# Delay-Independent Stability of Switched Linear Systems with

Authors: Yuangong Sun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is focused on delay-independent stability analysis for a class of switched
linear systems with time-varying delays that can be unbounded. When the switched system is
not necessarily positive, we first establish a delay-independent stability criterion under arbitrary
switching signal by using a new method that is different from the methods to positive systems in
the literature. We also apply this method to a class of time-varying switched linear systems with
mixed delays.


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## Body

## 1. Introduction

The theory of switched systems has historically assumed a position of great importance in systems theory and has been studied extensively in recent years [1–6]. A switched system is a type of hybrid dynamic system that consists of a family of continuous-time (discrete-time) subsystems and a switching signal, which determines the switching between subsystems. The stability of switched linear systems under arbitrary switching signal is a very important problem, which is usually studied by a common Lyapunov functional approach, especially a common quadratic Lyapunov functional approach [7–10].Very recently, the stability of positive switched linear system has attracted a lot of attention [11–16]. As usual, a system is said to be positive if its state and outputs are nonnegative whenever the initial condition and inputs are nonnegative. For stability of positive switched linear system under arbitrary switching signal, a common linear copositive Lyapunov function is usually applied [17–20]. A switched linear copositive Lyapunov function has been used in discrete-time positive switched systems in [21]. When the positive switched linear system involves multiple time-varying delays that can be unbounded, it has been proved in [22] that the stability of such systems under any switching signal does not depend on delays if the switched system shares a common linear copositive Lyapunov function, which generalizes the early results in [23, 24].For the general switched linear systems with unbounded time-varying delays, it is necessary to consider whether the similar delay-independent stability criterion under arbitrary switching signal can also be derived. Note that the system is not necessarily positive; the methods to positive systems in [22] usually do not hold. Consequently, to answer this problem, we need a new approach that is different from those methods to positive systems in the literature.The main purpose of this paper is to establish a delay-independent stability criterion under arbitrary switching signal for the general switched linear systems with time-varying delays that can be unbounded. Since the switched systems are not necessarily positive, a new method based on some smart techniques of real analysis is proposed. By using this method, we not only present a delay-independent stability criterion for the system, but also extend the main result to a class of time-varying switched system with mixed delays, where one kind of delays is time-varying state delay that can be unbounded and the others are bounded time-varying distributed delay. Another advantage of the new method used in this paper lies in that it imposes less constraint on unbounded state delays than that given in [22] (see the corresponding discussion in Section 2).Notations.
SayA≽0(≺0) if all elements of matrix A are nonnegative (negative). We write A≽B if and only if A-B≽0. Denote by 𝕄 the the set of Metzler matrices whose off-diagonal entries are nonnegative. ℝn is an n-dimensional real vector space, ℝ+n is the set of positive vectors, and ℝn×n is the set of real n×n-dimensional matrices. For x∈ℝn, denote ∥x∥=max1≤i≤n{|xi|}. For positive integers p, q, n, and m, denote p̲={1,2,…,p}, q̲={1,2,…,q}, n̲={1,2…,n}, and m̲={1,2,…,m}.


## 2. Problem Statements and Preliminaries

Consider the following switched linear system with time-varying delays:(2.1)ẋ(t)=A0ϱx(t)+∑l=1pAlϱx(t-τlϱ(t)),t≥0,x(t)=ϕ(t),t∈[-α,0],
where x∈ℝn is the state; the piecewise continuous function ϱ:[0,∞)→m̲ is the switching signal; Als∈ℝn×n are constant matrices for l∈p̲∪{0} and s∈m̲; time-varying delays τls(t)≥0, l∈p̲, s∈m̲, are continuous on [0,∞); ϕ(t) is the continuous vector-valued initial function on [-α,0] with α=maxl∈p̲,s∈m̲supt≥0{τls(t)-t}.Unlike the assumptions on the system matrices [22], we here do not require A0s∈𝕄 and Als≽0 for l∈p̲ and s∈m̲. What is more, we make a less restrictive assumption on time-varying delays τls(t) as follows:(H1)
limt→+∞t-τls(t)=+∞, l∈p̲, s∈m̲.We recall to introduce another assumption onτls(t) in [22] as follows:(H1′)
there existT>0 and a scaler 0≤θ<1 such that θ=supt>T,l∈p̲,s∈m̲τls(t)/t.We show that (H1) is less constrained than (H1′). In fact, it is not difficult to see that (H1′) implies (H1). However, (H1) does not yield (H1′). For example, let τls(t)=t-t≥0 for t≥0.Since(2.2)lim⁡t→+∞⁡t-τls(t)=lim⁡t→+∞t=+∞,lim⁡t→+∞τls(t)t=lim⁡t→+∞t-tt=1,
we see that (H1) holds while (H1′) does not hold.In the sequel, we say system (2.1) is asymptotically stable under arbitrary switching signal, if for any ϵ>0, there exists δ>0 such that any solution x(t,ϕ,ϱ) of system (2.1) under arbitrary switching signal satisfies ∥x(t,ϕ,ϱ)∥<ϵ when ∥ϕ∥<δ, and limt→+∞∥x(t,ϕ,ϱ)∥=0.Generally speaking, due to the less constraint on delaysτls(t) and system matrices Als≽0 for l∈p̲ and s∈m̲, the methods for positive systems in the literature usually become invalid. Consequently, a new method should be introduced to analyze the delay-independent stability for system (2.1) under arbitrary switching signal.


