Stabilization Analysis for the Switched Large-Scale Discrete-Time Systems via the State-Driven Switching

Abstract

A criterion of stabilization for the switched large-scale discrete-time system is deduced by employing state-driven switching method and the Lyapunov stability theorem. This particular method especially can be applied to cases when all individual subsystems are unstable.

1. Introduction

Recently, switched systems have attracted great research attention in academic and industrial applications. They also have been widely and successfully applied to a variety of industrial processes such as the automotive industry, chemical procedure control systems, navigation systems, automobile speed change systems, aircraft control systems, air traffic control, and many other fields [1–7]. Switched systems are a class of hybrid dynamical systems consisting of several continuous-time or discrete-time subsystems and a rule that orchestrates the switching sequences among them. In the study of switched systems, most of the works have been centralized on the problem of stability; see, for example, [8–17] and the references cited therein. Two important methods are used to construct the switching law for the stability analysis of the switched systems. One is the state-driven switching strategy [8–13], and the other is the time-driven switching strategy [14–17]. The time-driven switching method is based on the concept of a dwell time. If there exists at least an individual stable subsystem, then the switched system is stable with appropriate dwell time switching law. But in many applications, all individual subsystems are unstable. Whereas for systems with the state-driven switching strategy, there have been many choices of switching strategy to make the whole system stable even when all individual subsystems are unstable.

Recently, some stability conditions and stabilization approaches have been proposed for the switched discrete-time systems. Reference [18] studied the quadratic stabilization of discrete-time switched linear systems when a designed switching rule is imposed upon the feedback controller of subsystems and studied quadratic stabilization of switched system with norm-bounded time-varying uncertainties. The event-driven scheduling strategy for constructing switching law to stabilize the switched system is presented in [19]. Reference [20] studied stability property for the switched systems which are composed of a continuous-time LTI subsystem and a discrete-time LTI subsystem, when the two subsystems are Hurwitz and Schur stable, respectively. It was shown that if the subsystem matrices are symmetric, then a common Lyapunov function exists for these two subsystems and that the switched system is exponentially stable under arbitrary switching. There exists a switched quadratic Lyapunov function to check asymptotic stability of the switched discrete-time system in [21]. References [22, 23] studied robust stability analysis and control synthesis of uncertain switched systems. Reference [24] has shown that to achieve controllability for a switched linear system, it is sufficient to use cyclic and synchronous switching paths and constant control laws.

Large-scale systems, the mathematical models of many physical and engineering systems, are frequently of high dimension or possess interacting dynamic phenomena. Such systems consist of a number of interdependent subsystems which serve particular functions, share resources, and are governed by a set of interrelated goals and constraints [25, 26]. In real systems, the large-scale systems include electric power systems, nuclear reactors, aerospace systems, large electric networks, economic systems, process control systems, chemical and petroleum industries, different types of societal systems, and ecological systems. Recently, many approaches have been used to investigate the stability and stabilization of large-scale systems [27–33].

As we have indicated, very few research articles have ever discussed the stabilization of the switched large-scale discrete-time system. Therefore, we will investigate the stabilization analysis of this new class of switched large-scale discrete-time systems. In this paper, we will present novel and general approaches by Lyapunov stability theorem to construct a state-driven switching strategy such that the switched large-scale discrete-time system is asymptotically stable. Particularly, this method can be applied to cases when all individual subsystems are unstable.

Notation. The following notations will be used throughout the paper: 𝜆(𝐴) stands for the eigenvalues of matrix 𝐴, 𝐴_𝜆=𝜆_max(𝐴) (𝐴_𝜆=𝜆_min(𝐴)) means the maximum (minimum) eigenvalue in matrix 𝐴, ‖𝐴‖ denotes the norm of matrix 𝐴, that is, ‖𝐴‖=Max[𝜆(𝐴^𝑇𝐴)]^1/2, 𝐴^𝑇 is the transpose of matrix 𝐴, and diag{⋯} denotes a block-diagonal matrix.

