# Improved Results on Delay-Dependent RobustH∞ Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

Authors: Jingsha Zhang; Yongke Li; Xiaolin Ma; Zhilong Lin; Changlong Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the problem of the delay-dependent robustH∞ control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and H∞ controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.


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## Body

## 1. Introduction

Strictly speaking, some degrees of time delays are frequently encountered in almost all practical engineering systems such as [1–3], in which the existence of the delays may lower the performances of these engineering control systems and even result in the instability of such control systems. Then, it is an essential requirement to investigate the stability for systems with time delays. And quantities of results on stability for time-delay systems in actual applications have been obtained in the near past [4–6]. The stability analyses of the systems can be classified into delay-dependent ones and delay-independent ones. The stability analyses criteria in delay-dependent systems are always less conservative than the delay-independent ones. In addition, the stability criteria are developed by applying the LKF methodology. Otherwise, for some time-delay systems in practice such as air-feed system [7] and power converters system [8], the controllers are designed to make the systems more practical by employing advanced methods for the stability analyses. Though the time delay can be constant or varying, the former is a special case of the latter which is always more complex and typical. Thus, the stabilization problem of dynamical systems with time-varying delays has been taken into consideration.On the other hand, there are also many typical practical models of time-delay neutral dynamical systems such as partial element equivalent circuit model [9] and population genetics and community ecology [10], which are described by neutral functional differential equations containing time delays not only in the state but also in its derivative. Obviously, studying neutral systems with mixed delays (NSMDs) is more challenging than the system with a delay only in the state. Results about stability analysis for NSMD have been obtained in recent years. In the following, some classical methods are shown for NSMD. In [11], delay-dependent conditions were improved via utilizing the LMI method. In [12], by applying a candidate LKF, less conservative criteria were proposed. Besides, to reduce conservatism, an approach of piece-wise delay was expressed in [13]. Advanced integral inequalities were used in [14] to get the delay-dependent stability conditions, which were less conservative. NSMD with uncertainties were observed, and then the robust stability analyses were discussed in [15]. As we can see in [16], compared with an inequality which might not cause a less conservative result, a tighter inequality based on a suitable LKF could get the advantage we want. From [11–16], we can see that the methods of LKF and integral terms are the mainstream ones. By constructing an LKF with inequality terms and applying the LMI method, sufficient conditions are obtained.As known, uncertainties can lead to poor performance in many time-delay systems. Therefore, much attention has been paid to the analysis of stability and stabilization for systems with parametric uncertainties. And robust control can be used for the analysis and design of the systems including uncertainties which are complex to handle in dynamical models. The problem of robust stabilization for a class of uncertain neutral time-delay dynamical systems with the uncertain matrix coefficients of both the delayed state and derivative of the delayed state is considered so that we can ensure the discussed systems stable under specific conditions. In addition, to overcome the external disturbances of actual operations in engineering models such as in [17, 18] and make them stable artificially, H∞ control is also an important issue. In [17], robust H∞ control was designed for PFC rectifiers. For the cooperative driving system with external disturbances and communication delays in the vicinity of traffic signals, robust H∞ control