the transient are |μ1 | < Δ1 τ¯ 3 , |μ2 | < Δ2 τ¯ 2 , and |μ3 | < Δ3 τ¯ , where Δi > 0 with i = 1, . . . , 3 and τ¯ as the sample time. In the following sections, the control design is able to globally stabilize the origin of the RWP and the TORA systems will be presented.

4.3 RWP Controller Design Defining the needed sigmoidal function σ (κ(z 1 , z 2 )) (for simplicity we just write σ (z 1 , z 2 )) as (33) σ (z 1 , z 2 ) := −c0 tanh(c1 z 1 + c2 z 2 ), where 0 > c0 > π/2 and c1 , c2 > 0. Using the variables (25) the dynamics of system (15) becomes (we consider here β = 1) z˙ 1 = z 2 , z˙ 2 = W sin (σ (z 1 , z 2 ) + μ1 ),

μ˙ 1 = μ2 , μ˙ 2 = ν + ϕ(t),

where u = ν + σ¨ , σ¨ = −c0 (1 − tanh2 (c1 z 1 + c2 z 2 )) [c1 W sin(z 3 ) + c2 W cos(z 3 )z 4  −2 tanh(c1 z 1 + c2 z 2 ) (c1 z 2 + c2 W sin(z 3 ))2 ,

(34)

and the ν control can take the form of (28) or (29) to achieve the global stabilization of the RWP origin. Remark 4 The sigmoidal function (33) is similar to the one used in the control of the RWP in [37], with the difference that in this work we are controlling the RWP of fourth order and in the other work the RWP of third order (neglecting the position of the wheel).

4.4 TORA Controller Design Defining the same sigmoidal function σ (z 1 , z 2 ) as (33) and the same change of coordinates as (25), transform the system (19) to

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(35)

and the ν control can take the form of (28) or (29) to achieve the global stabilization of the origin of the TORA.

4.5 Simulation Results 4.5.1

RWP System

For the GAS, the control (14) and (34), the transformations (12) and (25), the 2DIA described by (29) with k1 = 4.16, k2 = 28.5, k I 1 = 6.3, k I 2 = 0, the 2-CTA described by (28) with k1 = 47.57, k2 = 19.84, k3 = 16.1, k4 = 7.7 were implemented in Matlab Simulink with the Runge–Kutta’s integration method of fixed step and a sampling time equal to 1 × 10−4 (s). Also, to perform a comparison with another kind of algorithm, a linear algorithm (2-LA) was tested: ν = −k1 μ1 − k2 μ2 ,

(36)

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For the GAS, the control (18) and (35), the transformations (17) and (25), the 2DIA described by (29) with k1 = 4.16, k2 = 28.5, k I 1 = 6.3, k I 2 = 0, the 2-CTA described by (28) with k1 = 47.57, k2 = 19.84, k3 = 16.1, k4 = 7.7, and the 2-LA described by (36) with k1 = k2 = 10, were implemented in Matlab Simulink with the Runge–Kutta’s integration method of fixed step and a sampling time equal to 1 × 10−4 (s). The simulations were made with the same unbounded perturbation ϕ(t) =

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2 sin(2t) + 2t + 2 and ς (t) = 1, the same initial condition x0 = [0, −1, 100π, −1]T and the same gains c0 = 1.5 and c1 = 3, c2 = 5. The results are shown in Figs. 16, 17, 18, and 19. Again, the sliding modes can compensate the matched unbounded disturbance and drive the μ1 , μ2 states to the origin in finite-time. Then the original states of the TORA are attracted to the origin asymptotically. The 2-LA behaves in the same way as the RWP system and drives the states near to the origin until it can not compensate the disturbance. Also, the control signal is continuous in the three algorithms.

4.6 Experimental Validation: RWP System The experiments were made again in the real RWP system. The control (14) and (34), the transformations (12) and (25), the 2-DIA described by (29) with k1 = 2.43, k2 = 14.62, k I 1 = 1.26, k I 2 = 0, c0 = 1.5, c1 = 0.17, and c2 = 7.5, the 2-CTA described by (28) with k1 = 36.3, k2 = 19.84, k3 = 4.02, k4 = 1.92, c0 = 1.5, c1 = 0.5, and c2 = 7, and the 2-LA described by (36) with k1 = 16, k2 = 24, c0 = 1.5, c1 = 1.6, and c2 = 3.6 were implemented in the real RWP system with the Runge–Kutta’s integration method of fixed step and a sampling time equal to 1 × 10−4 (s). The experiments were made with the same initial condition x0 = [π, 0, 0, 0]T . The results are shown in Figs. 20, 21, 22, and 23. The continuous sliding modes algorithms drive the states of the pendulum from the downward position close to the origin and stay around it, although the 2-CTA make it slower and with a larger control signal. The 2-LA can take the pendulum to the upright position, but due to the presence of uncertainties that it can not compensate, the wheel position can only be maintained around the origin and the pendulum falls in many times. Seeing the μ1 and μ2 plots, one can notice how the 2-DIA and the 2-CTA can maintain them in the origin and the 2-LA does not. Again, the chattering effect is diminished as seen in the continuous control signal of the sliding modes algorithms.

