# Robust Admissibilization for Discrete-Time Singular Systems with Time-Varying Delay

Authors: Dongdong Wang; Shengzhi Han; Jian Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The problems of the admissibility and state feedback stabilization for discrete-time singular systems with interval time-varying delay and norm-bounded uncertainty are studied. The system is equivalently transformed into a new comparison form by decomposition. By taking advantage of the Seuret summation inequality, the reciprocally convex inequality, and some relaxation techniques, a delay-dependent criterion that ensures the admissibility of the concerned systems is established. The result on robust stabilization is also obtained by fixing some parameters. It should be pointed out that the results are less dependent on the parameters so that some conservatism is reduced. A numerical example is included to illustrate the effectiveness and improvement of the proposed methods.


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## Body

## 1. Introduction

Discrete-time systems, which are the analogues of the continuous-time systems, have more advantages in flexibility, anti-interference, precision, and economy. The discrete-time systems inherit the dynamic behavior of the corresponding continuous-time systems, while there is a lot of difference of the stability analysis and control synthesis between them. Intensive results have been given to study the problems for various discrete-time systems.Most physical systems and processes in the real world can be represented as singular systems which contain not only differential equations but also nondynamic constraints and even improper parts of the systems. Considerable attention has been paid to singular systems [1–3]. It should be pointed out that the study of singular systems is much more complicated than that of state-space systems [4, 5]. In addition to stability, two more things called regularity and impulse-free (causality for the discrete-time cases) are introduced which are irrespective in state-space models [6, 7]. Time delay is common in the practical engineering and communication application [8–11]. It can be a source of instability of performance degradation which is needed to be eliminated [12]. Consequently, a lot of studies of time-delay systems have been carried out [13–20]. For discrete-time singular systems with time delay, several important results have been obtained. For example admissibility, new criteria are obtained by choosing the Lyapunov–Krasovskii functionals (LKFs) properly and using new inequality technology. In [21], delay-dependent stability criteria are proposed depending on the minimum and maximum delay bounds. By adopting an improved reciprocally convex combination approach, a less conservative criterion is built to ensure the system to be admissible [22]. Banu and Balasubramaniam [23] and Wu et al. [24] give sufficient condition via delay division approaches. The stability analysis for a class of discrete-time T-S fuzzy singular systems with time delays [25, 26] is studied. Furthermore, the filtering designs are studied in [27, 28]. Zheng and Bejarano [29] and Lin et al. [30] have considered the observer design problems. The admissibilization problem is studied in [31–39] by the LKF methods. The related stabilization conditions are usually nonlinear matrix inequalities. The static output feedback control problem is studied in [31, 32, 40, 41] by adopting new summation inequalities and establishing iterative algorithms. Zhu et al. [39] and Feng and Lam [34] solve the dissipative control problems under actuator saturation. Ma et al. [36] and Feng et al. [42] give the H∞ state feedback controller design. The delay-dependent stabilization for discrete-time interval TCS fuzzy systems is considered in [43]. Wu et al. [38] study the problem of robust state feedback control for discrete-time singular systems with time delays. By fixing some variables, the above criteria are linear matrix inequalities (LMIs) which are solvable by LMI toolbox [44], but they also bring conservatism. Cui et al. [33] give an iterative linear matrix inequality approach with initial condition optimisation which can reduce the conservatism. However, it has the shortcomings of the dependence on initial values and big calculation.In this paper, the problem of a robust controller design for discrete-time singular systems with time delay is investigated. By utilizing the Seuret summation inequality, the reciprocally convex technique, and some transformations, new sufficient condition which contains auxiliary matrices for the admissibility is derived in terms of LMIs. Moreover, the problem of robust admissibilization of uncertain systems is also studied and a new condition is proposed. If the parameterε is prescribed, the admissibilization condition is LMIs. For more auxiliary matrices added, the dependence on ε is reduced in the condition.Notation: throughout the paper, standard notations are used. The superscriptT stands for the matrix transpose, and −1 stands for the matrix inverse. “∗” denotes the elements that are induced by symmetry. I and 0 represent the identity matrix and zero matrix. diag⋅ denotes the block diagonal matrix.


