Switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems

doi:10.1016/j.automatica.2015.02.004

Automatica

Volume 54, April 2015, Pages 217-228

https://doi.org/10.1016/j.automatica.2015.02.004 Get rights and content

Abstract

This paper investigates the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary. A new switched adaptive control technique for the problem studied is set up by exploiting the generalized multiple Lyapunov functions method and the parameter separation technique. Different update laws are designed to reduce the conservativeness caused by adoption of a common update law for all subsystems. Also, the proposed technique permits removal of a common restriction in which the parameterization in the switched systems is restricted to a linear parameterization. As an application of the proposed design technique, stabilization for a class of non-triangular switched systems with nonlinear parameterization is achieved by design of adaptive controllers. Since no subsystem of such a switched system is assumed to be adaptively stabilizable, the problem under study cannot be handled by the existing methods. A two inverted pendulums as a practical example is also provided to demonstrate the effectiveness of the proposed design method.

Introduction

Adaptive control of non-switched nonlinear systems with parametric uncertainty has attracted considerable attention in the field of nonlinear control (Astolfi and Ortega, 2003, Huang, 2009, Karagiannis and Astolfi, 2008, Krstic et al., 1995, Tao, 2003). One of the reasons for the rapid growth and continuing popularity of adaptive control is its clearly defined goal: to control systems with uncertainties by estimating unknown system’s parameters. For about two decades, there is a vast amount of literature on design and analysis of various adaptive control systems using rigorous proofs (Back et al., 2007, Haddad et al., 2003, Hong et al., 2009, Spooner and Passino, 1999). In particular, adaptive control has proven its great capability in compensating for non-switched nonlinearly parameterized systems involving inherent nonlinearity on the basis of a parameter separation technique (Lin and Qian, 2002a, Lin and Qian, 2002b).

On the other hand, the study of switched systems has been extensively investigated in the last decade (Cao and Morse, 2010, Colaneri et al., 2008, Girard et al., 2010, Goebel et al., 2009, Mancilla-Aguilar and García, 2006, Qiao and Cheng, 2009), and the rapidly developing area of intelligent control, such as robotic, mechatronic and mechanical systems, gene regulatory networks, switching power converters, is an important source of motivation for this study (Hespanha, 2003, Lin and Antsaklis, 2009, Mojica-Nava et al., 2010, Serres et al., 2011). Meanwhile, many methodologies such as single Lyapunov function, multiple Lyapunov functions (MLFs), average dwell-time have been proposed based on some specified classes of switching laws in the study of switched systems (Branicky, 1998, Han et al., 2010, Liberzon, 2003, Liberzon and Morse, 1999, Long and Zhao, 2012, Long and Zhao, 2014b).

Recently, some results on adaptive control for switched linear systems have appeared (see, e.g., Bernardo, Montanaro, & Santini, 2008, Chiang & Fu, 2009, Di Bernardo, Montanaro, & Santini, 2013, Khalid, Osamah, & Kamal, 2005, Sang & Tao, 2012). However, there has been relatively little work for switched nonlinear systems with parametric uncertainty up to now. Adaptive stabilization for strict-feedback switched nonlinear systems under arbitrary switchings is achieved via backstepping which provides a common Lyapunov function (CLF) (Tamba & Leksono, 2010); in Han, Ge, and Lee (2009), an adaptive NN control scheme for strict-feedback switched nonlinear systems is proposed for switchings with a certain dwell-time; When the solvability of the disturbance rejection problem for subsystems is not assumed, an adaptive disturbance rejection problem for switched nonlinear systems in strict-feedback form with unknown exosystem is studied in Long and Zhao (2014a). It should be observed that the literature mentioned above has focused on adaptive control of switched nonlinear systems with linear parameterization, and a common update law is used to estimate all the vector parameters in different subsystems. To the best of our knowledge, no results in the switched nonlinearly parameterized systems have been reported. The crucial obstacle in the study is the complexity arising from the interaction among the system structure, uncertain parameters and switchings. There are two main issues to be addressed: When no adaptive stabilization problem for subsystems is solvable, how to solve the adaptive stabilization problem via design of a switching law and adaptive state-feedback controllers? In particular, how to design different update laws and a MLFs based switching signal to reduce the conservativeness caused by adoption of a common update law for all subsystems? Hence, the study of adaptive control of switched nonlinearly parameterized systems is of great significance and remains an open area.

Motivated by the above considerations, this paper studies the adaptive control problem for switched nonlinearly parameterized systems. None of individual subsystems is assumed to be globally adaptively stabilizable. Compared with the vast existing literature on switched nonlinear systems, the results of this paper have three distinct features. First of all, a main result about adaptive control for switched nonlinearly parameterized systems is established for the first time, which provides a tool for analyzing the behavior of switched nonlinearly parameterized systems. A sufficient condition for the adaptive stabilization problem is derived by exploiting the generalized multiple Lyapunov functions (GMLFs) method (Zhao & Hill, 2008) and the parameter separation technique (Lin & Qian, 2002a). Secondly, in order to reduce the conservativeness caused by adoption of a common update law for all subsystems, different update laws are designed. Also, the switched adaptive control technique permits removal of a common restriction in which the parametric uncertainty in the switched systems is restricted to a linear parameterization. Finally, an application of the design procedure to non-triangular switched systems with nonlinear parameterization is investigated. The dual design of adaptive controllers and switching laws are constructive by designing the MLFs and different update laws of all subsystems based on the GMLFs method and the parameter separation technique and the adding a power integrator technique (Lin and Qian, 2002a, Qian and Lin, 2001).

Notation: $‖ \cdot ‖$ denotes the standard Euclidean norm or the induced matrix 2-norm.

