# Stability Analysis andH∞ Model Reduction for Switched Discrete-Time Time-Delay Systems

Authors: Zheng-Fan Liu; Chen-Xiao Cai; Wen-Yong Duan
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper is concerned with the problem of exponential stability andH∞ model reduction of a class of switched discrete-time systems with state time-varying delay. Some subsystems can be unstable. Based on the average dwell time technique and Lyapunov-Krasovskii functional (LKF) approach, sufficient conditions for exponential stability with H∞ performance of such systems are derived in terms of linear matrix inequalities (LMIs). For the high-order systems, sufficient conditions for the existence of reduced-order model are derived in terms of LMIs. Moreover, the error system is guaranteed to be exponentially stable and an H∞ error performance is guaranteed. Numerical examples are also given to demonstrate the effectiveness and reduced conservatism of the obtained results.


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## Body

## 1. Introduction

Switched systems belong to a special class of hybrid control systems, which comprises a collection of subsystems described by dynamics differential or difference equations, together with a switching law that specifies the switching rule among the subsystems. Due to the theoretical development as well as practical applications, analysis and synthesis of switched systems have recently gained considerable attention [1–4].Furthermore, the time-delay phenomenon is frequently encountered in a variety of industrial and engineering systems [5–7], for instance, chemical process, long distance transmission line, communication networks, and so forth. Moreover, time-delay is a predominant source of the poor performance and instability. In the last two decades, there has been increasing interest in the stability analysis for the systems; see, for example, [8, 9] and the references cited there in. For switched delay systems, due to the impact of time-delays, the behavior of switched delay systems is usually much more complicated than that of switched systems or delay systems [10, 11].The average dwell time (ADT) technique [12] and multiple Lyapunov function approach [13] are two powerful and effective tools for studying the problems of stability for switched systems under controlled switching. By applying ADT scheme, the disturbance attenuation properties of time-controlled switched systems are investigated in [14]. In [15], the exponential stability and L2-gain of switched delay systems are studied by using ADT approach. Furthermore, based on ADT method, in [16–18] the stability of switched systems with stable and unstable subsystems co existing was considered. Using the ADT scheme, switching design for exponential stability was proposed in [19] for a class of switched discrete-time constant time-delay systems. By using the multiple Lyapunov function approach and ADT technique, the literature [20] studied the problem of state feedback stabilization of a class of discrete-time switched singular systems with time-varying state delay under asynchronous switching. However, many free weighing matrices were introduced, which made the stability result complicated. In [11], the problem of stabilization and robust H∞ control via ADT method switching for discrete switched system with time-delay was considered. However, the procedures given in [11] could not be applied to the case of asynchronous switching or the case of switching delay systems with stable and unstable subsystems co existing. This motivates the present study.On another research front line, it is well known that mathematical modeling of physical systems often results in complex high-order models. However, this causes the great difficulties in analysis and synthesis of the systems. Therefore, in practical applications it is desirable to replace high-order models with reduced-order ones for reducing the computational complexities in some given criteria without incurring much loss of performance or information. The purpose of model reduction is to obtain a lower-order system which approximates a high-order system according to certain criterion. Recently, much attention has been focused on the model reduction problem [21–25]. Many important results have been reported, which involve various efficient approaches, such as the balanced truncation method [24], the optimal Hanker norm reduction method [25], the cone complementarily linearization method [26], and sequential linear programming matrix method [27]. In terms of LMIs with inverse constraints or other non convex conditions, the model reduction of the discrete-time context has been investigated in [28, 29]. However, it is difficult to obtain the numerical solutions. In [30], the existence conditions for H∞ model reduction for discrete-time uncertain switched systems are derived in terms of strict LMIs by using switched Lyapunov function method. However, time delays are not taken into account. In the literature [31], a novel idea to approximate the original time-delay system by a reduced time-delay model has been proposed recently. However, the unstable subsystems are not taken into account.Motivated by the preceding discussion, the main contributions of this paper are highlighted as follows. The problem of exponential stability andH∞ model reduction for a class of switched linear discrete-time systems with time-varying delay have been investigated. To lessen the computation complexity and to reduce the conservatism, new discrete LKF are constructed and the delay interval is divided into two unequal subintervals by the delay decomposition method. The switching law is given by ADT scheme, such that even if one or more subsystem is unstable the overall switched system still can be stable. For the high-order systems, sufficient conditions for the existence of the desired reduced-order model are derived in terms of strict LMIs, which can be easily solved by using MATLAB LMI control toolbox. Finally, numerical examples are given to show the effectiveness of the proposed methods.The remainder of this paper is structured as follows. In Section2, the problem formulation and some preliminaries are introduced. In Section 3, the main results are presented on the exponential stability of switched discrete-time systems with time-varying delay. In Section 4, the main results on the H∞ model reduction for the high-order systems are presented. Numerical examples are given in Section 5. The last section concludes the work.Notations.We use standard notations throughout the paper. λmin(M)(λmax(M)) stands for the minimal (maximum) eigenvalue of M. MT is the transpose of the matrix M. The relation M>N  (M<N) means that the matrix M-N is positive (negative) definite. ∥x∥ denotes the Euclidian-norm of the vector x∈Rn. Rn represents the n-dimensional real Euclidean space. Rn×m is the set of all real n×m matrices. diag{⋯} stands for a block-diagonal matrix. In symmetric block matrices or long matrix expressions, we use an asterisk “*” to represent a term that is induced by symmetry. I denotes the identity matrix.


