# Robust Stability and Stabilization for Singular  Time-Delay Systems with Linear Fractional Uncertainties:  A Strict LMI Approach

Authors: Jinxing Lin; Lina Rong
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the problems of delay-dependent robust stability and stabilization for a class of continuous singular systems with time-varying delay in range and parametric uncertainties. The parametric uncertainties are assumed to be of a linear fractional form, which includes the norm bounded uncertainty as a special case and can describe a class of rational nonlinearities. In terms of strict linear matrix inequalities (LMIs), delay-range-dependent robust stability criteria for the unforced system are presented. Moreover, a strict LMI design approach is developed such that, when the LMI is feasible, a desired state feedback stabilizing controller can be constructed, which guarantees that, for all admissible uncertainties, the closed-loop dynamics will be regular, impulse free, and robustly asymptotically stable. Numerical examples are provided to demonstrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Singular time-delay systems, which are known as descriptor time-delay systems, implicit time-delay systems, or generalized differential-difference equations, often appear in various engineering systems, for example, aircraft attitude control, flexible arm control of robots, large-scale electric network control, chemical engineering systems, lossless transmission lines, and so forth [1, 2]. Since singular time-delay systems are matrix delay differential equations coupled with matrix difference equations, the study for such systems is much more complicated than that for standard state-space time-delay systems. Recently, a great deal of attention has been devoted to the study of such more general class of delay systems; see [3–27].The existing stability criteria for singular time-delay systems can be classified into two types: delay independent [3–5] and delay dependent [6–10]. Generally, delay-dependent conditions are less conservative than the delay-dependent ones, especially when the time delay is small. To obtain delay-dependent conditions, many efforts have been made in the literature, among which the model transformation and bounding technology for cross-terms [8–10] are often used. However, it is known that the bounding technology and the model transformation are the main source of conservation [28]. Recently, some improved stability conditions with less conservatism have been provided by utilizing the free weighting matrix method [11–13], the integral inequality [14], and the delay decomposition approach [15–17], in which neither the bounding technology nor model transformation is involved. However, these conditions in [6–17] were established under the assumption that the delay was time invariant. For the continuous singular systems with time-varying delay, Yue and Han investigated the delay-dependent stability condition by introducing the free weighting matrices [18]. In [19], a delay-dependent stability condition was presented by using the integral inequality method. But the range of the time-varying delay considered in [18, 19] is from 0 to an upper bound. In practice, a time-varying interval delay is often encountered; that is, the range of delay varies in an interval for which the lower bound is not restricted to 0. In this case, the stability criteria in [18, 19] are conservative because they do not take into account the information of the lower bound of delay. Moreover, when estimating the upper bound of the derivative of Lyapunov functional, some useful terms are ignored in [18, 19]. More recently, continuous singular systems with time-varying delay in a range have been extensively studied; see, for example, [20–27] and references therein.On the other hand, in recent years, more and more attention has been devoted to derive strict LMI conditions for stability analysis and controller design; see, for example [29, 30] and references therein. The strict LMI conditions, that is, definite LMIs without equality constraints, are highly tractable and reliable when checked by some recently developed algorithms for solving LMIs [31]. However, it should be pointed that the stability conditions derived in [20–27] are formulated in terms of nonstrict LMIs, whose solutions are difficult to calculate since equality constraints are often fragile and usually not met perfectly. Furthermore, up to now, to the best of the authors’ knowledge, for a continuous uncertain singular system with a time-varying interval delay, the problems of robust stability, stabilization, and feedback control have not been fully investigated yet [23]. Particularly, strict LMI-based condition has never been reported in the published works.In this paper, by using a strict LMI approach, we study the robust stability and stabilization problems for a class of singular systems with a time-varying interval delay and uncertainties. Different from the existing results in [13, 19, 21, 23], first, the criteria proposed in our paper do not contain any semidefinite matrix inequality and are expressed as strict LMIs. Second, the new criteria are obtained by only using a well-known integral-inequality and do not employ any free-weighting matrix, which makes our methods more efficient. Third, a new type of uncertainty, namely, linear fractional form, is considered in this paper. Three numerical examples are given to illustrate the effectiveness of the presented method.Notations.
R
n denotes the n-dimensional Euclidean space, and Rn×m denotes the sets of all n×m matrices. I is the identity matrix with appropriate dimensions. For a real symmetric matrix X, XT denotes its transpose, the notation X≥0 (X>0) means that the matrix X is positive semidefinite (positive-definite), and λmin(X)(λmax(X)) denotes the minimum (maximum) eigenvalue of X. Cn,τ≔C([-τ,0],Rn) denotes the Banach space of continuous vector functions mapping the interval [-τ,0] into Rn,  xt≔x(t+θ),  θ∈[-τ,0], and  t≥0 denotes the function family defined on [-τ,0] which is generated by n-dimensional real vector valued continuous function x(t), t∈[-τ,+∞). Obviously, xt∈Cn,τ. ∥·∥ refers to the Euclidean vector norm or spectral matrix norm, and ∥φ∥c≔sup-τ≤t≤0∥φ(t)∥ stands for the norm of a function φ∈Cn,τ. The symmetric terms in a symmetric matrix are denoted by *.