was studied in [18]. As presented in [19], the issue on robust H∞ control was discussed. Besides, H∞ control of neutral systems has been expressed in many papers up to now. In [20], the problem of robust H∞ control with state feedback was solved in the uncertain neutral system. For an uncertain neutral system in [21], H∞ performance for Markovian uncertain systems via the sliding mode control method was considered. Moreover, delay-dependent H∞ filtering conditions were proposed for uncertain neutral stochastic systems with Markovian switching and mode-dependent time delays [22]. By applying a H∞ state-feedback controller in [23], a neutral-type time-delay system was studied. And in another paper [24], the H∞ feedback controller was used to obtain the results.Taking all the discussed elements into consideration, a representative system named the teleoperation system was introduced in [25]. The teleoperation system is composed of a master manipulated by a human operator and a slave operating in a remote environment through a communication channel. Time-varying delays existed in the long-range or flexible communication channel [26]; thus, the problem of stability of teleoperation systems for time-varying delays by neutral LMI techniques was discussed in [25]. Besides, due to that no real system is pure linear, time-varying model uncertainties considered as perturbations existed in real implementation of bilateral teleoperation [27], and then the study on robust control design for teleoperation systems was done in [28]. Moreover, the stability criteria above were all given in the form of LMIs, which can be used to compute the maximum values of delays. And in order to obtain the controller parameters directly by solving LMIs, state-feedback control law was found to stabilize the system with guaranteeing performance in [29]. Above all, in our paper, both time-varying delays and uncertainties are considered in the neutral model to analysis the robust control with state-feedback controller by the technique of LMIs. Different from the general delay system such as [16, 23], this paper also considers delays in the neutral terms. And unlike the delays in [11–15, 20, 24], the delays shown in both discrete and distributed forms in this article are all varying and different instead of constant or the same delay function. Then, the model derived from the actual application is more complex but more representative. Moreover, appropriate Lyapunov function is constructed to deal with the complicated model terms technically and get the control gain directly without applying the traditional method in [30].Hence, motivated by the discussions above, the problem of delay-dependent robustH∞ control of uncertain neutral systems with mixed time-varying delays is explored in this article. After introducing an improved LKF and using the LMI method, a sufficient condition for the asymptotic stability of the system is established. Analyses on uncertainties and H∞ control are made then. Numerical examples are given to verify the results finally.The main contributions of this paper are summered as follows:(1)
In the LKF newly constructed, all terms are taken into account to obtain a larger delay bound.(2)
The methods including not only free-weighting matrices but also efficient convex optimization algorithms are applied technically to get results of less conservatism.(3)
Sufficient conditions are given all by LMIs, and the solutions can be solved by MATLAB Toolbox directly.(4)
Though similar papers concerning the delay-dependent robust control of uncertain neutral systems with mixed time-varying delays have already been many, there are some differences between the current submission and latest works. To the extent of the authors’ knowledge, none of the existing works including all the terms of robust control, neutral systems, uncertainties, and mixed time-varying delays derived from actual engineering problems. Besides, the Lyapunov functional proposed in this article can get better conservatism with larger delay bounds when compared with these in [31–33], which is shown in Example 3. Finally, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, such as in [30], the controller gain illustrated in Theorem 4 of this paper can be obtained more easily and simply.


