On Delay Independent Stabilization Analysis for a Class of Switched Large-Scale Time-Delay Systems

Abstract

In view of the state-driven switching method, the sufficient stability conditions with delay independence will be derived for the switched large-scale time-delay systems. A new stability criterion of switched large-scale time-delay systems is deduced by Lyapunov stability theorem. The method can be applied to cases when all individual switched systems are unstable. Finally, one example is exploited to illustrate the proposed schemes.

1. Introduction

Switched systems form an important class of hybrid systems. Such systems can be described by a family of continuous-time subsystems (or discrete-time subsystems) and a rule that orchestrates the switching among them. It is well known that a wide class of physical systems in power systems, chemical procedure control systems, navigation systems, automobile speed change systems, and so forth may be appropriately described by the switched model [1–10]. In the study of switched systems, most works have been centered on the problem of stability. In the last two decades, there have been increasing interests in the stability analysis for such switched systems; see, for example, [11] 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; the other is the time-driven switching strategy. The time-driven switching method is formed based on the main concept of a dwell time. If at least one stable subsystem exits, then the switched system is stable with a proper dwell time switching law. However, in reality, it is not unusual to encounter cases in which all subsystems are unstable. Therefore, [6] proposed that all subsystem can be unstable for time-driven switching method. We would have to turn to the state-driven switching strategy, from which many choices of switching laws ensuring stability exist, even if all the subsystems are unstable. Reference [5] proposed quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. For continuous-time switched linear systems, if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. The switching rules can be obtained by using the obtained common positive definite matrix.

Furthermore, the time-delay phenomenon is also unavoidable in practical systems, for instance, chemical process, long distance transmission line, hybrid procedure, electron network, and so forth. Time delays may cause instability and poor performance for practical systems [12]. In view of the aforementioned facts, the stability of switched systems with time delay is very worthy of research [13, 14]. Reference [13] studied stability and 𝐿₂ gain properties for a class of switched systems which are composed of a finite number of linear time-invariant symmetric systems with time delay and showed that, when all subsystems are in sense of satisfying an LMI, the switched system is asymptotically stable under arbitrary switching. Reference [14] proposed a delay-dependent robust 𝐻_∞ controller design method for the uncertain linear switched systems with time delay. Reference [15] investigated the problem of robust 𝐿_∞ reliable control for uncertain switched nonlinear systems with time delay and actuator failures. References [16, 17] concerned with the problem of delay-dependent stability analysis for a class of two-dimensional discrete switched systems described by the Roesser model with state delays. Besides, a large-scale system is often considered as a set of interconnected subsystems. The advantage of this aspect in stability analysis is to reduce complexity [18, 19]. Recently, many approaches have been used to investigate the stability and stabilization of large-scale time-delay systems [20, 21]. Therefore, the stability analysis of switched large-scale time-delay systems is very worthy to be researched.

Basically, current efforts to achieve stability in time-delay systems can be divided into two categories, namely, delay-independent criteria and delay-dependent criteria. For the delay-dependent criteria, the Lyapunov-Krasovskii functional technique is most suitable for application; the proposed criterion adopting this technique performs much better than several existing criteria [22–24]. In this paper, the delay-independent stability of switched large-scale time-delay system is considered. In view of the state-driven switching method, a new stability criterion of switched large-scale time-delay system is deduced by Lyapunov stability theorem [25]. This method can be applied to cases when all individual switched systems are unstable.

The following notations will be used throughout the paper: 𝜆(𝐴) stands for the eigenvalues of matrix 𝐴, 𝜆_max(𝐴) means the maximum eigenvalue in matrix 𝐴; that is, 𝜆_max(𝐴)=Max[𝜆(𝐴)], ‖𝐴‖ denotes the norm of matrix 𝐴; that is, ‖𝐴‖=Max[𝜆(𝐴^𝑇𝐴)]^1/2, and 𝐴^𝑇 is the transpose of matrix 𝐴.

2. System Description and Problem Statement

Consider the switched time-delay large-scale systems𝑥̇_𝑖(𝑡)=𝐴_{𝜎(𝑥)_𝑖𝑖}𝑥_𝑖(𝑡)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_{𝜎(𝑥)_𝑖𝑗}𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝐵_{𝜎(𝑥)_𝑖𝑗}𝑥_𝑗()𝑥𝑡−𝜏_𝑖(𝑡)=𝜓_𝑖(𝑡),𝑡∈[−𝜏,0], (1a)(1b)

where 𝑥_𝑖(𝑡)∈𝑅^𝑛𝑖 are the state vectors of the 𝑖th subsystem, 𝑖=1,2,…,𝑁,𝑛=∑^𝑁_𝑖=1𝑛_𝑖,𝐴_{𝜎(𝑥)_𝑖𝑖},𝐴_{𝜎(𝑥)_𝑖𝑗}, and 𝐵_{𝜎(𝑥)_𝑖𝑗} are some constant matrices of compatible dimensions, 𝑥(𝑡)=𝑥^𝑇₁(𝑡)𝑥^𝑇₂(𝑡)⋯𝑥^𝑇_𝑁(𝑡)^𝑇, and  𝜎(𝑥(𝑡))∶𝑅^𝑛→{1,2,…,𝑟} is a piecewise constant function of time, called a switch signal; that is, the matrix 𝐴_{𝜎(𝑥)_𝑖𝑖} switches between matrices 𝐴_{1_𝑖𝑖},𝐴_{2_𝑖𝑖},…,𝐴_{𝑟_𝑖𝑖} belonging to the set 𝐴≡{𝐴_{1_11},𝐴_{2_𝑖𝑖},…,𝐴_{𝑟_𝑖𝑖}} and 𝐴_{𝑙_𝑖𝑖}𝑙∈{1,2,…,𝑟}. The matrix 𝐵_{𝜎(𝑥)_𝑖𝑗} switches between matrices 𝐵_{1_𝑖𝑗},𝐵_{2_𝑖𝑗},…,𝐵_{𝑟_𝑖𝑗} belonging to the set 𝐵≡{𝐵_{1_𝑖𝑗},𝐵_{2_𝑖𝑗},…,𝐵_{𝑟_𝑖𝑗}} and 𝐵_{𝑙_𝑖𝑗},𝑖,𝑗∈{1,2,…,𝑁}.  𝐴_{𝜎(𝑥)_𝑖𝑖},  𝐴_{𝜎(𝑥)_𝑖𝑗}, and 𝐵_{𝜎(𝑥)_𝑖𝑗} are matrices of compatible dimensions. 𝜏>0 is the time-delay duration. 𝜓_𝑖(𝑡) is a vector-valued initial continuous function defined on the interval [−𝜏,0], and finally 𝜓_𝑖(𝑡), defined on −𝜏≤𝑡≤0, is the initial condition of the state, 𝑁 is the number of interconnected subsystems, 𝑛 is the whole systems dimension, and 𝑟 is the number of individual systems.