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In theory, the three algorithms can achieve global stabilization, but in practice, with the saturation of the real control signal (voltage) is no longer possible. Despite this, the sliding modes algorithms achieve the swing-up and stabilization in one step.

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5 Conclusions In this chapter, we have designed robust controllers for a Class of underactuated mechanical systems of two DOF, using a continuous Higher Order Sliding-Mode strategy. Two kinds of controller designs were presented: One generates a fifth-order sliding-mode and achieves Local Finite-Time Stability (LFTS). The other is a robust controller that provides Global Asymptotic Stability (GAS). These controllers compensate matched Lipschitz disturbances and/or uncertainties, cope with an uncertain control coefficient and generate a continuous control signal, possibly reducing the chattering effect. We provided evidence of the performance of the controllers using simulations for the RWP and the TORA systems, and by means of experiments on the RWP system. Acknowledgements The authors thank the financial support of CONACyT (Consejo Nacional de Ciencia y Tecnología): Project 282013, CVU’s 624679 and 705765; PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) IN115419 and IN110719.

Appendix 1: Homogeneity Let vector x ∈ Rn , its dilation operator is defined Δr := ( r1 x1 , . . . , rn xn ), ∀ > 0, where ri > 0 are the weights of the coordinates and r is the vector of weights. A function V : Rn → R (respectively, a vector field f : Rn → Rn , or vector-set F(x) ⊂ Rn ) is called r-homogeneous of degree m ∈ R if the identity V (Δr ) = m V (x) holds (or f (Δr x) = m Δr f (x), F(Δr x) = m Δr F(x)) [5, 33]. Suppose that the vector r is fixed. The homogeneous norm is defined by ||x||r, p :=  1p  p n ri |x | , ∀x ∈ Rn , for any p ≥ 1 and the set S = {x ∈ Rn : ||x||r, p = 1} is i i=1 the homogeneous unit sphere [5].

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Fig. 22 Control signal in the GAS for the RWP experiments

Some Properties of Homogeneous Functions Consider V1 and V2 two r-homogeneous functions (respectively, a vector field f 1 ) of degree m 1 , m 2 (and l1 ), then: (i) V1 V2 is homogeneous of degree m 1 + m 2 , (ii) 1 there exist a constant c1 > 0, such that V1 ≤ c1 ||x||m r, p , moreover, if V1 is positive 1 , (iii) ∂ V1 (x)/∂ xi is homogeneous definite, there exists c2 such that V1 ≥ c2 ||x||m r, p of degree m 1 − ri , (iv) L f V1 (x) is homogeneous of degree m 1 + l1 [5].

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0 1

-1 0

-2

-1 109

10

20

30

40

109.5

50

60

110

70

80

90

100

110

Fig. 23 Voltage control signal in the GAS for the RWP experiments

The following result is well known for continuous homogeneous functions (see [16, Theorem 4.4] or [2, 33]), and can be extended to semi-continuous functions [9]. Lemma 1 Let η : Rn → R and γ : Rn → R be two r-homogeneous and upper semi-continuous single-valued functions, with the same weights r = (r1 , . . . , rn ) and homogeneity degree m > 0. Suppose that γ (x) ≤ 0 in Rn . If {x ∈ Rn \ {0} : γ (x) = 0} ⊆ {x ∈ Rn \ {0} : η(x) < 0}, then there exists a real number λ∗ and a constant c > 0 so that, for all λ ≥ λ∗ and for all x ∈ Rn \ {0} the following inequality is satisfied: η(x) + λγ (x) ≤ −c ||x||m r, p .

The following Lemma is simple (just monotonicity) but useful Lemma 2 Consider the real variables x, y, it is always true that   sign x + yβ − yβ = sign(x), β > 0.

Appendix 2: Proof of Theorem 1 Using the back-stepping-like procedure to obtain the state-feedback controller and its Lyapunov Function used in [9], and introducing an additional term and a modification, we obtain the following homogeneous candidate Lyapunov function of degree m (with m ≥ 9 to render it differentiable)

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 m−2 m−2

5 5 5 5 2 m 5 5 V4 = γ3 V3 + |z 4 | 2 + k3 2 z 3  3 + k23 z 2  4 + k23 k14 ξ1 z4+ m   m m 5 5 5 2 1 5 5 5 1− k32 z 3  3 + k23 z 2  4 + k23 k14 ξ1 + |z 5 |m , m m where γ1 , γ2 , γ3 > 0 are arbitrary and − − − ξ1 = z 1 − kξ z 5 5 , ξ˙1 = α1 z 2 − 5kξ |z 5 |4 z˙ 5 , kξ = k4−5 k3 2 k2 3 k1 4 , 5

