# Novel Stabilization Conditions for Uncertain Singular Systems with Time-Varying Delay

Authors: Yuanchun Ding; Falu Weng; Ji Ge; Qi Liang; Xueyi Zhi
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The problem of delay-dependent robust stabilization for continuously singular time-varying delay systems with norm-bounded uncertainties is investigated in this paper. First, based on some mathematical transform, the uncertain singular system is described in a form which involves the time-delay integral items. Then, in terms of the delay-range-dependent Lyapunov functional and the LMI technique, the improved delay-dependent LMIs-based conditions are established for the uncertain singular systems with time-varying delay to be regular, causal, and stable. Furthermore, by solving these LMIs, an explicit expression for the desired state feedback control law can be obtained; thus, the regularity, causality, and stability of the closed-loop system are guaranteed. In the end, numerical examples are given to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

During the past several decades, singular systems, which are known as descriptor systems, implicit systems, and differential-algebraic systems, received a considerable attention because of their applications in many areas, such as engineering systems, social systems, economic systems, network analysis, biological systems, and time-series analysis [1, 2]. As we all know, it is required to consider not only stability but also regularity and absence of impulses (for continuous singular systems) or causality (for discrete singular systems) simultaneously for singular systems; thus, the study on singular systems is much more complicated than that on the regular ones [3, 4]. Recently, many scholars have applied themselves to the research of singular system and many stability and stabilization conditions have been established for singular systems; see, for example, [5–11] and the references therein.On the other hand, much attention has been paid to the study of time-delay systems in recent years, because time delays inevitably exist in a variety of practical systems, such as chemical processes, nuclear reactors, and biological systems, and lead to the instability and poor performance of systems [12–15]. Generally speaking, the existing results can be classified into two types: delay-independent results (see, e.g., [16, 17] and the references therein) and delay-dependent results (see, e.g., [18, 19] and the references therein). Furthermore, the delay-independent case is regarded as more conservative than the delay-dependent case, especially when the time delay is comparatively small. Thus, the delay-dependent stability and stabilization conditions for singular time-delay systems have received increasing attention during the past years. For example, by utilizing model transformation and bounding technique for cross-terms, Zhu et al. [20, 21] investigated the delay-dependent robust stabilization problem for uncertain singular time-delay systems. References [22, 23] also discussed the problem, and neither model transformation nor bounding technique for cross-terms is needed in the development of the results. Based on an improved Lyapunov functional, which includes some nonpositive items, Weng and Mao [24] discussed the delay-dependent robust stability and stabilization for uncertain singular time-delay systems, and some LMIs-based results were obtained. However, in the practical systems, most of those delays are time-varying because the external perturbances and uncertainties are always existing [25]. Thus, proposing some time-varying delay-tolerant results for the singular system is obviously more meaningful. In terms of Lyapunov stability theory and LMI technique, some results about the admissibility and dissipativity for discrete-time singular systems with mixed time-varying delays were proposed in [26]. Based on the probability idea, Weng and Mao [27] presented some delay-range-dependent and delay-distribution-independent stability criteria for discrete-time singular systems with time-varying delay, and several sufficient results were obtained. However, to the best of our knowledge, the stabilization conditions for singular time-varying delay systems still have not been fully investigated, and there is still much room for improvement.This paper is concerned with the problem of robust stabilization for continuously singular time-varying delay systems with norm-bounded and time-varying parametric uncertainties. The focus of this paper is to design a state feedback controller such that the closed-loop system is regular, causal, and stable for all admissible uncertainties. The proposed sufficient robust stabilization conditions of the considered system are described in terms of strict LMIs, which are formulated in terms of all the coefficient matrices of the original system. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed methods.Notation 1.
Throughout this paper, for real matricesX and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is semipositive definite (resp., positive definite). I is the identity matrix with appropriate dimensions; a superscript “T” represents transpose. ∥x∥ refers to the Euclidean norm of the vector x. For a symmetric matrix, * denotes the symmetric terms. M+MT is denoted as {M}H for simplicity.