## 2. Problem Formulation and Preliminaries

Consider the following singular time-delay system:(1)Ex˙(t)=(A+ΔA)x(t)+(Aτ+ΔAτ)x(t-τ(t))+(B+ΔB)u(t),x(t)=ϕ(t),t∈[-h2,0],
where x(t)∈Rn is the state vector, and u(t)∈Rm is the control input. E, A∈Rn×n, Aτ∈Rn×n, and B∈Rn×m are real constant matrices with appropriate dimensions, and 0<rankE=r≤n. τ(t) denotes the time-varying delay which satisfies h1<τ(t)≤h2, τ˙(t)≤μ<1. Note that h1 may not be equal to 0. ϕ(t) is a compatible continuous vector-valued initial function on [-h2,0]. ΔA, ΔAτ and ΔB are matrices with parametric uncertainties satisfying
(2)[ΔAΔAτΔB]=DΔ(t)[EaEτEb],(3)Δ(t)=F(t)(I-JF(t))-1,(4)I-JTJ>0,
where D, Ea, Eτ, Eb, and J are known real constant matrices of approximate dimensions and F(t) are unknown time-varying matrix function satisfying
(5)FT(t)F(t)≤I.
The parametric uncertainties ΔA, ΔAτ, and ΔB satisfying (2)–(5) are said to be admissible.Remark 1.
The above-structured linear fractional form includes the norm-bounded uncertainty as a special case whenJ=0 [3, 8–11, 13, 19, 23] and can describe a class of rational nonlinearities [32]. Note also that conditions (4) and (5) guarantee that I-JF(t) is invertible.The nominal unforced system of (1) can be written as
(6)Ex˙(t)=Ax(t)+Aτx(t-τ(t)),x(t)=ϕ(t),t∈[-h2,0].
The following notations are given.(i)
S0≔{ϕ(t)∈Cn,h2, ϕ(t) is the compatible initial function of system (6)}.(ii)
S≔{ϕ(t)∈Cn,h2, and there exists a uniquely continuous solution of system (6) on [0,+∞) for ϕ(t)}.(iii)
B(0,δ)≔{ϕ(t)∣ϕ(t)∈Cn,h2,∥ϕ∥c≤δ,δ>0}.Definition 2 (see [33]).
(
1
) The pair (E,A) is said to be regular if det(sE-A) is not identically 0.
(
2
) The pair (E,A) is said to be impulse-free if deg(det(sE-A))=rankE.Lemma 3 (see [3]).
If the pair(E,A) is regular and impulse free, then for any compatible initial function ϕ(t)∈Cn,h2, there exists a uniquely continuous solution of system (6) on [0,+∞) for ϕ(t).Definition 4 (see [3]).
The singular system (6) is said to be regular and impulse free, if the pair (E,A) is regular and impulse free.Definition 5 (see [11]).
(
1
) The system (6) is said to be stable, if for any ε>0, there exists a scalar δ(ε)>0 such that for any compatible initial function ϕ(t)∈B(0,δ)∩S, the solution x(t) of system (6) satisfies ∥x(t)∥≤ε, t>0.
(
2
) The system (6) is said to be asymptotically stable, if its zero solution is stable, and furthermore, there exists a b0>0 such that for any compatible initial function ϕ(t)∈B(0,b0)∩S, the solution x(t)→0 as t→∞.
The objective of this note is to develop delay-range-dependent robust stability conditions for system (1) with u(t)=0 and to design a state-feedback controller
(7)u(t)=Kx(t),K∈Rm×n
so that system (1) is closed-loop regular, impulse-free, and robustly asymptotically stable for admissible linear fractional form uncertainties. To this end, the following lemmas are needed.Lemma 6 (see [34]).
For any constant matrixW∈Rn×n, W=WT>0, scalar 0≤r(t)≤rM, and vector function x˙:[-rM,0]→Rn such that the following integration is well defined, then
(8)-rM∫-r(t)0x˙T(t+ξ)Wx˙(t+ξ)dξ≤(xT(t)xT(t-r(t)))(-WWW-W)(x(t)x(t-r(t))).Lemma 7 (see [35]).
Consider the functionφ:R+→R, if φ˙ is bounded on [0,∞); that is, there exists a scalar α>0 such that |φ˙(t)|≤α for all t∈[0,∞), and then φ is uniformly continuous on [0,∞).Lemma 8 (see [35]).
Consider the functionφ:R+→R, if φ(t) is uniformly continuous and ∫0∞φ(s)ds<∞, then limt→∞φ(t)=0.Lemma 9 (see [32]).
