# Exponential Admissibility andH∞ Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach

Authors: Jinxing Lin; Zhifeng Gao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper deals with the problems of exponential admissibility andH∞ control for a class of continuous-time switched singular systems with time-varying delay. The H∞ controllers to be designed include both the state feedback
(SF) and the static output feedback (SOF). First, by using the average dwell time scheme, the piecewise Lyapunov function, and the free-weighting matrix technique, an exponential admissibility criterion, which is not only delay-range-dependent but also decay-rate-dependent, is derived in terms of linear matrix inequalities (LMIs). A weighted H∞ performance criterion is also provided. Then, based on these, the solvability conditions for the desired SF and SOF controllers are established by employing the LMI technique, respectively. Finally, two numerical examples are
given to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Many real-world engineering systems always exhibit several kinds of dynamic behavior in different parts of the system (e.g., continuous dynamics, discrete dynamics, jump phenomena, and logic commands) and are more appropriately modeled by hybrid systems. As an important class of hybrid systems, switched systems consist of a collection of continuous-time or discrete-time subsystems and a switching rule orchestrating the switching between them and are of great current interest; see, for example, Decarlo et al. [1], Liberzon [2], Lin and Antsaklis [3], and Sun and Ge [4] for some recent survey and monographs. Switched systems have great flexibility in modeling parameter-varying or structure-varying systems, event-driven systems, logic-based systems, and so forth. Also, multiple-controller switching technique offers an effective mechanism to cope with highly complex systems and/or systems with large uncertainties, particularly in the adaptive context [5]. Many effective methods have been developed for switched systems, for example, the multiple Lyapunov function approach [6, 7], the piecewise Lyapunov function approach [8, 9], the switched Lyapunov function method [10], convex combination technique [11], and the dwell-time or average dwell-time scheme [12–15]. Among them, the average dwell-time scheme provides a simple yet efficient tool for stability analysis of switched systems, especially when the switching is restricted and has been more and more favored [16].On the other hand, time delay is a common phenomenon in various engineering systems and the main sources of instability and poor performance of a system. Hence, control of switched time-delay systems has been an attractive field in control theory and application in the past decade. Some of the aforementioned approaches for nondelayed switched systems have been successfully adopted to hand the switched time-delay systems; see, for example, Du et al. [17], Kim et al. [18], Mahmoud [19], Phat [20], Sun et al. [21], Sun et al. [22], Wang et al. [23], Wu and Zheng [24], Xie et al. [25], Zhang and Yu [26], and the references therein.Recently, a more general class of switched time-delay systems described by the singular form was considered in Ma et al. [27] and Wang and Gao [28]. It is known that a singular model describes dynamic systems better than the standard state-space system model [29]. The singular form provides a convenient and natural representation of economic systems, electrical networks, power systems, mechanical systems, and many other systems which have to be modeled by additional algebraic constraints [29]. Meanwhile, it endows the aforementioned systems with several special features, such as regularity and impulse behavior, that are not found in standard state-space systems. Therefore, it is both worthwhile and challenging to investigate the stability and control problems of switched singular time-delay systems. In the past few years, some fundamental results based on the aforementioned approaches for standard state-space switched time-delay systems have been successfully extended to switched singular time-delay systems. For example, by using the switched Lyapunov function method, the robust stability, stabilization, and H∞ control problems for a class of discrete-time uncertain switched singular systems with constant time delay under arbitrary switching were investigated in Ma et al. [27]; H∞ filters were designed in Lin et al. [30] for discrete-time switched singular systems with time-varying time delay. In Wang and Gao [28], based on multiple Lyapunov function approach, a switching signal was constructed to guarantee the asymptotic stability of a class of continuous-time switched singular time-delay systems. With the help of average dwell time scheme, some initial results on the exponential admissibility (regularity, nonimpulsiveness, and exponential stability) were obtained in Lin and Fei [31] for continuous-time switched singular time-delay systems. However, to the best of our knowledge, few work has been conducted regarding the H∞ control for continuous-time switched singular time-delay systems via the dwell time or average dwell time scheme, which constitutes the main motivation of the present study.In this paper, we aim to solve the problem ofH∞ control for a class of continuous-time switched singular systems with interval time-varying delay via the average dwell time scheme. Both the state feedback (SF) control and the static output feedback (SOF) control are considered. Firstly, based on the average dwell time scheme, the piecewise Lyapunov function, as well as the free-weighting technique, a class of slow switching signals is identified to guarantee the unforced systems to be exponentially admissible with a weighted H∞ performance γ, and several corresponding criteria, which are not only delay-range-dependent but also decay-rate-dependent, are derived in terms of linear matrix inequalities (LMIs). Next, the LMI-based approaches are proposed to design an SF controller and an SOF controller, respectively, such that the resultant closed-loop system is exponentially admissible and satisfies a weighted H∞ performance γ. Finally, two illustrative examples are given to show the effectiveness of the proposed approach.Notation 1.
Throughout this paper, the superscriptT represents matrix transposition. Rn denotes the real n-dimensional Euclidean space, and Rn×n denotes the set of all n×n real matrices. I is an appropriately dimensioned identity matrix. P>0 (P≥0) means that matrix P is positive definite (semi positive definite). diag{·,·,·} stands for a block diagonal matrix. λmin(P) (λmax(P)) denotes the minimum (maximum) eigenvalue of symmetric matrix P, L2[0,∞) is the space of square-integrable vector functions over [0,∞), ∥·∥ denotes the Euclidean norm of a vector and its induced norm of a matrix, and Sym{A} is the shorthand notation for A+AT. In symmetric block matrices, we use an asterisk (*) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Preliminaries and Problem Formulation

