Dwell-time approach to input–output stability properties for a class of discrete-time dynamical systems

doi:10.1016/j.sysconle.2011.01.007

Systems & Control Letters

Volume 60, Issue 6, June 2011, Pages 383-389

https://doi.org/10.1016/j.sysconle.2011.01.007 Get rights and content

Abstract

We consider a class of discrete-time dynamical systems that consist of switching of many linear time-invariant systems, and it can be used to cover a sampled-data version of Switched Linear Systems, which have been widely studied in the literature. For this class of dynamical systems, we establish the LMI (Linear Matrix Inequality) formulation of analyzing their input–output stability properties, including both $ℓ$ -2 stability and passivity, by constraining switching signals via the concept of dwell time. The LMI formulation of $ℓ$ -2 stability is applied to evaluating the closed-loop performance of an industrial refrigeration process that is regulated by several proportional-integral (PI) controllers in a switching structure.

Introduction

Switched control strategies have been wide used in the design of automatic control systems to deal with plant dynamics, handle constraints, and improve performance. Naturally, some simple and practical analytical/numerical tools, rather than exhaustive simulations or expensive experiments, are highly desired in verifying stability and robustness properties for switched control systems. While the asymptotic/exponential stability issue for switched linear systems has been extensively studied in the literature, the input–output stability analysis remains a serious challenge [1], [2], [3]. In this paper, by constraining switching signals via dwell time¹ we propose an LMI (Linear Matrix Inequality) formulation for analyzing $ℓ$ -2 stability for discrete-time linear hybrid systems (DLHS), a class of discrete-time dynamical systems that represents a sampled-data version of linear hybrid systems studied in [5], [6]. The linear hybrid system is generalized from a switched linear system by incorporating a state reset during mode switches (see [5, Section 4] for an industrial example, where integrator reset was adopted to guarantee bumpless transfer for actuator protection); in particular, a linear hybrid system without state reset reduces to a switched linear system.

The $ℓ$ -2 stability analysis for DLHS is in part motivated by an open problem posed in [2] on understanding the quantitative relationship between $ℓ$ -2 induced gains and dwell time (i.e. the computation of $ℓ$ -2 induced gains versus the dwell time, see Fig. 1, Fig. 2 in Section 6 for example) for switched linear systems. To the best of our knowledge, this open problem has not been completely solved, though some important advances have been reported in [7], [8], [9], [10], [11], [12]; to mention a few, in [10] the authors showed that $ℓ$ -2 stability can be characterized by the existence of a convex homogeneous (of degree two) Lyapunov function, though the construction of such a storage function (by solving the Hamilton–Jacobi inequalities) remains a theoretical challenge; in [12] the authors proposed a suboptimal LMI formulation of estimating $ℓ$ -2 induced gains versus dwell time, though the estimated gains may be conservative. Along this research path, there is also some relevant discrete-time work in [13] specialized for the case of the $ℓ$ -2 gains of switched linear systems being less than 1.

In this paper, we study a problem analogous to that in [2] and we propose an LMI formulation of optimally estimating $ℓ$ -2 induced gains versus dwell time for DLHS. As an important byproduct, we provide a numerical solution to a discrete-time version of the open problem in [2], since DLHS reduces to a discrete-time switched linear system when state reset is ignored. Moreover, the ideas of deriving LMI formulation of $ℓ$ -2 stability with respect to dwell time here can be easily extended to studying passivity with respect to dwell time; such an extension is natural as both $ℓ$ -2 stability and passivity share a similar LMI formulation for discrete-time LTI (Linear Time Invariant) systems.

The paper is organized as follows. Section 2 contains some preliminaries on LMI formulations of $ℓ$ -2 stability and passivity for discrete-time LTI systems. Section 3 introduces DLHS and stability notions. Sections 4 LMI formulation for, 5 LMI formulation for passivity analysis presents a LMI formulation of $ℓ$ -2 stability and passivity, respectively, for DLHS. Section 6 illustrates the application of the proposed $ℓ$ -2 stability analysis to evaluating the closed-loop performance for an industrial refrigeration process. The last section draws conclusions. We begin by listing basic definitions and notation:

•
Denote by $Z$ the set of integers; given any set $S \subset R$ , denote $Z_{S} ≔ Z \cap S$ .
•
Given a vector $v \in R^{n}, v^{'}$ denotes the transpose of $v$ , and $| v |$ denotes Euclidean norm of $v$ .
•
Let $M^{'}$ denote the transpose of $M \in R^{n \times m}$ . In the context of a symmetric matrix $[\begin{matrix} M_{11} & ★ \\ M_{21} & M_{22} \end{matrix}]$ , the symbol $★$ means the transpose of $M_{21}$ .