As we have indicated, very few research articles have ever discussed the stability of the switched time-delay large-scale system (1a) and (1b). To facilitate our analysis later, some helpful lemmas are given below first.

First, we consider the normal time-delay system 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏), (2) where 𝑥∈𝑅^𝑛, 𝐴 and 𝐵 are matrices in proper dimensions, and 𝜏 is the delay duration.

Lemma 1 (see [26]). The stability with delay independence of system (2) implies the stability with delay independence for the following systems: 𝑤̇(𝑡)=(𝐴+𝑧𝐵)𝑤(𝑡),∀|𝑧|≤1, (3) and vice versa.

Lemma 2. There exists a switching law for the switched system (1a) and (1b) such that the system is asymptotically stable if there exist positive constants 𝛼_𝑖(1≤𝑖≤𝑁) satisfying ∑^𝑁_𝑖=1𝛼_𝑖=1 such that the convex combination of the whole switched time-delay system 𝑥̇(𝑡)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏) (4) is an asymptotically stable system.

Proof. Since there exist positive numbers 𝛼_𝑖(1≤𝑖≤𝑁) such that 𝑥̇(𝑡)=∑^𝑁_𝑖=1𝛼_𝑖[𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏)] is asymptotically stable, there exists a Lyapunov function 𝑉(𝑥) such that 𝜕𝑉𝜕𝑥^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏)<0. (5) It follows that, for any 𝑡, there exists an 𝑖∈{1,2,…,𝑁} such that 𝜕𝑉𝐴𝜕𝑥^𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏)<0. (6) From (6), it implies that a convex combination of the corresponding Lyapunov function is negative along the trajectory and from, at least one must be negative. Thus, switched time-delay system (1a) and (1b) is asymptotically stable.

Remark 3. Condition (6) will be used to design switching laws. The technique is that the switching is forced so that the active system is always in the region corresponding to the Lyapunov function with negative derivative.

Lemma 4. In view of Lemma 1, the stability with delay independence of 𝑥̇(𝑡)=∑^𝑁_𝑖=1𝛼_𝑖[𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏)] implies the stability with delay independence for the following systems: 𝑤̇(𝑡)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖+𝑧𝐵_𝑖𝑤(𝑡),∀|𝑧|≤1, (7) and vice versa.

Thus, the system (7) is an asymptotically stable system; then, the system (4) is stable. That is, the switched time-delay system (1a) and (1b) is also stable.

A sufficient condition for stability of the switched time-delay large-scale system (1a) and (1b) is established in the following theorem by using the state-switched method.

3. Stabilization Analysis of Switched Large-Scale Time-Delay System

The system (1a) and (1b) can be rewritten as the following: 𝑋̇(𝑡)=𝐴_𝑙𝑋+𝐵_𝑙𝑋(𝑡−𝜏), (8) where 𝐴_𝑙=𝐴_{𝑙_11}𝐴_{𝑙_12}⋯𝐴_{𝑙_1𝑁}𝐴_{𝑙_21}𝐴_{𝑙_21}⋯𝐴_{𝑙_2𝑁}𝐴⋮⋮⋮⋮_{𝑙_𝑁1}𝐴_{𝑙_𝑁2}⋯𝐴_{𝑙_𝑁𝑁},𝐵_𝑙=𝐵_{𝑙_11}𝐵_{𝑙_12}⋯𝐵_{𝑙_1𝑁}𝐵_{𝑙_21}𝐵_{𝑙_21}⋯𝐵_{𝑙_2𝑁}𝐵⋮⋮⋮⋮_{𝑙_𝑁1}𝐵_{𝑙_𝑁2}⋯𝐵_{𝑙_𝑁𝑁},𝑋(𝑡)=𝑥^𝑇₁𝑥^𝑇₂⋯𝑥^𝑇_𝑁^𝑇. (9) First, we consider the switched large-scale time-delay system (1a) and (1b) with 𝑟=𝑁=2:𝑥̇₁(𝑡)=𝐴_{𝑙_11}𝑥₁(𝑡)+𝐴_{𝑙_12}𝑥₂(𝑡)+𝐵_{𝑙_11}𝑥₁(𝑡−𝜏)+𝐵_{𝑙_12}𝑥₂()𝑥̇𝑡−𝜏₂(𝑡)=𝐴_{𝑙_21}𝑥₁(𝑡)+𝐴_{𝑙_22}𝑥₂(𝑡)+𝐵_{𝑙_21}𝑥₁(𝑡−𝜏)+𝐵_{𝑙_22}𝑥₂(𝑡−𝜏); (10a)(10b) that is, 𝑋̇(𝑡)=𝐴_{𝑙_11}𝐴_{𝑙_12}𝐴_{𝑙_21}𝐴_{𝑙_22}𝑋(𝑡)+𝐵_{𝑙_11}𝐵_{𝑙_12}𝐵_{𝑙_21}𝐵_{𝑙_22}𝑋(𝑡−𝜏). (10c)