## 3. Main Result

In the sequel, we denoteÃls=[ãij(ls)] and Als=[aij(ls)] for l∈p̲∪{0} and s∈m̲, where(3.1)ãij(0s)=aij(0s),i=j,ãij(0s)=|aij(0s)|,i≠j,ãij(ls)=|aij(ls)|,l∈p̲.
It is easy to see that Ã0s∈𝕄 and Ãls≽0 for s∈m̲.We now present the main result of this paper.Theorem 3.1.
Assume that (H1) holds. If there exists a vectorξ∈ℝ+n such that
(3.2)Asξ≺0,s∈m̲,
where 𝒜s=∑l=0pÃls, then system (2.1) is asymptotically stable under arbitrary switching signal.Proof.
Denoteξ=[ξ1,ξ2,…,ξn]T and
(3.3)Asξ=-[η1s,η2s,…,ηns]T,s∈m̲,
where ηis>0 for i∈n̲  by (3.2). The remaining proof is divided into two parts.
(i) For any constantϵ>0, there exists a constant δ>0 such that ∥x(t,ϕ,ϱ)∥<ϵ when ∥ϕ∥<δ. In the sequel, we denote the ith element of the solution x(t,ϕ,ϱ) of system (2.1) by xi(t) for i∈n̲.
In fact, for any givenϵ>0, let δ=(d1/d2)ϵ, where
(3.4)d1=min⁡i∈n̲⁡{ξ1,ξ2,…,ξn},d2=max⁡i∈n̲⁡{ξ1,ξ2,…,ξn}.
When ∥ϕ∥<δ, we prove that
(3.5)|xi(t)|<ξid1δ,t≥0,i∈n̲.
Note that ξi/d1≥1 for i∈n̲; then
(3.6)|xi(0)|≤‖ϕ‖<δ≤ξid1δ,i∈n̲.
By the continuity of the solution of system (2.1), we have that there exists t′>0 such that
(3.7)|xi(t)|<ξid1δ,t∈[0,t′],i∈n̲.
We further show that (3.5) holds if ∥ϕ∥<δ. Otherwise, there exists t*>0 and at least one index k∈n̲ such that
(3.8)|xk(t*)|=ξkd1δ,|xi(t)|<ξid1δ,0≤t<t*,i∈n̲,
which implies D-|xk(t*)|≥0, where D- means the left derivative. Set the left limitation ϱ(t*-)=s1∈m̲. By (2.1), (3.1), (3.3), and (3.8), we get
(3.9)D-|xk(t*)|=D-xk(t*)sign⁡xk(t*)≤akk(0s1)|xj(t*)|+∑j=1,j≠kn|akj(0s1)||xj(t*)|+∑l=1p∑j=1n|akj(ls1)||xj(t*-τls1(t*))|≤δd1∑l=0p∑j=1nãkj(ls1)ξj=-δd1ηks1<0.
From (3.9), we get a contradiction with the fact D-|xk(t*)|≥0. Therefore, for any ϵ>0, by choosing δ=(d1/d2)ϵ and using (3.5), we have that
(3.10)‖x(t,ϕ,ϱ)‖<max⁡i∈n̲{ξid1δ}≤d2d1δ=ϵ,t≥0,
if ∥ϕ∥<δ. This completes the proof of part (i).
(ii) For any solutionx(t,ϕ,ϱ) of system (2.1), limt→+∞∥x(t,ϕ,ϱ)∥=0.
Letxi(t)=ξiyi(t) for i∈n̲. Denote the upper limitation of |yi(t)| by lim¯t→+∞|yi(t)|=β̅i and the lower limitation of |yi(t)| by lim̲t→+∞|yi(t)|=βi for i∈n̲. Set β̅k=maxi∈n̲{β̅i} for some k∈n̲ and
(3.11)cks=|ãkk(0s)|ξk+∑j=1,j≠knãkj(0s)ξj+∑l=1p∑j=1nãkj(ls)ξj.