Consider a class of switched singular time-delay system of the form(2.1)Ex˙(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t))+Bσ(t)u(t)+Bwσ(t)w(t),z(t)=Cσ(t)x(t)+Cdσ(t)x(t-d(t))+Dσ(t)u(t)+Dwσ(t)w(t),y(t)=Lσ(t)x(t),x(θ)=ϕ(θ),θ∈[-d2,0],
where x(t)∈Rn is the system state, u(t)∈Rm is the control input, z(t)∈Rq is the controlled output, y(t)∈Rp is the measured output, and w(t)∈Rl is the disturbance input that belongs to L2[0,∞); σ(t):[0,+∞)→ℐ={1,2,…,I} with integer I>1 is the switching signal; E∈Rn×n is a singular matrix with rankE=r≤n; for each possible value, σ(t)=i, i∈ℐ, Ai, Adi, Bi, Bwi, Ci, Cdi, Di, Dwi, and Li are constant real matrices with appropriate dimensions; ϕ(θ) is a compatible continuous vector-valued initial function on [-d2,0]; d(t) denotes interval time-varying delay satisfying
(2.2)d1≤d(t)≤d2,d˙(t)≤μ<1,
where 0≤d1<d2 and μ are constants. Note that d1 may not be equal to 0.SincerankE=r≤n, there exist nonsingular matrices P, Q∈Rn×n such that
(2.3)PEQ=[Ir000].
In this paper, without loss of generality, let
(2.4)E=[Ir000].
Corresponding to the switching signal σ(t), we denote the switching sequence by 𝒮:={(i0,t0),…,(ik,tk)∣ik∈ℐ,k=0,1,…} with t0=0, which means that the ik subsystem is activated when t∈[tk,tk+1). To present the objective of this paper more precisely, the following definitions are introduced.Definition 2.1 (see [2]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the number of switching of σ(t) over (T1,T2). If Nσ(T1,T2)≤N0+(T2-T1)/Ta holds for Ta>0, N0≥0, then Ta is called average dwell time. As commonly used in the literature [21, 26], we choose N0=0.Definition 2.2 (see [21, 29, 32]).
For any delayd(t) satisfying (2.2), the unforced part of system (2.1) with w(t)=0(2.5)Ex˙(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t)),xt0(θ)=x(t0+θ)=ϕ(θ),θ∈[-d2,0]
is said to be(1)
regular ifdet(sE-Ai) is not identically zero for each σ(t)=i, i∈ℐ,(2)
impulse ifdeg(det(sE-Ai))=rankE for each σ(t)=i, i∈ℐ,(3)
exponentially stable under the switching signalσ(t) if the solution x(t) of system (2.5) satisfies
(2.6)∥x(t)∥≤ιe-λ(t-t0)∥xt0∥c,∀t≥t0,
where λ>0 and ι>0 are called the decay rate and decay coefficient, respectively, and ∥xt0∥c=sup-d2≤θ≤0{∥x(t0+θ)∥},(4)
exponentially admissible under the switching signalσ(t) if it is regular, impulse free, and exponentially stable under the switching signal σ(t).Remark 2.3.
The regularity and nonimpulsiveness of the switched singular time-delay system (2.5) ensure that its every subsystem has unique solution for any compatible initial condition. However, even if a switched singular system is regular and causal, it still has inevitably finite jumps due to the incompatible initial conditions caused by subsystem switching [33]. For more details about the impulsiveness effects on the stability of systems, we refer readers to Chen and Sun [34], Li et al. [35], and the references therein. In this paper, without loss of generality, we assume that such jumps cannot destroy the stability of system (2.1). Nevertheless, how to suppress or eliminate the finite jumps in switched singular systems is a challenging problem which deserves further investigation.Definition 2.4.
For the givenα>0 and γ>0, system (2.1) is said to be exponentially admissible with a weighted H∞ performance γ under the switching signal σ(t), if it is exponentially admissible with u(t)=0 and w(t)=0, and under zero initial condition, that is, ϕ(θ)=0, θ∈[-d2,0], for any nonzero w(t)∈L2[0,∞), it holds that
(2.7)∫0t‍e-αszT(s)z(s)ds≤γ2∫0t‍wT(s)w(s)ds.Remark 2.5.
For switched systems with the average dwell time switching, the Lyapunov function values at switching instants are often allowed to increaseβ times (β>1) to reduce the conservatism in system stability analysis, which will lead to the normal disturbance attenuation performance hard to compute or check, even in linear setting [15, 36]. Therefore, the weighted H∞ performance criterion (2.7) [15, 21, 24] is adopted here to evaluate disturbance attenuation while obtaining the expected exponential stability.
This paper considers both SF control law(2.8)u(t)=Kσ(t)x(t)
and SOF control law
(2.9)u(t)=Fσ(t)y(t),
where Ki and Fi, σ(t)=i, i∈ℐ, are appropriately dimensioned constant matrices to be determined.
Then, the problem to be addressed in this paper can be formulated as follows. Given the switched singular time-delay system (2.1) and a prescribed scalar γ>0, identify a class of switching signal σ(t) and design an SF controller of the form (2.8) and an SOF controller of the form (2.9) such that the resultant closed-loop system is exponentially admissible with a weighted H∞ performance γ under the switching signal σ(t).