## 2. Problem Formulation and Dynamic Models

Consider an uncertain singular system with time-varying delay described by Wu and Zhou [28]:
(1)Ex˙(t)=(A+ΔA)x(t)+(Ad+ΔAd)x(t-d(t))+Bu(t),x(t)=Φ(t),t∈[-d2,0],
where x(t)∈Rn is the state variable and d(t) is a time-varying delay satisfying 0<d1≤d(t)≤d2 and d˙(t)≤μ. Φ(t) is a compatible initial value at t. The matrix E∈Rn×n may be singular and rank⁡E≤n is assumed.A, Ad, and B are real constant matrices with appropriate dimensions. ΔA andΔAd are norm-bounded parametric matrices and are assumed to be of the following form:
(2)[ΔAΔAd]=MF(t)[N1N2],
where F(t)∈Rk×s is an unknown parameter matrix satisfying F(t)TF(t)≤I.  M, N1, and N2 are known constant matrices with appropriate dimensions.The nominal unforced singular time-delay system of (1) can be described as
(3)Ex˙(t)=Ax(t)+Adx(t-d(t)),x(t)=Φ(t),t∈[-d2,0].The following definitions and lemmas will be used in the proof of the main results.Definition 1 (see [29]).
(i) The pair(E,A)is regular if det⁡(SE-A)is not identically zero; (ii) the pair(E,A)is said to be impulse free if it is regular and deg⁡{det⁡(SE-A)}=rank⁡E.Definition 2 (see [29]).
(i) The system (3) is said to be regular and impulse free if d(t) satisfies 0<d1≤d(t)≤d2; the pair (E,A) is regular and impulse free. (ii) The system (3) is said to be stable if, for any ε>0, there exists a scalar μ(ε), such that, for any compatible initial conditions sup-d2≤t≤0∥Φ(t)∥2<μ(ε), when t>0, the solution x(t) of the system (3) satisfies ∥x(t)∥≤ε. Furthermore, lim⁡t→0x(t)=0. (iii) The system (3) is said to be admissible if it is regular, impulse free, and stable.Definition 3.
(i) The uncertain singular time-varying delay system (1) is said to be robustly admissible if the system (1) with u(t)=0 is regular, impulse free, and stable for all admissible uncertainties satisfying (2) and any time-delay d(t) satisfying 0<d1≤d(t)≤d2. (ii) The singular time-varying delay system (1) is said to be stabilizable if there exists state feedback controlleru(t)=Kx(t) such that the closed-loop system is admissible for any time delay d(t) satisfying 0<d1≤d(t)≤d2. (iii) The uncertain singular time-varying delay system (1) is said to be robustly stabilizable if there exists state feedback controller u(t)=Kx(t) such that the closed-loop system is robustly admissible for all admissible uncertainties satisfying (2) and any time-delay d(t) satisfying 0<d1≤d(t)≤d2.
After some mathematical transform, the systems (1) and (3) can be described in the following forms:
(4)Ex˙(t)=(A+ΔA+Ad+ΔAd)x(t)-(Ad+ΔAd)∑i=0m-1∫t-((i+1)/m)d1t-(i/m)d1x˙(s)ds-(Ad+ΔAd)∫t-d(t)t-d1x˙(s)ds,x(t)=Φ(t),t∈[-d2,0],(5)Ex˙(t)=(A+Ad)x(t)-Ad∑i=0m-1∫t-((i+1)/m)d1t-(i/m)d1x˙(s)ds-Ad∫t-d(t)t-d1x˙(s)ds,x(t)=Φ(t),t∈[-d2,0].
For description in brevity, we define γ=d1/m  and  d12=d2-d1. This section is concluded by presenting a lemma, which will be used in the proof of our main results.Lemma 4 (see [30, 31]).
Given matricesχ, μ, and ν with appropriate dimensions and withχsymmetrical, then χ+μF(t)ν+νTF(t)TμT<0 holds for any F(t)∈Rk×s satisfying F(t)TF(t)≤I, if and only if there exists a scalar σ>0such that χ+σμμT+σ-1νTν<0.