We first show thatβ̅k=βk. Assume to the contrary that β̅k>βk. Choose a sufficiently small ϵ satisfying
(3.12)0<ϵ<min⁡{β̅k,min⁡s∈m̲{ηks}β̅kmax⁡s∈m̲{cks}}.
By the definition of β̅k, we have that |yi(t)|≤β̅k+ϵ, i∈n̲, hold for sufficiently large t. Since limt→+∞t-τls(t)=+∞ for l∈p̲ and s∈m̲, we have that there exists sufficiently large t1>0 such that
(3.13)|yi(t)|≤β̅k+ϵ,t≥t1,|yi(t-τls(t))|≤β̅k+ϵ,t≥t1,
where i∈n̲, l∈p̲ and s∈m̲.
On the other hand, by the assumption thatβ̅k>βk and the choice of ϵ, there exists a sufficiently large t2>t1>0 such that
(3.14)|yk(t2)|≥β̅k-ϵ>0,D+|yk(t2)|≥0,
where D+ means the right derivative. Otherwise, we have |yk(t)|<β̅k-ϵ or D+|yk(t)|<0 eventually, which contradicts with the assumption lim¯t→+∞yk(t)=β̅k>βk=lim̲t→+∞yk(t). By (3.3), it is easy to see that ãkk0s<0. Denote the right limitation ϱ(t2+)=s2∈m̲. By (3.3), (3.13), and (3.14), we get
(3.15)D+|yk(t2)|=ξk-1D+xk(t2)sign⁡xk(t2)≤ξk-1[∑j=1nãkj(0s2)ξj|yj(t2)|+∑l=1p∑j=1nãkj(ls2)ξj|yj(t2-τls2(t2))|]≤ξk-1[β̅k∑l=0p∑j=1nãkj(ls2)ξj+ϵcks2]=ξk-1[-β̅kηks2+ϵcks2]≤-ωkξk<0,
where ωk=β̅kmins∈m̲{ηks}-ϵmaxs∈m̲{cks}>0 by (3.12). This is a contradiction with the fact that D+|yk(t2)|≥0. Therefore, β̅k=βk.
Next, we show thatβ̅k=βk=0. Otherwise, β̅k=βk>0. Then, for sufficiently small ϵ satisfying (3.12), there exists t3>t1 such that (3.13) holds, and
(3.16)|yk(t)|≥β̅k-ϵ>0,t≥t3.
Here, (3.16) is concluded from the property of the lower limitation lim̲t→+∞|yk(t)|=βk=β̅k. Similar to the above analysis, we have
(3.17)D+|yk(t)|≤-ωkξk,t≥t3.
Integrating (3.17) from t3 to t on both sides, we get the following contradiction:
(3.18)|yk(t)|≤|yk(t3)|-ωkξk(t-t3)⟶-∞(t⟶+∞).
Thus, limt→+∞|yk(t)|=0. By the choice of k and the definition of yi(t), we have that limt→+∞|xi(t)|=0 for i∈n̲, which implies that limt→+∞∥x(t,ϕ,ϱ)∥=0.
By (i) and (ii), system (2.1) is asymptotically stable under arbitrary switching signal. This completes the proof of Theorem 3.1.Remark 3.2.
For the particular case whenm=1, condition (3.2) holds if and only if 𝒜1 is a Hurwitz matrix [25]. When m>1, it requires that all 𝒜s, s∈r̲, share a common ξ∈ℝ+n such that -𝒜sξ∈ℝ+n. This problem has been studied in [20], where necessary and sufficient conditions for the existence of such a vector ξ were established.