## 3. Exponential Admissibility andH∞ Performance Analysis

First, we apply the average dwell time approach and the piecewise Lyapunov function technique to investigate the exponential admissibility for the switched singular time-delay system (2.5) and give the following result.Theorem 3.1.
For prescribed scalarsα>0, 0≤d1≤d2 and 0<μ<1, if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi of the following form:
(3.1)Pi=[Pi110Pi21Pi22],
with Pi11∈Rr, Pi11>0, and Pi22 being invertible, such that
(3.2)[Φi11Φi12Φi13-Si1Ec1Ni1c12Si1c12Mi1AiTUi*Φi22Φi23-Si2Ec1Ni2c12Si2c12Mi2AdiTUi**Φi3300000***Φi440000****-c1Zi1000*****-c12Zi200******-c12Zi20*******-Ui]<0,
where
(3.3)Φi11=Sym{PiTAi+Ni1E}+∑l=13‍Qil+αETPi,Φi12=PiTAdi+(Ni2E)T+Si1E-Mi1E,Φi13=Mi1E-Ni1E,Φi22=-(1-μ)e-αd2Qi3+Sym{Si2E-Mi2E},Φi23=Mi2E-Ni2E,Φi33=-e-αd1Qi1,Φi44=-e-αd2Qi2,c1=1α(eαd1-1),c12=1α(eαd2-eαd1),d12=d2-d1,Ui=d1Zi1+d12Zi2.
Then, system (2.5) with d(t) satisfying (2.2) is exponentially admissible for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α, where β≥1 satisfies
(3.4)Pi11≤βPj11,Qil≤βQjl,Ziv≤βZjv,l=1,2,3,v=1,2,∀i,j∈ℐ.
Moreover, an estimate on the exponential decay rate is λ=(1/2)(α-(lnβ)/Ta).Proof.
The proof is divided into three parts: (i) to show the regularity and nonimpulsiveness; (ii) to show the exponential stability of the differential subsystem; (iii) to show the exponential stability of the algebraic subsystem.
(i) Regularity and nonimpulsiveness. According to (2.4), for each i∈ℐ, denote
(3.5)Ai=[Ai11Ai12Ai21Ai22],
where Ai11∈Rr. From (3.2), it is easy to see that Φi11<0, i∈ℐ. Noting Qil>0, l=1,2,3, we get
(3.6)Sym{PiTAi+Ni1E}+αETPi<0.
Substituting Pi and E given as (3.1) and (2.4) into this inequality yields
(3.7)[⋆⋆⋆Ai22TPi22+Pi22TAi22]<0,
where ⋆ denotes a matrix which is not relevant to the discussion. This implies that Ai22, i∈ℐ, is nonsingular. Then, by Dai [29] and Definition 2.1, system (2.5) is regular and impulse free.
(ii) Exponential stability of differential subsystem. Define the piecewise Lyapunov functional candidate for system (2.5) as the following:
(3.8)V(xt)=Vσ(t)(xt)=xT(t)ETPσ(t)x(t)+∑v=12‍∫t-dvt‍xT(s)eα(s-t)Qσ(t)vx(s)ds+∫t-d(t)t‍xT(s)eα(s-t)Qσ(t)3x(s)ds+∫-d10‍∫t+θt‍(Ex˙(s))Teα(s-t)Zσ(t)1(Ex˙(s))dsdθ+∫-d2-d1‍∫t+θt‍(Ex˙(s))Teα(s-t)Zσ(t)2(Ex˙(s))dsdθ.
Then, along the solution of system (2.5) for a fixed σ(t)=i, i∈ℐ, we have
(3.9)V˙i(xt)≤2xT(t)PiTEx˙(t)+∑v=12‍[xT(t)Qivx(t)-xT(t-dv)e-αdvQivx(t-dv)]+xT(t)Qi3x(t)-(1-μ)xT(t-d(t))e-αd2Qi3x(t-d(t))+(Ex˙(t))T(d1Zi1+d12Zi2)(Ex˙(t))-∫t-d1t‍(Ex˙(s))Teα(s-t)Zi1(Ex˙(s))ds-∫t-d2t-d1‍(Ex˙(s))Teα(s-t)Zi2(Ex˙(s))ds-α∑v=12‍∫t-dvt‍xT(s)eα(s-t)Qivξ(s)ds-α∫t-d(t)t‍xT(s)eα(s-t)Qi3x(s)ds-α∫-d10‍∫t+θt‍(Ex˙(s))Teα(s-t)Zi1(Ex˙(s))dsdθ-α∫-d2-d1‍∫t+θt‍(Ex˙(s))Teα(s-t)Zi2(Ex˙(s))dsdθ.
From the Leibniz-Newton formula, the following equations are true for any matrices Niv, Siv, and Miv, v=1,2, with appropriate dimensions
(3.10)2[xT(t)Ni1+xT(t-d(t))Ni2][Ex(t)-Ex(t-d1)-∫t-d1t‍Ex˙(s)ds],2[xT(t)Si1+xT(t-d(t))Si2][Ex(t-d(t))-Ex(t-d2)-∫t-d2t-d(t)‍Ex˙(s)ds],2[xT(t)Mi1+xT(t-d(t))Mi2][Ex(t-d1)-Ex(t-d(t))-∫t-d(t)t-d1‍Ex˙(s)ds].
On the other hand, the following equation is also true:
(3.11)-∫t-d2t-d1‍(Ex˙(s))Teα(s-t)Zi2(Ex˙(s))ds=-∫t-d2t-d(t)‍(Ex˙(s))Teα(s-t)Zi2(Ex˙(s))ds-∫t-d(t)t-d1‍(Ex˙(s))Teα(s-t)Zi2(Ex˙(s))ds.
By (3.8)–(3.11), we have
(3.12)V˙i(xt)+αVi(xt)≤ηT(t)[Φi+A~iT(d1Zi1+d12Zi2)A~i+c1N~iZi1-1N~iT+c12S~iZi2-1S~iT+c12M~iZi2-1M~iT]η(t)-∫t-d1t‍[ηT(t)N~i+(Ex˙(s))Teα(s-t)Zi1]eα(t-s)Zi1-1[ηT(t)N~i+(Ex˙(s))Teα(s-t)Zi1]Tds-∫t-d2t-d(t)‍[ηT(t)S~i+(Ex˙(s))Teα(s-t)Zi2]eα(t-s)Zi2-1[ηT(t)S~i+(Ex˙(s))Teα(s-t)Zi2]Tds-∫t-d(t)t-d1‍[ηT(t)M~i+(Ex˙(s))Teα(s-t)Zi2]eα(t-s)Zi2-1[ηT(t)M~i+(Ex˙(s))Teα(s-t)Zi2]Tds,
where η(t)=[xT(t)xT(t-d(t))xT(t-d1)xT(t-d2)]T, A~i=[AiAdi00], and
(3.13)Φi=[Φi11Φi12Φi13-Si1E*Φi22Φi23-Si2E**Φi330***Φi44],N~i=[Ni1Ni200],M~i=[Mi1Mi200],S~i=[Si1Si200].
By Schur complement, LMI (3.2) implies
(3.14)Φi+A~iT(d1Zi1+d12Zi2)A~i+c1N~iZi1-1N~iT+c12S~iZi2-1S~iT+c12M~iZi2-1M~iT<0.
Notice that the last three parts in (3.12) are all less than 0. So, if (3.14) holds, then
(3.15)V˙i(xt)+αVi(xt)<0.
For an arbitrary piecewise constant switching signal σ(t), and for any t>0, we let 0=t0<t1<⋯<tk<⋯, k=1,2,⋯, denote the switching points of σ(t) over the interval (0,t). As mentioned earlier, the ikth subsystem is activated when t∈[tk,tk+1). Integrating (3.15) from tk to tk+1 gives
(3.16)V(xt)=Vσ(t)(xt)≤e-α(t-tk)Vσ(tk)(xtk),t∈[tk,tk+1).
Let x(t)=[x1(t)x2(t)], where x1(t)∈Rr and x2(t)∈Rn-r. From (2.4) and (3.1), it can be deduced that for each σ(t)=i, i∈ℐ(3.17)xT(t)ETPix(t)=x1T(t)Pi11x1(t).
In view of this, and using (3.4) and (3.8), at switching instant ti, we have
(3.18)Vσ(ti)(xti)≤βVσ(ti-)(xti-),i=1,2,…,
where ti- denotes the left limitation of ti. Therefore, it follows from (3.16), (3.18), and the relation k=Nσ(t0,t)≤(t-t0)/Ta that
(3.19)Vσ(t)(xt)≤e-α(t-tk)βVσ(ti-)(xti-)≤⋯≤e-α(t-t0)βkVσ(t0)(t0)≤e-(α-(lnβ)/Ta)(t-t0)Vσ(t0)(xt0).
According to (3.8) and (3.19), we obtain
(3.20)λ1∥x1(t)∥2≤Vσ(t)(t),Vσ(t0)(xt0)≤λ2∥xt0∥c2,
where
(3.21)λ1=min∀i∈ℐλmin(Pi11),λ2=max∀i∈ℐλmax(Pi11)+1α(1-e-αd1)max∀i∈ℐλmax(Qi1)+1α(1-e-αd2)max∀i∈ℐ(λmax(Qi2)+λmax(Qi3))+1α2(αd1-1+e-αd1)max∀i∈ℐ(2λmax(Zi1)(∥Ai∥+∥Adi∥))+αd12-e-αd1+e-αd2α2max∀i∈ℐ(2λmax(Zi2)(∥Ai∥+∥Adi∥)).
Considering (3.19) and (3.20) yields
(3.22)∥x1(t)∥2≤1λ1Vσ(t)(xt)≤λ2λ1e-(α-(lnβ)/Ta)(t-t0)∥xt0∥c2
which implies
(3.23)∥x1(t)∥≤λ2λ1e-(1/2)(α-(lnβ)/Ta)(t-t0)∥xt0∥c.
(iii) Exponential stability of algebraic subsystem. SinceAi22, i∈ℐ, is nonsingular, we choose
(3.24)Gi=[Ir-Ai12Ai22-10Ai22-1],H=[Ir00In-r].
Then, it is easy to get
(3.25)E^:=GiEH=[Ir000],A^i:=GiAiH=[A^i110A^i21In-r],P^i:=Gi-TPiH=[P^i110P^i21P^i22],