In the light of Lemma 2, it is obvious that the stability of the switched time-delay system ((10a), (10b), and (10c)) is equivalent to that of the system

𝑥̇₁(𝑡)=²_𝑙=1𝛼_𝑙𝐴_{𝑙_11}𝑥₁(𝑡)+𝐴_{𝑙_12}𝑥₂(𝑡)+𝐵_{𝑙_11}𝑥₁(𝑡−𝜏)+𝐵_{𝑙_12}𝑥₂(𝑥̇𝑡−𝜏)₂(𝑡)=²_𝑙=1𝛼_𝑙𝐴_{𝑙_21}𝑥₁(𝑡)+𝐴_{𝑙_22}𝑥₂(𝑡)+𝐵_{𝑙_21}𝑥₁(𝑡−𝜏)+𝐵_{𝑙_22}𝑥₂(𝑡−𝜏); (11a)(11b)

that is, 𝑋̇(𝑡)=²_𝑙=1𝛼_𝑙𝐴_{𝑙_11}𝐴_{𝑙_12}𝐴_{𝑙_21}𝐴_{𝑙_22}𝑋(𝑡)+𝐵_{𝑙_11}𝐵_{𝑙_12}𝐵_{𝑙_21}𝐵_{𝑙_22}𝑋(𝑡−𝜏)=𝛼₁𝐴₁𝑋(𝑡)+𝐵₁𝑋(𝑡−𝜏)+𝛼₂𝐴₂𝑋(𝑡)+𝐵₂𝑋(𝑡−𝜏). (12)

To investigate the stability of system (11a) and (11b), we choose the Lyapunov function candidate as 𝑉_𝑖𝑥_𝑖(𝑡)=𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝑥_𝑖(𝑡), (13) where 𝑃=diag[𝑃₁,𝑃₂] is a unique real symmetric positive-definite matrix satisfying the Lyapunov equations: 𝐴^𝑇_{𝑖_𝑖𝑖}𝑃_𝑖+𝑃_𝑖𝐴_{𝑖_𝑖𝑖}=−𝐼,𝑖=1,2. (14)

It will be convenient throughout this section to use the following notations: 𝑆_{1_1}=1−𝛽𝑃²_𝜆1𝐵_{1_11}²+𝐵_{1_12}²+𝑃_𝜆1+𝑃_𝜆1𝐴_{1_12}+𝑃_𝜆2𝐴_{1_21}+𝛽𝑃_𝜆1𝑃_𝜆2𝐵_{1_11}𝐵_{1_21}+𝐵_{1_12}𝐵_{1_22},𝑆_{1_2}=2𝑃_𝜆1𝐴_{2_11}+𝛽𝑃²_𝜆1𝐵_{2_11}²+𝐵_{2_12}²+𝑃_𝜆1+𝑃_𝜆2𝐴_{2_21}+𝑃_𝜆1𝐴_{2_12}+𝛽𝑃_𝜆1𝑃_𝜆2𝐵_{2_11}𝐵_{2_21}+𝐵_{2_12}𝐵_{2_22},𝑆_{2_2}=1−𝛽𝑃²_𝜆2𝐵_{2_21}²+𝐵_{2_22}²+𝑃_𝜆2+𝑃_𝜆2𝐴_{2_21}+𝑃_𝜆1𝐴_{2_12}+𝛽𝑃_𝜆1×𝑃_𝜆2𝐵_{2_11}𝐵_{2_21}+𝐵_{2_12}𝐵_{2_22},𝑆_{2_1}=2𝑃_𝜆2𝐴_{1_22}+𝛽𝑃²_𝜆2𝐵_{1_21}²+𝐵_{1_22}²+𝑃_𝜆2+𝑃_𝜆1𝐴_{1_12}+𝑃_𝜆2𝐴_{1_21}+𝛽𝑃_𝜆1𝑃_𝜆2𝐵_{1_11}𝐵_{1_21}+𝐵_{1_12}𝐵_{1_22},𝑃_𝜆1=Max𝜆𝑃₁,𝑃_𝜆2=Max𝜆𝑃₂,𝛽=Max𝜆𝑃₁⁻¹𝜆𝑃₂⁻¹,𝛼=𝑆_{1_2}𝑆_{1_1}+𝑆_{1_2},𝑆𝛼=^2_2𝑆_{2_1}+𝑆_{2_2}. (15)

Theorem 5. Suppose the switched large-scale time-delay system (1a) and (1b) with 𝑟=𝑁=2. There exists a switching law such that the switched large-scale time-delay system (1a) and (1b) is asymptotically stable of delay independence, if constant 𝛼₁∈(𝛼,𝛼) and there matrices 𝑃₁,𝑃₂>0 satisfying

(i)𝑆_{1_1}()𝑆>0ii_{2_2}()𝑆>0iii_{1_1}𝑆_{2_2}>𝑆_{1_2}𝑆_{2_1}. (16a)(16b)(16c)