Section snippets

Preliminaries: LMI synthesis of $ℓ$ -2 stability and passivity for discrete-time LTI systems

Consider a discrete-time LTI system as follows, ${\begin{matrix} x (t + 1) = A x (t) + B u (t), \\ y (t) = C x (t) + D u (t), \end{matrix}$ where $x (t) \in R^{n}$ is the state, $u (t) \in R^{p}$ is the input, and $y (t) \in R^{q}$ is the output.

The following result describes the LMI formulation of $ℓ$ -2 stability for (1).

Lemma 2.1

[8, Lemma 2]

Let $γ > 0$ . The following statements are equivalent for the system (1):

(1)
it is Schur stable and has $ℓ$ -2 gain $γ$ , i.e. for each solution $x$ to(1) with zero initial condition, $0 < - \frac{1}{γ} \sum_{τ = 0}^{t} y^{'} (τ) y (τ) + γ \sum_{τ = 0}^{t} u^{'} (τ) u (τ) \forall t \in Z_{[0, \infty)};$
(2)
there exist $γ > 0$ and a symmetric positive

Time-varying dynamical systems and stability notion

A linear hybrid system studied in [5], [6] is a dynamical system by associating a switched linear system with a linear state reset during mode switches, and it reduces to a switched linear system when the state reset is an identity map (and hence the state reset can be ignored). In this paper, we consider a class of time-varying systems that represents a sampled-data version of linear hybrid systems, introduced as follows.

Let $I ≔ {1, 2, \dots, N}, I_{s} \subset I \times I$ , and $M_{A} ≔ {A_{i} \in R^{n \times n} : i \in I},$ $M_{B} ≔ {B_{i} \in R^{n \times p} : i \in I},$ $M_{C} ≔ {C_{i} \in R^{q \times n} :$

LMI formulation for $ℓ$ -2 stability analysis

We start with some observations and notation for (2). For each $T \in Z_{[1, \infty)}$ , all solutions $x (\cdot)$ to an LTI subsystem $[A_{i}, B_{i}, C_{i}, D_{i}]$ satisfy ${\begin{matrix} x (t + T) = A_{i} (T) x (t) + B_{i} (T) u (t, T), \\ y (t, T) = C_{i} (T) x (t) + D_{i} (T) u (t, T), \end{matrix} \forall t \in Z_{[0, \infty)}$ where the vectors $u (t, T), y (t, T)$ are defined as follows $u (t, T) ≔ [\begin{matrix} u (t + T - 1) \\ u (t + T - 2) \\ ⋮ \\ u (t + 1) \\ u (t) \end{matrix}], y (t, T) ≔ [\begin{matrix} y (t + T - 1) \\ y (t + T - 2) \\ ⋮ \\ y (t + 1) \\ y (t) \end{matrix}]$ where the matrices $A_{i} (T), B_{i} (T)$ , $C_{i} (T), D_{i} (T)$ are defined as follows, $A_{i} (T) ≔ A_{i}^{T},$ $B_{i} (T) ≔ [\begin{matrix} B_{i} & A_{i} B_{i} & \dots & A_{i}^{T - 1} B_{i} \end{matrix}],$ $C_{i} (T) ≔ [\begin{matrix} C_{i} A_{i}^{T - 1} \\ ⋮ \\ C_{i} A_{i} \\ C_{i} \end{matrix}],$ $D_{i} (T) ≔ [\begin{matrix} D_{i} & C_{i} B_{i} & C_{i} A_{i} B_{i} & \dots & C_{i} A_{i}^{T - 3} B_{i} & C_{i} A_{i}^{T - 2} B_{i} \\ 0 & D_{i} & C_{i} B \end{matrix}$

LMI formulation for passivity analysis

The ideas of establishing Theorem 4.1 is readily extended to studying passivity for the system (2).

