Systems, Automation, and Control 9783110591729, 9783110590241

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Nabil Derbel, Faouzi Derbel, and Olfa Kanoun (Eds.) Systems, Automation, and Control

Advances in Systems, Signals and Devices

| Edited by Olfa Kanoun, University of Chemnitz, Germany

Volume 9

Systems, Automation, and Control |

Edited by Nabil Derbel, Faouzi Derbel, and Olfa Kanoun

Editors of this Volume Prof. Dr.-Eng. Nabil Derbel University of Sfax Sfax National Engineering School Control & Energy Management Laboratory 1173 BP, 3038 Sfax Tunisia [email protected] Prof. Dr.-Ing. Faouzi Derbel Leipzig University of Applied Sciences Chair of Smart Diagnostic and Online Monitoring Wächterstrasse 13 04107 Leipzig Germany [email protected]

Prof. Dr.-Ing. Olfa Kanoun Technische Universität Chemnitz Chair of Measurement and Sensor Technology Reichenhainer Strasse 70 09126 Chemnitz Germany [email protected]

ISBN 978-3-11-059024-1 e-ISBN (PDF) 978-3-11-059172-9 e-ISBN (EPUB) 978-3-11-059031-9 ISSN 2365-7493 Library of Congress Control Number: 2019931927 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface of the Editors The ninth volume of the Series “Advances in Systems, Signals and Devices” (ASSD), is dedicated to fields related to “Systems, Automation and Control” (SAC). The scope of this issue encompasses all aspects of the research, development and applications of the science and technology in these fields. Topics of this issue concerns: system design, systemidentification, biological and economical models & control, modern control theory, nonlinear observers, control and application of chaos, adaptive/non-adaptive backstepping control techniques, advances in linear control theory, system optimization, multivariable control, large scale and infinite dimension systems, nonlinear control, distributed control, predictive control, geometric control, adaptive control, optimal and stochastic control, robust control, neural control, fuzzy control, intelligent control systems, diagnostics, fault tolerant control, robotics and mechatronics, navigation, robotics and humanmachine interaction, hierarchical and man-machine systems, etc. Authors are encouraged to submit novel contributions which include results of research or experimental work discussing new developments in the field of systems, automation and control. The journal can be also addressed for editing special issues for novel developments in specific fields. Guest editors are encouraged to make proposals to the editor in chief of the corresponding main field. The aims of this volume, in its own way, to promote an international scientific progress in the fields of systems, automation and control. It provides at the same time an opportunity to be informed about interesting results that have been reported during the international SSD conferences. The Editors Nabil Derbel, Faouzi Derbel and Olfa Kanoun

De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 1–14. https://doi.org/10.1515/9783110591729-201

Advances in Systems, Signals and Devices Editors in Chief: Systems, Automation and Control Nabil Derbel, Faouzi Derbel and Olfa Kanoun

Power Systems and Smart Energies Faouzi Derbel, Nabil Derbel and Olfa Kanoun

Communication, Signal Processing and Information Technology Faouzi Derbel, Nabil Derbel and Olfa Kanoun

Sensors, Circuits & Instrumentation Systems Olfa Kanoun, Faouzi Derbel and Nabil Derbel

Prof. Dr.-Eng. Nabil Derbel ENIS, University of Sfax, Tunisia [email protected] Prof. Dr.-Ing. Faouzi Derbel Leipzig Univ. of Applied Sciences, Germany [email protected] Prof. Dr.-Ing. Olfa Kanoun Technische Universität Chemnitz, Germany [email protected]

Editorial Board Members: Systems, Automation and Control Dumitru Baleanu, Çankaya University, Ankara, Turkey Ridha Ben Abdennour, Engineering School of Gabès, Tunisia Naceur Benhadj, Braïek, ESSTT, Tunis, Tunisia Mohamed Benrejeb, Engineering School of Tunis, Tunisia Riccardo Caponetto, Universita’ degli Studi di Catania, Italy Yang Quan Chen, Utah State University, Logan, USA Mohamed Chtourou, Engineering School of Sfax, Tunisia Boutaïeb Dahhou, Univ. Paul Sabatier Toulouse, France Gérard Favier, Université de Nice, France Florin G. Filip, Romanian Academy Bucharest Romania Dorin Isoc, Tech. Univ. of Cluj Napoca, Romania Pierre Melchior, Université de Bordeaux, France Faïçal Mnif, Sultan qabous Univ. Muscat, Oman Ahmet B. Özgüler, Bilkent University, Bilkent, Turkey Manabu Sano, Hiroshima City Univ. Hiroshima, Japan Abdul-Wahid Saif, King Fahd University, Saudi Arabia José A. Tenreiro Machado, Engineering Institute of Porto, Portugal Alexander Pozniak, Instituto Politecniko, National Mexico Herbert Werner, Univ. of Technology, Hamburg, German Ronald R. Yager, Mach. Intelligence Inst. Iona College USA Blas M. Vinagre, Univ. of Extremadura, Badajos, Spain Lotfi Zadeh, Univ. of California, Berkeley, CA, USA

Power Systems and Smart Energies Sylvain Allano, Ecole Normale Sup. de Cachan, France Ibrahim Badran, Philadelphia Univ., Amman, Jordan Ronnie Belmans, University of Leuven, Belgium Frdéric Bouillault, University of Paris XI, France Pascal Brochet, Ecole Centrale de Lille, France Mohamed Elleuch, Tunis Engineering School, Tunisia Mohamed B. A. Kamoun, Sfax Engineering School, Tunisia Mohamed R. Mékidèche, University of Jijel, Algeria Bernard Multon, Ecole Normale Sup. Cachan, France Francesco Parasiliti, University of L’Aquila, Italy Manuel Pérez, Donsión, University of Vigo, Spain Michel Poloujadoff, University of Paris VI, France Francesco Profumo, Politecnico di Torino, Italy Alfred Rufer, Ecole Polytech. Lausanne, Switzerland Junji Tamura, Kitami Institute of Technology, Japan

Communication, Signal Processing and Information Technology Til Aach, Achen University, Germany Kasim Al-Aubidy, Philadelphia Univ., Amman, Jordan Adel Alimi, Engineering School of Sfax, Tunisia Najoua Benamara, Engineering School of Sousse, Tunisia Ridha Bouallegue, Engineering School of Sousse, Tunisia Dominique Dallet, ENSEIRB, Bordeaux, France Mohamed Deriche, King Fahd University, Saudi Arabia Khalifa Djemal, Université d’Evry, Val d’Essonne, France Daniela Dragomirescu, LAAS, CNRS, Toulouse, France Khalil Drira, LAAS, CNRS, Toulouse, France Noureddine Ellouze, Engineering School of Tunis, Tunisia Faouzi Ghorbel, ENSI, Tunis, Tunisia Karl Holger, University of Paderborn, Germany Berthold Lankl, Univ. Bundeswehr, München, Germany George Moschytz, ETH Zürich, Switzerland Radu Popescu-Zeletin, Fraunhofer Inst. Fokus, Berlin, Germany Basel Solimane, ENST, Bretagne, France Philippe Vanheeghe, Ecole Centrale de Lille France

Sensors, Circuits & Instrumentation Systems Ali Boukabache, Univ. Paul, Sabatier, Toulouse, France Georg Brasseur, Graz University of Technology, Austria Serge Demidenko, Monash University, Selangor, Malaysia Gerhard Fischerauer, Universität Bayreuth, Germany Patrick Garda, Univ. Pierre & Marie Curie, Paris, France P. M. B. Silva Girão, Inst. Superior Técnico, Lisboa, Portugal Voicu Groza, University of Ottawa, Ottawa, Canada Volker Hans, University of Essen, Germany Aimé Lay Ekuakille, Università degli Studi di Lecce, Italy Mourad Loulou, Engineering School of Sfax, Tunisia Mohamed Masmoudi, Engineering School of Sfax, Tunisia Subha Mukhopadhyay, Massey University Turitea, New Zealand Fernando Puente León, Technical Univ. of München, Germany Leonard Reindl, Inst. Mikrosystemtec., Freiburg, Germany Pavel Ripka, Tech. Univ. Praha, Czech Republic Abdulmotaleb El Saddik, SITE, Univ. Ottawa, Ontario, Canada Gordon Silverman, Manhattan College Riverdale, NY, USA Rached Tourki, Faculty of Sciences, Monastir, Tunisia Bernhard Zagar, Johannes Kepler Univ. of Linz, Austria

Advances in Systems, Signals and Devices Volume 1 N. Derbel (Ed.) Systems, Automation, and Control, 2016 ISBN 978-3-11-044376-9, e-ISBN 978-3-11-044843-6, e-ISBN (EPUB) 978-3-11-044627-2, Set-ISBN 978-3-11-044844-3 Volume 2 O. Kanoun, F. Derbel, N. Derbel (Eds.) Sensors, Circuits and Instrumentation Systems, 2017 ISBN 978-3-11-046819-9, e-ISBN 978-3-11-047044-4, e-ISBN (EPUB) 978-3-11-046849-6, Set-ISBN 978-3-11-047045-1 Volume 3 F. Derbel, N. Derbel, O. Kanoun (Eds.) Power Systems & Smart Energies, 2017 ISBN 978-3-11-044615-9, e-ISBN 978-3-11-044841-2, e-ISBN (EPUB) 978-3-11-044628-9, Set-ISBN 978-3-11-044842-9 Volume 4 F. Derbel, N. Derbel, O. Kanoun (Eds.) Communication, Signal Processing & Information Technology, 2017 ISBN 978-3-11-044616-6, e-ISBN 978-3-11-044839-9, e-ISBN (EPUB) 978-3-11-043618-1, Set-ISBN 978-3-11-044840-5 Volume 5 F. Derbel, N. Derbel, O. Kanoun (Eds.) Systems, Automation, and Control, 2017 ISBN 978-3-11-046821-2, e-ISBN 978-3-11-047046-8, e-ISBN (EPUB) 978-3-11-046850-2, Set-ISBN 978-3-11-047047-5 Volume 6 O. Kanoun, F. Derbel, N. Derbel (Eds.) Sensors, Circuits and Instrumentation Systems, 2018 ISBN 978-3-11-044619-7, e-ISBN 978-3-11-044837-5, e-ISBN (EPUB) 978-3-11-044624-1, Set-ISBN 978-3-11-044838-2 Volume 7 F. Derbel, N. Derbel, O. Kanoun (Eds.) Power Systems & Smart Energies, 2018 ISBN 978-3-11-046820-5, e-ISBN 978-3-11-047052-9, e-ISBN (EPUB) 978-3-11-044628-9, Set-ISBN 978-3-11-047053-6 Volume 8 F. Derbel, N. Derbel, O. Kanoun (Eds.) Communication, Signal Processing & Information Technology, 2018 ISBN 978-3-11-046822-9, e-ISBN 978-3-11-047038-3, e-ISBN (EPUB) 978-3-11-046841-0, Set-ISBN 978-3-11-047039-0

Contents Preface of the Editors | V Safa Ziadi, Mohamed Njah, and Mohamed Chtourou PSO-CF2 Multi-Objective Mobile Robot Path Planning with PSO-PID end Position Regulation | 1 Ibtissem Malouche, Amira Kheriji, and Faouzi Bouani Automatic Model Predictive Control code-generation for real-time implementation in a high-performance microcontroller | 25 Hadil Soltani, Saloua Bel Hadj Ali Naoui, Rafika El Harabi, Abdel Aitouche, and Mohamed Naceur Abdelkrim Fault Tolerant Control for Uncertain State Time-delay Systems | 41 Wafa Jamel, Atef Khedher, Nasreddine Bouguila, and Kamel Ben Othman Observers Design for Takagi-Sugeno Models | 61 Sondess Mejdi, Anis Messaoud, Mouhib Allaoui, and Ridha Ben Abdennour Detection and Isolation of Sensor Faults for Nonlinear Systems: Robustness Against Disturbances | 83 Amina Ben Hmed, Messaoud Amairi, and Mohamed Aoun Stabilizing Fractional Order Integrator Design for Integrating and Unstable First Order Plants | 103 Essia Saidi, Yosra Hammi, and Ali Douik Modeling and Explicit Model-Predictive Control of a Two-Tank System by PWA Approach | 123 Amira Abdelkader, Moez Boussada, Koffi Fiaty, Ahmed Said Nouri, and Hassan Hammouri Comparison of Nonlinear Observers for a Nonlinear System | 139 Asma Atitallah, Saida Bedoui, and Kamel Abderrahim A Two-Step Procedure and Performance Analysis of Identification Algorithm for Hammerstein Time Delay System | 159

XIV | Contents Manel Atitallah, Rafika El Harabi, and Mohamed Naceur Abdelkrim Unknown Input Hamiltonian Observers-Based Fault Detection and Estimation | 177 Rihab Abdelkrim, Mohamed Chaabane, and Ahmed El Hajjaji Relaxed Stability and Stabilization Conditions for Linear Parameter Varying Systems | 197 O. Hrizi, B. Boussaid, A. Zouinkhi, C. Aubrun, and M. N. Abdelkrim On Fault Tolerant Control of Wheeled Mobile Robot Based on Fast Adaptive Fault Estimation Algorithm | 211 Majda Ltaief, Samah Ben Atia, and Saida Bedoui A Multi-Smith Predictor for Time-Delay Systems’ Control | 229 Asma Achnib, Tudor Bogdan Airimitoaie, Sergey Abrashov, Patrick Lanusse, and Mohamed Aoun Performances and Robustness of GPC Controller in an Anticipative Context | 249 Ismail Er Rachid, Redouane Chaibi, and Abdelaziz Hmamed Stability of 2-D Continuous T-S Fuzzy Systems Based on KYP Lemma | 267 Rakia Abdeljawad, Nesrine Bahri, and Majda Ltaief Stability Analysis and Stabilization of Discrete Singularly Perturbed System with Time-Delay | 281 Nesrine Montacer, Samah Ben Atia, Khadija Dehri, and Ridha Ben Abdennour Sliding mode observer synthesis for multivariable systems: An LMI approach | 303

Safa Ziadi, Mohamed Njah, and Mohamed Chtourou 2

PSO-CF Multi-Objective Mobile Robot Path Planning with PSO-PID end Position Regulation Abstract: In this paper, we propose the Particle Swarm Optimization (PSO) based approach of mobile robot path planning. It is the PSO based Canonical Force Field approach (PSO-CF2 ). The PSO optimization technique is applied here in the design of the shortest trajectory to the goal with the best collision avoidance. The PSO searches for the best combination of the CF2 parameters that minimizes the path length and maximizes the distance between the robot and obstacles. A PID position regulator is applied at the end of navigation to get an arrest as close as possible to the goal with the desired orientation. PSO is applied also in the research of the best coefficients of the PID regulator. Simulation results show the feasibility of our mobile robot path planning approach in different environments. Everywhere, it dresses a short and secure path to the goal without any sudden deviation while avoiding obstacles. Simulation results show also the quality of the robot arrest in the desired destination with the correct orientation. Keywords: Mobile robot path planning, Canonical Force Field (CF2 ), Particle Swarm Optimization (PSO), PID end position and orientation regulation MSC 2010: 65C05, 62M20, 93E11, 62F15, 86A22

1 Introduction Mobile robot navigation is a research domain interested to control the robot during its navigation in different environments and under different conditions. This domain can be divided into two sub-domains: First, the localization which refers in robotics to the definition of the accurate position in the search space according to the environmental perceptions gathered by sensors. Second, the path planning that refers to the search of a suitable collision-free path for a mobile robot to move from a start location to a target location. Path Planning is considered as the computation of an optimal path relatively to some criteria such as cost navigation, but essentially safe distance, path length and Safa Ziadi, Mohamed Njah, Digital Research Center of Sfax, Sfax, Tunisia; and Control & Energy Management Laboratory, Sfax, Tunisia, e-mails: [email protected], [email protected] Mohamed Chtourou, National Engineering School (ENIS), Control & Energy Management Laboratory, University of Sfax, Sfax, Tunisia, e-mail: [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 1–24. https://doi.org/10.1515/9783110591729-001

2 | S. Ziadi et al. navigation time. It can be divided into two classes according to the atomicity and availability of knowledge about the environment: Path planning in static environments and path planning in dynamic environments. In a static environment, all the obstacles are static, whereas in a dynamic environment, obstacles can be both static and dynamic and may move at arbitrary directions with varying speeds. This paper builds upon the PSO-CF2 robot path planning approach proposed in our previous work [21]. This approach combines PSO as a parameter optimization tool and CF2 as a robot path planning approach based on the concept of the Force Field (F 2 ) [5]. In this paper, PSO-PID is added to ensure the robot attains its destination with the correct orientation. This paper is organized as follows. In section 2, we remind some mobile robot path planning approaches among the largest reported ones in the literature. In a following subparagraph of the same section, we recall some of the most important works cited in the literature about the application of PSO in the optimization of mobile robot paths. In Section 3, we introduce the Canonical Force Field (CF2 ) the path planning method that we adopt in this work. We recall the principle of the PSO optimization technique and we describe the PSO-CF2 path planning method in the section 4. It is the classical CF2 method preceded with a PSO search of the best combination of CF2 parameters. In section 5, we introduce the PSO-PID position and orientation regulator. The different simulations of the presented approaches are discussed in section 6. Conclusion and future work are given in the last section.

2 State of the Art Motion planning of an intelligent mobile robot is the most interesting domain in the robotic field. Motion planning of a mobile robot, also known as path planning, refers to the search of a suitable collision-free path for a mobile robot to move from a start location to a target location. Since the mid-1970’s, and particularly after the important contribution of Wesley and Lozano-Pérez (1979), this domain pulled much attention [16]. It can be described as the definition of a collision avoidance path that manages the robot towards its destination. Many approaches have been proposed in the literature for the planning of robot trajectories. We select, for the first subparagraph, some approaches that we find have given advance to the research in the robot path planning domain. The PSO has been applied in the literature to get fast and efficient procedures of navigation. Some interesting applications will constitute the second subparagraph of this state of the art section.

PSO-CF2 Path Planning with PSO-PID Regulation

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2.1 Path Planning Approaches in the Literature In 1985, O. Khitib developed the Potential Field Method (PFM) [1]. In this method, a robot is treated as under the influence of a Potential Field (PF). This robot is attracted by the goal ‘qgoal ’ and is repulsed by the obstacles. In this approach, dynamic and kinematic characteristics aren’t considered in the motion planning operation. In this case, a local minimum problem exists where the robot gets stuck due to counteracting forces. In 1991, Borenstein and Koren developed the Vector Field Histogram (VFH) method [2]. This method constructs a local map of the environment around the robot based on the concept of certainty grid. This grid can be updated in real time using the information of the sensor readings which it translated into a polar histogram. If the polar obstacle density goes down a certain threshold value, the direction and the speed of the robot change. The drawback of this approach is that the robot can be trapped in local minima where there is no attractive force nor repulsive force. So, reaching the goal can’t be guarantied. In 1996, Reid Simmons developed the Curvature Velocity Method (CVM) [3]. This method is considered as a local obstacle avoidance method. The kinematic and dynamic constraints (acceleration and velocity bounds) of the mobile robot are accounted. This method is used to find the next suitable speed (linear speed v and rotational speed ω) for the mobile robot command. CVM is efficient for real-time obstacle avoidance. The disadvantages of this approach are the assumption of all obstacles as circular objects and the robot can get stuck in local minima such as U-shaped obstacles. In 1997, Dynamic Window Approach (DWA) is developed by Fox et al. [4]. This approach is certainly one of the most used approaches in the domain of mobile robot path planning. In DWA, a discrete velocity space is built around the robot’s velocity. A velocity is considered a solution if it adapts for the systems constraints (maximum speed and maximum acceleration). The motion direction is chosen by optimizing an objective function containing three terms: the fastest velocity, the greatest safety distance and the shortest path to the target point. This approach is effective in steering the robot to its goal and reacting adequately. The drawback is that it couldn’t be used in larger real environments. In 2005, D. Wang et al. suggested the Force Field (F 2 ) method [5] for multi-robot path planning. This method is defined as a virtual field of repulsive forces in the vicinity of a robot when it travels in a working space [6, 7]. This virtual repulsive force increases with the decrease of the distance to the robot. The basic concept of the F 2 method is to generate a force field for every robot that continuously changes according to the status: dimension, traveling velocity, priority, location and environmental factors, etc. A robot only reacts to obstacles or other robots that are in the coverage of its own force field and does not need to search the whole work space as many other

4 | S. Ziadi et al. methods require, which significantly increases the efficiency of motion planning and coordination. In the F 2 method, robot’s physical characteristics are used in the construction of its force field. Its dynamic and kinematic characteristics, such as linear velocity and angular velocity, are taken into consideration when determining a robot’s motion. These make the F 2 method suitable for real applications. For all these advantages, we adopt the Force Field (F 2 ) method for determining the robot trajectories.

2.2 Path Planning with PSO in the Literature In the last decade, PSO has been widely used, as a technique of optimization, in the resolution of the multi-objective mobile robot path planning problem. In [11] and [12], Particle Swarm Optimization method (PSO) searches in the solution space to find the appropriate parameter values, that satisfy the two objectives, the minimization of the robot path length and the maximization of the safe distance. The fitness function to minimize is a weighted sum of the path length and the inverse safe distance. These algorithms are very slow and the selected parameters depend strongly on the weights of the objectives in the fitness function. In [13], the modified Particle Swarm Optimization (MPSO) is proposed by N. Shiltagh et al. to solve the problem of mobile robot navigation in a working environment with obstacles. MPSO is capable of effectively guiding a robot moving from a start position to the goal in a complex environment and finding optimum/shortest path without colliding any obstacles in the environment. The function is based on two parameters: the first is the path length and the second is the travel time between two particles Pi and Pi+1 . Then, the fitness function is the inverse of the weighted sum of these two parameters. The drawback of this algorithm is that it does not give the safety distance due importance. So, the risk of collision is high. In order to improve the efficiency and the quality of the multi-objective path planning, a Chaos Immune Particle Swarm Optimization (CIPSO) algorithm is proposed in [14] by W. Hao et al. CIPSO combines chaos and PSO with immune network theory so as to enhance the searching velocity of the mobile robot and insure the safety of the space exploration. Chaos theory is used to generate the population for the first iteration. Immune network theory is included in calculating the degree of stimulated antibodies, network suppression and in updating network. Firstly, the PSO algorithm is used to optimize the N antibodies, that would be putted into the immune network. PSO is used to optimize three functions. The first is the length of the path of the robot’s navigation, the second is the smoothness of the path and the last is the safety of the path. For the environment model, they used the geometric map modeling method for its great favor that is the accurate and the concise expression. In the simulation, the authors used three algorithms, the CIPSO, the GA and the Immune-GA. The first test

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used 10 barriers, the second test used 20 barriers and in the last test they used 30 barriers. The simulation results demonstrate that the ability of CIPSO to solve mobile robot problem in complex environment is better than GA and Immune-GA. The disadvantage of CIPSO is the difficulty of its implementation. In [15], D. Wang et al. proposed the RPPSO (Ranked Pareto Particle Swarm Optimization) approach as an extension of the basic idea of PSO. This method is capable of solving multi-objective optimization problems efficiently. The RPPSO is successfully utilized in the multi-objective parameter optimization problem for the Variable Speed Force Field (VSF2 ) method of multi-robot motion planning and coordination. RPPSO is developed for a collision avoidance between mobile robots and obstacles. The difference between PSO and RPPSO is in the definition of the Global Best (Gbest ) variable and the definition of the Particle Best (Pbest ) variable. Gbest in PSO stores the current best location found by all particles but in RPPSO, it is a PN-dimensional vector which stores PN best locations found by all particles. Also, Pbest in PSO is the best position of the particle but in RPPSO it is a PN-dimensional vector of the best histories of the particle. F 2 is used to guide the robot from the start point until the destination and RPPSO is used to minimize the path length and to keep away the robot from obstacles and other robots as far as possible. The results of F 2 and RPPSO demonstrate the satisfactory performance in motion planning of multiple robots taking into account not only collision avoidance but also minimum travel lengths. The criticism that we can address to this algorithm is the difficulty of its implementation (like CIPSO).

2.3 Path Planning with PID in the Literature The movement of a wheeled mobile robot (WMR) is provided by motors and the control and the predict of the motors speed is hard. The presence of PID controller in [17] is to achieve the objective of the motors speed control. A differential drive wheeled mobile robot (DWMR) platform is used with a decision making algorithm (DMA) to test the capability of the controllers in different situations. The motors on the WMR can run with the desired speed thanks to these two PID controllers and the WMR can also move in a nearly perfect straight line without using any external sensors. A PID Controller, designed for the Robot based Agricultural System, was presented in [18]. The PID control system is used for the automatic driving and the speed control of the Robotic Vehicle. The PID control system continuously checked and corrected the direction of the travel to keep the vehicle on a desired track. PID control equations for Discrete Time Domain were derived and coded in Verilog RTL and validated by using MATLAB. Functional tests were carried out using ModelSim. The direction accuracy achieved was better than 0.01 %. K. Agarwal et al. [19] proposed an approach for mobile robots navigation. In this work, the PID controller was used to control the navigation of robots called Khepera 3 and iRobot Create. The most suitable set of values of PID parameters for safe and

6 | S. Ziadi et al. effective navigation of the robots was defined. In the navigation of Khepera, in the Go To Angle task was used a proportional control and in the Go to Goal task was applied an integral and a derivative control. In the case of iRobot Create, the robot traversed a rectangular path in anticlockwise direction using an adequate PID control. Matlab is used for the implementation of a PID controlled navigation system for mobile robots. The final results show that the proposed control system for both the robots, successfully provides a simple and effective method of mobile robot navigation.

3 Canonical Force Field CF2 Path Planning Approach 3.1 Presentation of the CF2 Approach D. Wang et al [3] developed the Force Field (F 2 ) method in 2005 for multi-robot path planning. In this F 2 method, the coverage of a robot’s force field is determined by parameters including the robot’s size, the travelling speed and the priority with respect to other robots. A robot only reacts to obstacles that are in the coverage of its own force field and does not need to search the whole work space as many other methods require, which significantly increases the efficiency of the motion planning and coordination. In the F 2 method, robot’s physical characteristics, such as size and geometry, are used in the construction of its force field. Its dynamic and kinematic characteristics, such as linear velocity and angular velocity, are taken into consideration when determining a robot’s motion. These make the F 2 method suitable for real applications. For all these advantages, we adopt the Force Field (F 2 ) method as a path planning strategy in our work. There exist four designs based on the concept of the Force Field (F 2 ): the Canonical Force Field (CF2 ) method [7], the Variable Speed Force Field (VSF2 ) method [8], the Subgoal-Guided Force Field (SGF2 ) method [9] and the Dynamic Variable Speed Force Field (DVSF2 ) method [10]. For our work, we chose the Canonical Force Field (CF2 ) method for its easiness in implementation and its effectiveness in finding a path to the goal with an acceptable collision avoidance capacity. A force field F 2 is defined as a virtual field of repulsive force in the vicinity of a robot when it travels in a working space. The magnitude and the orientation of a force field vary with the robot’s status. This virtual repulsive force increases with the decrease of the distance to the robot. The concept of the virtual repulsive force is based on some parameters: Rr is the radius of a robot, Vr is the absolute value of a robot’s velocity, Vmax is the maximum absolute value of a robot’s velocity and θr is the robot’s orientation in global coordinates. For any point (x,y) in the 2-D space (Figure 1), we need to define parameters θ, Er , Dmax and Dmin as follows: θ = θ0 − θr

(1)

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Figure 1: Illustration of robot parameters.

Er =

Dmax

Vr

Vmax ⋅ c k ⋅ Er ⋅ Rr = ⋅T 1 − Er cos(θ) p

Dmin = ρ0 ⋅ Dmax

(2) (3) (4)

Assume θr is the robot’s orientation in global coordinates, θ0 is the orientation of this point in global coordinates, thus θ denotes the relative angle of this point to the robot’s orientation. Er is a positive decimal fraction with 0 ≤ Er < 1. c is a positive number that assigns the influence of the environment on the force field with c > 1. k is a positive multiplier which delimits the coverage area of the force field. Dmax is the maximum active distance of a robot’s force field and Dmin is the distance at which this robot has the maximum repulsive force. Dmax shows how far this robot can affect others in its vicinity. Dmin provides a safe distance for the robot to prevent other objects from moving into this area. ρ0 is a positive fractional number with 0 < ρ0 < 1 that heavily influences how close the robot can be separated from obstacles. Tp represents the priority of a task undertaken by the robot with Tp ≤ 1. The variation of Dmax and Dmin depend on the variation of parameters k, c, P, Q and ρ0 . In our work, the optimization with PSO searches for the best set of these parameters that furnishes the shortest and safest path. Figure 1 shows the different parameters used in the calculation of the repulsive force. The repulsive force generated by a robot is defined by: 0 { { D −D |Frep−rob | = {P ⋅ D max−D max min { {Fmax

if D > Dmax if Dmin ≤ D < Dmax if D < Dmin

(5)

Where D is the shortest distance from point (x, y) to the perimeter of the robot. P is a positive constant scalar which limits the magnitude of the repulsive force. When D

8 | S. Ziadi et al. changes from Dmin to Dmax , the magnitude of the repulsive force changes from P to 0 gradually. Fmax is the maximum repulsive force which causes the maximum deceleration of the robot. P and Fmax should be selected based on the robot’s characteristics, with Fmax ≫ P.

3.2 The Attractive Force The attractive force Fatt is a virtual force applied by the goal on the center of the robot. It attracts the robot to the goal. It’s module is constant in the F 2 approach. The module and the orientation are given by (6) and (7) respectively. |Fatt | = Q θFatt = arctan(

(6)

Yt − Y ) Xt − X

(7)

3.3 The Reaction Force The reaction force Frep is a virtual force applied on the robot by the obstacles or other robots. It repulses the robot away from the obstacles or robots. Its module equals the module of the repulsive force as illustrated in equation (8): |Frep | = |Frep−rob |D=d

(8)

where d is the distance between the perimeter of the robot and the obstacle. Furthermore, the reaction force orientation is given by (9) θFrep = arctan(

Y − YA ) X − XA

(9)

where A(XA , YA ) is the obstacle point at which the largest repulsive force is attained. The Canonical Force Field method (CF2 ) is derived from the Force Field (F 2 ) approach. The particularity of the CF2 method is that a robot is assumed to travel in the working space with a constant speed. The direction of its motion is that of the sum of the attractive force applied by the goal and the repulsive forces generated by the obstacles. Equation (10) gives the formulation of this resultant force applied on the robot. n

m

i=1

j=1

FTotal = Fatt + ∑ Firep−rob + ∑ Fjrep−obs

(10)

Consider a robot located at (Xr , Yr ) with angle θr to the X-axis of the global coordinate at time t = s, its current moving direction θs is θr .

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| 9

In the CF2 method, this robot’s next moving direction θs+1 is the direction of FTotal . θs+1 = ∠FTotal

θ − θs ωs = s+1 Δt

(11) (12)

Where ω is the angular velocity.

4 PSO-CF2 Method 4.1 The Particle Swarm Optimization The Particle Swarm Optimization (PSO) is an evolutionary algorithm able to solve difficult multidimensional optimization problems in various fields. PSO is a new heuristic method of optimization, invented by Russell Eberhart (electrical engineer) and James Kennedy (social psychologist) in 1995 [5]. This algorithm is inspired from the real life. It is based on a model developed by Craig Reynolds in the late 1980s, to simulate the movement of a group of birds. The algorithm consists on a swarm of particles flying through the search space. Each individual i in the swarm is characterised by parameters of position Xi and velocity Vi , where Xi ∈ Rn , Vi ∈ Rn . n is the dimension of the search space. The position of every particle represents a potential solution of the problem of optimization. Vi (t + 1) = Vi (t) + c1 ∗ r1 (Xpbest;i − Xi ) + c2 ∗ r2 (Xgbest − Xi ) Xi (t + 1) = Xi (t) + Vi (t + 1)

(13) (14)

where c1 and c2 are acceleration constants, r1 and r2 are random factors in the [0, 1] interval, Xpbest is the personal best position and Xgbest is the global best position.

