# Exponential Admissibility and Dynamic Output Feedback Control of Switched Singular Systems with Interval Time-Varying Delay

Authors: Jinxing Lin; Chunxia Fan
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This paper is concerned with the problems of exponential admissibility and dynamic output feedback (DOF)
control for a class of continuous-time switched singular systems with interval time-varying delay. A full-order,
dynamic, synchronously switched DOF controller is considered. First, by using the average dwell time approach,
a delay-range-dependent exponential admissibility criterion for the unforced switched singular time-delay system is
established in terms of linear matrix inequalities (LMIs). Then, based on this criterion, a sufficient condition on the
existence of a desired DOF controller, which guarantees that the closed-loop system is regular, impulse free and
exponentially stable, is proposed by employing the LMI technique. Finally, some illustrative examples are given to
show the effectiveness of the proposed approach.


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## Body

## 1. Introduction

The past decades have witnessed an enormous interest in switched systems, due to their powerful ability in modeling of event-driven systems, logic-based systems, parameter- or structure-varying systems, and so forth; for details, see [1–4] and the references therein. Switched systems are a class of hybrid systems, which consist of a collection of continuous- or discrete-time subsystems and a switching rule specifying the switching between them. When focusing on the classification problems in switched systems, it is commonly recognized that there exist three basic problems [1]: (i) finding conditions for stability under arbitrary switching; (ii) identifing the limited but useful class of stabilizing switching signals, and (iii) construct a stabilizing switching signal. Many effective methods have been presented to tackle these three basic problems such as the multiple Lyapunov function approach [5], the piecewise Lyapunov function approach [6], the switched Lyapunov function approach [7], the convex combination technique [8], and the dwell time or average dwell time scheme [9–12]. On the other hand, time-delay is very common in engineering systems and is frequently a source of instability and poor performance [13]. Therefore, control of switched time-delay systems has received more and more attention in the past few years; see [14–23] and the references therein.As far as we know, singular systems (known also as descriptor, implicit or differential algebraic systems) also provide a natural framework for modeling of dynamic systems and describe a larger class of systems than the regular system models [24]. Switched singular systems have strong engineering background such as electrical networks [25], economic systems [26]. Recently, many results have been obtained in the literature for switched singular systems, such as stability and stabilization [27–30], reachability [31], H∞ control and filtering problems [32]. For switched singular time-delay (SSTD) systems, due to the coupling between the switching and the time-delay and because of the algebraic constraints in singular model, the behavior of such systems is much more complicated than that of regular switched time-delay systems or switched singular systems, and thus, to date, only a few results have been reported in the literature. In [33], the robust stability and H∞ control problems for discrete-time uncertain SSTD systems under arbitrary switching were discussed by using switched Lyapunov functions. In [34], a switching signal was constructed to guarantee the asymptotic stability of continuous-time SSTD systems. However, the aforementioned results are focused on the basic problem (i), see [33], and problem (iii), see [34], for SSTD systems. Problem (ii) is to identify stabilizing switching signals on the premise that all the individual subsystems are stable. Basically, we will find that stability is ensured if the switching is sufficiently slow [1], and it is well known that dwell time and average dwell time are two effective tools to define slow switching signals. In [9], it was shown that if all the individual subsystems are exponentially stable and that the dwell time of the switching signal is not smaller than a certain lower bound, then the switched systems is exponentially stable. This result was extended to both continuous-time switched linear time-delay systems [16] and discrete-time cases [17]. Unfortunately, so far, to the best of the authors' knowledge, the problem of solving the basic problem (ii) for SSTD systems via the dwell time or average dwell time scheme remains open and unsolved. On the other hand, the results in [33] are derived based on the state feedback controller. In fact, in many practical systems, state variables are not always available. In this case, the design of a controller that does not require the complete access to the state vector is preferable. An important example of such controller is the dynamic output feedback (DOF) controller. However, little attention has been paid to the DOF controller design problem for SSTD systems. This forms the motivation of this paper.In this paper, we are concerned with the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval time-varying delay. A full-order, dynamic, synchronously switched DOF controller is designed. First, by using the average dwell time approach and the piecewise Lyapunov function technique, a delay-range-dependent exponential admissibility criterion is derived in terms of LMIs, which guarantees the regularity, nonimpulsiveness, and exponential stability of the unforced system. A estimation of the convergence of the system is also explicitly given. Then, the corresponding solvability condition for the desired DOF controller is established by employing the LMI technique. Finally, some illustrative examples are given to show the effectiveness of the proposed approach.Notation 1.
Throughout this paper,Rn denotes the n-dimensional Euclidean space and Rn×n is the set of all n×n real matrices. P>0 (P≥0) means that matrix P is positive definite (semipositive definite). λmin(P) (λmax(P)) denotes the minimum (maximum) eigenvalue of symmetric matrix P. 𝒞n,d:=𝒞([-d,0],Rn) denotes the Banach space of continuous vector functions mapping the interval [-d,0] to Rn. Let xt∈𝒞n,d be defined by xt:=x(t+θ), θ∈[-d,0]. ∥·∥ denotes the Euclidean norm of a vector and its induced norm of a matrix and ∥xt∥d=sup-d≤θ≤0∥x(t+θ)∥. The superscript “T” represents matrix transposition, the symmetric terms in a matrix are denoted by “*”. diag{⋯} stands for a block-diagonal matrix and Sym{A} is the shorthand notation for A+AT.


