Generalized switching signals for input-to-state stability of switched systems

doi:10.1016/j.automatica.2015.11.027

Automatica

Volume 64, February 2016, Pages 270-277

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Abstract

This article deals with input-to-state stability (ISS) of continuous-time switched nonlinear systems. Given a family of systems with exogenous inputs such that not all systems in the family are ISS, we characterize a new and general class of switching signals under which the resulting switched system is ISS. Our stabilizing switching signals allow the number of switches to grow faster than an affine function of the length of a time interval, unlike in the case of average dwell time switching. We also recast a subclass of average dwell time switching signals in our setting and establish analogs of two representative prior results.

Introduction

A switched system comprises of two components—a family of systems and a switching signal. The switching signal selects an active subsystem at every instant of time, i.e., the system from the family that is currently being followed (Liberzon, 2003, Section 1.1.2). Stability of switched systems is broadly classified into two categories—stability under arbitrary switching (Liberzon, 2003, Chapter 2) and stability under constrained switching (Liberzon, 2003, Chapter 3). In the former category, conditions on the family of systems are identified such that the resulting switched system is stable under all admissible switching signals; in the latter category, given a family of systems, conditions on the switching signals are identified such that the resulting switched system is stable. In this article our focus is on stability of switched systems with exogenous inputs under constrained switching.

Prior study in the direction of stability under constrained switching primarily utilizes the concept of slow switching vis-a-vis (average) dwell time switching. Exponential stability of a switched linear system under dwell time switching was studied in Morse (1996). In Xie, Wen, and Li (2001) the authors showed that a switched nonlinear system is ISS under dwell time switching if all subsystems are ISS. A class of state-dependent switching signals obeying dwell time property under which a switched nonlinear system is integral input-to-state stable (iISS) was proposed in De Persis, De Santis, Morse (2003). The dwell time requirement for stability was relaxed to average dwell time switching to switched linear systems with inputs and switched nonlinear systems without inputs in Hespanha and Morse (1999). ISS of switched nonlinear systems under average dwell time was studied in Vu, Chatterjee, and Liberzon (2007). It was shown that if the individual subsystems are ISS and their ISS-Lyapunov functions satisfy suitable conditions, then the switched system has the ISS, exponentially-weighted ISS, and exponentially-weighted iISS properties under switching signals obeying sufficiently large average dwell time. Given a family of systems such that not all systems in the family are ISS, it was shown in the recent work (Yang & Liberzon, 2014) that it is possible to construct a class of hybrid Lyapunov functions to guarantee ISS of the switched system provided that the switching signal neither switches too frequently nor activates the non-ISS subsystems for too long. In Müller and Liberzon (2012) input/output-to-state stability (IOSS) of switched nonlinear systems with families in which not all subsystems are IOSS, was studied. It was shown that the switched system is IOSS under a class of switching signals obeying average dwell time property and constrained point-wise activation of unstable subsystems.

Given a family of systems, possibly containing non-ISS dynamics, in this article we study ISS of switched systems under switching signals that transcend beyond the average dwell time regime in the sense that the number of switches on every interval of time can grow faster than an affine function of the length of the interval. Our characterization of stabilizing switching signals involves pointwise constraints on the duration of activation of the ISS and non-ISS systems, and the number of occurrences of the admissible switches, certain pointwise properties of the quantities defining the above constraints, and a summability condition. In particular, our contributions are:

$\circ$
We allow non-ISS systems in the family and identify a class of switching signals under which the resulting switched system is ISS.
$\circ$
Our class of stabilizing switching signals encompasses the average dwell time regime in the sense that on every interval of time the number of switches is allowed to grow faster than an affine function of the length of the interval. Earlier in Kundu and Chatterjee (2015) we proposed a class of switching signals beyond the average dwell time regime for global asymptotic stability (GAS) of continuous-time switched nonlinear systems.
$\circ$
Although this is not the first instance when non-ISS subsystems are considered (see e.g., Müller & Liberzon, 2012 and Yang & Liberzon, 2014), to the best of our knowledge, this is the first instance when non-ISS subsystems are considered and the proposed class of stabilizing switching signals goes beyond the average dwell time condition.
$\circ$
We recast a subclass of average dwell time switching signals in our setting and establish analogs of an ISS version of Müller and Liberzon (2012, Theorem 2), and Vu et al. (2007, Theorem 3.1) as two corollaries of our main result.

The remainder of this article is organized as follows: In Section 2 we formulate the problem under consideration and catalog certain properties of the family of systems and the switching signal. Our main results appear in Section 3, and we provide a numerical example illustrating our main result in Section 4. In Section 5 we recast prior results in our setting. The proofs of our main results are presented in a consolidated manner in Section 7.