Section snippets

System description and preliminaries

In this section, we introduce an adaptive control problem for switched nonlinear systems with constant nonlinearly parameterized uncertainty. We consider the class of switched nonlinear systems of the form $\dot{x} = f_{σ (t)} (x, u_{σ (t)}, θ_{σ (t)}),$ where $x \in R^{n}$ is the system state, for each $k, u_{k} \in R^{n_{k}}$ is the control input of the $k$ th subsystem, $θ_{k} \in R^{q_{k}}$ is an unknown constant vector. $m$ is the number of subsystems of the switched system (1). The functions $f_{k} (\cdot), k \in M = {1, \dots, m}$ , are assumed to be $C^{1}$ with $f_{k} (0, 0, θ_{k}) = 0$ . The

Adaptive control

In this section, we establish a general theorem for the system (1) by exploiting the GMLFs method and the parameter separation technique, and that is fundamental to present our applications on adaptive control of non-triangular switched systems with nonlinear parameterization. For the system (1), a key point is to establish a switched adaptive control framework (see, Fig. 1).

In view of Remark 1, define ${\tilde{Θ}}_{k} = Θ_{k} - {\hat{Θ}}_{k}, k \in M$ , where ${\hat{Θ}}_{k} \in R$ is the estimate of $Θ_{k}$ .

Theorem 1

Consider the switched nonlinearly

Application to non-triangular switched systems

Recently, adaptive control of switched nonlinear systems in strict-feedback form under arbitrary switchings is achieved via backstepping which provides a CLF (Tamba & Leksono, 2010), and a common update law is used to estimate all the vector parameters in different subsystems. However, most switched systems do not possess a CLF and a common update law for all subsystems from both practical and theoretical points of view. Therefore, to handle adaptive control by design of appropriate switching

Examples

In this section, we show the applicability and effectiveness of our approach on two examples illustrating the main results of the paper.

Example 1

Let us illustrate the proposed switched adaptive control technique by means of a mechanical system as that has been dealt with in Spooner and Passino (1999). That is, we consider the control of two inverted pendulums connected by a spring as depicted in Fig. 2. Each pendulum may be positioned by a torque input $u_{i}$ applied by a servomotor at its base. The

Conclusions

This paper has studied the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems whose subsystems are not assumed to be globally adaptively stabilizable for the first time. A switched adaptive control technique based on the GMLFs method and the parameter separation technique was established to guarantee global stability in the sense of Lyapunov and global asymptotic regulation of the system state of the closed-loop system. In addition, the results obtained

Lijun Long received the B.S. and M.S. degrees in mathematics from Hunan Normal University, Changsha, China, and Southwest University, Chongqing, China, in 2003 and 2009, respectively, and the Ph.D. degree in control theory and applications from Northeastern University, Shenyang, China, in 2013.

He is currently an Associate Professor with the College of Information Science and Engineering, Northeastern University. His current research interests include switched systems, nonlinear systems, and

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Zhuo Wang received his B.E. degree in automation from Beihang University, China, and his Ph.D. degree in electrical and computer engineering from University of Illinois at Chicago, USA.

He worked as a Postdoctoral Fellow in the Department of Electrical and Computer Engineering at University of Alberta, Canada. He is currently a Research Assistant Professor at Fok Ying Tung Graduate School, Hong Kong University of Science and Technology, Hong Kong. Zhuo’s research interests include data-driven control and system analysis, adaptive control, artificial neural networks, and event-triggered control.

Jun Zhao received the B.S. and M.S. degrees in mathematics from Liaoning University, Shenyang, China, in 1982 and 1984, respectively, and the Ph.D. degree in control theory and applications from Northeastern University, Shenyang, China, in 1991.

He was a Post-Doctoral Fellow with Northeastern University from 1992 to 1993. Since 1994, he has been a Professor with the College of Information Science and Engineering, Northeastern University. He was a Senior Visiting Scholar with the Coordinated Science Laboratory, University of Illinois at Urbana–Champaign, Champaign, IL, USA, from 1998 to 1999, a Research Fellow with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, from 2003 to 2005, and a fellow of the School of Engineering with Australian National University, Melbourne, ACT, Australia, from 2007 to 2010. His current research interests include switched systems, nonlinear systems, and network synchronization.

^☆: This work was supported by the National Natural Science Foundation of China under Grants 61304058 and 61233002, and IAPI Fundamental Research Funds under Grant 2013ZCX03-02, and Fundamental Research Funds for the Central Universities under Grant N130404026, and China Postdoctoral Science Foundation under Grant 2013M540231. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.

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Brief paperSwitched adaptive control of switched nonlinearly parameterized systems with unstable subsystems☆

Abstract

Introduction

Section snippets

System description and preliminaries

Adaptive control

Application to non-triangular switched systems

Examples

Conclusions

Systems & Control Letters

Systems & Control Letters

Systems & Control Letters

Automatica

Systems & Control Letters

Automatica

Systems & Control Letters

Systems & Control Letters

Systems & Control Letters

Systems & Control Letters

Automatica

Automatica

Automatica

Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems

IEEE Transactions on Automatic Control

Novel switching model reference adaptive control for continuous piecewise affine systems

Multiple Lyapunov functions and other analysis tools for switched and hybrid systems

IEEE Transactions on Automatic Control

Variable structure based switching adaptive control for a class of unknown switching linear systems

Hybrid model reference adaptive control of piecewise affine systems

IEEE Transactions on Automatic Control

Approximately bisimilar symbolic models for incrementally stable switched systems

IEEE Transactions on Automatic Control

Hybrid dynamical systems

IEEE Control Systems Magazine

Brief paper
Switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems☆