## 3. Main Results

In this section, the delay-dependent conditions for system (1) to be stabilizable and robustly stabilizable are presented. As a basis, we first study system (5) and obtain the following Theorem 5.Theorem 5.
For the prescribed scalars satisfying0<d1≤d2, the singular time-delay system (5) is admissible for any time-delay d(t) satisfying 0<d1≤d(t)≤d2 if there exist positive symmetric matrices P, Ui(i=1,2,3,4), Q1, Q2, matrices Hi(i=1,2), Si(i=1,2,3,4,5,6,7), Gi(i=1,2,3,4,5,6,7,8), and Li(i=1,2,3,4,5) satisfying the following LMIs:(6)Ψi=[Ψ111Ψ112G1-G1Ψ115Ψ116-L1+L2TL5T*Ψi22G7-G7Ψ125Ψ126-L2T-L4-L5T**Ψ133Ψ134Ψ135Ψ136G2TG8T***Ψ44Ψ145Ψ146-G2T-G8T****Ψ155Ψ156LS2T-L3LS7T*****Ψ166-G2T-L2TΨ168******Ψi77Ψi78*******Ψi88]<0,i=1,2,where L∈Rn×(n-r) is any matrix with full columns and satisfies ETL=0 and
(7)Ψ111=U1+U2+U3+U4+{L1+H1(A+Ad)}H,Ψ112=-H1Ad-L1+L4T,Ψ116=-mH1Ad-G1-L1,Ψ115=ETP-H1+(H2(A+Ad))T+S1LT+L3T,Ψ122=-3ETQ2E-L4-L4T,Ψ126=-G7-L4,Ψ125=-(H2Ad)T+S6LT-L3T,Ψ133=-U1+{G3}H,Ψ134=-G3+G4T,Ψ135=S3LT+G5T,Ψ136=-G3+G6T,Ψ144=-U2-{G4}H,Ψ145=S4LT-G5T,Ψ146=-G4-G6T,Ψ155=γ2Q1+d122Q2-{H2}H,Ψ156=-mH2Ad+LS5T-G5-L3,Ψ166=-ETQ1E-{G6}H,Ψ168=-G8T-L5T,Ψ178=-L5T+ETQ2E,Ψ177=-(1-μ)U3-{L2}H-ETQ2E,Ψ188=-U4-ETQ2E,Ψ222=-ETQ2E-{L4}H,Ψ277=-(1-μ)U3-{L2}H-3ETQ2E,Ψ278=-L5T+3ETQ2E,Ψ288=-U4-3ETQ2E.Proof.
Under the conditions of Theorem5, it is first shown that the system (5) is regular and impulse free for any time-delay d(t) satisfying 0<d1≤d(t)≤d2. Define H=[InIn00AT000]. Then, by pre- and postmultiplying (6) by H and HT, respectively, it is possible to obtain
(8)-ETQ2E+ATPE+ETPA+ATL(S1+S6)T+(S1+S6)LTA<0.
Since rank⁡E=r≤n, there exist two nonsingular matrices M~ and N~ such that M~EN~=[Ir000]. Accordingly, denote M~AN~=[A1A2A3A4], N~T(S1+S6)=[S11S12], M~-TL=[0L-1], where L-1∈R(n-r)×(n-r) is any nonsingular matrix and Ir∈Rr×r is  an identity matrix. Then, by pre- and postmultiplying inequalities (8) by N~Tand N~, respectively, it is possible to obtain
(9)[⋄⋄*S12L1TA4+A4TL1S12T]<0.
Here, “⋄” representing the matrix blocks are irrelevant to the following discussion; the real expression of these two variables is omitted here. From (9), it is possible to obtain that
(10)S12L1TA4+A4TL1S12T<0.
It can be shown that A4 is nonsingular. Thus, the pair (E,A) is regular and impulse free [29]; that is to say, system (5) is regular and impulse free.
Then, we are in a position to show that system (5) is stable under the conditions of Theorem 5. Choose a Lyapunov-Krasovskii functional candidate as
(11)V(t)=V1(t)+V2(t)+V3(t),
where
(12)V1(t)=x(t)TETPEx(t)+∑i=0m-1∫t-iγtx(s)TU1x(s)ds+∑i=0m-1∫t-(i+1)γtx(s)TU2x(s)ds+∫t-dttx(s)TU3x(s)ds+∫t-d2tx(s)TU4x(s)ds,V2(t)=γ∑i=0m-1∫-γ0∫t-iγ+εt-iγx(s)TQ1x(s)dsdε,V3(t)=d12∫-d2-d1∫t+εtx˙(s)TETQ2Ex˙(s)dsdε.
The derivative of V(t) along the trajectories of (5) satisfies
(13)V˙(t)=V˙1(t)+V˙2(t)+V˙3(t),
where