Proof. The Lyapunov function for the overall system is 𝑉(𝑋(𝑡))=²_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝑥_𝑖(𝑡)=𝑋^𝑇(𝑡)𝑃𝑋(𝑡). (17) The differential of the Lyapunov function (17) is as follows: 𝑉̇(𝑋(𝑡))=²_𝑙=1𝛼_𝑙𝑋^𝑇(𝑡)𝐴^𝑇_𝑙𝑃+𝑃𝐴_𝑙𝑋(𝑡)+𝑋^𝑇(𝑡)×𝑃𝐵_𝑙𝑋(𝑡−𝜏)+𝑋^𝑇(𝑡−𝜏)𝐵^𝑇_𝑙(𝑡≤𝑃𝑋)_2𝑙=1𝛼_𝑙𝑋^𝑇(𝑡)𝐴^𝑇_𝑙𝑃+𝑃𝐴_𝑙+𝑃𝐵_𝑙𝑃⁻¹𝐵^𝑇_𝑙𝑃𝑋(𝑡)+𝑋^𝑇()(≤𝑡−𝜏𝑃𝑋𝑡−𝜏)_2𝑙=1𝛼_𝑙𝑋^𝑇(𝑡)𝐴^𝑇_𝑙𝑃+𝑃𝐴_𝑙+𝑃𝐵_𝑙𝑃⁻¹𝐵^𝑇_𝑙𝑃𝑋(𝑡)+𝑉(𝑋(𝑡−𝜏)). (18) By using Razumikhin theorem [27], we assume that there exists a real 𝜐>1 such that 𝑉(𝑋(𝑡−𝜃))<𝜐𝑉(𝑋(𝑡)),for𝜃∈[0,𝜏]; (19) then, 𝑉̇(𝑋(𝑡))<²_𝑙=1𝛼_𝑙𝑋^𝑇(𝑡)𝐴^𝑇_𝑙𝑃+𝑃𝐴_𝑙+𝑃𝐵_𝑙𝑃⁻¹𝐵^𝑇_𝑙𝑃𝑋(𝑡)+𝜐𝑉(𝑋(𝑡)). (20) By continuity, there exists a 𝜐=1+𝛿 with 𝛿>0 sufficiently small such that 𝑉̇(𝑋(𝑡))<0 for all 𝑖. Therefore, 𝑉̇(𝑋(𝑡))<²_𝑙=1𝛼_𝑙𝑋^𝑇(𝑡)𝐴^𝑇_𝑙𝑃+𝑃𝐴_𝑙+𝑃𝐵_𝑙𝑃⁻¹𝐵^𝑇_𝑙𝑃+𝑃𝑋(𝑡)<𝛼₁𝑋^𝑇(𝑡)𝐴_{1_11}𝐴_{1_12}𝐴_{1_21}𝐴_{1_22}^𝑇𝑃₁00𝑃₂+𝑃₁00𝑃₂𝐴_{1_11}𝐴_{1_12}𝐴_{1_21}𝐴_{1_22}𝑃+𝛽¹00𝑃₂𝐵_{1_11}𝐵_{1_12}𝐵_{1_21}𝐵_{1_22}×𝐵_{1_11}𝐵_{1_12}𝐵_{1_21}𝐵_{1_22}^𝑇𝑃₁00𝑃₂+𝑃₁00𝑃₂𝑋(𝑡)+𝛼₂𝑋^𝑇(𝑡)𝐴_{2_11}𝐴_{2_12}𝐴_{2_21}𝐴_{2_22}^𝑇𝑃₁00𝑃₂+𝑃₁00𝑃₂𝐴_{2_11}𝐴_{2_12}𝐴_{2_21}𝐴_{2_22}𝑃+𝛽¹00𝑃₂𝐵_{2_11}𝐵_{2_12}𝐵_{2_21}𝐵_{2_22}×𝐵_{2_11}𝐵_{2_12}𝐵_{2_21}𝐵_{2_22}^𝑇𝑃₁00𝑃₂+𝑃₁00𝑃₂𝑋(𝑡)<𝑥₁²𝛼₁−1+𝛽𝑃²_𝜆1×𝐵_{1_11}²+𝐵_{1_12}²+𝑃_𝜆1+𝑃_𝜆1×𝐴_{1_12}+𝑃_𝜆2𝐴_{1_21}+𝛽𝑃_𝜆1𝑃_𝜆2×𝐵_{1_11}𝐵_{1_21}+𝐵_{1_12}𝐵_{1_22}+𝛼₂2𝑃_𝜆1𝐴_{2_11}+𝛽𝑃²_𝜆1×𝐵_{2_11}²+𝐵_{2_12}²+𝑃_𝜆1+𝑃_𝜆2𝐴_{2_21}+𝑃_𝜆1𝐴_{2_12}+𝛽𝑃_𝜆1𝑃_𝜆2𝐵_{2_11}𝐵_{2_21}+𝐵_{2_12}𝐵_{2_22}+𝑥₂²𝛼₂−1+𝛽𝑃²_𝜆2×𝐵_{2_21}²+𝐵_{2_22}²+𝑃_𝜆2+𝑃_𝜆2𝐴_{2_21}+𝑃_𝜆1𝐴_{2_12}+𝛽𝑃_𝜆1𝑃_𝜆2𝐵_{2_11}𝐵_{2_21}+𝐵_{2_12}𝐵_{2_22}+𝛼₁2𝑃_𝜆2𝐴_{1_22}+𝛽𝑃²_𝜆2×𝐵_{1_21}²+𝐵_{1_22}²+𝑃_𝜆2+𝑃_𝜆1×𝐴_{1_12}+𝑃_𝜆2𝐴_{1_21}+𝛽𝑃_𝜆1𝑃_𝜆2×𝐵_{1_11}𝐵_{1_21}+𝐵_{1_12}𝐵_{1_22}. (21)
Furthermore, let 𝛼₁=𝛼,𝛼₂=1−𝛼; therefore, 𝑉̇(𝑋(𝑡))<𝑥₁²𝛼𝑆_{1_1}+(1−𝛼)𝑆_{1_2}+𝑥₂²(1−𝛼)𝑆_{2_2}+𝛼𝑆_{1_1} (22)
If inequalities ((16a), (16b) and, (16c)) are satisfied and constant 𝛼∈(𝛼,𝛼), then 𝑉̇(𝑋(𝑡))<0. Hence, there exists a switching law such that system (1a) and (1b) is asymptotically stable.