where A^i11=Ai11-Ai12Ai22-1Ai21, A^i21=Ai22-1Ai21, P^i11=Pi11, P^i21=Ai12TPi11+Ai22TPi21, and P^i21=Ai22TPi22. According to (3.25), denote
(3.26)A^di:=GiAdiH=[A^di11A^di12A^di21A^di22],Q^il:=HTQilH=[Q^il11Q^il12Q^il21Q^il22],Z^iv:=Gi-TZivGi-1=[Z^iv11Z^iv12Z^iv21Z^iv22],M^iv:=HTMivGi-1=[M^iv11M^iv12M^iv21M^iv22],N^iv:=HTNivGi-1=[N^iv11N^iv12N^iv21N^iv22],S^iv:=HTSivGi-1=[S^iv11S^iv12S^iv21S^iv22],l=1,2,3,v=1,2
and let
(3.27)ξ(t)=[ξ1(t)ξ2(t)]:=H-1x(t)=x(t),
where ξ1(t)∈Rr and ξ2(t)∈Rn-r. Then, for any σ(t)=i, i∈ℐ, system (2.5) is a restricted system equivalent (r.s.e.) to
(3.28)ξ˙1(t)=A^i11ξ1(t)+A^di11ξ1(t-d(t))+A^di12ξ2(t-d(t)),-ξ2(t)=A^i21ξ1(t)+A^di21ξ1(t-d(t))+A^di22ξ2(t-d(t)).
By (3.2) and Schur complement, we have
(3.29)[Φi11Φi12*Φi22]<0.
Pre- and postmultiplying this inequality by diag{HT,HT} and diag{H,H}, respectively, noting the expressions in (3.25) and (3.26), and using Schur complement, we have
(3.30)[P^i22T+P^i22+∑l=13‍Q^il22P^i22TA^di22T*-(1-μ)e-αd2Q^i322]<0.
Pre- and postmultiplying this inequality by [-A^di22TI] and its transpose, respectively, and noting Q^i122>0, Q^i222>0, and μ≥0, we obtain
(3.31)(e(1/2)αd2A^di22)TQ^i322(e(1/2)αd2A^di22)-Q^i322<0.
Then, according to Lemma 5 in Kharitonov et al. [37], we can deduce that there exist constants ℏi>1 and ηi∈(0,1) such that
(3.32)∥(e(1/2)αd2A^di22)l∥≤ℏie-ηil,l=0,1,…
Define
(3.33)t0=t,tj=tj-1-d(tj-1),j=1,2,…,∥A^21∥=max∀i∈ℐ∥A^i21∥,∥A^d21∥=max∀i∈ℐ∥A^di21∥,∥A^d22∥=max∀i∈ℐ∥A^di22∥,∀i∈ℐ.
Now, following similar line as in Part 3 in Theorem 1 of Lin and Fei [31], it can easily be obtained that
(3.34)∥ξ2(t)∥≤(χ1+χ2+χ3+χ4+χ5)e-(1/2)(α-(lnβ)/Ta)(t-t0)∥xt0∥c,
where
(3.35)χ1=∏j=0k‍ℏije-ηijTij,χ2=ℏikA^21λ2λ1eηikeηik-1,χ3=ℏike(1/2)αd2A^d21λ2λ1eηikeηik-1,χ4=A^21λ2λ1∑p=1k‍{ℏip-1[∏q=pk‍ℏiqe-ηiqTiq]eηip-1eηip-1-1},χ5=e(1/2)αd2A^d21λ2λ1∑p=1k‍{ℏip-1[∏q=pk‍ℏiqe-ηiqTiq]eηip-1eηip-1-1}.Tik,Tik-1,…,Ti0 are positive finite integers, respectively, satisfying
(3.36)tTik∈(tk-1,tk],tTik→tk,tTik+Tik-1∈(tk-2,tk-1],tTik+Tik-1→tk-1,⋮tTik+⋯+Ti0∈(-d2,t0],tTik+⋯+Ti0→t0.
Combining (3.27), (3.23) and (3.34) yields that system (2.5) is exponentially stable for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α. This completes the proof.Remark 3.2.
Theorem3.1 provides a sufficient condition of the exponential admissibility for the switched singular time-delay system (2.5). Note that due to the existence of algebraic constraints in system states, the stability analysis of switched singular time-delay systems is much more complicated than that for switched state-space time-delay systems [21–23, 25, 38]. Note also that the condition established in Theorem 3.1 is not only delay-range-dependent but also decay-rate-dependent. The delay-range-dependence makes the result less conservative, while the decay-rate-dependence enables one to control the transient process of differential and algebraic subsystems with a unified performance specification.Remark 3.3.
Different from the integral inequality method used in our previous work [31], the free-weighting matrix method [39] is adopted when deriving Theorem 3.1, and thus no three-product terms, for example, AiTZivAi, AdiTZivAdi, and so forth, are involved, which greatly facilitates the SF and SOF controllers design, as seen in Section 4.Remark 3.4.
Ifβ=1 in Ta≥Ta*=(lnβ)/α, which leads to Pi11≡Pj11, Qil≡Qjl, Ziv≡Zjv, l=1,2,3, v=1,2, for all i,j∈ℐ, and Ta*=0, then system (2.5) possesses a common Lyapunov function, and the switching signals can be arbitrary.
Now, the following theorem presents a sufficient condition on exponential admissibility with a weightedH∞ performance of the switched singular time-delay system (2.1) with u(t)=0.Theorem 3.5.
For prescribed scalarsα>0, γ>0, 0≤d1≤d2, and 0<μ<1, if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi with the form of (3.1) such that
(3.37)Φi=[Φ~i11Φ~i12Φi13-Si1EΦ~i15CiTc1Ni1c12Si1c12Mi1*Φ~i22Φi23-Si2EΦ~i25CdiTc1Ni2c12Si2c12Mi2**Φi33000000***Φi4400000****Φ~i55DwiT000*****-I000******-c1Zi100*******-c12Zi20********-c12Zi2]<0,
where
(3.38)Φ~i11=Φi11+AiTUiAi,Φ~i12=Φi12+AiTUiAdi,Φ~i15=PiTBwi+AiTUiBwi,Φ~i22=Φi22+AdiTUiAdi,Φ~i25=AdiTUiBwi,Φ~i55=-γ2I+BwiTUiBwi,
and Φi11, Φi12, Φi13, Φi22, Φi23, Φi33, Φi44, and Ui are defined in (3.2). Then, system (2.1) with u(t)=0 is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α, where β≥1 satisfying (3.4).Proof.
Choose the piecewise Lyapunov function defined by (3.8). Since (3.37) implies (3.2), system (2.1) with u(t)=0 and w(t)=0 is exponentially admissible by Theorem 3.1. On the other hand, similar to the proof of Theorem 3.1, from (3.37), we have that for t∈[tk,tk+1),
(3.39)V˙ik(xt)+αVik(xt)+Γ(t)≤0,
where Γ(t)=zT(t)z(t)-γ2wT(t)w(t). This implies that
(3.40)Vik(xt)≤e-α(t-tk)Vik(xtk)-∫tkt‍e-α(t-s)Γ(s)ds.
By induction, we have
(3.41)Vik(xt)≤βe-α(t-tk)Vik-1(xtk)-∫tkt‍e-α(t-s)Γ(s)ds⋮≤βke-αtVi0(x0)-∫tkt‍e-α(t-s)Γ(s)ds-∑p=1k-p‍βk-p∫tptp+1‍e-α(t-s)Γ(s)ds=e-αt+Nα(0,t)lnβVi0(x0)-∫0t‍e-α(t-s)+Nα(s,t)lnβΓ(s)ds.
Under zero initial condition, (3.41) gives
(3.42)0≤-∫0t‍e-α(t-s)+Nα(s,t)lnβΓ(s)ds.
Multiplying both sides of (3.42) by e-Nα(0,t)lnβ yields
(3.43)∫0t‍e-α(t-s)-Nα(0,s)lnβzT(s)z(s)ds≤γ2∫0t‍e-α(t-s)-Nα(0,s)lnβwT(s)w(s)ds.
Noting that Nα(0,s)≤s/Ta and Ta≥Ta*=(lnβ)/α, we get Nα(0,s)lnβ≤αs. Then, it follows from (3.43) that ∫0t‍e-α(t-s)-αszT(s)z(s)ds≤γ2∫0t‍e-α(t-s)wT(s)w(s)ds. Integrating both sides of this inequality from t=0 to ∞ leads to inequality (2.7). This completes the proof of Theorem 3.5.Remark 3.6.
Note that whenβ=1, which is a trivial case, system (2.1) with u(t)=0 achieves the normal H∞ performance γ under arbitrary switching.