## 2. Problem Description and Preliminaries

Consider a class of switched linear discrete-time systems with time-varying state delay of the form(1)x(k+1)=Aix(k)+Adix(k-d(k))+Biu(k),y(k)=Cix(k)+Cdix(k-d(k))+Diu(k),x(θ)=ϕ(θ),θ=-h,-h+1,…,1,
where x(k)∈Rn denotes the system state, y(k)∈Rm is the measured output, u(k)∈Rp is the disturbance input vector which belongs to l2[0,∞). ϕ(θ)∈Rn is a vector-valued initial function. The switching signal σ (denoting σ(k) for simplicity) :[0,∞)→N-={1,2,…,T} is a piecewise constant function and depends on time. σ=i means that the ith subsystem is activated. T is the number of subsystems. The system matrices Ai,Adi,Bi,Ci,Cdi, and Di are a set of known real matrices with appropriate dimensions.For a given finite positive integerh>0, d(k) is time-varying delay and satisfies the following condition
(2)0≤d(k)≤h,∀k∈N+.
To facilitate theoretical development, we introduce the following definitions and lemmas.Definition 1 (see [19]).
The system (1) with disturbance input u(k)=0 is said to be exponentially stable if there exist a switching function σ(·) and positive number c such that every solution x(k,ϕ) of the system satisfies
(3)∥x(k)∥≤cλk-k0∥ϕ∥s,∀k≥k0,
for any initial conditions (k0,ϕ)∈R+×Cn. c>0 is the decay coefficient, 0<λ≤1 is the decay rate, and ∥ϕ∥s=sup{∥ϕ(l)∥,l=k0-h,k0-h+1,…,k0}.Definition 2 (see [11]).
Consider the system (1) with the following conditions.(1)
Withu(k)=0, the system (1) is exponentially stable with convergence rate λ>0.(2)
TheH∞ performance ∥y(k)∥2<γ∥u(k)∥2 is guaranteed for all nonzero u(k)∈L2[0,∞) and a prescribed κ>0 under the zero condition.In the above conditions, the system (1) is exponentially stabilizable with H∞ performance γ and convergence rate λ.Hereγ characterizes the disturbance attenuation performance. The smaller the γ is, the better the performance is.Definition 3 (see [12]).
For a switching signalσ(k) and any T2>k>T1≥0, let Nσ(T1,T2) denote the number of switching of σ(k) over (T1,T2). If for any given N0≥1 and Ta>0, we have Nσ(T1,T2)≤N0+(T2-T1)/Ta, then Ta and N0 are called the ADT and the chatter bound, respectively.Lemma 4 (see [9]).
For any matrixR=RT>0, integers a≤b, vector function ξ(k):{-b,-b+1,…,-a}→Rn, then
(4)(a-b)∑s=k-bk-a-1‍zT(s)Rz(s)≤ξT(k)[-RR-R]ξ(k).
Here
(5)z(k)=x(k+1)-x(k),ξT(k)=[xT(k-a)xT(k-b)].Lemma 5 (Schur complement [32]).
LetM,P,Q be given matrices such that Q>0. Then
(6)[PM*-Q]<0⟺P+MQ-1MT<0.The aim of this paper is to find a class of time-based switching signals for the discrete-time switched time-delay systems (1), whose subsystem is not necessarily stable, to guarantee the system to be exponentially stable. For a high-order system, we are interested in constructing a reduced-order switched system to approximate the system.


