# Robust Delay Independent Stability Analysis for the Switched Interval Time-Delay Systems with Time-Driven Switching Strategy

Authors: Juing-Shian Chiou; Jen-Hsing Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Some new criteria of delay independent stability for the switched interval time-delay systems are deduced. The switching structure does depend on time-driven switching strategies. The total activation time ratio of the switching law can be determined to guarantee that the switched interval time-delay system is exponentially stable.


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## Body

## 1. Introduction

Switched systems constitute 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 between them. It is well known that a wide class of physical systems in power systems, chemical process control systems, navigation systems, automobile speed change system, and so forth may be appropriately described by the switched model [1–7]. In the study of switched systems, most works have been centralized on the problem of stability. In the last two decades, there has been increasing interest in the stability analysis for such switched systems; see, for example, [8, 9] 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 [9]; the other is the time-driven switching strategy [8].The state-driven switching method is that if all subsystems have the common Lyapunov function or the multiple Lyapunov functions, there are many choices of switching strategy to make the whole system stable. However, using these kinds of methods, the system must meet conditions completely. Therefore, the common Lyapunov function or the multi-Lyapunov function is difficult to construct for practical systems; even if we can construct the function, it is more complicated and not easy to implement on practical systems.The time-driven switching method is based on the concept of dwell time [2] that when all subsystem matrices are Hurwitz stable, then the entire switched system is exponentially stable for any switching signal if the time between consecutive switching (dwell time) is sufficiently large. [10] that switching among stable linear systems results in a stable system provided that switching is slow-on-the-average. But in many applications, unstable subsystems of switched systems cannot be avoided in fact [11]. If the average dwell time is chosen sufficiently large, and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then exponential stability of a desire degree is guaranteed.Furthermore, the time-delay phenomenon also cannot be avoided in practical systems, for instance, chemical process, long distance transmission line, hybrid procedure, electron network, and so forth. The problem of time-delay may cause instability and poor performance of practical systems [12–14]. Therefore, the stability analysis of switched systems with time delay is very worthy to be researched. In a control system, uncertainties may be due to measure errors, modeling errors, linearization approximations, and so forth. There seem to be some alternatives in formulating uncertainties or perturbations. [13, 15] shared the formulation which systems matrices are assumed to be perturbation. However, it does not take the position that the system matrices are expressed in as the sum of the two terms, but that the bounds for them are somehow known. These systems are called interval time-delay systems. The goal of this paper is to derive some robust sufficient stability conditions for the switched interval time-delay system.Basically, current efforts to achieve stability in time-delay systems can be divided into two categories, namely, delay-independent criteria and delay-dependent criteria. In this paper, in view of delay-independent analysis, we expect to aid in studying stability and designing time-driven switching law to achieve and implement in a practical switched interval time-delay system.The following notations will be used throughout the paper:λ(A) stands for the eigenvalues of matrix A, ∥A∥ denotes the norm of matrix A; that is, ∥A∥=Max[λ(ATA)]1/2, and μ(A) means the matrix measure of matrix A; that is, μ(A)=Max[λ((A+AT)/2)]. ∥A∥=Max[λ(ATA)]1/2.