Given matricesQ=QT, Γ, and Ξ of approximate dimensions, then
(9)Q+ΓΔ(t)Ξ+ΞTΔT(t)ΓT<0,
where Δ(t) is as in (3), if and only if there exists scalar ε>0 such that
(10)Q+[ε-1ΞTεΓ][I-J-JTI]-1[ε-1ΞεΓT]<0.


## 3. Stability Issue

In this section, first of all, we will present new delay-range-dependent stability conditions that guarantee system (6) to be regular, impulse free, and asymptotically stable in terms of LMI, which will play a key role in obtaining the robust stability criterion for the uncertain system (1).Theorem 10.
Given scalars0≤h1<h2 and μ, the singular system (6) is regular, impulse free, and asymptotically stable if there exist positive-definite matrices P, Qi, i=1,2,3, Zj, j=1,2 and matrix S with appropriate dimensions such that
(11)Λ=[Λ11Λ1200h2ATZ1h12ATZ2*Λ22Λ23Λ24h2AτTZ1h12AτTZ2**Λ33000***Λ4400****-h2Z10*****-h12Z2]<0,
where
(12)Λ11=ETPA+ATPE+SRTA+ATRST+∑i=13Qi-h2-1ETZ1E,Λ12=ETPAτ+SRTAτ+h2-1ETZ1E,Λ22=-(1-μ)Q3-h2-1ETZ1E-h12-1ET(Z1+Z2)E-h12-1ETZ2E,Λ23=h12-1ETZ2E,Λ24=h12-1ET(Z1+Z2)E,Λ33=-Q1-h12-1ETZ2E,Λ44=-Q2-h12-1ET(Z1+Z2)E,h12=h2-h1,
and R∈Rn×(n-r) is any matrix with full column rank and satisfies ETR=0.Proof.
SincerankE=r≤n, there must exist two invertible matrices G- and H- such that
(13)E-=G-EH-=[Ir000].
Then, R can be parameterized as
(14)R=G-T[0Φ-],
where Φ-∈R(n-r)×(n-r) is any nonsingular matrix.
Similar to (13), we define
(15)A-=G-AH-=[A-11A-12A-21A-22],P-=G--TPG--1=[P-11P-12P-21P-22],S-=H-TS=[S-1S-2],R-=G--TR=[0Φ-],Z-1=G--TZ1G--1=[Z-1,11Z-1,12Z-1,21Z-1,22].
Since Λ11<0 and Qi>0, i=1,2,3, we can formulate the following inequality easily:
(16)Ψ=ETPA+ATPE+SRTA+ATRST-h2-1ETZ1E<0.
Pre- and postmultiplying Ψ<0 by H-T and H-, respectively, yields
(17)Ψ-=H-TΨH-=[Ψ-11Ψ-12*A-22TΦ-S-2T+S-2Φ-TA-22]<0,
where Ψ-11 and Ψ-12 represent the matrices not relevant in the following discussion. From (17), it is easy to see that
(18)A-22TΦ-S-2T+S-2Φ-TA-22<0
which gives that A-22 is nonsingular.
Define(19)G~=[I-A-12A-22-10I]G-,H~=H-[I0-A-22-1A-21A-22-1].
After some algebraic manipulations, we can obtain
(20)E~=G~EH~=[Ir000],A~=G~AH~=[A~1100I],
where A~11=A-11-A-12A-22-1A-21. Then, it can be shown that
(21)det(sE-A)=det(G~-1)det(sE~-A~)det(H~-1)=det(G~-1)(-1)(n-r)det(sIr-A~11)det(H~-1)
which implies that det(sE-A) is not identically zero and deg(det(sE-A))=r=rankE. Then, the pair of (E,A) is regular and impulse-free, which shows that system (6) is regular and impulse-free. In the following, we will prove that system (6) is also asymptotically stable.
Denote(22)A~τ=G~AτH~=[A~τ11A~τ12A~τ21A~τ22],P~=G~-TPG~-1=[P~11P~12P~21P~22],R~=G~-TR=[0Φ~],S~=H~TS=[S~1S~2],Q~i=H~TQiH~=[Q~i,11Q~i,12Q~i,21Q~i,22],i=1,2,3,Z~j=G~-TZjG~-1=[Z~j,11Z~j,12Z~j,21Z~j,22],j=1,2.
By using Schur complement and noting thatQi>0, i=1,2, μ<1, it follows from (11) that
(23)[Λ-11Λ12*Λ-22]<0,
where
(24)Λ-11=ETPA+ATPE+SRTA+ATRST+Q3-h2-1ETZ1E,Λ-22=-Q3-h2-1ETZ1E-h12-1ET(Z1+Z2)E-h12-1ETZ2E.
Pre- and post-multiplying (23) by diag{H~T,H~T} and diag{H~,H~}, respectively, yields
(25)[Λ-~11Λ~12*Λ-~22]<0,
where
(26)Λ-~11=E~TP~A~+A~TP~E~+S~R~TA~+A~TR~S~T+Q~3-h2-1E~TZ~E~,Λ~12=E~TP~A~τ+S~R~TA~τ+h2-1E~TZ~1E~,Λ-~22=-Q~3-h2-1E~TZ~1E~-h12-1E~T(Z~1+Z~2)E~-h12-1E~TZ~2E~.