## 2. Preliminaries and Problem Formulation

Consider a class of SSTD systems of the form(2.1)Eẋ(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t))+Bσ(t)u(t),y(t)=Cσ(t)x(t)+Cdσ(t)x(t-d(t)),x(t)=ϕ(t),t∈[-d1-d2,0],
where x(t)∈Rn is the system state, u(t)∈Rm is the control input, and y(t)∈Rp is the measured output; σ(t):[0,+∞)→ℐ={1,2,…,N} with integer N>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, Ci and Cdi are constant real matrices with appropriate dimensions; ϕ(t)∈𝒞n,d1+d2 is a compatible vector valued initial function; d(t) is an interval time-varying delay satisfying(2.2)d1≤d(t)≤d1+d2,ḋ(t)≤μ,
where d1≥0, d2>0 and 0≤μ<1 are constants.Remark 2.1.
Model (2.1) can describe many practical time-delay systems (e.g., chemical engineering systems, lossless transmission lines, partial element equivalent circuit, etc.) with time-varying parameters or structures, which may be caused by random failures and repairs of the components, sudden environment changes, and varying of the operating point of a system [13, 35]. In real application, the importance of the study of controller design problem for model (2.1) also arises from the extensive applications in networked control [36].SincerankE=r≤n, there exist nonsingular transformation matrices P, Q∈Rn×n such that PEQ=diag{Ir,0}. In this paper, without loss of generality, let(2.3)E=diag⁡{Ir,0}.
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 2.2 (see [16, 37]).
For the switching signalσ(t) and any delay d(t) satisfying (2.2), the unforced part of system (2.1)
(2.4)Eẋ(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t)),xt0(θ)=x(t0+θ),θ∈[-d1-d2,0]
is said to be(1)
regular ifdet(sE-Ai) is not identically zero for each σ(t)=i, i∈ℐ,(2)
impulse free 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 (2.4) satisfies ∥x(t)∥≤ιe-λ(t-t0)∥xt0∥d1+d2, for all t≥t0, where λ>0 and ι>0 are called the decay rate and decay coefficient, respectively;(4)
exponentially admissible if it is regular, impulse free and exponentially stable under the switching signalσ(t).Definition 2.3 (see [9]).
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, we choose N0=0.This paper considers the full-order DOF controller of the following form:(2.5)Ecẋc(t)=Acσ(t)xc(t)+Bcσ(t)y(t),u(t)=Ccσ(t)xc(t)+Dcσ(t)y(t),
where xc(t)∈Rn is the controller state vector, and Ec, Aci, Bci, Cci, and Dci, 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 SSTD system (2.1), identify a class of switching signal σ(t) and design a DOF controller of the form (2.5) such that the resultant closed-loop system is exponentially admissible under the switching signal σ(t).Before ending this section, we introduce the following lemma, which is essential for the development of our main results.Lemma 2.4.
For any constant matrixZ∈Rn×n, Z=ZT>0, positive scalar α, and vector function ẋ:[-τ,∞)→Rn such that the following integration is well defined, then
(2.6)eατ-1α∫t-d(t)teα(s-t)ẋT(s)ETZEẋ(s)ds≥(∫t-d(t)tEẋ(s)ds)TZ(∫t-d(t)tEẋ(s)ds),t≥0,
where 0≤d(t)≤τ.Proof.
The proof is almost the same as Lemma 1 in [20]. Using Schur complement, we have
(2.7)[eα(s-t)ẋT(s)ETZEẋ(s)ẋT(s)ET*eα(t-s)Z-1]≥0.
Integrating it from t-d(t) to t, we get
(2.8)[∫t-d(t)teα(s-t)ẋT(s)ETZEẋ(s)ds∫t-d(t)tẋT(s)ETds*(eατ-1)Z-1α]≥0.
Using Schur complement again, we find that Lemma 2.4 holds.


## 3. Main Results

In this section, we first apply the average dwell time approach to investigate the exponential admissibility for SSTD system (2.4), and give the following result.Theorem 3.1.
For prescribed scalarsα>0, d1≥0, d2>0, and 0≤μ<1, if for each i∈ℐ, there exist matrices Qil>0, Zil>0, l=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)Φi=[Φi11PiTAdi+AiTUiAdic1ETZi1E*Φi22c2ETZi2E**Φi33]<0,
where Φi11=Sym{PiTAi}+∑l=12Qil+αETPi-c1ETZi1E+AiTUiAi, Φi22=-(1-μ)e-α(d1+d2)Qi2-c2ETZi2E+AdiTUiAdi, Φi33=-e-αd1Qi1-c1ETZi1E-c2ETZi2E, c1=(αd1)/(eαd1-1), c2=(αd2)/(eαd2-1) and Ui=d12Zi1+d22eαd1Zi2. Then, system (2.4) 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
(3.3)Pi11≤βPj11,Qil≤βQjl,Zil≤βZjl,l=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; and (iii) to show the exponential stability of the algebraic subsystem.