## 2. Problem Formulation and Preliminaries

Consider the discrete-time singular system with time delay which can be represented by(1)Exk+1=A+ΔAxk+Ad+ΔAdxk−dk+Buk,xk=φk,k=−d2,−d2+1,…,0,where xk∈ℝn is the state vector of the system, uk∈ℝp is the control input, φk is a given initial condition, and dk denotes the time delay satisfying d1≤dk≤d2, where d1 and d2 are positive integers representing the lower and upper bounds of the delay, respectively.E
∈
ℝ
n
×
n is a known matrix, and we assume that rankE=r<n. A, Ad, and B are known real constant matrices. ΔA and ΔAd are real-valued unknown matrices representing time-varying parameter uncertainties, respectively, and are assumed as(2)ΔA=MFkN,ΔAd=MdFkNd,where M, Md, N, and Nd are known real constant matrices and F⋅:ℕ⟶ℝl1×l2 are known time-varying matrix function satisfying(3)FikTFik≤I,∀k.For the discrete time-delay system:(4)Exk+1=Axk+Adxk−dk,we need to introduce some definitions.Definition 1 (see [12]).
(i) The pairE,A is said to be regular if detzE−A is not identically zero. (ii) The pair E,A is said to be causal if degdetzE−A=rankE.Definition 2 (see [12]).
(i) For given integersd1 and d2, the discrete time-delay system (4) is said to be regular and causal, if the pair E,A is regular and causal. (ii) System (4) is said to be stable if for any scalar ϵ>0, there exists a scalar δϵ>0 such that, for any compatible initial conditions φk satisfying sup−d2≤k≤0φk≤δϵ, the solution xk to system (4) satisfies xk≤ϵ for any k≥0; moreover, limk⟶∞xk=0. (iii) System (4) is said to be admissible if it is regular, causal, and stable.
Moreover, for the pairE,A+ΔA, there exist invertible matrices G and H, such that(5)E¯≔GEH=Ir000,A˜=A¯+ΔA¯=GA+ΔAH=A11A12A21A22,where A11∈Rr×r.(6)A˜d=A¯d+ΔA¯d=GAd+ΔAdH=Ad1100Ad22.
In this paper, the state feedback controller with the following form is used:(7)uk=Kxk,where K is the controller gains to be determined. The closed-loop system is given by(8)Exk+1=A+ΔA+BKxk+Ad+ΔAdxk−dk.
Now, we will use some lemmas in the proof which should be introduced first.Lemma 1.
LetM, N, and Fk be real matrices of appropriate dimensions with Fk satisfying FkTFk≤I. Then, for any matrix U, we have(9)MFkN+NTFkTMT≤MUMT+NTU−1N.Lemma 2 (see [12]).
LetΩ=Ω11Ω12Ω21Ω22, where Ω11, Ω12, Ω21, and Ω22 are any real matrices with appropriate dimensions such that Ω22 is invertible and Ω+ΩT<0. Then, we have(10)Ω11+Ω11T−Ω12Ω22−1Ω21−Ω21TΩ22−TΩ12T<0.Lemma 3 (see [44]).
Consider the following inequality in the variableX:(11)BXC+BXCT+Ω<0,which has a solution X if and only if(12)NBΩNBT<0,NCTΩNC<0,where NB and NC denote bases of the null spaces of B and C, respectively.Lemma 4 (see [45]).
For any matrixR>0, integer r1<r2, and vector function xi:k−r2+1,k−r1⟶ℝn, there holds(13)∑j=−r2+1−r1xj−xj−1TRxj−xj−1≥1rx−r1−x−r2TRx−r1−x−r2+3rx−r1+x−r2−2r+1∑−r2+1−r1xjT×Rx−r1+x−r2−2r+1∑−r2+1−r1xj.Lemma 5 (see [46]).
For matricesZ, W, vectors η1, η2, and real scalars μ≥0, ν≥0 satisfying ZW∗Z≥0 and μ+ν=1, the following inequality holds(14)−1μη1TZη1−1νη2TZη2≤−η1η2TZW∗Zη1η2.