Substituting (20), (22) into (25), we have
(27)[S~2Φ~T+Φ~S~2T+Q~3,22S~2Φ~TA~τ22T*-Q~3,22]<0.
Pre- and post-multiplying (27) by ξ=[Φ~-TS~2-1-A~τ22Φ~-TS~2-1] and ξT, respectively, yields
(28)A~τ22Θ~A~τ22T-Θ~<0,
where
(29)Θ~=-Φ~-TS~2-1(S~2Φ~T+Φ~S~2T+Q~3,22)S~2-TΦ~-1>0.
Therefore,
(30)ρ(A~τ22)<1.
Now, let(31)x~(t)=[x~1(t)x~2(t)]=H--1x(t),
where x~1(t)∈Rr and x~2(t)∈Rn-r. Using the expressions in (20), (22), and (31), system (6) can be decomposed as
(32)x~˙1(t)=A~11x~1(t)+A~τ11x~1(t-τ(t))+A~τ12x~2(t-τ(t)),(33)0=x~2(t)+A~τ21x~1(t-τ(t))+A~τ22x~2(t-τ(t)),
or equivalently rewritten as
(34)E~x~˙(t)=A~x~(t)+A~τx~(t-τ(t)).
It is easy to see that the stability of system (6) is equivalent to that of system (34).
Construct the Lyapunov-Krasovskii functional for system (34) as
(35)V~(x~t)=x~T(t)E~TP~E~x~(t)+∑i=12∫t-hitx~T(s)Q~ix~(s)ds+∫t-τ(t)tx~T(s)Q~3x~(s)ds+∫-h20∫t+θtx~˙T(s)E~TZ~1E~x~˙(s)dsdθ+∫-h2-h1∫t+θtx~˙T(s)E~TZ~2E~x~˙(s)dsdθ.
By Lemma 6, the following inequalities are true
(36)-h2∫t-h2tx~˙T(s)E~TZ~1E~x~˙(s)ds=-h2∫t-τ(t)tx~˙T(s)E~TZ~1E~x~˙(s)ds-h2∫t-h2t-τ(t)x~˙T(s)E~TZ~1E~x~˙(s)ds≤(x~T(t)x~T(t-τ(t)))×(-E~TZ~1E~E~TZ~1E~E~TZ~1E~-E~TZ~1E~)(x~(t)x~(t-τ(t)))+h12-1(x~T(t-τ(t))x~T(t-h2))×(-E~TZ~1E~E~TZ~1E~E~TZ~1E~-E~TZ~1E~)(x~(t-τ(t))x~(t-h2)),-h12∫t-h2t-h1x~˙T(s)E~TZ~2E~x~˙(s)ds=-h12∫t-h2t-τ(t)x~˙T(s)E~TZ~2E~x~˙(s)ds-h12∫t-τ(t)t-h1x~˙T(s)E~TZ~2E~x~˙(s)ds≤(x~T(t-τ(t))x~T(t-h2))×(-E~TZ~2E~E~TZ~2E~E~TZ~2E~-E~TZ~2E~)(x~(t-τ(t))x~(t-h2))+(x~T(t-h1)x~T(t-τ(t)))×(-E~TZ~2E~E~TZ~2E~E~TZ~2E~-E~TZ~2E~)(x~(t-h1)x~(t-τ(t))).
On the other hand, noticing that ETR=0, we can deduce that
(37)0=2x~T(t)S~R~TE~x~˙(t),
where S~ is any matrix with appropriate dimensions.
Taking the derivative ofV~(x~t) with respect to t along the trajectory of system (34) and using (36) and (37), we have
(38)V~˙(x~t)=x~˙T(t)E~TP~E~x~(t)+x~T(t)E~TP~E~x~˙(t)+∑i=12{x~T(t)Q~ix~(t)-x~T(t-hi)Q~ix~(t-hi)}+x~T(t)Q~3x~(t)-(1-τ˙(t))x~T(t-τ(t))Q~3x~(t-τ(t))+h2x~˙T(t)E~TZ~1E~x~˙(t)-∫t-h2tx~˙T(s)E~TZ~1E~x~˙(s)ds+(h2-h1)x~˙T(t)E~TZ~2E~x~˙(t)-∫t-h2t-h1x~˙T(s)E~TZ~2E~x~˙(s)ds≤ψ~T(t)Λ~ψ~(t),
where
(39)ψ~(t)=[x~T(t)x~T(t-τ(t))x~T(t-h1)x~T(t-h2)]T,Λ~=[Λ~11Λ~1200*Λ~22Λ~23Λ~24**Λ~330***Λ~44],
with
(40)Λ~11=E~TP~A~+A~TP~E~+S~R~TA~+A~TR~S~T+∑i=13Q~i+h2A~TZ~1A~+h12A~TZ~2A~-h2-1E~TZ~1E~,Λ~12=E~TP~A~τ+S~R~TA~τ+h2A~TZ~1A~τ+h12A~TZ~2A~τ+h2-1E~TZ~1E~,Λ~22=-(1-μ)Q~3+h2A~τTZ~1A~τ+h12A~τTZ~2A~τ-h2-1E~TZ~1E~-h12-1E~T(Z~1+Z~2)E~-h12-1E~TZ~2E~,Λ~23=h12-1E~TZ~2E~,Λ~24=h12-1E~T(Z~1+Z~2)E~,Λ~33=-Q~1-h12-1E~TZ~2E~,Λ~44=-Q~2-h12-1E~T(Z~1+Z~2)E~.