Switching Law. Switched large-scale time-delay system ((1a), (1b)) with 𝑟=𝑁=2 is switched to or stay at mode 𝑖 at time 𝑡 if inequality (23) is satisfied at time 𝑡: ²_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐴_{𝑙_𝑖𝑖}𝑥_𝑖(𝑡)+²_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡)+²_𝑗=1𝐵_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡−𝜏)<0𝑙=1,2. (23)

Furthermore, we can extend to the case of 𝑟=𝑁 switched systems. In the light of Lemma 2, it is obvious that the stability of the switched large-scale time-delay system (1a) and (1b) is equivalent to that of the system 𝑋̇(𝑡)=^𝑁_𝑙=1𝛼_𝑙𝐴_𝑙𝑋(𝑡)+𝐵_𝑙𝑋(𝑡−𝜏). (24)

To investigate the stability of system (24), we choose the Lyapunov function candidate as 𝑉(𝑋(𝑡))=𝑋^𝑇(𝑡)𝑃𝑋(𝑡), (25) where 𝑃=diag(𝑃₁,𝑃₂,…,𝑃_𝑁) are unique real symmetric positive-definite matrices satisfying the Lyapunov equations 𝐴^𝑇_{𝑖_𝑖𝑖}𝑃_𝑖+𝑃_𝑖𝐴_{𝑖_𝑖𝑖}=−𝐼,1≤𝑖≤𝑁. (26) Without loss of generality, 𝐴_{𝑖_𝑖𝑖} are Hurwitz matrices.

Now, another sufficient condition for the stabilization of switched large-scale time-delay system (1a) and (1b) with 𝑟=𝑁 switched subsystems is addressed as the following theorem.

Theorem 6. Suppose the switched large-scale time-delay system (1a) and (1b) with 𝑁 switched subsystems, and then there exists a switching law such that switched large-scale time-delay system (1a) and (1b) is asymptotically stable of delay independence if constants 𝛼_𝑙 satisfy

^𝑁_𝑙=1𝛼_𝑙𝛼=1,_𝑙−1+𝑃²_{𝜆𝑙1≤𝑖≤𝑁}𝐵_{𝑙_𝑙𝑖}²+𝑃_𝜆𝑙+𝑃_{𝜆𝑙1≤𝑗≤𝑁𝑗≠𝑙}𝐴_{𝑙_𝑙𝑗}+𝑃_{𝜆𝑙1≤𝑖≤𝑁𝑖≠𝑙}𝑃_{𝜆𝑖1≤𝑗≤𝑁}𝐵_{𝑙_𝑙𝑗}𝐵_{𝑙_𝑖𝑗}+_{1≤𝑗≤𝑁𝑗≠𝑙}𝛼_𝑗2𝑃_𝜆𝑖𝐴_{𝑗_𝑖𝑖}+𝑃²_{𝜆𝑙1≤𝑖≤𝑁}𝐵_{𝑗_𝑙𝑖}²+𝑃_𝜆𝑙+𝑃_𝜆𝑗×𝐴_{𝑗_𝑗𝑙}+𝑃_{𝜆𝑙1≤𝑖≤𝑁𝑖≠𝑙}𝑃_{𝜆𝑖1≤𝑘≤𝑁}𝐵_{𝑗_𝑙𝑘}𝐵_{𝑗_𝑖𝑘}+_{1≤𝑖≤𝑁𝑖≠𝑗}𝑃_{𝜆𝑖1≤𝑘≤𝑁𝑘≠𝑖𝑘≠𝑗}𝐴_{𝑗_𝑖𝑘}<0. (27a)(27b)

Proof. With the same as proof of Theorem 5, Theorem 6 can be proved.

Switching Law. The switched time-delay system (1a) and (1b) is switched to or stay at mode 𝑖 at time 𝑡 if inequality (28) is satisfied at time 𝑡: ^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐴_{𝑙_𝑖𝑖}𝑥_𝑖(𝑡)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝐵_𝑖𝑗𝑥_𝑗(𝑡−𝜏)<0,𝑙=1,2,…,𝑁. (28)

Corollary 7. The nominal large-scale switched systems (1a) and (1b) without time-delay, that is, 𝐵_{𝑙_𝑖𝑗}=0, and then there exists a switching law such that switched large-scale system is asymptotically stable if constants 𝛼_𝑙 satisfy ^𝑁_𝑙=1𝛼_𝑙𝛼=1,_𝑙−1+𝑃^𝑁_{𝜆𝑙𝑗≠𝑙𝑗=1}𝐴_{𝑙_𝑙𝑗}+^𝑁_{𝑗=1𝑗≠𝑙}𝛼_𝑗2𝑃_𝜆𝑙𝐴_{𝑗_𝑙𝑙}+𝑃_𝜆𝑗𝐴_{𝑗_𝑗𝑙}++^𝑁_{𝑖=1𝑖≠𝑗}𝑃^𝑁_{𝜆𝑖𝑘=1𝑘≠𝑖𝑘≠𝑗}𝐴_{𝑗_𝑖𝑘}<0𝑓𝑜𝑟𝑙=1,2,…,𝑁. (29)(30)

Furthermore, we can develop another method to investigate the stability of switched large-scale time-delay systems (1a) and (1b).

In the light of Lemma 1, it is obvious that the stability of the switched time-delay system (1a) and (1b) is equivalent to that of the switched large-scale system 𝑤̇_𝑖(𝑡)=𝐴_{𝑙_𝑖𝑖}𝑤_𝑖(𝑡)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑤_𝑗(𝑡),∀|𝑧|≤1, (31) where 𝐴_{𝑙_𝑖𝑖}=𝐴_{𝑙_𝑖𝑖}+𝑧𝐵_{𝑙_𝑖𝑖} and 𝐴_{𝑙_𝑖𝑗}=𝐴_{𝑙_𝑖𝑗}+𝑧𝐵_{𝑙_𝑖𝑗}.