## 3. Main Results

In this section, a new condition of admissibility for discrete-time singular systems with time-varying delay is derived and it is extended to design the state feedback controllers. For simplicity, in rest of the section, we will consider the transformation of system (1) as(15)E¯x¯k+1=A˜x¯k+A˜dx¯k−dk+B¯uk=A¯+ΔA¯x¯k+A¯d+ΔA¯dx¯k−dk+B¯uk.Now, letx¯k=x¯1kx¯2k, where x¯k=H−1xk, x¯1k∈ℝr, x¯2k∈ℝn−r, B¯=GB, M¯=GM, Md¯=GMd, N¯=NH, and Nd¯=NdH.We first consider the stability of system (15). If A22 is invertible, by (15) and (5), the singular delay system can be decomposed as(16)x¯1k+1=A11−A12A22−1A21x¯1k+Ad11x¯1k−dk−A12A22−1Ad22x¯2k−dk+B1−A12A22−1B2uk,(17)0=A21x¯1k+A22x¯2k+Ad22x¯2k−dk+B2uk.It is easy to see that the stability of the singular delay system (15) is equivalent to that of system (16).The closed-loop system is(18)E¯x¯k+1=A¯+ΔA¯+B¯K¯x¯k+A¯d+ΔA¯dx¯k−dk,where K¯=KH.For system (15) with uk=0, we have the following theorem.Theorem 1.
The discrete-time delay singular system (15) with uk=0 is admissible, if there exist matrices P>0, U1>0, U2>0, Q1=Q11∗Q12Q13>0, Q2=Q21∗Q22Q23>0, Q3=Q31∗Q32Q33>0, R1=R11∗R12R13>0, R2=R21∗R22R23>0, S, T, W1, W2, Y1, Y2, Y3, and Y4, such that(19)−12Y4+Y4T∗∗∗∗∗∗−12Y3+Y2T−12Y1+Y1T∗∗∗∗∗Π1Π2X∗∗∗∗−Y3−12Y2T−Y1−12Y1T+PΠ2T−Y1−Y1T∗∗∗−Y4−12Y4T+d12R11+d2−d12R21−Y2−12Y3TΠ1T−Y2−Y3T−Y4−Y4T∗∗M¯TΠ3M¯TΠ4M¯TΠ5M¯TΠ4M¯TΠ3−U1∗M¯dTΠ3M¯dTΠ4M¯dTΠ5M¯dTΠ4M¯dTΠ30−U2<0,(20)diagR2,3R2∗TTdiagR2,3R2>0,where(21)X=e1TQ1e1−e2TQ2e2+e1TQ2e1−e4TQ2e4+d2−d1+1e1TQ3e1−e3TQ3e3−e1−e2TE¯TR1E¯e1−e2−3E¯e1+E¯e2−2e5TR1E¯e1+E¯e2−2e5−E¯e1−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7TdiagR2,3R2TTTdiagR2,3R2E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7T+e1T000SA¯e1+A¯de3+A¯e1+A¯de3T000Se1−e1TP000e1+N¯e1TU1N¯e1+N¯de3TU2N¯de3.Π1=e1TA¯T+e3TA¯dTY3+Y4W1T−e1TE¯TY40T,Π2=e1TA¯T+e3TA¯dTY1+Y2W2T−e1TE¯TY20T,Π3=Y3+Y4W1T,Π4=Y1+Y2W2T,Π5=000STe1,ei=0n×i−1nIn0n×n−in,i=1,2,…,7.Proof.
By Lemma 1 and equations (2) and (3), we obtain that(22)ΔA¯e1+ΔA¯de3TΠ3Π4Π5Π4Π3+∗≤N¯e1TU1N¯e1+Π3Π4Π5Π4Π3TM¯U1−1M¯TΠ3Π4Π5Π4Π3+N¯de3TU2N¯de3+Π3Π4Π5Π4Π3TM¯dU2−1M¯dTΠ3Π4Π5Π4Π3,where U1>0 and U2>0. By (22) and Schur’s complement, (19) gives(23)−12Y4+Y4T∗∗∗∗−12Y3+Y2T−12Y1+Y1T∗∗∗Π¯1Π¯2X¯∗∗−Y3−12Y2T−Y1−12Y1T+PΠ¯2T−Y1−Y1T∗−Y4−12Y4T+d12R1+d2−d12R2−Y2−12Y3TΠ¯1T−Y2−Y3T−Y4−Y4T<0,where(24)Π¯1=e1TA˜T+e3TA˜dTY3+Y4W1T−e1TE¯TY40T,Π¯2=e1TA˜T+e3TA˜dTY1+Y2W2T−e1TE¯TY20T,X¯=X+e1T000SΔA¯e1+ΔA¯de3+ΔA¯e1+ΔA¯de3T000STe1−N¯e1+N¯de3TUN¯e1+N¯de3.