2. System Description and Problem Statement

Consider the switched large-scale discrete-time systems with 𝑟 individual systems and each individual system composed of 𝑁 interconnected subsystems: 𝑥_𝑖(𝑘+1)=𝐴_{𝜎(𝑥(𝑘))𝑖𝑖}𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_{𝜎(𝑥(𝑘))𝑖𝑗}𝑥_𝑗(𝑘), (1)

where 𝑥_𝑖(𝑘)∈𝑅^𝑛𝑖 is the state vectors of the 𝑖th subsystem, 𝑖=1,2,…,𝑁, 𝑛=∑^𝑁_𝑖=1𝑛_𝑖, 𝐴_{𝜎(𝑥(𝑘))𝑖𝑖} and 𝐴_{𝜎(𝑥(𝑘))𝑖𝑗} are some constant matrices of compatible dimensions, 𝑥(𝑘)=[𝑥^𝑇₁(𝑘)𝑥^𝑇₂(𝑘)⋯𝑥^𝑇_𝑁(𝑘)]^𝑇∈𝑅^𝑛, 𝜎(𝑥(𝑘))∶𝑅^𝑛→{1,2,…,𝑟} is a piecewise constant scalar function of state, called a switch signal; that is, 𝜎(𝑥(𝑘))=𝑙 implies the subsystem matrices 𝐴_𝑙𝑖𝑖 and 𝐴_𝑙𝑖𝑗, 𝑙∈{1,2,…,𝑟}. Essentially, the switched discrete-time system (1) can be described as follows.

Individual system 𝑙: 𝑥_𝑖(𝑘+1)=𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘). (2) For the stabilization of switched large-scale discrete-time system (2), some helpful lemmas are given subsequently.

Lemma 1 (see [34]). For any matrices 𝐴 and 𝐵 with appropriate dimensions, one has 𝐴^𝑇𝐵+𝐵^𝑇𝐴≤𝛾𝐴^𝑇1𝐴+𝛾𝐵^𝑇𝐵 (3) for any constant 𝛾>0.

Lemma 2. For any matrices 𝐴₁,𝐴₂,…,𝐴_𝑁 with the same dimensions, the following inequality holds for any positive constant 𝜀: ^𝑁_𝑖=1𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖≤(1+𝜀)𝐴^𝑇₁𝐴₁+1+𝜀⁻¹(1+𝜀)𝐴^𝑇₂𝐴₂+1+𝜀⁻¹²(1+𝜀)𝐴^𝑇₃𝐴₃+⋯+1+𝜀^−1𝑁−2(1+𝜀)𝐴^𝑇_𝑁−1𝐴_𝑁−1+1+𝜀^−1𝑁−1𝐴^𝑇_𝑁𝐴_𝑁. (4)

Proof. For a positive constant 𝜀, 𝐴 and 𝐵 have the same dimension, such that (𝐴+𝐵)^𝑇(𝐴+𝐵)≤(1+𝜀)𝐴^𝑇𝐴+1+𝜀⁻¹𝐵^𝑇𝐵. (5) In view of inequality (5), we have ^𝑁_𝑖=1𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖≤(1+𝜀)𝐴^𝑇₁𝐴₁+1+𝜀^−1𝑁_𝑖=2𝐴_𝑖^𝑇𝑁_𝑖=2𝐴_𝑖≤(1+𝜀)𝐴^𝑇₁𝐴₁+(1+𝜀)1+𝜀⁻¹𝐴^𝑇₂𝐴₂+1+𝜀^−12𝑁_𝑖=3𝐴_𝑖^𝑇𝑁_𝑖=3𝐴_𝑖⋮≤(1+𝜀)𝐴^𝑇₁𝐴₁+(1+𝜀)1+𝜀⁻¹𝐴^𝑇₂𝐴₂+⋯(1+𝜀)1+𝜀^−1𝑁−2𝐴^𝑇_𝑁−1𝐴_𝑁−1+1+𝜀^−1𝑁−1𝐴^𝑇_𝑁𝐴_𝑁. (6)

Lemma 3. Let 𝛼_𝑙=1/(1+𝜀⁻¹)^𝑙−1(1+𝜀), 𝑙=1,2,…,𝑟−1, and 𝛼_𝑟=1/(1+𝜀⁻¹)^𝑟−1; then 𝛼_𝑙∈[0,1] (1≤𝑙≤𝑟) and ∑^𝑟_𝑙=1𝛼_𝑙=1 for any positive constant 𝜀.