Part (i) regularity and nonimpulsiveness. According to (2.3), for each i∈ℐ, denote(3.4)Ai=[Ai11Ai12Ai21Ai22],Zi1=[Zi111Zi112Zi112TZi122],
where Ai11∈Rr and Zi111∈Rr. From (3.2), it is easy to see that Φi11<0, i∈ℐ. Noting Qil>0 and Zil>0, l=1,2, we get Sym{PiTAi}+αETPi-c1ETZi1E<0. Substituting Pi, Ai, Zi1 and E given as (3.1), (3.4) and (2.3) into this inequality and using Schur complement, we have Sym{Ai22TPi22}<0, which implies that Ai22, i∈ℐ, is nonsingular. Then by [24] and Definition 2.2, system (2.4) is regular and impulse free.
Part (ii) exponential stability of the differential subsystem. Define the piecewise Lyapunov functional candidate for system (2.4) as follows(3.5)V(xt)=Vσ(t)(xt)=xT(t)ETPσ(t)x(t)+∫t-d1teα(s-t)xT(s)Qσ(t)1x(s)ds+∫t-d(t)teα(s-t)xT(s)Qσ(t)2x(s)ds+d1∫-d10∫t+θteα(s-t)(Eẋ(s))TZσ(t)1(Eẋ(s))dsdθ+d2∫-d1-d2-d1∫t+θteα(s-t+d1)(Eẋ(s))TZσ(t)2(Eẋ(s))dsdθ.
As mentioned earlier, the ikth subsystem is activated when t∈[tk,tk+1). Then, along the solution of system (2.4) under the switching sequence 𝒮, for t∈[tk,tk+1), we have
(3.6)V̇ik(xt)+αVik(xt)≤2xT(t)PikTEẋ(t)+xT(t)Qik1x(t)-e-αd1xT(t-d1)Qik1x(t-d1)+xT(t)Qik2x(t)-(1-μ)e-α(d1+d2)xT(t-d(t))Qik2x(t-d(t))+(Eẋ(t))T(d12Zik1+d22eαd1Zik2)(Eẋ(t))+αxT(t)ETPikx(t)-d1∫t-d1teα(s-t)(Eẋ(s))TZik1(Eẋ(s))ds-d2∫t-d(t)t-d1eα(s-t+d1)(Eẋ(s))TZik2(Eẋ(s))ds.
By replacing Eẋ(t) with Aikx(t)+Adikx(t-d(t)) and using Lemma 2.4 and Schur complement, LMI (3.2) yields
(3.7)V̇ik(xt)+αVik(xt)<0.
Integrating (3.7) from tk to tk+1 gives
(3.8)Vσ(t)(xt)=Vik(xt)≤e-α(t-tk)Vik(xtk),t∈[tk,tk+1).
Let x(t)=[x1(t)x2(t)], where x1(t)∈Rr and x2(t)∈Rn-r. From (2.3) and (3.1), it can be seen that for each i, i∈ℐ, xT(t)ETPix(t)=x1T(t)Pi11x1(t). Noting this, and using (3.3) and (3.5), at switching instant tk, we have
(3.9)Vik(xtk)≤βVσ(tk-)(xtk-)=βVik-1(xtk-),k=1,2,…,
where tk- denotes the left limitation of tk. Therefore, it follows from (3.8), (3.9) and the relation k=Nσ(t0,t)≤(t-t0)/Ta that
(3.10)Vσ(t)(xt)≤e-α(t-tk)βVσ(tk-)(xtk-)≤⋯≤e-α(t-t0)βkVσ(t0)(t0)≤e-(α-(ln⁡β)/Ta)(t-t0)Vσ(t0)(xt0).