4.2 The Fitness Function Our optimization goals are minimizing the travel length (f1 (xi )) and keeping away the robot from obstacles (f2 (xi )). Consequently, the PSO-CF2 algorithm is bi-objective: the first objective is the minimization of the length of the path followed by the robot to the goal and the second is the maximization of the distance between the nearest obstacle and the robot Dnearest−robot . The fitness function is a weighted sum of the first term and the inverse of the second term. The CF2 parameters to optimize and their domain of variation are as follows: P ∈ [5, 20], c ∈ [1.5, 3], k ∈ [2, 10], Q ∈ [5, 20] and ρ0 ∈ [0.2, 1]. Consider xi the ith particle of the swarm. xi = [k, P, Q, c, ρ0 ]

i = 1, 2, . . . ., n

(15)

10 | S. Ziadi et al. The mathematical formulation of f1 and f2 and the fitness function are as follows: f1 (xi,t ) = FF(xi )

(16)

where FF(xi ) is the length of the path obtained by applying the CF2 approach with the xi parameters. n

f2 (xi,t ) = ∑ j=1

1 √(Xxi,t − Xobs,j

)2

+ (Yxi,t − Yobs,j )2

(17)

where n is the number of obstacles and (Xxi,t , Yxi,t ) is the robot position. fitness = w1 ⋅ f1 (xi ) + w2 ⋅ f2 (xi )

(18)

where w1 + w2 = 1. When the robot travels to the goal, the Canonical Force Field (CF2 ) is called to drive the robot through a suitable collision-free path. When the robot arrives at its destination, FF(xi ) returns the real resultant path length. If a robot moves close to obstacles, f2 (xi ) gives a high value. On the contrary, when it travels far from the robot obstacles, f2 (xi ) will result in a lower value.

4.3 The Steps of the PSO-CF2 Method The framework of the PSO-CF2 algorithm is given as follows: Step 1: Initialization: Initialize parameters and population with random positions and velocities. Step 2: For each particle i, evaluate the fitness value. Step 3: Find the pbest : If the Fitness value of a particle is better than its best fitness value (pbest ) in history, then set the current fitness value as its new pbest value. Step 4: Find the gbest : If any pbest is updated and becomes better than the current gbest , then affect the current value to gbest . Step 5: Update velocity and position: Update velocity and move to next position according to equations (19, 20): Vi (t + 1) = Vi (t) + c1 ⋅ r1 (Xpbest;i − Xi ) + c2 ⋅ r2 (Xgbest − Xi ) Xi (t + 1) = Xi (t) + Vi (t + 1)

(19) (20)

Step 6: If the number of particles is met, then go to step 7; otherwise go back to step 2. Step 7: Stopping Criterion: If the predefined number of iterations is met, then go to step 8; otherwise go back to step 2. Step 8: Effect the gbest parameters to the robot and start navigation.

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| 11

5 Regulation of the Stopping Position In this section, we are interested to solve the stopping problem. The robot stopping problem can be divided into two control problems, the position control problem and the orientation control problem. The position control facilitates the robot to achieve the target position and the orientation control facilitates the robot to achieve the desired orientation.

5.1 PID Position and Orientation Regulation In our work, two PID controllers are proposed for the regulation of the stopping position [20]. The first one is the PID position controller. This controller is used to control the linear velocity. The second one is the PID orientation controller. This controller is used to control the angular velocity of the mobile robot. Figure 2 illustrates the block diagram of the stopping control. This control system is assumed to be equipped with a global positioning system that measures the Cartesian coordinates and the orientating angle.

Figure 2: Robot Control System.

Figure 3 presents the stopping problem, where Δl is the distance between the robot and the desired position (Xd , Yd ) and θd is the desired orientation in the Cartesian space. The robot will not go directly to the final position with a fixed orientation. First, we define the position of the desired point linked to point C by the following equations (21, 22): Δl = √(Xd − X)2 + (Yd − Y)2 = √ΔX 2 + ΔY 2 α = arctan(

ΔY ) ΔX

(21) (22)

The proposed control strategy is based on a move to a false target. This target is called R. The point R is relocated at the same distance Δl related to the point C and

12 | S. Ziadi et al.

Figure 3: Robot Position and Orientation scheme.

is clockwise rotated by the angle θd − α related to the final position. The distance between the robot and the point R is measured in the θ alignment: Δs = Δl cos(2α − θd − θ)

(23)

5.2 Robot Position Control The robot position control problem is resolved by completing the condition when Δs → 0 then Δl → 0 and the robot attains the desired position. The position error is calculated as follows (24): es = Δs

(24)

A PID controller (25) is used to calculate us the linear velocity: t

de (t) us (t) = Kps ⋅ es (t) + Kds ⋅ s + Kis . ∫ es (τ)dτ dt

(25)

0

5.3 Robot Orientation Control The orientation error is defined by the following equation: eθ = 2α − θd − θr

(26)

The robot orientation controller has to assure the convergence of the error eθ to zero. A PID controller is used to calculate uθ the angular velocity: t

de (t) uθ (t) = Kpθ ⋅ eθ (t) + Kdθ ⋅ θ + Kiθ ⋅ ∫ eθ (τ)dτ dt 0

(27)

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5.4 PSO Optimization of the PID Coefficients The PSO algorithm is used to define the optimal parameters for the two PID controllers, the position and the orientation controllers. The fitness function to minimize by the PSO algorithm is the Mean Square Error (MSE) function (28). MSE =

1 n 1 n ∑(eθ (k))2 + ∑(es (k))2 n k n k

(28)

Where n is the number of samples, k is a sample time, eθ (k) is the orientation error and es (k) is the position error. The PSO algorithm is used to find the optimal parameters for the two PID controllers, the position controller and the orientation controller [Kps , Kis , Kds , Kpθ , Kiθ , Kdθ ]. Figure 4 shows the block diagram of the PSO-PID controller.

Figure 4: PSO-PID controller.

6 Simulations For all simulations in this paper, we used Matlab commands and we assumed that all robots are identical and that they are characterized by the parameters given in Table 1. In the first subsection of this paragraph, the simulation results presented show the importance of selecting optimal parameters on the quality of the generated path. The three following subsections present simulation results of the application of PSO-CF2 in different environments, respectively, with one static obstacle, with two static obstacles and with multi-static-obstacles. In the last subsection (5), we use dynamic obstacles.

14 | S. Ziadi et al. Table 1: Parameters of the mobile robot and PSO. Parameter

value

Unit

0.3 0.2 1 0.3 10 30 5 2 2 0.3 0.7 50 100 6 pi/2

m/s m/s – m – – – – – – – – – – rad

vmax vr Tp Rr Size of the swarm (PSO-CF2 ) Maximum iterations number (PSO-CF2 ) Dimension (PSO-CF2 ) c1 c2 w1 w2 Size of the swarm (PSO-PID) Maximum iterations number (PSO-PID) Dimension (PSO-PID) θd

6.1 A Comparative Study 6.1.1 Between the Results of the Application of CF2 and those of PSO-CF2 In this subsection, we use a simple case to prove the importance of selecting adequate parameters for the CF2 path planning approach. The robot is supposed to travel between (1.5, −1) and (9, 8.5) in an environment containing only one circular obstacle. The results of the selected parameters are given in the Table 2. Table 2: Influence of CF2 parameters on the quality of the generated path. Success 1 2 3 3 4 5 6

yes no yes yes yes yes yes

k

P

Q

c

ρ0

length (m)

Safe Distance (m)

6.741 4.55 4.88 8.547 5.5 4.89 9.89

17.716 5.55 15.05 10.05 17 10 15

5.05 17.318 10 17.55 7.55 5.05 15.05

2.521 1.5 2.5 1.788 2 3 1.55

0.674 0.2 0.55 0.455 0.7 0.258 0.5

15 – 13,8 13,40 14,6 13,80 16,4

116.9 – 106,497 116,864 117,15 102,95 162,872

Table2 shows clearly that the path generated by the CF2 algorithm depends strongly on the selected set of parameters. Also, we remark that, in the 2nd group of parameters, the robot fails to attain its destination and to avoid the collision. Then, in this case, the selection of parameters is unsuccessful. Consequently, there exists a set of parameters

PSO-CF2 Path Planning with PSO-PID Regulation

| 15

that furnishes a path with optimal performances and this explains our PSO-CF2 path planning approach.

6.1.2 Precision of the Stop Position with PSO-PID and without PSO-PID CF2 isn’t suitable to stop the mobile robot in the exact destination. In order to stop the robot, in the first experience, we use the underlying model that lies within some acceptable interval with ε is around 0. xt − ε ≤ x ≤ xt + ε yt − ε ≤ y ≤ yt + ε

(29) (30)

The CF2 parameters used in this example are k = 10, P = 15, Q = 17, c = 2.5 and ρ0 = 0.3. The path followed by the robot in these conditions is given in the Figure 5. It’s a short path that avoids successfully the obstacle. However the stop in the accurate position wasn’t clear. For this we applied the PSO-PID position regulator to ensure the arrival of the robot to its destination. The Figure 6 proves clearly the efficiency of this regulator: The robot arrives to its precise destination.

Figure 5: Navigation Without PSO-PID.

16 | S. Ziadi et al.

Figure 6: Navigation With PSO-PID.

6.2 Environment with One Static Obstacle In this simulation, a robot is supposed to travel between (1.5, −1) and (9, 8.5) in an environment containing only one obstacle.

Figure 7: Simulation 1: The resultant path in a one obstacle environment.

The algorithm converges after 30 iterations. Figure 7 shows the trajectory of the robot and proves the efficiency of the PSO-CF2 in avoiding the obstacle and reaching the destination at the precise position and the precise orientation thanks to the PSO-PID end position regulation.

PSO-CF2 Path Planning with PSO-PID Regulation

| 17

Figure 8: Simulation 1: The robot orientation in a one obstacle environment.

Figure 9: Evolution of the PSO parameters with respect to iterations.

Figure 8 shows the orientation of the robot throughout the navigation and proves the efficiency of the PSO-PID end orientation to ensure the arrival of the robot with its precise orientation θd . Figure 9 presents the evolution of the values of parameters k, P, Q, c, ρ0 of CF2 . These parameters have been introduced in equations (2, 3, 4, 5). Figure 10 and Figure 11 present the evolution of the path length and the reverse of the safe distance respectively. With a good selection of the weighting coefficients, we give more importance to the maximization of the safe distance with respect to the minimization of the path length.

18 | S. Ziadi et al.

Figure 10: Evolution of the path length with respect to iterations.

Figure 11: Evolution of the reverse safe distance with respect to iterations.

In this case, after 21 iterations, PSO-CF2 converges to the optimal solution. Table 3 presents the optimal parameters of the PSO-CF2 path planning algorithm, the optimal parameters of the PSO-PID end position controller and the PSO-PID end orientation controller: Table 3: PSO-CF2 and PSO-PID parameters. PSO-CF2 result k P Q c ρ0 length (m) safe distance (m)

10 13.44 14.86 1.5 0.3224 13,931 148,85

PSO-PID result Kps Kis Kds Kpθ Kiθ Kdθ MSE

0,215 0 0,613 1 0,618 0 0,175

PSO-CF2 Path Planning with PSO-PID Regulation

| 19

6.3 Environment with Two Static Obstacles In Figure 12, two different simulations are illustrated. These simulations are undertaken in environments containing two obstacles. Figure 12 shows clearly that the robot navigation is achieved with the shortest path and with a good safety distance. In these two cases, the robot starts from its initial position (1.5, −1), travels between two circles in the case (a) and between two walls in the case (b) until it reaches the arrival point (5.8, 7.5). The results prove the effectiveness of PSO-CF2 to find the optimal path whatever the environment and the effectiveness of the PSO-PID to stop in the specified destination.

Figure 12: Simulation 2: The resultant path in a two-obstacle environment.

Table 4 presents the optimal parameters of PSO-CF2 and PSO-PID. Table 4: PSO-CF2 and PSO-PID parameters.

k P Q c ρ0 length (m) safe distance (m)

PSO-CF2 case (a)

PSO-CF2 case (b)

5.225 15.61 19.59 2.178 0.7042 98,67 220,487

6.97 18.98 14.66 1.5 0.2 113,95 202,658

Kps Kis Kds Kpθ Kiθ Kdθ MSE

PID result case (a)

PID result case (b)

0,499 0,648 0,306 0,852 0,2217 0,507 2, 011 ∗ 10−5

0,405 0,264 0,563 1 0,573 0 0,018

20 | S. Ziadi et al.

6.4 Environment with Multi-Static-Obstacle In these simulations, a robot moves from the initial position (10, 1) to the specified goal (2, 6.7). The robot travels between three static obstacles. The resultant path is shown in the Figure 13.

Figure 13: Simulation 3: The resultant path in a multi-static-obstacle environment.

Table 5 contains the parameters of the calculated PSO-CF2 parameters and the PSOPID coefficients (the position control parameters and the orientation control parameters).

Table 5: PSO-Ct!F2 and PSO-PID parameters. PSO-CF2 k P Q c ρ0 length (m) safe distance (m)

5,568 16,125 20 3 0,652 11.4 285,332

PID result Kps Kis Kds Kpθ Kiθ Kdθ MSE

0.198 0.541 0.321 1 0 0.0876 0,000398

PSO-CF2 Path Planning with PSO-PID Regulation

| 21

6.5 Environment with Dynamic Obstacles Two simulations have been realized in environments with dynamic obstacles. The first simulation is with one dynamic obstacle and the second is with two dynamic obstacles. The result of these simulations are presented in the Figure 14. The obstacles move in the time interval [s, s + 1]. Their location and orientation at time s + 1 is expressed as a function of the current location and orientation (31): xs + vs ⋅ cos(θs ) ⋅ Δt xs+1 ] [ [ ] [ys+1 ] = [ys + vs ⋅ sin(θs ) ⋅ Δt ] [θs+1 ] [θs + ωs ⋅ Δt ]

(31)

In these two cases, the robot starts from its initial position (1.5, 0), travels in the environment until it reaches the arrival point (3.2, 10). In the first case (Figure 14 a), the environment contains only one dynamic obstacle (A blue moving robot). In the second case (Figure 14 b), the environment contains two dynamic obstacles (two blue moving robots). The results prove the effectiveness of the PSO-CF2 and the PSO-PID to guide the robot in dynamic environments until the precise destination.

Figure 14: Simulation 4: The resultant path in an environment with dynamic obstacles: (a) one dynamic obstacle, (b) two dynamic obstacles.

Table 6 contains the parameters of the calculated PSO-CF2 parameters and the PSOPID coefficients (the position control parameters and the orientation control parameters).

22 | S. Ziadi et al. Table 6: PSO-CF2 and PID results.

k P Q c ρ0 length (m) safe distance (m)

PSO-CF2 case (a)

PSO-CF2 case (b)

10 20 20 1,5 0.982 13,31 174,8472

10 20 20 3 0.82 13,024 352,018

Kps Kis Kds Kpθ Kiθ Kdθ MSE

PID result case (a)

PID result case (b)

0,228 0,681 0,2561 1 0 0,140 2.371 ∗ 10−4

0,361 0,305 0,161 1 0 0,472 5,873 ∗ 10−5

7 Conclusion In this paper, we presented an evolutionary approach of mobile robot path planning based on the PSO optimization technique and the Canonical Force Field (CF2 ) path planning approach. In this approach, PSO is used to optimize the parameters of the CF2 approach. It’s a bi-objective optimization. The first objective is to find the shortest path to the destination and the second is to maximize the safe distance. To ensure the arrival of the robot to its destination, we applied a PID end position regulation. The parameters of the PID controller are tuned using PSO. We used our PSO-CF2 approach and PSO-PID regulator in the definition of a robot trajectory to its precise destination in different known environments, with static obstacles and dynamic obstacles. Simulation results shows the quality of the selected Force field parameters to find the shortest and the safest trajectory to the goal and to assure the arrival of the robot to its correct destination with the desired orientation θd . In a future work, we will use PSO in the optimization of the parameters of the F 2 approach, safety distance and navigation time with varying robot speed.

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D. Wang, N. M. Kwok, D. K. Liu, H. Lau, G. Dissanayake, PSO-Tuned F 2 Method for Multi-Robot Navigation, International Conference on Intelligent Robots and Systems, pp. 3765–3770, IEEE, 2007. D. K. Liu, D. Wang, G. Dissanayake, A Force Field, Method Based Multi-Robot Collaboration, Robotics, Automation and Mechatronics, IEEE Conference, pp. 1–6, 2006. D. Wang, D. Liu, G.Dissanayake, A Variable Speed Force Field Method for Multi-Robot Collaboration, International Conference on Intelligent Robots and Systems, pp. 1–6, IEEE, 2006. D. Wang, D. K. Liu, N. M. Kwok, K. J. Waldron, A Subgoal-Guided Force Field Method for Robot Navigation, Mechtronic and Embedded Systems and Applications, pp. 488–493, IEEE, 2008. J. V.Miro, T. Taha, D. Wang, G. Dissanayake, D. Liu, An efficient strategy for robot navigation in cluttered environments in the presence of dynamic obstacles, The Eighth International Conference on Intelligent Technologies, 2007. S. Ahmadzadeh, M.Ghanavati, Navigation of mobile robot using the PSO particle swarm optimization, Journal of Academic and Applied Studies, Vol. 2, pp. 32–38, 2012. B. Deepak, D. R. Parhi, PSO based path planner of an autonomous mobile robot, Central European Journal of Computer Science, Vol. 2, pp 152–168, 2012. N. Adnan Shiltagh, L. Dalawr Jalal, Optimal Path Planning For Intelligent Mobile Robot Navigation Using Modified Particle Swarm Optimization, International Journal of Engineering and Advanced Technology, Vol. 2, pp 2249–8958, 2013. W. Hao, S. Qin, Multi-objective Path Planning for Space Exploration Robot Based on Chaos Immune Particle Swarm Optimization Algorithm, Lecture Notes in Computer Science, Vol. 7003, pp. 42–52, 2011. D. Wang, N. M. Kwok, D. K. Liu, Q. P. Ha, Ranked Pareto Particle Swarm Optimization for Mobile Robot Motion Planning, Studies in Computational Intelligence, Vol. 177, pp. 97–118, 2009. M. Shahab Alam and M.Usman Rafique, Mobile Robot Path Planning in Environments Cluttered with Non-convex Obstacles Using Particle Swarm Optimization, International Conference on Control, Automation and Robotics(ICCAR), Singapore, May (2015). Y. C. Koo and E. A. Bakar, Motor Speed Controller For Differential Wheeled Mobile Robot, ARPN Journal of Engineering and Applied Sciences, Vol. 10, No. 22, pp. 10698–10702, 2015. C. S. Mala and S. Ramachandran, Design of PID Controller for Direction Control of Robotic Vehicle, Global Journal of Researches in Engineering: F Electrical and Electronics Engineering, Vol. 14, Issue 3, 2014. K. Agarwal, S. Mahtab, S. Bandyopadhyay and S. Das Gupta, A Proportional-Integral-Derivative Control Scheme of Mobile Robotic platforms using MATLAB, IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE), Vol. 7, Issue 6, pp. 32–39, 2013. A.Bara and S. Dale, Dynamic modeling and stabilization of wheeled mobile robot, WSEAS International Conference. Proceedings. Mathematics and Computers in Science and Engineering, 2009. Safa Ziadi, Mohamed Njah, and Mohamed Chtourou, PSO-CF2 : A New Method for the Path Planning of a Mobile Robot, In IEEE International Multi-Conference on Systems, Signals and Devices (SSD’15), Mahdia-Tunisia, Mars (2015).

24 | S. Ziadi et al.

Biographies Safa Ziadi received the Engineering diploma in Electrical Engineering in 2011 and the M.S degree in New Information Technologies and Dedicated Systems in 2012 from the National Engineering School (ENIS), University of Sfax. She is currenlty, a PhD student in Electrical Engineering. Her research interests include mobile robot path planning and Particle Swarm Optimization.

Mohamed Njah received the Engineering diploma in Electrical Engineering from the National Engineering School (ENIS), University of Sfax in 1990, the DEA degree in Automatic Control from the ESSTT, University of Tunis in 1995 and the PhD degree in Electrical Engineering from ENIT University Elmanar Tunis in 2003. He was at SITEX textiles Company as an industrial informatics engineer from 1990 to 2001. He was a university assistant at FSEG Mahdia from 2001 to 2004 and assistant professor at ENIG Gabès from 2004 to 2009. Since 2009, he is assistant professor at ENIS. His research interests include neural networks and evolutionary algorithms. Mohamed Chtourou was born in Sfax (Tunisia) in 1963. He received the Engineering Diploma in electrical engineering from the Ecole Nationale d’Ingénieurs de Sfax-Tunisia in 1989, the Diplôme d’Etudes Aprofondies in Automatic Control from the Institut National des Sciences Appliquées de Toulouse-France in 1990, and the Doctorat in Process Engineering from the Institut National Polytechnique de Toulouse-France in 1993 and the Habilitation Universitaire in Automatic Control from the Ecole Nationale d’Ingénieurs de Sfax-Tunisia in 2002. He is currently a professor in the Department of Electrical Engineering of National School of Engineers of SfaxTunisia. His current research interests include learning algorithms, artificial neural networks and their engineering applications, fuzzy systems, and intelligent control. He is author and co-author of more than seventy papers in international journals and of more than 100 papers published in national and international conferences.

Ibtissem Malouche, Amira Kheriji, and Faouzi Bouani

Automatic Model Predictive Control code-generation for real-time implementation in a high-performance microcontroller Abstract: It is well-known that the implementation of Model Predictive Controller (MPC) on embedded system is a challenging task due to computational complexities associated whith optimization problem solving. The main contribution in this paper is an automatic framework for Model Predictive Control (MPC) implementation on Systems-on-a-Chip applications. This is especially interesting when dealing with high performances devices since it increases the number of addressable control applications in the industrial field and particularly in fast systems. Aiming to allow the implementation of such a computationally expensive controller on chip, we propose optimization hints satisfying trade-offs between code size and computation speed versus digital precision and effectiveness of the computed control action. The illustration of the proposed implementation is tested on a high performance STM32F407 microcontroller. Analysis of the accuracy of the digital implementation of MPC algorithm are given, and it shows that the proposed framework controls successfully the process with a good set-point tracking and a low computational burden with a high speed. Keywords: Microcontroller, Model Predictive Control, Embedded C, Optimization hints

1 Introduction Model predictive control has been developed considerably over the last years both within the research control community and in industry [1]. The major advantage of MPC is its ability to anticipate future events by computing the suitable control actions. Moreover, its aptitude to yield high performance control systems capable of operating without expert intervention for long period time is one of its key of success. Besides, its ability in considering, in a straightforward way, system control, state and output constraints is a major reason of its popularity. MPC does not designate a specific control strategy but a very ample range of control methods which make an explicit use of a model of the process to obtain the control Ibtissem Malouche, Faouzi Bouani, Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis, Laboratoire Analyse, Conception et Commande des Systèmes (LR11ES20), Tunis, Tunisia, e-mails: [email protected], [email protected] Amira Kheriji, Université de Tunis El Manar, Institut Supérieur d’Informatique de Tunis (Tunisie), Ecole Nationale d’Ingénieurs de Tunis, Laboratoire Analyse, Conception et Commande des Systèmes (LR11ES20), Tunis, Tunisia, e-mail: [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 25–40. https://doi.org/10.1515/9783110591729-002

26 | I. Malouche et al. signal. Its formulation integrates optimal control, stochastic control, control of processes with dead time, networked control system, multivariable control and control of non-linear processes [1], [2], [3], [5], [17] and [18]. Although MPC has been found to be quite a robust type of control in most reported applications [1], its implementation on low-cost system on chip solutions has been historically hindered by many restrictions and constraints [1], [4], [5], [6], [7], [8] and [11]. Among these constraints are mathematical complexities, which are not a problem in general for the research control community but represent a drawback for the use in practice, the time-to-market delays, and possible design errors if the algorithm is manually written in embedded C language on one side and the high computing and associated memory demands of the algorithm on the other side. There are many existing works focusing on the MPC hardware implementation [1], [4], [5], [6], [9], [10]. Some of these works are based on manual implementation which can take days even weeks as outlined in [5] and [6]. In [4] an interesting MPC automated workflow is presented but the high computing results obtained does not allow its implementation on fast systems. The goal of this work is to present a completely automatic framework allowing implementation of the MPC algorithm on fast systems with no manual embedded C coding effort and with consideration of trade-offs between data size and computation speed, versus numerical precision and effectiveness of the computed control action. This objective is challenging especially in case of system with fast dynamics, system described with Multiple Input Multiple Output (MIMO) and time varying models since the MPC control would be computationally expensive. To achieve this target; we present an original idea which consists in the re-use of the simulation MPC algorithm code for implementation. This is based on conversion from MATLAB script files to embedded C code for Cortex M4 architecture. Optimization hints are also presented which results in a faster and more efficient system-development workflow. The outline of this paper is as follows: In section 2, a theoretical background of the MPC method based on output deviation algorithm is presented. In section 3, we present the software and hardware environment used and outlined steps of the implementation method. Optimization techniques applied to the generated code which aims to satisfy the trade-offs of code size and execution speed are highlighted in section 4. Finally, in section 5, we present the simulation results of this method using MATLAB tool for Single Input Single Output (SISO) and (MIMO) systems and highlight the effectiveness of the generated code through a comparison between the simulation and the implementation results.

2 MPC Output Deviation Method We consider the following discrete state space model: Δx(k + 1) = FΔx(k) + GΔu(k), Δy(k) = HΔx(k)

(1)

Automatic MPC Implementation in a High-Performance Microcontroller | 27

In which the operator Δx(k) ∈ Rn is the state deviation vector, Δu(k) is the input deviation and Δy(k) is the measured output deviation. As mentioned in [2], the closed loop predicted estimate of x(k + j) with respect to the deviatory control input Δu(k) becomes: j−1

j−1

j−1

i=0

i=0

l=i

̂ + j|k) = [ ∑ F j−1−i ] FΔx(k) + x(k) + ∑ [∑ F j−1−l ]GΔu(k + i) x(k

(2)

Using equations (1) and (2), we can obtain the j-step ahead output predictor value described in equation (3) which is written as follows at time (k + j): j−1

j−1

j−1

i=0

i=0

l=i

̂ + j|k) = [ ∑ HF j−1−i ] FΔx(k) + Hx(k) + ∑ [∑ HF j−1−l ]GΔu(k + i) y(k

(3)

The MPC based on a state space model aims to minimize the quadratic criteria given by: Hp

2

Hc

̂ + j|k) − w(k + j)] + λ ∑ [Δu(k + j − 1)]2 J = ∑ [y(k j=1

j=1

(4)

Here, Hp is the prediction horizon, Hc is the control horizon, λ is the weighting factor and w(k + j) denotes the set-point at time (k + j). It is easier to use the matrix form. As in [2], the output sequence on Hp can be written as follows: Ŷ = LΔU + Mx(k),

(5)

in which: T Ŷ = [ŷ (k + 1|k) , ŷ (k + 2|k) , . . . , ŷ (k + Hp |k)] ,

ΔU = [Δu(k), Δu(k + 1), . . . , Δu(k + Hc − 1)].

It is assumed that there is no control action after time (k + Hc − 1), i. e. Δu(k + i) = 0 for i > (Hc − 1). Since the MPC is a receding horizon approach, only the first computed control increment Δu(k) is implemented. The L matrix with the (Hp , Hc ) dimension and M which is an (Hp , n) dimensional matrix are given by [2]: HG HFG + HG L=( .. . HF Hp−1 G + ⋅ ⋅ ⋅ + HG

0 HG .. . HF Hp−2 G + ⋅ ⋅ ⋅ + HG

H HF + H M=( ) .. . HF Hp−1 + HF Hp−1 + ⋅ ⋅ ⋅ + HF + H

0 0 ) .. . . . . HG (6)

28 | I. Malouche et al. The objective function can be expressed as: J = (Ŷ − W)T (Ŷ − W) + λΔU T ΔU,

(7)

in which: W = [w(k + 1), . . . , w(k + Hp )]T . After this objective function minimization, we obtain the following recursive form of the control input: u(k) = u(k − 1) + L̃ [W − MFΔx(k) − H ∗ x(k|k)] ,

(8)

where L̃ denotes the first row element of the matrix: L̃ = [LT L + Λ]−1 LT

(9)

In which Λ = diag(λ ⋅ ⋅ ⋅ λ): dimensional matrix and H ∗ is given by: H H = ( ... ) H ∗

(*) In case of system with time varying model, steps 3 and 4 are online treatments. Table 1 describes the MPC controller simulation steps. These steps are then implemented on embedded hardware starting from M-File MATLAB script in a completely automatic way. Table 1: MPC controller.

Offline treatment

Online treatment

1. Enter F , G, H matrices 2. Enter Hc , Hp , λ and initial vector of W , 3. Compute L, M using (6) and compute H∗ matrix, 4. Compute L̃ matrix using (9), (*) 5. Compute the output of the system y(k) 6. Compute Δu(k) using (8).

3 Automatic MPC Algorithm Implementation Workflow Since systems are subject to simulation before deployment, translating automatically the simulation code to implementation C code will increase productivity, improve quality, and foster innovation.

Automatic MPC Implementation in a High-Performance Microcontroller | 29

3.1 Hardware and Software Environments Figure 1 presents the hardware and software environments used to simulate and then implement the MPC controller.

Figure 1: MPC simulation and implementation environments.

STM32F407 32-bit is a high performance microcontroller. It includes ARM Cortex™-M4 core, floating point unit and built-in single-cycle multiply-accumulate (MAC) instructions. The Adaptive Real-Time Accelerator™ combined with STMicroelectronics 90 nm technology provides linear performance up to 168 MHz. This MCU includes 1MB of onchip Flash memory and 192 KB of SRAM [13]. These features expand the number of addressable real time control applications. STM32F4DISCOVERY is a low cost kit based on the STM32F407 and designed to help design engineers developing their applications easily. MDK-ARM is a software development and debugging environment for ARM-based microcontroller devices. It is specifically designed for microcontroller applications [14]. Its C Compiler is the only compilation tool co-developed with the ARM processors, and specifically designed to optimally support the ARM architectures [14]. Embedded MATLAB Coder™ works with Real-Time Workshop to convert code from a dynamically typed language (MATLAB) to a statically typed language (C) [15].

30 | I. Malouche et al.

3.2 MPC Implementation Steps The conversion from M-File MATLAB script to C code generation steps are provided in Table 2. Table 2: Steps of C code generation workflow. Step 1 Step 2

Step 3

Step 4

Step 5

Build MATLAB Script file The outcome of this step is a MATLAB m script file build for the system simulation. Convert MATLAB script to Embedded MATLAB code The simulation model of the control system may require further adaptation to the embedded execution environment before it is used as a source for the embedded code. Create Simulink model Since Embedded C Code generation for STM32F407 is only supported via Simulink Coder and TLC patch (available and downloadable for free since 2013 from [16]), a conversion from MATLAB script file to Simulink model is required. Generate C code The Real-Time Workshop toolbox is invoked yielding the functionality of MATLAB and Simulink extension with the ability to automatically generate C code and MDK-ARM pre-configured project [14], [15]. Implement and execute generated C mode The MDK-ARM project’s code is then optimized, compiled, linked, loaded and executed from STM32F407 on-chip flash memory.