In the light of Lemma 4, a convex combination of switched large-scale systems (31) can be written as follows: 𝑤̇_𝑖(𝑡)=^𝑟_𝑙=1𝛼_𝑙𝐴_{𝑙_𝑖𝑖}𝑤_𝑖(𝑡)+^𝑁_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑤_𝑗(𝑡). (32)

First, we consider the switched system (1a) and (1b) with 𝑟=𝑁=2. In the light of Lemma 4, it is obvious that the stability of the switched time-delay system is equivalent to that of the system𝑊̇(𝑡)=𝛼𝐴₁+𝑧𝐵₁+(1−𝛼)𝐴₂+𝑧𝐵₂𝑊(𝑡)=𝛼𝐴_{1_11}+𝑧𝐵_{1_11}𝐴_{1_12}+𝑧𝐵_{1_12}𝐴_{1_21}+𝑧𝐵_{1_21}𝐴_{1_22}+𝑧𝐵_{1_22}+()×𝐴1−𝛼_{2_11}+𝑧𝐵_{2_11}𝐴_{2_12}+𝑧𝐵_{2_12}𝐴_{2_21}+𝑧𝐵_{2_21}𝐴_{2_22}+𝑧𝐵_{2_22}𝑊(𝑡), (33) where 0≤𝛼≤1.

To investigate the stability of system (33), we choose the Lyapunov function candidate as 𝑉_𝑖𝑤_𝑖(𝑡)=𝑤^𝑇_𝑖𝑃_𝑖𝑤_𝑖, (34) where 𝑊(𝑡)=[𝑤^𝑇₁𝑤^𝑇₂]^𝑇 and 𝑃=diag[𝑃₁,𝑃₂] are unique real symmetric positive-definite matrices satisfying the Lyapunov equations (14).

It will be convenient throughout the remainder of this section to use the following notations: 𝑓_{2_11}=𝐴_{2_11}+𝐵_{2_11},𝑓_{1_22}=𝐴_{1_22}+𝐵_{1_22},𝑓_{1_12}=𝐴_{1_12}+𝐵_{1_12},𝑓_{2_21}=𝐴_{2_21}+𝐵_{2_21},𝑓_{2_12}=𝐴_{2_12}+𝐵_{2_12},𝑓_{1_21}=𝐴_{1_21}+𝐵_{1_21},𝑃_𝜆1=Max𝜆𝑃₁,𝑃_𝜆2=Max𝜆𝑃₂,𝛼=2𝑓_{2_11}+𝑓_{2_12}𝑃_𝜆1+𝑓_{2_21}𝑃_𝜆2×2𝑓_{2_11}+𝑓_{2_12}𝑃_𝜆1+𝑓_{2_21}𝑃_𝜆2+1−2𝐵_{1_11}+𝑓_{1_12}𝑃_𝜆1+𝑓_{1_21}𝑃_𝜆2⁻¹,𝛼=1−2𝐵_{2_22}+𝑓_{2_21}𝑃_𝜆2+𝑓_{2_12}𝑃_𝜆1×2𝑓_{1_22}+𝑓_{1_21}𝑃_𝜆2+𝑓_{1_12}𝑃_𝜆1−2𝐵+1_{2_22}+𝑓_{2_21}𝑃_𝜆2+𝑓_{2_12}𝑃_𝜆1⁻¹. (35)

Now, another sufficient condition for the stabilization of switched large-scale time-delay system (1a) and (1b) with 𝑟=𝑁=2 is addressed as the following theorem.

Theorem 8. Suppose the switched time-delay system (1a) and (1b) with 𝑟=𝑁=2. These exists a switching law such that the switched time-delay system (1a) and (1b) is asymptotically stable of delay independence if constant 𝛼∈(𝛼,𝛼) and there matrices 𝑃₁,𝑃₂>0 satisfying

(i)1−2𝐵_{1_11}+𝑓_{1_12}𝑃_𝜆1+𝑓_{1_21}𝑃_𝜆2()2𝐵>0ii1−_{2_22}+𝑓_{2_21}𝑃_𝜆2+𝑓_{2_12}𝑃_𝜆1(2𝐵>0iii)1−_{1_11}+𝑓_{1_12}𝑃_𝜆1+𝑓_{1_21}𝑃_𝜆2×1−2𝐵_{2_22}+𝑓_{2_21}𝑃_𝜆2+𝑓_{2_12}𝑃_𝜆1>2𝑓_{2_11}+𝑓_{2_12}𝑃_𝜆1+𝑓_{2_21}𝑃_𝜆2×2𝑓_{1_22}+𝑓_{1_21}𝑃_𝜆2+𝑓_{1_12}𝑃_𝜆1. (36a)(36b)(36c)

Proof. The Lyapunov function for the overall system is 𝑉(𝑊(𝑡))=²_𝑖=1𝑤^𝑇_𝑖(𝑡)𝑃_𝑖𝑤_𝑖(𝑡)=𝑊^𝑇(𝑡)𝑃𝑊(𝑡). (37)
The differential of the Lyapunov function (37) is as follows 𝑉̇(𝑊(𝑡))=²_𝑖=1𝑤̇^𝑇_𝑖(𝑡)𝑃_𝑖𝑤_𝑖(𝑡)+𝑤^𝑇_𝑖(𝑡)𝑃_𝑖𝑤̇_𝑖(𝑡=𝑊̇_𝑇)(𝑡)𝑃𝑊(𝑡)+𝑊^𝑇𝑃𝑊̇≤𝑤₁²2𝛼𝐴_{1_11}+𝐵_{1_11}+(1−𝛼)𝑓_{2_11}𝑃_𝜆1+12𝛼𝑓_{1_12}+(1−𝛼)𝑓_{2_12}𝑃_𝜆1+𝛼𝑓_{1_21}+(1−𝛼)𝑓_{2_21}𝑃_𝜆2+12𝑤₂²𝛼𝑓_{1_12}+(1−𝛼)𝑓_{2_12}𝑃_𝜆1+𝛼𝑓_{1_21}+(1−𝛼)𝑓_{2_21}𝑃_𝜆2+𝑤₂²2𝑓_{1_22}+()×𝐴1−𝛼_{2_22}+𝐵_{2_22}𝑃_𝜆2+12𝛼𝑓_{1_12}+(1−𝛼)𝑓_{2_12}𝑃_𝜆1+𝛼𝑓_{1_21}+(1−𝛼)𝑓_{2_21}𝑃_𝜆2+12𝑤₁²𝛼𝑓_{1_12}+(1−𝛼)𝑓_{2_12}𝑃_𝜆1+𝛼𝑓_{1_21}+(1−𝛼)𝑓_{2_21}𝑃_𝜆2≤𝑤₁²{−𝛼1−2𝐵_{1_11}+𝑓_{1_12}𝑃_𝜆1+𝑓_{1_21}𝑃_𝜆2+()×1−𝛼2𝑓_{2_11}+𝑓_{2_12}𝑃_𝜆1+𝑓_{2_21}𝑃_𝜆2+𝑤₂²(𝛼−1)1−2𝐵_{2_22}+𝑓_{2_21}𝑃_𝜆2+𝑓_{2_12}𝑃_𝜆1+𝛼2𝑓_{1_22}+𝑓_{1_21}𝑃_𝜆2+𝑓_{1_12}𝑃_𝜆1. (38)
If inequalities ((36a), (36b), and (36c)) are satisfying and constant 𝛼∈(𝛼,𝛼), then 𝑉̇(𝑊(𝑡))<0. Hence, there exists a switching law such that system (1a) and (1b) is asymptotically stable.