According to (3.5) and (3.10), we obtain
(3.11)λ1‖x1(t)‖2≤Vσ(t)(xt),Vσ(t0)(xt0)≤λ2‖xt0‖d1+d22
where λ1=min∀i∈ℐλmin(Pi11), and λ2=max∀i∈ℐλmax(Pi11)+(1/α)(1-e-αd1)max∀i∈ℐλmax(Qi1)+(1/α)(1-e-α(d1+d2))max∀i∈ℐλmax(Qi2)+(d1/α2)(αd1-1+e-αd1)max∀i∈ℐ(2λmax(Zi1)(∥Ai∥+∥Adi∥))+(1/α2)(-d2+αd22eαd1+d2e-αd2)max∀i∈ℐ(2λmax(Zi2)(∥Ai∥+∥Adi∥)). Then, combining (3.10) with (3.11) yields
(3.12)‖x1(t)‖≤λ2λ1e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Part (iii) exponential stability of the algebraic subsystem. SinceAi22, i∈ℐ, is nonsingular, set Gi=[Ir-Ai12Ai22-10Ai22-1] and H=[Ir00In-r]. Then, it is easy to get(3.13)Ê:=GiEH=[Ir000],Âi:=GiAiH=[Âi110Âi21In-r],P̂i:=Gi-TPiH=[P̂i110P̂i21P̂i22],
where Âi11=Ai11-Ai12Ai22-1Ai21, Âi21=Ai22-1Ai21, P̂i11=Pi11, P̂i21=Ai12TPi11+Ai22TPi21, and P̂i22=Ai22TPi22. According to (3.13), denote
(3.14)Âdi:=GiAdiH=[Âdi11Âdi12Âdi21Âdi22],Q̂il:=HTQilH=[Q̂il11Q̂il12Q̂il21Q̂il22],Ẑil:=Gi-TZilGi-1=[Ẑil11Ẑil12Ẑil21Ẑil22],l=1,2.
Let
(3.15)ξ(t)=[ξ1(t)ξ2(t)]:=H-1x(t)=x(t),
where ξ1(t)∈Rr and ξ2(t)∈Rn-r. Then, for any fixed σ(t)=i, i∈ℐ, system (2.4) is restricted system equivalent (r.s.e.) to
(3.16)ξ̇1(t)=Âi11ξ1(t)+Âdi11ξ1(t-d(t))+Âdi12ξ2(t-d(t)),(3.17)-ξ2(t)=Âi21ξ1(t)+Âdi21ξ1(t-d(t))+Âdi22ξ2(t-d(t)).
By (3.2) and Schur complement, we have [Φi11′PiTAdi*Φi22′]<0, where Φi11′=Sym{PiTAi}+∑l=12Qil+αETPi-c1ETZi1E and Φi22′=-(1-μ)e-α(d1+d2)Qi2-c2ETZi2E. Pre- and postmultiplying this inequality by diag{HT,HT} and diag{H,H}, respectively, noting the expressions in (3.13) and (3.14), and using Schur complement, we have
(3.18)[Sym{P̂i22T}+∑l=12Q̂il22P̂i22TÂdi22*-(1-μ)e-α(d1+d2)Q̂i222]<0.
Pre- and postmultiplying this inequality by [-Âdi22TI] and its transpose, respectively, and noting Q̂i122>0 and 0≤μ<1, we obtain Âdi22TQ̂i222Âdi22-e-α(d1+d2)Q̂i222<0. Then, according to Lemma 7 in [38], we can deduce that there exist constants ℏi>1 and ηi>0 such that
(3.19)‖(e(1/2)α(d1+d2)Âdi22)l‖≤ℏie-ηil,l=0,1,…,∀i∈I.
Define(3.20)t0=t,tj=tj-1-d(tj-1),j=1,2,…,(3.21)‖Â21‖=max⁡∀i∈I‖Âi21‖,‖Âd21‖=max⁡∀i∈I‖Âdi21‖,‖Âd22‖=max⁡∀i∈I‖Âdi22‖,∀i∈I.
As mentioned earlier, under the switching sequence 𝒮, for t∈[tk,tk+1), the ikth subsystem is activated. Then, from (3.17) and (3.20), we have
(3.22)ξ2(t)-Âik21ξ1(t0)-Âdik21ξ1(t1)-Âdik22ξ2(t1).
Similarly, it can be obtained that ξ2(t1)=-Âik21ξ1(t1)-Âdik21ξ1(t2)-Âdik22ξ2(t2). Substituting this into (3.22), we get ξ2(t)=(-Âdik22)2ξ2(t2)-∑j=01(-Âdik22)j(Âik21ξ1(tj)+Âdik21ξ1(tj+1)). Continuing in the same manner and noting that tj<tj-1, then there exists a finite positive integer Tik such that
(3.23)ξ2(t)=(-Âdik22)Tikξ2(tTik)-∑jik=0Tik-1(-Âdik22)jik(Âik21ξ1(tjik)+Âdik21ξ1(tjik+1)),
where tTik∈(tk-1,tk] and tTik→tk. When t∈[tk-1,tk), the ik-1th subsystem is activated. Then, following a similar procedure as the above, there exists a finite positive integer Tik-1 such that
(3.24)ξ2(tTik)=(-Âdik-122)Tik-1ξ2(tTik+Tik-1)-∑jik-1=TikTik+Tik-1-1(-Âdik-122)jik-1-Tik(Âik-121ξ1(tjik-1)+Âdik-121ξ1(tjik-1+1)),
where tTik+Tik-1∈(tk-2,tk-1] and tTik+Tik-1→tk-1. After k-times iterative manipulations, t belongs to [t0,t1), and there exists a finite positive integer Ti0 such that
