# Robust Stability Analysis and Synthesis for Switched Discrete-Time Systems with Time Delay

Authors: Liguo Zhang; Hongfeng Li; Yangzhou Chen
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

The problems of robust stability analysis and synthesis for a class of uncertain switched time-delay systems with polytopic type uncertainties are addressed. Based on the constructive use of an appropriate switched Lyapunov function, sufficient linear matrix inequalities (LMIs) conditions are investigated to make such systems a uniform quadratic stability with anL2-gain smaller than a given constant level. System synthesis is to design switched feedback schemes, whether based on state, output measurements, or by using dynamic output feedback, to guarantee that the corresponding closed-loop system satisfies the LMIs conditions. Two numerical examples are provided that demonstrate the efficiency of this approach.


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## Body

## 1. Introduction

The stability analysis and synthesis problem of switched systems is one of the fundamental and challenging research topics, and various approaches has been obtained so far. For arbitrary switching law, a common Lyapunov function gives stability [1–3]. Liberzon and Hespanha have studied the global uniform asymptotic stability problem from the viewpoint of Lie algebra [4, 5]. On the other hand, Branicky [6],  Johansen and Rantzer [7], respectively, have proposed the multiple Lyapunov function method for analysis and synthesis of switched systems with prescribed switching law. Furthermore, average dwell time technique [8–12] is an effective tool of choosing such switching law. In the recent literatures [13–16], stability analysis and synthesis of discrete time switched system were studied. A survey of switched systems problems have been proposed by Liberzon and Morse [5].On the other hand, in many industrial hybrid systems such as power systems [17] and network control systems [13], time delay often occurs in the transition of the discrete states and the interior running of each subsystem. Therefore, more recently, much research attention has been devoted to the study of switched systems with time delay. In [18], L2-gain properties under arbitrary switching for a class of switched symmetric delay systems were studied. In [19], sufficient conditions of asymptotical stability were established for switched linear delay systems. Based on switched Lyapunov function approach, H∞ filtering problem of discrete-time switched systems with state delay was developed. In these papers mentioned above, only stable analysis is considered and state feedback results are proposed. In addition, output measurement and dynamic output feedback that are important synthesis methods for switched systems without delay are also expected to be effective for switched delay systems. However, no such results have been available up to now.This paper studies the robust stability analysis and synthesis problems for uncertain switched delay systems with polytopic type uncertainties. Compared with the existing results on switched delay systems, the results of this paper have two features. Firstly, we give design of a novel switched Lyapunov function while the existing works commonly aim at the design of the common Lyapunov function. Secondly, dynamic output feedback is achieved while the existing literature usually addresses the state feedback control.The rest of the paper is organized as follows. Section2 briefly presents the uncertain switched time delay systems with polytopic type uncertainties and robust stability analysis with L2-gain, based on the switched Lyapunov function. Section 3 presents the robust synthesis with the switched state feedback or output feedback design schemes, while Section 4 reports the results for switched dynamic output feedback. Finally, the main conclusions are summarized in Section 5.Notation
The notation used in this paper is fairly standard. The superscriptT stands for the matrix transposition, the notation ∥·∥ refers to the Euclidea vector norm, Rndenotes the n dimensional Euclidean space. In addition, in the symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry, and diag{⋯} stands for a block-diagonal matrix. A symmetric matrix P>0(≥0) means that P is a positive (semipositive) definite matrix.