(14)V˙1(t)=x˙(t)TETPEx(t)+mx(t)TU1x(t)-∑i=0m-1x(t-iγ)TU1x(t-iγ)+mx(t)TU2x(t)-∑i=0m-1x(t-(i+1)γ)TU2x(t-(i+1)γ)+x(t)TU3x(t)-(1-μ)x(t-d(t))TU3x(t-d(t))+x(t)TU4x(t)-x(t-d2)TU4x(t-d2),V˙2(t)≤γ2x˙(t)TETQ1Ex˙(t)-∑i=0m-1∫t-(i+1)γt-iγx˙(s)TdsETQ1E∫t-(i+1)γt-iγx˙(s)ds,V˙3(t)=d122x˙(t)TETQ2Ex˙(t)-d12∫t-d(t)t-d1x˙(s)TETQ2Ex˙(s)ds-d12∫t-d2t-d(t)x˙(s)TETQ2Ex˙(s)ds≤d122x˙(t)TETQ2Ex˙(t)-d12d(t)-d1∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-d12d2-d(t)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-dtx˙(s)ds.
We have -d12/(d(t)-d1)-d12/(d2-d(t))≤4; thus, when d1≤d(t)≤(d1+d2)/2, it is easy to obtain
(15)-d12d(t)-d1∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds≤(-4+d12d2-d(t))∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds.
By defining θ1(t)=2-d12/(d2-d(t)) and according to d1≤d(t)≤(d1+d2)/2, we have 0≤θ1(t)≤1. Thus, based on the convex theory, it is easy to obtain that
(16)(-4+d12d2-d(t))∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-d12d2-d(t)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds=(12+θ1(t)2)×(-3∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds)+(1-(12+θ1(t)2))×(-∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-3∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds).
When (d1+d2)/2≤d(t)≤d2, we have
(17)-d12d2-d(t)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds≤(-4+d12d(t)-d1)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds.
By defining θ2(t)=2-d12/(d(t)-d1) and according to (d1+d2)/2≤d(t)≤d2, we have 0≤θ2(t)≤1. Based on convex theory, we achieve
(18)-d12d(t)-d1∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds+(-4+d12d(t)-d1)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds=(12+θ2(t)2)×(-∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-3∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds)+(1-(12+θ2(t)2))×(-3∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds).
It is easy to obtain 0≤(1+θ1(t))/2≤1 and 0≤(1+θ2(t))/2≤1. Then, based on the analysis mentioned above, we can obtain that when θ(t) satisfies 0≤θ(t)≤1, there exists
(19)-d12d(t)-d1∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-d12d2-d(t)∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds≤θ(t)(-3∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds)+(1-θ(t))×(-∫t-d(t)t-d1x˙(s)TdsETQ2E∫t-d(t)t-d1x˙(s)ds-3∫t-d2t-d(t)x˙(s)TdsETQ2E∫t-d2t-d(t)x˙(s)ds).
By considering (5), it is obvious that
(20)(x(t)TH1+x˙(t)TETH2)×∑i=0m-1((A+Ad)x(t)-mAd∫t-((i+1)/m)d1t-(i/m)d1x˙(s)ds-Ad∫t-d(t)t-d1x˙(s)ds-Ex˙(t)∫t-((i+1)/m)d1t-(i/m)d1x˙(s)ds)=0.
Noting ETL=0, we obtain
(21)2∑i=0m-1(∫t-dtt-d1x˙(s)Tdsx(t)TS1+xt-d(t)TS2+xt-iγTS3+xt-(i+1)γTS4+∫t-(i+1)γt-iγx˙(s)TdsS5+∫t-dtt-d1x˙(s)TdsS6+xt-d2TS7)LTEx˙(t)=0.
Furthermore, according to the free-weighting-matrix method, we have
(22)(∫t-dtt-d1x˙(s)Tdsx(t)TG1+xt-d(t)TG2+xt-iγTG3+xt-(i+1)γTG4+x˙(t)TETG5+∫t-(i+1)γt-iγx˙(s)TdsG6+∫t-dtt-d1x˙(s)TdsG7+xt-d2TG8)×(x(t-iγ)-x(t-(i+1)γ)-∫t-(i+1)γt-iγx˙(s)ds)=0,(x(t)TL1+xt-d(t)TL2+x˙(t)TETL3+∫t-dtt-d1x˙(s)TdsL4+xt-d2TL5)×(x(t)-x(t-dt)-∫t-dtt-d1x˙(s)ds-∑i=0m-1∫t-(i+1)γt-iγx˙(s)ds)=0.