(3.25)ξ2(tTik+⋯+Ti1)=(-Âdi022)Ti0ξ2(tTik+⋯+Ti0)-∑ji0=Tik+⋯+Ti1Tik+⋯+Ti0-1(-Âdi022)ji0-Tik-⋯-Ti1(Âi021ξ1(tji0)+Âdi021ξ1(tji0+1)),
where tTik+⋯+Ti0∈(-d1-d2,t0] and tTik+⋯+Ti0→t0. By a simple induction, we have
(3.26)ξ2(t)=[∏j=0k(-Âdij22)Tij]ξ2(tTik+⋯+Ti1+Ti0)-∑jik=0Tik-1(-Âdik22)jik(Âik21ξ1(tjik)+Âdik21ξ1(tjik+1))-∑p=1k{[∏q=pk(-Âdiq22)Tiq]∑jip-1=Tik+⋯+TipTik+⋯+Tip-1-1(φ1(t)+φ2(t))},
where
(3.27)φ1(t)=(-Âdip-122)jip-1-Tik-⋯-TipÂip-121ξ1(tjip-1),φ2(t)=(-Âdip-122)jip-1-Tik-⋯-TipÂdip-121ξ1(tjip-1+1)
Therefore, from (3.15), (3.21), and (3.26), and noting tTik+⋯+Ti0∈(-d1-d2,t0], we obtain
(3.28)‖ξ2(t)‖≤Δ1+Δ2+Δ3+Δ4+Δ5,
where
(3.29)Δ1=[∏j=0k‖(Âdij22)Tij‖]‖xt0‖d1+d2,Δ2=Â21∑jik=0Tik-1‖(Âdik22)jik‖‖ξ1(tjik)‖,Δ3=Âd21∑jik=0Tik-1‖(Âdik22)jik‖‖ξ1(tjik+1)‖,Δ4=Â21∑p=1k{[∏q=pk‖(Âdiq22)Tiq‖]∑jip-1=Tik+⋯+TipTik+⋯+Tip-1-1‖φ1(t)‖},Δ5=Âd21∑p=1k{[∏q=pk‖(Âdiq22)Tiq‖]∑jip-1=Tik+⋯+TipTik+⋯+Tip-1-1‖φ2(t)‖}.
Note
(3.30)t0≥tTik+⋯+Ti0=t-∑j=0Tik+⋯+Ti0-1d(tj)≥t-(Tik+⋯+Ti0)(d1+d2).
Using (3.19) and the relation Ta≥Ta*=(lnβ)/α, the first term in (3.28) can be estimated as
(3.31)Δ1=[∏j=0k‖(e(1/2)α(d1+d2)Âdij22)Tij‖e-(1/2)α(Tik+⋯+Ti0)(d1+d2)]‖xt0‖d1+d2≤[∏j=0kℏije-ηijTij]e-(1/2)α(t-t0)‖xt0‖d1+d2≤[∏j=0kℏije-ηijTij]e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2:=χ1e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
By (3.15), (3.12), (3.21), (3.20) and (3.19), we get
(3.32)‖(Âdik22)jik‖‖ξ1(tjik)‖≤‖(Âdik22)jik‖λ2λ1e-(1/2)(α-(ln⁡β)/Ta)(tjik-t0)‖xt0‖d1+d2≤λ2λ1‖e(1/2)α(d1+d2)(Âdik22)jik‖e-(1/2)(α-(ln⁡β)/Ta)(tjik-1-t0)‖xt0‖d1+d2≤⋯≤λ2λ1‖(e(1/2)α(d1+d2)Âdik22)jik‖e-(1/2)(α-(ln⁡β)/Ta)(t0-t0)‖xt0‖d1+d2≤λ2λ1(ℏike-ηikjik)e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Then, the second term in (3.28) can be estimated as
(3.33)Δ2≤Â21λ2λ1[∑jik=0Tik-1ℏike-ηikjik]e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2≤ℏikÂ21λ2λ1eηikeηik-1e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2:=χ2e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Similarly, the third term in (3.28) can be bounded by
(3.34)Δ3≤ℏike(1/2)α(d1+d2)Âd21λ2λ1eηikeηik-1e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2:=χ3e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
In addition, following a similar deduction as that in (3.32), we obtain
(3.35)‖φ1(t)‖≤λ2λ1(e(1/2)α(d1+d2))Tik+⋯+Tip(ℏip-1e-ηip-1(jip-1-Tik-⋯-Tip))e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Then, considering this and (3.19), the fourth term in (3.28) can be estimated as
(3.36)Δ4≤Â21λ2λ1∑p=1k{[∏q=pk‖(e(1/2)α(d1+d2)Âdiq22)Tiq‖]∑jip-1=Tik+⋯+TipTik+⋯+Tip-1-1ℏip-1e-ηip-1(jip-1-Tik-⋯-Tip)}×e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2≤Â21λ2λ1∑p=1k{ℏip-1[∏q=pkℏiqe-ηiqTiq]eηip-1eηip-1-1}e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2:=χ4e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Similarly, the fifth term in (3.28) can be bounded by
(3.37)Δ5≤e(1/2)α(d1+d2)Âd21λ2λ1∑p=1k{ℏip-1[∏q=pkℏiqe-ηiqTiq]eηip-1eηip-1-1}e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2:=χ5e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Therefore, using (3.31) and (3.33)–(3.37), ∥ξ2(t)∥ can be estimated as
(3.38)‖ξ2(t)‖≤(χ1+χ2+χ3+χ4+χ5)e-(1/2)(α-(ln⁡β)/Ta)(t-t0)‖xt0‖d1+d2.