Multiplying (23) left and right by 0n−r×3rIn−r0n−r×6+2r and its transpose, we obtain that Q13+Q23+d2−d1+1Q33+A22TST+SA22<0, which implies that A22TST+SA22<0. Therefore, system (15) with uk=0 is regular and causal.
Next, we will consider the stability of the system. We choose the Lyapunov–Krasovskii functional as follows:(25)Vk=V1k+V2k+V3k,V1k=x¯1TkPx¯1k,V2k=∑j=k−d1k−1x¯TjQ1x¯j+∑j=k−d2k−1x¯TjQ2x¯j+∑j=−d2+1−d1+1∑i=k−1+jk−1x¯TiQ3x¯i,V3k=d1∑j=−d1+10∑i=k−1+jk−1ηTiE¯TR1E¯ηi+d2−d1∑j=−d2+1−d1∑i=k−1+jk−1ηTE¯TR2E¯ηi,where ηi=x¯k+1−x¯k, P>0, Qi>0, i=1,2,3, and Ri>0, i=1,2.
Calculating the forward difference ofVk, we obtain(26)ΔVk=Vk+1−Vk=ΔV1k+ΔV2k+ΔV3k,(27)ΔV1k=x¯1Tk+1Px¯1k+1−x¯1TkPx¯1k,(28)ΔV2k≤x¯TkQ1x¯k−x¯Tk−d1Q1x¯k−d1+x¯TkQ2x¯k−x¯Tk−d2Q1x¯k−d2+d2−d1+1x¯TkQ3x¯k−x¯Tk−dkQ3x¯k−dk,(29)ΔV3k=d12ηkTE¯TR1E¯ηk+d2−d12ηkTE¯TR2E¯ηk−d1∑j=k−d1k−1ηjTE¯TR1E¯ηj−d2−d1∑j=k−d2k−d1−1ηjTE¯TR2E¯ηj.
By Lemmas 4 and 5, the cross terms in (29) are bounded as (30)−d1∑j=k−d1k−1ηjTE¯TR1E¯ηj≤−x¯k−x¯k−d1TE¯TR1E¯x¯k−x¯k−d1−3x¯k+x¯k−d1−2d1+1∑j=k−d1kx¯jTE¯TR1E¯x¯k+x¯k−d1−2d1+1∑j=k−d1kx¯j,(31)−d2−d1∑j=k−d2k−d1−1ηjTE¯TR2E¯ηj≤−x¯k−d1−x¯k−dkx¯k−d1+x¯k−dk−2dk−d1+1∑j=k−dkk−d1x¯jx¯k−dk−x¯k−d2x¯k−dk+x¯k−d2−2d2−dk+1∑j=k−d2k−dkx¯jTdiagR2,3R2∗TTdiagR2,3R2×x¯k−d1−x¯k−dkx¯k−d1+x¯k−dk−2dk−d1+1∑j=k−dkk−d1x¯jx¯k−dk−x¯k−d2x¯k−dk+x¯k−d2−2d2−dk+1∑j=k−d2k−dkx¯j,where(32)diagR2,3R2∗TTdiagR2,3R2>0.
Noted that(33)x¯Tk000S¯E¯x¯k+1+x¯Tk+1E¯T000S¯Tx¯k=0.
Combining (26)–(33), we have(34)ΔVk≤ξTkΞξk,where(35)Ξ=ATPA−e11TPe11+A−e11Td12R11+d2−d12R21A−e11+e1T000S¯A˜e1+A˜de3+A˜e1+A˜de3T000S¯Te1+e1TQ1e1−e2TQ1e2+e1TQ2e1−e4TQ2e4+d2−d1+1e1TQ3e1−e3TQ3e3−E¯e1−E¯e2TR1E¯e1−E¯e2−3E¯e1+E¯e2−2e5TR1E¯e1+E¯e2−2e5E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7TdiagR2,3R2∗TTdiagR2,3R2E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7,A=A11−A12A22−1A21Ir0r×n−re11+Ad11−A22−1Ad22e3,e11=E¯0n×6n,ξkT=x¯Tkx¯Tk−d1x¯Tk−dkx¯Tk−d21d1+1∑j=k−d1kx¯TjE¯T1dk−d1+1∑j=k−dkk−d1x¯TjE¯T1d2−dk+1∑j=k−d2k−dkx¯TjE¯T.
It is known thatΔVk<0 if the following inequality holds:(36)Ξ<0.
Hence, there exists a scalarϵ>0 such that ΔVk≤−ϵx¯k2. Then, we have −V0≤Vk+1−V0=∑i=0kΔVi≤−ϵ∑i=0kx¯i2≤0, which implies 0≤∑i=0kΔVi≤V0/ϵ. Thus, the series ∑i=0kΔVi converge, which implies that limk⟶∞x¯k=0. According to Definition 2, system (15) with uk=0 is stable.