Notations: Let $R$ denote the set of real numbers, $‖ \cdot ‖$ denote the Euclidean norm, and for any interval $I \subset [0, + \infty [$ we denote by ${‖ \cdot ‖}_{I}$ the essential supremum norm of a map from $I$ into some Euclidean space. For measurable sets $A \subset R$ we let $| A |$ denote the Lebesgue measure of $A$ .

Section snippets

Preliminaries

We consider the switched system $\dot{x} (t) = f_{σ (t)} (x (t), v (t)), x (0) = x_{0} (given) , t \geq 0$ generated by

$\circ$
a family of continuous-time systems with exogenous inputs $\dot{x} (t) = f_{i} (x (t), v (t)), x (0) = x_{0} (given) , i \in P, t \geq 0,$ where $x (t) \in R^{d}$ is the vector of states and $v (t) \in R^{m}$ is the vector of inputs at time $t$ , $P = {1, 2, \dots, N}$ is a finite index set, and
$\circ$
a piecewise constant function $σ : [0, + \infty [⟶ P$ that selects, at each time $t$ , the index of the active system from the family (2); this function $σ$ is called a switching signal. By

Main results

We are now in a position to present our main results.

Theorem 1

Consider the family of systems (2). Let $P_{S}, P_{U} \subset P$ and $E (P)$ be as described in Section 2.1. Suppose that Assumption 1, Assumption 2 hold. Let there exist constants $c_{1}$ and $c_{2}$ , and a class $F K_{\infty}$ function $ρ : {[0, + \infty [}^{2} \to [0, + \infty [$ satisfying $ρ (0, 0) = 0$ such that the following conditions hold: $- \sum_{j \in P_{S}} | λ_{j} | ρ_{j}^{S} (r, s) + \sum_{k \in P_{U}} | λ_{k} | ρ_{k}^{U} (r, s) + \sum_{(m, n) \in E (P)} (ln μ_{m n}) ρ_{m n} (r, s) \leq c_{1} - ρ (r, s)$ $for every interval] r, r + s] \subset [0, + \infty [of time, and lim_{t \to + \infty} \sum_{i = 0}^{N_{σ} (0, t)} exp (- ρ (τ_{i}, t - τ_{i})) \leq c_{2} .$

Numerical example

A. The family of systems: We consider a family with $P = {1, 2, 3, 4}$ with $f_{1} (x, v) = (\begin{matrix} - x_{1} + sin (x_{1} - x_{2}) \\ - x_{2} + 0.8 sin (x_{2} - x_{1}) + 0.5 v \end{matrix}),$ $f_{2} (x, v) = (\begin{matrix} x_{2} \\ - x_{1} - x_{2} + v \end{matrix}),$ $f_{3} (x, v) = (\begin{matrix} x_{1} + sin (x_{1} - x_{2}) \\ x_{2} + sin (x_{2} - x_{1}) + 0.5 v \end{matrix}),$ $f_{4} (x, v) = (\begin{matrix} x_{2} \\ - x_{2} + v \end{matrix}) .$ Clearly, $P_{S} = {1, 2}$ and $P_{U} = {3, 4}$ . Let $E (P) = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (4, 1), (4, 2)}$ . We choose $V_{1} (x) = 0.5 (x_{1}^{2} + 1.25 x_{2}^{2})$ , $V_{2} (x) = x_{1}^{2} + x_{1} x_{2} + x_{2}^{2}$ , $V_{3} (x) = 0.5 (x_{1}^{2} + x_{2}^{2})$ , $V_{4} (x) = x_{1}^{2} + x_{2}^{2}$ , and obtain the following estimates: $λ_{1} = 1.75, λ_{2} = 0.5, λ_{3} = - 2.1667, λ_{4} = - 0.6378$ , $μ_{12} = 6, μ_{13} = 1, μ_{14} = 2, μ_{21} = 4$ , $μ_{23} = 1, μ_{24} =$

Discussion

In this section we recast a subclass of average dwell time switching signals in our setting and establish analogs of the prior results: an ISS version of Müller and Liberzon (2012, Theorem 2), and Vu et al. (2007, Theorem 3.1), with the aid of our main result. Our first result of this section is:

Proposition 1

Consider the family of systems (2). Suppose that Assumption 1 holds with $| λ_{j} | = λ_{S}$ for all $j \in P_{S}$ and $| λ_{k} | = λ_{U}$ for all $k \in P_{U}$ , and Assumption 2 holds with $μ_{m n} = μ$ for all $(m, n) \in E (P)$ . Let $\bar{ρ}$ and $τ_{a}$ be