Combining (3.15), (3.12) and (3.38) yields that system (2.4) is exponentially stable for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α. This completes the proof.Remark 3.2.
In terms of LMIs, Theorem3.1 presents a delay-range-dependent exponential admissibility condition for the switched singular systems with interval time-varying delay. It is noted that this condition is obtained by using the integral inequality (Lemma 2.4); no additional free-weighting matrices are introduced to deal with the cross-term. Therefore, the condition proposed here involves much less decision variables than those obtained by using the free-weighting matrices method [16, 19, 21, 22] if the same Lyapunov function is chosen.Remark 3.3.
Equation (3.26) plays an important role in analyzing the exponential stability of the algebraic subsystem, which can be seen as a generalization of the iterative equation in [37] for nonswitched singular time-delay system to switched case.Remark 3.4.
Ifβ=1 in Ta≥Ta*=(lnβ)/α, which leads to Pi11≡Pj11, Qil≡Qjl, Zil≡Zjl, l=1,2, for all i,j∈ℐ, and Ta*=0, then system (2.4) possesses a common Lyapunov function and the switching signals can be arbitrary.In the following, we are to deal with the design problem of DOF controller for the SSTD system (2.1). Applying the DOF controller (2.5) to system (2.1) gives the following closed-loop system(3.39)E¯η̇(t)=A¯σ(t)η(t)+A¯dσ(t)η(t-d(t)),
where η(t)=[xT(t)xcT(t)]T, and(3.40)E¯=[E00Ec],A¯σ(t)=[Aσ(t)+Bσ(t)Dcσ(t)Cσ(t)Bσ(t)Ccσ(t)Bcσ(t)Cσ(t)Acσ(t)]A¯dσ(t)=[Adσ(t)+Bσ(t)Dcσ(t)Cdσ(t)0Bcσ(t)Cdσ(t)0].
The following Theorem presents a sufficient condition for solvability of the DOF controller design problem for system (2.1).Theorem 3.5.
For prescribed scalarsα>0, γ>0, d1≥0, d2>0 and 0≤μ<1, if for each i∈ℐ, and given scalars ξi1 and ξi2, there exist matrices Yi, Âci, B̂ci, Ĉci, D̂ci, Qil11, Qil12, Qil22, Zil11, Zil12, Zil22, l=1,2, and Ri and Ui of the following form
(3.41)Ri=[Ri110Ri21Ri22],Ui=[Ui110Ui21Ui22]
with Ri11∈Rr×r, Ui11∈Rr×r, Ri11>0, Ui11>0, Ri22∈R(n-r)×(n-r), Ui22∈R(n-r)×(n-r), and Ri22 and Ui22 being invertible, such that
(3.42)[Υi11Υi12Υi13Υi14Υi150c1ETZi111Ec1ETZi112E*Υi22Υi23Υi24Υi250c1ETZi112TEc1ETZi122E**Υi33Υi34Υi35000***Υi44Υi45000****Υi55Υi56c2ETZi211Ec2ETZi212E*****Υi66c2ETZi212TEc2ETZi222E******Υi77Υi78*******Υi88]<0,(3.43)Q¯il=[Qil11Qil12*Qil22]>0,Z¯il=[Zil11Zil12*Zil22]>0,l=1,2,
where Υi11=Sym{ξi1Ai+ξi1BiD̂ciCi+B̂ciCi}+∑l=12Qil11+αETRi-c1ETZi111E, Υi12=Âci+ξi1BiĈci+ξi2AiT+ξi2CiTD̂ciTBiT+CiTB̂ciT+∑l=12Qil12-c1ETZi112E, Υi13=-ξi1I+RiT+ξi1AiT+ξi1CiTD̂ciTBiT+CiTB̂ciT, Υi14=-Yi+ξi2AiT+ξi2CiTD̂ciTBiT+CiTB̂ciT, Υi15=Υi35=ξi1Adi+ξi1BiD̂ciCdi+B̂ciCdi, Υi22=Sym{Âci+ξi2BiĈci}+∑l=12Qil22+αETUi-c1ETZi122E, Υi23=-ξi2I+ÂciT+ξi1ĈciTBiT, Υi24=-Yi+ÂciT+ξi2ĈciTBiT+UiT, Υi25=Υi45=ξi2Adi+ξi2BiD̂ciCdi+B̂ciCdi, Υi33=-2ξi1I+d12Zi111+d22eαd1Zi211, Υi34=-ξi2I-Yi+d12Zi112+d22eαd1Zi212, Υi44=-Sym{Yi}+d12Zi122+d22eαd1Zi222, Υi55=-(1-μ)e-α(d1+d2)Qi211-c2ETZi211E, Υi56=-(1-μ)e-α(d1+d2)Qi212-c2ETZi212E, Υi66=-(1-μ)e-α(d1+d2)Qi222-c2ETZi222E, Υi77=-(1-μ)e-αd1Qi111-c1ETZi111E-c2ETZi211E, Υi78=-(1-μ)e-αd1Qi112-c1ETZi112E-c2ETZi212E, Υi88=-(1-μ)e-αd1Qi122-c1ETZi122E-c2ETZi222E. Then, there exists a DOF controller in the form of (2.5), such that system (3.39) is exponentially admissible for any switching sequence 𝒮 with average dwell time Ta≥Ta*=lnβ/α, where β≥1 satisfies
(3.44)Ri11≤βRj11,Ui11≤βUj11,Q¯il≤βQ¯jl,Z¯il≤βZ¯jl,l=1,2,∀i,j∈I.