Switching Law. Switched time-delay system (1a) and (1b) with 𝑟=𝑁=2 is switched to or stay at mode 𝑖 at time 𝑡 if inequality (39) is satisfied at time 𝑡: ²_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐴_{𝑙_𝑖𝑖}𝑥_𝑖(𝑡)+²_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡)+²_𝑗=1𝐵_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡−𝜏)<0𝑙=1,2. (39)

Now, a sufficient condition for the stabilization of switched large-scale time-delay system (1a) and (1b) with 𝑟=𝑁 switched systems is addressed as the following theorem.

Theorem 9. Suppose the switched large-scale time-delay system (1a) and (1b) with arbitrary 𝑁 switched systems, and then there exists a switching law such that switched large-scale time-delay system (1a) and (1b) is asymptotically stable of delay independence if constants 𝛼_𝑙 satisfy^𝑁_𝑙=1𝛼_𝑙=1,−𝛼_𝑙1−2𝐵_{𝑙_𝑙𝑙}𝑃_𝜆𝑙+^𝑁_{𝑖≠𝑙𝑖=1}𝑓_{𝑙_𝑙𝑖}𝑃_𝜆𝑙+^𝑁_{𝑖≠𝑙𝑖=1}𝑓_{𝑙_𝑖𝑙}𝑃_𝜆𝑖+^𝑁_{𝑗≠𝑙𝑗=1}𝛼_𝑗𝑓_{𝑗_𝑙𝑗}𝑃_𝜆𝑙+^𝑁_{𝑖≠𝑗𝑖=1}𝑓_{𝑗_𝑙𝑖}𝑃_𝜆𝑙+^𝑁_𝑖=1𝑓_{𝑗_𝑖𝑙}𝑃_𝜆𝑖<0, (40a)(40b)where 𝑓_{𝑙_𝑙𝑖}=‖𝐴_{𝑙_𝑙𝑖}‖+‖𝐵_{𝑙_𝑙𝑖}‖,   𝑓_{𝑗_𝑙𝑖}=‖𝐴_{𝑗_𝑙𝑖}‖+‖𝐵_{𝑗_𝑙𝑖}‖,   𝑓_{𝑗_𝑖𝑙}=‖𝐴_{𝑗_𝑖𝑙}‖+‖𝐵_{𝑗_𝑖𝑙}‖,   𝑓_{𝑗_𝑙𝑗}=‖𝐴_{𝑗_𝑙𝑗}‖+‖𝐵_{𝑗_𝑙𝑗}‖,𝑓_{𝑙_𝑖𝑙}=‖𝐴_{𝑙_𝑖𝑙}‖+‖𝐵_{𝑙_𝑖𝑙}‖, and 𝜆_max(𝑃_𝑖)=𝑃_𝜆𝑖,𝑖=1,2,…,𝑁.

Proof. With the same as proof of Theorem 8, Theorem 9 can be proved.

Corollary 10. The nominal switched large-scale systems (1a) and (1b) without time-delay, that is, 𝐵_{𝑙_𝑖𝑗}=0, and then there exists a switching law such that switched large-scale system is asymptotically stable if constants 𝛼_𝑙 satisfy

^𝑁_𝑙=1𝛼_𝑙=1,−𝛼_𝑙1−^𝑁_{𝑖≠𝑙𝑖=1}𝐴_{𝑙_𝑙𝑖}𝑃_𝜆𝑙+^𝑁_{𝑖≠𝑙𝑖=1}𝐴_{𝑙_𝑖𝑙}𝑃_𝜆𝑖+^𝑁_{𝑗≠𝑙𝑗=1}𝛼_𝑗𝐴_{𝑗_𝑙𝑗}𝑃_𝜆𝑙+^𝑁_{𝑖≠𝑗𝑖=1}𝐴_{𝑗_𝑙𝑖}𝑃_𝜆𝑙+^𝑁_𝑖=1𝐴_{𝑗_𝑖𝑙}𝑃_𝜆𝑖<0𝑙=1,2,…,𝑁. (41a)(41b)

Remark 11. The main contribution of the results is that this method can be applied to cases when all individual switched systems are unstable in this paper.