It is easy to see that (17) guarantees V~˙(x~t)<0 and
(41)λ1∥x~1(t)∥2-V~(x~(0))≤-V~(x~(t))-V~(x~(0))=∫0tV~˙(x~(s))ds≤-λ2∫0t∥x~(s)∥2ds≤-λ2∫0t∥x~1(s)∥2ds<0,
where λ1=λmin(P~11)>0, λ1=-λmax(Λ~)>0.
Taking into account (41), we can deduce that
(42)λ1∥x~1(t)∥2+λ2∫0t∥x~1(s)∥2ds≤V~(x~(0)).
Therefore,
(43)∥x~1(t)∥2≤m1,(44)∫0t∥x~1(s)∥2ds≤m2,
where m1=(1/λ1)V~(x~(0))>0, m2=(1/λ2)V~(x~(0))>0. Thus, ∥x~1(t)∥ is bounded. Considering this and (30), it can be deduced from (33) that ∥x~2(t)∥ is bounded;, hence, it follows that from (32) that ∥x~˙1(t)∥ is bounded. Therefore, (d/dt)∥x~˙1(t)∥2 is bounded too. By Lemma 7, we obtain that ∥x~˙1(t)∥2 is uniformly continuous. Therefore, noting (44) and using Lemma 8, we get
(45)limt→∞∥x~1(t)∥=0.
This, together with (33), implies that
(46)limt→∞∥x~2(t)∥=0.
Thus, according to Definition 5, system (34) is stable. This completes the proof.Remark 11.
From the proof of Theorem10, it is clear to see that neither model transformation nor bounding technique for cross-terms is involved. Hence, the conservatism inherited from these ideas will no longer exist in Theorem 10.Remark 12.
Free-weighting matrices in [11–13, 22, 23, 25] plays an important role to reducing the conservatism of delay-dependent stability conditions. However, too many free-weighting matrices will complicate the system analysis and increase the computational demand. It is worth pointing out that no free-weighting matrix is involved in Theorem 10.Remark 13.
Recently, the delay-partitioning or delay-fractioning method [36] was widely used to reduce the conservatism of the delay-dependent results of standard time-delay systems. This method can be extended to study the stability of singular time-delay system (1). Suppose we decompose the delay interval [h1,h2] into Nequidistant subintervals. Defining hi=h1+i(h2-h1)/N, i=1,2,…,N-1, and constructing the following Lyapunov-Krasovskii functional:
(47)V~(x~t)=x~T(t)E~TP~E~x~(t)+∑i=12∫t-hitx~T(s)Q~ix~(s)ds+∑j=1N-1∫t-hjtxT(s)Q~j′x(s)ds+∫-h20∫t+θtx~˙T(s)E~TZ~1E~x~˙(s)dsdθ+∫-h1-h1∫t+θtx~˙T(s)E~TZ~2E~x~˙(s)dsdθ+∑l=1N-2∫-hl+1-hl∫t+θtx~˙T(s)E~TZ~l+2E~x~˙(s)dsdθ+∫-h2-hN-1∫t+θtx~˙T(s)E~TZ~N+1E~x~˙(s)dsdθ
with Q~1>0, Q~2>0, Q~j′>0, j=1,2,…,N-1, Z~1>0, Z~2>0 and Z~l+2>0, l=1,2,…,N-2. Then, by checking the variation of V~(x~t) for the case when τ(t)∈[h1,h1] or τ(t)∈[hj,hj+1] (j=1,2,…,N-1) or τ(t)∈[hN-1,h2], respectively, we can derive the delay-dependent condition, which can guarantee that V~(x~t)<0. Generally, increasing N may result in the reduction of conservatism of the obtained results. However, the corresponding computational complexity will be increased greatly since the dimensions and matrix variables of the involved LMIs will be sharply expanded. For example, in [36], a numerical example has shown that, with N changing from 1 to 3, the allowable upper bounds of h2 increased 12.9%, but the consumed CPU time increased 9 times.Theorem10 presents a delay-range-dependent criterion for system (6) with time-varying delay τ(t) in a range. If we set Q1=0 and Z2=0, Theorem 10 yields the following delay-dependent stability criterion.Corollary 14.