Moreover, a desired DOF controller realisation is given by
(3.45)Aci=Yi-1Âci,Bci=Yi-1B̂ci,Cci=Ĉci,Dci=D̂ci,∀i∈I.Proof.
From Theorem3.1, we known that system (3.39) is exponentially admissible for any switching sequence 𝒮 with average dwell time Ta≥Ta*=(lnβ)/α, where β≥1 satisfying (3.3), if for each i∈ℐ, there exist matrices Q¯il>0, Z¯il>0, l=1,2, and P¯i with the form of (3.1) such that the inequality (3.2) with E, Ai and Adi instead of E¯, A¯i and C¯i, respectively, holds. By decomposing Φi in (3.2), we obtain that for each i∈ℐ(3.46)Φi=ΠiΛiΠiT<0,
where J¯i is any invertible matrix with compatible dimension, and
(3.47)Πi=[I00A¯iA¯di00I000I]T,Λi=[Λi11P¯iT-J¯i+A¯iTJ¯iTJ¯iA¯dic1E¯TZ¯i1E¯*-J¯i-J¯iT+U¯iJ¯iA¯di0**Λi33c2E¯TZ¯i2E¯***Λi44]
with Λi11=Sym{J¯iA¯i}+∑l=12Q¯il+αE¯iTP¯i-c1E¯TZ¯i1E¯, Λi33=-(1-μ)e-α(d1+d2)Q¯i2-c2E¯TZ¯i2E¯, and Λi44=-e-αd1Q¯i1-c1E¯TZ¯i1E¯-c2E¯TZ¯i2E¯. Hence, Φi<0 holds if
(3.48)Λi<0.
LetE¯=diag{E,E}. For each i∈ℐ, define(3.49)P¯i=diag⁡{Ri,Ui}.
By (2.3) and (3.41), we have
(3.50)ETRi=RiTE≥0,ETUi=UiTE≥0,
which combining (3.49) yields
(3.51)E¯TP¯i=P¯iTE¯≥0.
Then, from (3.41), (3.44) and (3.51), it can be deduced that
(3.52)E¯TP¯i≤βE¯TP¯j,∀i,j∈I.
Denote(3.53)J¯i=[ξi1IYiξi2IYi].
Substituting (3.43), (3.49) and (3.53) into (3.48), and defining
(3.54)Âci=YiAci,B̂ci=YiBci,Ĉci=Cci,D̂ci=Dci,∀i∈I
we can easily obtain (3.42). This completes the proof.Remark 3.6.
Note that conditionΦi of Theorem 3.1 involves some product terms between the Lyapunov matrices and the system matrices, which complicates the DOF control synthesis problem. To solve this problem, in the proof of Theorem 3.5, we have made a decoupling between the Lyapunov matrices and the system matrices by introducing a slack matrix J¯i in condition Λi. Compared with the variable change method used in [39, 40], the decoupling technique proposed here simplifies the DOF controller design problem greatly, which decreases the conservatism in some sense.Remark 3.7.
Scalarsξi1 and ξi2, i∈ℐ, in Theorem 3.5 are tuning parameters which need to be specified first. The optimal values of these parameters can be found by applying some optimization algorithms such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm [41].Remark 3.8.
It is noted that in this paper, the derivative matrixE is assumed to be switch-mode-independent. If E is also switch-mode-dependent, then E is changed to Ei, i∈ℐ. In this case, the transformation matrices P and Q should become Pi and Qi so that PiEiQi=diag{Iri,0}, and the state of the transformed system becomes x̃(t)=Qi-1(t)=[x̃i1T(t)x̃i2T(t)]T with x̃i1T(t)∈Rri and x̃i1T(t)∈Rn-ri, which implies that there does not exist one common state space coordinate basis for all subsystems. Then, some assumptions for Ei (e.g., Ei, i∈ℐ, have the same right zero subspace [27]) should be made so that Qi remains the same; in this case, the method presented in this paper is also valid. How to investigate the general SSTD system with E being switch-mode-dependent is an interesting problem for future work via other approaches.