Then, combining manipulations (13)–(22) yields
(23)V˙(t)≤∑i=0m-1(θ(t)ξ(t)TΨ1ξ(t)+(1-θ(t))ξ(t)TΨ2ξ(t)),
where(24)ξ(t)=[x(t)T∫t-d(t)t-d1x˙(s)Tdsxt-iγTxt-(i+1)γTx˙T(t)ET∫t-(i+1)γt-iγx˙(s)Tdsx(t-dt)Tx(t-d2)T]T.Then, we can obtain V˙(t)<0 from (6). Thus, we can deduce that
(25)V˙(t)≤-λ2∥ζ(t)∥2≤-λ2∥x(t)∥2,
where λ2=-max⁡{λmax⁡(Ψ1),λmax⁡(Ψ2)}>0. Therefore, the system (5) is stable based on Definition (2). This completes the proof.Theorem 6.
For the prescribed scalars satisfying0<d1≤d2, the singular system (1) is stabilizable for any time-delay d(t) satisfying 0<d1≤d(t)≤d2 if there exist positive symmetric matrices P, Ui(i=1,2,3,4), Q1, Q2 matrices H1=Z-1[H-1110H-121H-22]Z, H2=βZ-1[β-1H-2110β-1H-221H-22]Z, Si(i=1,2,3,4,5,6,7), Gi(i=1,2,3,4,5,6,7,8), Li(i=1,2,3,4,5), T∈Rp×n, and scalar β satisfying the following LMIs:(26)Ψ~i=[Ψ~111Ψ112G1-G1Ψ~115Ψ116-L1+L2TL5T*Ψi22G7-G7Ψ125Ψ126-L2T-L4-L5T**Ψ133Ψ134Ψ135Ψ136G2TΨ138***Ψ44Ψ145Ψ146-G2T-G8T****Ψ155Ψ156LS2T-L3LS7T*****Ψ166-G2T-L2TΨ168******Ψi77Ψi78*******Ψi88]<0,i=1,2,where L∈Rn×(n-r)  is any matrix with full columns and satisfies ETL=0, H-22∈Rp×p is any nonsingular matrix, Z∈Rn×nis any nonsingular constant matrix satisfying ZB=[0B1T]T, where B1∈Rp×p is nonsingular, and Ψ~111=U1+U2+U3+U4+{L1+H1(A+Ad)+BT}H, Ψ~115=(H2(A+Ad)+βBT)T+ETP-H1+S1LT+L3T. Then, a suitable state feedback controller is described as K=(H-22B1)-1B1T.Proof.
By replacingA with A+BK and choosing K=(H-22B1)-1B1T, H1=Z-1[H-1110H-121H-22]Z, and H2=βZ-1[β-1H-2110β-1H-221H-22]Z, it is easy to obtain H1BK=BT and H2BK=βBT. Thus, we can obtain (6) from LMIs (26). This completes the proof.Theorem 7.
For the prescribed scalars satisfying0<d1≤d2, the singular system (1) is robustly stabilizable for any time-delay d(t) satisfying 0<d1≤d(t)≤d2 if there exist positive symmetric matrices P, Ui(i=1,2,3,4), Q1, Q2, matrices H1=Z-1[H-1110H-121H-22]Z, H2=βZ-1[β-1H-2110β-1H-221H-22]Z, Si(i=1,2,3,4,5,6,7), Gi(i=1,2,3,4,5,6,7,8), Li(i=1,2,3,4,5), T∈Rp×n, and scalars β, λ>0 satisfying the following LMIs:
(27)[Ψ~iΠ~2TλΛ~2T*-λI0**-λI]<0,i=1,2,
where Ψ~1,Ψ~2, and L follow the same definitions as in Theorem 6 and H-22∈Rp×p is any nonsingular matrix, Z∈Rn×n is any nonsingular constant matrix satisfying ZB=[0B1T]T, where B1∈Rp×p is nonsingular, and Π~2=[MTH1T000MTH2T000], Λ~2=[N1+N2-N2000-mN200]. Then, a suitable state feedback control law is described as K=(H-22B1)-1B1T.Proof.
ReplacingA and Ad with A+ΔA and Ad+ΔAd, respectively, (26) can be expressed as
(28)Ψ~i+Λ~2TF(t)Π~2+Π~2TF(t)TΛ~2<0,i=1,2.
By Lemma 4, LMIs (28) hold for any F(t) satisfying F(t)TF(t)≤I if and only if there exists scalar λ>0 such that
(29)Ψ~i+λΛ~2TΛ~2+λ-1Π~2TΠ~2<0,i=1,2.
Applying the Schur complement, (27) is equivalent to (29). This completes the proof.