## 2. Problem Formulation

Given a class of linear discrete time switched systems with time delay(2.1)x(k+1)=Aσx(k)+Adσx(k-d)+Bσu(k)+Γσw(k),(2.2)z(k)=Cσx(k)+Cdσx(k-d)+Dσu(k)+Φσw(k),(2.3)y(k)=Lσx(k),σ∈{1,…,N}.x(k)∈Rn, u(k)∈Rm, w(k)∈Rq, y(k)∈Rp, z(k)∈Rr are the system state vector, control input, exogenous disturbance, measured output, and controlled output, respectively, and d presents the time delay. The particular mode σ at any given time instant may be a selective procedure characterized by a switching rule of the following form:
(2.4)σ(k+1)=δ(σ(k),x(k)).
The function δ(·) is usually defined using a partition of the continuous state space. Let S denote the set of all selective rules. Therefore, the linear discrete time switched systems under consideration are composed of N subsystems, each of which is activated at particular switching instant.For a switching modej∈{1,…,N}, the associated matrices Aj,…,Φj contain uncertainties represented by a real convex-bounded polytopic model of the type
(2.5)[AjAdjBjΓjLjCjCdjDjΦj0]≜{∑p=1Mjλip[AjpAdjpBjpΓjpLjpCjpCdjpDjpΦjp0]},
where λj=(λj1,λj2,…,λjMj)∈Λj belongs to the unit simplex of Mj vertices
(2.6)Λj≜{λj:∑p=1Mjλjp=1,λjp≥0},
where Ajp,…,Φjp,p=1,…,Mj are known as real constant matrices of appropriate dimensions.Distinct from (2.1)–(2.3) is the free switched system
(2.7)x(k+1)=Aσx(k)+Adσx(k-d)+Γσw(k),(2.8)z(k)=Cσx(k)+Cdσx(k-d)+Φσw(k),σ∈{1,…,N}.We have the following definitions.Definition 2.1.
Switched system (2.7)-(2.8) is Uniform Quadratic Stable (UQS), if there exist a Lyapunov functional V(x,k)>0, a constant ε>0, such that for all admissible uncertainties satisfying (2.5)-(2.6) and arbitrary switching rule σ(·) activating subsystem j∈{1,…,N} at instant k+1 and subsystem i∈{1,…,N} at instant k, the Lyapunov functional difference ΔV(x(k),k) satisfies
(2.9)ΔV(x(k),k)≜V(x(k+1),k+1)-V(x(k),k)≤-εxT(k)x(k),∀x(k)≠0.Definition 2.2.
Given a scalarγ≥0, the L2-gain Υ of switched system (2.7)-(2.8) over S is
(2.10)Υ≜inf{γ≥0:∥z(k)∥2<γ2∥w(k)∥2,∀σ∈S,∀λj∈Λj}.Definition 2.3.
Switched system (2.7)-(2.8) is UQS with an L2-gainΥ<γ, if for all switching signal σ∈S and for all admissible uncertainties satisfying (2.5)-(2.6), it is UQS and
(2.11)∀w(k)≠0,∥z(k)∥2<γ2∥w(k)∥2.
We consider the following switching quadratic Lyapunov function:
(2.12)Vσ(x(k),k)≜V1σ(x(k),k)+V2(x(k),k),
where
(2.13)V1σ(x(k),k)=xT(k)Pσx(k),V2(x(k),k)=∑l=1dxT(k-l)Qσx(k-l),Pσ=PσT>0,Qσ=QσT>0.Theorem 2.4.
The following statements are equivalent.(1)
There exist a switching Lyapunov function of the type (2.12) with σ∈S and a scalar γ>0 such that switched system (2.7)-(2.8) with polytopic representation (2.5)-(2.6) is UQS with L2-gainΥ<γ.(2)
There exist matricesPi=PiT>0, Qi=QiT>0, Xj=XjT>0, and a scalar γ>0 satisfying the LMIs
(2.14)[-Pi+Qi00AipTCipT*-Qi0AdipTCdipT**-γ2IΓipTΦipT***-Xj0****-I]<0,(i,j)∈{1,…,N}×{1,…,N}, p∈{1,…,Mi}.Proof.
(1)⇒(2) Suppose that there exist a constant γ>0 and a switching Lyapunov function of the type (2.12). Let the switching rule σ(·) activates subsystem j at instant k+1and subsystem i at instant k. Thus,
(2.15)ΔVσ(x(k),k)=Vσ(k+1)(x(k+1),k+1)-Vσ(k)(x(k),k)=xT(k+1)Pjx(k+1)-xT(k)Pix(k)+∑l=1dxT(k+1-l)Qjx(k+1-l)-∑l=1dxT(k-l)Qix(k-l),
Then, it can be shown that
(2.16)ΔVσ(x(k),k)+zT(k)z(k)-γ2wT(k)w(k)=xT(k+1)Pjx(k+1)-xT(k)Pix(k)+xT(k)Qjx(k)-xT(k-d)Qix(k-d)+zT(k)z(k)-γ2wT(k)w(k)=[Aix(k)+Adix(k-d)+Γiw(k)]TPj[Aix(k)+Adix(k-d)+Γiw(k)]-xT(k)Pix(k)+xT(k)Qjx(k)-xT(k-d)Qi×x(k-d)-γ2wT(k)w(k)+[Cix(k)+Cdix(k-d)+Φiw(k)]T[Cix(k)+Cdix(k-d)+Φiw(k)]=[x(k)x(k-d)w(k)]TΘ̃[x(k)x(k-d)w(k)]<0,
where
(2.17)Θ̃=[-Pi+Qj+AiTPjAi+CiTCiAiTPjAdi+CiTCdiAiTPjΓi+CiTΦi*-Qi+AdiTPjAdi+CdiTCdiAdiTPjΓi+CdiTΦi**-γ2I+ΓiTPjΓi+ΦiTΦi].
Since ∥x(k)x(k-d)w(k)∥≠0, then Θ̃<0.
By the Schur complement,Θ̃<0 is equivalent to
(2.18)[-Pi+Qj00AiTPjCiT*-Qi0AdiTPjCdiT**-γ2IΓiTPjΦiT***-Pj0****-I]<0.
Applying the congruent transformation
(2.19)diag{I,I,I,Xj,I},Xj=Pj-1,
we obtain
(2.20)[-Pi+Qj00AiTCiT*-Qi0AdiTCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Upon using the vertex representation (2.5)-(2.6), we get (2.14) from (2.20).
(2)⇒(1) Follow by reversing the steps in the proof and applying Definitions 2.1–2.3 to the system (2.7)-(2.8) for all modes (i,j)∈{1,…,N}×{1,…,N} and using (2.5)-(2.6).
The proof is complete.Consider switched system (2.7)-(2.8) with w(k)=0, a special case of Theorem 2.4 is provided.Corollary 2.5.
The following statements are equivalent.(1)
There exists a switching Lyapunov function of type (2.12) with σ∈S such that switched system (2.7)-(2.8) with w(k)=0, and polytopic representation (2.5)-(2.6) is UQS.(2)
There exist matricesPi=PiT>0,Qi=QiT>0, and Xj=XjT>0 satisfying the LMIs
(2.21)[-Pi+Qj0AipTCipT*-QiAdipTCdipT**-Xj0***-I]<0,(i,j)∈{1,…,N}×{1,…,N}, p∈{1,…,Mi}.Since the exogenous disturbancew(k)=0 in the switched system (2.7)-(2.8), we could let the coefficients Γi, Φi be zero matrices in the inequality (2.20). Directly deleting the third row and the third column from the matrix inequality (2.20), that is, deleting the term relevant to w(k), we get the inequality (2.21).