## 4. Numerical Examples

In this section, some numerical examples are presented to demonstrate the effectiveness of the proposed methods.Example 4.1.
Consider the switched system (2.4) with E=I, N=2 (e.g., there are two subsystems) and the following parameters, which are borrowed from [21]:
(4.1)A1=[-200-0.9],Ad1=[-100-1],A2=[-200-0.7],Ad2=[-10-1-1].
For μ=0.4, α=0.5 and β=1.1, employing the LMIs in [21, 22] and those in Theorem 3.1 yields an allowable upper bound d̃2 (in this paper d̃2=d1+d2) of the delay d(t) that guarantees the stability of system (2.4). Table 1 shows the values of the upper bound for various d1 and the number of involved variables by using different methods. It is easily seen from Table 1 that Theorem 3.1 of this paper not only provides better results than those criteria in [21, 22] but also reduces the computational overhead to some extent.Table 1
Comparison of allowable upper boundd̃2 for different d1 in Example 4.1.
Methodsd̃2Number of variablesLemma 1 [21]1.130 (d1=0.1)1.099 (d1=0.3)1.084 (d1=0.7)84Theorem 1 [22]1.130 (d1=0.1)1.099 (d1=0.3)1.084 (d1=0.7)84Theorem3.11.134 (d1=0.1)1.133 (d1=0.3)1.133 (d1=0.7)30Example 4.2.
Consider the switched system (2.4) with N=2 and the related parameters are given as follows:
(4.2)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=0.2, μ=0.4 and α=0.5. It can be verified that both of the above two subsystems are stable. Let β=1; it can be found that there is no feasible solution to this case, which implies that there is no common Lyapunov function for the above two subsystems (see Remark 3.4). Now, we consider the average dwell time scheme, and set β=1.2. Solving the LMIs (3.2) gives the following solutions:
(4.3)P1=[19.7719017.9611819.3011],Q11=[0.7438-1.2478-1.247868.3107],Q12=103×[0.00020.01020.01021.0735],Z11=[624.44250.88100.8810379.5848],Z12=[332.4246-0.4902-0.4902373.4311],P2=[23.37110-26.9375642.4532],Q21=[0.7501-0.8269-0.826966.5062],Q22=[0.22018.93168.9316917.3497],Z21=[560.83840.76430.7643379.5975],Z22=[294.2097-0.4003-0.4003373.4858]
which means that the above switched system is exponentially admissible. Moreover, by further analysis, it can be found that the allowable minimum of β is βmin=1.046 when α=0.5; in this case Ta*=(lnβmin)/α=0.0899.Example 4.3.
Consider the switched system (2.1) with N=2 and
(4.4)E=[1000],A1=[0.901-5],Ad1=[0.50.110.1],B1=[0.30.1],C1=[0.30.1],Cd1=[0.10.1],A2=[0.50.12-5],Ad2=[0.20.51.50.1],B2=[0.40.1],C2=[0.10.3],Cd2=[0.10.1],
and d(t)=0.3+0.2sin(1.5t). A simple calculation yields d1=0.1, d2=0.4 and μ=0.3. By simulation, it can be checked that both of the above two subsystems with u(t)=0 are 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 a DOF control u(t) in the form of (2.5) such that the closed-loop system is exponentially admissible.Figure 1
State trajectories of the open-loop subsystem 1.Figure 2
State trajectories of the open-loop subsystem 2.Setα=0.5, β=1.05 (thus Ta≥Ta*=(lnβ)/α=0.0976), and choose ξ11=0.9255, ξ12=0.0067, ξ13=0.9811, ξ14=0.0016. Solving the LMIs (3.41)–(3.44), the corresponding gain matrices of the DOF controller are computed as(4.5)Ac1=[-26.3829-0.4700-0.8920-0.8529],Ac2=[-67.74800.64239.5961-0.8803],Bc1=[0.6811-0.2272],Bc2=[4.0654-1.1668],Cc1=[33.63910.1324],Cc2=[23.4740-0.5180],Dc1=-22.4825,Dc2=-17.9093
To show the effectiveness of the obtained DOF controller, giving a random switching signal with the average dwell time Ta≥0.13 as shown in Figure 3, we get the state trajectories of the closed-loop system as shown in Figure 4, for the given initial condition ϕ(t)=[12]T, t∈[-0.5,0]. It is clear that the designed controller is feasible and ensures the stability of the closed-loop system despite the switching and the time-varying delay.Figure 3
Switching signal with the average dwell timeTa>0.13.Figure 4
State trajectories of the closed-loop system under DOF control.


## 5. Conclusions

In this paper, the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval time-varying delay have been investigated. A class of switching signals has been identified for the switched singular time-delay systems to be exponentially admissible under the average dwell time scheme. The DOF controller has been designed, and the corresponding solvability condition has been established by using the LMI technique. Numerical examples have been provided to illustrate the effectiveness of the proposed methods.



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*Source: 680382-2010-12-02.xml*