## 2. System Description and Problem Statement

First, consider the following switched time-delay system(1)x˙(t)=Aσ(t)x(t)+Bσ(t)x(t-τ),x(t0)=x0,x(t)=ψ(t),t∈[-τ,0],
where x(t)∈Rn is state, Aσ(t)∈Rn×n, Bσ(t)∈Rn×n, t0≥0 is the initial time, x0 is the initial state, and σ(t):[t0,∞)→{1,2,…,N} is a piecewise constant function of time, called a switch signal; that is, the matrix Aσ(t) switches between matrices A1,A2,…,AN belonging to the set Α≡{A1,A2,…,AN} and Ai, i∈{1,2,…,N}; the matrix Bσ(t) switches between matrices B1,B2,…,BN belonging to the set B≡{B1,B2,…,BN} and Bi, i∈{1,2,…,N}. τ>0 is the time-delay duration. ψ(t) is a vector-valued initial continuous function defined on the interval [-τ,0], and finally ψ(t), defined on -τ≤t≤0, is the initial condition of the state.Let us consider the switched interval time-delay system described by(2)x˙(t)=AiIx(t)+BiIx(t-τ),i∈{1,2,…,N},
where AiI and BiI are matrices whose elements vary in prescribed defined as
(3)AiI=[akl_i],BiI=[bkl_i],
where k,l=1,2,…,n. a_kl_i≤akl_i≤a-kl_i and b_kl_i≤bkl_i≤b-kl_i.Denote:(4)A_i=[a_kl_i],A-i=[a-kl_i],B_i=[b_kl_i],B-i=[b-kl_i],
and let
(5)Ai=A_i+A¯i2,Bi=B_i+B¯i2,
where Ai and Bi are the average matrices between A_i, A¯i, and B_i, B¯i, respectively. Furthermore,
(6)Aib=AiI-Ai,Bib=BiI-Bi,
where Aib and  Bib are the bias matrices between A-i, Ai, and B-i, Bi, respectively. Also,
(7)Aim=A-i-Ai,Bim=B-i-Bi,
where Aim and Bim are the maximal bias matrices between A-i, Ai, and B-i, Bi, respectively.From the properties of matrix norm, we have(8)∥Aib∥≤∥Aim∥=αi,∥Bib∥≤∥Bim∥=βi,
and denote
(9)αI=max(αi),βI=max(βi),
where 1≤i≤N.In this paper, we study the robust stability analysis and switching law design for the switched interval time-delay systems.


## 3. Delay-independent Stability Analysis

Some helpful lemmas and definitions are given below.Lemma 1 (see [16]).
Consider the time-delay system:(10)x˙(t)=Ax(t)+Bx(t-τ),
where x∈Rn, A and B are matrices in proper dimensions, and τ is the delay duration. The stability of the time-delay system implies the stability for the following systems:
(11)w˙(t)=(A+zB)w(t),∀|z|=1,
and vice versa.In the light of Lemma1, for the switched time-delay system (1), all individual subsystems can be implied
(12)w˙(t)=(Ai+zBi)w(t)=A-i(z)w(t),∀|z|=1.
Therefore, the system (12) is exponentially stable if and only if the switched time-delay system (1) is exponentially stable.Lemma 2 (see [17]).
For matricesA∈Rn×n and  B∈Rn×n, the following relation holds:
(13)∥exp[(A+zB)t]∥≤exp[μ(A+zB)t]≤exp[(μ(A)+∥B∥)t],∀|z|=1.Without loss of generality, we assume that the switched interval time-delay system (2) at least has one individual subsystem whose μ(Ai)+∥Bi∥+αi+βi values are less than zero, the that of remaining individual system are not less than zero; that is,(14a)μ(Ai)+∥Bi∥+αi+βi<0,1≤i≤r,(14b)μ(Ai)+∥Bi∥+αi+βi≥0,r+1≤i≤N.Definition 3.
Consider(15a)λdoi-=min(|μ(Ai)+∥Bi∥+αi+βi|),1≤i≤r,(15b)λdoi+=max(μ(Ai)+∥Bi∥+αi+βi),r+1≤i≤N.
Furthermore, we assume thatT+(t) (or T-(t)) is the total activation time of individual subsystems whose μ(Ai)+∥Bi∥+αi+βi values are not less than zero (total activation time of individual subsystems whose μ(Ai)+∥Bi∥+αi+βi values are less than zero). The total activation time ratio between T-(t) and T+(t) can be called a switching law of the switched interval time-delay system (2). Therefore, we will find the ratio for the total activation time such that the switched interval time-delay system (2) is globally and exponentially stable with stability margin λ.Theorem 4.
Suppose that the switched interval time-delay system (2) exists in at least one individual subsystem whose μ(Ai)+∥Bi∥+αi+βi value is less than zero. The switched interval time-delay system (2) is globally and exponentially stable with stability margin λ, if the system (2) satisfies the following switching law:
(16)inft≥t0[T-(t)T+(t)]≥(λdoi++λ*)(λdoi--λ*),
where λ∈(0,λdoi-) and λ*∈(λ,λdoi-).Proof.
By Lemma1, the stability of the switched interval time-delay system (2) can be transformed into the following system:
(17)w˙(t)=(AiI+zBiI)w(t)=A-iIw(t).
The trajectory response of system (17) is written as follows:
(18)w(t)=eA-pi+1I(t-ti)eA-piI(ti-ti-1)⋯eA-p1I(t1-t0)w(t0).
In view of Lemma 2, we can obtain the inequality
(19)∥w(t)∥≤e(Api+1I+zBpi+1I)(t-ti)·e(ApiI+zBpiI)(ti-ti-1)⋯eAp1I+zBp1I(t1-t0)∥w(t0)∥≤e[μ(Api+1I)+∥Bpi+1I∥](t-ti)·e[μ(ApiI)+∥BpiI∥](ti-ti-1)⋯e[μ(Ap1I)+∥Bp1I∥](t1-t0)∥w(t0)∥.
From the properties of matrix measure, we have
(20)μ(AiI)≤μ(Aib+Ai)≤μ(Aib)+μ(Ai)≤μ(Ai)+αi,∥BiI∥≤∥Bi∥+βi.
Hence, the inequality (19) can be written as
(21)∥w(t)∥≤e[μ(Api+1)+αpi+1+∥Bpi+1∥+βpi+1](t-ti)⋯e[μ(Api)+αpi+∥Bpi∥+βpi](ti-ti-1)⋯e[μ(Ap1)+αp1+∥Bp1∥+βp1](t1-t0)∥w(t0)∥≤eλdoi+T+-λdoi-T-∥w(t0)∥.
Furthermore, the switching law (16) means that
(22)λdoi+T+(t)-λdoi-T-(t)≤-λ*(T+(t)+T-(t))=-λ*(t-t0).
Finally, if we choose λ∈(0,λdoi-) and λ*∈(0,λdoi-), the following inequality can be obtained:
(23)∥w(t)∥≤e-λ*(t-t0)∥w(t0)∥≤e-λ(t-t0)∥w(t0)∥.
From the previous inequality (23), the system (17) is globally and exponentially stable with stability margin λ and implies that the system (2) is also stable as the systems (17) and (2) have same stability as properties. Hence, the switched interval time-delay system (2) is also globally and exponentially stable with stability margin λ.Remark 5.
By Theorem4, the stability condition of the switched interval time-delay system (1) is independent of time-delay.