## 3. Switched Control Synthesis

Extending on Section2, we examine here the problem of switched control synthesis using either switched state feedback or output feedback design schemes.


### 3.1. Switched State Feedback

With reference to system (2.1)-(2.2), we consider that the arbitrary switching rule σ(·) activates subsystem i at an instant k. Our objective herein is to design a switched state feedback u(k)=Kix(k),i∈{1,…,N} such that the closed-loop system
(3.1)x(k+1)=(Ai+BiKi)x(k)+Adix(k-d)+Γiw(k)=A̅ix(k)+Adix(k-d)+Γiw(k),(3.2)z(k)=(Ci+DiKi)x(k)+Cdix(k-d)+Φiw(k)=C̅ix(k)+Cdix(k-d)+Φiw(k)
is UQS with an L2-gain Υ<γ.Theorem 3.1.
Switched system (3.1)-(3.2) is UQS with an L2-gainΥ<γ, if there exist matrices Pi=PiT>0,Qi=QiT>0,Xj=XiT>0,Zi,Ti=XiQXi, and a scalar γ>0, such that the following LMIs hold:
(3.3)[-Xi+Tj00XiAipT+ZiTBipTXiCipT+ZiTDipT*-Ti0XiAdipTXiCdipT**-γ2IΓipTΦipT***-Xj0****-I]<0.
Moreover, the gain matrix is given byKi=ZiXi-1.Proof.
It follows from Theorem2.4 that switched system (3.1)-(3.2) is UQS with an L2-gainΥ<γ, if, for ∀(i,j)∈{1,…,N}×{1,…,N},  there exist matrices Pi=PiT>0, Q=QT>0, andXj=XjT>0 holding the following LMIs:
(3.4)[-Pi+Qj00A̅iTC̅iT*-Qi0AdiTCdiT**-γ2IΓiTΦiT***-Xj0****-I]=[-Pi+Qj00AiT+KiTBiTCiT+KiTDiT*-Qi0AdiTCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Applying the congruent transformationdiag{Xi,Xi,I,I,I} to LMIs (3.4), and let Xi=Pi-1,Zi=KiXi, we obtain
(3.5)[-Xi+Tj00XiAiT+ZiTBiTXiCiT+ZiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Upon using vertex representation (2.5)-(2.6), we get (3.3) from (3.5).