## 4. Controller Design

In this section, based on the results of the previous section, we are to deal with the design problems of both SF and SOF controllers for the switched singular time-delay system (2.1).


### 4.1. SF Controller Design

Applying the SF controller (2.8) to system (2.1) gives the following closed-loop system:
(4.1)Ex˙(t)=A-σ(t)x(t)+Adσ(t)x(t-d(t))+Bwσ(t)w(t),z(t)=C-σ(t)x(t)+Cdσ(t)x(t-d(t))+Dwσ(t)w(t),x(θ)=ϕ(θ),θ∈[-d2,0],
where
(4.2)A-σ(t)=Aσ(t)+Bσ(t)Kσ(t),C-σ(t)=Cσ(t)+Dσ(t)Kσ(t).
The following theorem presents a sufficient condition for solvability of the SF controller design problem for system (2.1).Theorem 4.1.
For prescribed scalarsα>0, γ>0, 0≤d1≤d2, and 0<μ<1, if for each i∈ℐ, and given scalars ϵif, f=1,2,…,6, ϵi7>0, and ϵi8>0, there exist matrices Ril>0, l=1,2,3, Ziv>0, Ti, and Xi of the following form:
(4.3)Xi=[Xi110Xi21Xi22],
with Xi11∈Rr, Xi11>0, and Xi22 being invertible, such that
(4.4)[Ψi11Ψi12Ψi13Ψi14BwiΨi16Ψi17c12ϵi5Ic12ϵi3IΨi110Ψi111Ξi*Ψi22Ψi23Ψi240Ψi26Ψi27c12ϵi6Ic12ϵi4IΨi210Ψi2110**Ψi33000000000***Ψi4400000000****-γ2IDwiT000d1BwiTd12BwiT0*****-I000000******-c1Zi100000*******-c12Zi20000********-c12Zi2000*********Ψi101000**********Ψi11110***********-Γi]<0,
where
(4.5)Ψi11=Sym{AiXi+BiTi+ϵi1EXi}+αXiTET,Ψi12=AdiRi3+ϵi2XiTET+ϵi5ERi3-ϵi3ERi3,Ψi13=ϵi3ERi1-ϵi1ERi1,Ψi14=-ϵi5ERi2,Ψi16=TiTDiT+XiTCiT,Ψi17=c1ϵi1I,Ψi110=d1TiTBiT+d1XiTAiT,Ψi111=d12TiTBiT+d12XiTAiT,Ψi22=-(1-μ)e-αd2Ri3+Sym{ϵi6ERi3-ϵi4ERi3},Ψi23=ϵi4ERi1-ϵi2ERi1,Ψi24=-ϵi6ERi2,Ψi26=Ri3CdiT,Ψi27=c1ϵi2I,Ψi210=d1Ri3AdiT,Ψi211=d12Ri3AdiT,Ψi33=-e-αd1Ri1,Ψi44=-e-αd2Ri2,Ψi1010=-2d1ϵi7I+d1ϵi72Zi1,Ψi1111=-2d12ϵi8I+d12ϵi82Zi2,Ξi=[XiTXiTXiT],Γi=diag{Ri1,Ri2,Ri3}.
Then, there exists an SF controller (2.8) such that the closed-loop system (4.1) with d(t) satisfying (2.2) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α, where β≥1 satisfies
(4.6)Xi11≥β-1Xj11,Ril≥β-1Rjl,Ziv≤βZjv,l=1,2,3,v=1,2,∀i,j∈ℐ.
Moreover, the feedback gain of the controller is
(4.7)Ki=TiXi-1,i∈ℐ.Proof.
According to Theorem3.5, the closed-loop system (4.1) is exponentially admissible with a weighted H∞ performance γ if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi with the form of (3.1) such that inequality (3.37) holds with Ai and Ci instead of A-i and C-i, respectively. By Schur complement, (3.37) is equivalent to
(4.8)[Φi11′Φi12Φi13-Si1EPiTBwiC-iTc1Ni1c12Si1c12Mi1d1A-iTd12A-iT*Φi22Φi23-Si2E0CdiTc1Ni2c12Si2c12Mi2d1AdiTd12AdiT**Φi3300000000***Φi440000000****-γ2IDwiT000d1BwiTd12BwiT*****-I00000******-c1Zi10000*******-c12Zi2000********-c12Zi200*********-d1Zi1-10**********-d12Zi2-1]<0,
where Φi12, Φi13, Φi22, Φi23, Φi33, and Φi44 are defined in (3.2), and
(4.9)Φi11′=Sym{PiTA-i+Ni1E}+∑l=13‍Qil+αETPi.
Since Pi11>0 and Pi22 is invertible, then Pi is invertible. Let
(4.10)Xi=Pi-1,Ri1=Qi1-1,Ri2=Qi2-1,Ri3=Qi3-1.
By (3.1), Xi has the form of (4.3). Pre- and postmultiplying (4.8) by diag{XiT,Ri3,Ri1,Ri2,I,I,I,I,I,I,I} and its transpose, respectively, and noting (4.10), we obtain
(4.11)[Φi11′′Φi12′′Φi13′′Φi14′′BwiΦi16′′Φi17′′Φi18′′Φi19′′Φi110′′Φi111′′*Φi22′′Φi23′′Φi24′′0Φi26′′Φi27′′Φi28′′Φi29′′Ψi210Ψi211**Ψi3300000000***Ψi440000000****-γ2IDwiT000d1BwiTd12BwiT*****-I00000******-c1Zi10000*******-c12Zi2000********-c12Zi200*********-d1Zi1-10**********-d12Zi2-1]<0,
where
(4.12)Φi11′′=Sym{A-iXi+XiTNi1EXi}+XiT∑l=13‍QilXi+αXiTET,Φi12′′=AdiRi3+XiT(Ni2E)TRi3+XiTSi1ERi3-XiTMi1ERi3,Φi13′′=XiT(Mi1E-Ni1E)Ri1,Φi14′′=-XiTSi1ERi2,Φi16′′=XiTC-iT,Φi17′′=c1XiTNi1,Φi18′′=c12XiTSi1,Φi19′′=c12XiTMi1,Φi110′′=d1XiTA-iT,Φi111′′=d12XiTA-iT,Φi22′′=-(1-μ)e-αd2Ri3+Sym{Ri3Si2ERi3-Ri3Mi2ERi3},Φi23′′=Ri3(Mi2E-Ni2E)Ri1,Φi24′′=-Ri3Si2ERi2,Φi26′′=Ri3CdiT,Φi27′′=c1Ri3Ni2,Φi28′′=c12Ri3Si2,Φi29′′=c12Ri3Mi2.
Now, introducing change of variables
(4.13)Ni1=ϵi1Xi-T,Ni2=ϵi2Ri3-1,Mi1=ϵi3Xi-T,Mi2=ϵi4Ri3-1,Si1=ϵi5Xi-T,Si2=ϵi6Ri3-1,Ti=KiXi,
where ϵif,f=1,2,…,6 are scalars, noting the fact that
(4.14)-Zi1-1≤-2ϵi7I+ϵi72Zi1,-Zi2-1≤-2ϵi8I+ϵi82Zi2,
where ϵi7 and ϵi8 are positive scalars, and using Schur complement on (4.11), we can easy obtain (4.4). In addition, by (3.4) and (4.10), it is easily to verify that the condition (4.6) is equivalent to (3.4). This completes the proof.Remark 4.2.
Scalarsϵih, h=1,2,…,8, i∈ℐ, in Theorem 4.1 are tuning parameters which need to be specified first; such tuning parameters are frequently encountered when dealing with the SF control problem of singular time-delay systems; see, for example, Ma et al. [27], Shu and Lam [40], and Wu et al. [38]. A simple way to choose these tuning parameters is using the trial-and-error method. In fact, (4.4) for fixed ϵih, is bilinear matrix inequality (BMI) regarding these tuning parameters. Therefore, if one can accept more computation burden, better results can be obtained by directly applying some existing optimization algorithms, such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm [41], and the branch-and-cut algorithm [42].