## 3. Stability Analysis

With the preliminaries given in the previous section we are ready to state the exponential stability andH∞ performance of switched systems (1). To obtain the exponential stability of switched systems (1), we construct following discrete LKF:
(7)Vi(k)=Vi1(k)+Vi2(k)+Vi3(k),∀i∈N-.
Here
(8)Vi1(k)=xT(k)Pix(k),Vi2(k)∑s=k-ϑk-1‍(1+αi)k-1-sxT(s)Qi1x(s)+∑s=k-hk-ϑ-1‍(1+αi)k-1-sxT(s)Qi2x(s)+∑s=k-d(k)k-1‍(1+αi)k-1-sxT(s)Qi3x(s),Vi3(k)∑θ=-ϑ-1‍∑s=k+θk-1‍(1+αi)k-1-szT(s)Ri1z(s)+∑θ=-h-ϑ-1‍∑s=k+θk-1‍(1+αi)k-1-szT(s)Ri2z(s)+∑θ=-d(k)-1‍∑s=k+θk-1‍(1+αi)k-1-szT(s)Ri3z(s),
where Pi,Qim,Rim(i∈N-, m=1,2,3) are symmetric positive definite matrices with appropriate dimensions, z(k)=x(k+1)-x(k), and integer ϑ∈(0,h) and αi are given constants.Remark 6.
In order to derive less conservative criteria than the existing ones, the delay interval[0,h] is divided into two unequal subintervals: [0,ϑ] and [ϑ,h], where ϑ∈(0,h) is a tuning parameter. The information about x(t-ϑ) can be taken into account. This plays a vital role in deriving less conservative results. Thus, for any k∈Z+, we have d(k)∈[0,ϑ] or d(k)∈[ϑ,h].
Firstly, we will provide a decay estimation of the system LKF in (7) along the state trajectory of switched system (1) without disturbance input (i.e., u(k)=0).Lemma 7.
Given constants-1<αi≤0, h>0 and ϑ∈(0,h), if there exist some symmetric positive definite matrices Pi,Qim,Rim(i∈N-, m=1,2,3) such that the following LMIs hold:
(9)[Ψi11Ψi12Ψi22]<0,(10)[Φi11Φi12Φi22]<0,
where
(11)Ψi11=[ψ11iψ12i00ψ22iψ23i0*ψ33iψ34i**ψ44i],Φi11=[ψ11i0ϕ13i0ϕ22iϕ23iϕ24i*ϕ33i0**ψ44i],Ψi12=[AiTPi(Ai-I)TW1iTAdiTPiAdiTW1iT0000],Φi12=[AiTPi(Ai-I)TW2iTAdiTPiAdiTW2iT0000],Ψi22=diag{-Pi-W1i},Φi22=diag{-Pi-W2i},ψ11i=-(1+αi)Pi-(1+αi)ϑϑ(Ri1+Ri3)+Qi1+Qi3,ψ12i=(1+αi)ϑϑ(Ri1+Ri3),ψ22i=-(1+αi)ϑQi3-(1+αi)ϑϑ(2Ri1+Ri3),ψ23i=(1+αi)ϑϑRi1,ψ33i=(1+αi)ϑ(Qi2-Qi1)-(1+αi)hh-ϑRi2-(1+αi)ϑϑRi1,ψ34i=(1+αi)hh-ϑRi2,ψ44i=-(1+αi)hQi2-(1+αi)hh-ϑRi2,ϕ13i=(1+αi)ϑϑ(Ri1+Ri3),ϕ22i=-(1+αi)hQi3-(1+αi)hh-ϑ(2Ri2+Ri3),ϕ23i=(1+αi)hh-ϑ(Ri2+Ri3),ϕ24i=(1+αi)hh-ϑRi2,ϕ33=-(1+αi)hh-ϑ(Ri2+Ri3)-(1+αi)ϑϑ(Ri1+Ri3)-(1+αi)ϑ(Qi1-Qi2),W1i=(h-ϑ)Ri2+ϑRi1+ϑRi3,W2i=(h-ϑ)Ri2+ϑRi1+hRi3.
Then, by means of LKF (7), along the trajectory of the systems (1) without disturbance input, one has
(12)ΔVi(k)=Vi(k+1)-Vi(k)≤αiVi(k).Proof.
Let us choose the system LKF (7). Define
(13)Vi(k+1)-(1+αi)Vi(k)=∑m=13‍Δ~Vim(k),
where
(14)Δ~Vim(k)=Vim(k+1)-(1+αi)Vim(k).
Therefore, the following equality holds along the solution of (1):
(15)Δ~Vi1(k)=xT(k+1)Pix(k+1)-(1+αi)xT(k)Pix(k),(16)Δ~Vi2(k)=xT(k)(Qi1+Qi3)x(k)-(1+αi)hxT(k-h)Qi2x(k-h)-(1+αi)d(k)xT(k-d(k))Qi3x(k-d(k))-(1+αi)ϑxT(k-ϑ)(Qi1-Qi2)x(k-ϑ),(17)Δ~Vi3(k)=zT(k)((h-ϑ)Ri2+ϑRi1+d(k)Ri3)z(k)-∑s=k-ϑk-1‍(1+αi)k-szT(s)Ri1z(s)-∑s=k-hk-ϑ-1‍(1+αi)k-szT(s)Ri2z(s)-∑s=k-d(k)k-1‍(1+αi)k-szT(s)Ri3z(s).
For any k∈Z+, we have d(k)∈[0,ϑ] or d(k)∈[ϑ,h].(1)
Ifd(k)∈[0,ϑ], it gets
(18)-∑s=k-ϑk-1‍(1+αi)k-szT(s)Ri1z(s)=-∑s=k-ϑk-1-d(k)‍(1+αi)k-szT(s)Ri1z(s)-∑s=k-d(k)k-1‍(1+αi)k-szT(s)Ri1z(s).So (17) could be
(19)Δ~Vi3(k)≤zT(k)((h-ϑ)Ri2+ϑRi1+ϑRi3)z(k)-∑s=k-hk-ϑ-1‍(1+αi)k-szT(s)Ri2z(s)-∑s=k-τ(k)k-1‍(1+αi)k-szT(s)(Ri1+Ri3)z(s)-∑s=k-ϑk-1-τ(k)‍(1+αi)k-szT(s)Ri1z(s).
From Lemma 4, we have
(20)-∑s=k-hk-ϑ-1‍(1+αi)k-szT(s)Ri2z(s)≤(1+αi)hh-ϑξ1T(t)[-Ri2Ri2-Ri2]ξ1(t),(21)-∑s=k-d(k)k-1‍(1+αi)k-szT(s)(Ri1+Ri3)z(s)≤(1+αi)τ(k)τ(k)ξ2T(k)[-Ri1-Ri3Ri1+Ri3-Ri1-Ri3]ξ2(k)≤(1+αi)ϑϑξ2T(k)[-Ri1-Ri3Ri1+Ri3-Ri1-Ri3]ξ2(k),(22)-∑s=k-ϑk-1-d(k)‍(1+αi)k-szT(s)Ri1z(s)≤(1+αi)ϑϑ-τ(k)ξ3T(k)[-Ri1Ri1-Ri1]ξ3(k)≤(1+αi)ϑϑξ3T(k)[-Ri1Ri1-Ri1]ξ3(k),
where
(23)ξ1T(k)=[xT(k-ϑ)xT(k-h)],ξ2T(k)=[xT(k)xT(k-d(k))],ξ3T(k)=[xT(k-d(k))xT(k-ϑ)].
Combining (13)–(22), it yields
(24)Vi(k+1)-(1+αi)Vi(k)≤ξT(k)Ψi11ξ(k)+xT(k+1)Pix(k+1)+zT(k)W1iz(k),
where
(25)ξT(k)=[xT(k)xT(k-d(k))xT(k-ϑ)xT(k-h)].
Multiplying (9) both from left and right by diag{0000Pi-1Wi-T}, by Schur Complement, further, considering (24), one can infer that (12) holds.(2)
Ifd(k)∈[ϑ,h], it gets
(26)-∑s=k-d(k)k-1‍(1+αi)k-szT(s)Ri3z(s)=-∑s=k-ϑk-1‍(1+αi)k-szT(s)Ri3z(s)-∑s=k-d(k)k-ϑ-1‍(1+αi)k-szT(s)Ri3z(s).  One obtains
(27)Δ~Vi3(k)≤zT(k)((h-ϑ)Ri2+ϑRi1+hRi3)z(k)-∑s=k-ϑk-1‍(1+αi)k-szT(s)(Ri1+Ri3)z(s)-∑s=k-d(k)k-ϑ-1‍(1+αi)k-szT(s)(Ri2+Ri3)z(s)-∑s=k-hk-d(k)-1‍(1+αi)k-szT(s)Ri2z(s).
Similarly, it is easy to get that
(28)Vi(k+1)-(1+αi)Vi(k)≤ξT(k)Φi11ξ(k)+xT(k+1)Pix(k+1)+zT(k)W2iz(k).