V3 = γ2 V2 +

  m−3  m  m−3 5 5 m 5 3 3 |z 3 | 3 + k2 3 z 2  4 + k14 ξ1 k23 z3 + 1 − m m

5

5

m 5 5 z 2  45 + k 4 ξ1 , 1

  m−4 m 5 4 4 m m m−4 m 4 5 4 5 k14 |ξ1 | 5 . V2 = γ1 |ξ1 | + |z 2 | + k1 ξ1  z2 + 1 − m m m Using Young’s inequality, it is possible to show that the function V4 is positive definite. Its derivative along the trajectories of (22) is given by   V˙4 = γ3 γ2 Fk2,3 + Fk3,4 + β(t, z)k4 G k4 + Fk4 + z 5 m−1 z˙ 5 ,

(37)

where  G k4 = − z 4  

m−2 2

m−2 2

+ k3



5 3

5 3

5 4

5 3

5 4

z 3  + k2 z 2  + k2 k1 ξ1

 m−2  5

×

 1  5 5 5 5 5 5 5 5 5 5 , − k4−1 z 5 + z 4  2 + k32 z 3  3 + k32 k23 z 2  4 + k32 k23 k14 ξ1 + k4−5 z 5 5

 m−7 m−2 5 5 5 m−2 5 5 5 k3 2 z 3  3 + k23 z 2  4 + k23 k14 ξ1 × 5   5 5   5 5 5 2 1 |z 3 | 3 z 4 + k23 α2 |z 2 | 4 z 3 + k23 k14 α1 z 2 − 5kξ |z 5 |4 z˙ 5 × 3 4  

 25 5 5 5 5 5 3 3 4 3 4 z 4 + k3 z 3  + k2 z 2  + k2 k1 ξ1 , 

Fk4 =

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Fk3,4 = z 3  

m−3 3

m−3 3



5 4

5 4

 m−3  5

z 2  + k1 ξ1

+ k2

 z4 +

m−8 5 5 z 2  45 + k 4 ξ1 × 1

 m−3 m−3 k2 3 5

 3   5 5  5 5 1 5 α2 |z 2 | 4 z 3 + k14 α1 z 2 − 5kξ |z 5 |4 z˙ 5 , z 3 + k2 z 2  4 + k14 ξ1 4

  m−4  m−5  m−4 m−4 Fk2,3 = γ1 ξ1  5 α1 z 2 − 5kξ |z 5 |4 z˙ 5 + α2 z 2  4 + k1 4 ξ1  5 z 3     m−4  m−4 m−9 4 k1 4 |ξ1 | 5 z 2 + k1 ξ1  5 α1 z 2 − 5kξ |z 5 |4 z˙ 5 . + 5 Using Lemma 2 we conclude that the term G k4 is negative semi-definite and it vanishes only on the set



5 3

5 3

5 3

5 4

5 4

S1 = z 4 = −k3 z 3  + k2 z 2  + k2 k1 ξ1

 25

.

Evaluating V˙4 on this set, and noting that Fk4 S1 = 0 and Fk3,4 S1 = k3 G k3 + Fk3 , we obtain   V˙4 S1 = γ3 γ2 Fk2,3 + k3 G k3 + Fk3 + z 5 m−1 z˙ 5 | S1 , where  G k3 = − z 3 

m−3 3

m−3 3

+ k2



5 4

5 4

z 2  + k1 ξ1



 m−3 5

5

5

5

5

5

z 3  3 + k23 z 2  4 + k23 k14 ξ1

 25

,

   m−8

 35 m−3 5 5 5 m−3 5 5 k2 3 z 2  4 + k14 ξ1 Fk3 = z 3 + k2 z 2  4 + k14 ξ1 × 5   5   5 1 α2 |z 2 | 4 z 3 + k14 α1 z 2 − 5kξ |z 5 |4 z˙ 5 | S1 . 4 

We note that the term G k3 is negative semi-definite and it is zero only on the set



5 4

5 4

S2 = z 3 = −k2 z 2  + k1 ξ1

 35

.