### 3.2. Example1

In this example, we design a switched state feedback control for this system based on Theorem3.1, switching occurs between two modes described by the following coefficients.Mode 1: consider(3.6)A11=[0.30.1-0.40.2],A12=[0.20.10.60.3],B11=[210.61],B12=[20.90.72],Ad11=[00.10.20],Ad12=[0-0.1-0.10],Γ11=[0.20.3],Γ12=[0.20.8],C11=[0.10.3],C12=[0.70.3],D11=[0.10.3],D12=[0.90.3],Cd11=[02],Cd12=[0-1],Φ11=0.6,Φ12=0.1.Mode 2: consider(3.7)A21=[0.10.20.30.4],A22=[0.20.10.60.3],B21=[1431],B22=[20.90.72],Ad21=[00.10.20],Ad22=[0-0.1-0.10],Γ21=[0.10.5],Γ22=[0.20.8],C21=[0.60.2],C22=[0.90.3],D21=[0.80.3],D22=[0.90.3],Cd21=[-0.10],Cd22=[-10],Φ21=0.3,Φ22=0.1.
We set time delay d=1, exogenous disturbance w(k)=0.02sin(0.02πk), then the feasible solution of LMIs (3.3) is given by γ=2.1, and
(3.8)X1=[0.20310.01760.01760.1763],X2=[0.2056-0.3407-0.34071.6231],Z1=[0.0024-0.0065-0.0473-0.0334],Z2=[0.0079-0.0993-0.0017-0.0844],T1=[0.06610.01990.01990.1558],T2=[0.1276-0.2793-0.2793-0.9434],Q1=[1.5700-0.0322-0.03225.0029],Q2=[2.88440.23200.23200.3284].
Since Ki=ZiXi-1, the control gains are
(3.9)K1=[0.0154-0.0386-0.2182-0.1678],K2=[-0.0964-0.0814-0.1450-0.0825].
With the initial state x(0)=[-0.02-0.71]T, and the following switching signal (Figure 1), the system state, control, and output signal are shown in  Figures 2, 3, and 4, respectively.Figure 1
The switching signal of the switched systems.Figure 2
State response of the closed-loop system.Figure 3
Control input of the switched discrete-time systems.Figure 4
Controlled output signal of the closed-loop switched systems.Figure 5
State response of the closed-loop switched discrete-time systems.Figure 6
Control input of the switched discrete time systems.Figure 7
Output signal of the closed-loop switched discrete time systems.From the simulation results, it can be clearly seen that the proposed control law guarantees the asymptotic stability of the closed-loop system.