If (10) holds, by Schur Complement, then we have (12). This completes the proof.Remark 8.
Our LKF does not include free-weighing matrices as in previous investigations, and this may lead to reduce the computational complexity and get less conservation results.Remark 9.
In order to get less conservative results, the delay interval[0,h] can be divided into much more subintervals. However, when the number of dipartite numbers increases, the matrix formulation becomes more complex and the time-consuming grows bigger.Now we have the following theorem.Theorem 10.
If there exist some constants-1<αi<0 and positive definite symmetric matrices Pi,Qim,Rim(i∈N-, m=1,2,3) and μ≥1 such that (9), (10), and the following inequalities hold:
(29)Pi≤μPj,Qim≤μQjm,Rim≤μRjm,∀i,j∈N-.
Then, the switched system (1) with u(k)=0 and ADT satisfies τa>-lnμ/lnα which is exponentially stable.Proof.
By Lemma7, we have
(30)ΔVi(k)=Vi(k+1)-Vi(k)≤αiVi(k),∀i∈N-.
Therefore,
(31)Vi(k0+n)≤(αi+1)nVi(k0).
There exists μi≥1(i∈N-), such that
(32)Vi(k)≤μiVj(k),∀i,j∈N-.
We let τ1,…,τNσ(k0,k0+k) denote the switching times of σ in (k0,k0+k), and let Nσ(k0,k+k0) be the switching number of σ in (k0,k+k0), by (31) and (32), one obtains
(33)Vσ(k+k0)(k+k0)≤μσ(τ1)⋯μσ(τNσ(k0,k0+k))(ασ(k+k0)+1)m1⋯(ασ(k0)+1)mNσ(k0,k+k0)Vσ(k0)(k0),
where m1+⋯+mNσ(k0,k+k0)=k.
By-1<αi<0, for all i∈N-, we know that there exists α≜maxi∈N-{αi+1}∈(0,1). Let μ=maxi∈N-{μi}; from (33), one obtains
(34)Vi(k+k0)≤αkμNσVj(k0)=αk+Nσ(lnμ/lnα)Vj(k0).
By Definition 2, for any k0<k, it follows that
(35)Vi(k)≤αk+Nσ(lnμ/lnα)Vj(k0)≤αk(1+(lnμ/Talnα))Vj(k0).
By the system LKF (7), there always exist two positive constants c1,c2 such that
(36)c1∥x(k)∥2≤Vi(k),Vi(k0)≤c2∥x(k0)∥s2,
where
(37)c1=mini∈N-{λmin(Pi)},c2=maxi∈N-{λmax(Pi)+∑m=13‍(λmax(Qim)+λmax(Rim))}.
Therefore,
(38)∥x(k)∥2≤c2c1αk(1+(lnμ/Talnα))∥x(k0)∥s2.
If the average dwell time τa satisfies τa>-lnμ/lnα for μ≥1, then the switched system (1) with u(k)=0 is exponentially stable with λ=α1/2=maxi∈N-{(αi+1)1/2}∈(0,1) stability degree.Remark 11.
The caseα=0 implies the asymptotic stability.The following theorem provides exponential stability analysis withH∞ performance of the system (1).Theorem 12.
For given constantsγ>0, λ>0 and -1<αi<0, if there exist positive definite symmetric matrices Pi,Qim,Rim(i∈N-,m=1,2,3) and μ≥1 such that (29) and the following LMIs hold:
(39)[Ψi110Ψi13-γ2IΨi23*Ψi33]<0,(40)[Φi110Φi13-γ2IΦi23*Φi33]<0,
where
(41)Ψi13=[AiTPi(Ai-I)TW1iTCiTAdiTPiAdiTW1iTCdiT000000],Φi13=[AiTPi(Ai-I)TW2iTCiTAdiTPiAdiTW2iTCdiT000000],Ψi33=diag{-Pi-W1i-I},Φi33=diag{-Pi-W2i-I},Ψi23=[BiTPiBiTW1iTDiT],Φi23=[BiTPiBiTW2iTDiT].
Then, the system (1) with average dwell time satisfies τa>-lnμ/lnα  which is globally exponentially stable with convergence rate λ and H∞ performance γ.Proof.
Choose the LKF (7); the result is carried out by using the techniques employed for proving Lemma 7 and Theorem 10. If d(k)∈[0,ϑ], by (24), we have
(42)Vi(k+1)-αVi(k)+yT(k)y(k)-γ2uT(k)u(k)≤xT(k+1)Pix(k+1)+zT(k)Wiz(k)+yT(k)y(k)+ζ1T(k)[Ψi1100-γ2]ζ1(k),
where
(43)ζ1T(k)=[ξT(k)uT(k)].
If d(k)∈[ϑ,h], by (28), we have
(44)Vi(k+1)-αVi(k)+yT(k)y(k)-γ2uT(k)u(k)≤xT(k+1)Pix(k+1)+zT(k)Wiz(k)+yT(k)y(k)+ζ1T(k)[Φi1100-γ2]ζ1(k).
Combining (39) and (40), by Schur Complement, one can obtain
(45)Vi(k+1)-αVi(k)+yT(k)y(k)-γ2uT(k)u(k)≤0.
Let
(46)J(k)=yT(k)y(k)-γ2uT(k)u(k);
we have
(47)Vi(k+1)≤αVi(k)-J(k).
By Definition 2 and Theorem 10, it is sufficient to show that ∑k=0∞J(k)<0 for any nonzero u(k). Combining (35) and (47), it can be shown that
(48)Vσ(k)≤αkμNσ(0,k)Vσ(0)-∑s=0k-1‍αk-s-1μNσ(s,k)J(s).
Under the zero initial condition, we have
(49)V(0)=0,V(∞)≥0.
Combining (48), we have
(50)∑s=0k-1‍αk-s-1μNσ(s,k)J(s)=∑s=0k-1‍α-1elnα+lnμ/τaJ(s)≤0.
Now, we consider
(51)∑k=1∞‍∑s=0k-1‍α-1elnα+lnμ/τaJ(s).
Exchanging the double-sum region, by τa>-lnμ/lnα and α∈(0,1), one can easily get
(52)∑k=1∞‍∑s=0k-1‍α-1elnα+lnμ/τaJ(s)=∑s=0∞‍J(s)∑k=s+1∞‍α-1elnα+lnμ/τa=elnα+lnμ/τaα-11-elnα+lnμ/τa∑s=1∞‍J(s)≤0,
which means that ∑s=1∞J(s)≤0. Then, by Definition 2, the system (1) with average dwell time satisfies τa>-lnμ/lnα  which is globally exponentially stable with convergence rate λ and H∞ performance γ. This completes the proof.If there exist some unstable subsystems in the switched system (1) with u(k)=0, in this case, we need to estimate the growth rate of the system LKF in (7) along the state trajectory of switched system (1). And the corresponding αj>0(j∈N-). By using the techniques employed for proving Lemma 7, one can easily obtain the following Lemma.Lemma 13.