Next, we will decompose (23) by (5). Multiplying (23) left and right by(37)I2rIrI6n+2rIn−rand its transpose, we obtain that(38)Φ+ΦT<0,where(39)Φ=−12Y400000−12Y3+Y2T−12Y10000Π¯11Π¯2112X¯1Π¯21Π¯11Π6−Y3−12Y2T−Y1−12Y1T+P0−Y100−Y4−12Y4+d12R1+d2−d12R2−Y2−12Y3T00−Y40Π¯12Π¯220Π¯22Π¯12Π7,Π¯11=e¯11TA11TY3+Y4T+A21TW1T+e¯3TA˜dTY3+Y4W1T−e¯11TY40T,Π¯21=e¯11TA11TY1+Y2T+A21TW2T+e¯3TA˜dTY1+Y2W2T−e¯11TY20T,Π¯12=A12TY3+Y4T+A22TW1T,Π¯22=A12TY1+Y2T+A22TW2T,X¯1=e¯11TQ11e¯11−e¯2TQ1e¯2+e¯11TQ21e¯11−e¯4TQ2e¯4+d2−d1+1e¯11TQ31e¯11−e¯3TQ3e¯3−4e¯11TR11e¯11−e¯2TE¯TR1E¯e¯2−2e¯11TIr0R1E¯e¯2−2e¯2TE¯TR1Ir0Te¯11−6e¯11TIr0R1e¯5−6e¯5TR1Ir0Te¯11−3E¯e¯2−2e¯5TR1E¯e¯2−2e¯5,−E¯e¯2−E¯e¯3E¯e¯2+E¯e¯3−2e¯6E¯e¯3−E¯e¯4E¯e¯3+E¯e¯4−2e¯7TdiagR2,3R2TTTdiagR2,3R2E¯e¯2−E¯e¯3E¯e¯2+E¯e¯3−2e¯6E¯e¯3−E¯e¯4E¯e¯3+E¯e¯4−2e¯7−e¯11TPe¯11,Π6=e¯11TQ12T+Q22T+d2−d1+1Q23T+e¯11TA21TST+e¯3T0Ad22TST,Π7=A22TST+12Q13+Q23+d2−d1+1Q33,e¯11=Ir0r×n−r0r×6n,e¯i=0n×r0n×i−1In0n×n−i,i=2,3,…,6.
For the special structure of the matrixΦ, it is easy to find out matrices Γ1∈ℝ3r×3r, Γ2∈ℝ6n+2r×6n+2r, and Z, such that(40)Φ+ΦT=Σ1+Σ2<0,(41)Σ1≔Φ¯+Φ¯T−diagΓ1,Γ2,0n−r<0,(42)Σ2≔diagΓ1,Γ2,0n−r+06n+5r∗Π¯62TΠ¯72<0,where(43)Φ¯=−12Y400000−12Y3+Y2T−12Y10000Π¯11Π¯2112X¯1Π¯21Π¯11Π¯61−Y3−12Y2T−Y1−12Y1T+P0−Y100−Y4−12Y4+d12R1+d2−d12R2−Y2−12Y3T00−Y40Π¯12Π¯220Π¯22Π¯12Π¯71,Π¯61=e¯11TA21TSTZT+e¯3T0Ad22TSTZT,Π¯62=0n−r×2rQ12+Q22+d2−d1+1Q32e¯11+e¯11TA21TSTI−ZT+e¯3T0Ad22TSTI−ZT0n−r×2r,Π¯71=A22TSTZT,Π¯72=A22TSTI−ZT+Q13+Q23+d2−d1+1Q33.
Since the matrixA22 is invertible, applying Lemma 2 to (41) and combining (42), we obtain that(44)−12Y4+Y4T∗∗∗∗−12Y3+Y2T−12Y1+Y1T∗∗∗Π˜1Π˜2X˜∗∗−Y3−12Y2T−Y1−12Y1T+PΠ˜2T−Y1−Y1T∗−Y4−12Y4T+d12R11+d2−d12R21−Y2−12Y3TΠ˜1T−Y2−Y3T−Y4−Y4T<0,where(45)X˜=e1TQ1e1−e2TQ1e2+e1TQ2e1−e4TQ2e4+d2−d1+1e1TQ3e1−e3TQ3e3−e1−e2TE¯TR1E¯e1−e2−3E¯e1+E¯e2−2e5TR1E¯e1+E¯e2−2e5−E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7diagR2,3R2TTTdiagR2,3R2E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7+e1T000S¯A˜e1+A˜de3+A˜e1+A˜de3T000S¯Te1−e1TP000e1.Π˜1=AY3+Y40T−e1TE¯TY40T,Π2=AY1+Y20T−e1TE¯TY20T,S¯=ZS.
Next, we will give that inequality (36) can be satisfied if (44) holds.
Noted that equation (44) can be rewritten as(46)ϒ+ζTY1Y2Y3Y4Tψ+ψTY1Y2Y3Y4ζ<0,where(47)ϒ=0000d12R1+d2−d12R2000P000X˜000P000d12R1+d2−d12R20000,ζ=0−12IrA−Ir0−12Ir0A−e110−Ir,ψ=0Ir0Ir0Ir000Ir.