### 3.3. Switched Static Output Feedback

The objective of this subsection is to design switched output feedbacku(k)=Giy(k), i∈{1,…,N}, such that the closed-loop system
(3.10)x(k+1)=(Ai+BiGiLi)x(k)+Adix(k-d)+Γiw(k)=Âix(k)+Adix(k-d)+Γiw(k),z(k)=(Ci+DiGiLi)x(k)+Cdix(k-d)+Φiw(k)=Ĉix(k)+Cdix(k-d)+Φiw(k)
is UQS with an L2-gain Υ<γ.Before developing our main results, we give the following lemma.Lemma 3.2 (see [14]).
Given a matrixLi∈Rp×n,rank[Li]=p, and having the singular value decomposition form
(3.11)Li=Ui[Λpi0]ViT,
where Ui∈Rp×p,Vi∈Rn×n are unitary matrices and Λpi∈Rp×p is a diagonal positive matrix. Let Xi=XiT>0,Xi∈Rn×n, then there exists a matrix 0<Ei∈Rp×p, such that
(3.12)LiXi=EiLi
if and only if
(3.13)Xi=Vi[Xiu0*Xiv]ViT,Xiu∈Rp×p,Xiv∈R(n-p)×(n-p).
Thus, we get Ei-1=UiΛpiXiu-1Λpi-1Ui-1.Theorem 3.3.
Consider switched system (3.1)-(3.2) with polytopic representation (2.5)-(2.6) subject to the output feedback control u(k)=Giy(k), and output matrix  Li  having the SVD form Li=Ui[Λpi0]ViT,Λpi∈Rp×p. The resulting closed-loop system is UQS with an L2-gain  Υ<γ,  if  there exist matrices 0<Xiu=XiuT∈Rp×p, 0<Xiv=XivT∈R(n-p)×(n-p), 0<Xj=XjT, 0<Ti=TiT, Ei>0, Ri, LiXi=EiLi, and a scalar γ>0, such that the LMIs
(3.14)[-Xi+Tj00XiAipT+LiTRiTBipTXiCipT+LiTRiTDipT*-Ti0XiAdipTXiCdipT**-γ2IΓipTΦipT***-Xj0****-I]<0
have a feasible solution. Moreover, the feedback gain is given by Gi=RiUiΛpiXiu-1Λpi-1Ui-1, i∈{1,…,N}.Proof.
In view of Theorem2.4, we have
(3.15)[-Xi+Tj00XiAiT+XiLiTGiTBiTXiCiT+XiLiTGiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Let Ri=GiEi, then (3.15) can be rewritten as follows:
(3.16)[-Xi+Tj00XiAiT+LiTRiTBiTXiCiT+LiTRiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Upon using vertex representation (2.5)-(2.6), we get (3.14) from (3.16). According to Lemma 3.2, we obtain that Gi=RiUiΛpiXiu-1Λpi-1Ui-1,i∈{1,…,N}.


### 3.4. Example2

Similar to Example1, we wish to design a switched static output feedback control law based on Theorem 3.3. Using all coefficients in Example 1 that adapt to Example 2.Mode 1: consider(3.17)L11=[0.5000.5],L12=[0.6000.4].Mode 2: consider(3.18)L21=[0.4000.6],L22=[0.6000.4].
In this example, we set time delay d=1, the exogenous disturbance w(k)=0.02sin(0.02πk), then the feasible solution of LMIs (3.14) is given by γ=2 and
(3.19)X1=[0.19550.02540.02540.1803],X2=[0.1655-0.1082-0.10820.5388],R1=[0.0115-0.0072-0.0927-0.0927],R2=[-0.0071-0.0789-0.0425-0.0504],E11=[0.19550.02540.02540.1803],E12=[0.19550.03810.01690.1803],E1=[0.1955-0.0343-0.01950.1803],E21=[0.1655-0.0721-0.16230.5388],E22=[0.1655-0.1623-0.07210.5388],E2=[0.1655-0.1352-0.09920.5388],T1=[0.06880.02180.02180.1611],T2=[0.0920-0.1057-0.10570.1611],Q1=[0.06880.00780.03690.1631],Q2=[0.1176-0.2333-0.06650.2616].
Since Gi=RiEi-1, the control gains are
(3.20)G1=[0.0642-0.0521-0.4312-0.4320],G2=[-0.1537-0.1849-0.3685-0.1860].
With the initial state x(0)=[-0.54-0.56]T, and the switching signal as Example 1, the system state, control, and output signal are shown as follows.