Given constantsαj>0, h>0 and ϑ∈(0,h), if there exist some symmetric positive definite matrices Pj,Qjm,Rjm(j∈N-, m=1,2,3) such that the following LMIs hold:
(53)[Ψ-j11Ψj12Ψj22]<0,[Φ-j11Φj12Φj22]<0,
where
(54)Ψ-j11=[ψ-11jψ-12j00ψ-22jψ-23j0*ψ-33jψ-34j**ψ-44j],Φ-j11=[ψ-11j0ϕ-13j0ϕ-22jϕ-23jϕ-24j*ϕ-33j0**ψ-44j],ψ-11j=-(1+αj)Pj+Qj1+Qj3-1ϑ(Rj1+Rj3),ψ-12j=1ϑ(Rj1+Rj3),ψ-22j=-Qj3-1ϑ(2Rj1+Rj3),ψ-23j=1ϑRj1,ψ-34j=(1+αj)ϑh-ϑRj2,ψ-33j=(1+αj)ϑ(Qj2-Qj1)-(1+αj)ϑh-ϑRj2-1ϑRj1,ψ-44j=-(1+αj)hQj2-(1+αj)ϑh-ϑRi2,ϕ-11j=ψ-11j,ϕ-13j=(1+αj)ϑ(Rj1+Rj3),ϕ-22j=-(1+αj)ϑQj3-(1+αj)ϑh-ϑ(2Rj2+Rj3),ϕ-23j=(1+αj)ϑh-ϑ(Rj2+Rj3),ϕ-24j=(1+αj)ϑh-ϑRj2,ϕ-33j=-(1+αj)ϑh-ϑ(Rj2+Rj3)-(1+αj)ϑ(Rj1+Rj3)-(1+αj)ϑ(Qj1-Qj2).
Then, by means of LKF (7), along the trajectory of the systems (1) without disturbance input, one has
(55)ΔVj(k)=Vj(k+1)-Vj(k)≤αjVj(k).Remark 14.
The proof of Lemma13 is similar to that of Lemma 7 and is thus omitted here. Based on Lemmas 7 and 13, one can easily design the stabilizing switching law to guarantee the system (1) with u(k)=0 to be exponentially stable, although some subsystems are unstable.
Without loss of generality, we can assume thatN-u={j1,j2,…,js} is the set of all unstable subsystems and N-s={is+1,is+2,…,ip} is the set of all stable subsystems. For simplicity, the LKF (7) is defined as Vi(αi,k)≜Vi(k). Choose the LKF Vi(αi,k)  (-1<αi<0, i∈N-s) for the stable subsystem and choose the LKF Vj(αj,k)  (αj>0, j∈N-u) for the unstable subsystem. Then, we have the following conclusion.Theorem 15.
If there exist some constants-1<αi<0, αj>0  (j≠i, i∈N-s, j∈N-u) and positive definite symmetric matrices Pi,Qim,Rim,Pj,Qjm,Rjm  (m=1,2,3) and μ≥1 such that Lemmas 7 and 13 and the following LMIs hold:
(56)Pl≤μPs,Qlm≤μQsm,Rlm≤μRsm,∀l,s∈N-.
Then, the switched system (1) with u(k)=0 and the average dwell time satisfies τa>lnμ/-κ, Tk0,n+k0α/Tk0,n+k0β≥(lnβ-κ)/(-lnα+κ), κ∈(lnα,0)  which is exponentially stable.Proof.
Consider the following LKF candidate:(57)Vσ(k)(k)={Vi(αi,k),σ(k)=i∈N-s,Vj(αj,k),σ(k)=j∈N-u.
By Lemmas 7 and 13, we have
(58)Vσ(k+1)(k+1)≤(ασ(k+1)+1)Vσ(k+1)(k).
Let Tk0,n+k0α be the total activity time in which all subsystems satisfied 0>αi>-1 on the interval (k0,n+k0) and Tk0,n+k0β≜n-Tk0,n+k0α the total activity time in which all subsystems satisfied αj>0 on the interval (k0,n+k0). By using the techniques employed for proving Theorem 10, combining (56) and (58), we derive that
(59)Vσ(n+k0)(n+k0)≤μNσ(n+k0)αTk0,n+k0αβTk0,n+k0βVσ(k0)(k0)=eTk0,n+k0αlnα+Tk0,n+k0βlnβ+Nσ(k0,n+k0)lnμVσ(k0)(k0),
where
(60)α≜maxi∈N-s{αi+1}∈(0,1),β≜maxj∈N-u{αj+1}>1.
By Tk0,n+k0α/Tk0,n+k0β≥(lnβ-κ)/(-lnα+κ), κ∈(lnα,0), one obtains
(61)Tk0,n+k0αlnα+Tk0,n+k0βlnβ≤κn.
So we have
(62)Vσ(n+k0)(n+k0)≤eκn+Nσ(k0,n+k0)lnμVσ(k0)(k0).
By Definition 2, for any n+k0>k0, it follows that
(63)Vσ(n+k0)(n+k0)≤eκn+Nσ(k0,n+k0)lnμVσ(k0)(k0)≤en(κ+(lnμ/τa))Vσ(k0)(k0).
By τa>lnμ/-κ, we have limk→∞Vσ(k)=0. Moreover, the overall system is exponentially stable. This completes the proof.Remark 16.
From the proof of Theorem15, one can see that the obtained exponential stability for the switched system (1) with u(k)=0 is exponential stable with e-1/2 stability degree. In order to get a free decay rate, we can replace the condition τa>lnμ/-κ, Tk0,n+k0α/Tk0,n+k0β≥(lnβ-κ)/(-lnα+κ),κ∈(lnα,0) by τa>logϵμ/-κ, Tk0,n+k0α/Tk0,n+k0β≥(logϵβ-κ)/(-logϵα+κ), κ∈(logϵα,0), ϵ>1; then the switched system (1) with u(k)=0 is exponentially stable with ϵ-1/2 stability degree.Theorem 17.
For given constantsγ>0, -1<αi<0, αj>0  (j≠i, i∈N-s, j∈N-u), if there exist positive definite symmetric matrices Pi,Qim,Rim,Pj,Qjm,Rjm  (m=1,2,3) and μ≥1 such that (56), (39), (40), and the following LMIs hold:
(64)[Ψ-j110Ψj13-γ2IΨj23*Ψj33]0,[Φ-j110Φj13-γ2IΦj23*Φj33]<0,
and Tk0,n+k0α/Tk0,n+k0β≥(lnβ-κ)/(-lnα+κ), κ∈(lnα,0), and the average dwell time satisfies τa>lnμ/-κ; then the switched system (1) is exponentially stable and with H∞ performance γ.Remark 18.
The proof of Theorem17 is similar to that of Theorems 12 and 15 and is thus omitted here.