Proof. Obviously, if there exists a positive constant 𝜀 then 0<𝛼_𝑙<1, 𝑙=1,2,…,𝑟, and ^𝑟_𝑙=1𝛼_𝑙=1+11+𝜀1+𝜀⁻¹()11+𝜀+⋯+1+𝜀^−1𝑟−2()+11+𝜀1+𝜀^−1𝑟−1=1/(1+𝜀)−1/(1+𝜀)1+𝜀^−1𝑟−11−1/1+𝜀⁻¹+11+𝜀^−1𝑟−1𝛼=1₁+𝛼₂+⋯+𝛼_𝑟−1isageometricprogression. (7)

Remark 4. By Lemmas 2 and 3, it is an obvious fact that ^𝑟_𝑙=1𝛼_𝑙𝐴^𝑇_𝑙^𝑟_𝑙=1𝛼_𝑙𝐴_𝑙≤^𝑟_𝑙=1𝛼_𝑙𝐴^𝑇_𝑙𝐴_𝑙. (8) First, we discuss the stability of the nominal switched discrete-time system. For the nominal switched discrete-time system, 𝑥(𝑘+1)=𝐴_𝑙𝑥(𝑘),𝑙=1,2,…,𝑟. (9)

Lemma 5. There exists a switching law for the nominal switched discrete-time system (9) such that the system (9) is asymptotically stable if there exist a symmetric matrix 𝑃>0 and positive constants 𝛼_𝑙 (1≤𝑙≤𝑟) satisfying ∑^𝑟_𝑙=1𝛼_𝑙=1 such that ^𝑟_𝑙=1𝛼_𝑙𝐴^𝑇_𝑙𝑃𝐴_𝑙−𝑃<0. (10)

Proof. If there exist a symmetric matrix 𝑃>0 and positive constants 𝛼_𝑙 (1≤𝑙≤𝑟) satisfying ∑^𝑟_𝑙=1𝛼_𝑙=1 such that the inequality (10) holds, the inequality (10) is equivalent to the following inequality: ^𝑟_𝑙=1𝛼_𝑙𝐴^𝑇_𝑙𝑃𝐴_𝑙−𝑃<0,𝑙=1,2,…,𝑟. (11) Then, for all 𝑥(𝑘)∈𝐑^𝑛, 𝑥(𝑘)≠0, 𝑥^𝑇(𝑘)^𝑟_𝑙=1𝛼_𝑙𝐴^𝑇_𝑙𝑃𝐴_𝑙−𝑃𝑥(𝑘)<0. (12) Therefore, it follows that, for any 𝑘, at least there exists an 𝑙∈{1,2,…,𝑟} such that 𝑥^𝑇(𝑘)𝐴^𝑇_𝑙𝑃𝐴_𝑙−𝑃𝑥(𝑘)<0. (13) From (13), it implies that a convex combination of the corresponding Lyapunov function (12) is negative along the trajectory, and from (13) at least one must be negative. Thus, the nominal switched discrete-time system (9) is asymptotically stable.

Lemma 6 (see [34]). Chebyshev inequality holds for any matrix 𝑊_𝑖∈𝑅^𝑛×𝑛: ^𝑁_𝑖=1𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖≤𝑁^𝑁_𝑖=1𝐴^𝑇_𝑖𝐴_𝑖. (14)

In the study of switched large-scale discrete-time systems, the main problems are how to constructively design a switching rule and how to derive sufficient stability conditions which can guarantee the stability of the switched large-scale discrete-time systems under the switching rule.

3. Stabilization Analysis

For the switched large-scale discrete-time system (2) with 𝑟 individual systems and each individual system composed of 𝑁 interconnected subsystems, assume that ∑^𝑟_𝑙=1√2𝛼_𝑙^1/2𝐴_𝑙𝑖𝑖 are Schur matrices. To investigate the stability of system (1), we denote 𝑥^𝑇(𝑘)=[𝑥^𝑇₁𝑥^𝑇₂⋯𝑥^𝑇_𝑁]^𝑇 and 𝑃=diag{𝑃₁,𝑃₂,…,𝑃_𝑁}, where 2^𝑟_𝑙=1𝛼_𝑙^1/2𝐴_𝑙𝑖𝑖^𝑇𝑃_𝑖^𝑟_𝑙=1𝛼_𝑙^1/2𝐴_𝑙𝑖𝑖−𝑃_𝑖=−𝐼. (15)