Theorem 5.1

Let $T \in Z_{[1, \infty)}$ . If there exists a collection of symmetric positive definite matrices ${P_{1}, P_{2}, \dots, P_{N}}$ of compatible dimensions such that Eqs. (21), (22) given in Box II, then the system (2) under switching signals $S_{[T, \infty)}$ is passive.

Proof

Let $σ \in S_{[T, \infty)}$ , associated with a sequence ${t_{k}}_{k = 0}^{K}$ , be given. Let $x$ be a solution to (2) under the switching signal $σ$ . Consider the Lyapunov function $V (x (t)) = x {(t)}^{'} P_{σ (t)} x (t) .$

An industrial switched control system

In this section we show how the $ℓ$ -2 stability analysis derived in Section 4 can be applied to a practical example. The model data of the example are derived from an industrial process reported in [5, Section 4], which is regularized by four PI controllers in a switching structure, by discretizing its closed-loop model with sample time one. Consequently, we consider the system (2) with $x \in R^{5}, y \in R, u \in R$ , $I = {1, 2, 3, 4}$ and $A_{1} = [\begin{matrix} 0.9962 & 0 & 0 & 0 & - 0.0002 \\ 0.0676 & 0.9940 & 0 & 0 & 0.0034 \\ 0.2246 & 0 & 0.8251 & 0 & 0.0112 \\ - 0.0041 & 0 & 0 & 0.9835 & - 0.0002 \end{matrix}$

Conclusions

By constraining switching signals via dwell time, we establish both LMI formulation of $ℓ$ -2 stability and that of passivity for DLHS (discrete-time linear hybrid systems), which covers both a sampled-data version of linear hybrid systems and that of switched linear systems. As a byproduct, we have provided a numerical solution to estimate nonconservative $ℓ$ -gains versus dwell time for discrete-time switched linear systems. The results reported here are readily applicable to analyzing input–output

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Citation Excerpt :
It should be mentioned that the controller of the subsystem and switching sequence can be designed in a co-ordinated way, such that even if one or more subsystem is unstable, the overall switched system still can be stable. It should be pointed out that the average dwell time method becomes an important and attractive method to find a suitable switching signal to guarantee the switched system stability or improve other performances [2–4]. Recently, the stability analysis problem for neural networks has been well recognized and a great number of results on this problem have been reported in the literature [5–11].
This paper provides a novel result on robust exponential stability for a class of uncertain discrete-time switched fuzzy neural networks (DSFNNs) with time-varying delays and parameter uncertainties. By implementing an average dwell time approach with a new Lyapunov–Krasovskii functional, we obtain some delay-dependent sufficient conditions guaranteeing the robust exponential stability of the considered switched fuzzy neural networks. In other words, a class of switching signals specified by the average dwell time is identified to guarantee the exponential stability of the considered DSFNNs. The obtained conditions are formulated in terms of Linear Matrix Inequalities (LMIs) which can be easily verified via the LMI toolbox. Finally, numerical examples with simulation results are provided to illustrate the applicability and usefulness of the obtained results.
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^☆: This work was done while the author was at United Technologies Research Center, CT.

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Dwell-time approach to input–output stability properties for a class of discrete-time dynamical systems☆

Abstract

Introduction

Section snippets

Preliminaries: LMI synthesis of ℓ-2 stability and passivity for discrete-time LTI systems

[8, Lemma 2]

Time-varying dynamical systems and stability notion

LMI formulation for ℓ-2 stability analysis

LMI formulation for passivity analysis

An industrial switched control system

Conclusions

Automatica

Automatica

Switching in Systems and Control

L2-induced gains of switched linear systems

Stability criteria for switched and hybrid systems

SIAM Review

Supervisory control of families of linear set-point controllers part I: exact matching

IEEE Transactions on Automatic Control

Root-mean-square gains of switched linear systems

IEEE Transactions on Automatic Control

Preliminaries: LMI synthesis of $ℓ$ -2 stability and passivity for discrete-time LTI systems

LMI formulation for $ℓ$ -2 stability analysis

$L_{2}$ -induced gains of switched linear systems