### 1.1. Notation

The notations throughout this paper are fairly standard.A−1 and AT denote, respectively, the matrix inverse and transpose of A. ℜn and ℜn×m are the n-dimensional Euclidean space and the set of n×m real matrices. B>C means that B−C is positive definite. I is a unit matrix with compatible dimensions.MW∗N denotes MWWTN. Let f2=∫0∞f2dt, ft∈L20,∞. f expresses the Euclidean norm of function ft. L20,∞ refers to the space of square integral vectors.


## 1.1. Notation

The notations throughout this paper are fairly standard.A−1 and AT denote, respectively, the matrix inverse and transpose of A. ℜn and ℜn×m are the n-dimensional Euclidean space and the set of n×m real matrices. B>C means that B−C is positive definite. I is a unit matrix with compatible dimensions.MW∗N denotes MWWTN. Let f2=∫0∞f2dt, ft∈L20,∞. f expresses the Euclidean norm of function ft. L20,∞ refers to the space of square integral vectors.


## 2. Description of the Problem

Consider the NSMD with uncertainties:(1)x˙t=Atxt+A1txt−dt+A2tx˙t−ht+A3t∫t−dttxsds+A4t∫t−httxsds+But+Bwwt,zt=Ctxt+C1txt−dt+Dtwt,xt=φt,x˙t=ϕt,t∈−H,0,with xt∈ℜn, ut∈ℜm, wt∈ℜp, and zt∈ℜq as the state vector, the control input, the disturbance which belongs to L20,∞, and the controlled output, respectively; φt and ϕt are given continuous initial condition functions. Continuous functions dt and ht are time-varying delays which satisfy(2)0≤dt≤dM,d˙t≤u1<1,0≤ht≤hM,h˙t≤u2<1,H=maxdM,hM,where dM,hM,u1,u2 are constants. At,A1t,A2t,A3t,A4t,Ct,C1t and Dt are matrices with uncertainties satisfying(3)At=A+ΔAt,A1t=A1+ΔA1t,A2t=A2+ΔA2t,A3t=A3+ΔA3t,A4t=A4+ΔA4t,Ct=C+ΔCt,C1t=C1+ΔC1t,Dt=D+ΔDt,where A,A1,A2,A3,A4,C,C1 and D are known constant matrices with suitable dimensions. The time-varying unknown matrices ΔAt,ΔA1t,ΔA2t,ΔA3t,ΔA4t,ΔCt,ΔC1t, and ΔDt satisfy the following form:(4)ΔAtΔA1tΔA2tΔA3tΔA4tΔCtΔC1tΔDt=MFtNANA1NA2NA3NA4NCNC1ND,where M,NA,NA1,NA2,NA3,NA4,NC,NC1 and ND are known constant matrices with appropriate dimensions and Ft as an unknown matrix function satisfies FTtFt≤I for any t≥0.For system (1), the nominal form is given as follows:(5)x˙t=Axt+A1xt−dt+A2x˙t−ht+A3∫t−dttxsds+A4∫t−httxsds+But+Bwwt,zt=Cxt+C1xt−dt+Dwt,xt=φt,x˙t=ϕt,t∈−H,0.For system (5), a state-feedback controller is designed by(6)ut=Kxt,where K∈ℜm×n is a constant matrix to be designed.Combining (1) with (3) and the controller expression (6), we can get the closed-loop system of (1):(7)x˙t=A+ΔAt+BKxt+A1+ΔA1txt−dt+A2+ΔA2tx˙t−ht+A3+ΔA3t∫t−dttxsds+A4+ΔA4t∫t−httxsds+Bwwt,zt=C+ΔCtxt+C1+ΔC1txt−dt+D+ΔDtwt,xt=φt,x˙t=ϕt,t∈−H,0.Remark 1.
The mixed time-varying delays, respectively, distribute in the state and the derivative of the state, and the system could not always be stable with all values of delays. Then, we must find the bound of each time-varying delay, the larger the boundary and the better conservative system indicating that the broader time scope can be got to keep the system stable.Remark 2.
The two kinds of both discrete and distributed delays exist in the actual systems. And the large delay always causes the instability of the system. And either of the large delay can cause the instability of the system. Thus, considering the bounds of the discrete and distributed delays is a must.Lemma 1.
Schur complement [34]: given matrices X,Y,Z with Y>0, then we have(8)XZ∗−Y<0⇔X+ZY−1ZT<0.Lemma 2.
Jensen’s inequality [35]: for any scalar η>0 and any constant matrix M∈ℜn×n, M=MT>0, the following inequality holds(9)∫0ηwsdsTM∫0ηwsds≤η∫0ηwTsMwsds.Lemma 3.
Cauchy matrix inequality [36]: let N∈ℜn×n be a positive define matrix, then(10)−2yTx≤xTNx+yTN−1y,∀x,y∈ℜn.Lemma 4 (see [37]).
For any constant matrixQ∈ℜn×n,Q=QT>0, and free-weighting matrix V, we have(11)−∫t−dttx˙TsQx˙sds≤dtξTtVQ−1VTξt+2ξTtVxt−xt−dt,where(12)ξTt=xTtxTt−dt∫t−dttxsdsTxTt−dMxTt−ht∫t−httxsdsTxTt−hMx˙Ttx˙Tt−dtx˙Tt−ht.Lemma 5 (see [38]).
Given matricesD,E with suitable dimensions and Ft satisfying FTtFt≤I, then for any constant ε>0, the inequality holds(13)DFtE+ETFTtDT≤εDDT+ε−1ETE.