Given scalarsh2>0, h1=0 and μ, system (6) is regular, impulse free, and asymptotically stable if there exist positive-definite matrices P, Qi, i=2,3, Z1, and matrix S with appropriate dimensions such that
(48)[T11T120h2ATZ1*T22h2-1ETZ1Eh2AτTZ1**Τ330***-h2Z1],
where
(49)Τ11=ETPA+ATPE+SRTA+ATRST+∑i=23Qi-h2-1ETZ1E,Τ12=ETPAτ+SRTAτ+h2-1ETZ1E,Τ22=-(1-μ)Q3-2h2-1ETZ1E,Τ33=-Q2-h2-1ETZ1E,
and R∈Rn×(n-r) is any matrix with full column rank and satisfies ETR=0.Now, we will present the delay-range-dependent robust stability conditions for the uncertain singular time-delay system (1) with u(t)=0 via Theorem 10.Theorem 15.
Given scalars0≤h1<h2 and μ, the uncertain singular time-delay system (1) with u(t)=0 is regular, impulse free, and robustly asymptotically stable if there exist positive-definite matrices P, Qi, i=1,2,3, Zj, j=1,2, and matrix S with appropriate dimensions and a scalar η>0 such that the following LMI holds:
(50)[Λ11Λ1200h2ATZ1h12ATZ2ηEaT(ETP+SRT)D*Λ22Λ23Λ24h2AτTZ1h12AτTZ2ηEτT0**Λ3300000***Λ440000****-h2Z100h2Z1D*****-h12Z20h12Z2D******-ηIηJ*******-ηI]<0,
where Λ11, Λ12, Λ22, Λ23, Λ24, Λ33, Λ44, and h12 are defined in (11).Proof.
Suppose (50) to be true. Let ε=1/η. Pre- and postmultiplying the left-hand side matrix of (50) by diag{I,I,I,I,I,I,ε,ε} and its transpose, respectively, we obtain
(51)Λ+[ε-1ΞTεΓ][I-J-JTI]-1[ε-1ΞεΓT]<0,
where
(52)Γ=[DT(PE+RST)000h2DTZ1h12DTZ2]T,Ξ=[EaEτ0000],
and Λ is defined in (11). Thus, Θ=Λ+ΓΔ(t)Ξ+ΞTΔT(t)ΓT<0 holds according to Lemma 9. It can be verified that Θ is exactly the left-hand side of (11) when Aand Aτ are replaced with A+DΔ(t)Eaand Aτ+DΔ(t)Eτ in (11), respectively. The result then follows from Theorem 10.


## 4. Control Design

On the basis of the previous stability conditions, we will present a design method of robustly stabilizing controllers in this section. For simplicity, we first consider system (6).Theorem 16.
Given scalars0≤h1<h2 and μ, if there exist scalar α>0, positive-definite matrices P, Qi, i=1,2,3, Zj, j=1,2, X, and matrices R, S, U with appropriate dimensions such that
(53)Ψ=[Ψ11Ψ12Ψ13000Ψ170Ψ19*Ψ22XAτT00h2Z1-h2Xh12Z2-h12X**Ψ33Ψ34Ψ350h2AτX0h12AτX***Ψ4400000****Ψ550000*****-h2Z1000******-h2X00*******-h12Z20********-h12X]<0,
where
(54)Ψ11=AX+XAT+BU+UTBT+∑i=13Qi-h2-1EZ1ET,Ψ12=EP+SRT-X+AX+BU,Ψ13=XAτT+h2-1EZ1ET,Ψ17=h2AX+h2BU,Ψ19=h12AX+h12BU,Ψ22=-2X+3aI,Ψ33=-(1-μ)Q3-h2-1EZ1ET-h12-1E(Z1+Z2)ET-h12-1EZ2ET,Ψ34=h12-1EZ2ET,Ψ35=h12-1E(Z1+Z2)ET,Ψ44=-Q1-h12-1EZ2ET,Ψ55=-Q2-h12-1E(Z1+Z2)ET
and R1∈Rn×(n-r) is any matrix with full column rank and satisfying ETR1=0, then there exists a state feedback controller (7) such that the resulting closed-loop system of system (6) is regular, impulse free, and asymptotically stable. In this case, a suitable controller gain is given by
(55)u(t)=UX-1x(t).Proof.
With the control lawu(t)=Kx(t), the resultant closed-loop system of system (6) is
(56)Ex˙(t)=(A+BK)x(t)+Aτx(t-τ(t)).
Following the same philosophy as that in [37], we represent system (56) as the following form,
(57)[E000][x˙(t)y˙(t)]=[0IA+BK-I][x(t)y(t)]+[00Aτ0][x(t-τ(t))y(t-τ(t))],
where y(t)=Ex˙(t).
For notational convenience, we introduce(58)E-=[E000],A-=[0IA+BK-I],A-τ=[00Aτ0].