### 4.2. SOF Controller Design

Connecting the SOF controller (2.9) to system (2.1) yields the closed-loop system
(4.15)Ex˙(t)=A-σ(t)′x(t)+Adσ(t)x(t-d(t))+Bwσ(t)w(t),z(t)=C-σ(t)′x(t)+Cdσ(t)x(t-d(t))+Dwσ(t)w(t),
where
(4.16)A-σ(t)′=Aσ(t)+Bσ(t)Fσ(t)Lσ(t),C-σ(t)′=Cσ(t)+Dσ(t)Fσ(t)Lσ(t).
The following theorem presents a sufficient condition for solvability of the SOF controller design problem for system (2.1).Theorem 4.3.
For prescribed scalarsα>0, γ>0, 0≤d1≤d2, and 0<μ<1, if for each i∈ℐ, and a given matrix Ji, there exist matrices Qil>0, l=1,2,3, Ziv>0, and Pi of the form (3.2) such that
(4.17)[Λi11Λi12Λi13Λi14-Si1EJiTBwiC-i′Tc1Ni1c12Si1c12Mi1*Λi22Λi2300JiTBwi0000**Λi33Λi34-Si2E0CdiTc1Ni2c12Si2c12Mi2***Λi44000000****Λi5500000*****-γ2IDwiT000******-I000*******-c1Zi100********-c12Zi20*********-c12Zi2]<0,
where c1, c12, and Ui are defined in (3.2), and
(4.18)Λi11=Sym{JiA-i′+Ni1E}+∑l=13‍Qil+αETPi,Λi12=-Ji+A-i′TJiT+PiT,Λi13=JiAdi+ETNi2T+Si1E-Mi1E,Λi14=Mi1E-Ni1E,Λi22=-JiT-Ji+Ui,Λi23=JiAdi,Λi33=-(1-μ)e-αd2Qi3+Sym{Si2E-Mi2E},Λi34=Mi2E-Ni2E,Λi44=-e-αd1Qi1,Λi55=-e-αd2Qi2.
Then, there exists an SOF controller (2.9) such that the closed-loop system (4.15) with d(t) satisfying (2.2) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α, where β≥1 satisfying (3.4).Proof.
From Theorem3.5, we know that system (4.15) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α, where β≥1 satisfying (3.4), if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi with the form of (3.1) such that the inequality (4.10) with Ai and Ci instead of A-i′ and C-i′, respectively, holds. By decomposing Φi in (4.10), we obtain that for each i∈ℐ,
(4.19)Φi=Π(i)ΛiΠiT<0,
where Λi is exactly the left half of the inequality (4.17), and
(4.20)Π(i)=[IA-i′T000000000AdiTI0000000000I0000000000I000000BwiT000I0000000000I0000000000I0000000000I0000000000I].
Hence, the condition (4.17) implies Φi<0. This completes the proof.Remark 4.4.
Note that there exist product terms between the Lyapunov and system matrices in inequality (3.37) of Theorem 3.5, which will bring some difficulties in solving the SOF controller design problem. To resolve this problem, in the proof of Theorem 4.3, we have made a decoupling between the Lyapunov and system matrices by introducing a slack matrix variable Ji and then obtained a new inequality (4.17). It should be pointed that in Haidar et al.  [32], a sufficient condition for solvability of the SOF controller design problem for the deterministic singular time-delay system has been proposed. However, the controller gain was computed by using an iterative LMI algorithm, which was complex. Although the new inequality (4.17) may be conservative mainly due to the introduction of matrix variable Ji, the introduced decoupling technique enables us to obtain a more easily tractable condition for the synthesis of SOF controller.Remark 4.5.
MatricesJi, i∈ℐ, in Theorem 4.3 can be specified by the algorithm stated in Remark 3.6.Remark 4.6.
In this paper, we have only discussed a special case of the derivative matrixE having no switching modes. If E also has switching modes, then E is changed to Ei, i∈ℐ. In this case, the transformation matrices P and Q should become Pi and Qi, and we have PiEiQi=diag{Iri,0}. Accordingly, the state of the transformed system becomes x~(t)=[x~i1T(t)x~i2T(t)]T with x~i1T(t)∈Rri, which means that there does not exist one common state space coordinate basis for all subsystems, and thus it is complicated to discuss the transformed system. Hence, some assumptions for the matrices Ei (e.g., Ei, i∈ℐ, have the same right zero subspace [43]) should be given so that the matrices Qi remain the same; in this case, the method presented here is also valid. However, the general case of E with switching modes is an interesting problem for future investigation via other methods.


## 4.1. SF Controller Design

Applying the SF controller (2.8) to system (2.1) gives the following closed-loop system:
(4.1)Ex˙(t)=A-σ(t)x(t)+Adσ(t)x(t-d(t))+Bwσ(t)w(t),z(t)=C-σ(t)x(t)+Cdσ(t)x(t-d(t))+Dwσ(t)w(t),x(θ)=ϕ(θ),θ∈[-d2,0],
where
(4.2)A-σ(t)=Aσ(t)+Bσ(t)Kσ(t),C-σ(t)=Cσ(t)+Dσ(t)Kσ(t).
The following theorem presents a sufficient condition for solvability of the SF controller design problem for system (2.1).Theorem 4.1.
For prescribed scalarsα>0, γ>0, 0≤d1≤d2, and 0<μ<1, if for each i∈ℐ, and given scalars ϵif, f=1,2,…,6, ϵi7>0, and ϵi8>0, there exist matrices Ril>0, l=1,2,3, Ziv>0, Ti, and Xi of the following form:
(4.3)Xi=[Xi110Xi21Xi22],
with Xi11∈Rr, Xi11>0, and Xi22 being invertible, such that
(4.4)[Ψi11Ψi12Ψi13Ψi14BwiΨi16Ψi17c12ϵi5Ic12ϵi3IΨi110Ψi111Ξi*Ψi22Ψi23Ψi240Ψi26Ψi27c12ϵi6Ic12ϵi4IΨi210Ψi2110**Ψi33000000000***Ψi4400000000****-γ2IDwiT000d1BwiTd12BwiT0*****-I000000******-c1Zi100000*******-c12Zi20000********-c12Zi2000*********Ψi101000**********Ψi11110***********-Γi]<0,
where
(4.5)Ψi11=Sym{AiXi+BiTi+ϵi1EXi}+αXiTET,Ψi12=AdiRi3+ϵi2XiTET+ϵi5ERi3-ϵi3ERi3,Ψi13=ϵi3ERi1-ϵi1ERi1,Ψi14=-ϵi5ERi2,Ψi16=TiTDiT+XiTCiT,Ψi17=c1ϵi1I,Ψi110=d1TiTBiT+d1XiTAiT,Ψi111=d12TiTBiT+d12XiTAiT,Ψi22=-(1-μ)e-αd2Ri3+Sym{ϵi6ERi3-ϵi4ERi3},Ψi23=ϵi4ERi1-ϵi2ERi1,Ψi24=-ϵi6ERi2,Ψi26=Ri3CdiT,Ψi27=c1ϵi2I,Ψi210=d1Ri3AdiT,Ψi211=d12Ri3AdiT,Ψi33=-e-αd1Ri1,Ψi44=-e-αd2Ri2,Ψi1010=-2d1ϵi7I+d1ϵi72Zi1,Ψi1111=-2d12ϵi8I+d12ϵi82Zi2,Ξi=[XiTXiTXiT],Γi=diag{Ri1,Ri2,Ri3}.
Then, there exists an SF controller (2.8) such that the closed-loop system (4.1) with d(t) satisfying (2.2) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α, where β≥1 satisfies
(4.6)Xi11≥β-1Xj11,Ril≥β-1Rjl,Ziv≤βZjv,l=1,2,3,v=1,2,∀i,j∈ℐ.
Moreover, the feedback gain of the controller is
(4.7)Ki=TiXi-1,i∈ℐ.Proof.
According to Theorem3.5, the closed-loop system (4.1) is exponentially admissible with a weighted H∞ performance γ if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi with the form of (3.1) such that inequality (3.37) holds with Ai and Ci instead of A-i and C-i, respectively. By Schur complement, (3.37) is equivalent to
(4.8)[Φi11′Φi12Φi13-Si1EPiTBwiC-iTc1Ni1c12Si1c12Mi1d1A-iTd12A-iT*Φi22Φi23-Si2E0CdiTc1Ni2c12Si2c12Mi2d1AdiTd12AdiT**Φi3300000000***Φi440000000****-γ2IDwiT000d1BwiTd12BwiT*****-I00000******-c1Zi10000*******-c12Zi2000********-c12Zi200*********-d1Zi1-10**********-d12Zi2-1]<0,
where Φi12, Φi13, Φi22, Φi23, Φi33, and Φi44 are defined in (3.2), and
(4.9)Φi11′=Sym{PiTA-i+Ni1E}+∑l=13‍Qil+αETPi.
Since Pi11>0 and Pi22 is invertible, then Pi is invertible. Let
(4.10)Xi=Pi-1,Ri1=Qi1-1,Ri2=Qi2-1,Ri3=Qi3-1.
By (3.1), Xi has the form of (4.3). Pre- and postmultiplying (4.8) by diag{XiT,Ri3,Ri1,Ri2,I,I,I,I,I,I,I} and its transpose, respectively, and noting (4.10), we obtain
(4.11)[Φi11′′Φi12′′Φi13′′Φi14′′BwiΦi16′′Φi17′′Φi18′′Φi19′′Φi110′′Φi111′′*Φi22′′Φi23′′Φi24′′0Φi26′′Φi27′′Φi28′′Φi29′′Ψi210Ψi211**Ψi3300000000***Ψi440000000****-γ2IDwiT000d1BwiTd12BwiT*****-I00000******-c1Zi10000*******-c12Zi2000********-c12Zi200*********-d1Zi1-10**********-d12Zi2-1]<0,
where
(4.12)Φi11′′=Sym{A-iXi+XiTNi1EXi}+XiT∑l=13‍QilXi+αXiTET,Φi12′′=AdiRi3+XiT(Ni2E)TRi3+XiTSi1ERi3-XiTMi1ERi3,Φi13′′=XiT(Mi1E-Ni1E)Ri1,Φi14′′=-XiTSi1ERi2,Φi16′′=XiTC-iT,Φi17′′=c1XiTNi1,Φi18′′=c12XiTSi1,Φi19′′=c12XiTMi1,Φi110′′=d1XiTA-iT,Φi111′′=d12XiTA-iT,Φi22′′=-(1-μ)e-αd2Ri3+Sym{Ri3Si2ERi3-Ri3Mi2ERi3},Φi23′′=Ri3(Mi2E-Ni2E)Ri1,Φi24′′=-Ri3Si2ERi2,Φi26′′=Ri3CdiT,Φi27′′=c1Ri3Ni2,Φi28′′=c12Ri3Si2,Φi29′′=c12Ri3Mi2.
Now, introducing change of variables
(4.13)Ni1=ϵi1Xi-T,Ni2=ϵi2Ri3-1,Mi1=ϵi3Xi-T,Mi2=ϵi4Ri3-1,Si1=ϵi5Xi-T,Si2=ϵi6Ri3-1,Ti=KiXi,
where ϵif,f=1,2,…,6 are scalars, noting the fact that
(4.14)-Zi1-1≤-2ϵi7I+ϵi72Zi1,-Zi2-1≤-2ϵi8I+ϵi82Zi2,
where ϵi7 and ϵi8 are positive scalars, and using Schur complement on (4.11), we can easy obtain (4.4). In addition, by (3.4) and (4.10), it is easily to verify that the condition (4.6) is equivalent to (3.4). This completes the proof.Remark 4.2.
Scalarsϵih, h=1,2,…,8, i∈ℐ, in Theorem 4.1 are tuning parameters which need to be specified first; such tuning parameters are frequently encountered when dealing with the SF control problem of singular time-delay systems; see, for example, Ma et al. [27], Shu and Lam [40], and Wu et al. [38]. A simple way to choose these tuning parameters is using the trial-and-error method. In fact, (4.4) for fixed ϵih, is bilinear matrix inequality (BMI) regarding these tuning parameters. Therefore, if one can accept more computation burden, better results can be obtained by directly applying some existing optimization algorithms, such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm [41], and the branch-and-cut algorithm [42].


