Stability, stabilization and L2-gain analysis of periodic piecewise linear systems

doi:10.1016/j.automatica.2015.08.024

Automatica

Volume 61, November 2015, Pages 218-226

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Abstract

In this paper, the stability, stabilization and $L_{2}$ -gain problems are investigated for periodic piecewise linear systems, in which not all subsystems are Hurwitz. First, some sufficient and necessary conditions for the exponential stability are established. By employing a discontinuous Lyapunov function with time-varying Lyapunov matrix, stabilization and $L_{2}$ -gain conditions of periodic piecewise linear systems are proposed by allowing the corresponding Lyapunov function to be possibly non-monotonically decreasing over a period. A state-feedback periodic piecewise controller is developed to stabilize the system, and the corresponding algorithm is proposed to compute the controller gain. The $L_{2}$ -gain criteria with continuous time-varying Lyapunov matrix and piecewise constant Lyapunov matrices are studied as well. Numerical examples are given to show the validity of the proposed techniques.

Introduction

Continuous-time periodic systems can be described by ordinary differential equations with periodic coefficients. As a special class of time-varying systems, general periodic linear systems exhibit similar behaviour to linear time-invariant systems with the Floquet theory (Bittanti & Colaneri, 2008). However, comparing with discrete-time periodic linear systems, control problems for continuous-time periodic linear systems are more difficult to solve since, with the lifting technique, discrete-time periodic linear systems can be transformed into time-invariant systems. The literature on periodic systems is vast, and numerous references can be found (Bittanti & Colaneri, 2008).

Periodic piecewise system can be regarded as a special class of periodic systems, in that it combines several time-invariant constant subsystems. In practice, many engineering systems can be described by periodic piecewise system models, such as conveying machine with periodic-varying loads, isolation suspension with piecewise spring. Moreover, a periodic piecewise system can be regarded as an approximated system of the continuous-time periodic system by dividing the period into several intervals where the subsystems are approximated by the mean value model of each sub-interval (Selstad & Farhang, 1996). Based on a Lyapunov function with continuous time-varying Lyapunov matrix, the stability and $L_{2}$ -gain problems of periodic piecewise systems are studied in Li, Lam, Chen, Cheung, and Niu (2015).

From another perspective, periodic piecewise system is a special case of switched systems, of which the switching signal is periodic (that is, both the switching sequence and the dwell time are fixed). Many analyses and control problems of general switched systems have been investigated (Liu and Lam, 2012, Wu and Lam, 2009). The stability of switched systems with all and partial non-Hurwitz subsystems are studied by Xiang and Xiao (2014), Zhai, Yasuda, and Michel (2001), the result has also been extended to $L_{2}$ -gain problem (Zhai, Hu, Yasuda, & Michel, 2001). The integral input-to-state stability problem of the hybrid impulsive delayed systems is considered with (Sun & Wang, 2012). Average dwell time approach is broadly used in switched system analysis as found in many references mentioned above. A novel notion of mode-dependent average dwell time is proposed in Zhao, Zhang, Shi, and Liu (2012), which relaxes the restrictions of average dwell time. The stability, stabilization and $H_{\infty}$ problems of switched systems based on mode-dependent average dwell time approach are discussed in Zhao, Liu, and Wang (2013) and Zhao et al. (2012). Among switched systems, some are special because of their switching laws, which may result from the system dynamic requirements or be designed to obtain a better system performance. The stability and control problems of special switched systems can be found in Sun (2006), Tokarzewski (1987) and Xiong, Lam, Shu, and Mao (2014).

With the techniques used in switched systems, in this paper, we study the stability, stabilization and $L_{2}$ -gain problems for a class of continuous-time periodic piecewise systems with possibly non-Hurwitz subsystems. Some sufficient, and also necessary conditions, of the exponential stability are first established. In order to conveniently study the stabilization and $L_{2}$ -gain performance, a possibly discontinuous time-varying Lyapunov function is adopted here. Motivated by the mode-dependent idea (Zhao et al., 2012), a sufficient exponential stability condition is presented by allowing the proposed Lyapunov function to be possibly non-monotonically decreasing over a period. On the basis of this stability result, the stabilization and $L_{2}$ -gain criteria are investigated, a corresponding algorithm of the stabilizing controller is proposed to compute the controller gain. The $L_{2}$ -gain characterizations with continuous time-varying Lyapunov matrix and piecewise constant Lyapunov matrices are provided as well. Numerical examples are used to demonstrate the effectiveness of the proposed techniques. The paper is organized as follows. The problems are formulated in Section 2. The stability analysis and stabilizing controller synthesis results are given in Section 3, and $L_{2}$ -gain problems is studied in Section 4. Numerical examples are given in Section 5. The paper is concluded in Section 6.