Then, by Theorem 10, we can show that system (57) is regular, impulse free, and asymptotically stable if (11) holds, where E, A, Aτ, P, Qi, and i=1,2,3, Zj, j=1,2, R, S are replaced by E-, A-, A-τ, P-, Q-i, i=1,2,3, Z-j, and j=1,2, R-, S-, respectively. For a special issue, we choose P-, Q-i, Z-j, R-, and S- as
(59)P-=[P00αI],Q-1=[Q100αI],Q-2=[Q200αI],Q-3=[Q300αI],Z-1=[Z100X],Z-2=[Z200X],R-=[R00X],S-=[SI0I],
where P∈Rn×n,   Qi∈Rn×n,  i=1,2,3,  Zj∈Rn×n,  and j=1,2, X∈Rn×n are symmetrical positive-definite matrices, R∈Rn×(n-r) is with full column rank and satisfies ETR=0, S∈Rn×(n-r) is any matrix and α>0 is a scalar. It is easy to verify that R- is with full column rank and satisfies E-TR-=0. Then, the following LMI can be obtained:
(60)[Φ11Φ12Φ13000Φ170Φ19*Φ22XAτ00h2Z1-h2Xh12Z2-h12X**Φ33Φ34Φ350h2AτTX0h12AτTX***Φ4400000****Φ550000*****-h2Z1000******-h2X00*******-h12Z20********-h12X]<0,
where
(61)Φ11=X(A+BK)+(A+BK)TX+∑i=13Qi-h2-1ETZ1E,Φ12=ETP+S1R1T-X+(A+BK)TX,Φ13=XAτ+h2-1ETZ1E,Φ17=h2(A+BK)TX,Φ19=h12(A+BK)TX,Φ22=-2X+3aI,Φ33=-(1-μ)Q3-h2-1ETZ1E-h12-1ET(Z1+Z2)E-h12-1ETZ2E,Φ34=h12-1ETZ2E,Φ35=h12-1ET(Z1+Z2)E,Φ44=-Q1-h12-1ETZ2E,Φ55=-Q2-h12-1ET(Z1+Z2)E.
Note that the pairs(E,(A+BK)) and (E,(A+BK+Aτ)) are regular, causal if and only if the pairs (ET,(A+BK)T) and (ET,(A+BK+Aτ)T) are regular, causal. Furthermore, det(sE-(A+BK)-e-τ(t)sAτ)=det(sET-(A+BK)T-e-τ(t)sAτT). Then, as long as the regularity, causality, and stability problems are concerned, we can consider the following system instead of (56):
(62)ETζ˙(t)=(A+BK)Tζ(t)+AτTζ(t-τ(t)),
where ζ(t)∈Rn is the state vector.
In this sense, (53) can be obtained by replacing E, (A+BK), Aτ in (60) by ET, (A+BK)T, AτT, respectively, and introducing a matrix U=KX.The robust stabilizability result for uncertain singular system (1) is presented in the following theorem.Theorem 17.
Given scalars0≤h1<h2 and μ, if there exist scalar α>0, positive-definite matrices P, Qi, i=1,2,3, Zj, j=1,2, X, and matrices R, S, U with appropriate dimensions and scalars η1>0, η2>0 such that (63)[Ψ11Ψ12Ψ13000Ψ170Ψ19Ψ110η1DΨ1120*Ψ22XAτT00h2Z1-h2Xh12Z2Ψ29Ψ2100Ψ2120**Ψ33Ψ34Ψ350h2AτX0Ψ39000η2D***Ψ44000000000****Ψ5500000000*****-h2Z10000000******-h2X00Ψ7100Ψ7120*******-h12Z200000********-h12XΨ9100Ψ9120*********-η1Iη1J00**********-η1I00***********-η2Iη2J************-η2I]<0,where Ψ11, Ψ12, Ψ13, Ψ17, Ψ19, Ψ22, Ψ33, Ψ34, Ψ35, Ψ44, and Ψ55 are defined in (53), Ψ29=-h12X, Ψ110=Ψ210=XEaT+UTEbT, Ψ112=Ψ212=XEτT, Ψ39=h12AτX, Ψ710=h2(XEaT+UTEbT), Ψ910=h12(XEaT+UTEbT), Ψ712=h2XEτT, Ψ912=h12XEτT, and R1∈Rn×(n-r) is any matrix with full column rank and satisfying ETR1=0; then there exists a state feedback controller (7) such that the resulting closed-loop system of system (1) is regular, impulse free, and robustly asymptotically stable. In this case, a suitable controller gain is given by
(64)u(t)=UX-1x(t).Proof.
ReplacingA by A+DΔ(t)Ea, Aτby Aτ+DΔ(t)Eτ and B by B+DΔ(t)Eb in (53), respectively, results in the following condition:
(65)Ψ+Γ1Δ(t)Ξ1+Ξ1TΔT(t)Γ1T+Γ2Δ(t)Ξ2+Ξ2TΔT(t)Γ2T<0,
where(66)Γ1=[DT00000000]T,Ξ1=[EaX+EbUEaX+EbU0000h2(EaX+EbU)0h12(EaX+EbU)],Γ2=[00DT000000]T,Ξ2=[EτUEτU0000h2EτX0h12EτX],and Ψ is defined in (53).