Evaluating V˙4 S1 on this set, and noting that Fk3 S2 = 0 and Fk2,3 S2 = Fk2 + k2 G k2 , we get   = γ3 γ2 Fk2 + k2 G k2 + z 5 m−1 z˙ 5 | S ∩S , V˙4 S1 ∩S2

where

1

2

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 35 m−4 5 m−4 m−4 5 z 2  4 + k14 ξ1 , G k2 = −α2 z 2  4 + k1 4 ξ1  5   m−4  m−4 m−5  m−9 4 5 Fk2 = γ1 ξ1  k1 4 |ξ1 | 5 × α1 z 2 − 5kξ |z 5 | z˙ 5 | S1 ∩S2 + 5    4 4 z 2 + k1 ξ1  5 α1 z 2 − 5kξ |z 5 | z˙ 5 | S1 ∩S2 . G k2 is negative semi-definite and it vanishes only on the set   4 S3 = z 2 = −k1 ξ1  5 . Evaluating V˙4 S1 ∩S2 on this set we get V˙4 S1 ∩S2 ∩S3 ∈ γ2 γ3 Fk2 S3 + z 5 m−1 z˙ 5 S1 ∩S2 ∩S3   m−1 m−5 ∈ −α1 k1 γ3 γ2 γ1 |ξ1 | 5 − k I 1 z 5 m−1 − 5kξ γ3 γ2 γ1 ξ1  5 |z 5 |4 ×    0  m  ¯ L¯ . 1 − k I 2 k14 ξ1 + kξ z 5 5 + − L, Since k1 > 0 the first term in the latter expression is non-positive and it is zero only on the set S4 = {ξ1 = 0} . Evaluating V˙4 S1 ∩S2 ∩S3 on this set we get    ¯ L¯ . V˙4 S1 ∩S2 ∩S3 ∩S4 ∈ −k I 1 z 5 m−1 z 5 0 + − L, The latter expression is negative if L¯ =

L kI 1

< 1, that is, for

L < kI 1 ,

(38)

since z 5 0 is a multivalued function, defined as ⎧ if z 5 > 0 , ⎨ +1 z 5 0 = [−1, +1] if z 5 = 0 , ⎩ −1 if z 5 < 0 . Lemma 1 implies that it is possible to render V˙4 S1 ∩S2 ∩S3 < 0 selecting k I 1 > 0 small. Applying Lemma 1 once more, we conclude that V˙4 S1 ∩S2 < 0 selecting k2 > 0 sufficiently large. Again, Lemma 1 shows that we can get V˙4 S1 < 0 selecting k3 > 0 sufficiently large. Finally, Lemma 1 implies that V˙4 < 0 if k4 > 0 is sufficiently large.

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Since V˙4 is negative definite by appropriate selection of the gains (what is always feasible), then the origin of system (22) is asymptotically stable. Moreover, since the system is homogeneous of negative degree, the origin is finite-time stable (see, e.g., [24]). Since (22) is a local homogeneous approximation of system (11), we conclude that it is locally finite-time stable. 

Appendix 3: Gain Selection From the previous proof we derive conditions for the gains to render V˙4 < 0. The following inequalities are obtained for the gain selection: k −1 I 1 > max z∈Ω1

⎧ ⎨

m

−k14 γ3 γ2 γ1 |ξ1 |

⎫ ⎬

m−1 5

 , ⎩ z m−1 − 5k γ γ γ ξ  m−5 4 z˙ 5 | S1 ∩S2 ∩S3 ⎭ 5 ξ 3 2 1 1 5 |z 5 |

k2 > max z∈Ω2

k3 > max z∈Ω3



Fk2 + γ3−1 γ2−1 z 5 m−1 z˙ 5 S1 ∩S2 G k2

,

γ2 Fk2,3 + Fk3 + γ3−1 z 5 m−1 z˙ 5 S1 G k3

(39)

(40)

,



 γ3 γ2 Fk2,3 + Fk3,4 + Fk4 + z 5 m−1 z˙ 5 1 max , k4 > bm z∈Ω4 G k4

(41)

(42)

  1 where the homogeneous unit spheres Ωi are given by Ω1 = |ξ1 | 5 + |z 5 | = 1 ,     1 1 1 1 1 Ω2 = |ξ1 | 5 + |z 2 | 4 + |z 5 | = 1 , Ω3 = |ξ1 | 5 + |z 2 | 4 + |z 3 | 3 + |z 5 | = 1 , and   1 1 1 1 Ω4 = |ξ1 | 5 + |z 2 | 4 + |z 3 | 3 + |z 4 | 2 + |z 5 | = 1 . Functions (39)–(42) are shown to have a maximum value. Since they are homogeneous of degree d = 0 the maximum can be found on the homogeneous unit sphere Ωi .