## 4.H∞ Model Reduction

In this section, we will approximate system (1) by a reduced-order switched system described by
(65)x^(k+1)=Arix^(k)+Ardix^(k-d(k))+Briu(k),y^(k)=Crix^(k)+Crdix^(k-d(k))+Driu(k),
where x^(k)∈Rq is the state vector of the reduced-order system with q<n and y^(k)∈Rm is the output of reduced-order system. Ari,Ardi,Cri,Crdi,Bri, and Dri are the matrices with compatible dimensions to be determined. The system (65) is assumed to be switched synchronously by switching signal σ(k) in system (1).Augmenting the model of system (1) to include the states of (65), we can obtain the error system as follows:
(66)x~(k+1)=A~ix~(k)+A~dix~(k-d(k))+B~iu(k),e~(k)=C~ix~(k)+C~dix~(k-d(k))+D~iu(k).
Here
(67)A~i=[Ai00Ari],A~di=[Adi00Ardi],B~i=[BiBri],x~(k)=[x(k)x^(k)],C~i=[Ci-Cri],C~di=[Cdi-Crdi],D~i=Di-Drdi,e~(k)=y(k)-y^(k).The following theorem gives a sufficient condition for the existence of an admissibleH∞ reduced-order model (65) for system (1).Theorem 19.
Given constants0<α<1, γ>0, μ≥1, h>0, and ϑ  (0<ϑ<h), if there exist some symmetric positive definite matrices P~i,Q~im,R~im(m=1,2,3) and matrices Xi,Yi,Li,Hi,Fi(i∈N-) such that the following LMIs hold
(68)[Πi1Πi2Πi3]<0,(69)[Π-i1Πi2Π-i3]<0,(70)P~i≤μP~j,Q~im≤μQ~jm,R~im≤μR~jm,∀i,j∈N-.
Then system (66) with the average dwell time τa satisfies τa>-lnμ/lnα  which is exponentially stable with an H∞ norm bound γ.
Here(71)Πi1=[φ11iφ12i000φ22iφ23i00*φ33iφ34i0**φ44i0***φ55i],Π-i1=[φ11i0φ-13i00φ-22iφ-23iφ-24i0*φ-33i00**φ44i0***φ55i],Πi2=[φi16Tφi17Tφi18Tφi26Tφi27Tφi28T000000φi56Tφi57Tφi58T],Πi3=diag{P~i-2U~iW~i-2U~i-I},Π-i3=diag{P~i-2U~iW^-2U~i-I},φ11i=Q~i1+Q~i3-αP~-αϑϑ(R~i1+R~i3),φ12i=αϑϑ(R~i1+R~i3),φ22i=-αϑQ~i3-αϑϑ(2R~i1+R~i3),φ23i=αϑϑR~i1,φ34i=αhh-ϑR~i2,φ33i=αϑ(Q~i2-Q~i1)-αhh-ϑR~i2-αϑϑR~i1,φ44i=-αhQ~i2-αhh-ϑR~i2,φ55i=-γ2I,φ-13i=φ12i,φ-22i=-αhQ~i3-αhh-ϑ(2R~i2+R~i3),φ-23i=αhh-ϑ(R~i2+R~i3),φ-24i=φ34i,φ-33=αϑ(Q~i2-Q~i1)-αϑϑ(R~i1+R~i3)-αhh-ϑ(R~i2+R~i3),W~i=(h-ϑ)R~i2+ϑR~i1+ϑR~i3,W^i=(h-ϑ)R~i2+ϑR~i1+hR~i3,φi16T=[AiTXiTAiTETYi0LiT],φi17T=[AiTXiT-XiTAiTETYi-ETY0LiT-YiT],φi18T=[CiT-CriT],φi26T=φi27T=[AidTXiTAidTETYi0HiT],φi28T=[CdiT-CrdiT],φi56=φi57=[XiBiFi+YiTEBi],φi58=Di-Drdi.
Furthermore, if a feasible solution to the above LMIs (68), (69), and (70) exists, then the system matrices of an admissible H∞ reduced-order model in the form of (65) are given by
(72)Ari=Yi-1Li,Ardi=Yi-1Hi,Bri=Yi-1Fi.Proof.
Consider the following LKF for the switched system (66):
(73)Vi(k)=Vi1(k)+Vi2(k)+Vi3(k).
Here
(74)Vi1(k)=x~T(k)P~ix~(k),Vi2(k)∑s=k-ϑk-1‍αk-1-sx~T(s)Q~i1x~(s)+∑s=k-hk-ϑ-1‍αk-1-sx~T(s)Q~i2x~(s)+∑s=k-d(k)k-1‍αk-1-sx~T(s)Q~i3x~(s),Vi3(k)∑θ=-ϑ-1‍∑s=k+θk-1‍αk-1-sz~T(s)R~i1z~(s)+∑θ=-h-ϑ-1‍∑s=k+θk-1‍αk-1-sz~T(s)R~i2z~(s)+∑θ=-d(k)-1‍∑s=k+θk-1‍αk-1-sz~T(s)R~i3z~(s),
where z~(k)=x~(k+1)-x~(k) and P~i,Q~im,R~im(i∈N-, m=1,2,3) are symmetric positive definite matrices with appropriate dimensions; integer ϑ and α are given constants.
By using the techniques employed for proving Lemma7, one can easily obtain the result. Calculate the difference of Vi(k) in (73) along the state trajectory of system (66).(1)
Ifd(k)∈[0,ϑ], it gets
(75)Vi(k+1)-αVi(k)+e~T(k)e~(k)-γ2uT(k)u(k)≤ξ~T(k)Πi1ξ~(k)+x~T(k+1)P~ix~(k+1)+zT(k)W~iz(k)+e~T(k)e~(k),where(76)ξ~T(k)=[x~T(k)x~T(k-d(k))x~T(k-ϑ)x~T(k-h)uT(k)].For any appropriately dimensioned matrices P~i>0 and nonsingular matrices U~i, we have
(77)(P~i-U~i)TP~i-1(P~i-U~i)≥0.
Thus
(78)-U~iTP~i-1U~i≤P~i-2U~i.
If (68) holds, we have