4 Optimization Techniques Applied to the Generated Embedded C Code Presentation of the optimization techniques, allowing the efficiency of the generated C code in execution time (speed) and need memory space (size), is the objective of this section.

4.1 MATLAB Coder and MDK-ARM Optimization Options For Code generation, Embedded Simulink Coder embeds many configuration options and advanced optimization for fine-grain control of the generated code’ functions based on the processor architecture. These options allow control function boundaries, preserve expressions and apply optimization on multiple blocks to further reduce code size. The MDK-ARM tool integrates an ARM C compiler including a number of compiler optimization allowing code generation based on chosen microcontroller device and application area. The maximum level of optimization is chosen (−O3), and the goal of the optimization chosen is performance (Optimize for time option is enabled).

Automatic MPC Implementation in a High-Performance Microcontroller | 31

4.2 Code Optimization Hints Today, most of the compiler research is done in the optimization phase. However, the increasing size and complexity of software products and the use of these products in embedded systems results in the demand for more optimized versions of the source code [21]. In this context, after generated code analysis, size and speed execution optimization techniques are made by hand allowing maximum optimization of the generated code. Table 3 provides summary of the optimization techniques applied. Table 3: Code optimization hints. Dead store elimination Stack size adjustment Data Initialization Local variables Loop Invariant IF In-lining Functions Single floating point usage

Eliminate code that cannot be reached or code whose results are not subsequently used. Stack depth is adjusted based on the generated code. The whole section of data is copied instead of variable by variable copy. Favor local variables usage since global variables are stored in the memory, whereas, local variables are stored in the registers and stack memory. ‘IF’ statement with conditions that do not change from iteration to iteration are moved out of the loop. function calls are replaced with program code which has benefice for execution speed. Simple precision floating type usage rather than double precision types (generated by default by MATLAB Embedded Ceder) is an optimization hint for performance increase. In fact a simple addition of two variables, declared as double, requires 82 CPU cycles (4.88 10−7 s) due to double software library call whereas in simple precision case, it requires only 2 CPU cycles (1.19 10−8 s) to be executed thanks to the hardware FPU usage. To avoid the unnecessary type conversion or confusion, the letter “f” is assigned following the numeric value.

These techniques are applied to MPC embedded C generated code in Section 5 which results on portable, compact and high execution performances C code. Thus, the generated code can be executed on any Cortex-M4 based MCU even though with smaller on chip flash and RAM memories and with lower CPU frequencies. Hence, wide range of microcontrollers can be addressed.

5 Simulations and Implementation Examples In this section, we illustrate the control results of the proposed MPC algorithm through two real systems models. Simulations and implementation results are provided in or-

32 | I. Malouche et al. der to show the effectiveness of the presented implementation workflow and the contribution of the optimization hints on both schemes.

5.1 Study Case 1: The Rotating Antenna System The rotating antenna discrete model is represented as following: x(k + 1) = (

1 0

y(k) = ( 1

0.1 0 ) x(k) + ( ) u(k) 0.9 0.0787 0 ) x(k)

(10)

The motivation to consider this SISO system is that it is considered as one of the fastest reported embedded MPC application [4]. Therefore, simulating this system and then validating the MPC implantation workflow described in section 3.B on such a fast system will confirm its effectiveness for fast systems control. Comparison of code size and execution speed results before and after considering the optimization hints will outline their important contribution to control such a fast system even in case of high prediction and control horizons values. The closed loop simulation results presented in Fig 2(a) consider different values of prediction horizon Hp . The control horizon Hc is equal to 3 and the weighting factor λ is equal to 0.01 for all tests.

Figure 2: Study case 1: Simulation results.

Figure 2 outlines the closed loop simulation results for different control horizon Hc . Figures 2(a) and 2(b) illustrate the predicting behavior of the output with respect to the changes of the set-point. The example used in the simulation section is tested in run mode and executed from the STM32F407 on-chip flash memory. Results provided in Table 4 and Table 5

Automatic MPC Implementation in a High-Performance Microcontroller | 33 Table 4: Without code optimization hints. (a) Hc = 3, λ = 0.01, varying Hp Prediction Horizon 5 Flash code size (KBytes) 42.76 RAM data size (KBytes) 5.63 Execution time per sample (μs) 57.85 (b) Hp = 20, λ = 0.01, varying Hc Control Horizon 3 Flash code size (KBytes) 43.26 RAM code size (KBytes) 8.13 Execution time per sample (μs) 257.61

10 42.91 6.13 132

20 43.26 8.13 257.61

6 43.30 9.38 257.61

10 43.72 12.13 260.04

Table 5: With code optimization hints. (a) Hc = 3, λ = 0.01, varying Hp Prediction Horizon Flash code size (KBytes) RAM data size (KBytes) Execution time per sample (μs) (b) Hp = 20, λ = 0.01, varying Hc Control Horizon Flash code size (KBytes) RAM code size (KBytes) Execution time per sample (μs)

5 4.78 1.09 0.90

10 4.70 1.34 4.22

20 4.94 2.11 4.79

3 4.94 2.11 4.79

6 5.68 4.09 4.81

10 5.79 4.84 4.91

respectively focuses on code size and execution speed results found before and after implementation of code optimization techniques. Embedded MATLAB Coder and ARM compiler optimization are enabled for both cases. From Table 4 and Table 5, it is deduced that generated code is subject to very interesting optimization study. The average contribution of this optimization is 10 times more compact code and 50 times faster system. From Fig 2(a), 2(b), 3(a), 3(b), 3(c) and 3(d), it is easily noticed that the implementation results are perfectly in line with simulation results. Accordingly, we can conclude that the implemented MPC algorithm controls successfully the processes with a good set-point tracking.

5.2 Study Case 2: Wheeled Mobile Robot (WMR) The following linear, discrete-time, time varying model of the WMR dynamics is obtained by using a successive linearization approach around the point (xr , ur ) [17], [18],

34 | I. Malouche et al.

Figure 3: Study case 1: Implementation results.

[19] and [20]: 1 ̄ + 1) = ( 0 x(k 0

0 1 0

−vr (k)T sin(θr (k)) cos(θr (k))T ̄ + ( sin(θr (k))T vr (k)T cos(θr (k)) ) x(k) 1 0

1 y(k) = ( 0 0

0 1 0

0 ̄ 0 ) x(k) 1

0 ̄ 0 ) u(k) T (11)

Where, x̄ = x − xr represents the error with respect to the reference car and ū = u − ur is its associated error control input. u = [vwa ]T is the control input, where v and wa are the linear and the angular velocities, respectively. The state vector describes the configuration (abscissa xa (k), ordinate yo (k) and angular position θ(k)) of the center of the axis of the wheels with respect to a global inertial frame (OXY). The robot set-point (S) describes the configuration (abscissa xar (k), ordinate yor (k) and angular position θr (k)) of the center of the axis of the wheels with respect to a global inertial frame (OXY). T is the sampling period and vr is the reference linear velocity.

Automatic MPC Implementation in a High-Performance Microcontroller | 35

Figure 4: Study case 2: Simulation results.

36 | I. Malouche et al. Despite the apparent simplicity of this WMR MIMO kinematic model, the design of stabilizing control laws for this system can be considered as a challenge [18]. MPC appears therefore as an interesting and promising approach for overcoming these problems [18]. Nevertheless, the use of MPC for real-time control of systems with fast dynamics such as a WMR has been hindered for some time due to its numerical intensive nature [18]. Besides, the WMR in our case is described by MIMO and time-varying model; therefore, the optimization problem must be solved at each sampling time which increases the computational burden. However, with the consideration of fast processors and the usage of proposed approach to generate embedded C code, the use of MPC in such demanding application becomes possible. On the other hand, the adjustment of the MPC prediction horizon Hp parameter on the real system is crucial for trajectory tracking [18], [19] and [20]. Hence, the contribution of the proposed automatic workflow is major in this case since it allows fine tuning this parameter very easily by generating MPC embedded C code for different Hp values in few minutes allowing a wise choice of this parameter. Figure 3 and 4 show respectively the closed loop simulation and implantation results using two prediction horizon values: Hp = 2 and Hp = 20. Hc and the λ values are equal respectively to 3 and 0.01 for all tests, T is equal to 5.26 ms and the linear velocity vr is equal to 12 cm/s. For each Hp value, the robot abscissa, ordinate, angular position simulation over time are presented in Fig 4(a), 4(b), 4(c), 4(d), 4(e), 4(f). The linear velocity and angular velocity control input evolution over time are outlined at Fig 4(g) and 4(h). It is clearly noticed from Fig 4(a), 4(b), 4(c), 4(d), 4(e), 4(f), 4(g), 4(h), 5(a) and 5(b) that the robot trajectory on (OXY) plan is tracked successfully using both Hp values but it can be seen that there are significant difference between both cases in term of trajectory tracking performance. In Fig 6(a), 6(b), 6(c), 6(d), 6(e), 6(d), 6(f), 6(g) and 6(h), it is clearly noticeable that the implementation results are perfectly in line with simulation ones which confirms, again, the effectiveness of the implementation workflow.

Figure 5: Robot trajectory in (OXY) plan.

Automatic MPC Implementation in a High-Performance Microcontroller | 37

Figure 6: Study case 2: Implementation results.

38 | I. Malouche et al. Since the model is MIMO and time varying, the added value of optimization hints is important. Applying them allows reaching 90µs execution time per sample which is a very interesting value and far from results obtained on existing works [18] at same working conditions. Note: The logic analyzer of MDK-ARM toolchain is used to plot MPC signals changes overtime. This tool graphically displays the signals evolution collected from the Serial wire Viewer (SWV) asynchronous trace pin. Since this gathering is limited by the one data pin trace usage and the ST-LINK hardware debug probe limited trace frequency, a data loss is expected when using 168 MHz MCU frequency [12], [13]. To overcome this debugging problem, 1 MHz is used as MCU system frequency to plot all implementation figures. This limitation concerns only debugging, it has no impact on real time execution which runs successfully with 168 MHz.

6 Conclusion A framework for embedding MPC control algorithm on a high performed STM32F407 microcontroller has been presented. This new approach results in a faster and more efficient system-development workflow since the generated code is very close to simulated model. In order to obtain compact and high preferment C code easily portable even for wide range of system on-chip, optimization techniques have been applied to the generated code. The rotating antenna system and the Wheeled Mobile Robots (WMR) were used to test the MPC firmware. Hence, an efficient implementation of these codes yields a low computational burden with a high speed. Indeed, based on the simulation and implementation results, we noticed that the generated C code controls successfully the processes with a good set-point tracking. There are still much detailed analysis and tests to be done, which should consider the input-output constraints of the process.

Bibliography [1] [2]

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E. Camacho and C. Bordons. Model Predictive Control. Springer, 2004. K. Watanabet, K. Ikeda, T. Fukuda, and S. Tzafestas. Adaptive generalized predictive control using a state space approach. International Workshop on Intelligent Robots and Systems IROS Osaka, Japan, 1609–1614, 1991. D. W. Clarket, C. Mohtadit and P. S. Tuffs. Generalized Predictive Control Part I. The Basic Algorithm. Automatica, 23(2):137–148, 1987. J. Currie, A. Prince-Pike, and D. I. Wilson. Auto-Code Generation for Fast Embedded Model Predictive Controllers. 19th International Conference on Mechatronics and Machine Vision in Practice (M2VIP12) Auckland, New-Zealand, 122–128, 28-30th Nov 2012.

Automatic MPC Implementation in a High-Performance Microcontroller | 39

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A. Kheriji, F. Bouani, and M. Ksouri. Efficient implementation of constrained robust model predictive control using a state space model. International Conference on Informatics in Control, Automation and Robotics (ICINCO) Madeira, Portugal, 116–121, June 2010. A. Kheriji, F. Bouani, and M. Ksouri. A Microcontroller Implementation of Constrained Model Predictive Control. International Journal of Electrical and Electronics Engineering, 5(4):272–279, 2011. D. I. Wilson and B. R. Young. The Seduction of Model Predictive Control. Electrical and Automation Technology, January 2006. L. G. Bleris, J. Garcia, M. V. Kothare, and M. Arnold. Towards embedded model predictive control for System-on-a-Chip applications. Journal of Process Control, 16(3):255–264, 2006. K. Ling, S. Yue, and J. Maciejowski. A FPGA implementation of model predictive control. Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 2006. K. Ling, B. Wu, and J. Maciejowski. Embedded model predictive control (mpc) using a FPGA. Proceedings of the 17th IFAC World Congress Seoul, Korea, 6–11, July 2008. M. G. Forbes, R. S. Patwardhan, H. Hamadah and R. B. Gopaluni. Model Predictive Control in Industry: Challenges and Opportunities. 9th International Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control (IFAC) 2015, Whistler, British Columbia, Canada, July 2015. A. Vikstrom. A study of automatic translation of MATLAB code to C code using software from MathWorks. Master’s Thesis, Lulea University, 2009. STMicroelectronics – RM0090 Reference manual. www.st.com/resource/en/ reference-manual/dm00031020.pdf, May 2016. KEIL Tools by ARM – uvision4 user guide. http://www.keil.com/support/man/docs/uv4/, 2014. MathWorks-Real-Time Workshop Embedded Coder Reference. http://www.mathworks.com/ help/releases/R14sp2/pdf-doc/ecoder/ecoder-ug.pdf. STMicroelectronics. www.st.com/stm32-mat-target, 2013. F. Kuhne, W. F. Lages, J. M. Gomes, and Jr. da Silva. Point stabilization of mobile robots with nonlinear model predictive control. International Conference on Mechatronics and Automation Canada, 1163–1168, 2005. F. Kuhne, W. F. Lages, J. M. Gomes, and Jr. da Silva. Model Predictive Control of a Mobile Robot Using Linearization. Mechatronics and Robotics Aachen, Germany, 2004. L. Pacheco and N. Luo. Mobile robot local trajectory tracking with dynamic predictive control technics. International Journal of innovative Computing, Information and control, 7(6):3457–3483, 2011. G. Klancar and I.Skrjanc. Tracking-error model-based predictive control for mobile robots in real time. Robotics and Autonomous Systems, 55:460–469, 2007. A. K. Sarma. New trends and Challenges in Source Code Optimization. International Journal of Computer Applications, 131(16):27–32, December 2015.

40 | I. Malouche et al.

Biographies Ibtissem Malouche was born in Kairouan, Tunisia, in 1980. She received the Eng. Degree in Electrical Engineering and the Master degree in the Automatic and Signal Processing from the National School of Engineering of Tunis (ENIT), in 2003 and 2004 respectively. Since 2003, she is working in STMicroelectronics in the microcontroller division. She received her PhD degree in 2018 in the Laboratory of Analysis, Design and Control of Systems at ENIT. Her main research interests include constrained predictive control, real time systems and implementation of the control algorithms.

Amira Kheriji was born in Tunis, Tunisia, in 1982. She received the Eng. Degree in Electrical Engineering and the Master degree in the Automatic and Signal Processing from the National School of Engineering of Tunis (ENIT), in 2006 and 2008 respectively. From 2006 to 2008, she worked in STMicroelectronics of Tunis as application and support engineer in the microcontroller division and she was responsible of the ST microcontroller low power aspect. She received her Ph. D. degree in 2011 in the Laboratory of Analysis, Design and Control of Systems at ENIT. Her main research interests include constrained predictive control, robust predictive control, real time systems and implementation of the control algorithms. Faouzi Bouani is a professor at the National School of Engineering of Tunis (ENIT). He received his B. Sc. and M. Sc. degrees, respectively in 1990 and 1992, from the Ecole Normale Supérieure de l’Enseignement Technique of Tunis. He received his Ph. D. degree in 1997 and the Habilitation universitaire in 2007, in Electrical Engineering both from (ENIT). His research interests include nonlinear predictive control, robust predictive control, and the computational intelligence technique.

Hadil Soltani, Saloua Bel Hadj Ali Naoui, Rafika El Harabi, Abdel Aitouche, and Mohamed Naceur Abdelkrim

Fault Tolerant Control for Uncertain State Time-delay Systems Abstract: This paper deals with robust passive and active fault tolerant control strategies for uncertain systems with state time delays. First, the control law is formulated as H∞ model-matching problem in order to guarantee dynamic performances and to compensate fault effects. Indeed, a new stability condition, based on the LyapunovKrasovskii theory is developed. Further, a sufficient condition for solvability of the robust control gains is obtained in terms of Linear Matrix Inequality feasibility conditions. Second, the Fault Tolerant Control strategy based on Frobenius norm is designed by minimizing the distance between the closed-loop model of the faulty system and the reference one. Therefore, stability ensured by the Modified Pseudo-Inverse Method. The effectiveness of the designed methodology is verified based on a twostage chemical reactor with delay recycle streams. Keywords: Uncertain time-delay systems, H∞ Fault Tolerant Control, Stability, Modified Pseudo-Inverted Method, LMI MSC 2010: 65C05, 62M20, 93E11, 62F15, 86A22

1 Introduction Uncertainties and time delay occur in the dynamic response of many physical processes [1–6]. They may induce complex behaviors for the control scheme, which are sources of dysfunction and oscillations, making systems difficult to analyze with classical methods, especially, for checking the stability and designing stabilizing controllers. Indeed, Fault Tolerant Control (FTC) is aimed at achieving acceptable performance and stability for the safe system, i. e. fault-free system as well as for the faulty system. Two classes of FTC can be distinguished in literature. The first one, known as the passive FTC approach [7]. It is robust against faults and guarantees the stability even with an acceptable performance degradation. The second one is the active

Hadil Soltani, Saloua Bel Hadj Ali Naoui, Rafika El Harabi, Mohamed Naceur Abdelkrim, University of Gabes, Gabès, Tunisia, e-mails: [email protected], [email protected], [email protected], [email protected] Abdel Aitouche, CRISTAl-HEI, Research Center of Informatics, Signal and Automation, Lille, France, e-mail: [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 41–60. https://doi.org/10.1515/9783110591729-003

42 | H. Soltani et al. approach based on the reconfiguration of the controller. Then, the main challenge is now to provide nominal performance in presence of faults and uncertainties timedelay system subject to disturbances. A large number of works deals with the stability analysis [8–12] and the control synthesis of uncertain linear state time delay systems [13–16]. The robust H∞ control case of uncertain systems with interval constant time-varying delay have been reported in [17]. Qiu et al. [18] developed a New approach to delay-dependent H∞ control for continuous-time Markovian jump systems with time-varying delay based on the Linear Matrix Inequality (LMI) approach. Wang et al. have addressed the issue of the robust H∞ controller design for a class of linear quantum systems with time delay in [19]. Doyle et al. [20] derived simple state-space formulas for all controllers solving a standard H∞ problem based on the approximation that neglects some of the timedelay terms. Moreover, the model matching and the Pseudo-Inverse Method (PIM) has been first introduced in flight control systems. As [21, 22], classical and modified PIM (MPIM) have been extended, by using a set of admissible models, rather than searching for an optimal one which does not provide any guarantee about the post-fault system behavior. The PIM principle is to modify the constant feedback gain so that the reconfigured system approximates the nominal system in some sense. These methods do not use Fault Detection Identification mechanism. To solve this stability problem a MPIM [23] in which the difference between the closed-loop matrices is minimized subject to stability constraints, whilst recovering the performance as much as possible is proposed. The scientific contribution of the presented paper concerns the development of a reliable state feedback control law taking into account actuator and sensor failures, so that the desired performances and stability will be sustained. The two proposed frameworks (active and passive FTC) are predicated on a good choice of a Lyapunov Krasovskii function. This latter is able to give new delay independent and parameter dependent stability conditions. The problem is later performed through two ways, firstly, by a standard H∞ model matching optimization problem solved under LMI constraints and secondly, using an active fault tolerant control for uncertain linear time delay system based on an existing technique PIM which is adapted and changed into MPIM so as to get the new strategy reconfiguration that ensures the stability. Further, a comparative study of the improved FTC schemes is given at the end of this work, those different techniques are analyzed and assessed both in view of a numerical simulation on a two-stage chemical reactor with delay recycle streams. This paper is structured as follows: Section 2 presents some essential notations and the problem statement is herein formulated. Section 3 focuses on the design of the active and passive FTC for linear uncertain time delay systems. Afterward, Section 4 is dedicated to show the performances of the developed fault tolerant controllers by means of a tow stage chemical reactor with delay. Finally, Section 5 draws some remarks as a conclusion.

Fault Tolerant Control for Uncertain State Time-delay Systems |

43

2 Preliminaries and Problem Statement 2.1 Preliminaries Notation: ⬦ In a block matrix, the notation * stands for the terms induced by symmetry. ℝn is the n dimensional Euclidean space, (.)T is the matrix transposition, ℝn×n is the space of all real matrices and P > 0 (< 0) for P ∈ ℝn×n means that P is symmetric and positivedefinite (negative-definite). The space of function in ℝp that is square integrable over [0, ∞[ is denoted by Lp2 [0, ∞[. I is the identity matrix with appropriated dimension. All matrix if their dimensions are not explicitly stated, are assumed to be compatible. ⬦ The norm of the vector function h(t) ∈ Lp2 [0, ∞[ is ∞

1/2 T

‖h(t)‖2 = [ ∫ h (t) h(t) dt ] ] [0

.

⬦ The H∞ norm of a transfer function G(s) is denoted by ‖G(s)‖2 = supw δmax [G(jw)], where δmax (.) is the largest singular value of (.) ⬦ This norm ‖.‖F is called the Frobenius norm or the Hilbert-Schmidt norm. It can be defined in various ways as a matrix norm of an m × n matrix A: m n

󵄨 󵄨2 ‖A‖F = √∑ ∑ 󵄨󵄨󵄨aij 󵄨󵄨󵄨 i=1 j=1

min{m,n}

= √trace(A∗ A) = √ ∑ i=1

σi2

where A denotes the conjugate transpose of A, σi are the singular values of A, and the trace function is used. The following lemma is used to design the expected robust H∞ fault tolerant controller for the uncertain linear time-delay system. ∗

Lemma 1 ([24]). Let Ω, Σ and F be real matrices of appropriate dimension. where: Σ = diag{Σ1 , Σ2 , Σ3 } with: ΣTi Σi < I,

i = 1, 2, 3.

Then, for any real matrix Λ = diag{ε1 , ε2 , ε3 }, we have the following inequality: Ω1 + Ω2 diag{Σi Fi } + diag{FiT ΣTi }ΩT2 < Ω1 + Ω2 ΛΩT2 + diag{FiT }Λ−1 diag{FiT }

44 | H. Soltani et al.

2.2 Problem Statement Consider the following linear uncertain time-delay system: ̇ = (A + ΔA)x(t) + (Ad + ΔAd )x(t − h) + (B + ΔB)u(t) x(t) { { y(t) = Cx(t) + Du(t) { { x(0) = 0 ∀t ∈ [−h, 0] {

(1)

where x(t) ∈ ℝn is the state space vector, y(t) ∈ ℝq denotes the measurement output vector, u(t) ∈ ℝp is the control input vector. The matrices A, Ad , B, C, D are known with appropriate dimensions and h is a known constant time-delay. Without loss of generality, we assume (A, C) to be observable and omit the control input. ΔA = E1 Σ1 F1 , ΔAd = E2 Σ2 F2 and ΔB = E3 Σ3 F3 are real-valued matrix functions representing the norm-bounded parameter uncertainties. E1 , E2 , E3 , F1 , F2 and F3 are known real constant matrices of appropriate dimensions which specify how uncertain parameters in Σ1 , Σ2 and Σ3 enter the nominal matrices A, Ad and B. Σ1 , Σ2 and Σ3 which may be time-varying, are real matrices with Lebesgue measurable elements and satisfy Σ1 ΣT1 ≤ I, Σ2 ΣT2 ≤ I and Σ3 ΣT3 ≤ I. Consider the following faulty system given by: { { { { {

̇ = (A + ΔA)x(t) + (Ad + ΔAd )x(t − h) + (B + ΔB)u(t) + Bf f (t) + Bd d(t) x(t) y(t) = Cx(t) + Du(t) + Df f (t) + Dd d(t) x(0) = 0 ∀t ∈ [−h, 0]

(2)

where d(t) ∈ ℝm is the unknown input vector satisfying d(t) ∈ Lp2 [0, ∞[, f (t) ∈ ℝl is the fault affects the system. Bf , Bd , Df and Dd are real known matrices. The contribution of this paper is to design a stabilizing control law for the faulty uncertain system (2) independently of the state time delay. In this paper, we assume that all states are known. In the case that some states are not measured, an observer can be built in order to estimate them. The selected state-feedback control is defined as follows: u(t) = −Kx(t) + Pv(t)

(3)

where K(n, m) and P(n, n) are matrices to be determined, v(t) ∈ ℝm ∗ 1 is the reference signal.

3 Fault Tolerant Control Design 3.1 Passive Fault Tolerant Control Design The overall design of FTC systems is summarized and illustrated in Fig. 1.

Fault Tolerant Control for Uncertain State Time-delay Systems | 45

Figure 1: The passive fault control scheme.

The main objective is to find the robust delay-dependent conditions of the existence of robust fault tolerant control law for uncertain linear systems with time-delay solved using LMIs by obtaining the closed-loop gain K. The design problem is formulated as parameterizing the control gain matrices such that the closed-loop control system is stable. While for some prescribed positive γ, the H∞ performance bound constraint ‖Hyf ‖∞ < γ is guaranteed. H∞ control problem of uncertain time-delay systems is achieved if and only if the H∞ stabilization problem of a corresponding augmented system is solvable. Consider the parameter uncertainties described by the following representation: Ā = A + ΔA, Ā d = Ad + ΔAd , B̄ = B + ΔB. The passive fault tolerant control is choosing as follows: u(t) = −Kx(t) + v(t)

(4)

(P = I identity matrix with size m). The closed-loop system obtained by combining (2) and (3) can be described as follows: ̄ ̄ ̇ = (Ā − BK)x(t) x(t) + Ā d x(t − h) + Bv(t) + Bf f (t) + Bd d(t) { { { y(t) = (C − DK)x(t) + Dv(t) + Df f (t) + Dd d(t) { { { { x(0) = 0 ∀t ∈ [−h, 0]

(5)

The main goal is to establish the robust delay-dependent sufficient conditions of the existence of robust fault tolerant control laws for uncertain linear systems with time-delay using the LMI technique. Thus, to find a suitable matrix gain which stabilizes the system and ensures fault compensation and robustness against perturbation, a new condition is given. The following proposed theorem offers the theoretical basis for achieving the desired design goal. It provides the closed-loop gain K by solving the optimization problem under LMI constraints.

46 | H. Soltani et al. Theorem 1. Consider the linear uncertain time-delay system (5). Then given scalers γ > 0 and h,̄ the system is robustly stable for the constant h which satisfies 0 ≤ h ≤ h,̄ if there exist positive definite matrices P, Q, K and matrix Y, α1 , α2 , α3 , ε1 , ε2 and ε3 positive scalars such the following LMIs are satisfied. M11 (P) + E1 α1 E1T + E2 α2 E2T + E3 α3 E3T + F1T ε1−1 F1 AT P T D C − DT DK + BT P DTf C − DTf DK + BTf P −DTd DK + DTd C + BTd P [ [ [ [ [ [ [ [

PA −Q + F2T ε2−1 F2 0 0 0

C T D − KDT D + PB 0 T D D + F3T ε3−1 F3 DTf D DTd D

C T Df − K T DT Df + PBf 0 DT Df DTf Df − γ 2 I DTd Df

C T Dd − K T DT Dd + PBd 0 DT Dd DTf Dd DTd Dd

(6)

] ] ] ] 70s

After synthesizing the proportional multiple integral observer according to the Theorem 1 and the proportional integral observer with unknown inputs according to the Theorem 2, the simulation results are depicted in the Fig. (10) to (15).

Figure 10: Actuator fault and their estimates.

Figure 11: Actuator fault and their estimates.

Observers Design for Takagi-Sugeno Models | 77

Figure 12: Sensor fault and their estimates.

Figure 13: Sensor fault and their estimates.

Figure (10) shows the actuator fault and their estimations with PMI and PIUI observers. The sensor fault and their estimations are given in Fig. (12). The results provided by the two kinds of observers are good (Fig. (14) and (15)). According to the Fig. (11) and (13), we note that the PMI observer responds faster than the PIUI observer. So, the PMI observer provides a more precise estimation of states and faults.

5 Conclusion The design of proportional multiple integral observer and proportional integral observer with unknown inputs are studied in this paper. The results presented in this

78 | W. Jamel et al.

Figure 14: State estimation errors PIUI.

Figure 15: Sate estimation errors PMI.

work contribute to estimate simultaneously state, actuator and sensor faults for nonlinear systems represented by Takagi-Sugeno models. By using a mathematical transformation, sensor faults are considered as unknown inputs of an augmented system. Existence conditions of the two observers have been established using the Lyapunov theory. The convergence conditions of the state and the unknown inputs estimation errors are given in the LMI formulation. The examples have illustrated the efficiency of the proposed approaches. It can be concluded that the proposed observers provide good estimates of the system state. Indeed, the simulation results show that we can estimate well the faults affecting the system even in the case of time varying faults.

Observers Design for Takagi-Sugeno Models | 79

The future works will concern the use of the proposed observers in the context of

fault tolerant control.

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[16] Jamel W., Khedher A., Ben Othman K., Design of Unknown Inputs Multiple Observer for Uncertain Takagi-Sugeno Multiple Model, International Journal of Engineering and Advanced Technology (IJEAT), ISSN: 2249–8958, Vol. 2, No. 6, p. 431–438, August 2013. [17] Jamel W., Bouguila N., Khedher A., Ben Othman K., Observer design for nonlinear systems represented by Takagi-Sugeno models, WSEAS Transactions on Systems, Vol. 9, No. 7, p. 804–813, July 2010. [18] Jamel W., Khedher A., Bouguila N., Ben Othman K., State estimation via observers with unknown inputs: application to a particular class of uncertain Takagi-Sugeno systems, Studies in Informatics and Control, Vol. 19, No. 3, p. 219–228, September 2010. [19] Khedher A., Ben Othman K., Maquin D. and Benrejeb M., Adaptive observer for fault estimation in nonlinear systems described by a Takagi-Sugeno model, 18th Mediterranean Conference on Control and Automation, MED’10, June 24–26, Marrakech, Morroco, 2010. [20] Khedher A., Ben Othman K., Proportional Integral Observer Design for State and Faults Estimation: Application to the Three Tanks System, International Review of Automatic Control, Vol. 3, No. 2, p. 115–124, March 2010. [21] Khedher A., Ben Othman K., Benrejeb M. and Maquin D., State and unknown input estimation via a proportional integral observer with unknown inputs, 9Th International Confereence on Science and Techniques of Automatic Control and Computer Engineering STA’2008, December 20–23, Sousse, Tunisia, 2008. [22] Orjuela R., Marx B., Maquin D., Ragot J., Fault diagnosis for nonlinear systems represented by heterogeneous multiple models, Conference on Control and Fault-Tolerant Systems, SysTol’10, October 6–10, 2010. [23] R.Orjuela, B. Marx, J. Ragot and D. Maquin, On the simultaneous state and unknown inputs estimation of complex systems via a multiple model strategy, IET Control Theory & Applications, Vol. 3, No. 7, p. 877–890, 2009. [24] Ming L., Xibin C., Peng S., Fuzzy-Model-Based Fault-Tolerant Design for Nonlinear Stochastic Systems Against Simultaneous Sensor and Actuator Faults, IEEE Transactions on Fuzzy Systems, vol. 21, no. 5, p. 789–799, 2013. [25] Murray-Smith R., T. A. Johansen, Multiple model approaches to modelling and control, Taylor and Francis, London, 1997. [26] Patton R., Frank P. and Clark R., Fault diagnosis in dynamic systems: Theory and application, Prentice Hall international, 1989. [27] Ruiyun Q., Gang T., Bin J., Chang T., Adaptive Control Schemes for Discrete-Time T-S Fuzzy Systems with Unknown Parameters and Actuator Failures, IEEE Transactions on Fuzzy Systems, vol. 20, no. 3, p. 471–486, 2012. [28] Sename O., Unknown input robust observer for time delay system, IEEE Conference on Decision and Control, 2, p. 1629–1630, 1997. [29] Shaocheng T., Baoyu H., Yongming L., Observer-Based Adaptive Decentralized Fuzzy Fault-Tolerant Control of Nonlinear Large-Scale Systems with Actuator Failures, IEEE Transactions on Fuzzy Systems, vol. 22, no. 1, p. 1–15, 2014. [30] Sharma R., Aldeen M., Estimation of unknown disturbances in nonlinear systems, Control 2004, University of Bath, UK, September 2004. [31] Tanaka K., Ikeda T. and Wang H., Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs, IEEE Transactions on Fuzzy Systems, vol. 6, no. 2, p. 250–265, 1998.