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Industry-Grade Robust Controller Design for Constant Voltage Arc Welding Process Arun Kumar Paul, Manas Kumar Bera, Mangesh Waman, and Bijnan Bandyopadhyay

Abstract Arc welding is an indirectly controlled process where the electrical variables of arc are controlled to create joints of requisite mechanical and metallurgical characteristics after maintaining workable environment for the welder. Constant voltage arc types are mostly used in industry because they are energy efficient, more productive and welder friendly, these processes could be easily automated. However, the nature of arc types for CV processes appears drastically different for different metal transfer modes, input conditions, and process settings. The nonlinear arc itself poses as altogether different loads to power controller. Moreover, the process consists of several dynamic activities, each having different response time, their interaction with control variables add extra burden to controller. To simplify controller design, the input conditions of the process are defined in such a way that minimally defined two-input process could be controlled by controlling two independent dynamic activities. To meet that purpose with increased robustness this article proposes to use two independent easily implementable yet superior SOSM controllers—one for control of arc and another for control of wire feed speed. Robust control of wire feeding would reduce disturbance burden on the arc controller. The approach would be experimentally validated by practically designing both controllers along with creating requisite joints for wide range use suitable for ready-to-use in industrial applications.

A. K. Paul (B) · M. Waman Electronics Devices World Wide Private Limited, MIDC, Mumbai, India e-mail: [email protected] M. Waman e-mail: [email protected] M. K. Bera Electronics and Instrumentation Engineering, National Institute of Technology, Silchar, India e-mail: [email protected] B. Bandyopadhyay Systems and Control, Indian Institute of Technology Bombay, Mumbai, India e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2021 A. Mehta and B. Bandyopadhyay (eds.), Emerging Trends in Sliding Mode Control, Studies in Systems, Decision and Control 318, https://doi.org/10.1007/978-981-15-8613-2_16

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Keywords Flux cored arc welding (FCAW) · Gas metal arc welding (GMAW) · Second-order sliding mode control (SOSMC) · Submerged arc welding (SAW) · Super-twisting control (STC) · Pulse width modulation (PWM)

1 Introduction ARC WELDING PROCESS [1–16] is an extremely popular method of joining similar/dissimilar metals by using the energy generated in the electric arc. The process is versatile and complex. In most cases, the filler metal from electrode tip is added to the weld pool to create the joint. The process is continuously evolving to cater new applications using new materials and also for creating superior joint features. Depending upon the nature of arc control, it is majorly divided into either a constant current (CC) or a constant voltage (CV) process. Based on heat input and metal transfer mode the characteristics of CV arc are further sub-divided into different arc types [6]. The arc power or energy is appropriately controlled to ensure melting of requisite quantity of metal and their subsequent transfer to the weld pool. Though, the controllable features of CC and CV processes are quite different, still, with the help of modern power control techniques, one equipment is capable of performing welding of any arc type [5]. The comparative statement on performance of a few arc welding processes is listed in Table 1. In CV processes the welding current I a is flexible. To cover wide range applications the welding methods under CC and CV arcs are further sub-divided to cater new type of joints using new materials. The physical dimension of the workpiece have influence on the behavior of arc. For making joints Table 1 Capability of different arc welding methods Constant current process

Constant voltage process

Metal type

SMAW

GMAW

Steel

TIG

FCAW

SAW

Yes

Yes

Yes

Yes

Yes

Stainless steel Yes

Yes

Yes

Yes

Yes

Al, Mg

Yes

Yes

NP

NP

NP

Cast iron

Yes

NP

Yes

Yes

Yes

Cu, Brass, Ti

NS

Yes

Yes

NP

NP

Job thickness

≥2 mm

≥0.3 mm

≥0.5 mm

≥2 mm

≥2.5 mm

Deposition rate

Moderate

Poor

High

High

Highest

Arc efficiency 0, let the bounds of ψf and ψ˙ f are defined as    |ψf | < δ |Sf | and ψ˙ f  < δ1 .

(22)

(23)

One typical problem product designer faces in practical implementation is what method to be adopted to choose the gains and what would be their quantified value. Multiple approaches are available. It is difficult to quantify worst-case perturbations for wide range (speed, torque, and torque characteristics) applications [18]. The product evaluation would be difficult, its optimal design could be even complicated.

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Depending upon the considered disturbance characteristics, the bounds of both ψf and ψ˙ f could vary differently because the torque characteristics are wide. Broadly, two popular but drastically different approaches [28, 29] are used for finding the gains. The aim  of the first approach [28] is to reduce chattering where the gains are based on ψ˙ f  alone, their values are,  ρ1 = 1.5 δ1 and ρ2 = 1.1δ1

(24)

Small value of ρ2 could introduce certain wind up even in SOSM control. In Approach 2 [29], both ψf and ψ˙ f (23) are considered where the gains are ρ1 > 2δ and ρ2 > ρ1

5ρ3 δ + 6δ1 + 4(δ + δ1 /2ρ1 )2 2(ρ1 − 2δ)

(25)

Equations (24) and (25) do not clearly define the upper limit of gains. How to optimize the control design, is it eliminating chattering alone? Chattering is not acceptable at the electrode tip where, often, the pendent molten droplet resides. Figure 4 shows the response of SOSM controller for a step reference. When compared with performance of PI control (Fig. 3), there was sharp improvement in response time. However, under light load condition, the high-gain system generated chattering. Persistent oscillations were due to fractional order term in (21), its impact was more when the value of error was small, particularly, for sensitive pancake PMDC motors. Fig. 4 Performance of original SOSM function at light load condition generated chattering at no load, the ripple in actual speed was large. [ρ 1 = 40, ρ 2 = 7500]