## 3.1. Switched State Feedback

With reference to system (2.1)-(2.2), we consider that the arbitrary switching rule σ(·) activates subsystem i at an instant k. Our objective herein is to design a switched state feedback u(k)=Kix(k),i∈{1,…,N} such that the closed-loop system
(3.1)x(k+1)=(Ai+BiKi)x(k)+Adix(k-d)+Γiw(k)=A̅ix(k)+Adix(k-d)+Γiw(k),(3.2)z(k)=(Ci+DiKi)x(k)+Cdix(k-d)+Φiw(k)=C̅ix(k)+Cdix(k-d)+Φiw(k)
is UQS with an L2-gain Υ<γ.Theorem 3.1.
Switched system (3.1)-(3.2) is UQS with an L2-gainΥ<γ, if there exist matrices Pi=PiT>0,Qi=QiT>0,Xj=XiT>0,Zi,Ti=XiQXi, and a scalar γ>0, such that the following LMIs hold:
(3.3)[-Xi+Tj00XiAipT+ZiTBipTXiCipT+ZiTDipT*-Ti0XiAdipTXiCdipT**-γ2IΓipTΦipT***-Xj0****-I]<0.
Moreover, the gain matrix is given byKi=ZiXi-1.Proof.
It follows from Theorem2.4 that switched system (3.1)-(3.2) is UQS with an L2-gainΥ<γ, if, for ∀(i,j)∈{1,…,N}×{1,…,N},  there exist matrices Pi=PiT>0, Q=QT>0, andXj=XjT>0 holding the following LMIs:
(3.4)[-Pi+Qj00A̅iTC̅iT*-Qi0AdiTCdiT**-γ2IΓiTΦiT***-Xj0****-I]=[-Pi+Qj00AiT+KiTBiTCiT+KiTDiT*-Qi0AdiTCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Applying the congruent transformationdiag{Xi,Xi,I,I,I} to LMIs (3.4), and let Xi=Pi-1,Zi=KiXi, we obtain
(3.5)[-Xi+Tj00XiAiT+ZiTBiTXiCiT+ZiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Upon using vertex representation (2.5)-(2.6), we get (3.3) from (3.5).


## 3.2. Example1

In this example, we design a switched state feedback control for this system based on Theorem3.1, switching occurs between two modes described by the following coefficients.Mode 1: consider(3.6)A11=[0.30.1-0.40.2],A12=[0.20.10.60.3],B11=[210.61],B12=[20.90.72],Ad11=[00.10.20],Ad12=[0-0.1-0.10],Γ11=[0.20.3],Γ12=[0.20.8],C11=[0.10.3],C12=[0.70.3],D11=[0.10.3],D12=[0.90.3],Cd11=[02],Cd12=[0-1],Φ11=0.6,Φ12=0.1.Mode 2: consider(3.7)A21=[0.10.20.30.4],A22=[0.20.10.60.3],B21=[1431],B22=[20.90.72],Ad21=[00.10.20],Ad22=[0-0.1-0.10],Γ21=[0.10.5],Γ22=[0.20.8],C21=[0.60.2],C22=[0.90.3],D21=[0.80.3],D22=[0.90.3],Cd21=[-0.10],Cd22=[-10],Φ21=0.3,Φ22=0.1.
We set time delay d=1, exogenous disturbance w(k)=0.02sin(0.02πk), then the feasible solution of LMIs (3.3) is given by γ=2.1, and
(3.8)X1=[0.20310.01760.01760.1763],X2=[0.2056-0.3407-0.34071.6231],Z1=[0.0024-0.0065-0.0473-0.0334],Z2=[0.0079-0.0993-0.0017-0.0844],T1=[0.06610.01990.01990.1558],T2=[0.1276-0.2793-0.2793-0.9434],Q1=[1.5700-0.0322-0.03225.0029],Q2=[2.88440.23200.23200.3284].
Since Ki=ZiXi-1, the control gains are
(3.9)K1=[0.0154-0.0386-0.2182-0.1678],K2=[-0.0964-0.0814-0.1450-0.0825].
With the initial state x(0)=[-0.02-0.71]T, and the following switching signal (Figure 1), the system state, control, and output signal are shown in  Figures 2, 3, and 4, respectively.Figure 1
The switching signal of the switched systems.Figure 2
State response of the closed-loop system.Figure 3
Control input of the switched discrete-time systems.Figure 4
Controlled output signal of the closed-loop switched systems.Figure 5
State response of the closed-loop switched discrete-time systems.Figure 6
Control input of the switched discrete time systems.Figure 7
Output signal of the closed-loop switched discrete time systems.From the simulation results, it can be clearly seen that the proposed control law guarantees the asymptotic stability of the closed-loop system.