Observers Design for Takagi-Sugeno Models | 81

Biographies Wafa Jamel was born in Tunisia in 1982. She obtained the Engineer degree in electrical engineering from the “Ecole Nationale d’Ingénieurs de Monastir (ENIM)”, Tunisia, in 2007 and obtained the master degree in automatic and diagnosis from the “Ecole Nationale d’Ingénieurs de Monastir (ENIM)” in 2009. She obtained the Ph. D. degree in the electrical engineering within the framework of LARATSI-ENIM in January 2015. Her research is related to diagnosis, synthesis of observers for Takagi-sugeno systems, States and faults estimations and active faults tolerant control strategies.

Atef Khedher was born in Tunisia in 1980. He obtained the Engineer degree in electro-Mechnical engineering from the “ENIS” in 2003 and obtain the master degree in automatic and industrial Maintenance from the “ENIM” in 2005. He obtained his Phd Degree in automatic and computer science in 2011 from the “ENIT”. He is currently associate professor at ISSAT Gafsa. His research is related to Diagnosis, state and fault estimation for continus systems and Discretes events systems.

Nassreddine Bouguila was born in Tunisia in 1961. He received the engineering degree from the Tunis School of Engineering, Tunisia, in 1986. He received the Ph. D. degree in physics from SUPELEC, France, in 1990 and also obtained HDR from the Monastir University – Tunisia in 2013. He is currently Professor at the Monastir School of Engineering, Tunisia. His research is related to Reliability, fuzzy systems and diagnosis of complex systems.

Kamel Ben Othman was born in Tunisia in 1958. He obtained the Engineer degree in Mechanical and Energetic engineering from the “Université de Valencienne” in 1981 and obtain the PhD degree in automatic and signal processing from the “Université de Valencienne” in 1984 and the HDR from the “Ecole Nationale d’Ingénieurs de Tunis” in 2008. He is currently professor at “ENI Monastir”. His research is related to Reliability, fuzzy systems and Diagnosis of complex systems.

Sondess Mejdi, Anis Messaoud, Mouhib Allaoui, and Ridha Ben Abdennour

Detection and Isolation of Sensor Faults for Nonlinear Systems: Robustness Against Disturbances Abstract: In this paper, a sensor fault detection and isolation scheme based on discrete unknown input multiobservers for a disturbed nonlinear systems is proposed. The proposed multiobservers are based on an uncoupled state multimodel. Sensor faults can be isolated using the generated structured residuals. Nevertheless, the presence of disturbances make the isolation task difficult. Then, a robust sensor fault detection is investigated to make it easier. A bank of unknown input multiobservers is designed to generate structured residuals to accomplish the isolation task. The multiobserver gains are provided in terms of Linear Matrix Inequalities (LMI)s ensuring the convergence of the estimation error. A numerical example is presented to prove the effectiveness of the proposed approach. Keywords: Nonlinear systems, uncoupled multimodel, robust fault detection, fault isolation, discrete unknown input multiobserver, residual generation

1 Introduction Physically and under some conditions no system can work perfectly when many faults can occur. Indeed, they can damage its stabilization and may cause unacceptable process performance degradation [8]. Therefore, Fault Detection and Isolation (FDI) have become crucial methods for complex systems. The FDI procedure is carried out in three important steps: fault detection, fault isolation and fault identification. The first step is the most important as it can inform about the presence or not of faults. After that, the fault isolation is directly fulfilled when the fault detection is achieved. In the literature, the problem of model-based fault detection and isolation has been introduced. Thus, many FDI techniques have been introduced in survey papers [23]. It is necessary that a fault detection and diagnosis scheme can be developed in order to detect and identify occurring faults as early as possible. Once the fault detection is successfully achieved thus its task to locate (isolate) and determine the faulty actuator or the faulty sensor [1, 2]. However, fault detection is extensively investigated in literature [8, 10, 14, 18], fault isolation has received less attention especially for disturbed non Sondess Mejdi, Anis Messaoud, Mouhib Allaoui, Ridha Ben Abdennour, Research Laboratory: CONPRI (Numerical Control of Industrial Processes), National Engineering School of Gabes, University of Gabes, Gabès, Tunisia, e-mails: [email protected], [email protected], [email protected], [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 83–102. https://doi.org/10.1515/9783110591729-005

84 | S. Mejdi et al. linear system represented by an uncoupled state multimodel. Indeed, the fault isolation is more difficult then fault detection and very crucial for fault diagnosis. Among the various methods which have been used in applications fields in the literature the model based approaches as the analytical redundancy and the use of observer schemes have been mainly exploited in [6, 9, 23] and which are widely used to generate structured residuals. Then many types of observers have been constructed and various methods have been developed to treat the considered problem of detection and isolation of faults [1–3, 5, 9, 10, 16, 21]. The unknown input observer is the most efficient one which generates structured residuals for the detection and the isolation of faults [16]. The fault isolation task is achieved by introducing two schemes based on bank of observers generating structured residuals named Generalized Observer Scheme (GOS) and Dedicated Observer Scheme (DOS), which are designed for some special cases and are only used to detect and isolate faults [3, 23]. This paper is organized as follows. The sensor faults detection and isolation with generated residuals using a bank of a discrete Unknown Input MultiObservers (UIMO) for a disturbed nonlinear system are presented in section 2. In section 3, we propose a discrete Unknown Input MultiObserver synthesis for disturbed nonlinear systems to prove a robust detection of sensors faults regardless of the disturbances. Then, a numerical example is given in section 4 to illustrate the effectiveness of proposed approach. Section 5 concludes this paper.

2 Sensor Faults Detection and Isolation Based on a Discrete Uncoupled State Unknown Input Multiobserver The multimodel approach represents an interesting mathematical representation [15] inspired by fuzzy models Takagi-Sugeno (T-S) which represented nonlinear systems in the form of an interpolation between local linear models. This approach is based on the fragmentation of a complex system into a set of linear partial systems [3, 4, 14, 19, 20]. In this work, an uncoupled multimodel representation for disturbed nonlinear system subject to sensor faults is retained (1): xi (k + 1) = Ai xi (k) + Bi u(k) + Wi w(k) { { { { yi (k) = Ci xi (k) { Nm { { { yMM (k) = ∑ μi (νk−1 )yi (k) + Fc fc (k) i=1 { where: – Nm is the number of partial models.

(1)

Detection and Isolation of Sensor Faults for Nonlinear Systems | 85

– – – – –

xi (k) ∈ ℜni and yi (k) ∈ ℜp denote respectively the state and the measured output vectors of the ith partial model. yMM (k) ∈ ℜp and u(k) ∈ ℜm are respectively the multimodel output vector and the input vector. fc (k) ∈ ℜp is the sensor faults vector, w(k) ∈ ℜd is an external disturbance. Ai ∈ ℜni ×ni , Bi ∈ ℜni ×m , Ci ∈ ℜp×ni , Wi ∈ ℜni ×d are known matrices and appropriately dimensioned. Fc ∈ ℜp×p represents the influence matrix of the sensor faults.

However, μi (νk−1 ) are the activation functions using a decision variable νk−1 which can be a measurable or non-measurable variable [19]. Assuming that the decision variable νk−1 is measurable and available in real time. We have chosen almost the input or the output of the nonlinear system [15, 20]. These activation functions satisfy the following properties [4, 14, 19, 20]: Nm

{ { ∑ μi (νk−1 ) = 1 { { i=1 { 0 ≤ μi (νk−1 ) ≤ 1

∀ i = 1, . . . , Nm

(2)

In order to obtain a compact form of multimodel structure, we define the vector xcf (k) as being the augmented state vector: xcf (k) = [ x1T (k)

xiT (k)

...

Nm

T

T xNm (k) ] ∈ ℜn ,

...

n = ∑ ni

(3)

i=1

The uncoupled state multimodel (1) can be rewritten as follows: {

xcf (k + 1) = Acf xcf (k) + Bcf u(k) + Wcf w(k) yMM (k) = Ccf (k)xcf (k) + Fc fc (k)

where: [ [ [ [ [ Acf = [ [ [ [ [

A1 0 .. .

0 .. .

Ai

[ 0

⋅⋅⋅

Bcf = [ B1 T

⋅⋅⋅

0 .. .

⋅⋅⋅

..

. 0

Bi T

] ] ] ] ] n×n ]∈ℜ , ] ] ] ]

0 ANm ]

⋅⋅⋅

T

BNm T ] ∈ ℜn×m ,

Ccf (k) = [μ1 (νk−1 )C1 . . . μi (νk−1 )Ci . . . μNm (νk−1 )CNm ] ∈ ℜp×n ,

Wcf = [ W1 T

⋅⋅⋅

Wi T

⋅⋅⋅

T

WNm T ] ∈ ℜn×d ,

(4)

86 | S. Mejdi et al. f c11 ⋅ ⋅ ⋅ .. .. . . ⋅⋅⋅ [ 0 fc1 (k) [ .. [ . [ [ (k) f fc (k) = [ [ cj [ .. [ . [ [ Fc = [ [

[

fcp (k)

0 .. . f cpp

] ] ∈ ℜp×p ]

and

]

] ] ] ] ] ∈ ℜp represents the sensor faults vector. ] ] ] ] ]

Assuming that one fault appears at a time when each output can be affected by a sensor fault [1, 3, 16], the uncoupled state multimodel can be rewritten as follows: {

xcf (k + 1) = Acf xcf (k) + Bcf u(k) + Wcf w(k) yMM (k) = Ccf (k)xcf (k) + Fcj fcj (k) + F̄cj fc̄ j (k)

(5)

where: – Fcj is jth column of the matrix Fc . – F̄cj is a new created sensor fault distribution matrix from Fc when its jth column is null and f ̄ denotes the new sensor fault vector extracted from f (k) when its jth row is null.

c

cj

This new state representation (5) is equivalent to (4). The dynamic of a sensor fault can be described as follows [7, 16]: fcj (k + 1) = fcj (k) + Tuc (k),

∀j = 1 . . . p

(6)

where uc (k) is the sensor error input. T represents the sampling time and j ∈ {1 . . . p} represents the number of sensor faults. The state representation given by equation (5) is no longer appropriate for discrete unknown input multiobserver synthesis dedicated to detect and isolate sensor faults. To overcome this problem, a new augmented multimodel representation can be defined [1, 2, 7, 16]: x (k) x (k + 1) A on×1 B { { ] = [ cf ] + [ cf ] u(k) ] [ cf [ cf { { (k) f f + 1) (k o 1 0 { c c 1×n j j { { { { on×1 Wcf +[ ] uc (k) + [ ] w(k) { { T 0 { { { { x (k) { { { ] + F̄cj fc̄ j (k) yMM (k) = [ Ccf (k) Fcj ] [ cf fcj (k) {

(7)

Let us define the new augmented state vector [7]: x̄cf (k) = [

xcf (k) ] ∈ ℜn+1 fcj (k)

(8)

Detection and Isolation of Sensor Faults for Nonlinear Systems | 87

The new state representation of the new faulty system, introducing the new matrices, can be expressed as: {

̄ c (k) + W̄ cf w(k) x̄cf (k + 1) = Ā cf x̄cf (k) + B̄ cf u(k) + Tu ̄ yMM (k) = Ccf (k)x̄cf (k) + F̄cj fc̄ j (k)

(9)

Where: A Ā cf = [ cf o1×n

on×1 ] ∈ ℜ(n+1)×(n+1) , 1

B B̄ cf = [ cf ] ∈ ℜ(n+1)×m , 0 C̄ cf (k) = [ Ccf (k)

Fcj ] ∈ ℜp×(n+1) ,

W W̄ cf = [ cf ] ∈ ℜ(n+1)×d 0 and

o T̄ = [ n×1 ] ∈ ℜ(n+1)×1 . T

The matrix C̄ cf (k) has time varying parameters [3, 14, 19]. Taking into account the properties of the convex sum, it is rewritten as a weighted sum of matrices: Nm

̃̄ C̄ cf (k) = ∑ μj (νk−1 )C cf j j=1

(10)

̃̄ is a block matrix in the form: where C cfj ̃̄ = [ 0 C cfj

⋅⋅⋅

C̄ j

⋅⋅⋅

0 ] ∈ ℜp×n

With this new state representation, the detection and isolation of sensor fault is treated as an actuator fault detection and isolation [1, 2, 7, 10, 16]. According to (9), the discrete unknown input multiobserver has the following new compact structure: ̄ (k) + Ḡ cf u(k) + L̄ cf yMM (k) z̄ (k + 1) = N̄ cf zcf { { cf ̄ (k) − Ē cf yMM (k) x̂ (k) = zcf { { cf ̂ ̄ { ŷMM (k) = Ccf (k)xcf (k)

(11)

̄ (k) ∈ ℜn+1 is the state of the multiobserver and x̂ cf (k) ∈ ℜn+1 is the estimated Where zcf state vector. The matrices N̄ cf ∈ ℜ(n+1)×(n+1) , L̄ cf ∈ ℜ(n+1)×p , Ḡ cf ∈ ℜ(n+1)×m and Ē cf ∈ ℜ(n+1)×p . Substituting x̂ (k) and y (k) by their expressions, the state estimation error is given as:

cf

MM

̄ (k) + Ē cf F̄cj fc̄ j (k) ēx (k) = (In+1 + Ē cf C̄ cf (k)) x̄cf (k) − zcf where In+1 is an identity matrix of dimension n + 1.

(12)

88 | S. Mejdi et al. Posing P̄ cf (k) = In+1 + Ē cf C̄ cf (k), P̄ cf (k) ∈ ℜ(n+1)×(n+1)

(13)

The expression of the estimation error is equivalent to: ̄ (k) + Ē cf F̄cj fc̄ j (k) ēx (k) = P̄ cf (k)xcf (k) − zcf

(14)

Thereafter, the dynamic of estimation error is described as: ēx (k + 1) = N̄ cf ēx (k) + [P̄ cf (k + 1)Ā cf − N̄ cf − K̄ cf C̄ cf (k)] x̄cf (k) + [P̄ cf (k + 1)B̄ cf − Ḡ cf ] u(k) + P̄ cf (k + 1)W̄ cf w(k) ̄ c (k) − K̄ cf F̄c fc̄ (k) + Ē cf F̄c fc̄ (k + 1) + P̄ cf (k + 1)Tu j j j j

(15)

K̄ cf is defined as K̄ cf = L̄ cf + N̄ cf Ē cf K̄ cf ∈ ℜ(n+1)×p

(16)

The following conditions have to be fulfilled after decoupling the dynamic of the estimation error from the state, the input of system and the sensor error input. P̄ (k + 1)Ā cf − N̄ cf − K̄ cf C̄ cf (k) = 0 { { cf P̄ (k + 1)B̄ cf − Ḡ cf = 0 { { ̄ cf ̄ { Pcf (k + 1)T = 0

(17)

Then, the dynamic of the estimation error is often established: ēx (k + 1) = N̄ cf ēx (k) + P̄ cf (k + 1)W̄ cf w(k) − K̄ cf F̄cj fc̄ j (k) + Ē cf F̄cj fc̄ j (k + 1)

(18)

From the evolution of the estimation error, it can be seen, in presence of disturbances, that all sensor faults are totally detected except the jth one. For the existence of the discrete Unknown Input MultiObserver (11), it is necessary and sufficient to check the following conditions of Theorem 1. Theorem 1. The multiobserver (11), which is an UIMO for the uncoupled state multimodel defined in (9), exists [13, 16, 22] if: ̃̄ T)̄ = rank(T), ̄ ∀i = 1 . . . Nm; – rank(C cfi ̄ – T is full rank. The analysis of the multiobserver stability, based on the Lyapunov approach, is ensured in terms of LMIs [4]. It provides the exponential convergence of the estimation error to zero. Then, the existence of a symmetric positive definite matrix X̄ = X̄ T > 0 satisfying Theorem 2 is sufficient.

Detection and Isolation of Sensor Faults for Nonlinear Systems | 89

Theorem 2. The state estimation error is exponentially convergent towards zero, if there exists a symmetric positive definite matrix X̄ = X̄ T > 0, S̄cf and W̄ ccf of appropriate dimensions satisfying the following LMIs [4, 14]: [

(1 − 2α) X̄ (X̄ Ā cf + θ1 Ā cf − φ1 )

(X̄ Ā cf + θ1 Ā cf − φ1 ) X̄

T

]>0

X̄ T̄ + S̄cf C̄ cfi T̄ = 0

(19) (20)

∀i, j = 1 . . . Nm

where ̃̄ θ1 = S̄cf C cfi ̃̄ φ1 = W̄ ccf C cf

j

These inequalities have solutions for a chosen decay rate, 0 < α < 0.5 Proof. To study the exponential convergence of the estimation errors, we use the second method of Lyapunov. In this case, the exponential stability is established if there exists a Lyapunov function V(k) > 0 such that: ∃X̄ = X T̄ > 0;

α > 0;

ΔV(k) + 2αV(k) < 0

(21)

with ΔV(k) = V(k + 1) − V(k)

(22)

The Lyapunov function, which guarantees the convergence towards zero of the estimation errors, is defined by: V(k) = ēx (k)T X̄ ēx (k)

(23)

Including (18) and (22), inequality (21) is equal to: ēx (k)T {

T [P̄ cf (k + 1)Ā cf − K̄ cf C̄ cf (k)] X̄ } ēx (k) < 0 [P̄ cf (k + 1)Ā cf − K̄ cf C̄ cf (k)] + (2α − 1) X̄

(24)

Then, it is necessary to verify: T [P̄ cf (k + 1)Ā cf − K̄ cf C̄ cf (k)] X̄ [P̄ cf (k + 1)Ā cf − K̄ cf C̄ cf (k)] + (2α − 1) X̄ < 0

(25)

90 | S. Mejdi et al. Taking into account (2), (10) and (13) the previous inequality becomes: T

̃̄ ̃̄ Ā − K̄ C ̄ [Ā cf + Ē cf C cf cfj ] X cfi cf ̃̄ ̃̄ Ā − K̄ C ̄ [Ā cf + Ē cf C cf cfj ] + (2α − 1) X < 0 cfi cf ̃̄ T̄ = 0 T̄ + Ē cf C cfi

(26)

∀i, j = 1 . . . Nm Posing: W̄ ccf = X̄ K̄ cf ,

S̄cf = X̄ Ē cf

(27)

Then, it can be easy to have: ̃̄ T ̄ −1 ̃̄ Ā − W̄ C (1 − 2α) X̄ − [X̄ Ā cf + S̄cf C ccf cfj ] X cfi cf ̃̄ ̃̄ Ā − W̄ C [X̄ Ā cf + S̄cf C ccf cfj ] > 0 cfi cf

(28)

̃̄ T̄ = 0 X̄ T̄ + S̄cf C cfi ∀i, j = 1 . . . Nm

Using the Schur complement applied to the inequality (28), the following LMI(s) are obtained: [

(1 − 2α) X̄ ̄ ̄ (X Acf + θ1 Ā cf − φ1 )

(X̄ Ā cf + θ1 Ā cf − φ1 ) X̄

T

]>0

(29)

The resolution of the generated LMIs permit to compute the multiobserver gains: { { { { { { { { { { { { {

K̄ cf Ē cf Ḡ cf N̄ cf L̄ cf

= X̄ −1 W̄ ccf = X̄ −1 S̄cf = (In+1 + Ē cf C̄ cf (k + 1)) B̄ cf = (In+1 + Ē cf C̄ cf (k + 1)) Ā cf − K̄ cf C̄ cf (k) = K̄ cf − N̄ cf Ē cf

(30)

Since the residuals are generated (the faults detection is accomplished), it is task to determine or locate which sensor is faulty [17]. The simplest evaluation of residuals allows the faults isolation. Structured residuals generated by a bank of “s” multiobservers (s ∈ {1 . . . p}), where each one is driven by one of the considered j outputs, are exploited for faults isolation. The well known Dedicated Observer Scheme (DOS), which is used for faults isolation, can be designed to generate structured residuals sensitive to a special sensor fault and insensitive to all other ones. In this case, each multiobserver “s” is driven by all inputs and the jth system output.

Detection and Isolation of Sensor Faults for Nonlinear Systems | 91

For each sensor, a discrete unknown input multiobserver is synthesized to detect and isolate faults. Once the jth sensor fault occurs, the localization of the jth sensor fault is established if all residuals are different from zero except its relative residual. A structured residual vector of the sth multiobserver, relative to the sensor fault detection and isolation, RIs,j (k) is defined: [ [ [ [ RIs,j (k) = [ [ [ [ [ [

rIs,1 (k) .. . rIs,j (k) .. . rIs,p (k)

] ] ] ] ] ∈ ℜp ] ] ] ] ]

It is expressed as: RIs,j (k) = yMMj (k) − ŷMMj (k) = C̄ cf (k)ēx (k) + F̄cj fc̄ j (k)

(31)

Thereafter: RIs,j (k + 1) = C̄ cf (k + 1)N̄ cf ēx (k) − C̄ cf (k + 1)K̄ cf F̄cj fc̄ j (k) + C̄ cf (k + 1)P̄ cf (k + 1)W̄ cf w(k) + C̄ cf (k + 1) Ē cf F̄cj fc̄ j (k + 1) + F̄cj fc̄ j (k + 1)

(32)

We note, from (32), that the generated residuals relative to sensor fault j is sensitive to the disturbance and the others sensor faults. Moreover, the p components of the residual vector RIs,j (k) react in the same way when the jth sensor fault is occur. In this condition, we define a binary logic variable RI,s (k) characterizing the logic evolution of residuals which performs the sensors isolation. To determine which sensor is in fault, after ensuring the decoupling conditions by referring to (18) and (32), the localization task of the sth sensor is achieved if: {

RI,j (k) ≠ 0 RI,j (k) = 0

for j ≠ s for j = s

The evolution of the structured residual RIs,j (k) indicates that the proposed multiobserver is able to isolate sensors regardless of the presence of disturbances. Thus, the disturbances enact the isolation task difficult causing false alarms. Then, the goal is to discriminate between disturbances and sensor faults to locate successfully the faulty sensors. To overcome this problem, a robust unknown multiobserver is synthesized to completely decouple the impact of disturbances with no need to thresholds.

92 | S. Mejdi et al.

3 Robust Sensor Faults Detection Against Disturbances Any practical system can be normally affected by disturbances and other unknown inputs [11] which is why it is essential that a reliable and a robust fault diagnosis must be achieved taking these effects into consideration. The aim of the robustness is to differentiate between faults and external disturbances. To fulfill a successful robust fault detection in the presence of disturbances, it is indispensable to use the decoupling principle [8] that can provide a residual insensitive to disturbances and sensitive to all occurring faults. In this section, a robust sensor faults detection is achieved based on the proposed unknown input multiobserver.

3.1 Design of an Unknown Input MultiObserver: Robustness Against Disturbances The reconstruction of the multimodel states and the outputs can be achieved by the proposed discrete multiobserver using a decoupling condition. We used the unknown input multiobserver which is considered the best generator of residuals and a the suitable solution to systems with unknown inputs [12]. This provides additional decoupling techniques compared to the classical ones. Based on the uncoupled state multimodel, given by equation (4), a discrete unknown input multiobserver, under a compact form structure, can be expressed as follows [4, 13, 14, 22]: z (k + 1) = Ncf zcf (k) + Gcf u(k) + Lcf yMM (k) { { cf x̂ (k) = zcf − Ecf yMM (k) { { cf y { ̂MM (k) = Ccf (k)x̂cf (k)

(33)

The estimation error can be defined by the following expression: ex (k) = xcf (k) − x̂cf (k)

(34)

Referring to (4) and (33) the error state estimation is expressed as follows: ex (k) = Pcf xcf (k) − zcf (k) + Ecf Fc fc (k)

(35)

Substituting (4) and (33), the dynamic state estimation error is obtained: ex (k + 1) = Ncf ex (k) + [Pcf (k + 1)Bcf − Gcf ] u(k)

+ [Pcf (k + 1)Acf − Ncf − Kcf Ccf (k)] xcf (k) + Pcf (k + 1)Wcf w(k) − Kcf Fc fc (k)

+ Ecf Fc fc (k + 1)

(36)

Detection and Isolation of Sensor Faults for Nonlinear Systems | 93

Ensuring the decoupling property of the estimation error from the state, the nonlinear system input and the disturbance, it is necessary to verify the conditions: P (k + 1)Acf − Ncf − Kcf Ccf (k) = 0 { { cf P (k + 1)Bcf − Gcf = 0 { { cf { Pcf (k + 1)Wcf = 0

(37)

If the previous conditions (37) hold, equation (36) becomes: ex (k + 1) = Ncf ex (k) − Kcf Fc fc (k) + Ecf Fc fc (k + 1)

(38)

The solution of this equation Pcf (k + 1)Wcf = 0 depends on the existence of Ecf [13, 14] under the conditions given in Theorem 3. Theorem 3. The multiobserver (33) is an UIMO for the uncoupled state multimodel defined in (4) and exists [13, 22] if: – rang(C̃ cfi Wcf ) = rang(Wcf ), ∀i = 1 . . . Nm; – Wcf is full rank. To provide exponential convergence of the estimation error near zero a particular study, based on the Lyapunov approach, is developed in terms of Linear Matrix inequalities LMI(s) described in Theorem 4. Theorem 4. The states estimation error converges exponentially to zero, if there exists a symmetric positive definite matrix X = X T > 0, Scf and Wccf of appropriate dimensions satisfying the following LMIs [14]: [

(1 − 2α) X (XAcf + θAcf − φ)

T

(XAcf + θAcf − φ) X

]>0

(39)

XWcf + Scf C̃ cfi Wcf = 0,

(40)

θ = Scf C̃ cfi φ = Wccf C̃ cfj

(41)

∀ i, j = 1 . . . Nm

where: {

The solution of these inequalities exist for a given rate decay, 0 < α < 0.5 Then, the resolution of the Linear Matrix Inequalities allows to determine the multiobserver gains: { { { { { { { { { { { { {

Kcf Ecf Gcf Ncf Lcf

= X −1 Wccf = X −1 Scf = (In + Ecf Ccf (k + 1)) Bcf = (In + Ecf Ccf (k + 1)) Acf − Kcf Ccf (k) = Kcf − Ncf Ecf

(42)

94 | S. Mejdi et al. Let us define the residual vector relative to robust detection: [ [ [ [ RD,j (k) = [ [ [ [ [ [

rD,1 (k) .. . rD,j (k) .. . rD,p (k)

] ] ] ] ] ∈ ℜp ] ] ] ] ]

which can be expressed by: RD,j (k) = yMMj (k) − ŷMMj (k) = Ccf (k)ex (k) + Fc fc (k)

(43)

Thereafter: RD,j (k + 1) = Ccf (k + 1)Ncf ex (k) + Fc fc (k + 1) + Ccf (k + 1)Ecf Fc fc (k + 1) − Ccf (k + 1)Kcf Fc fc (k)

(44)

From (44), it can be concluded that the residual depend only on all the sensor faults. We can note RD (k) the binary logic variable which is set to “1” if the robust detection is achieved at the occurrence of the jth sensor fault.

4 Numerical Example Consider the discrete-time Single-Input Multiple-Output (SIMO) nonlinear system [11] subject to sensor faults defined by the following matrices. −0.4 [ A1 = [ 0.3 [ 0.4

0.2 −0.6 0.2

0.4 [ ] W1 = [ −0.4 ] , [ 0.4 ] −0.45 0.375 [ A2 = [ 0.15 −0.45 0.75 [ 0.75 0.4 [ ] W2 = [ 0.4 ] , [ −0.4 ]

0.3 ] 0.3 ] , 0.6 ]

1 [ ] B1 = [ −0.5 ] , [ −0.5 ]

0.375 ] 0 ], 0.75 ]

0 F̄c1 = [ 0

0 ] 1

C1 = [

−0.5 ] 1 ], [ −0.5 ]

[ B2 = [

−1 and F̄c2 = [ 0

0 1

1 0

C2 = [ 0 ]. 0

0 1

1 ], 1

1 0

1 ], 1

Detection and Isolation of Sensor Faults for Nonlinear Systems | 95

Two sudden sensor faults occur at different instants. The first sensor fault appears at 1000 < k < 1200, however the second sensor fault is considered at time 1600 < k < 1800. To get a robust detection, a multiobserver driven by the system input and the two outputs is designed which generate two residuals rD,1 (k) and rD,2 (k) completely decoupled from disturbances. Figure 1 illustrates the evolution of the generated residuals acting in the same way.

Figure 1: Robust generated residuals against disturbances.

Due to the disturbances decoupling principle of the unknown input multiobserver, it can be easily seen in figures 2, 3, 4, 5, 6 and 7 a good accuracy of the estimation errors. It can be seen that the proposed robust unknown input multiobserver performs a good quality of estimation. Indeed, it provides estimation errors not affected by the disturbances. The robust unknown input multiobserver provides good accuracy even that disturbances appear in the model. The scheme represented in fig. 8 is used for a disturbed nonlinear system with two outputs (p = 2). We have to synthesize a bank of two multiobservers (s = 2). Each of the two multiobservers uses one of the two outputs. This bank allows to generate four residuals. The first multiobserver of the bank detect the second sensor fault and the second multiobserver can detect the first sensor fault.

96 | S. Mejdi et al.

Figure 2: Partial model 1: Evolution of the estimation error of the state 1.

Figure 3: Partial model 1: Evolution of the estimation error of the state 2.

Figure 4: Partial model 1: Evolution of the estimation error of the state 3.

Figure 5: Partial model 2: Evolution of the estimation error of the state 1.

Detection and Isolation of Sensor Faults for Nonlinear Systems | 97

Figure 6: Partial model 2: Evolution of the estimation error of the state 2.

Figure 7: Partial model 2: Evolution of the estimation error of the state 3.

Figure 8: Sensor Faults Detection and Isolation scheme.

Each multiobserver generates two similar residuals. Thereafter, we have to use only one residual of each multiobserver. We can use here the binary logic variable RD (k) and RI,s (k). Therefore, three residuals are presented here when the two sensors faults occur.

98 | S. Mejdi et al.

Figure 9: Generated residuals.