## 4. Extension to Time-Varying Switched Systems with Mixed Delays

We now extend Theorem3.1 to a class of time-varying switched system with mixed delays:(4.1)ẋ(t)=A0ϱ(t)x(t)+∑l=1pAlϱ(t)x(t-τlϱ(t))+∑r=1qBrϱ(t)∫t-σrϱ(t)tx(u)du,t≥0,x(t)=ϕ(t),t∈[-γ,0],
where Als(t)=[aij(ls)(t)], Brs(t)=[bij(rs)(t)] are continuous matrix function on [0,∞), delays σrs(t) are continuous on [0,∞), and γ=maxl∈p̲,r∈q̲,s∈m̲{supt≥0{τls(t)-t},supt≥0{σrs(t)-t}}.Assume that(H2)
there exist constantsθrs>0 such that 0≤σrs(t)≤θrs for r∈q̲ and s∈m̲;(H3)
there exist constant matricesÂ0s=[âij(0s)]∈𝕄 and Âls=[âij(ls)]≽0,B̂rs=[b̂ij(rs)]≽0 such that, for t≥0 and i,j∈n̲,(4.2)aij(0s)(t)≤âij(0s),i=j,|aij(0s)(t)|≤âij(0s),i≠j,|aij(ls)(t)|≤âij(ls),l∈p̲,|bij(rs)(t)|≤b̂ij(rs),r∈q̲.
When xi(t)≠0, a straightforward computation based on (4.2) yields that(4.3)D±|xi(t)|=sign⁡xi(t)[∑j=1naij(0ϱ±)(t)xj(t)+∑l=1p∑j=1naij(lϱ±)(t)xj(t-τlϱ±(t))+∑r=1q∑j=1nbij(rϱ±)(t)∫t-σrϱ±(t)txj(u)du]≤∑j=1nâij(0ϱ±)|xj(t)|+∑l=1p∑j=1nâij(lϱ±)|xj(t-τlϱ±(t))|+∑r=1qθrϱ±∑j=1nb̂ij(rϱ±)max⁡t-θrϱ±≤u≤t⁡{|xj(u)|},
where ϱ±=ϱ(t±). Then, similar to the analysis in Theorem 3.1, it is not difficult to get the following stability criterion for system (4.1).Theorem 4.1.
Assume that (H1)–(H3) hold. If there exists a vectorξ∈ℝ+n such that
(4.4)Bsξ≺0,s∈m̲,
where ℬs=∑l=0pÂls+∑r=1qθrsB̂rs, then system (4.1) is asymptotically stable under arbitrary switching signal.Consider the following uncertain switched system:(4.5)ẋ(t)=[A0ϱ+ΔA0ϱ(t)]x(t)+∑l=1p[Alϱ+ΔAlϱ(t)]x(t-τlϱ(t))+∑r=1q[Brϱ+ΔBrϱ(t)]∫t-σrϱ(t)tx(u)du,t≥0x(t)=ϕ(t),t∈[-γ,0],
where ΔAls(t)=[Δaij(ls)(t)] and ΔBrs(t)=[Δbij(ls)(t)] are uncertain matrices satisfying(4.6)Δ|aij(ls)(t)|≤Δaij(ls),|Δbij(ls)(t)|≤Δbij(ls),t≥0.
Set(4.7)âij(0s)=aij(0s)+Δaij(0s),i=j,âij(0s)=|aij(0s)|+Δaij(0s),i≠j,âij(ls)=|aij(ls)|+Δaij(ls),l∈p̲,b̂ij(rs)=|bij(rs)|+Δbij(rs),r∈q̲.
Then, based on the same analysis as above, we have the following result for the uncertain system (4.5).Theorem 4.2.
Assume that (H1) and (H2) hold. If there exists a vectorξ∈ℝ+n such that (4.4) holds, then system (4.5) is asymptotically stable under arbitrary switching signal.


## 5.  A Numerical Example

To illustrate Theorem3.1, we present a simple numerical example of system (2.1) with(5.1)A01=[-3-0.50.5-1.5],A11=[-10.5-0.50.5],A02=[-1.50.25-0.25-1.5],A12=[0.5-0.250.25-0.5],
and τ11(t)=τ12(t)=0.4t+1 for t≥0. By (3.1), we have that(5.2)A1=[-211-1],A2=[-10.50.5-1].
It is not difficult to verify that (H1) holds and there exists a vector ξ=[1,1.5]T such that 𝒜sξ≺0 for s=1,2. Therefore, by Theorem 3.1, we know that system (2.1) is asymptotically stable under arbitrary switching signal. Since system (2.1) is not positive, Theorem  2 in [22] is invalid for this case. The state of the system is given in Figure 1.Figure 1
The state of the system under stochastic initial condition and switching.It is not difficult to work out an example of Theorem4.1. We omit it here due to the similarity with the above example.


## 6. Conclusion

In this paper, we investigate the delay-independent stability of the nonpositive switched linear systems with time-varying delays. By using a new method that is different from those methods to positive systems, we show that the stability of the system is also independent of delays if the switched system shares a common linear copositive Lyapunov function. We also apply this method to a class of time-varying switched linear systems with mixed delays, which generalizes some existing results in the literature.



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