(79)[Πi1Πi2Θi3]<0,
where
(80)Θi3=diag{-U~iTP~i-1U~iU~iTW~i-1U~i-I}.
Let
(81)U~i=[Xi0YiTEYi],E=[I0],YiAri=Li,YiArdi=Hi,YiBri=Fi.
Multiplying (79) both from left and right by diag{00000U~i-TU~i-T-I}, by Schur Complement, further, considering (75), one can infer
(82)Vi(k+1)-αVi(k)+e~T(k)e~(k)-γ2uT(k)u(k)≤0.
Similarly, for the case of d(k)∈[ϑ,h], the fact that (69) holds means that (82) is true. Set
(83)Γ(k)=e~T(k)e~(k)-γ2uT(k)u(k),
we have
(84)Vi(k+1)≤αVi(k)-Γ(k).
Let Nσ(k0,k) be the number of switching times in (k0,k). From (84) and (70), we can obtain
(85)Vi(k+k0)≤αkμNσ(k0,k)Vi(k0)-∑s=k0k-1‍αk-s-1μNσ(s,k)Γ(s)≤αk+Nσ(k0,k)(lnμ/lnα)Vj(k0)-∑s=k0k-1‍αk-s-1+Nσ(s,k)(lnμ/lnα)μNσ(s,k)Γ(s).Assume the zero disturbances input u(k)=0 to the state equation of system (66). By Definition 2, for any k0<k, it follows that
(86)Vi(k)≤αk+Nσ(lnμ/lnα)Vj(k0)≤αk(1+(lnμ/τalnα))Vj(k0).
From τa>-lnμ/lnα, one obtains limk→∞Vi(k)=0. There exist cn>0,  n=1,2, such that
(87)c1∥x~(k)∥2≤Vi(k),Vi(k0)≤c2∥x~(k0)∥s2.
Here
(88)∥x~(k)∥s=maxθ=-h,…,0∥x~(k+θ)∥,c1=λmin(Pi),c2=λmax(Pi)+∑k=13‍(λmax(Qik)+λmax(Rik)).
Therefore
(89)∥x~(k)∥2≤c2c1αk(1+(lnμ/τalnα))∥x~(k0)∥s2.
If the average dwell time τa satisfies τa>-lnμ/lnα, then the switched system (66) is exponentially stable with λ=α1/2 stability degree. For any nonzero u(k)∈l2[0,∞), under zero initial condition, combining (68), (69), (70), (85), and (89), one can easily obtain
(90)J=∑k=0∞‍[e~T(k)e~(k)-γ2uT(k)u(k)]≤0.
Therefore ∥e~(k)∥2≤γ∥u(k)∥2. This completes the proof.Remark 20.
Recently, authors in [30, 31] have studied the problem of model reduction for discrete-time switched systems. In those papers, time delays are not taken into account. However, in most of the cases in engineering problems, there always exist unknown time-varying delays; moreover, the case of stable and unstable subsystems co exists. Motivated by this, in this paper, we discussed the problem of H∞ model reduction for switched linear discrete-time systems with time-varying delays via delay decomposition approach [10–12]. Accordingly, numerical results are given for time-varying delay cases.
If there exist some unstable subsystems in the switched system (1), we have the following conclusion.Theorem 21.
Given constants0<α<1, β>1, γ>0, μ≥1, h>0, and ϑ  (0<ϑ<h), if there exist some symmetric positive definite matrices P~i,Q~im,R~im(m=1,2,3) and matrices Xi,Yi,Li,Hi,Fi(i∈N-) such that (68), (69), (70), and the following LMIs hold:
(91)[Π~i1Πi2Πi3]<0,[Π^i1Πi2Π-i3]<0.
And Tk0,n+k0α/Tk0,n+k0β≥(lnβ-κ)/(-lnα+κ), κ∈(lnα,0); then system (66) with the average dwell time τa satisfies τa>-lnμ/lnα  which is exponentially stable with an H∞ norm bound γ. Furthermore, if a feasible solution to the above LMIs (68), (69), (70), and (91) exists, then the system matrices of an admissible H∞ reduced-order model in the form of (65) are given by (72).
Here,(92)Π~i1=[φ~11iφ~12i000φ~22iφ~23i00*φ~33iφ~34i0**φ~44i0***φ~55i],Π^i1=[φ~11i0φ^13i00φ^22iφ^23iφ^24i0*φ^33i00**φ~44i0***φ~55i],φ~11i=Q~i1+Q~i3-βP~-1ϑ(R~i1+R~i3),φ~12i=1ϑ(R~i1+R~i3),φ~22i=-Q~i3-1ϑ(2R~i1+R~i3),φ~23i=1ϑR~i1,φ~33i=βϑ(Q~i2-Q~i1)-βϑh-ϑR~i2-1ϑR~i1,φ~34i=βϑh-ϑR~i2,φ~44i=-βhQ~i2-βϑh-ϑR~i2,φ~55i=-γ2I,φ^13i=βϑ(R~i1+R~i3),φ^22i=-βϑQ~i3-βϑh-ϑ(2R~i2+R~i3),φ^23i=βϑh-ϑ(R~i2+R~i3),φ^24i=βϑh-ϑ(R~i2),φ^33i=βϑ(Q~i2-Q~i1)-βϑ(R~i1+R~i3)-βϑh-ϑ(R~i2+R~i3).Remark 22.
The proof of Theorem21 is carried out by using the techniques employed in the previous section and is thus omitted here.