Figure 9 shows the evolution of the residuals generated by the multiobserver reserved to robust detection of all sensor faults and by a bank of the proposed discrete unknown input multiobservers. Simulation results show that the proposed scheme is successful in detecting and isolating both sensor faults. It can be seen in figure (9.a), all sensor faults are detected. Furthermore, we can conclude that the faults are robustly detected even in the presence of disturbances. Indeed, we can remark that the first residual is decoupled from the impact of disturbance. The residual rD,1 (k) shows that the detection is successfully

Detection and Isolation of Sensor Faults for Nonlinear Systems | 99

achieved regardless of the presence of disturbances. This can inform about the presence of faults and there is faults in system. Then, the main goal is to determine their sources. At k = 1000, by comparison with rD,1 (k) the residual rI2,1 (k) has not change only rI1,1 (k) which stay at zero, so the first sensor, given by the second multiobserver (first Multiobserver of the bank s = 1), is in fault. We can note that the first sensor is isolated. At k = 1600, we can see that rI1,1 (k) leaves zero while the rI2,1 (k) becomes equal to zero which explains that the second sensor is faulty. Since the disturbance is presented and when no faults have been occurred, it can note false alarms [16, 17]. However, with the robust detection already achieved using the discrete unknown input multiobserver the faults isolation is easily done without using any threshold. Moreover, it is necessary to make decision logic to decide if a fault is occurring. At first, from the robust detection residual rD,1 (k), we are sure that sensor faults are detected. We can note that the binary logic variable RD (k) = 1. Then, a final decision is made to decide which sensor is faulty. After a simple comparison, since the first and the second sensor faults are applied, between rI1,1 (k) and rI2,1 (k), let’s use the binary logic variable. Indeed: R (k) = 1 { { D R (k) = 0 { { I,1 { RI,2 (k) = 1 It is clear that the first sensor, which is faulty, is isolated. The detection and isolation of the second sensor is achieved if: R (k) = 1 { { D R (k) = 1 { { I,1 { RI,2 (k) = 0 By using the robust detection and the proposed bank of multiobservers, it is precisely that the sensor faults are totally detected and completely localized. For more precision, a signature table 1 associated to the proposed scheme of the discrete multiobservers is elaborated to characterize the residuals binary evolution. Table 1: Logic signature table. Detection and Isolation sensor fault “1” sensor fault “2”

UIMO1 RD (k)

UIMO2 RI,1 (k)

UIMO3 RI,2 (k)

1 1

0 1

1 0

100 | S. Mejdi et al.

5 Conclusion In this paper, sensor faults detection and isolation for disturbed nonlinear systems is proposed. A discrete uncoupled state multimodel is retained to represent the considered systems. A discrete Unknown Input MultiObservers based on the proposed structure are developed and the convergence of the estimation error is often established by a LMI(s) formulation. Structured residuals generated by a bank of the proposed multiobservers are exploited for a robust detection, against disturbances, and for isolation of sensor faults. A numerical example is realized to show good performances in terms of precision, robust detection in presence of external disturbances, and reliable isolation.

Bibliography [1]

D. Fragkoulis, G. Roux and B. Dahhou. Sensor fault detection and isolation for single, multiples and simultaneous faults: Application to a waste water treatment process. IFAC Proceedings, (42)11:934–939, 2009. [2] D. Fragkoulis, G. Roux and B. Dahhou. A global scheme for multiple and simultaneous faults in system actuators and sensors. In 6th International Multi-Conference on Systems, Signals and Devices, 2009. [3] D. Ichalal, B. Marx, J. Ragot and D. Maquin. Diagnostic des systèmes non linéaires par approche multimodèle. Workshop Surveillance, Sûreté et Sécurité des Grands Systèmes, 3SGS’08, Jun 2008. [4] D. Ichalal, B. Marx, J. Ragot and D. Maquin. Multi-observateurs à entrées inconnues pour ̀ un système de Takagi-Sugeno à variables de décision non mesurables. In 5eme Confèrence Internationale Francophone d’Automatique, CIFA’2008, Bucharest, Roumanie, September 2008. [5] G. Paviglianiti and F. Pierri. Sensor Fault Detection and Isolation for Chemical Batch Reactors. In International Conference On Control Applications, Munich, Germany, October 4–6, 2006. [6] I. Hwang, S. Kim, Y. Kim, and C. E. Seah. A survey of fault detection, isolation, and reconfiguration methods. IEEE Transactions on Control Systems Technology, (18)3:636–653, 2010. [7] J. Park, G. Rizzoni and W B. Ribbens. On the representation of sensor faults in fault detection filters. Automatica, (30)11:1793–1795, 1994. [8] J. Zarei and J. Poshtan. Sensor fault detection and diagnosis of a process using unknown input observer. Mathematical and Computational Applications, (16)1, 2011. [9] K. Bouibed, L. Seddiki, K. Gueltonb and H. Akdaga. Actuator and sensor fault detection and isolation of an actuated seat via nonlinear multi-observers. Systems Science and Control Engineering: An Open Access Journal, (2)1:150–160, 2014. [10] K. Zhang, S. Hu and B. Jiang. Sliding mode integral observers for sensor faults detection and isolation in nonlinear systems. In IEEE International Conference on Control and Automation, Guangzhou, China, May 30–1 June, 2007. [11] M. Chadli and H. R. Karimi. Robust observer design for unknown inputs Takagi-Sugeno models. IEEE Transactions on Fuzzy Systems, (21)1, February 2013.

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[12] M. Darouach. On the novel approach to the design of unknown input observers. IEEE Transactions on Automatic Control, (39), Mar 1994. [13] M. Darouach, M. Zasadzinski and S. J. Xu. Full-order observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, (3), 1994. [14] M. Allaoui. Des multiobservateurs à entrées inconnues et une commande prédictive-répétititve multimodèle pour les systèmes non linéaires PhD thesis, National Engineering School of Gabes, 2017. [15] M. Allaoui, A. Messaoud and R. Ben Abdennour. Robust design of non-stationary sinusoidal unknown inputs multiobserver based on decoupled state multimodel. In 15th international conference on Sciences and Techniques of Automatic control and computer engineering STA’2014, 2014. [16] M. H. Sobhani and J. Poshtan. Fault detection and isolation using unknown input observers with structured residual generation. International Journal of Instrumentation and Control Systems (IJICS), (2)2, April 2012. [17] M. H. Sobhani and J. Poshtan. Observer-based fault detection and isolation of three-tank benchmark system. In 2nd Conference on Control, Instrumentation,and Automation, Shiraz, Iran, 26–28 Nov, 2011. [18] R. N. Clark. Instrument fault detection. IEEE Transactions on Aerospace and Electronic Systems, May 1978. [19] R. Orjuela, B. Marx, J. Ragot and D. Maquin. Fault diagnosis for nonlinear systems represented by heterogeneous multiple models. In Conference on Control and Fault-Tolerant Systems, SysTol 2010, Nice, France, 2010. [20] S. Ben Atia, A. Messaoud and R. Ben Abdennour. Robust multiobserver design for discrete uncertain nonlinear systems with time-varying delay. Transactions of the Institute of Measurement and Control, TIMC, July 21, 2016. [21] W. Chen and M. Saif. An actuator fault isolation strategy for linear and nonlinear systems. In American Control Conference, June 8–10, Portland, OR, USA, 2005. [22] Y. Shan-Ju, C. Wei and W. Wen-June. Unknown input based observer synthesis for uncertain Takagi- Sugeno fuzzy system. IET Control Theory and Applications, (9)5:729–735, 2015. [23] Z. Gao, C. Cecati and S. X. Ding. A survey of fault diagnosis and fault- tolerant techniques Part I: Fault diagnosis with model based and signal based approaches. IEEE Transactions on Industrial Electronics, (62)6:3757–3767, 2015.

Biographies Sondess Mejdi received her Engineering Diploma in Electrical and Automatic Engineering, in July 2016, from National School of Engineers of Gabes-Tunisia. Currently, she is a Ph. D. candidate at Research Unit: CONPRI (Numerical Control of Industrial Processes). Her areas of interest deal with faults diagnosis, multimodel, unknown inputs multiobservers and representation of complex systems.

102 | S. Mejdi et al.

Anis Messaoud received the Engineering and Master degrees in electrical Engineers and automatic control from National School of Engineers of Gabes-Tunisia, in 2004 and 2006 respectively. In 2010, he obtained his Ph. D. degree in electrical-automatic engineering from the National School of Engineers of Gabes, Tunisia. His specific research interests are in the area predictive control of complex systems, Multimodel and Multicontrol approaches. He is currently Associate Professor in the Electric Engineering Department in National School of Engineers of Gabes-Tunisia.

Mouhib Allaoui received his Engineering Diploma in Electrical and Automatic Engineering, in June 2012, and the Master degree in Automatic Control and Intelligent Techniques, in December 2012, from National School of Engineers of Gabes-Tunisia. In 2017, he obtained his Ph. D. degree in electrical-automatic engineering from the National School of Engineers of Gabes, Tunisia. Currently, he is a Ph. D. researcher at Research Unit -CONPRI (Numerical Control of Industrial Processes) and a Contractual Assistant Professor at the Higher Institute of Sciences and Energy Technology of Gafsa. His areas of interest deal with multimodel and supervised multicontrol approaches, unknown inputs multiobservers and representation of complex systems. Ridha Ben Abdennour received his PhD (Doctoral degree) in Automatic Control from Higher School of Technical Education in 1987, and his State Doctoral degree in Electrical Engineering from the National School of Engineers of Tunis-Tunisia, in 1996. He is Professor in Automatic Control at the National School of Engineers of Gabes-Tunisia. He is the founder and He was the Head of the Electrical Engineering Department and the Director of the High Institute of Technological Studies of Gabes. Ridha BEN ABDENNOUR is the Head of the Research Unit of Numerical Control of Industrial Processes and he is the founder and the honorary President of the Tunisian Association of Automatic Control and Digitalization. His research is on Identification, Multimodel and Multicontrol approaches, Neural Networks, Observers, Sliding mode, Numerical Control and Fuzzy supervision. He is the co-author of a book on Identification and Numerical Control of Industrial Processes and he is the author of more than 350 publications. Ridha BEN ABDENNOUR participated in the organization of many Conferences and he was member of some scientific committees of congresses.

Amina Ben Hmed, Messaoud Amairi, and Mohamed Aoun

Stabilizing Fractional Order Integrator Design for Integrating and Unstable First Order Plants Abstract: The paper deals with the stabilization problem of the first order system with and without integration. In this work, we present a simple stabilization method to guarantee stability and desired performances. Analytic expressions are developed to determine the stabilizing controller parameters by describing the stability region. Moreover, the time domain and the frequency curves of the non-commensurate fractional system are used to fulfil the desired specifications. Some numerical examples are presented to show the useful and the reliability of the proposed method. Finally, the fractional integrator controller can be used combined with Smith Predictor to control a first order system with time delay and to achieve some desired specifications. Keywords: Fractional calculus, control design, fractional controller, stabilization, Smith Predictor

1 Introduction Recently, fractional order calculus has gained a considerable importance in various fields as physics and engineering (see [1–3] and references therein for more details). In fact, the dynamic behavior of many physical systems can be described by a fractional order system theory which has been used in several applications [4], such as viscoelasticity, diffusion, modeling, and control [5]. Proportional-integral-derivative (PID) controllers still the most adopted controllers in industry [6]. In spite of this, there are some limitations in control design as in the trade-off between stability and rapidity. In fact, to obtain a suitable performances, many tuning method has been proposed [7–9]. Recently, Podlubny [10] has proposed a generalisation of the conventional PI and PID controllers, namely the PI α , PDβ and PI α Dβ controllers by introducing two extra freedom degrees α the order of the integrator and β the order of differentiator. In control design process, there are new effective tuning methods for fractional PID controller given in several research activities [10] where an easier achieving of control requirements and flexible tuning are provided. Moreover, in case of rational system, a better flexibility in parameters adjustment can be achieved with the fractional conAcknowledgement: This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia. Amina Ben Hmed, Messaoud Amairi, Mohamed Aoun, University of Gabès, Gabès, Tunisia, e-mails: [email protected], [email protected], [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 103–122. https://doi.org/10.1515/9783110591729-006

104 | A. Ben Hmed et al. troller. In addition, Xue et al. has demonstrated that the fractional controller outperforms the classical PID one in terms of stability and control specifications [11]. Processes encountered in the industry are frequently unstable. Accordingly, many frameworks deals with the tuning of controller for unstable and integrating systems [12–14]. A tuning rules for fractional PI controller is proposed in [15] to obtain a tradeoff between dynamic performances and stability for first order system with integration. Moreover, in [16] a useful stabilization method to obtain a stabilizing fractional-order PID controllers for a given fractional-order time delay system is given. This method can be also used to guarantee only frequency specifications as gain and phase margins. In this work, we propose a simple design of a stabilizing fractional order integrator controller for first order systems with and without integration. In which many industrial processes can be modelled by these systems see for example [17, 16]. The designed I α controller provide a stable closed-loop with some desired time-domain and frequency specifications. By using Smith Predictor, this method is used to control a first order system with time delay. The rest of the paper is organized as follows. Section 2 presents the problem statement and introduces some important results concerning stability of non-commensurate fractional order system. The design of a fractional I α controller to stabilize and control first order system is presented in Section 3. Then, this method is extended to control a first order system with time delay. Section 4 is devoted to stabilize and control a first order system with integration. An application to DC-Motor is done to illustrate the proposed method.

2 Problem Statement Consider the unity-feedback control scheme presented in Fig. 1.

Figure 1: Unity Output Feedback Control System.

C(s) and G(s) represent respectively the controller and the plant transfer function. The signals y and yr denotes respectively the reference input and the output. The considered controller is a simple fractional order integrator I α described by C(s) =

ki , sα

where ki ∈ ℝ∗ and 0 < α < 2 are respectively the gain and the fractional order.

(1)

Stabilizing Fractional Order Integrator Design

| 105

The objective is to design a stabilizing fractional order integrator (1) based on a desired closed-loop system described by Fd (s) =

ωα+1 n , sα+1 + 2ξ ωn sα + ωα+1 n

(2)

where ωn , ξ and α represent respectively the natural frequency, the pseudo-damping factor and the non-commensurate fractional order (ωn > 0, ξ ∈ ℝ and 0 < α < 2). The desired closed-loop system (2) is called non-commensurate fractional order system of the second kind and it is studied in [18, 19]. Therefore, stability, resonance conditions and several time-domain and frequency curves are presented to achieve a desired behavior. Theorem 1. The desired closed-loop transfer function given by (2) is stable if and only if one of the following conditions is satisfied [18]: – 0 < α ⩽ 1 and ξ ⩾ 0.

–

0 < α < 1, ξ < 0 and

–

1 < α < 2, ξ > 0 and

−2α+1 ξ ))α < 1. (−ξ tan( απ 2 cos( απ ) 2 2α+2 4ξ 2 ω2n ω2α − ω2α+2 u + ωu n

). > 0 with ωu = 2ξ ωn tan( (2−α)π 2

According to Theorem 1, three resonance and stability regions of the desired closed-loop system Fd (s) are determined as shown in Fig. 2. Region A represents combinations of α and ξ for no resonant systems. Region B represents combinations of α and ξ for systems with only one resonance and Region C defines values of ξ and α for unstable systems.

Figure 2: Stability and resonance regions of the desired closed-loop system Fd (s).

106 | A. Ben Hmed et al. In this paper the stability of an unstable first order system with and without integration using a fractional I α controller is treated. The proposed controller must fulfil some time-domain performances in terms of the settling time, the first overshoot and the steady state error, and some frequency specifications as the gain in dB and phase at resonance.

3 I α Controller Design for Stabilization and Control of a First Order System 3.1 Stabilization of an Unstable First Order System In the following, a simple method is presented to obtain all stabilizing values of the parameters controller I α , namely ki and α, in the (ki , α)-space. The adopted unity feedback control scheme is presented in Fig. 1. The objective is tuning parameters controller to stabilize and control a first order system described by G(s) =

b , s+a

(3)

where b and a are respectively the gain and the pole of G(s). The obtained closed-loop system is given by F(s) =

ki b sα+1 + asα + ki b

(4)

Based on the desired transfer function (2), the closed-loop parameters are given by {

a = 2ξ ωn bki = ωα+1 n

Hence, if ωn = 1 rad/s, relation (5) gives ki = expressed in terms of ki , ωn and b as follows α=

1 b

ln(bki ) − 1, ln(ωn )

(5)

∀α, else the fractional order α can be ωn ≠ 1

(6)

Referring to relation (5), the stability and resonance regions presented in Fig. 2 can be given in the 2ωa versus α. Therefore, when a < 0, the unstable system G(s) can be n stabilized only in the hatched region as shown in Fig. 3. Moreover, from theorem (1), the closed-loop system F(s) is stable when 0 < α < 1 and a < 0 if condition (7) is verified. −a2α+1 1

(

−a

1

) 2(ki b) α+1 2(ki b) α+1 cos( απ 2

tan(

α

απ )) < 1 2

(7)

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| 107

Figure 3: Stability and resonance regions of the closed-loop system F (s) when 0 < α < 1.

Remark 1. When a ⩾ 0 (i. e. ξ ⩾ 0) and 0 < α ≤ 1 the stability region shows that the closed-loop system is always stable. – –

From the previous study, the stabilization steps are as follows: If a ⩾ 0, then the closed-loop system is always stable so no stabilization problem. If a < 0, then the I α is designed to stabilize the closed loop system with 0 < α < 1 by following steps: – Determine the stabilizing region (ki , α) from relation (7). – Choose an appropriate couple (ki , α) to stabilize the loop.

For more illustration, we consider an unstable first order system without integration described by 1 s−1

(8)

ki sα+1 − sα + ki

(9)

G(s) = The closed-loop system is presented by F(s) =

In this case, it is important to choose a fractional order 0 < α < 1 as indicated above. Then, applying relation (7), the stability limit of the closed-loop system (9) is determined numerically in terms of ki and α and plotted in Fig. 4. To ensure that the closed-loop system is stable, it is important to choose a gain ki above the stability limit for each fractional order α. For example, to stabilize the system (8), we can choose a fractional order α = 0.75 and a gain ki > 5.08. To verify the stability of the closed loop system (9) two arbitrary cases: ki = 4 and ki = 6 are taken. The Nyquist diagrams of the open-loop system β(s) = are plotted in Fig. 5.

ki sα (s − 1)

(10)

108 | A. Ben Hmed et al.

Figure 4: Stability region of an unstable first order system.

Figure 5: Nyquist diagrams of the open-loop system β(s) for ki = 4 and ki = 6.

For the first case ki = 4, the Nyquist diagram enclose the critical point in the clockwise (N = 1). Moreover, the corresponding open-loop system β(s) has only one unstable pole different to zero (P = 1). Then, according to the modified Nyquist criterion, the

Stabilizing Fractional Order Integrator Design

| 109

closed-loop system (9) is unstable (Z = P + N ≠ 0). However, for the other case, the Nyquist diagram enclose the critical point in the anticlockwise (N = −1), then the closed-loop system (9) is stable (Z = P + N = 0).

3.2 Control Design for a First Order System The objective is to tune the controller parameters ki and α in order to obtain a stable closed-loop system with a specified time-domain or frequency-domain performances. Thus, the controller design can be done based on the iso-first overshoot curves and the minimum settling time of the desired closed-loop system presented respectively by Fig. 7 and Fig. 6 or on the gain in dB and the phase at resonance curves presented in Fig. 8. Moreover, according to the desired performances, the fractional order controller α can be determined.

Figure 6: Iso first overshoot curves for the desired closed-loop system.

– –

–

Therefore, the control design steps are as follows: Specifying the desired ωn . Setting the value of the fractional order α from the iso first overshoot curves and minimum settling time (Fig. 7 and Fig. 6) or the frequency curves (Fig. 8), according to the value of ξ = 2ωa and the desired performances. n

Determining the value of the controller gain ki =

ωα+1 n . b

110 | A. Ben Hmed et al.

Figure 7: Minimum settling time versus α for the desired closed-loop system when ωn = 1 rad/s.

Figure 8: Gain in dB and phase at resonance curves of the desired closed-loop system Fd (s).

To show the use of these curves to achieve the desired time-domain and frequency performances, an example of a first order system without integration is considered 1 s + 0.8

(11)

ki sα+1 + 2 × 0.4sα + ki

(12)

G(s) = The closed-loop system is described by F(s) =

We want to get a closed-loop response with the following desired performances: – A unit natural frequency. – A zero steady-state error for a step input. – An overshoot around 5 %. – A settling time less than 3s. From relation (5), to obtain a unit natural frequency, the controller gain is fixed to 1 (ki = 1). Moreover, to provide a first overshoot D1 % ≈ 5 with ξ = 0.4, the fractional

Stabilizing Fractional Order Integrator Design | 111

Figure 9: Step and control responses of the closed-loop system when α = 0.8 and ki = 1.

order controller α can be obtained graphically from Fig. 6 (i. e. α = 0.8). With these setting parameters, Fig. 7 shows that the desired settling time is less than 3s. Then, the transfer function of the controller and the closed-loop system are given respectively by: C(s) = F(s) =

1

s0.8

1 s1.8 + 2 × 0.4s0.8 + 1

(13) (14)

The step response of the closed-loop system (14) shows that the desired time-domain performances are satisfied with a tolerable control as shown in Fig. 9 .

3.3 Control Design of a First Order System with Time Delay By using the previous results for first order system without time delay, the objective now is to design a fractional I α controller combined with a Smith Predictor to stabilize and satisfy some time-domain or frequency specifications for the following time-delay systems: G(s) =

b −Ls e s+a

(15)

where a, b and L are the nominal parameters. Here, the Smith Predictor is used to remove the time delay term L from the closedloop system characteristic polynomial as shown in Fig. 10. The obtained closed-loop is given by F(s) =

G(s)C(s) −Ls e 1 + G(s)C(s)

(16)

112 | A. Ben Hmed et al.

Figure 10: Closed-loop scheme using Smith Predictor.

The closed loop transfer function can be rewritten as F(s) =

sα+1

bki e−Ls + asα + bki

(17)

The controller parameters ki and α can be determined likewise the case of system without time delay which makes it easier to fulfil its desired performances. Therefore, the parameters of the controller I α can be designed using the characteristic polynomial of the closed-loop system F(s) and the desired characteristic polynomial sα+1 + 2ξ ωn sα + ωα+1 n . So, the obtained controller can stabilize and satisfy some desired closed-loop specifications. A numerical example is taken to illustrate the proposed technique. Let’s consider a first order system with time delay G(s) =

1 e−3s s + 0.8

Figure 11: Step response of the closed-loop system (19).

(18)

Stabilizing Fractional Order Integrator Design | 113

The obtained closed-loop is presented by F(s) =

ki e−3s sα+1 + 2 × 0.4sα + ki

(19)

The proposed strategy should satisfy the following performances: – a unit natural frequency; – a zero steady-state error for a step input; – an overshoot around 5 %. From relation (5), to obtain a unit natural frequency, the controller gain is fixed to ki = 1. To provide a first overshoot D1 % ≈ 5, the fractional order controller is taken α = 0.8 from Fig. 6. With these tuning parameters, the step response of the closedloop system shows that the desired performances are satisfied.

4 I α Controller Design for Stabilization and Control of a First Order System with Integration The first order system with integration discussed in this section is G(s) =

b s(s + a)

(20)

The objective is to obtain a stable closed-loop system with a desired time-domain or frequency-domain specifications. Therefore, the problem is to determine the set of all stabilizing controller parameters for the system G(s). The obtained closed-loop transfer function is given by F(s) =

ki b sβ+1 + asβ + ki b

(21)

where β = α + 1. From the stability and resonance regions of the desired closed-loop system Fd (s), we can draw these regions for the closed-loop system F(s) as shown in Fig. 12. Since the closed-loop system is always unstable when 1 < β < 2 (i. e. 0 < α < 1) and a ≤ 0 as shown in Fig. 12, the following study is restricted to the case a > 0. Therefore, using Theorem (1), the closed loop system (21) is stable when 0 < α < 1 and a > 0 if the following condition is verified: a2 (a × tan( 2

− (ki b) > 0 where β = α + 1.

2β

2β+2

(2 − β)π (2 − β)π )) + (a × tan( )) 2 2

(22)

114 | A. Ben Hmed et al.

Figure 12: Stability and resonance regions of the closed-loop system F (s) when 1 < β < 2.

From the previous study, the stabilization steps are as follows: – If a < 0, then the controlled system cannot be stabilized using a fractional I α controller. – If a ⩾ 0, then the I α controller is designed to stabilize the closed loop system with 0 < α < 1 by following steps: (i) Determine the stabilizing region (ki , α) from relation (22). (ii) Choose an appropriate couple (ki , α) to stabilize the loop. For more illustration, consider a first order system with integration described by G(s) =

1 s(s + 4)

(23)

The obtained closed-loop system is given by F(s) =

ki sα+2 + 2 × 2sα+1 + ki

(24)

The stability limit of the closed-loop system (24) is determined numerically in terms of ki and α and plotted in Fig. 13. To verify the stability limit of the closed loop system, three arbitrary cases for α = 0.5 are taken: – on the stability region: ki = 30; – at the stability limit: ki ≃ 45; – on the instability region: ki = 60.

Stabilizing Fractional Order Integrator Design | 115

Figure 13: Stability limit of the unstable first order system with integration.

Figure 14: Nyquist diagrams of the open-loop system for ki = 30, ki ≃ 45 and ki = 60 with α = 0.5. k

i are plotted in the Bode and The corresponding open-loop system β(s) = sα+1 (s+4) Nyquist diagrams (Fig. 14). For the first case (ki = 30), the critical point is never enclosed by the Nyquist diagram (N = 0). Moreover, the open-loop β(s) has no unstable pole different from zero (P = 0). Then, according to the modified Nyquist criterion, the corresponding closed-

116 | A. Ben Hmed et al. loop system is stable. However, for the last case (ki = 60), the Nyquist plot enclose the critical point in the clockwise (N = 1), then the closed loop system is unstable (Z = P + N ≠ 0).

4.1 Control Design for First Order System with Integration Now, the objective is to tune the controller parameters ki and α to obtain a stable closed-loop system with a specified time-domain or frequency performances. Hence, the control design steps are as follows: – Setting the value of the fractional order α from the iso first overshoot curves and minimum settling time (Fig. 15 and Fig. 16) or the frequency curves (Fig. 17), according to the value of ξ = 2ωa and the desired time-domain performances. –

n

Determining the value of the controller gain ki =

ωα+2 n b

β+1

=

ωn b

.

Figure 15: Iso-first overshoot curves when 1 < β < 2 with β = α + 1.

Figure 16: Minimum settling time versus β = α + 1 for the desired closed-loop system when ωn = 1 rad/s.

Stabilizing Fractional Order Integrator Design | 117

Figure 17: Gain in dB and phase at resonance curves of the desired closed-loop system Fd (s).

Figure 18: Mathematical model of a DC motor.

To illustrate the proposed method, a general DC-motor model is considered. The mathematical model of a DC-motor is presented by Fig. 18, where the applied voltage Va controls the angular velocity ω(t). θ is the angular position, J the rotor inertia, L and R are respectively the armature resistance and inductance. The transfer function of a DC-motor is described by GDC (s) =

Km θ(s) = Va (s) s((Ls + R)(Js + Kf ) + Kb Km )

(25)

For the majority of DC motors the constant time of the armature is negligible. Therefore, the simplified model of (25) is given by GDC (s) = = where the gain KDC =

Km RKf +Kb Km

Km θ(s) = Va (s) s(R(Js + Kf ) + Kb Km ) Km /(RKf + Kb Km ) s(τs + 1)

=

KDC s(τs + 1)

and the time constant τ =

RJ . RKf +Kb Km

(26)

For the considered DC-motor the physical constants are [12]: R = 6 Ω, Km = Kb = 0.1, Kf = 0.2 N ms and J = 0.01 kg m2 /s2 .

118 | A. Ben Hmed et al. The simplified transfer function of (26) is described by GDC (s) =

0.08 s(0.05s + 1)

(27)

As a desired closed-loop control, we want a closed-loop peak resonance GdB = 2.14 dB with a fractional order controller α = 0.4 and a steady state gain of 0 dB. To verify these performances, the pseudo-damping factor can be setting to ξ = 2.5 with β = 1.4 as shown in Fig. 17(a). So, from relation (5), the natural frequency of the desired closed-loop system is given by ωn = 2ξa = 4. Hence, one can deduce that the controller ωα+2

β+1

gain ki = bn = ωbn = 17.41. The Bode diagrams of the considered closed-loop system satisfy the frequency performances as shown in Fig. 19.

Figure 19: Bode Diagrams of the closed-loop system with α = 0.4.

4.2 Control Design for Integrator First Order System with Time Delay By using the obtained results of a first order system without time delay, the objective now is to design a fractional I α controller combined with a Smith Predictor to stabi-

Stabilizing Fractional Order Integrator Design | 119

lize and guarantee some time-domain or frequency specifications for the time-delay system b e−Ls s(s + a)

G(s) =

(28)

where a, b and L are the nominal parameters. The Smith Predictor is tuned to remove the time delay term L from the closed-loop system characteristic polynomial as shown in Fig. 10. The obtained closed-loop is given by F(s) =

G(s)C(s) −Ls e 1 + G(s)C(s)

(29)

The closed loop F(s) can be rewritten as F(s) =

sα+2

bki e−Ls + asα+1 + bki

(30)

The controller parameters can be determined likewise the case of system without time delay which makes it easier to achieve its desired performances. The controller parameters ki and α can be designed using the characteristic polynomial of the closed-loop β+1 system and the desired characteristic polynomial sβ+1 + 2ξ ωn sβ + ωn where β = α + 1. α As results, the fractional I controller can stabilize and fulfill some desired closed-loop performances. To illustrate the proposed strategy, a first order system with time delay is considered: G(s) =

1 e−2s s(s + 3)

(31)

The obtained closed-loop is presented by F(s) =

ki e−2s sα+2 + 2 × 1.5sα+1 + ki

(32)

Here, the fractional controller I α must stabilize and satisfy the following desired closed-loop behavior: – a zero steady-state error for a step input; – a unit natural frequency; – a pseudo-damping ξ > 0.707; – a first overshoot around 3 %. From relation (5), to provide a unit natural frequency, the controller gain is fixed to ki = 1. Moreover, to achieve a first overshoot D1 % ≈ 3 with ξ = 1.5, the fractional order controller can be obtained graphically from Fig. 15. Thus α = 0.1 (β = 1.1).

120 | A. Ben Hmed et al.

Figure 20: Step response of the closed-loop system (32).

5 Conclusion This paper presents a simple method of stabilization and control of certain first order system with and without integration. Analytical expressions are developed to describe the controller stability region. Therefore, a fractional I α controller is designed to guarantee the stability and to satisfy some desired time domain and frequency performances for these systems. Moreover, the time-domain and the frequency curves of the desired closed-loop system are used to show the desired specifications. Illustrative examples have been taken to demonstrate the usefulness and efficiency of the proposed method. Then, to extend this method for system with time delay, the I α controller with Smith Predictor is used in control design. However, this strategy is applied only for the certain system. The present studies can be extended for the problem of stabilisation of an uncertain first order system with and without time delay.

Bibliography [1] [2] [3] [4]

D. Mehdi and B. Majid. Applications of fractional calculus. Appl. Math. Sci, 21(4): 1021–1032, 2010. S. Manabe. The non-integer integral and its application to control systems. Research Laboratory, Mitsubishi Electric Manufacturing Co, 2(2), 1961. K. Oldham and J. Spanier. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 111, 1974. Rudolf Hilfer. Applications of fractional calculus in physics. World Scientific, 2000.