Control voltage ω: 31.4 rad/s/D

ωref: 31.4 rad/s/D Start pulse

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4.3 Modified SOSM Function and Its Implementation To improve dynamic performance, in industrial motor speed control, instead of PI, switchable PI control function [20] is used. Here, the wire feeding approaches the target speed with large value of K P and around zero error the I-action is dominant. It was noticed that first term of (21) could make SOSMC less suitable for light load applications [18]. Moreover, in all real-life control applications, due to the integration term, each control function gets saturated till the desired set point is achieved. The comparative behavior of control function could be noticed only around zero error. Therefore, retaining robust features around zero error, (21) is modified to improve on following features: 1. 2. 3. 4.

Achieve accurate control in finite time Zero chattering or oscillations Simple design rule and Easy implementation criteria. The modified SOSM function could be expressed as  u f−MSM ≈ ρ1 Sf + ρ2

sgn(Sf )dτ .

(26)

Whether the approximate formula (26) meets the SOSMC criteria could be judged by practical means by testing or comparing following features [9]: 1. Both sliding surface and its derivative would always be zero and 2. Meeting or exceeding the repeatability and reproducibility criteria [30] of original system (21) 3. Less chattering at wide range load conditions could make it even superior. Though, (26) could be implemented in embedded digital system. For realizing superior real-time control action, here, it was implemented in hardware domain (see Fig. 5) where the gains could be decided as ρ1 =

R5 + R9 1 and ρ2 = . R5 R10 C2

(27)

Retaining the gains of (25), (26) was practically implemented. Its performance is shown in Fig. 6. Its dynamic response for step input was superb. Most importantly, the speed controller did not generate any chattering at light load condition. Therefore, for industrial applications (26) should be preferred. The control function (26) was extensively tested. As shown in Fig. 7, its performance for tracking oscillatory reference under fixed load torque was excellent. It was superb for wide range speed tracking under wide range load torque (Fig. 8). The speed error was negligible, its accuracy was less than 0.5%.

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+ R10

-1

e

+

-

-

+

PWM

R11

1V 1V

C2

+ R9

R

R -

R

+

R5

uf-MSM

R

Fig. 5 Control circuit for simplified SOSM function is simple, like PI function

Fig. 6 Excellent performance at no load for step response for set speed at 145 rad/s. [ρ 3 = 40, ρ 4 = 7500]

Error e: 78.5 rad/s/D ω: 78.5 rad/s/D

ωref: 78.5 rad/s/D Start

5 Controller Design for Control of Current in CV Process Presuming constant arc parameters (2), then any drift in V arc would be reflected as

Varc = E la + Ra Ia .

(28)

The drift V arc could be compensated either by change in arc length la or current I a . Energy impact due to change in la would be felt only in arc plasma, but change in I a would be reflected in heating of electrode (both in arc heat at tip and joule heat

Industry Grade Robust Controller Design … Fig. 7 Excellent tracking ability with oscillatory reference at T L = 0.12 Nm. [ρ 3 = 40, ρ 4 = 7500]

407

Control voltage Error: negligible

ω: 157 rad/s/D ωref: 157 rad/s/D

Fig. 8 Excellent tracking ability with wide range speed reference under variable load, at maximum load, T L = 0.12 Nm

Control voltage

ω: 78.5 rad/s/D

ωref: 78.5 rad/s/D

in l s ), workpiece and the arc plasma. Superior energy distribution through current control makes much larger impact on the process behavior. Moreover, compensating any drift V arc through I a would be prompt and accurate because its response time is many times faster (Table 2) than that of wire-feeder controller. Lastly, effecting control by la makes the process multi-input, complex. Therefore, current control is preferred for CV process.

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Let the sliding variable for control of arc current be constructed as S1 = e1 = Iref − Ia .

(29)

I ref is the reference value for I a and e1 is the current error. In SOSMC, both S 1 and its derivative must converge to zero in finite time i.e., S1 = Iref − Ia = 0.

(30)

The derivatives of S 1 is 1 d 1 RS + RC + ρels Varc + Ia S˙1 = − Voc + (Iref ) + L1 dt L1 L1 1 = − k1 u 1 + ψ1 (Ia , ls , t) L1

(31)

S 1 has relative degree of one. In arc welding, due to non-linear characteristics of arc the condition Voc (max) i.e. k1 u 1 (max) >> L 1 ψ1 (Ia , ls , t) is always true. Hence, ψ 1 (.) is bounded, particularly, when I a is present. Therefore, S 1 can be steered into sliding manifold using super twisting algorithm, with following input function:  u 1 = ρ3 |S1 | sgn(S1 ) + ρ4 0.5

sgn(S1 )dτ ,

(32)

where ρ 3 and ρ 4 are gains. Without compromising on performance, with similar logic as applied in the previous section for (26), (32) could as well be simplified like  u 1 = ρ3 S1 + ρ4

sgn(S1 )dτ .