Notation: $R^{n}$ stands for the $n$ -dimensional Euclidean space, $‖ \cdot ‖$ stands for the Euclidean vector norm, the superscript ‘ $T$ ’ refers to matrix transposition, $α (\cdot)$ denotes the spectral abscissa, $ρ (\cdot)$ represents the spectral radius, and $\bar{λ} (\cdot), \underline{λ} (\cdot)$ stand for the maximum, minimum eigenvalues of a real symmetric matrix, respectively. ‘ $diag$ ’ represents the matrix with diagonal elements. In addition, $P > 0 (\geq 0)$ means $P$ is real symmetric and positive definite (semi-positive definite) matrix.

Section snippets

Problem formulation

Consider a continuous-time periodic piecewise linear system given as $\dot{x} (t) = A (t) x (t) + B (t) u (t) + B_{w} (t) w (t),$ $z (t) = C (t) x (t) + D (t) u (t) + D_{w} (t) w (t),$ where $x (t) \in R^{n}, z (t) \in R^{m}$ are the state vector and output vector, $u (t), w (t)$ are the control input and disturbance input vectors, respectively. For all $t \geq 0$ , $A (t) = A (t + T_{p}), B (t) = B (t + T_{p}), B_{w} (t) = B_{w} (t + T_{p}), C (t) = C (t + T_{p}), D (t) = D (t + T_{p}), D_{w} (t) = D_{w} (t + T_{p})$ , where $T_{p}$ is the fundamental period. Suppose the interval $[0, T_{p})$ is partitioned into $S$ subintervals $[t_{i - 1}, t_{i}), i \in N, N = {1, 2, \dots, S}$ ,

Stability analysis and stabilization

In this section, some sufficient, and also necessary conditions, of exponential stability analysis are given first. Since the stabilizing controller is hard to be extended from the obtained stability conditions, a sufficient stability condition based on a discontinuous time-varying Lyapunov function is proposed as well. Using such an approach, a stabilizing controller is developed, and the corresponding algorithm is given to compute the controller gain.

$L_{2}$ -gain analysis

In this section, the aim is to study and characterize the effect of disturbance attenuation. The $L_{2}$ -gain criterion established based on the Lyapunov function (11) is given as follows.

Theorem 4

Consider periodic piecewise linear system (1) with $u = 0$ . Given $λ^{*} > 0, μ_{i} \geq 1, i = 1, 2, \dots, S$ . If there exist $λ_{i}^{♢}, i = 1, 2, \dots, S$ , with $λ_{i}^{♢} > 0 (< 0)$ for $i \in N^{↓} (N^{↑})$ , and matrices $P_{i, i - 1} > 0, P_{i, i + 1} > 0, i = 1, 2, \dots, S$ , such that $[\begin{matrix} Φ_{i} (P_{i, i - 1}) + \frac{P_{i, i + 1} - P_{i, i - 1}}{T_{i}} & P_{i, i - 1} B_{w i} & C_{i}^{T} \\ * & - γ^{2} I & D_{w i}^{T} \\ * & * & - I \end{matrix}] < 0,$ $[\begin{matrix} Φ_{i} (P_{i, i + 1}) + \frac{P_{i, i + 1} - P_{i, i - 1}}{T_{i}} & P_{i, i + 1} B_{w i} & C_{i}^{T} \\ * & - γ^{2} I & D_{w i}^{T} \\ * & * & - I \end{matrix}] < 0,$ $P_{1, 0} \leq μ_{1}$