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J. Sabatier, M. Aoun, A. Oustaloup, G. Grégoire, F. Ragot, and P. Roy. Fractional system identification for lead acid battery state of charge estimation. Signal processing, 86(10): 2645–2657, 2006. H. Panagopoulos, K. Astrom, and T. Hagglund. Design of pid controllers based on constrained optimization. IEEE, 149(1):32–40, January 2002. R. Caponetto, L. Fortuna, and D. Porto. A new tuning strategy for a non integer order pid controller. In First IFAC workshop on fractional differentiation and its application, 2004, 168–173. A. Ben Hmed, M. Amairi, S. Najar, and M. Abdelkrim. Design of fractional pi controller with guaranteed time and frequency-domain performances. In 10th International Multi-Conference on Systems, Signals & Devices (SSD), IEEE, 2013, 1–6. B. Saidi, M. Amairi, S. Najar, and M. Aoun. Bode shaping-based design methods of a fractional order pid controller for uncertain systems. Nonlinear Dynamics, 80(4): 1817–1838, 2015. I. Podlubny. Fractional-order systems and pIλ Dμ controller. IEEE, 44(1): 208–214, January 1999. D. Xue, C. Zhao, and Y. Chen. Fractional order pid control of a dc-motor with elastic shaft: a case study. Proceedings of American control conference, 7: 3182–3187, 2006. I. Petráš. Fractional-order feedback control of a dc motor. Journal of electrical engineering, 60(3): 117–128, 2009. F. Padula and A. Visioli. On the stabilizing pid controllers for integral processes. IEEE Transactions on Automatic Control, 57(2): 494–499, 2012. Y.-C. Cheng and C. Hwang. Stabilization of unstable first-order time-delay systems using fractional-order pd controllers. Journal of the Chinese Institute of Engineers, 29(2): 241–249, 2006. G. Maione and P. Lino. New tuning rules for fractionalpiα controllers. Nonlinear Dynamics, 49(1): 251–257, 2007. S. Hamamci. An algorithm for stabilization of fractional-order time delay systems using fractional-order pid controllers. IEEE transactions on automatic control, 52(10): 1964–1969, 2007. T. Umeno and Y. Hori. Robust speed control of dc servomotors using modern two degrees-of-freedom controller design. IEEE Transactions on Industrial Electronics, 38(5): 363–368, 1991. A. B. Hmed, M. Amairi, and M. Aoun. Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind. Communications in Nonlinear Science and Numerical Simulation, 22(1): 842–865, 2015. A. Ben Hmed, M. Amairi, S. Najar, and M. Aoun. Resonance study of an elementary fractional transfer function of the third kind. In International Conference on Fractional Differentiation and Its Applications (ICFDA), 2014. IEEE, 2014, 1–6.

122 | A. Ben Hmed et al.

Biographies Amina Ben Hmed was born in Tunisia. She received the electrical engineering diploma in 2012 from the National Engineering School of Gabes in Tunisia. She is currently Ph. D. student in Electrical Engineering and she is a member of the Modeling, Analysis and Control of Systems (MACS) laboratory. Her current research interests include automatic control and fractional differentiation.

Messaoud Amairi was born in Tunisia. He received his Ph. D. in Electrical Engineering in 2011. He is currently an Associate Professor in automatic control at the National Engineering School of Gabes (Tunisia) and he is a member of the Modeling, Analysis and Control of Systems (MACS) laboratory. His current research interests include automatic control, system identification, and fractional differentiation.

Mohamed Aoun was born in Tunisia. He received his Ph. D in automatic control in 2005 from the University of Bordeaux, France. He is currently a Professor in automatic control, electrical engineering, and computer engineering at ENIG, Tunisia and a member of its MACS research laboratory. His research interests include automatic control, system identification, and fractional differentiation.

Essia Saidi, Yosra Hammi, and Ali Douik

Modeling and Explicit Model-Predictive Control of a Two-Tank System by PWA Approach Abstract: In this paper, we consider the solution of the control problem using the piecewise affine approach for a two-tank system. This approach is well adapted to some hybrid systems because of its ability to address several issues such as explicit model predictive control. Two topics are addressed: First, we start by proposing piecewise affine formalism for the two-tank system. Then, we aim to determine the explicit model predictive control law for the resultant system under constraints by solving an appropriate linear criterion. Keywords: Model predictive control, piecewise affine systems, explicit control, twotank system

1 Introduction Hybrid dynamical systems called (SDH) are systems that simultaneously involve continuous and discrete phenomena [1]. They are governed by continuous differential equations and discrete variables [2, 3]. In fact, they usually result from the interaction between discrete algorithms of planning and continuous control algorithms (see Fig. 1). Many examples of hybrid systems can be found in real life, such as embedded systems, where it is noted an interaction between the discrete part (the program) and the physical part (the system), industrial production lines, telecommunication networks, air traffic management systems, industrial process control, automated systems, etc. Different approaches have been proposed to model hybrid systems such as jumped linear systems (JLS), switched linear systems (SLS), piecewise affine systems (PWA) [4], mixed logical dynamical systems (MLD) [5]. In fact, different techniques are proposed to control hybrid systems, such as optimal control, model predictive control (MPC) [6]. The control structures are based on optimization techniques for which we aim to minimize a linear or quadratic cost function [5]. However, many such problems are unsolvable online with MPC. To overcome this problem, approximation techniques may be used.

Essia Saidi, National Engineering School of Monastir, Monastir, Tunisia, e-mail: [email protected] Yosra Hammi, National Engineering School of Tunis, Tunis, Tunisia, e-mail: [email protected] Ali Douik, National Engineering School of Sousse, Sousse, Tunisia, e-mail: [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 123–137. https://doi.org/10.1515/9783110591729-007

124 | E. Saidi et al.

Figure 1: Principle of hybrid dynamical systems.

In this paper, we investigate the use of explicit model predictive control for the class of piecewise affine systems [7] by applying it to a two-tank system. The paper is organized as follows. In Section 2, we will present the class of piecewise affine systems illustrated by an academic example. Afterward, the PWA modeling of a two-tank system will be studied in Section 3. Then, the model predictive control of PWA systems is presented in Section 4. The explicit model predictive control of the given system will be discussed in Section 5. Finally, conclusions are given in Section 6.

2 Piecewise Affine Systems 2.1 Definition A piecewise affine system (PWA) is a collection of linear / affine systems sharing a same continuous state and connected by switches which are determined by a polyhedral partition of the state-input space. A PWA is used to model a large number of physical processes, or to approximate a nonlinear dynamic via the linearization of nonlinear behaviour in different operating points. In fact, modeling PWA systems can be derived from the state-space representation of classic linear-systems, such systems can be defined as follows: { x

xk+1 = Ai xk + Bi uk + fi yk = Ci xk + gi

(1)

for [ ukk ] ∈ Xi , while xk , uk and yk denote the state, the input and the output of the system, respectively. {Xi }1≤i≤s represents the polyhedral partition of the state-input space defined by x {Qi [ ukk ] ≤ qi }. Ai , Bi , fi , Ci , gi , Qi and qi are matrices of suitable dimensions.

Model-Predictive Control of a Two-Tank System | 125

A PWA system (1) is well-posed if xk+1 and yk have a unique solution for a state xk and a given input uk , that is: Xi ∩ Xj = 0,

for i ≠ j

Therefore, the model (1) can be considered as a collection of linear/affine models based on the continuous state xk and indexed by the discrete state i. The discrete state i indexes the active model at each instant k and takes its values in a finite set {1, . . . , s}, where s represents the number of sub-models. For a more detailed description of PWA systems, the reader is referred to [8, 9] and the references therein.

2.2 Application Example Consider the following example of a PWA system: cos αk { { xk+1 = 0.8 [ sin αk { { y = [0 1]x k k {

− sin αk 0 ] uk ] xk + [ 1 cos αk

where: 󵄨󵄨 π 󵄨 αk = 󵄨󵄨󵄨󵄨 π3 󵄨󵄨 − 3

if [1 if [1

0]xk ≥ 0 0]xk < 0

with xk ⊂ [−10, 10] and uk ∈ [−1.1, 1.1]. We note that the system evolves according two modes. Thus, it can be defined upon two polyhedral regions (Fig. 2).

Figure 2: PWA system with two polyhedral regions.

126 | E. Saidi et al.

3 Hybrid Modeling of a Two-Tank System by PWA Approach The two-tank system shown in Fig. 3 has been used in many fields of research. In fact, it was adopted as a standard benchmark of the scientific action “Diagnosing hybrid systems” (AS193) which is supported by the National Center for Scientific Research (CNRS) [10]. We propose in this section a PWA model for such process.

Figure 3: Two-tank system.

The considered system consists of two cylindrical tanks T1 and T2 , that have identical sections, which are interconnected by two conduits, an upper one C2 and a lower one C1 , equipped respectively by two off valves V4 and V3 (controlled all or nothing). In fact, C2 is located at the bottom of the tanks while C1 is located at a height hv of 0.3 m. The tanks are filled by pump acting on the tank T1 with a flow rate Qp , which can be manipulated continuously from 0 to a maximum flow rate Qmax . In fact, the valve V1 is used to model failings in tank R1 , and the flow through the second tank is controlled by the valve V2 . In our case, V1 is kept closed for the normal behavior (V1 = 0), and to simplify the calculation, the valves V3 and V2 are kept opened (V3 = 1 and V2 = 1). The details and the parameters of the system are given in Tab. 1, [11]. If we apply the law of conservation of mass in the tanks, we get: V̇ = Aḣ = ∑ Qin − ∑ Qout

(2)

Qin , Qout , A, h, and V are the incoming flows, outflows, the section of the tank, the height and the volume of liquid in the tank, respectively. Thus, we obtain the following

Model-Predictive Control of a Two-Tank System |

127

Table 1: Parameters of the two-tank system. Symbol

Signification

Value

A S g hv hmax Qmax Ts

Section of tanks R1 and R2 Cross section of valves V1 , V2 , V3 and V4 gravity constant Height of the valve V4 maximum level of liquid in each tank Maximum pump flow Sampling period

0.0154 m2 3.6 × 10−5 m2 9.81 m/s2 0.3 m 0.63 m 10−4 m3 /s 10 s

differential equations: 1 ̇ { { h1 = A (Qp − Q1V1 − Q4V4 − Q3V3 ) { 1 { ̇ + Q3V3 − Q2V2 ) h = (Q { 2 A 4V4

(3)

Assuming that the flow follows the Torricelli law, we obtain: Q1V1 = V1 S√2gh1

(4)

Q2V2 = V2 S√2gh2

(5)

Q3V3 = V3 S sign(h1 − h2 )√2g|h1 − h2 |

(6)

Q4V4 = V4 S√2g[ max(h1 , hv ) − max(h2 , hv )]

(7)

Derivation of the PWA Model of the Two-Tank System We propose a method of deriving the PWA model of a two-tank system. The following steps should be followed: – Linearization of nonlinear relationships. For this system, there are flows through the pipes given by equations (4), (5), (6) and (7). – Define the different operating modes. – Define the continuous dynamics for each mode. – Description of the system in the discrete time domain. First, we begin by linearizing the flows Q1V1 , Q2V2 , Q3V3 and Q4V4 [11]: Q1V1 = k1 V1 h1

Q2V2 = k1 V2 h2

Q3V3 = k1 V3 (h1 − h2 )

Q4V4 = k2 V4 [ max(h1 , hv ) − max(h2 , hv )]

(8) (9) (10) (11)

128 | E. Saidi et al. with: k1 = S√

2g hmax

(12)

k2 = S√

2g hmax − hv

(13)

According to the level of the liquid h1 in the tank R1 , we can infer that the system may be formulated in the form of a piecewise affine system which progress according two modes: 1. Mode 1: If h1 ≤ hv then the valve V4 is closed (V4 = 0). 2. Mode 2: If h1 < hv then the valve V4 is opened (V4 = 1). Thereafter, the first-order discretization technique (Euler) is applied to equations derived from the system (3) by replacing ḣ by: k(k + 1) − h(k) ḣ ≃ T Supposing that: x(k) = [

h1 (k) ], h2 (k)

u(k) = Qp

the following discrete PWA system is obtained: 1. Model 1: T 1 − k1 { { { [ A { { xk+1 = [ T { { [ A k1 { { { { yk = [0 1]xk

T T k ] A 1 [ x + ] k A ] uk T 1 − 2 k1 ] [ 0 ] A

with: 1 0 −1 0 0 [ 0 [ [ [ [ [ [ [ [ [ 2.

0 1 0 −1 0 0

0 hv ] [ ] [ ] [ 0 ] [ hv ] [ ] [ ] [ 0 ] [ ] xk + [ ] uk ≤ [ 0 ] [ 0 ] [ 0 ] [ ] [ ] [ ] [ ] [ 1 ] [ Qmax ] [ −1 ] [ 0

] ] ] ] ] ] ] ] ] ]

Model 2: T 1 − (k1 + k2 ) { { { [ A { { xk+1 = [ T { { [ A (k1 + k2 ) { { { { yk = [0 1]xk

T T k [ ] A 1 ] xk + [ A ] uk − [ T 1 − 2 k1 ] [ 0 ] [ A

T kh A 2 v ] ] T k2 hv ] A

Model-Predictive Control of a Two-Tank System |

129

with: −1 0 −1 0 0 [ 0 [ [ [ [ [ [ [ [ [

0 1 0 −1 0 0

0 −hv ] [ ] [ ] [ 0 ] [ hv ] [ ] [ ] [ 0 ] [ ] xk + [ ] uk ≤ [ 0 ] [ 0 ] [ 0 ] [ ] [ ] [ ] [ ] [ 1 ] [ Qmax ] [ −1 ] [ 0

] ] ] ] ] ] ] ] ] ]

4 Model Predictive Control of PWA Systems 4.1 Principle of MPC of PWA Systems The model predictive control (MPC) is a control methodology that offers interesting solutions for control of systems under linear or non-linear constraints and, more recently, for the regulation of hybrid systems, for more details, see [12]. It is based on the following principles [13]: – The use of an explicit model of the process to calculate forecasts of future behavior (future process outputs) on a prediction horizon. – Minimize a cost criterion which depends, generally, on the difference between the predicted trajectory of the considered process and the reference output, while respecting certain constraints to be satisfied, in order to obtain an optimal sequence of commands that will be applied to the system [10]. – Once the control sequence is obtained, only the first element of it will be applied to the system, and, at each sampling period, the entire calculation procedure will be repeated [7]. Sometimes, the time required to solve the optimization problem in real time is greater than the sampling period of the system, therefore, this causes problems when implanting the regulator obtained from the predictive control, especially for systems with fast dynamics. To overcome this limitation, the control law is “offline” computed using methods of multiparametric programming. Considering the PWA system (1), and assuming that the current state xk is known, MPC requires solving the following criteria each time step [14]: N

N−1

i=1

i=0

J = ∑ ||Q(yk+1/k − wk+i )||p + ∑ ||Ruk+i ||p

(14)

where: umin ≤ uk+i ≤ umax , ymin ≤ yk+i ≤ ymax . Q, R, N, p and wk+i represent the weighting matrices, the prediction horizon, the cost function norm and the output reference, respectively.

130 | E. Saidi et al. umin , umax , ymin and ymax represent the minimum and the maximum values of the input and the output, respectively. According to the value of p we will treat either a linear programming problem, or a quadratic programming problem. – Using the norm p = 2, the cost function (14) becomes quadratic, so that ||Qw||2 = wT Qw, so, the MPC problem is solved by a quadratic programming (QP) for each potential sequence. N

N−1

i=1

i=0

J = ∑ Q(yk+1/k − wk+i )2 + ∑ R(uk+i )2 –

(15)

Using the norm p = 1, the cost function (14) becomes linear, that is ||Qw||1 = |Qw|, so, the MPC problem is solved by linear programming (LP). The following optimization problem is defined: N

N−1

i=1

i=0

J = ∑ |Q(yk+1/k − wk+i )| + ∑ |Ruk+i |

(16)

The explicit control law obtained by multiparametric programming methods is a piecewise linear function, dependent on the value of the state at each sampling period. The structure of this control law can be stored in three tables, the first contains the partitioned regions on which the control law is defined, while the second and third contain the gains associated with each region, so, the control law can be defined as follow [15]: u = Fi xk + Gi ,

xk ∈ Ri

(17)

where Ri is a polyhedral partition of the state space such that Ri ∩ Rj = 0, and Fi , Gi are the gains.

4.2 Application Example Considering the PWA system of section 2.2. The purpose of this section is to design a model predictive control law able of stabilizing the state x2 to a reference point, that is x2 = 2, while manipulating the input value u continuously from −1.1 to 1.1. The controller must face the following constraints: −1.1 ≤ u ≤ 1.1 −10 ≤ x2 ≤ 10 Thereafter, we assume that: Q100,

R = 0.1 [

1 0

0 ], 1

p = 1,

N=3

Model-Predictive Control of a Two-Tank System | 131

Thus, the following optimization problem must be resolved: 3

2

i=1

i=0

J = ∑ |Q(y − 2)| + ∑ |Ru| The control law obtained is distributed over 96 regions (Fig. 4).

Figure 4: Control law distributed over 96 regions.

5 Simulations and Results The purpose of this section is to design a control law (MPC) capable of stabilizing the liquid level in the Tank 2 to a reference point, that is h2 = 0.2 m, while manipulating the flow rate Qp continuously from 0 to a maximum flow Qmax . We must first define the model of the dynamical system. We can do it by importing the model from HYSDEL source [16] and convert it into an equivalent PWA model. The controller must face the following constraints: 0 ≤ Qp ≤ Qmax

0 ≤ h2 ≤ 0.3 Thereafter, we assume that: Q100,

R = 0.1 [

1 0

0 ] 1

We should note that simulation results will be obtained using the Multi-Parametric Toolbox (MPT), see [17]. Using the norm p = 1, the MPC problem is solved by linear

132 | E. Saidi et al. programming (LP) [18]. For N = 5, the following optimization problem is defined: 5

4

i=1

i=0

J = ∑ |Q(y − 0.2)| + ∑ |Ru| In this case, the control law obtained is distributed over 113 regions (Fig. 5).

Figure 5: Control law distributed over 113 regions.

The complexity of explicit solutions has always been a major issue. In fact, we can distinguish two levels of complexity associated with multi-parametric solutions of predictive control problems [17]. – 1st complexity level: Complexity in terms of execution time of the resulting solution. – 2nd complexity level: complexity in terms of number of regions of the resulting solution. Indeed, the first level is essential for calculating the explicit solution, while the importance of the second level is mostly revealed during the implementation of the explicit control law in real applications. Obviously, when the number of regions increases, it takes more time to identify an appropriate control law as well as a memory to store all the obtained regions with the associated control laws. In fact, the simplification is established by merging the regions that have the same control law, while retaining the same features and performances of the original control law. Therefore, the control law obtained can be further simplified to reduce the number of regions (Fig. 6) and the execution time (Tab. 2). In Tab. 2, t1 represents the execution time of the control law before simplification, while t2 represents the execution time of the control law after simplification. Once this explicit solution is obtained, it can be applied to the considered system. Thereafter, we simulate the system from the initial value x0 = [0.2, 0.094], the closed

Model-Predictive Control of a Two-Tank System |

133

Figure 6: Control law distributed over 41 regions.

Table 2: Comparison of the execution time before and after simplification. Execution time N=5

t1 (seconds)

t2 (seconds)

78.86

7.02

Figure 7: Closed loop trajectory for x0 = [0.2, 0.094].

loop path and the evolution of the states, the input and the output relative to this value are shown in (Fig. 7, Fig. 8, Fig. 9 and Fig. 10). At t = 2 s, h1 exceeds 0.3, and it is noted that the valve V4 changes from 0 to 1 (see Fig. 8 and Fig. 10), thereafter, at t = 7 s, h1 becomes equal to 0.342 and the liquid in the tank 2 achieves the desired value (0.2) exactly (Fig. 9), this is confirmed by Fig. 7, noting that the closed loop path reaches a target reference defined by h1 = 0.342 and h2 = 0.2.

134 | E. Saidi et al.

Figure 8: Evolution of state variables.

Figure 9: Evolutions of the output and the reference.

Figure 10: Evolution of inputs.

As we can see in this example, the liquid level in each tank and the liquid inflow do not exceed the maximum values already imposed, which proves the ability of MPC to handle constraints on inputs, states and outputs. For different horizons, we can compare the time execution of the control low as well the relative regions as it is shown in Tab. 3. As we see, the choice of prediction horizon is very important, in fact, when it increases, the system becomes more complicated in terms of number of regions and execution time when computing the control law.

Model-Predictive Control of a Two-Tank System |

135

Table 3: Comparison of the execution time for different horizons. Prediction horizon Number of control low regions before simplification Number of control low regions after simplification t1 (seconds) t2 (seconds)

N=3

N=4

N=5

N = 10

38 16 9.81 2.01

86 32 26.21 4.68

113 36 65.11 7.12

236 81 572.97 25.44

6 Conclusion In this paper, we have presented a methodology of hybrid modeling of a two-tank system in order to perform an efficient control law that will be applied to the considered system. The used modeling methodology is based on piecewise affine (PWA) approach. On the other hand, the used control strategy is based on explicit model predictive control so that the liquid level in the second tank could be optimally controlled. In this framework, a linear cost function was considered. Thereafter, an appropriate horizon prediction was found out in order to improve the system performance in terms of computation time.

Bibliography [1]

[2] [3] [4] [5]

[6] [7] [8] [9]

F. Elguezar. Modélisation et simulation des Systèmes dynamiques hybrides affines par morceaux. Exemples en électronique de puissance. PhD thesis, University of Toulouse, December, 2009. F.Hamdi. Contribution à la Synthèse d’Observateurs Pour les Systèmes Hybrides. PhD thesis, University of Batna, July, 2010. M. Kurovszky. Etude des systèmes dynamiques hybrides par représentation d’état discrète et automate hybride. PhD thesis, University of Grenoble, December 2010. H. Lin and P. J. Antsaklis. Hybrid Dynamical Systems: An Introduction to Control and Verification. Foundations and Trends® in Systems and Control, 1:1–172, 2014. E. D. Santis, M. D. Benedetto, S. D. Gennaro and P. Giordano. Hybrid observer design methodology, IST-2001-32460 of European Commission. Distributed control and stochastic analysis of hybrid systems supporting safety critical real-time system design (HYBRIDGE), 2003. C. E. Garcia, D. M. Prett and M. Morari. Model predictive control:theory and practice – a survey. Automatica, 25:335–348, 1989. D. Q. Mayne and S.Rakovic. Model predictive control of constrained piecewise affine discrete-time systems. Int. Journal of Robust and Nonlinear Control, 13:261–279, 2003. G. Ferrari-Trecate, F. A. Cuzzola, D. Mignone and M. Morari. Analysis of discrete-time piecewise affine and hybrid systems. Automatica, 38:2139–2146, 2002. E. D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Trans. on Automatic Control, 346–358, 1981.

136 | E. Saidi et al.

[10] D.Lefebvre, D.Aubry, H.Chafouk, C.Combastel, E. A.Domlan, G.Graton, G.Hoblos, R.Kajdan, F.Kratz, A.Lalami, M.Lebbal, D.Maquin and J.Ragot. Diagnostic des SDH. Quelques contributions issues des approches continues. European Journal of Systems, 885–912, 2007. [11] Y. Hammi, N. Zanzouri and M. Ksouri. Modeling and Predictive Control of Tow Tank System by MLD approach, The 4th Int. Conf. on Circuits, Systems and Signals (CSS’10), 140–145, 2010. [12] M. Lazar. Model Predictive Control of Hybrid Systems: Stability and Robustness. PhD thesis, Eindhoven University, 2006. [13] A. Bemporad and M. Morari. Robust Model Predictive Control: A Survey, Robustness in identification and control. 207–226, 1999. [14] J. Thomas, D. Dumur, J. Buisson and H. Guéguen. Model predictive control for hybrid systems under a state partition based MLD approach (SPMLD). Informatics in Control, Automation and Robotics, 217–224, 2006. [15] M. Baotic, F. J. Christophersen and M. Morari. A new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index. European Control Conf., Cambridge, UK, 2003. [16] F. Torrisi and A. Bemporad. Hysdel – a tool for generating computational hybrid models for analysis and synthesis problems. IEEE Trans Control Systems Technology, 235–249, 2004. [17] M. Kvasnica, P. Grieder, M. Baotic and F. J. Christophersen. Multi-Parametric Toolbox (MPT), 2005. [18] E. Saidi, Y.Hammi and A. Douik. Explicit model-predictive control of hybrid dynamical systems: application to a two-tank system. IEEE 12th Int. Multi-Conf. on Systems, Signals & Devices (SSD’15), 1–6, 2015.

Biographies Essia Saidi received the MA degree in Automatic from National Engineering School of Gabes (ENIG) in 2011, the Ph. D. degree in Electrical Engineering from the National Engineering School of Monastir (ENIM) in 2016, Tunisia. Her researches are in the fields of hybrid systems and control.

Yosra Hammi received the MA degree in Electronic from the university of science of Tunis, Tunisia, in 2006, the Ph. D. degree in Electrical Engineering from the National Engineering School of Tunis (ENIT) in 2013, Tunisia. She is currently an Associate professor at National Engineering School of Gabes. Her research interests are in the area of Hybrid systems and control.

Model-Predictive Control of a Two-Tank System |

137

Ali Douik was born in Tunis, Tunisia. He received the Masterdegree from the Higher Institute of training technical teachers of Tunis in 1990, the Ph. D. degree in Electrical Engineering from the High School of Science and Technique in Tunis in 1996 and University Habilitation in Electrical Engineering from Monastir University in 2010. He is currently an Associate professor at National Engineering School of Sousse. His researches are in Automatic Control, robust control, evolutionary optimization control and Digital Image Processing.

Amira Abdelkader, Moez Boussada, Koffi Fiaty, Ahmed Said Nouri, and Hassan Hammouri

Comparison of Nonlinear Observers for a Nonlinear System

Abstract: This paper present the esterification operation of an olive oil waste taking place in a Semi Continuous Stirred Tank Reactor (SCSTR). The plant model, presents a high level of non linearities and interconnections between states. The necessary steps required to move from differential algebraic model to a differential model is given. In this paper a comparative study of three techniques for observing the states of non-linear systems (SCSTR) is presented. The first and the second present two different versions of nonlinear high gain observer under two different canonic form of observability and different gain of observer. The third observer is the Extended Kalman Filter (EKF). The performance of the three algorithms is compared through the simulation results. We show in this work that it is possible to synthesis two versions of high gain observer for a MIMO system and ensure good applicability of a such observer. The results are validated on a real measurements of the plant. The second part is devoted to performances’ comparison between the three observers. Keywords: Nonlinear system, Chemical reactor, High gain observer, Extended Kalman Filter

1 Introduction Compared to the other process in the chemical industry, the chemical reactor is one of the most complex. In fact, it’s a nonlinear system with a strong interconnection between states and parameters. Moreover, the estimation of different state is a difficult task in chemical process, due the complexity of the related physico-chemical phenomena. Many studies have been elaborated in the topic of nonlinear observer [1], [2], [3], [4], [5], [6], [7] and [8] but the problem is still open seen the levels and the forms of the nonlinearities presents in system model or its structure. Acknowledgement: This work was supported by the Ministry of Higher Education and Scientific Research in Tunisia. Amira Abdelkader, Moez Boussada, Ahmed Said Nouri, CONPRI, National School of Engineering of Gabes, Gabes university, Gabes, Tunisia, e-mails: [email protected], [email protected], [email protected] Koffi Fiaty, Hassan Hammouri, LAGEP, Université de Lyon 1, Lyon, France, e-mails: [email protected], [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 139–157. https://doi.org/10.1515/9783110591729-008

140 | A. Abdelkader et al. Nowadays, there is a growing interest in the development of software sensors for chemical processes. Many works in this area are based on an Extended Kalman Filter approach which leads to complex non-linear algorithms. This paper aims to develop a two nonlinear high gain observer versions for an olive oil waste esterification process in a semi continuous stirred tank reactor compared with the EKF observer. This paper is organized as follows; in the next section, we introduce the reactor model. In section 3, we show how one can design simple observers for a class of nonlinear systems which include this reactor model, two versions of high gain observer is presented for estimating concentrations, volume and reactor temperature, and their corresponding convergences are established. Simulation results and comparison between observers are reported throughout section 4. Finally we close the paper with some remarks.

2 Model Presentation From a chemical point of view, the esterification reaction of the fatty acids presents in the olive oil waste is formulated by equation (1). Acid + Alcohol 󴀕󴀬 Ester + Water RCO2 H + C2 H5 OH 󴀕󴀬 RCO2 C2 H5 + H2 O

(1)

The schematic of this reactor is shown in figure 1. The model obtained from the material and energy balances is based on the following assumptions: – the reaction mixture is assumed to be perfectly stirred, – the thermodynamic equilibrium at the liquid-vapor interface is reached and is described by the Raoult law, – saturation vapor pressure of the different compounds are described by Antoine equation,

Figure 1: Schema of the chemical reactor.

Comparison of Nonlinear Observers for a Nonlinear System |

141

Under the above assumptions the mathematical model is given by [9]: { { { { { { { { { { { { { { { { { { {

dCi dt

=

Fe (Cie V e e

−

ρe C) ρ i

=

F ρ ρ

dTr dt

=

F e ρe (C e T e VρCp p r

=

Fc (T e Vc j

dTj dt

ρlv C) ρ i

− Cp Tr ) − F lv

ρlv ΔHv ρ VCp

F lv (Ci∗ V

+ υi r,

i = 1..4

lv lv

dV dt

−

−

−

F ρ ρ

− Tj ) −

UA (T Vc ρc Cp j

− r ΔρCr H + p

UA (T VρCp j

− Tr )

(2)

− Tr )

With C1 , C2 , C3 and C4 , are respectively the concentration of the fatty acids, ethanol, water and esters. F e is the input flow rate. F lv is the vaporization flow rate. V is the liquid phase volume in the reactor. Tr is the temperature of the reaction mixture and Tj is the jacket temperature. The reaction rate is given by [11]: E )C1 C2 , with a and b are constants. – r = (a + bCcat ) exp(− RT r – Ccat the catalysor concentration, defined as: acat ∗ mcat , with acat is its activity and mcat is its mass, – with a = 1.15 m3 /mol/s, b = 0.579 m3 /mol/s ∗ (1/meqH+ ), acat = 3.2 and mcat = 10 g. The concentration of the different components in the liquide film are calculated from the following relations by: Ci∗ = yi∗

ρlv ∑i yi∗ Mi

with yi∗ =

Pi0 (Tr ) Ci ; P ∑i Ci

ln(Pi0 (Tr )) = αi −

βi Tr + γi

and ρlv = ∑ i

yi∗ Mi ∑k

yk∗ Mk ρk

The coefficients αi , βi and γi were determined in [12]. Now we have to define the parameter F lv seen that this parameter plays a very important role and influences the behavior of the process since it is present in the chemical and thermodynamic aspect. This equations’ set can be presented in a state space form as an algebraicdifferential model: ẋ = f (x, u, θ) { { g(x) =0 { { y = h(x) {

(3)

142 | A. Abdelkader et al. Where g(x) = (∑ yi∗ − 1)

(4)

is an algebraic equation. With – xi ∈ ℜn , the state vector xT = [C1..4 V Tr Tj ] – u ∈ ℜm , control vector uT = [F e Tje ] – y ∈ ℜp , output vector yT = [C1 V Tr Tj ] – θ ∈ ℜnp , parameter vector θ = [F lv ].