Theorem 7. Consider the switched large-scale discrete-time system (1), and assume that ∑^𝑟_𝑙=1𝛼_𝑙^1/2𝐴_𝑙𝑖𝑖 are Schur matrices. There exists a switching law such that the switched large-scale discrete-time system (1) is asymptotically stable, if matrices 𝑃₁,𝑃₂,…,𝑃_𝑁>0 satisfy (15) and the following inequality (16) holds: ^𝑟𝑁_{𝑙=1𝑗≠𝑖𝑗=1}𝛼_𝑙𝐴^𝑇_𝑙𝑗𝑖𝑃_𝑗𝐴_𝑙𝑗𝑖≤0.5(𝑁−1)⁻¹, (16) where 𝛼_𝑙=1/(1+𝜀⁻¹)^𝑙−1(1+𝜀), (𝑙=1,2,…,𝑟−1), 𝛼_𝑟=1/(1+𝜀⁻¹)^𝑟−1 and 𝜀>0.

Proof. By inequality (12), we obtain ^𝑁_𝑖=1^𝑟_𝑙=1𝛼_𝑙𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇×𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)−𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑥_𝑖(𝑘)≤^𝑁_𝑖=1^𝑟_𝑙=1𝛼_𝑙𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)^𝑇𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑟_𝑙=1𝛼_𝑙𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)^𝑇𝑃_𝑖^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)+^𝑟_𝑙=1𝛼_𝑙^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑟_𝑙=1𝛼_𝑙^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇𝑃_𝑖^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)−𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑥_𝑖(𝑘)≤^𝑁_𝑖=1^𝑟_𝑙=1𝛼_𝑙2𝐴_𝑙𝑖𝑖(𝑘)^𝑇𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑟_𝑙=12𝛼_𝑙^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇×𝑃_𝑖^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)−𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑥_𝑖(𝑘)≤^𝑁_𝑖=1^𝑟_𝑙=12𝛼_𝑙𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)^𝑇𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑟_𝑙=12𝛼_𝑙()×𝑁−1_{𝑁𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇𝑃_𝑖𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)−𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑥_𝑖(𝑘)≤^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)()×−𝐼+2𝑁−1_{𝑟𝑁𝑙=1𝑗≠𝑖𝑗=1}𝛼_𝑙𝐴^𝑇_𝑙𝑗𝑖𝑃_𝑗𝐴_𝑙𝑗𝑖𝑥_𝑖(𝑘). (17) If conditions (16) are satisfied, then the inequality (17) is smaller than zero for 𝑖=1,2,…,𝑁 and 𝑙=1,2,…,𝑟. Therefore, the switched large-scale discrete-time system is asymptotically stable.

Remark 8. For the switched large-scale discrete-time system (1) with arbitrary 𝑟 individual systems and 𝑁 state vectors, Theorem 7 can be applied to cases when all individual subsystems are unstable. The particular method also can be applied to cases when 𝑟>𝑁, or 𝑟=𝑁, 𝑟<𝑁.

Switching Law. Switched large-scale discrete-time system (1) is switched to or stays at mode 𝑙 at sampling step 𝑘 if (18) is satisfied at 𝑘: ^𝑁_𝑖=1𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)^𝑇×𝑃_𝑖𝐴_𝑙𝑖𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_𝑙𝑖𝑗𝑥_𝑗(𝑘)−𝑃_𝑖<0,𝑙=1,2,…,𝑟. (18)

4. Conclusion

We have adopted the Lyapunov stability theorem to deal with the stabilization analysis of switched large-scale discrete-time systems. The results are straightforward. By a simple switching law, the sufficient stability conditions have been derived for the switched large-scale discrete-time systems via the state-driven switching. The results can be applied to cases when 𝑟>𝑁, 𝑟=𝑁, or 𝑟<𝑁 for the switched large-scale discrete-time system with arbitrary 𝑟 individual systems and 𝑁 state vectors. In addition, the particular method can be applied to cases when all individual subsystems are unstable.

Acknowledgment

This work is supported by the National Science Council, Taiwan, under Grants nos. NSC101-2221-E-218-027 and NSC100-2632-E-218-001-MY3.