4. Numerical Example

Example 12. Consider a switched time-delay system composed of three individual switched systems given as follow.
Switched System 1:𝑥̇₁(𝑡)=−2.5𝑥₁(𝑡)+0.1𝑥₁()𝑥̇𝑡−𝜏₂(𝑡)=0.1𝑥₂(𝑡)−0.12𝑥₁(𝑡−𝜏)−0.06𝑥₂(𝑡−𝜏)+0.08𝑥₃()𝑥̇𝑡−𝜏₃(𝑡)=0.2𝑥₃(𝑡)+0.06𝑥₁(𝑡−𝜏)+0.08𝑥₂(𝑡−𝜏)+0.06𝑥₃(𝑡−𝜏). (42a)Switched System 2: 𝑥̇₁(𝑡)=0.1𝑥₁(𝑡)+0.1𝑥₁(𝑡−𝜏)+0.02𝑥₂(𝑡−𝜏)+0.14𝑥₃()𝑥̇𝑡−𝜏₂(𝑡)=−4𝑥₂(𝑡)+0.04𝑥₁(𝑡−𝜏)+0.176𝑥₂(𝑡−𝜏)+0.132𝑥₃()𝑥̇𝑡−𝜏₃(𝑡)=0.2𝑥₃(𝑡)−0.02𝑥₁(𝑡−𝜏)−0.068𝑥₂(𝑡−𝜏)−0.176𝑥₃(𝑡−𝜏). (42b)Switched System 3: 𝑥̇₁(𝑡)=0.2𝑥₁(𝑡)+0.08𝑥₁(𝑡−𝜏)−0.1𝑥₂()𝑥̇𝑡−𝜏₂(𝑡)=0.2𝑥₂(𝑡)+0.044𝑥₁(𝑡−𝜏)+0.18𝑥₂(𝑡−𝜏)+0.02𝑥₃()𝑥̇𝑡−𝜏₃(𝑡)=−3𝑥₃(𝑡)−0.092𝑥₁(𝑡−𝜏)−0.04𝑥₂(𝑡−𝜏)+0.04𝑥₃(𝑡−𝜏). (42c)
Via normal tests of stability for the large-scale time-delay system, three individual switched systems ((42a), (42b), and (42c)) are all unstable. Furthermore, according to Lyapunov equation (26), the 𝑃_𝑖(𝑖=1,2,3) can be obtained as the following: 𝑃₁=0.2,𝑃₂=0.125,𝑃₃=0.1667. (43) In view of the stability conditions of Theorem 6, the conditions (27a) and (27b) can be written as follows:(i)𝛼₁+𝛼₂+𝛼₃()=1ii−0.7874𝛼₁+0.2472𝛼₂+0.3023𝛼₃()<0iii0.1607𝛼₁−0.8668𝛼₂+0.1924𝛼₃()<0iv0.2410𝛼₁+0.2408𝛼₂−0.8119𝛼₃<0. (44a)(44b)(44c)(44d)Furthermore, let 𝛼₃=1−𝛼₁−𝛼₂; the above conditions ((44a), (44b), and (44c)) can be integrated as−1.0897𝛼₁−0.0551𝛼₂+0.3023<0−0.0318𝛼₁−1.0592𝛼₂+0.1924<01.0528𝛼₁+1.0526𝛼₂−0.8119<0. (45a)(45b)(45c)Therefore, 𝛼₁ and 𝛼₂ are selected inside the admissible region of the triangle as shown in Figure 1, and the switched time-delay system ((42a), (42b), and (42c)) is asymptotically stable by the following switching law: ³_𝑖=1𝑥^𝑇_𝑖(𝑡)𝑃_𝑖𝐴_{𝑙_𝑖𝑖}𝑥_𝑖(𝑡)+³_{𝑗≠𝑖𝑗=1}𝐴_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡)+³_𝑗=1𝐵_{𝑙_𝑖𝑗}𝑥_𝑗(𝑡−𝜏)<0,𝑙=1,2,3. (46)

Figure 1 

The admissible region 𝛼1 and 𝛼2 (𝑥_label: 𝛼1, 𝑦_label: 𝛼2 and 𝛼1+𝛼2+𝛼3=1).

In view of the stability conditions of Theorem 9, in addition, the conditions (40a) and (40b) can be written as follows:(i)𝛼₁+𝛼₂+𝛼₃()=1ii−0.9350𝛼₁+0.1203𝛼₂+0.1528𝛼₃()<0iii0.0783𝛼₁−0.9192𝛼₂+0.1297𝛼₃()<0iv0.1200𝛼₁+0.1845𝛼₂−0.9622𝛼₃<0. (47a)(47b)(47c)(47d)Furthermore, let 𝛼₃=1−𝛼₁−𝛼₂; the above conditions ((47a), (47b), and (47c)) can be integrated as−1.0878𝛼₁−0.0325𝛼₂+0.1528<0−0.0513𝛼₁−1.0488𝛼₂+0.1297<01.0822𝛼₁+1.1467𝛼₂−0.9622<0. (48a)(48b)(48c)

Therefore, 𝛼₁ and 𝛼₂ are selected inside the admissible region of the triangle as shown in Figure 2.

(a)

(b)

(c)

Figure 2 

(a) State response for 𝑥1. (b) State response for 𝑥2. (c) State response for 𝑥3.

The trajectories of the switched large-scale time-delay system ((42a), (42b), and (42c)) and the switching during period [0.5, 0.6] sec are shown in Figure 3 with initial value 𝑋(0)=2−13^𝑇 and 𝜏=0.5 sec.

Figure 3 

The switching times of the system during [1, 1.1] sec.

5. Conclusion

We have shown that the Lyapunov stability theorem can deal with the stabilization of switched large-scale time-delay systems. In addition, it is shown that the main advantages of our approach are that it can be applied to cases when all individual switched systems are unstable, quantify the region of stability, extend to arbitrary subsystems of switched large-scale time-delay systems, and develop the simple switching rule to stabilize the switched large-scale time-delay systems. For future research, we are interested in topics such as the stability problems of designing the switched laws with time dependency for the switched large-scale time-delay systems?

Acknowledgment

This work is supported by the National Science Council, Taiwan, under Grand nos. NSC 102-2221-E-218 -017 and NSC100-2632-E-218-001-MY3.