2.1 The Liquid-Vapor Flow Rate The determination of the evaporation flow rate is an important step in the model resolution. Actually, this flow includes a considerable amount of information from the fact that it is closely related to two physical aspects present in the reactor, mass transfer aspect and a heat transfer aspect. In process control, it is important to use parametric estimation techniques. In our case, the rate is a variable quantity over time which strongly influences the evolution of the esterification process. Thus, a wrong value of this flow rate might distort the calculation. Therefore, it would be interesting to update it at each step providing an adaptive character model. In [9], [10], the authors used an optimization based approach to solve an algebraicdifferential model performing an estimated value of the evaporation flow rate. The results were interesting but needed a relatively strong amount of calculation and then time consuming which does not be the perfect approach to be implemented in real time configuration. We present below another technique based on the index reduction of the original system (algebraic-differential system). This technique provides an explicit relationship of the evaporation flow rate which will reduce considerably the time allowed to solve the model. So that we pass of an algebraic-differential model as present in (3) to a differential model it is necessary to find an explicit expression for the parameter θ. Indeed to determine the evaporation flow rate we must derive the algebraic relationship g(x) (4) with respect to time. The expression of F lv can be written as follows. F lv =

− ∑ AAi ∑ BBi

with AAi = xi ATr Qpi ∑ Ci2 + ACi ∑ Ci − Ci ∑ ACi BBi = xi BTr Qpi ∑ Ci2 + BCi ∑ Ci − Ci ∑ BCi

(5)

Comparison of Nonlinear Observers for a Nonlinear System |

143

and Qpi =

−Bpi

(Cpi + Tr )2

with ATr =

ΔH F e ρe e e UA (C T − Cp Tr ) − r r + (T − Tr ) VρCp p r ρCp VρCp j BTr = ACi =

ρlv ΔHv ρ VCp

F e e ρe (Ci − Ci ) + υi r V ρ

and ρlv 1 C) BCi = − (Ci∗ − V ρ i

with i = 1..4

For more details see [13].

2.2 Experimental Design and Model Validation As the simulation model shown in Fig 2, the reaction occurs after a certain delay. Indeed it starts with taking the heating period reaction at a temperature which exceeds the boiling temperature of the water medium. This allows to shift the equilibrium towards formation of the ester and evacuation of water formed by the spray reaction.

Figure 2: Evolution of Acid, Ester, Ethanol and water concentration.

144 | A. Abdelkader et al. Once this temperature was reached, the catalyst is introduced and the alcohol feed starts. The feed flow rate F e should allow the presence of a small amount of alcohol in the reaction medium after the establishment of the liquid-vapor equilibrium of binary water – alcohol causing the esterification of fatty acids. This moment is taken as the start time of the reaction. Monitoring the progress of the reaction is provided by control of the concentration of the acid in the reaction medium. Thus it presents in Fig 3 the evolution of different parameters, the input rate of flow F e , the vaporization rate of flow F lv , and the condensate Dc is weighed using a balance (an experimental value). Finally the evolution of yi∗ (liquid-vapor composition) can be seen in Fig 4 which verifies the coast function g(x) (4).

Figure 3: Evolution of F e , F lv and Dc .

Figure 4: Evolution of yi ∗ .

Comparison of Nonlinear Observers for a Nonlinear System | 145

3 Version 1: High Gain Observer Now, we take the expression of F lv (5) so the algebro-differential model (3) becomes a differential model. The esterification plant can now be described by the following equations: {

ẋ = f (x, u) y = h(x)

(6)

With { { { { { { { { { f (x, u) = { { { { { { { { { { – – –

fi =

Fe (Cie V e e

−

F ρ ρ

f6 =

F e ρe (C e T e VρCp p r

f7 =

c

F (T e Vc j

ρlv C) ρ i

− Cp Tr ) − F lv

ρlv ΔHv ρ VCp

−

F lv (Ci∗ V

+ υi r,

i = 1..4

lv lv

f5 =

−

−

ρe C) ρ i

F ρ ρ

− Tj ) −

UA (T Vc ρc Cp j

− r ΔρCr H + p

UA (T VρCp j

− Tr )

− Tr )

xi ∈ ℜn , the state vector xT = [C1..4 V Tr Tj ]. u ∈ ℜm , control vector uT = [F e , Tje ]. y ∈ ℜp , output vector yT = [C1 V Tr Tj ].

We consider now the following assumptions: assumption 1: The function f (x, u) is globally lipschitz with respect to x(t), uniformly with respect to u(t). assumption 2: The input u(t) is bounded. Therefore, system (2) admits a high gain observer which can be written as follows: {

ẋ̂ = f (x,̂ u) + ρΔθ S−1 C T (C x̂ − y)

̂ − SA(u, x)̂ + ρC T C Ṡ = −λS − AT (u, x)S

(7)

where ρ, λ and θ are observer adjusting parameters. Notes X1 = [x1 x5 x6 ]T and X2 = [x2 x3 x4 ]T . We note that we don’t take account of the jacket temperature x7 = Tj in order to ease the calculation process. The model can then be put in the following form: {

Ẋ 1 = F1 (u, X1 ) Ẋ 2 = F2 (u, X1 , X2 )

(8)

Let A(u, X) = ( with A1 (u, X) =

𝜕F1 (u,X1 ) 𝜕X2

0 0

A1 (u, X) ) 0

(9)

146 | A. Abdelkader et al. –

𝜕F1 𝜕X2

is the jacobian described by: 𝜕F1 𝜕X2

–

=(

𝜕ẋ1 𝜕x2 𝜕ẋ5 𝜕x2 𝜕ẋ6 𝜕x2

𝜕ẋ1 𝜕x3 𝜕ẋ5 𝜕x3 𝜕ẋ6 𝜕x3

𝜕ẋ1 𝜕x4 𝜕ẋ5 𝜕x4 𝜕ẋ6 𝜕x4

)

(10)

𝜕F1 ) = 3 full rank. verifying Rang( 𝜕X 2 Let θ > 0 real number and let Δθ is a block diagonal matrix defined as follows:

Δθ = (

θI3 0

0 ) θ2 I3

(11)

– Results and discussion The performances of the observer were tested using real measurements in the simulation environment. Thus, the values of the vector X1 are experimental values. The initial conditions of the model and the observer are the following: For the simulations, the system has been considered with initial conditions given by: x1 (0) = 0.09 mol/l,

x5 (0) = 1.4 l and

x6 (0) = 294 K

On the other hand, the observer has been initialized with: x̂1 (0) = 0.1 mol/l,

x̂2 (0) = 0 mol/l,

x̂5 (0) = 1.54 l,

x̂3 (0) = 0 mol/l

x̂6 (0) = 323.6 K,

and x̂4 (0) = 0 mol/l

Finally, the observer has been tuned with: ρ = 3, λ = 2 and θ = 5. In figure 5, estimation of ethanol concentration (x2 ), water concentration (x3 ) and ester concentration (x4 ) are presented.

Figure 5: Estimation of ethanol concentration (x2 ), water concentration (x3 ) and ester concentration (x4 ).

Comparison of Nonlinear Observers for a Nonlinear System |

147

Figure 6: Estimation of acid concentration C1 (x1 ).

Figure 7: Estimation of volume V (x5 ).

Estimation results are shown in Figures 6, 7 and 8 where estimations x1 , x5 and x6 are compared with their real values (derived from experimental values). It can be seen from these figures that the observer design (7) provides a successful estimation results.

4 Version 2: High Gain Observer Let us consider here the model (2) described by the following state space representation: {

ẋ = f (x) + g(x)u y = Cx

(12)

148 | A. Abdelkader et al.

Figure 8: Estimation of reactor temperature Tr (x6 ).

with: lv

− FV (Ci∗ −

[ [ [ [ f (x) = [ [ F lv ρlv [ − [ ρ [ [ [ [ [ [ g(x) = [ [ [ [ [

ρlv C) ρ i

+ υi r

] ] ] ] ] ] ΔHv Δr H UA ] − r + (T − T ) ] j r VCp ρCp VρCp ] UA (Tj − Tr ) VρC ] −

Cie V

lv lv

F ρ ρ

j j p

−

ρe C Vρ i

0 0 0 Fc

ρe ρ

ρe (C e T e VρCp p r

0

− Cp T r )

] ] ] ] ] ] ] ] ]

with – xi ∈ ℜn , state vector xT = [C1..4 V Tr Tj ] – u ∈ ℜm , control vector uT = [F e , Tje ] u { ẋ = f (x) + g(x) [ 1 { { { u2 { { h (x) 1 { { ] [ [ { { { { y = [ h2 (x) ] = [ { [ h3 (x) ] [

] x1 ] x5 ] x6 ]

(13)

In [5], authors show that if the system is uniformly observable, it can be transformed via the following change of variable:

149

Comparison of Nonlinear Observers for a Nonlinear System |

x1 x5 x6 Lf [x1 ] Lf [x5 ] [ Lf [x6 ]

[ [ [ [ h(x) x → ψ(x) = [ ]=[ [ Lf [h(x)] [ [ [

] ] ] ] ] = [ z1 ] ] z2 ] ] ]

(14)

]

𝜕h with z1 = [x1 , x5 , x6 ]T , z2 = [Lf [x1 ], Lf [x5 ], Lf [x6 ] ]T , Lf (h(x)) = ∑ 𝜕x fi i Under the following canonic form observability:

0 1 B1 (z, u) z1̇ z1 0 { { ̇ { { z = [ z2̇ ] = [ 0 0 ] [ z2 ] + [ φ(z) ] [ B2 (z, u) ] ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ { { B(z,u) Φ(z) A { { { y = z1

(15)

with Lg [x1 ] ] [ B1 (z, u) = [ Lg [x5 ] ] , [ Lg [x6 ] ]

Lf Lg [x1 ] ] [ B2 (z, u) = [ Lf Lg [x5 ] ] [ Lf Lg [x6 ] ]

L2f [x1 ] [ 2 ] and φ(z) = [ Lf [x5 ] ] 2 [ Lf [x6 ] ]

we have L1f h(x(t)) =

dh(x(t)) 𝜕h(x(t)) dx(t) 𝜕h(x(t)) = . = .f (x(t)) dt 𝜕x dt 𝜕x

with L1f h(x(t)) is the Lie derivative of a function h with respect to function f to order one. In fact, we define the Lie successive derivative of a function h with respect to function f as: Lkf h(x(t)) = Lf [Lk−1 f h(x(t))],

with L0f h(x(t)) = h(x(t))

We have so: 1 dh(x(t)) d2 h(x(t)) d[ dt ] d[Lf h(x(t))] = = dt dt dt 2 1 1 𝜕[Lf h(x(t))] dx(t) 𝜕[Lf h(x(t))] . = .f (x(t)) = 𝜕x dt 𝜕x

L2f h(x(t)) = Lf [L1f h(x(t))] =

We now state the following assumptions: Assumption 1: The function Φ(z) is globally lipschitz with respect to z(t). Assumption 2: The function B(z, u) is globally lipschitz with respect to z(t), uniformly with respect to the input u(t).

150 | A. Abdelkader et al.

4.1 The Observer Equation A high gain observer for system (13) is described by the following dynamical system: ż̂ = Aẑ + Φ(z)̂ + B(z,̂ u) − Gθ (C ẑ − y)

(16)

Where – θ > 0 is the setting parameter of the observer. – Sθ is the unique solution of the algebraic equation of Lyapunov: Ṡθ = −θSθ + AT Sθ − Sθ A + Cz CzT –

θG

vector gain given by: Gθ = Sθ−1 G = [ θ2 G1 ]. Where G is a constant matrix so that 2 (A − GC) is stable [2].

̂ ̂ can be obtained by x(t) ̂ = Ψ−1 (z(t)). Ones the state vector z(t) is estimated, x(t) Some−1 times, the function Ψ can not be expressed in terms of z and one way to overcome this difficulty is expressing the equation of the observer directly in the original coordx(t) dinates x. In fact, taking account that: dz(t) = 𝜕Φ(x) . dt 𝜕x dt The observer equation in the original coordinate system is given by: {

−1 ̂ − [ 𝜕Ψ(x) ] Gθ (C x̂ − y) ẋ̂ = f (x)̂ + g(x)u 𝜕x ŷ = Cx x̂

(17)

For more details (Proof), see [14]. Here we use to estimate the states the values of experimental outputs, without taking account for the jacket temperature output Tj one hand to reduce the calculation and on the other hand to use the matrix Ψ (14) and its inverse one needs to have a square matrix. – Results and discussion The performances of the observer is tested in the simulation, we give simulation results below. The initial conditions for the system and observer are: x1 (0) = 0.09 mol/l, x̂1 (0) = 0.093 mol/l, x̂2 (0) = 0 mol/l,

x5 (0) = 1.4 l and x̂5 (0) = 1.45 l,

x̂3 (0) = 0 mol/l,

x6 (0) = 294 K.

x̂6 (0) = 304.5 K, x̂4 (0) = 0 mol/l.

Finally, the observer has been tuned with: θ = 2.2. The proposed observer in this section is able to estimate ethanol concentration (x2 ), water concentration (x3 ) and ester concentration (x4 ) as can be seen in figure 9.

Comparison of Nonlinear Observers for a Nonlinear System | 151

Figure 9: Evolution of Ester, Ethanol and water concentration.

5 Extended Kalman Filter The Kalman filter permits estimation of a linear system [15]. When a system is nonlinear, Kalman filter is applied to the linearized model at each instant, so the nonlinear filtering algorithm is called Extended Kalman filter [16]. It is essentially a set of mathematical equations that allow to implement a estimator that is optimal in the sense that it minimizes the estimated error covariance. Let our model (2) is in the form (3), begins as follows: {

ẋ = f (x, u) + v(t) y = h(x) + w(t)

(18)

with v(t) and w(t) Gaussian noise. In our case, we consider the noise as errors of experimental measurement. The EKF can be expressed as follows: {

ẋ̂ = f (x,̂ u) − PH T R−1 (h(x)̂ − y) Ṗ = FP + PF T − PH T R−1 HP + Q

(19)

𝜕f with F = 𝜕x and H = 𝜕h . 𝜕x The jacobian matrix is defined by:

F=(

𝜕f1 𝜕x1

.. .

𝜕f6 𝜕x1

⋅⋅⋅ .. . ⋅⋅⋅

𝜕f1 𝜕x6

.. .

𝜕f6 𝜕x6

),

H=(

𝜕h1 𝜕x1

.. .

𝜕h3 𝜕x1

⋅⋅⋅ .. . ...

𝜕h1 𝜕x6

.. .

)

(20)

𝜕h3 𝜕x6

Matrix P0 (initial value vector and error covariance matrix), matrix Q (process noise covariance matrix) and matrix R (measurement noise covariance matrix) represent the tuning parameters for the extended Kalman filter. As there is no general rule for the choice of these tuning parameters besides some restrictions as positive

152 | A. Abdelkader et al. definiteness and an adequate dimension, they have to be adjusted experimentally. It is worth noting that in literature it is common to choose them as diagonal matrices. The matrix R and Q values are as follows: Q = 10−2 × I6

R = 7.10−2 × I3

5.1 Comparison Simulation of the Three Observers Figure 10 show the evolution of different concentration (Ester, Ethanol and water) using EKF observer. Comparison of the estimation performance of different observers and their relative errors are shown in Figures 11, 12, 13, 14, 15 and 16 respectively.

Figure 10: Evolution of Ester, Ethanol and water concentration.

Figure 11: Estimation of Fatty acids concentration.

Comparison of Nonlinear Observers for a Nonlinear System | 153

Figure 12: Estimation and relative errors of fatty acids concentration x1 .

Figure 13: Estimation of Volume x5 .

Figure 14: Estimation and relative errors of volume x5 .

154 | A. Abdelkader et al.

Figure 15: Estimation of x6 Reactor temperature.

Figure 16: Estimation and relative errors of reactor temperature x6 .

We note that the results show that the first version of the observer gives a good estimate this is due to the synthesis of the observer and the choice of tuning parameters.

6 Conclusion This study is intended to investigate the effectiveness of various nonlinear observers, both versions of high gain observers that we have proposed in this paper for a MIMO SCSTR are designed using a differential model after a transformation from algebraicdifferential model.

Comparison of Nonlinear Observers for a Nonlinear System | 155

The objectives are to underline the different parameters synthesis of three observers The second version proposed in this study, uses a tuning parameter θ requires a good set of this parameter, but we have a constant gain and there is not an update to each calculation, which explains the less satisfactory results compared to the first version. In fact, in the first version we used three tuning parameters where ρ is the parameter who control the speed of convergence. The correction gain is computed at the same time as the estimated state, and therefore constantly update. The observers of the kind presented here have been tested successfully for our chemical reactor presented in this paper. In the simulation the two version of High Gain Observer and the EKF have quite a good results. Whatever the method used to estimate the state variables, the performances depend on the choices of tuning parameters such as initial values, weighting matrices, the matrix gain, etc... The EKF is widely used and give a good perfermances in practice. The main disadvantage for the EKF algorithm is that it requires an approximate knowledge of the initials conditions. The good behavior of the first version of high gain observer compared with than the second version and the EKF is clear.

Bibliography [1]

G. Bornard and H. Hammouri. A high gain observer for a class of uniformly observable systems. In Proc. 30th IEEE Conference on Decision and Control, 122, Brighton, England, 1991. [2] M. Farza, K. Busawon, and H. Hammouri. Simple nonlinear observers for on line estimation of kinetic rates in bioreactors. Automatica J. IFAC, 34(3):301–318, 1998. [3] H. Hammouri, G. Bornard, and K. Busawon. High gain observer based on a structured nonlinear systems. Transactions on Automatic Control, 55(4): 987–992, 2010. [4] H. Hammouri and M. Farza. Nonlinear observers for locally uniformly observable systems. Control, Optimization and Calculus of Variation; ESAIM COCV, 9: 353–370, March 2003. [5] J. P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systems application to bioreactors. IEEE Transaction Automatique Control, 37: 875–880, 1992. [6] R. Aguilar, R. Martinez-Guerra, and A. Poznyak. Reaction heat estimation in continuous chemical reactors using high gain observers. Chemical Engineering Journal, 87: 351–356, 2002. [7] N. Boizot. Nonlinear Observers and Applications, 363. Springer, 2007. [8] G. Wang, S. Peng, and H. Huang. A sliding observer for nonlinear process control. Chemical Engineering Science, 52(5): 787–805, 1997. [9] M. Boussaada, K. Fiaty, and G. Gilles, A moving state estimator of an olive oil waste esterification in a semi-batch reactor: experimental validation. ISCRE Hong Kong, 2002. [10] M. Boussaada. Modélisation, estimation d’etat et commande non linéaire d’un réacteur semi continu d’estrification des acides gras contenus dans les huiles de grignons d’olives. Thesis, LAGEP, Lyon, 2002.

156 | A. Abdelkader et al.

[11] M. H. Frikha, M. Benzina, and S. Gabsi. Equation Cinétique de l’estérification des acides libres de l’huile de grignons d’olive par l’éthanol au dessus de l’azéotrope eau éthanol. Entropie 206, 48–53, 1997. [12] R. C. Reid, J. M. Prausnitz, and B. E. Poling. The properties of gases and liquids. Mc Graw-Hill, 4Ed., 1988. [13] A. Abdelkader, M. Boussada, A. Nouri, H. Hammouri, and K. Fiatty. A nonlinear high gain observer for an olive oil waste esterification in a Continuous Stirred Tank Reactor. International conference on Sciences and Techniques of Automatic control & computer engineering, 1049–1054, 2014. [14] M. Bouhamida, B. Daaou, A. Mansouri, and M. Chenafa. Observer-based input-output linearization control of a multivariable continuous chemical reactor. J. Korean Math. Soc, 49(3):641–658, 2012. [15] E. Kalman and R. S. Bucy. New results in linear filtering and prediction theory. Trans. ASME, 83(1), 95–108, 1961. [16] D. I. Wilson, M. Agarwal, and D. W. T. Rippin. Experiences implementing the extended Kalman filter on an industrial batch reactor. Computers Chem. Engng, Elsevier Science Ltd, 22(11), 1653–1672, 1998.

Biographies Amira Abdelkader received the PhD in 2016 in electrical engineering from National School of Engineers of Gabes co-supervised with claude Bernard University Lyon 1 (France) and his automatic-electrical engineering diploma, and the Master in automatic and smart technologies from the National School of Engineers of Gabes, Tunisia in 2008 and 2010. His research interest include nonlinear système, observer, diagnostic.

Moez Boussada received the DEA in 1997 and a PhD in 2002 in industrial automation from the University of Claude Bernard Lyon 1 France. From 2003 to 2011, he was an Assistant Professor at the high institute of computer sciences and multimedia (Gabès Tunisia) and since 2012 he is an assistant professor at the high school of engineers of Gabèes Tunisia. Dr Boussada’s research interests include, the state estimation and control algorithms for nonlinear systems, FDI, FTC in the theoretical side and he is interested in application in chemical plants control and renewable energy.

Comparison of Nonlinear Observers for a Nonlinear System | 157

Koffi Fiaty received his PhD from Claude Bernard Lyon-1 University in 1991 after his chemical engineering diplomain 1987 from the Industrial high school of chemistry in Lyon. Presently his is Associate Professor at the Claude Bernard Lyon-1 University. His research field concerns transport phenomena coupled or not with chemical reaction, through micro porous systems loaded or not in membranes. In recent years he is interested in problems of pollution particularly the treatment of greenhouse gases, and in the enhancement of agricultural wastes. At this level he initiates collaborations with some universities in Africa (University of Lomé in Togo, University Marien N’Gouabi of Brazzaville in Congo, National Engineering School of Gabes in Tunisia). Hassan Hammouri received the Ph. D. degrees of mathematics from the University Joseph Fourier in 1983 and the D. Sc. degree in control from the Institut National Polytechnique de Grenoble in 1991. Since 1992, he is professor of automatic control in the department of Electrical and Chemical engineering of University Lyon 1. His research interests include stability of nonlinear systems, nonlinear observer, fault diagnosis, control of distributed parameter systems, and applications to control of Electrical, Chemical and Biological systems.

Ahmed Said Nouri received the DEA. and the Ph. D. degrees in automatic from National Institute for Applied Sciences, France in 1989 and 1994, respectively. In 2008, he obtained the University Delegation in electric automatic and industrial informatics engineering from National School of Engineers of Sfax, Tunisia. He is professor in automatic control at National School of Engineering of Sfax (previously at Gabes), Tunisia. He was the chairman of the Electrical Engineering Department of National School of Engineering of Gabes. He is a member of the Research Unit of Numerical Control of Industrial Processes and is the general secretary of the Tunisian Association of Automatic and Numerisation. He is the co-author of the book The Elementary and Practical Theory of the Sliding Mode Control edited by Springer.

Asma Atitallah, Saida Bedoui, and Kamel Abderrahim

A Two-Step Procedure and Performance Analysis of Identification Algorithm for Hammerstein Time Delay System Abstract: It is quite challenging to identify in one step a nonlinear model with time delay due to the nonlinear relationship between the time delay and the other model parameters. Therefore, a two step procedure for estimating the system parameters and the time delay is proposed in this paper: The first step is to estimate the parameter vector using the gradient approach. Assuming that the lower and upper bounds of the time delay are known in advance, the Variable Regression Estimation technique is applied as the second step to identify the time delay. The convergence analysis of the proposed algorithm indicates that the parameter estimation errors converge to zero under persistent excitation conditions. Simulation results are presented to illustrate the performance of the proposed method. Keywords: Identification, Hammerstein systems, Time delay, Gradient method, Variable Regression Estimation technique, Convergence analysis

1 Introduction The behavior of dynamic systems can be described by models. Such models are exploited for the purposes of explanation, prediction, fault detection, control, etc. Block oriented models [1–4] are often used to model many nonlinear dynamical systems (for example control valves, chemical processes, power amplifiers). The dynamics and the nonlinearity of a system are modeled using connections of groups of linear dynamic and static nonlinear blocks [2, 4]. Series, parallel and feedback connections are possible ways to interconnect these blocks [2]. This flexibility provides the block-oriented modeling approach with an ability to capture a large class of nonlinear systems. In fact, the most well known block oriented models are the Hammerstein and Wiener models. They consist of a single linear dynamic block and static nonlinear subsystem. These models are used to decompose a complex system in simpler subsystems: the Hammerstein structure is used to model nonlinear systems where the nonlinear beAcknowledgement: This work was supported by the ministry of Higher Education and Scientific Research in Tunisia. Asma Atitallah, Saida Bedoui, Kamel Abderrahim, University of Gabes, National Engineering school of Gabes, Research Unit Numerical Control of Industrial Processes, Gabes, Tunisia, e-mails: [email protected], [email protected], [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 159–175. https://doi.org/10.1515/9783110591729-009

160 | A. Atitallah et al. haviour is present at the input of the system [5–8]. Concerning the Wiener structure, it is often used to model systems for which the static nonlinearity is present at the output of the system [9]. Due to their structured nature, parsimonious and fairly easy to analyze, Hammerstein models are chosen in this paper. System identification which contains generally parameter and time delay estimations is a technique that can be used to construct system models from experimental data [10–12]. Numerous approaches have been presented in the literature to estimate only the parameters of Hammerstein systems [13–19]. However, time delay estimation remains until now a discussed problem in this kind of nonlinear systems and this due to many reasons. Indeed, identifying time delay is not an easy task for systems with input, state or/and output delays. Thus due to their specific structure parameter that differs both from the system order and from the system parameters. Time delay estimation techniques have been used in various branches of physical science and technology [20–22]. Traditionally these are used in the area of signal processing for communication [20] and controlled processes [23–25]. In many real processes, the time delay is a priori unknown or time varying, in some circumstance, process parameters may be time-varying also, therefore real time delay and parameter estimation approaches are necessary. Convergence analysis of algorithms is also a basic topic for system identification. The martingale convergence theorem is used recently to analyse the performance of many algorithms such as stochastic gradient and least squares [26–31]. In the following, we propose an approach to estimate the system parameters and the time delay assuming that the lower and the upper bounds of the time delay are known in advances [11]. This assumption guarantee the convergence of the estimated time delay to a constant value. In fact, the main contribution of this paper is to discuss the performance analysis of the proposed approach which proved that the parameter estimation errors converge to zero by using the martingale convergence theorem. This paper is organized as follows. Section 2 presents the model and its assumptions. In Section 3, we present the proposed method for identification of Hammerstein time delay systems. In section 4, we discuss the convergence analysis of the proposed method. Section 5 provides a numerical example to show the effectiveness of the proposed algorithm. Finally, some concluding remarks are given in section 6.

2 System Description and Problem Statement Consider the Hammerstein time delay systems [13, 11] depicted in Figure 1: ̄ + v(k), A(q−1 )y(k) = q−d B(q−1 )u(k) nc

̄ u(k) = f (u(k)) = ∑ cl fl (u(k)), l=1

(1) (2)

A Two-Step Procedure and Performance Analysis of Identification Algorithm

| 161

Figure 1: Hammerstein time delay system.

where u(k) is the system input, y(k) is the system output, v(k) is a white noise, the ̄ of the nonlinear block is a linear combination of a known basis functions output u(k) f := {f1 , f2 , . . . , fnc } with coefficients (c1 , c2 , . . . , cnc ), d is the unknown time delay and it satisfies the following condition: dmin ≤ d ≤ dmax with dmin and dmax are constant positive scalars denoting, respectively, the lower and the upper time delays. A(q−1 ) and B(q−1 ) are two polynomials in shift operator q−1 with: A(q−1 ) := 1 + a1 q−1 + a2 q−2 + ⋅ ⋅ ⋅ + ana q−na , B(q−1 ) := b1 q−1 + b2 q−2 + ⋅ ⋅ ⋅ + bnb q−nb , Define the parameter vectors θ: a ] [ θ := [ b ] ∈ ℝna +nb +nc [ c ] where a := [a1 , a2 , . . . , ana ]T ∈ ℝna ,

b := [b1 , b2 , . . . , bnb ]T ∈ ℝnb , c := [c1 , c2 , . . . , cnc ]T ∈ ℝnc ,

The output information vector ϕ(k) and the input information matrix F(k) are defined respectively as: T

ϕ(k) := [−y(k − 1), −y(k − 2), . . . , −y(k − na )] ∈ ℝna ,

(3)

F(k) := q−d F(k)

(4)

and

162 | A. Atitallah et al. where f1 (u(k − 1)) .. . f (u(k − nb )) 1 [

[ F(k) = [ [

⋅⋅⋅ .. . ⋅⋅⋅

fnc (u(k − 1)) .. . fnc (u(k − nb ))

] ] ∈ ℝnb ×nc , ]

(5)

]

and q−d is the delay operator. Thus, the system (1) can be rewritten in this form: y(k) = ϕT (k)a + bT F(k)c + v(k).

(6)

Note that for any pair αb and c/α (α ≠ 0), the system (6) has the same input-output relationship. To avoid this problem, we need to fix a constraint of the nonlinear block. So, without loss of generality, we assume that ||c|| = 1, and the first nonzero entry of c is positive, i. e., c1 > 0, refer to [32]. For the identification model in (1)–(2), assume that: A1. u(k) = 0, y(k) = 0 and v(k) = 0 for k ≤ 0 A2. The nonlinear function fl is a polynomial, A3. The orders na , nb and nc are known, A4. The disturbance v(k) is a white noise with zero mean and finite variance σ 2 , A5. ||c|| = 1 and the first nonzero entry of c is positive, A6. The time delay is bounded, i. e. dmin ≤ d ≤ dmax . Generally, classical prediction error minimization methods are widely used for model identification where the cost function to be minimized is defined as: ̂ J(k, θ, d) = e2 (k) = (y(k) − y(k))

2

(7)

This cost function form is used thereafter where the prediction error e(k) is given by: ̂ e(k) = y(k) − ϕT (k)â − b̂ T q−d F(k)c.̂

(8)

In fact, it is quite challenging to identify in one step a Hammerstein model with time delay due to the nonlinear relationship between the delay parameter and the other model parameters. Therefore, the criterion J(k, θ, d) will be minimized in two steps with respect to the parameters and the time delay as follows: 𝜕J(k, θ,̂ d) = 0, 𝜕θ̂

J(k, θ, d)̂ = min[J(k, θ, di )] ∀ di ∈ [dmin , dmax ],

(9) (10)

where equations (9) and (10) are, respectively, the parameter and the time delay estimations equations.

A Two-Step Procedure and Performance Analysis of Identification Algorithm

| 163

Remark that equation (10) is not written in the partial derivative form because the time delay d̂ can take only integer values. Problem statement: In the following, an algorithms to handle the identification problem of a Hammerstein time delay systems is proposed by using the input/output measurement data {u(k), y(k)} and the convergence analysis of this algorithm is also developed.

3 The Two Stage Identification Algorithm In this section, an algorithm for the identification of system parameters and time delay is developed by applying respectively the gradient search and the Variable Regression Estimation technique.

3.1 Parameter Estimation Typically, gradient search is a useful tool for nonlinear optimization problems. It is used to minimize J(k, θ,̂ d) which leads to the following expression: ̂ ̂ ̂ − 1) − μ1 (k) 𝜕J(k, θ, d) , θ(k) = θ(k ̂ − 1) 2 𝜕θ(k

(11)

where μ1 (k) is the step size or convergence factor. Subsequently, ̂ ̂ − 1) + μ (k)χ(k)e (k), θ(k) = θ(k 1 1

(12)

where χ(k) is the generalized information vector defined by: ϕ(k) [ ̂ n +n +n ̂ − 1) ] c(k χ(k) = [ F(k) ] ∈ ℝ a b c, ̂ ̂T [ F (k)b(k − 1) ]

(13)

and the prediction error e1 (k) is defined as: ̂ c(k ̂ − 1) − b̂ T (k − 1)F(k) ̂ − 1). e1 (k) = y(k) − ϕT (k)a(k

(14)

We adopt the normalization constraint ĉ with the first positive element, i. e., ̂ c(k) = sgn(ĉ1 (k)) where sgn(.) represents the operator sign.