Numerical example

Consider a periodic piecewise system with $T_{p} = 2 s$ and $T_{1} = 1.2 s$ , $T_{2} = 0.4 s$ , $T_{3} = 0.4 s$ , this system consists of three subsystems described by $A_{1} = [\begin{matrix} - 2.1 & 0.6 \\ 0 & 1.5 \end{matrix}], B_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}], B_{w 1} = [\begin{matrix} 0.1 \\ 0.2 \end{matrix}],$ $C_{1} = [\begin{matrix} 0.8 & 0.2 \end{matrix}], D_{1} = [0], D_{w 1} = [0],$ $A_{2} = [\begin{matrix} 0.6 & 1.8 \\ 3 & 0.8 \end{matrix}], B_{2} = [\begin{matrix} 1 \\ 2 \end{matrix}], B_{w 2} = [\begin{matrix} - 0.3 \\ 0.3 \end{matrix}],$ $C_{2} = [\begin{matrix} 0.4 & - 0.6 \end{matrix}], D_{2} = [0], D_{w 2} = [0],$ $A_{3} = [\begin{matrix} - 2.4 & 0.8 \\ - 0.4 & 0.6 \end{matrix}], B_{3} = [\begin{matrix} - 0.5 \\ 2 \end{matrix}], B_{w 3} = [\begin{matrix} - 0.1 \\ 0.05 \end{matrix}],$ $C_{3} = [\begin{matrix} 0.5 & 0.1 \end{matrix}], D_{3} = [0], D_{w 3} = [0],$ it can be observed that all the subsystems are unstable and the 1st subsystem is non-stabilizable. For the above periodic piecewise system, $M = e^{1.2 A_{1}} e^{0.4 A_{2}} e^{0.4 A_{3}}$

Conclusion

The stability, stabilization and $L_{2}$ -gain analysis problems for continuous-time periodic piecewise linear systems with possibly non-Hurwitz subsystems are studied in this paper. First, some sufficient and necessary conditions of the exponential stability are proposed. By employing a discontinuous time-varying Lyapunov function, a sufficient exponential stability condition is developed, and state feedback periodic piecewise controllers are designed to stabilize the system. A schematic algorithm

Panshuo Li received her B.Sc. and M.Sc. in Mechanical Engineering from Dong Hua University and Shanghai Jiao Tong University, Shanghai, China in 2009 and 2012, respectively. She is currently pursuing her Ph.D. in the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong. Her current research interests include periodic systems, robust control, vibration control and active vehicle suspensions.

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James Lam received a B.Sc. (1st Hons.) degree in Mechanical Engineering from the University of Manchester, and was awarded the Ashbury Scholarship, the A.H. Gibson Prize, and the H. Wright Baker Prize for his academic performance. He obtained the M.Phil. and Ph.D. degrees from the University of Cambridge. He is a recipient of the Croucher Foundation Scholarship and Fellowship, the Outstanding Researcher Award of the University of Hong Kong, and the Distinguished Visiting Fellowship of the Royal Academy of Engineering. Prior to joining the University of Hong Kong in 1993 where he is now Chair Professor of Control Engineering. He held lectureships at the City University of Hong Kong and the University of Melbourne. He is a Chartered Mathematician, Chartered Scientist, Chartered Engineer, Fellow of Institute of Electrical and Electronic Engineers, Fellow of Institution of Engineering and Technology, Fellow of Institute of Mathematics and Its Applications, and Fellow of Institution of Mechanical Engineers. He is Editor-in-Chief of IET Control Theory and Applications and Journal of The Franklin Institute, Subject Editor of Journal of Sound and Vibration, Editor of Asian Journal of Control, Section Editor of Cogent Engineering, Associate Editor of Automatica, International Journal of Systems Science, International Journal of Applied Mathematics and Computer Science, Multidimensional Systems and Signal Processing, and Proc. IMechE Part I: Journal of Systems and Control Engineering. He is a member of the IFAC Technical Committee on Networked Systems. His research interests include model reduction, robust synthesis, delay, singular systems, stochastic systems, multidimensional systems, positive systems, networked control systems and vibration control.

Kie Chung Cheung received his primary and secondary education in Hong Kong before going abroad to higher studies. He received his B.Sc. degree in 1986 from the University of Birmingham, UK. He was awarded a Croucher Foundation Scholarship to do research and obtained his Ph.D. in 1990 from the University of Wales, UK. He then worked in Dunlop Topy Wheels Limited of Coventry, UK, for two years as Technical Services Engineer before returning to Hong Kong. His current research interests include intelligent automatic inspection, neural networks and fuzzy expert systems.

^☆: This work was partially supported by General Research Fund (GRF) HKU 7140/11E and HKU CRCG 201409176081. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen.

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Brief paperStability, stabilization and L2-gain analysis of periodic piecewise linear systems☆

Abstract

Introduction

Section snippets

Problem formulation

Stability analysis and stabilization

L2-gain analysis

Numerical example

Conclusion

Automatica

Automatica

Automatica

Journal of The Franklin Institute

Periodic systems: filtering and control

Brief paper
Stability, stabilization and $L_{2}$ -gain analysis of periodic piecewise linear systems☆

$L_{2}$ -gain analysis