By Lemma 3, (46) is equivalent to(48)χζTϒχζ<0,χψTϒχψ<0,where(49)χζT=Ir000−12Ir0Ir0−12Ir000I7nAA−e11,χψT=00I7n000−Ir0Ir0−Ir000Ir,which can be rewritten as(50)X˜−2P−2d12R1+d2−d12R2<0,(51)−d12R1+d2−d12R20d12R1+d2−d12R2A−e110−PPAA−e11Td12R1+d2−d12R2ATP−X˜<0.
By Schur’s complement, (51) is equivalent to Ξ<0, which ensures the stability of the system. This completes the proof.Remark 1.
Theorem 1 gives a new criterion for discrete-time singular systems with time-varying delay. By Lemmas 2 and 3, the admissible condition (36) can be solved by (19). Compared with condition (36) and some of the existing conditions, some auxiliary matrices Y1, Y2, Y3, and Y4 are added in (19) in addition to the Lyapunov matrix P. The auxiliary matrices bring extra relaxation for the criterion.
Next, we will further deal with the state feedback control problem for discrete-time systems (15) and the following theorem is obtained.Theorem 2.
The closed-loop system (18) is admissible, if there exist matrices P>0, U1>0, U2>0, Q1>0, Q2>0, Q3>0, R1>0, R2>0, S, T, W, L, Y1, Y2, Y4, and ε, such that(52)−12Y4+Y4T∗∗∗∗∗∗−12Y3+Y2T−12Y1+Y1T∗∗∗∗∗Π˜1Π˜2X˜∗∗∗∗−Y3−12Y2T−Y1−12Y1T+PΠ˜2T−Y1−Y1T∗∗∗−Y4−12Y4T+d12R11+d2−d12R21−Y2−12Y3TΠ˜1T−Y2−Y3T−Y4−Y4T∗∗N¯Π3N¯Π4N¯Π5N¯Π4N¯Π3−U1∗N¯dΠ3N¯dΠ4N¯dΠ5N¯dΠ4N¯dΠ30−U2<0,(53)diagR2,3R2∗TTdiagR2,3R2>0,where(54)X˜=e1TQ1e1−e2TQ1e2+e1TQ2e1−e4TQ2e4+d2−d1+1e1TQ3e1−e3TQ3e3−e1−e2TE¯TR1E¯e1−e2−3E¯e1+E¯e2−2e5TR1E¯e1+E¯e2−2e5−E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7TdiagR2,3R2TTTdiagR2,3R2E¯e2−E¯e3E¯e2+E¯e3−2e6E¯e3−E¯e4E¯e3+E¯e4−2e7+e1T000SA¯e1+A¯de3T+A¯e1+A¯de3000STe1+e1T000In−rLTBTe1+e1TBL000In−re1−e1TP000e1+e1TM¯U1M¯Te1+e3TM¯dU2M¯dTe3,Π˜1=e1TA¯+e3TA¯dεY1+Y2WT+εe1TBLIr0T−e1TE¯TY40T,Π˜2=e1TA¯+e3TA¯dY1+Y2WT+e1TBLIr0T−e1TE¯TY20T,Y3=εY1+Y2−Y4,Π3,Π4,Π5,ei,i=1,2,…,7,are defined in Theorem 1. If it is the case, the controller gains can be solved as(55)K=LY1+Y2W0S−TH−1.Proof.
ReplacingAT, ΔAT, AdT, and ΔAdT into A+BK, ΔA, Ad, and ΔAd, respectively, and letting Y3=εY1+Y2−Y4, W2=W, W1=εW, and L=KY1+Y2W0ST. Using the similar procedure as the proof in Theorem 1, this theorem can be easily proved.Remark 2.