## 3. Main Results

In this section, a new sufficient condition criterion of asymptotic stability for system (14) is presented by using the LKF method and LMI technique:(14)x˙t−A2x˙t−ht=Axt+A1xt−dt+A3∫t−dttxsds+A4∫t−httxsds,xt=φt,x˙t=ϕt,t∈−H,0.Theorem 1.
For the given constantsdM,u1,hM, and u2, system (14) is asymptotically stable if there exist positive-definite matrices Q1,Q2,Qa,Qb,Zi,i=1,2,3,4,5,6,7,8,Rj,j=1,2,3,4,Sk,k=1,2,P1 and matrices with appropriate dimensions Pll=2,…,9,Uα,Vα,Gα,Hα,Lα,Jαα=1,2,W satisfying the following LMI:(15)Ω=Ω1+Ω2+Ω2T,Ω1=Ω11−Ω12Ω22−1Ω12T,φ11=P2TA+ATP2+Q1+Q2+Qa11+Qb11+dMZ5+Z6+hMZ7+Z8+h1U1+h1U1T+h2V1+h2V1T+dM2R3+hM2R4,φ12=P2TA1+ATP3−h1U1+h1U2T,φ13=P2TA3+ATP4,φ14=ATP5,φ15=ATP6−h2V1+h2V2T,φ16=P2TA4+ATP7,φ17=ATP8,φ18=P1−P2T+ATP9+Qa12+Qb12,φ22=P3TA1+A1TP3−h1Qa11−h1U2−h1U2T,φ23=P3TA3+A1TP4,φ24=A1TP5,φ25=A1TP6,φ26=P3TA4+A1TP7,φ27=A1TP8,φ28=−P3T+A1TP9,φ33=P4TA3+A3TP4−h1R3,φ34=A3TP5,φ35=A3TP6,φ36=P4TA4+A3TP7,φ37=A3TP8,φ38=−P4T+A3TP9,φ44=−Q1,φ46=P5TA4,φ48=−P5T,φ55=−h2Qb11−h2V2−h2V2T,φ56=P6TA4,φ58=−P6T,φ66=P7TA4+A4TP7−h2R4,φ67=A4TP8,φ68=−P7T+A4TP9,φ77=−Q2,φ78=−P8T,φ88=−P9−P9T+Qa22+Qb22+dMZ1+Z2+hMZ3+Z4+dMR1+hMR2+dM2S1+hM2S2,Ω11=φ11φ12φ13φ14φ15φ16φ17φ180P2TA2∗φ22φ23φ24φ25φ26φ27φ28−h1Qa12P3TA2∗∗φ33φ34φ35φ36φ37φ380P4TA2∗∗∗φ440φ460φ480P5TA2∗∗∗∗φ55φ560φ580P6TA2−h12Qb12∗∗∗∗∗φ66φ67φ680P7TA2∗∗∗∗∗∗φ77φ780P8TA2∗∗∗∗∗∗∗φ880P9TA2∗∗∗∗∗∗∗∗−h1Qa220∗∗∗∗∗∗∗∗∗−h12Qb22,Ω12=h1dMU1h2hMV1h1dMU2000000h2hMV20000000000,Ω22=−h1dMR10∗−h2hMR2,h1=1−u1,h2=1−u2,Ω2=G1+L1+G2+L2+2dMJ1+2hMJ2H1−G10−H1−L1H2−G20−H2−L2000+W−A−A1−A300−A40I0−A2,G1=G01G02⋮G10,H1=H01H02⋮H10,L1=L01L02⋮L10,J1=J01J02⋮J10,G2=G11G12⋮G20,H2=H11H12⋮H20,L2=L11L12⋮L20,J2=J11J12⋮J20,W=W01W02⋮W10,Ξ=ΩdMG1dMH1dML1hMG2hMH2hML22dMJ1dMJ12hMJ2hMJ2dM2J1hM2J2∗−dMZ100000000000∗∗−dMZ10000000000∗∗∗−dMZ2000000000∗∗∗∗−hMZ300000000∗∗∗∗∗−hMZ30000000∗∗∗∗∗∗−hMZ4000000∗∗∗∗∗∗∗−2dMZ500000∗∗∗∗∗∗∗∗−dMZ60000∗∗∗∗∗∗∗∗∗−2hMZ7000∗∗∗∗∗∗∗∗∗∗−hMZ800∗∗∗∗∗∗∗∗∗∗∗−dM2S10∗∗∗∗∗∗∗∗∗∗∗∗−hM2S2<0.Proof.