(33)

Signal u1 generated in control circuit is translated to V oc in power domain as Voc = k1 u 1 =

dpwm VDC Voc (max) , where k1 = n u 1 (max)

Voc (max) =

dpwm (max)VDC n

(34) (35)

V DC is rectified voltage, V oc (max) and u1 (max), respectively, are maximum value of V oc and u1 and n is turns ratio of TR1 , d pwm (d pwm (max): its maximum value) is PWM duty cycle of inverter. The chosen value of u1 (max) in control circuit is 10 V. Like previous section, the performance of simplified SOSMC was compared with that of PI function for control of current in manual arc welding using E6010 electrodes. For control of I a , same 20 kHz full bridge inverter was used for both PI and robust control functions. It was clear (Fig. 9) that, when PI control was used, the arc current was not rigidly controlled, ripple content was large. It would result arc

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Fig. 9 Performance of PI control for constant current manual arc welding at 75 A: the ripple content was large

Varc: 20V/D

Ripple in Ia: 50A/D

Ia: 50A/D

stability problem [8]. Under SOSMC (33) the performance was robust (Fig. 10). The ripple content was negligible. There was no problem for arc stability. For control of current in CV process, (33) would as well be used. Negligible current error as well as zero current ripple meant basic SOSM conditions were fulfilled. Fig. 10 Performance of simplified SOSMC was robust for constant current manual arc welding at 120 A. [ρ 3 = 10, ρ 4 = 4000]

Control voltage Varc: 20V/D

Ia: 100A/D Ripple in Ia: 20A/D

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Fig. 11 A simple reference profile of current in a droplet transfer cycle for waveform control of current in GMAW process

Arcing phase

Voltage Current controlled controlled Droplet cycle

6 Implementation of Robust Current Control Module for CV Process For practical validation one 400A power controller using simplified SOSM control function (Fig. 5) was developed. It was full-bridge inverter switching at 20 kHz having V oc (max) at 80 V and d pwm (max) was 0.9. Generating real-time feedback for V arc directly from arc was difficult, often, it could as well be far away from the controller, therefore, instead the terminal voltage V MT (Fig. 1) was used. The measurement of V arc for spray mode of transfer was relatively easy where methods such as multi-rate feedback could be used [10]. However, the same could not be used for short-circuiting mode of transfer where frequency of transfer, wave shape of V arc , etc. vary widely. Weld pool oscillations, unwanted short-circuits, shielding gas, etc. could also invite measurement error. It generated satisfactory control performance when ten or more consecutive samples (interval: 2 ms) of V MT were processed (kth sample was included and (k-10)th sample was discarded in each measurement). The measured average value of V arc at the kth sampling instance was   k 1  Varc (k) ≈ VM T . 10 k−9

(36)

Using (33) several joints were made using different input conditions and welding methods. Welding conditions for each one is listed in Table 4. Table 4 Input conditions for control implementation Parameter Process

Gas 1 (%) Gas 2 (%) d (mm) V f (cm/s) V arc (V)

Figure 12 MAG

Ar: 0

CO2 : 100 1.2

7.0

Figure 14 MIG

Ar: 80

CO2 : 20

1.2

7.0

19.0

Figure 15 Joint in FCAW

–

CO2 : 100 1.2

13.3

28.0

Figure 16 Joint in FCAW

–

32.0

22.0

CO2 : 100 1.2

16.7

Figure 18 Joining two dissimilar metals Ar: 98

O2 : 2

1.2

11.7

26.0

Figure 20

–

1.2

13.3

28.0

Ar: 100

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Control voltage Ia: 200A/D

Iexp(t): 180A/D

Varc: 20V/D

Fig. 12 Performance of SOSM control is robust for making welding joint using MAG method at V arc of 22 V and wire feeding speed 7.0 cm/s

Fig. 13 T-Joint on two 8 mm thick MS slabs using 100% CO2 shielding using welding conditions of Fig. 12

Execution of metal transfer from tip of electrode to weld pool makes control of CV process complex. In industrial controllers normally three different types of transfer mode are used, they are 1. Short circuit mode 2. Spray mode, and its derivatives, such as, 3. Pulsed gas metal arc welding (P-GMAW). Here, short-circuit and spray mode would be practically validated. In the first case, the droplet is transferred when the power source is shorted through the droplet. Disturbances such as weld pool oscillations affect this short-arc process greatly. Large short-circuit current generates spatters, fumes, the joint quality gets affected. The process is better controlled by implementing proper waveform control approaches [2, 15, 31], whose basic objectives are as follows:

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Fig. 14 Performance of SOSM control is robust for making welding joint using MIG method at V arc of 19 V and wire feeding speed 7.0 cm/s

Control voltage

Ia: 100A/D

Iexp(t): 90A/D

Varc: 20V/D

Fig. 15 Performance of SOSM control is robust for making welding joint using FCAW method at V arc of 28 V and wire feeding speed 13.3 cm/s

Control voltage

Ia: 200A/D

Iref: 180A/D

Varc: 20V/D

1. 2. 3. 4. 5. 6.