## 3.3. Switched Static Output Feedback

The objective of this subsection is to design switched output feedbacku(k)=Giy(k), i∈{1,…,N}, such that the closed-loop system
(3.10)x(k+1)=(Ai+BiGiLi)x(k)+Adix(k-d)+Γiw(k)=Âix(k)+Adix(k-d)+Γiw(k),z(k)=(Ci+DiGiLi)x(k)+Cdix(k-d)+Φiw(k)=Ĉix(k)+Cdix(k-d)+Φiw(k)
is UQS with an L2-gain Υ<γ.Before developing our main results, we give the following lemma.Lemma 3.2 (see [14]).
Given a matrixLi∈Rp×n,rank[Li]=p, and having the singular value decomposition form
(3.11)Li=Ui[Λpi0]ViT,
where Ui∈Rp×p,Vi∈Rn×n are unitary matrices and Λpi∈Rp×p is a diagonal positive matrix. Let Xi=XiT>0,Xi∈Rn×n, then there exists a matrix 0<Ei∈Rp×p, such that
(3.12)LiXi=EiLi
if and only if
(3.13)Xi=Vi[Xiu0*Xiv]ViT,Xiu∈Rp×p,Xiv∈R(n-p)×(n-p).
Thus, we get Ei-1=UiΛpiXiu-1Λpi-1Ui-1.Theorem 3.3.
Consider switched system (3.1)-(3.2) with polytopic representation (2.5)-(2.6) subject to the output feedback control u(k)=Giy(k), and output matrix  Li  having the SVD form Li=Ui[Λpi0]ViT,Λpi∈Rp×p. The resulting closed-loop system is UQS with an L2-gain  Υ<γ,  if  there exist matrices 0<Xiu=XiuT∈Rp×p, 0<Xiv=XivT∈R(n-p)×(n-p), 0<Xj=XjT, 0<Ti=TiT, Ei>0, Ri, LiXi=EiLi, and a scalar γ>0, such that the LMIs
(3.14)[-Xi+Tj00XiAipT+LiTRiTBipTXiCipT+LiTRiTDipT*-Ti0XiAdipTXiCdipT**-γ2IΓipTΦipT***-Xj0****-I]<0
have a feasible solution. Moreover, the feedback gain is given by Gi=RiUiΛpiXiu-1Λpi-1Ui-1, i∈{1,…,N}.Proof.
In view of Theorem2.4, we have
(3.15)[-Xi+Tj00XiAiT+XiLiTGiTBiTXiCiT+XiLiTGiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Let Ri=GiEi, then (3.15) can be rewritten as follows:
(3.16)[-Xi+Tj00XiAiT+LiTRiTBiTXiCiT+LiTRiTDiT*-Ti0XiAdiTXiCdiT**-γ2IΓiTΦiT***-Xj0****-I]<0.
Upon using vertex representation (2.5)-(2.6), we get (3.14) from (3.16). According to Lemma 3.2, we obtain that Gi=RiUiΛpiXiu-1Λpi-1Ui-1,i∈{1,…,N}.


## 3.4. Example2

Similar to Example1, we wish to design a switched static output feedback control law based on Theorem 3.3. Using all coefficients in Example 1 that adapt to Example 2.Mode 1: consider(3.17)L11=[0.5000.5],L12=[0.6000.4].Mode 2: consider(3.18)L21=[0.4000.6],L22=[0.6000.4].
In this example, we set time delay d=1, the exogenous disturbance w(k)=0.02sin(0.02πk), then the feasible solution of LMIs (3.14) is given by γ=2 and
(3.19)X1=[0.19550.02540.02540.1803],X2=[0.1655-0.1082-0.10820.5388],R1=[0.0115-0.0072-0.0927-0.0927],R2=[-0.0071-0.0789-0.0425-0.0504],E11=[0.19550.02540.02540.1803],E12=[0.19550.03810.01690.1803],E1=[0.1955-0.0343-0.01950.1803],E21=[0.1655-0.0721-0.16230.5388],E22=[0.1655-0.1623-0.07210.5388],E2=[0.1655-0.1352-0.09920.5388],T1=[0.06880.02180.02180.1611],T2=[0.0920-0.1057-0.10570.1611],Q1=[0.06880.00780.03690.1631],Q2=[0.1176-0.2333-0.06650.2616].
Since Gi=RiEi-1, the control gains are
(3.20)G1=[0.0642-0.0521-0.4312-0.4320],G2=[-0.1537-0.1849-0.3685-0.1860].
With the initial state x(0)=[-0.54-0.56]T, and the switching signal as Example 1, the system state, control, and output signal are shown as follows.