̂ c(k) , ̂ ‖c(k)‖

(15)

164 | A. Atitallah et al. Referring to the stochastic gradient algorithm for the linear systems [32], we can take the convergence factor to be: μ1 (k) =

1 , r(k)

(16)

with 󵄩2 󵄩 r(k) = r(k − 1) + 󵄩󵄩󵄩χ(k)󵄩󵄩󵄩 ,

r(0) = 1,

(17)

3.2 Time Delay Estimation In the following, we develop an algorithm for time delay identification systems using the Variable Regression Estimation technique when the lower dmin and upper dmax bounds of the time delay are known in advance. Using assumption A6 (dmin ≤ d ≤ dmax ) and solving the optimizing problem (18) leads to the algorithm of minimising J(k, θ, di ) as follows: J(k, θ, d)̂ = min[J(k, θ, di )] ∀ di ∈ [dmin , dmax ],

(18)

where J(k, θ, di ) = μ2 J(k − 1, θ, di ) + [e2 (k, θ, di )]

2

di ∈ [dmin , dmax ],

(19)

and μ2 is the forgetting factor and satisfying: μ2 ∈ [0, 1].

(20)

For iteration k, the error e2 (k, di ) is defined by: ̂ ̂ e2 (k, di ) = y(k) − ϕT (k)a(k) − b̂ T (k)F(k, di )c(k),

(21)

and F(k, di ) is given by: F(k, di ) = q−di F(k)

(22)

Notice that the regression matrix structure F(k, di ), at iteration k, changes when the time delay di browse of dmin to dmax . That is why the name “Variable Regression Estimation technique” is given to call this method. The time delay estimation is adjusted to minimize the criterion J(k, θ, di ) and the estimated time delay is given by: ̂ d(k) = arg min[J(k, θ, di )], di

∀di ∈ [dmin , dmax ].

(23)

A Two-Step Procedure and Performance Analysis of Identification Algorithm

| 165

4 Convergence Analysis and Algorithm The martingale theory is an important tool for analysing the convergence of recursive algorithms. In this section, we establish the convergence property of the proposed algorithm where the following Lemma and Theorem are required: Lemma 1 (Martingale convergence theorem [26, 29, 30]). Assume that the non-negative sequences {T(k)}, {η(k)} and {ζ (k)} satisfy the inequality: T(k) ≤ T(k − 1) + η(k) − ζ (k) ∞ and ∑∞ k=1 η(k) < ∞, then we have ∑k=1 ζ (k) < ∞ and ζ (k) → 0, T(k) is bounded.

The proof of Lemma 1 is straightforward and hence omitted [33]. Theorem 1. From assumption A4., we have: E[v(k)] = 0, E[v(k)2 ] = σ 2 , E[v(k)v(i)] = 0,

k ≠ i

where E denotes the expectation operator. For the system in (6) and the proposed algorithm in (12) and (16)–(17), there exist a positive constant m such that the following persistent excitation condition holds: A7.

k−1

∑

j=0

χ(j)χ T (j) ≥ mI, r(j)

a.s.

Then the parameter estimation error given by the proposed algorithm converge to zero, i. e., 󵄩󵄩 ̃ 󵄩󵄩2 󵄩󵄩θ(k)󵄩󵄩 → 0

a.s.

Proof. Define the parameter error vector: ̃ ̂ −θ θ(k) := θ(k) Inserting (14) and (16) into (12), we have: ̃ ̃ − 1) + χ(k) [y(k) − ϕT (k)a(k ̂ c(k ̂ − 1) − b̂ T (k − 1)F(k) ̂ − 1)] θ(k) = θ(k r(k) ̂ Assuming that F(k) = F(k) and using (6), equation (24) becomes: ̃ ̃ − 1) θ(k) = θ(k +

χ(k) T ̂ − 1) − b̂ T (k − 1)F(k)c(k ̂ − 1)] [ϕ (k)a + bT F(k)c + v(k) − ϕT (k)a(k r(k)

(24)

166 | A. Atitallah et al. ̃ − 1) + χ(k) [ϕT (k)(a − a(k ̂ − 1)) + (bT − b̂ T (k − 1))F(k)c = θ(k r(k) ̂ − 1)) + v(k)] + b̂ T (k − 1)F(k)(c − c(k ̃ − 1) = θ(k

χ(k) T ̃ ̃ − 1) − (F(k)c) b(k ̃ − 1) + v(k)] [−ϕT (k)a(k − 1) − b̂ T (k − 1)F(k)c(k r(k) ̃ − 1) + v(k)] (25) ̃ − 1) + χ(k) [− [ ϕT (k) (F(k)c)T b̂ T (k − 1)F(k) ] θ(k = θ(k r(k) +

where ̃ − 1) = a(k ̂ − 1) − a ∈ ℝna a(k ̃ − 1) = b(k ̂ − 1) − b ∈ ℝnb b(k

̃ − 1) = c(k ̂ − 1) − c ∈ ℝnc c(k Let consider:

ϕ(k) ] n +n +n F(k)c ]∈ℝ a b c T ̂ [ F (k)b(k − 1) ]

[ ξ1 (k) = [

(26)

and ̃ − 1) T1 (k) = ξ1T (k)θ(k

(27)

Hence, equation (25) can be rewritten as: ̃ ̃ − 1) + χ(k) [−T (k) + v(k)] θ(k) = θ(k 1 r(k)

(28)

Taking the norm of both sides of (28), we obtain: 󵄩󵄩2 󵄩 χ(k) 󵄩 󵄩󵄩 ̃ 󵄩󵄩2 󵄩󵄩󵄩 ̃ [−T1 (k) + v(k)]󵄩󵄩󵄩 󵄩󵄩θ(k)󵄩󵄩 = 󵄩󵄩θ(k − 1) + 󵄩󵄩 󵄩󵄩 r(k) χ(k) ‖χ(k)‖2 2 󵄩2 󵄩 ̃ [−T1 (k) + v(k)] − 1)󵄩󵄩󵄩 + 2θ̃ T (k − 1) = 󵄩󵄩󵄩θ(k [−T1 (k) + v(k)] + 2 r(k) r (k) θ̃ T (k − 1)χ(k) 󵄩2 󵄩 ̃ − 1)󵄩󵄩󵄩 + 2 = 󵄩󵄩󵄩θ(k [−T1 (k) + v(k)] r(k) +

‖χ(k)‖2 2 [T1 (k) − 2T1 (k)v(k) + v2 (k)] r 2 (k)

Or, we have ϕ(k) Ona ] [ n +n +n ̃ − 1) ] F(k)c ] + [ F(k)c(k ] = ξ1 (k) + ξ2 (k) ∈ ℝ a b c T ̂ Onc ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [ F (k)b(k − 1) ] ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [ ]

[ X(k) = [

ξ1 (k)

ξ2 (k)

(29)

A Two-Step Procedure and Performance Analysis of Identification Algorithm 0

0

0

0

| 167

where Ona = [ .. ] ∈ ℝna and Onc = [ .. ] ∈ ℝnc . . Hence, equation (29) becomes:

T 2 (k) ‖χ(k)‖2 T12 (k) θ̃ T (k − 1)χ(k)v(k) 󵄩2 󵄩󵄩 ̃ 󵄩󵄩2 󵄩󵄩 ̃ + +2 󵄩󵄩θ(k)󵄩󵄩 = 󵄩󵄩θ(k − 1)󵄩󵄩󵄩 − 2 1 2 r(k) r(k) r (k) 2 T ̃ ‖χ(k)‖ T1 (k)v(k) θ (k − 1)ξ2 (k)T1 (k) ‖χ(k)‖2 v2 (k) −2 −2 + 2 r(k) r (k) r 2 (k) ‖χ(k)‖2 2 θ̃ T (k − 1)χ(k)v(k) 2 󵄩2 󵄩 ̃ − 2 − 1)󵄩󵄩󵄩 − [ = 󵄩󵄩󵄩θ(k ]T1 (k) + 2 r(k) r(k) r (k) −2

‖χ(k)‖2 T1 (k)v(k) θ̃ T (k − 1)ξ2 (k)T1 (k) ‖χ(k)‖2 v2 (k) + −2 2 r(k) r (k) r 2 (k)

(30)

Using (17), we have: −[

‖χ(k)‖2 1 2 ] 0

(6)

with: Pi ∈ Rnxn . This function allows relaxing the constraints imposed by the quadratic approach. Indeed, finding a Lyapunov matrix for each local model is easier than find a common Lyapunov matrix for all ones. To find the matrices Pi , a convex optimization procedure was proposed by Johansson [19] in the case of nonlinear systems continuously differentiable. Otherwise, Aouani et al. [12] propose a theorem using this kind of Lyapunov function as: Theorem 1 ([12]). The continuous LPV polytopic system (1) is stable such that the time derivative of the uncertainty satisfies (4), if there exist symmetric and positive matrices P1 ⋅ ⋅ ⋅ Pr , and a matrix G with appropriate dimensions, that allows separation between Pi and Ai and if a positive real α verify v < 2α, the following LMI is required: (

vPi − 2αPi Pi + GT Ai + αGT

(∗)T ) 0, P2i > 0, Qi > 0, i = 1, 2, . . . , N, ATλ Pi + Pi Aλ + FλT (Ψi ⊗ Qi )Fλ < 0,

λ = 1, . . . , r

(25)

where Aλ = [

A1λ A3λ

A2λ ], A4λ

Fλ = [

A1λ I

A2λ ], 0

and

with

ωci =

Pi = diag{P1i , P2i }.

For i = 2, 3, . . . , N − 1, Ψi = [

jωci ], −ωi−1 ωi

−1 −jωci

(ωi−1 + ωi ) . 2

(26)

0 ] −ω2N−1

(27)

For i = 1 and i = N, Ψ1 = [

0 ] ω21

−1 0

and ΨN = [

1 0

respectively. Proof. For i = 2, . . . , N − 1, i = 1 and i = N, according to [9] the matrix Ψi should be taking as (26) and (27), respectively. The condition in (25), can be written as F T (h)(Φ ⊗ P1i + Ψi ⊗ Qi )F(h) + Θi (h) < 0

(28)

where F(h) = [

A1 (h) I

A2 (h) ], 0

Φ=[

0 1

1 ] 0

(29)

and Θi (h) = [

T

A3 (h) 0

A4 (h) A (h) ] (Φ ⊗ P2i ) [ 3 I 0

A4 (h) ] I

(30)

Denote G(jω, h) = (jωI − A1 (h))−1 A2 (h), by the lemma 4, the following inequality follows: ∗

[

G(jω, h) G(jω, h) ] Θi (h) [ ] < 0, I I

∀ω ∈ Ω+

note that S(jω, h) = A4 (h) + A3 (h)(jωI − A1 (h))−1 A2 (h) = A4 (h) + A3 (h)G(jω, h).

(31)

274 | I. Er Rachid et al. Thus, substituting the expression of Θi (h) into (31) results in ∗

S(jω, h) S(jω, h) [ ] (Φ ⊗ P2i ) [ ] < 0, I I

∀ω ∈ Ω+

(32)

or in a more compact form, S(jω, h)∗ P2i + P2i S(jω, h) < 0,

P2i > 0,

∀ω ∈ Ω+

(33)

So Reλ(S(jω, h)) < 0 is finally guaranteed for all ω ∈ Ω+ . Combining Reλ(A1 (h)) < 0, Reλ(A4 (h)) < 0 and Reλ(S(jω, h)) < 0, we conclude that system (1) is asymptotically stable based on Lemma 2. The proof is completed. Remark 3. When N = 1, if letting Qi = 0, P1i > 0 and P2i > 0 be real, then (25) reduces to (17), that is, Lemma 6 is a special case of Theorem 1. Theorem 2. For a given positive integer N, define frequency intervals Ω+ as in (24). System (1) is asymptotically stable if there exist P1i > 0, P2i > 0, Qi > 0, Y1i , Y2i , i = 2, 3, . . . , N − 1, such that ϒ11i [ ∗ [ Ξiλ = [ [ ∗ [ ∗

ϒ12iλ ϒ22iλ ∗ ∗

0 ϒ23iλ ϒ33iλ ∗

ϒ14iλ ϒ24iλ ϒ34iλ ϒ44iλ

] ] ] < 0, ]

λ = 1, . . . , r

(34)

]

ϒ11i = −Qi − Y1i − Y1iT , ϒ12iλ = P1i + jωci Qi − Y1iT + Y1i A1λ , ϒ14iλ = Y1i A2λ , ϒ22iλ = −ωi−1 ωi Qi + AT1λ Y1iT + Y1i A1λ , ϒ23iλ = AT3λ Y2iT , ϒ24iλ = AT3λ Y2iT + Y1i A2λ , ϒ33i = −Y2i − Y2iT , ϒ34iλ = P2i − Y2iT + Y2i A4λ , ϒ44iλ = Y2i A4λ + AT4λ Y2iT . For i = 1, we replace ϒ12i , ϒ22iλ in (34) by T ϒ121λ = P11 − Y11 + Y11 A1λ , 2 T ϒ221λ = ω1 Q1 + AT1λ Y11 + Y11 A1λ , respectively. For i = N, we replace ϒ11i , ϒ12iλ and ϒ22iλ in (34) by T ϒ11N = QN − Y1N − Y1N , T ϒ12Nλ = P1N − Y1N + Y1N A1λ , T ϒ22Nλ = −ω2N−1 QN + AT1λ Y1N + Y1N A1λ , respectively. Proof. From Theorem 1, let Γ=[

Φ ⊗ P1i + Ψi ⊗ Qi ∗

0 ]. Φ ⊗ P2i

(35)

Stability of 2-D Continuous T-S Fuzzy Systems Based on KYP Lemma

| 275

According to [9], for i = 2, . . . , N − 1, i = 1 and i = N, Ψi as in (26) and (27) respectively, and Φ as in (29). Let Y1i [ Y [ Y = [ 1i [ 0 [ 0

0 0 Y2i Y2i

] ] ], ]

Z(h) = [

A1 (h) A3 (h)

−I 0

0 −I

A2 (h) ], A4 (h)

X = I,

]

(34) is equivalent to sym(X T YZ(h)) + Γ < 0

(36)

since one can choose X ⊥ = 0, the first inequality in (16) vanishes, and then by lemma 5, T (36) hold for some Y if and only if Z(h)⊥ ΓZ(h)⊥ < 0. Note that Z(h)⊥ can be selected as A1 (h) [ I [ Z(h)⊥ = [ [ A3 (h) [ 0

A2 (h) 0 A4 (h) I

] ] ], ] ] T

and then by calculation, we can obtain the equivalence between Z(h)⊥ ΓZ(h)⊥ < 0 and (25). Consequently (25) is equivalent to (34).

4 Numerical Example In this part, we provide an example to demonstrate the effectiveness of the proposed method. Consider system (1), where matrices are obtained by a [19]: Ac = [

A1 A3

A2 ], A4

Ac = (Ad − 1)(Ad + 1)−1

The matrices in the original problem are the following [15]: Ad = [

Ad1 Ad3

Ad2 ], Ad4

Ad3 = [

−0.1 −0.2

Ad1 = [

0.5 0.1

−0.1 ], 0.6

0.5 ], −0.1 Ad4 = [

Ad2 = [

−0.5 −0.1

0.4 0.6

1.1 ], 0.1

−0.5 ] −0.7

we obtain A1 = [

5.2979 4.2128

−16.0426 ], −12.7447

A2 = [

20.5957 16.4255

23.9149 ], 16.5106

276 | I. Er Rachid et al.

A3 = [

−5.7872 −7.4894

16.2553 ], 22.2128

A4 = [

−22.5745 −26.9787

−23.4894 ]. −30.5745

Theorem 2 with N = 1 don’t decide the stability of the system. By increasing N, it is found that Theorem 2 with N = 2, 4, 8 succeeds, note that, the above system is asymptotically stable only for i = 2, . . . , N. But for i = 1, whatever N, and whatever the way of partitioning the entire interval (fig. 1), always system is not stable. This is due to the quick variation of the curve of Reλmax (S(jω, h)) in the vicinity of ω0 = 0.

Figure 1: Reλmax (S(jω)) and γi∗ for Ω = [0, 1].

Denote γi∗ as the minimum value of −γi that satisfies sup Reλmax (S(jω, h)) < −γi < 0

ω∈Ωi

0 P

γi∗ could be computed from (34) by replacing Φ ⊗ P2i in (35) by [ P2i γ2ii ] and minimizing −γi . Fig. 1 shows Reλmax (S(jω, h)) and the executed γi∗ by Theorem 2 with N = 1, 2, 4, 8. The stability of the above system is obtained, since Reλmax (S(jω, h)) < 0 is evident. With N growing, it is further shown that −γi tends to the value of Reλmax (S(jω, h)) over Ω+ . Remark 4. In the literature, the authors of [15] and [16] get almost the similar situation for the discrete and the continuous-discrete cases, respectively, in which they note that the way of partitioning the entire frequency domain play a role in the conservativeness of the used method. Means that there is no systematic way to reduce this conservativeness.

Stability of 2-D Continuous T-S Fuzzy Systems Based on KYP Lemma

| 277

Remark 5. In the field [16, +∞], ℜe[λmax (S(jω, h))] remains relatively stationary to the value Reλmax (S(j∞)) = ℜe[λmax (A4 (h))] = −1.0850, then γi∗ also tends to this value throughout the domain. Even if it decomposed, we find very similar values to −1.0850. That’s why we worked on just the domain [0, 16], (Fig. 1).

5 Conclusions In this paper, the stability problem of 2-D continuous T-S fuzzy systems in Roesser model has been studied. Sufficient condition of stability for 2-D continuous T-S fuzzy system is proposed via LMIs formulation. By combining the GKYP lemma with the frequency-partitioning approach and introducing a piecewise constant matrix function, the conservativeness of the problem has been reduced. Finally, the advantage of the proposed approach have been demonstrated through numerical example.

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Biographies Ismail Er Rachid Was born in Fes, Morocco, in 1989. He received his Master degree in signals systems and informatics from University Sidi Mohammed Ben Abdellah, faculty of sciences, Fez, Morocco in 2013, he is currently a Ph. D. student in the same faculty. His current research interests include stability theory and performance of systems in the frequency domain.

Redouane Chaibi received the Master in Signals Systems and Computing from University of Sidi Mohammed Ben Abdellah, Faculty of Sciences, Fez, Morocco in 2014. His research interests include stability and stabilization of T-S fuzzy system.

280 | I. Er Rachid et al.

Abdelaziz Hmamed was born in Sefrou, Morocco, in 1951. He received his Doctorate of State degree in Electrical Engineering from the Faculty of Sciences, Rabat, Morocco, in 1985. Since 1986, he has been with the Department of Physics, Faculty of Sciences, Dhar El Mehraz at Fes, where he is currently a Full Professor. His research interests are delay systems, stability theory, systems with constraints, 2D systems and applications.

Rakia Abdeljawad, Nesrine Bahri, and Majda Ltaief

Stability Analysis and Stabilization of Discrete Singularly Perturbed System with Time-Delay Abstract: In this paper, the problems of asymptotic stability and stabilization for linear discrete singularly perturbed systems (SPS) with time delay are investigated. First, using an appropriate Lyapunov functional, new Linear Matrix Inequalities (LMIs) are proposed to analyse the asymptotic stability of discrete SPS with time delay. Delaydependent stability criterion and delay-independent stability criterion are treated. Afterward, a state feedback stabilization controller design based on an adequate Lyapunov functional is proposed. Sufficient conditions formulated in terme of LMIs, are given to guarantee the system closed loop stability. A set of numericals examples is given to illustrate the effectiveness of the proposed methods. Keywords: Discrete Singularly Perturbed System, Time-delay, Stability Bound, Stabilization, State feedback MSC 2010: 65C05, 62M20, 93E11, 62F15, 86A22

1 Introduction Time-delay is frequently present in many practical application such as biological systems, chemical process, nuclear reactors and aircraft system. It has undesirable influence on the system. It can lead to instability and poor performances of the system. In the last three decades, the stability analysis and synthesis of time-delay system have been widely studied [1–10]. On the other hand, many chemical and physical system are characterized by different dynamic phenomena: Slow and fast dynamic. These systems with different dynamics are modeled by SPSs. SPSs are characterized by small positive parameter which called singular perturbation parameter. This parameter is used to determine the degree of separation between the slow and fast modes of the system. The stability analysis of SPSs is totally different from that of the ordinary system. It is more complex due to the presence of different dynamics. It is known as stability bound. It consists of determining the upper bound of singular perturbation parameter ϵ∗ such that the system is stable for ∀ϵ ∈ [0, ϵ∗ ]. Acknowledgement: This work was supported by the ministry of Higher Education and Scientific Research in Tunisia. Rakia Abdeljawad, Nesrine Bahri, Majda Ltaief, University of Gabes, Gabes, Tunisia, e-mails: [email protected], [email protected], [email protected] De Gruyter Oldenbourg, ASSD – Advances in Systems, Signals and Devices, Volume 9, 2019, pp. 281–301. https://doi.org/10.1515/9783110591729-016

282 | R. Abdeljawad et al. The stability analysis and stabilization of continuous SPSs with or without time delay have been extensively studied by many researches [1–8]. In [1, 2] the stability bound of SPSs with time delay was treated via a time scale separation technique. In addition, Fengqi et al., Kang et al. and Liu et al. were studied the stability of continuous SPSs with time delay based on Lyapunov functional [3–5]. Moreover in [6], the stability of delay SPSs was examined by using state transformation and Lyapunov function. For the stabilization study, Chiou designed a controller for continuous SPSs with time delay via reduction technique [7]. In [8], the problem of exponential stabilization of delay SPSs was examined by using Lyapunov functional. In discrete time, singularly perturbed system can be represented by three models. The first model is pure singularly perturbed discrete system which is inherently discrete in nature. The two other models are obtained by slow and fast sampling of continuous time systems. Recently, the stability analysis and control design of discrete SPSs have been extensively studied by many researchers [9–16]. Most of the proposed methods were based on decomposition technique of SPSs. It consist to conclude the stability of the SPSs from the stability of the decomposed system (slow and fast subsystems) [10]. In addition, Li et al. proposed a method to analyse the stability bound of discrete SPSs in which the time delay has not yet been considered. This method is based on decomposition technique [11]. Moreover, Phillips studied the stability of discrete SPSs without time delay by using reduction technique [12]. However, this technique does not work when the slow and fast states cannot be separated completely. In [13], the exponential stability of discrete SPSs with time delay was examined by using Lyapunov-Krasovskii functional. But this method was under the assumption of the stability of the slow and fast subsystem. The stability analysis of SPSs is used to define the upper bound of singular perturbation parameter ϵ∗ to get a stable system. Several studies have been done in order to improve the stability bound of singularly perturbed system. In [14], Malloci et al. proposed a method to design a controllers for SPSs based on decomposition technique and Lyapunov function. But, this method was devoted to fast sampling model. This model does not always retain the time scale property, whereas the slow sampling model is characterized by its preservation of time scale character [15]. In [16], Dong et al. developed a method for designing H∞ controllers given in terms of solutions to a set of LMIs for slow sampling standard model. But, the time delay wasn’t considered. It should be pointed out that the stability and stabilization problem of discrete SPSs with time delay has not been fully investigated. This paper presents novel results on stability analysis and control synthesis for slow sampling SPSs with time delay. Our method is dedicated to discrete singularly perturbed delay system which are represented by R_model and C_model. It is based on an appropriate Lyapunov functional. It is developed without using technique decomposition or assuming the stability of the decomposed system. The obtained results are given as a set of LMIs. This paper is organized as follows; The second section gives description of the considered systems. In the third section, the asymptotic stability problem of discrete

Stability Analysis and Stabilization of Discrete Singularly Perturbed System

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SPSs with time delay is studied. In first time, delay independent stability criterion is treated. After that delay dependent stability criterion is examined by using Lyapunov function and LMIs technique. In the fourth section, the problem of stabilizing controller design is investigated using an appropriate Lyapunov function. Sufficient conditions are given to guarantee the asymptotic stability of the open and closed loop system. Several simulation examples are given to illustrate the application of the derived results. Finally, a conclusion finishes this paper. Notation: (*) is used for the blocks induced by symmetry. The superscript T stands for matrix transposition.

2 System Description In this paper, we will focus on the delayed discrete singularly perturbed systems obtained by slow sampling. The obtained discrete models can be represented by two equivalents models (R_model and C_model) given as follows [17, 18]: – R_model: x(k + 1) = A(ε)x(k) + Ad (ε)x(k − d) + B(ε)u(k)

(1)

Where x(k) ∈ Rn is the system state. u(k) ∈ Rp is the control input. d: positive integer, is the time delay. x(k) = [ B(ε) = [ –

x1 (k) ], x2 (k)

B1 ] εB2

A(ε) = [

A11 εA21

and Ad (ε) = [

Ad11 εAd21

A12 ] εA22 Ad12 ] εAd22

C_model: x(k + 1) = A(ε)x(k) + Ad (ε)x(k − d) + Bu(k) A(ε) = [

A11 A21

εA12 ], εA22

B=[

B1 ] B2

and Ad (ε) = [

(2) Ad11 Ad21

εAd12 ] εAd22

A11 ∈ Rn1 ∗n1 , A12 ∈ Rn1 ∗n2 , A21 ∈ Rn2 ∗n1 , A22 ∈ Rn2 ∗n2 , Ad11 ∈ Rn1 ∗n1 , Ad12 ∈ Rn1 ∗n2 , Ad21 ∈ Rn2 ∗n1 and Ad22 ∈ Rn2 ∗n2 are known constant matrices. n = n1 + n2 : order of system. ε is a small positive parameter, called singular perturbation parameter.

284 | R. Abdeljawad et al. The following Lemma will be used in establishing our main results. Lemma 1 ([19]). Let x(.) be a vector function. Let η(i) = x(i+1)−x(i), then for any constant matrix W ∈ Rn×n , W = W T > 0, the following inequality holds: −d(k)

k−1

∑

i=k−d(k)

ηT (i)Wη(i) ≤ [

T

x(k) −W ] [ x(k − d(k)) W

W x(k) ][ ] −W x(k − d(k))

Lemma 2 ([20]). For a given positive scalar ε∗ , if the following conditions are satisfied: F1 ≥ 0 ∗2

ε F1 + ε∗ F2 + F3 < 0 F3 < 0

Then ε2 F1 + εF2 + F3 < 0,

for ε ∈ [0, ε∗ ]

3 Stability Analysis In this section, stability bound of discrete unforced SPS with time delay (u = 0). Two methods are developed: The first method is based on delay-independent stability criterion. It determine the upper bound of ε such that the system is stable for any d ∈ ]0, ∞[. Whereas, the second method is based on delay-dependent stability criterion which gives the upper bound of singularly perturbed system parameter ε and the interval of time delay d that guarantee the stability of the unforced system. Unforced SPS with time delay (R_model and C_model) are given as follows: – For R_model and C_model: x(k + 1) = A(ε)x(k) + Ad (ε)x(k − d)

(3)

3.1 Delay Independent Stability Criterion Theorem 1. Consider the discrete singularly perturbed system with time delay (3). Given ε∗ > 0, the singularly perturbed system (3) is asymptotically stable for all ε ∈ [0, ε∗ ] and ∀ d = 1, 2, ..∞, if there exist symmetric positive-definite matrix, P > 0 and Q > 0 satisfying the LMIs (4) and (5): −P + Q ∗ ∗ [ [ [

0 −Q ∗

AT (0)P ] ATd (0)P ] < 0 −P ]

(4)

Stability Analysis and Stabilization of Discrete Singularly Perturbed System

−P + Q ∗ ∗ [ [ [

| 285

AT (ε∗ )P ] ATd (ε∗ )P ] < 0 −P ]

0 −Q ∗

(5)

Proof. Define a quadratic Lyapunov-Krasovskii functional as follows: k−1

v(k) = xT (k)Px(k) + ∑ xT (i)Qx(i)

(6)

i=k−d

Where P and Q are symmetric positive-definite matrix. Δv = v(k + 1) − v(k) = xT (k + 1)Px(k + 1) − xT (k)Px(k) + T

T

k

k−1

i=k−d+1 T T

i=k−d

∑ xT (i)Qx(i) − ∑ xT (i)Qx(i)

= x (k)[A (ε)PA(ε) − P + Q]x(k) + x (k)A (ε)PAd (ε)x(k − d) + xT (k − d) ATd (ε)PA(ε)x(k) + xT (k − d)(ATd (ε)PAd (ε) − Q)x(k − d)

= ξ T (k)Φ(ε)ξ (k).

(7)

Where ξ (k) = [

x(k) ] x(k − d)

and

Φ(ε) = [

AT (ε)PA(ε) − P + Q ∗

AT (ε)PD(ε) ] D (ε)PD(ε) − Q T

Δv < 0 ∀ε ∈ [0, ε∗ ] if and only if ϕ(ε) < 0, ∀ε ∈ [0, ε∗ ] i. e., [

AT (ε)PA(ε) − P + Q ∗

AT (ε)PAd (ε) T Ad (ε)PAd (ε) − Q

] 0 and d > 0, the singularly perturbed system with time delay is asymptotically stable for all ε ∈ [0, ε∗ ] and d > 0, if there exist symmetric positive-definite matrix, P > 0, Q > 0, and M > 0 satisfying the LMIs (10) and (11): −P + Q − M ∗ ∗ ∗ [ [ [ [ [

−P + Q − M ∗ ∗ ∗ [ [ [ [ [

M −Q − M ∗ ∗ M −Q − M ∗ ∗

d(AT (0) − I)M dATd (0)M −M ∗ T

d(A (ε ) − I)M dATd (ε∗ )M −M ∗ ∗

AT (0)P ATd (0)P 0 −P T

A (ε )P ATd (ε∗ )P 0 −P

] ] ] 0 and M̄ > 0 and matrix K̄ satisfying the LMIs (24) and (25): [ [ [ [

Q̄ − X − M̄ M̄

∗ −Q̄ − M̄ Ad (ε∗ )X dAd (ε∗ )X

A(ε∗ )X + B(ε∗ )K̄ ∗ ∗ ̄ [ dA(ε )X + dB(ε )K − dX ∗ Q̄ − X − M̄ [ ̄ ̄ M − Q − M̄ [ [ ̄ [ A(0)X + B(0)K Ad (0)X ̄ [ dA(0)X + dB(0)K − dX dAd (0)X

∗ ∗ −X 0 ∗ ∗ −X 0

∗ ∗ ∗ −2X + M̄ ∗ ∗ ∗ −2X + M̄

] ] ] 0 and matrix K̄ satisfying the LMIs (41) and (42): [ [ [ [ [

Q̄ − X − M̄ M̄ ∗ A(ε )X + BK̄ dA(ε∗ )X + dBK̄ − dX

∗ −Q̄ − M̄ Ad (ε∗ )X dAd (ε∗ )X

∗ ∗ −X 0

∗ ∗ ∗ −2X + M̄

] ] ] 0 , if ỹi (k) < 0

i = 1..p

(4)

The state reconstruction error is defined as follows: ̃ ̂ x(k) = x(k) − x(k)

(5)

̂ Replacing x(k) and x(k) by its expression, we obtain the dynamical reconstruction error system ̃ ̃ + 1) = (A − LH)x(k) ̃ − Msign(y(k)) x(k

(6)

The main objective of this section is to obtain the observer gain matrix L without degrading the stability of the proposed observer. In the present work, the proposed method in the determining of the observer gain matrix is the LMI approach based on the standard Lyapunov function. Let consider the following lemma and Schur complement that will be used in the theorem proof. Lemma. For any matrices X and Y with appropriate dimensions, the following property holds for any positive scalar α [22]: X T Y + Y T X < α−1 X T X + αY T Y

(7)

Schur Complement ([23]). Suppose R = RT , G = GT and S of appropriate dimensions, the following Linear Matrices Inequalities (LMIs) are equivalent: i)

(

G ST

S ) 0 satisfying (A − LH)T P(A − LH) + α−1 (A − LH)T PP(A − LH) − P < −Q

(16)

Using the Schur complement applied to (16), the following inequality is obtained [

−P + α−1 (A − LH)T PP(A − LH) + Q P(A − LH)

(A − LH)T P ]