## 4. Illustrative Examples

Example 1.
Consider that the continuous singular time-varying delay system (5) has the system matrices of E=[9362], A=[-13.1-13.7-15.4-23.8], and Ad=[-18.6-10.4-25.2-16.8]. Set μ=0.2 and select L=[2-3]T. For the deferent lower bounds d1, the upper bounds of d2 for the system to be admissible are shown in Table 1. It is obvious that the delay-dependent stability results obtained in this paper are better than those in [32–34].Table 1
Comparisons of the upper bounds ofd2 with different d1 of the system without uncertainties.
d
1
0.5
1
1.5
2
[32]
2.9056
2.9073
2.9103
2.9163
[33]
3.0401
3.0416
3.0441
3.0490
[34]
3.0401
3.0416
3.0441
3.0490
Theorem5
3.0501
3.0540
3.0604
3.0724Example 2.
Consider that the continuous singular time-varying delay system has the following system matrices [25]:
(30)E=[1101-11201],A=[211-1010.501],Ad=[-1.50.5-0.8110.50.70.51],B=[121.5001].
Set μ=0 and select L=[-112]T; Theorem 5 yields that the system is stable for any constant delay d satisfying 0≤d≤5, which has less conservatism than 0≤d≤3.1 which was obtained in [28]. Then, set μ=0.2 and select L=[-112]T. For any time-varying delay d(t) satisfying 0.5≤d(t)≤2, the LMIs (26) are feasible, and a controller gain law can be obtained as follows:
(31)K=[-0.6466-0.1275-0.26810.20720.5078-0.1450].Now, we consider the system uncertainties, and the uncertain system matrices have the following forms ofM=[0.40.30.1]T, N1=[0.20.40.5], and N2=[0.30.70.5]. Set μ=0.2  and choose L=[-112]T. For any time-varying delay d(t) satisfying 0.5≤d(t)≤2, the LMIs (27) are feasible, and a robust controller can be obtained as
(32)K=[-2.2790-0.4928-0.94420.76541.6355-0.4224].


## 5. Conclusion

In this research, the robustly delay-dependent stabilization for continuously singular time-varying delay systems with norm-bounded uncertainties is investigated. Based on Lyapunov stability theory and LMI technique, the new delay-dependent LMIs-based conditions are established for the singular time-varying delay system to be regular, impulse free, and stable. By solving these LMIs, the desired state feedback control law can be obtained, and the regularity, causality, and stability of the closed-loop system are guaranteed. Finally, simulation results are given to show the effectiveness of the proposed method.



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*Source: 796596-2014-06-22.xml*