By Lemma9, (65) holds for Δ(t) satisfying (3), if there exist scalars ε1>0, ε2>0 such that
(67)Ψ+[ε1-1Ξ1Tε1Γ1][I-J-JTI]-1[ε1-1Ξ1ε1Γ1T]+[ε2-1Ξ2Tε2Γ2][I-J-JTI]-1[ε2-1Ξ2ε2Γ2T]<0.
Suppose (63) to be true. Let εi=ηi, i=1,2. Pre- and postmultiplying the left-hand side matrix of (63) by diag{I,I,I,I,I,I,ε1-1,ε1-1,ε2-1,ε2-1} and its transpose, respectively, and using Schur complement equivalence to (63) yields (67). The result then follows from Theorem 16.


## 5. Numerical Examples

In this section, some examples are provided to illustrate the benefits of our results.Example 1.
Consider the nominal unforced part of system (1) with
(68)E=[1000],A=[0.500-1],Aτ=[-1.1100.5].
The case for h1=0 and μ=0 has been studied in [12]. We choose R=[01]T. The comparison among Corollary 14 in this note and those in [6–9, 11, 12, 19] is listed in Table 1 for h1=0. It should be pointed out that the results of [7] fail to deal with the system because the matrix describing the relationship between the fast and slow variables cannot be chosen beforehand. It can be seen that our method is less conservative than those in [6, 9] and gives the same results as that in [8, 11, 12, 19]. However, when the time delay is a varying delay, our method gives better results than that in [19] for μ=0.3 and μ=0.75 since the relationship among h2, τ(t), h2-τ(t), and τ(t)-h1 has been taken into account in Corollary 14. On the other hand, for system with time-varying delay in a range, Table 1 also lists the allowable maximum upper bounds of h2 for different μ with h1=0.3 and h1=0.6.Table 1
Allowable upper bounds ofh2 with given h1 for different μ.
h
1
Methods
μ
=
0
μ
=
0.3
μ
=
0.75
0
[7]
—
—
—
[9]
0.5567
—
—
[6]
0.9680
—
—
[8, 11, 12]
1.0660
—
—
[19]
1.0660
1.0130
0.6496
Corollary14
1.0660
1.0263
0.7141
0.3
Theorem10
1.0660
1.0315
0.7376
0.6
Theorem10
1.0660
1.0421
0.7426Example 2.
Consider the unforced part of uncertain system (1) with
(69)E=[1000],A=[0.40.20-1],Aτ=[-1100.3],D=[0.1000.1],Ea=[-0.05000.05],Eτ=[0.0500-0.05].
We choose R=[01]T. For h1=0, the comparison among Theorem 15 in this note and that in [19] is listed in Table 2. It is clear that the result in this note is better than that in [19]. The corresponding maximum upper bounds of h2 for different h1 and μ derived by Theorem 15 are also listed in Table 2.Table 2
Allowable upper bounds ofh2 with given h1 for different μ.
h
1
Methods
μ
=
0
μ
=
0.3
μ
=
0.75
0
[19]
1.1836
1.1217
1.0307
Theorem15 (J=0)
1.1836
1.1348
1.1095
0.3
Theorem15 (J=0)
1.1836
1.1394
1.1197
Theorem15 (J=0.1)
1.1842
1.1400
1.1200
0.6
Theorem15 (J=0)
1.1836
1.1489
1.1317
Theorem15 (J=0.1)
1.1842
1.1494
1.1321Example 3.
Consider system (1) with
(70)E=[1101-11201],A=[1.50.51-1010.501],Aτ=[-1011-10.50.30.5-1],B=[111001],D=[0.10.10.1]T,Ea=[0.20.10.3],Eτ=[0.10.20.1],Eb=[0.10.1],μ=0.1.
We choose R=[-112]T. According to Theorem 17, Table 3 shows the allowed maximum upper bounds of h2 and the corresponding state feedback gain K for different h1.Table 3
Allowable upper bounds ofh2 with given h1.
h
1
Methods
h
2
K
0
Theorem17 (J=-0.1)
0.9062
[
-
0.5515
-
0.1800
-
0.7849
-
1.0962
-
0.7997
-
0.2503
]
0.3
Theorem17 (J=-0.1)
1.0482
[
-
0.5845
-
0.1547
-
0.8065
-
1.1257
-
0.7953
-
0.2363
]
0.6
Theorem17 (J=-0.1)
1.1919
[
-
0.6093
-
0.1360
-
0.8218
-
1.1582
-
0.7942
-
0.2136
]


## 6. Conclusions

In this note, the delay-range-dependent robust stability and stabilization for singular time-delay systems with linear fractional uncertainty and time-varying delay in a range are studied. The results are obtained by using the strict LMI approach and constructing an appropriate Lyapunov-Krasovskii functional. Numerical examples have been given to demonstrate the effectiveness of the presented criteria and their improvement over existing results.



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