Usually, an additional equationY1xk+Y2E¯xk+1T−E¯xk+1+A˜+BKxk+A˜dxk−dk+∗=0 is added in the derivations of criteria for the state feedback controller design for discrete-time singular systems with time delay [34, 36, 38, 42]. It is obvious that it leads to a nonlinear problem for controller synthesis. The usual way in the existing controller design is to set Y2=εY1, where ε is fixed, such that the conditions can be transformed into LMIs. However, the dealing methods bring much conservatism and the results depend extremely on the parameter ε. In our method, by introducing auxiliary matrices Y1, Y2, Y3, and Y4 in Theorems 1 and 2 and letting Y3=εY1+Y2−Y4, the dependence on ε is weakened and obviously the conservatism is reduced. In general, the value of ε can be taken in the interval of 0,2.Remark 3.
The auxiliary matrices which are added in the criteria can really increase the computational complexity.


## 4. Numerical Example

Consider the discrete time-delay singular system (1) with(56)E=1000,A=1110.3,Ad=0.100.30.1,B=−20,M=Md=0.20.10−0.3,N=0.1000.1.It is proved in [33] that the system is unstable at least for a particular choice of dk. If 1≤dk≤1000, the method in [33, 38] is infeasible. By Theorem 2, set ε=1, the controller gain to make the closed-loop system robustly admissible is designed as K=0.48240.5194. Figure 1 depicts the closed-loop system responses with the control law uk. It is observed that the closed-loop system is robust stable with the uncertainties.Figure 1
Control results of example 1. (a) State response curves for the closed-loop system. (b) Input signals.
(a)
(b)


## 5. Conclusion

In this paper, the state feedback admissibilization for discrete-time singular systems with time-varying delay is under consideration. By using the Seuret summation inequality, the reciprocally convex inequality, and some relaxation techniques, a new admissible condition for the considered system is obtained. And the state feedback controller design problem can be solved by the linear matrix inequalities (LMIs) such that the closed-loop system is admissible. Finally, a simulation is provided to demonstrate the effectiveness of the proposed methods. Furthermore, research topics can include the extension of our results to more complex cases with performance control and performance analysis and to further reduce the conservatism.



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