Ensure undisturbed heating and melting of electrode tip and base plate Stabilize the weld pool oscillations Allow formation of spherical shape droplet Guide the droplet to touch the stable weld pool at small current Ensure the droplet transfer at small current under surface tension Alternately, for transfer using electro-magnetic force, the droplet should be transfer ready before touching the pool.

CV processes, particularly for short-circuit mode of transfer, needed to have proper profile of reference current waveform. Here, just to verify the capability of simplified SOSM controller (33) for controlling the arc current for CV processes, one simple

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Control voltage

Ia: 200A/D

Iref: 180A/D

Varc: 20V/D

Fig. 16 Performance of SOSM control is robust for making welding joint using FCAW method at V arc of 32 V and wire feeding speed 16.7 cm/s

Fig. 17 T-Joint on two 15 mm thick MS slabs using FCAW method using welding conditions of Fig. 16

waveform pattern was chosen (Fig. 11). In the arcing phase, the reference waveform consisted of one exponentially decaying term (I exp ) and background current (I bg ). The peak of I exp occurred when the droplet transfer was just completed. Here, the time constant of decay was kept unchanged for a particular wire diameter and the peak of I exp along with the back ground current I bg were generated by microcontroller by using (28). The expression of current reference I ref (t) during each arcing phase could be expressed as

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Control voltage Ia: 200A/D

Iref: 180A/D

Varc: 20V/D

Fig. 18 Performance of SOSM control is robust for making MIG welding joint using 98% Ar at V arc of 26 V and wire feeding speed 11.7 cm/s

6mm SS slab

15mm MS s lab Fig. 19 T-Joint using two dissimilar metals (using 15 mm MS slab and 6 mm SS slab) using welding conditions of Fig. 18

Ir e f (t) = Iexp (t) + Ibg , where Iexp (0) >> Ibg .

(37)

Figures 12 and 14 are waveforms of controller for creating two joints under two different welding conditions (Table 4). From controller point of view, the current or arc control was proper. The welding joint corresponding to the welding conditions of Fig. 12 is shown in Fig. 13. Perceptible spatters were present. This was due to

Industry Grade Robust Controller Design … Fig. 20 Performance of SOSM control is robust for making MIG welding joint using 100% Ar at V arc of 28 V and wire feeding speed 13.3 cm/s

415

Control voltage Ia: 200A/D

Iref: 180A/D

Varc: 20V/D

improper waveform feeding for welding. The soft landing of droplet on the weld pool did not happen because that time the background current was large. If proper current waveform was fed to the controller, the controller would generate quality welding joints which would be validated while making joints in spray mode. The control of spray mode of transfer could be closely regarded as directly controlled process where joint quality is directly attributed to parameters of arc, V f and R. Here, the arc length is always positive, naturally, the disturbance from weld pool is small. Control of this mode is relatively simple because both V arc and I a could be measured accurately and fast. Welding conditions for results shown in Figs. 15, 16, 18 and 20 are listed in Table 4. The waveforms in these figures clearly showed that current I a was robustly controlled using simplified SOSMC function. Two welding joints are shown in Figs. 17 and 19.

7 Conclusion In this article, the design of SOSMC-based industry grade product for robust control of complex constant voltage nonlinear arc welding process was addressed. Like industrial arc controllers, when documented welding procedure was followed, the multi-input, nonlinear, and coupled CV process was converted into two singleinput independently controlled dynamic activities—control of wire feeding rate and welding current. Though, it needed robust control performance, the control of wire feeding rate at pre-fixed value was simple and straight forward. For control of arc voltage the current was controlled where the reference current profile was changed dynamically, it became like waveform control for efficient execution of droplets in short-circuiting mode. For control of both dynamic activities, the suitability of a

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control function was judged after comparing the results of PI, SOSM and simplified SOSM input functions. It was concluded that simplified SOSM was a superior practicable robust control approach—not only for performance under wide range load, but also for its simplicity in realizing the industrial product development. Finally, using simplified SOSMC function one industrial grade arc welding controller was developed whose results were comparable with the quality of modern industrial arc welding controllers. With superior results in two drastically different dynamic applications it could be concluded that for wide range applications the simplified SOSM control function would be a preferred robust control idea.

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