## 4.2. SOF Controller Design

Connecting the SOF controller (2.9) to system (2.1) yields the closed-loop system
(4.15)Ex˙(t)=A-σ(t)′x(t)+Adσ(t)x(t-d(t))+Bwσ(t)w(t),z(t)=C-σ(t)′x(t)+Cdσ(t)x(t-d(t))+Dwσ(t)w(t),
where
(4.16)A-σ(t)′=Aσ(t)+Bσ(t)Fσ(t)Lσ(t),C-σ(t)′=Cσ(t)+Dσ(t)Fσ(t)Lσ(t).
The following theorem presents a sufficient condition for solvability of the SOF controller design problem for system (2.1).Theorem 4.3.
For prescribed scalarsα>0, γ>0, 0≤d1≤d2, and 0<μ<1, if for each i∈ℐ, and a given matrix Ji, there exist matrices Qil>0, l=1,2,3, Ziv>0, and Pi of the form (3.2) such that
(4.17)[Λi11Λi12Λi13Λi14-Si1EJiTBwiC-i′Tc1Ni1c12Si1c12Mi1*Λi22Λi2300JiTBwi0000**Λi33Λi34-Si2E0CdiTc1Ni2c12Si2c12Mi2***Λi44000000****Λi5500000*****-γ2IDwiT000******-I000*******-c1Zi100********-c12Zi20*********-c12Zi2]<0,
where c1, c12, and Ui are defined in (3.2), and
(4.18)Λi11=Sym{JiA-i′+Ni1E}+∑l=13‍Qil+αETPi,Λi12=-Ji+A-i′TJiT+PiT,Λi13=JiAdi+ETNi2T+Si1E-Mi1E,Λi14=Mi1E-Ni1E,Λi22=-JiT-Ji+Ui,Λi23=JiAdi,Λi33=-(1-μ)e-αd2Qi3+Sym{Si2E-Mi2E},Λi34=Mi2E-Ni2E,Λi44=-e-αd1Qi1,Λi55=-e-αd2Qi2.
Then, there exists an SOF controller (2.9) such that the closed-loop system (4.15) with d(t) satisfying (2.2) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α, where β≥1 satisfying (3.4).Proof.
From Theorem3.5, we know that system (4.15) is exponentially admissible with a weighted H∞ performance γ for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α, where β≥1 satisfying (3.4), if for each i∈ℐ, there exist matrices Qil>0, l=1,2,3, Ziv>0, Miv, Niv, Siv, v=1,2, and Pi with the form of (3.1) such that the inequality (4.10) with Ai and Ci instead of A-i′ and C-i′, respectively, holds. By decomposing Φi in (4.10), we obtain that for each i∈ℐ,
(4.19)Φi=Π(i)ΛiΠiT<0,
where Λi is exactly the left half of the inequality (4.17), and
(4.20)Π(i)=[IA-i′T000000000AdiTI0000000000I0000000000I000000BwiT000I0000000000I0000000000I0000000000I0000000000I].
Hence, the condition (4.17) implies Φi<0. This completes the proof.Remark 4.4.
Note that there exist product terms between the Lyapunov and system matrices in inequality (3.37) of Theorem 3.5, which will bring some difficulties in solving the SOF controller design problem. To resolve this problem, in the proof of Theorem 4.3, we have made a decoupling between the Lyapunov and system matrices by introducing a slack matrix variable Ji and then obtained a new inequality (4.17). It should be pointed that in Haidar et al.  [32], a sufficient condition for solvability of the SOF controller design problem for the deterministic singular time-delay system has been proposed. However, the controller gain was computed by using an iterative LMI algorithm, which was complex. Although the new inequality (4.17) may be conservative mainly due to the introduction of matrix variable Ji, the introduced decoupling technique enables us to obtain a more easily tractable condition for the synthesis of SOF controller.Remark 4.5.
MatricesJi, i∈ℐ, in Theorem 4.3 can be specified by the algorithm stated in Remark 3.6.Remark 4.6.
In this paper, we have only discussed a special case of the derivative matrixE having no switching modes. If E also has switching modes, then E is changed to Ei, i∈ℐ. In this case, the transformation matrices P and Q should become Pi and Qi, and we have PiEiQi=diag{Iri,0}. Accordingly, the state of the transformed system becomes x~(t)=[x~i1T(t)x~i2T(t)]T with x~i1T(t)∈Rri, which means that there does not exist one common state space coordinate basis for all subsystems, and thus it is complicated to discuss the transformed system. Hence, some assumptions for the matrices Ei (e.g., Ei, i∈ℐ, have the same right zero subspace [43]) should be given so that the matrices Qi remain the same; in this case, the method presented here is also valid. However, the general case of E with switching modes is an interesting problem for future investigation via other methods.