## 4. Switched Dynamic Output Feedback

We consider the more general case and employ at every modei∈{1,…,N}, a switched dynamic output feedback scheme of the following form:
(4.1)ξ(k+1)=Aciξ(k)+Bciy(k),u(k)=Cciξ(k).
Applying controller (4.1) to system (2.1)-(2.3) and letting ηT(k)=[xT(k)ξT(k)], we get the closed-loop system
(4.2)η(k+1)=Ãiη(k)+Ãdiη(k-d)+Γ̃iw(k),z(k)=C̃iη(k)+C̃diη(k-d)+Φiw(k),
where
(4.3)Ãi=[AiBiCciBciLiAci],Ãdi=[Adi000],Γ̃i=[Γi0],C̃i=[CiDiCci],C̃di=[Cdi0].Theorem 4.1.
Consider switched system (4.2) with polytopic representation (2.5)-(2.6) and output matrix Li having the SVD form Li=Ui[Λpi0]ViT,Λpi∈Rp×p, LiXi=ẼiLi. This system is UQS with an L2-gain Υ<γ, if there exist
(4.4)P̃i=[P̃si0*P̃ci],T̃i=[T̃si0*T̃ci]>0,X̃i≜P̃i-1=[X̃si0*X̃ci],0<X̃siu=X̃siuT,0<X̃siv=X̃sivT,Ỹj=[Ỹsj0*Ỹcj],0<Ỹsj=ỸsjT,0<Ỹcj=ỸcjT,Ωci,Πci,Ψci,(i,j)∈{1,…,N}×{1,…,N},
and a scalar γ>0 satisfying the systems LMIs
(4.5)[-X̃si+T̃si0000X̃siAipTLipTΩciTX̃siCipT*-X̃ci+T̃ci000ΠciTBipTΨciTΠciTDipT**-T̃si00X̃siAdipT0X̃siCdipT***-T̃ci0000****-γ2IΓipT0ΦipT*****-Ỹsj00******-Ỹcj0*******-I]<0.
Moreover, the gain matrix is given by Aci=ΨciX̃ci-1,Cci=ΠciX̃ci-1, and Bci=ΩciẼi-1=ΩciUi×ΛpiX̃siu-1Λpi-1Ui-1.Proof.
In view of Theorem2.4, we have
(4.6)[-X̃i+T̃i00X̃iÃiTX̃iC̃iT*-T̃i0X̃iÃdiTX̃iC̃diT**-γ2IΓ̃iTΦiT***-Ỹj0****-I]<0.
By applying (4.3)-(4.4) to (4.6), we obtain
(4.7)[-X̃si+T̃si0000X̃siAiTX̃siLiTBciTX̃siCiT*-X̃ci+T̃ci000X̃ciCciTBiTX̃ciAciTX̃ciCciTDiT**-T̃si00X̃siAdiT0X̃siCdiT***-T̃ci0000****-γ2IΓiT0ΦiT*****-Ỹsj00******-Ỹcj0*******-I]<0.
Let Ψci=AciX̃ci,Ωci=BciẼi,Πci=CciX̃ci, and using vertex representation (2.5)-(2.6), we get (4.5) from (4.7).


## 5. Conclusions

We have examinedL2-gain analysis and control synthesis for a class of switched time-delay systems with convex-bounded parameter uncertainties in the system matrices using an appropriate switched Lyapunov functional. LMI-based feasibility conditions have been developed to ensure that the linear switched time-delay system with polytopic uncertainties is UQS with an L2-gain. Switched feedback schemes have been designed using state output measurements and by using dynamic output feedback to guarantee that the corresponding closed-loop system enjoys the uniform quadratic stability with an L2-gain smaller than a prescribed constant level.



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*Source: 408105-2010-08-25.xml*