## 4. Example

Example 1.
Consider the switched interval time-delay system with interval matrices.
Subsystem 1. Consider(24a)A1I=[[-0.3,0.3][0.8,1.2][0.8,1.2][-0.3,0.3]],B1I=[[-0.1,0.3][-0.1,0.1][-0.1,0.1][-0.1,0.3]].Subsystem 2. Consider
(24b)A2I=[[-3.1,-2.9][-0.2,0.2][0.9,1.1][-3.3,-2.7]],B2I=[[-0.2,0][-0.1,0.1][-0.4,0][-0.4,-0.2]].
From (5) and (7), we obtain the average matrices and maximal bias matrices.
Subsystem 1. Consider
(25)A1=[0110],B1=[0.1000.1],A1m=[0.30.20.20.3],B1m=[0.20.10.10.2].Subsystem 2. Consider
(26)A2=[-301-3],B2=[-0.10-0.2-0.3],A2m=[0.10.20.10.3],B2m=[0.10.10.20.1].
From (15a) and (15b), we can calculate λdoi+=1.9 and λdoi-=1.4867. Finally, the total activation time ratio for the switching law is (with λ=0.3, λ*=0.6)
(27)T-(t)T+(t)≥(λdoi++λ*)(λdoi--λ*)=2.8194.In order to satisfy the switching law (27), we choose the total activation time ratio 3 : 1. The activation time of subsystem 1 is 0.1 sec, and the activation time of subsystem 2 is 0.3 sec, respectively. The trajectory of the switched interval time-delay system (for the average matrices A1, B1, A2, and B2) is shown in Figure 1 with initial state [12]T and time-delay 0.1 sec.Figure 1
Trajectory response in Example.


## 5. Conclusion

We have developed methodologies for the delay-independent stability criteria of switched interval time-delay systems with time-driven switching strategy. On delay-independent stability analysis, the sufficient conditions of the switched laws are presented, and the total activation time ratio under the switching laws is required to be not less than a specified constant, such that the switched interval time-delay system is delay-independent and exponentially stable with stability margin. In addition, the main advantages of our approach showed that we can quantify the region of stability, extend to arbitrary subsystems of switched time-delay systems, and develop the simple time-driven switching rule to stabilize the switched interval time-delay systems.



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