Concluding remarks

In this article we presented a class of switching signals under which a continuous-time switched system is uniformly ISS. We utilized multiple ISS-Lyapunov-like functions for our analysis and our characterization of stabilizing switching signals allowed the number of switches on any interval of time to grow faster than an affine function of the length of the interval unlike in the case of average dwell time switching. We also discussed two representative prior results: an ISS version of Müller

Proofs

Proof of Theorem 1

Fix $t > 0$ . Let $0 ≕ τ_{0} < τ_{1} < \dots < τ_{N_{σ} (0, t)}$ be the switching instants before (and including) $t$ . In view of (5), $V_{σ (t)} (x (t)) \leq exp (- λ_{σ (τ_{N_{σ} (0, t)})} (t - τ_{N_{σ} (0, t)})) V_{σ (t)} (x (τ_{N_{σ} (0, t)})) + γ ({‖ v ‖}_{[0, t]}) \int_{τ_{N_{σ} (0, t)}}^{t} exp (- λ_{σ (τ_{N_{σ} (0, t)})} (t - s)) d s .$ Applying (6) and iterating the above, we obtain the estimate $V_{σ (t)} (x (t)) \leq ψ_{1} (t) V_{σ (0)} (x_{0}) + γ ({‖ v ‖}_{[0, t]}) ψ_{2} (t),$ where $ψ_{1} (t) ≔ exp (- \sum_{\binom{i = 0}{τ_{N_{σ} (0, t) + 1} ≔ t}}^{N_{σ} (0, t)} λ_{σ (τ_{i})} S_{i + 1} + \sum_{i = 0}^{N_{σ} (0, t) - 1} ln μ_{σ (τ_{i}) σ (τ_{i + 1})}),$ $ψ_{2} (t) ≔ \sum_{\binom{i = 0}{τ_{N_{σ} (0, t) + 1} ≔ t}}^{N_{σ} (0, t)} (exp (- \sum_{\binom{k = i + 1}{τ_{N_{σ} (0, t) + 1} ≔ t}}^{N_{σ} (0, t)} λ_{σ (τ_{k})} S_{k + 1} + \sum_{k = i + 1}^{N_{σ} (0, t) - 1} ln μ_{σ (τ_{k}) σ (τ_{k + 1})}) \times 1$

Atreyee Kundu received Ph.D. in Systems and Control Engineering from the Indian Institute of Technology, Bombay, in March 2015. She is currently a Postdoctoral researcher at the Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands. Her research interests include hybrid systems and its applications, robust and optimal control.

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Debasish Chatterjee was educated at the Indian Institute of Technology, Kharagpur (B.Tech, 2002), and the University of Illinois at Urbana-Champaign (Ph.D., 2007). He is currently an Associate Professor in Systems & Control Engineering, Indian Institute of Technology, Bombay. Prior to joining IIT Bombay he was a postdoc at ETH Zurich. His research interests are optimization based control, hybrid systems, and applications of probability to engineering systems.

Daniel Liberzon was born in the former Soviet Union in 1973. He did his undergraduate studies in the Department of Mechanics and Mathematics at Moscow State University from 1989 to 1993.

In 1993 he moved to the United States to pursue graduate studies in mathematics at Brandeis University, where he received the Ph.D. degree in 1998 (supervised by Prof. Roger W. Brockett of Harvard University).

Following a postdoctoral position in the Department of Electrical Engineering at Yale University from 1998 to 2000 (with Prof. A. Stephen Morse), he joined the University of Illinois at Urbana-Champaign, where he is now a professor in the Electrical and Computer Engineering Department and the Coordinated Science Laboratory.

His research interests include nonlinear control theory, switched and hybrid dynamical systems, control with limited information, and uncertain and stochastic systems. He is the author of the books Switching in Systems and Control (Birkhäuser, 2003) and Calculus of Variations and Optimal Control Theory: A Concise Introduction (Princeton Univ. Press, 2012).

His work has received several recognitions, including the 2002 IFAC Young Author Prize and the 2007 Donald P. Eckman Award. He delivered a plenary lecture at the 2008 American Control Conference. He has served as Associate Editor for the journals IEEE Transactions on Automatic Control and Mathematics of Control, Signals, and Systems. He is a fellow of IEEE.

^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Graziano Chesi under the direction of Editor Richard Middleton.

¹: Tel.: +31 40 247 4850; fax: +31 40 243 4970.

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Brief paperGeneralized switching signals for input-to-state stability of switched systems☆

Abstract

Introduction

Section snippets

Preliminaries

Main results

Numerical example

Discussion

Concluding remarks

Proofs

Systems & Control Letters

Systems & Control Letters

Automatica Journal of IFAC

Systems & Control Letters

Automatica

Brief paper
Generalized switching signals for input-to-state stability of switched systems☆