## 5. Numerical Examples

In this section, we present two illustrative examples to demonstrate the applicability and effectiveness of the proposed approach.Example 5.1.
Consider the switched system (2.5) with I=2 (i.e., there are two subsystems) and the related parameters are given as follows:
(5.1)E=[1000],A1=[0.7300-1],Ad1=[-1.1100.5],A2=[0.40-0.1-1],Ad2=[-10.100.1]
and d1=0.1, d2=0.3, μ=0.4, and α=0.5. It can be checked that the previous two subsystems are both stable independently. Consider the quadratic approach (see Remark 3.3, β=1, and we know that it requires a common Lyapunov functional for all subsystems); by simulation, it can be found that there is no feasible solution to this case, that is to say, there is no common Lyapunov functional for all subsystems. Now, we consider the average dwell time scheme, and set β=1.25, and solving the LMIs (3.2) gives the following solutions:
(5.2)P1=103×[0.035400.02561.3047],Q11=[1.2978-1.6836-1.683678.9994],Q12=[0.4821-1.6749-1.674978.5452],Q13=103×[0.00020.00930.00931.8512],Z11=[128.95002.76822.7682308.4441],Z12=[128.11434.40054.4005323.5358],M11=[-81.4620-2.820111.23290.2679],M12=[59.25442.0497-62.6780-1.9627],N11=[-82.1186-1.781011.32730.1165],N12=[58.70981.2698-62.8565-1.1401],S11=[-0.15400.01830.02970.2129],S12=[-0.2125-0.0163-0.0270-0.3559],P2=[43.67000-66.4319953.1992],Q21=[1.3435-0.9553-0.955376.7387],Q22=[0.4965-1.3638-1.363876.0801],Q23=103×[0.00020.00780.00781.5081],Z21=[116.44802.37022.3702308.2446],Z22=[110.60463.61343.6134323.0948],M21=[-73.9202-2.42557.42570.2466],M22=[52.52671.7299-2.7859-0.1158],N21=[-74.6177-1.52407.49180.1543],N22=[51.95731.0641-2.7465-0.0661],S21=[-0.1625-0.00420.01500.0002],S22=[-0.2236-0.00850.01500.0022],
which means that the aforementioned switched system is exponentially admissible. Moreover, by further analysis, we find that the allowable minimum of β is βmin=1.046 when α=0.5 is fixed; in this case, Ta*=(lnβmin)/α=0.0899. By the previous analysis, we know that the average dwell time approach proposed in this paper is less conservative than the quadratic approach.Example 5.2.
Consider the switched system (2.1) with I=2 and
(5.3)E=[1000],A1=[0.901-5],Ad1=[0.50.110.1],B1=[-3-1],Bw1=[0.50.03],C1=[0.10.3],Cd1=[0.10.1],D1=1.1,Dw1=0.15A2=[0.50.15-2],Ad2=[0.20.51.50.1],B2=[-4-1],Bw2=[0.30.03],C2=[0.10.3],Cd2=[0.10.1],D2=1,Dw2=0.1,
and d(t)=0.3+0.2sin(1.5t). A straightforward calculation gives d1=0.1, d2=0.5, and μ=0.3. By simulation, it can be checked that the previous two subsystems with u(t)=0 are both unstable, and the state responses of the corresponding open-loop systems are shown in Figures 1 and 2, respectively, with the initial condition given by ϕ(t)=[12]T, t∈[-0.5,0]. In view of this, our goal is to design an SF control u(t) in the form of (2.8) and an SOF control u(t) in the form of (2.9), such that the closed-loop system is exponentially admissible with a weighted H∞ performance γ=1.5.
For SF control law, setα=0.4, β=1.05 (thus Ta≥Ta*=(lnβ)/α=0.122), and choose ϵ11=0.2, ϵ12=0.1, ϵ13=0.1, ϵ14=0.02, ϵ15=0.004, ϵ16=0.03, ϵ17=1.9, ϵ18=1, ϵ21=0.06, ϵ22=0.1, ϵ23=0.14, ϵ24=0.17, ϵ25=0.1, ϵ26=0.1, ϵ27=0.4, and ϵ28=0.1. Solving the LMIs (4.4), we obtain the following solutions:
(5.4)X1=[0.09300-0.02970.2059],R11=[1.04440.00080.0008191.0844],R12=[0.44610.00090.0009200.5541],R13=[0.0889-0.1241-0.12410.8811],Z11=[0.89790.00100.00100.8617],Z12=[1.76090.01940.01941.4794],X2=[0.093200.02080.0847],R21=[1.03970.00080.0008191.0844],R22=[0.44420.00090.0009200.5541],R23=[0.0900-0.1233-0.12330.8771],Z21=[0.90170.00110.00110.8640],Z22=[1.78820.02200.02201.4824],T1=[-0.20000.0046],T2=[-1.6983-0.0889].
Therefore, from (4.7), the gain matrices of an SF controller can be obtained as
(5.5)K1=[-2.14370.0221],K2=[2.48250.1243].
For SOF control law, letα, β be the same as in the SF control case, and choose
(5.6)J1=diag{1.08,3.07},J2=diag{3.95,1.58}.
By solving the LMIs (4.17), we obtain the following solutions:
(5.7)P1=[0.11550-5.884217.7617],Q11=[3.31160.52390.52392.1174],Q12=[1.50550.52310.52312.1194],Q13=[1.12411.30091.30093.5238],Z11=[5.9711-0.2713-0.271310.1082],Z12=[4.0344-0.1822-0.18227.5545],M11=[-13.11220.59310.1486-0.0065],M12=[13.8147-0.6248-0.00150.0001],N11=[-15.31400.69700.1666-0.0070],N12=[13.1090-0.5965-0.00140.0001],S11=[-0.00920.0004-0.00280.0006],S12=[-1.10160.05000.00020.0001],P2=[0.11530-10.14944.3434],Q21=[3.31320.52420.52422.1163],Q22=[1.50610.52310.52312.1182],Q23=[1.12431.30071.30073.5222],Z21=[5.9744-0.2714-0.271410.1049],Z22=[4.0385-0.1824-0.18247.5509],M21=[-13.23350.59850.0221-0.0010],M22=[13.9337-0.6301-0.00860.0004],N21=[-15.44200.70250.0237-0.0011],N22=[13.2333-0.6018-0.01020.0003],S21=[-0.00380.0002-0.00070.0000],S22=[-1.10760.0501-0.0001-0.0000],K1=15.1634,F2=1.6543.
To show the effectiveness of the designed SF and SOF controllers, giving a random switching signal with the average dwell time Ta≥0.13 as shown in Figure 3, we get the state responses using the SF and SOF controllers for the system as shown in Figures 4 and 5, respectively, for the given initial condition ϕ(t)=[12]T, t∈[-0.5,0]. It is obvious that the designed controllers are feasible and ensure the stability of the closed-loop systems despite the interval time-varying delays.Figure 1
The state trajectories of the open-loop subsystem 1.Figure 2
The state trajectories of the open-loop subsystem 2.Figure 3
Switching signal with the average dwell timeTa>0.13.Figure 4
The state trajectories of the closed-loop system under SF control.Figure 5
The state trajectories of the closed-loop system under SOF control.


## 6. Conclusions

In this paper, the problems of exponential admissibility andH∞ control for a class of continuous-time switched singular systems with interval time-varying delay have been investigated. A class of switching signals specified by the average dwell time has been identified for the unforced systems to be exponentially admissible with a weighted H∞ performance. The state feedback and static output feedback controllers have been designed, and their corresponding solvability conditions have been established by using the LMI technique. Simulation results have demonstrated the effectiveness of the proposed design method.



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*Source: 482792-2012-12-13.xml*