## 5. Examples

In this section, we consider some numerical examples to illustrate the benefits of our results.Example 1 (see [20]).
Consider the discrete-time switched system (1) with u(k)=0 and the following parameters:
(93)A1=[00.3-0.20.1],Ad1=[00.100.2],A2=[00.3-0.2-0.1],Ad2=[00.100].
For this system, we choose μ=1.1 and λ=0.931. Applying Theorem 10, by solving the LMIs (9) and (10) and (29), we can obtain the allowable delay upper bound h=20. It is reported, with decay rate λ=0.931, that the upper bound h can be obtained as 14 in [19] and 16 in [20]. Therefore, the result in this brief can indeed provide larger delay bounds than the results in [19, 20]. This supports the effectiveness of the proposed idea in Theorem 10 in reducing the conservatism of stability criteria.Example 2.
Consider the discrete-time switched system (1) with u(k)=0 and parameters as follows:
(94)A1=[00.3-0.20.1],Ad1=[00.100.2],A2=[00.3-0.2-0.1],Ad2=[1.30.100.9].
It is easy to check that the A2+Ad2 is unstable. In this case, we need to find a class of switching signals to guarantee the overall switched system to be exponentially stable. Set d(k)=[|3sin(kπ/6)|] and α=0.5329, according to Theorem 15 and by solving the LMIs (9), (10), (53), and (29), set ϑ=1; we have μ=2.4 and β=2.01. Choosing γ′=-0.18, we have Tk0,n+k0α/Tk0,n+k0β≥(lnβ-γ′)/(-lnα+γ′)=1.953 and τa>lnμ/-γ′=4.9. The simulation result of the switched system is shown in Figure 1, where the initial condition ϕ(θ)=[1.1-0.8]T and the switching law is shown in Figure 2. It can be seen from Figure 1 that the designed switching signals are effective although one subsystem is unstable. However, the results in [20] cannot find any feasible solution to guarantee the exponential stability of system (1).Figure 1
The state response.Figure 2
Switching law.Example 3 (see [31]).
Consider the system (1) with parameters as follows:(95)A1=[0.130.22-0.130.080.05-0.030.190.06-0.07-0.05-0.04-0.12-0.170.210.030.28],A2=[0.110.22-0.130.080.05-0.030.150.06-0.07-0.03-0.04-0.12-0.170.210.030.2],Ad1=Ad2=[0.020.010000.0200000.020.010000.02],B1=[0.19-0.180.16-0.08]T,B2=[0.23-0.130.16-0.04]T,C1=C2=[1.20.50.030.28],Cd1=Cd2=[0.020.050.010.09],D1=D2=0.1.
When the decay rate α is fixed, the maximum value of the time-delay h and the minimum value of the performance index γ can computed by solving the LMIs (68)–(70) procedure in Theorem 19, which is listed in Table 1 via different methods. Here, we choose μ=1.001. Assume that decay rate α=0.9; we can compute the maximum value of allowed delay h=42 and the minimum value of the performance index γ=1.67. From ADT τa>-lnμ/lnα, we have τa>0.0095. When h=2 and α=0.9, we can compute the minimum value of performance index γ=0.53. On the other hand, assume that maximum allowed delay h=2 and performance index γ=2; we can compute the minimum value of the decay rate α=0.59 and τa>0.0019.Table 1
Comparison of parameters via different methods.
α
γ
h
τ
a
[31]
0.9
2
2
>1.7305
Theorem19
0.9
1.67
42
>0.0095
Theorem19
0.9
0.53
2
>0.0095
Theorem19
0.59
2
2
>0.0019
Theorem19
0.6
1.8
2
>0.002Letα=0.9; here, we are interested in designing a q-order (q<4) system (65) and choose the ADT τa=2 switching signals such that the model error system (66) is exponentially stable with H∞ norm bound γ=2. By solving the corresponding LMIs (68)–(70) procedure in Theorem 19. For comparison with [31], we set the delay d(k)=2, and the following reduced-order models can be given.Third Order Model
(96)
A
r
1
=
[
0.2753
0.0282
-
0.0033
0.0097
0.2507
-
0.0033
-
0.0045
-
0.0124
0.2569
]
,
A
r
2
=
[
0.2799
0.0259
-
0.0058
0.0074
0.2581
-
0.0025
-
0.0051
-
0.01
0.2611
]
,
A
r
d
1
=
[
-
0.005
0.0069
-
0.0023
0.0037
-
0.0046
0.003
-
0.0011
0.0002
-
0.0006
]
,
A
r
d
2
=
[
-
0.001
0.0066
-
0.0025
0.0039
-
0.0044
0.0033
-
0.0018
0.001
-
0.0024
]
,
B
r
1
=
[
-
0.171
0.1795
-
0.111
]
T
,
B
r
2
=
[
-
0.191
0.148
-
0.1285
]
T
,
C
r
1
=
[
-
0.3016
-
0.1328
-
0.0149
]
,
C
r
2
=
[
-
0.2987
-
0.1265
-
0.0173
]
,
D
r
1
=
-
0.1754
,
C
r
d
1
=
[
-
0.0314
-
0.0047
-
0.0182
]
,
C
r
d
2
=
[
-
0.0361
-
0.0011
-
0.0199
]
,
D
r
2
=
-
0.2396
.Second Order Model
(97)
A
r
1
=
[
0.2419
0.0355
0.015
0.2141
]
,
A
r
d
1
=
[
-
0.0028
0.0088
0.0052
-
0.007
]
,
B
r
1
=
[
-
0.1528
0.1617
]
,
C
r
1
=
[
-
0.3109
-
0.1453
]
T
,
A
r
2
=
[
0.2382
0.0324
0.0147
0.2183
]
,
A
r
d
2
=
[
-
0.0023
0.0076
0.0046
-
0.006
]
,
B
r
2
=
[
-
0.1667
0.1203
]
,
C
r
2
=
[
-
0.3076
-
0.1362
]
T
,
C
r
d
1
=
[
-
0.0488
0.0034
]
,
D
r
1
=
-
0.2422
,
C
r
d
2
=
[
-
0.05
0.0057
]
,
D
r
2
=
-
0.3605
.First Order Model
(98)
A
r
1
=
0.2528
,
A
r
d
1
=
-
0.0057
,
B
r
1
=
-
0.1498
,
C
r
1
=
-
0.2769
,
C
r
d
1
=
0.0301
,
D
r
1
=
-
0.1792
,
A
r
2
=
0.2606
,
A
r
d
2
=
-
0.005
,
B
r
2
=
-
0.1787
,
C
r
2
=
-
0.2851
,
C
r
d
2
=
-
0.04
,
D
r
2
=
-
0.2624
.
To illustrate the model reduction performances of the obtained reduced-order models, let the initial condition be zero; the exogenous input is given as u(k)=1.8exp(-0.4k). The output errors between the original system and the corresponding three reduced models obtained in this paper (shown by the blue line) and the literature [31] (shown by the red line) are displayed in Figures 3, 4, and 5. The switching signal is shown in Figure 6. The simulation result of the switched system is shown in Figures 3–5. It can be seen from Figures 3–5 that the output errors between the original system and the reduced-order models obtained in this paper are smaller than that in [31].Figure 3
Output errors between the original system and the 3rd model.Figure 4
Output errors between the original system and the 2nd model.Figure 5
Output errors between the original system and the 1st model.Figure 6
Switching law.


## 6. Conclusions

The problem of exponential stability withH∞ performance and H∞ model reduction for a class of switched linear discrete-time systems with time-varying delay have been investigated in this paper. The switching law is given by ADT technique, such that even if one or more subsystem is unstable the overall switched system can still be exponentially stable. Sufficient conditions for the existence of the desired reduced-order model are derived and formulated in terms of strict LMIs. By solving the LMIs, the system of reduced-order model can be obtained, which also provides an H∞ gain for the error system between the original system and the reduced-order model. Finally, numerical examples are provided to illustrate the effectiveness and less conservativeness of the proposed method. A potential extension of this method to nonlinear case deserves further research.



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*Source: 101473-2014-01-02.xml*