Choose an LKF candidate as(16)Vxt=∑i=17Vixt,where(17)V1xt=ζTtEPζt,V2xt=∫t−dMtxTsQ1xsds+∫t−hMtxTsQ2xsds,V3xt=∫t−dttxsx˙sTQaxsx˙sds+∫t−httxsx˙sTQbxsx˙sdsV4xt=∫−dM0du∫t+utx˙TsZ1+Z2x˙sds+∫−hM0du∫t+utx˙TsZ3+Z4x˙sds+∫−dM0du∫t+utxTsZ5+Z6xsds+∫−hM0du∫t+utxTsZ7+Z8xsds,V5xt=∫t−dtts−t−dtx˙TsR1x˙sds+∫t−htts−t−htx˙TsR2x˙sds,V6xt=dM∫t−dtts−t−dtxTsR3xsds+hM∫t−htts−t−htxTsR4xsds,V7xt=2∫−dM0du∫u0dv∫t+vtx˙TsS1x˙sds+2∫−hM0du∫u0dv∫t+vtx˙TsS2x˙sds,with(18)E=I000000000000000T,P=P10000000P2P3P4P5P6P7P8P9,ζt=xTtxTt−dt∫t−dttxsdsTxTt−dMxTt−ht∫t−httxsdsTxTt−hMx˙TtT.
The time derivative ofVxt is given by(19)V˙1xt=2ζTtEPζ˙t=2ζTtP1x˙t0⋮0=2ζTtPTx˙t0=2ζTtPTx˙t−x˙t+Axt+A1xt−dt+A2x˙t−ht+A3∫t−dttxsds+A4∫t−httxsds,(20)V˙2xt=xTtQ1xt−xTt−dMQ1xt−dM+xTtQ2xt−xTt−hMQ2xt−hM,(21)V˙3xt=xtx˙tTQaxtx˙t−1−d˙txt−dtx˙t−dtTQaxt−dtx˙t−dt+xtx˙tTQbxtx˙t−1−h˙txt−htx˙t−htTQbxt−htx˙t−ht≤xtx˙tTQa11+Qb11Qa12+Qb12∗Qa22+Qb22xtx˙t−1−u1xt−dtx˙t−dtTQa11Qa12∗Qa22xt−dtx˙t−dt−1−u2xt−htx˙t−htTQb11Qb12∗Qb22xt−htx˙t−ht,(22)V˙4xt=dMx˙TtZ1+Z2x˙t−∫t−dttx˙TsZ1x˙sds−∫t−dMt−dtx˙TsZ1x˙sds−∫t−dMtx˙TsZ2x˙sds+hMx˙TtZ3+Z4x˙t−∫t−httx˙TsZ3x˙sds−∫t−hMt−htx˙TsZ3x˙sds−∫t−hMtx˙TsZ4x˙sds+dMxTtZ5+Z6xt−∫t−dttxTsZ5xsds−∫t−dMt−dtxTsZ5xsds−∫t−dMtxTsZ6xsds+hMxTtZ7+Z8xt−∫t−httxTsZ7xsds−∫t−hMt−htxTsZ7xsds−∫t−hMtxTsZ8xsds,(23)V˙5xt=tx˙TtR1x˙t−1−d˙tt−dtx˙Tt−dtR1x˙t−dt−1−d˙t∫t−dttx˙TsR1x˙sds−t−dtx˙TtR1x˙t−1−d˙tx˙Tt−dtR1x˙t−dt+tx˙TtR2x˙t−1−h˙tt−htx˙Tt−htR2x˙t−ht−1−h˙t∫t−httx˙TsR2x˙sds−t−htx˙TtR2x˙t−1−h˙tx˙Tt−htR2x˙t−ht=dtx˙TtR1x˙t−1−d˙t∫t−dttx˙TsR1x˙sds+htx˙TtR2x˙t−1−h˙t∫t−httx˙TsR2x˙sds≤dtx˙TtR1x˙t−1−u1∫t−dttx˙TsR1x˙sds+htx˙TtR2x˙t−1−u2∫t−httx˙TsR2x˙sds.
By applying Lemma4,(24)V˙5xt≤dMx˙TtR1x˙t+1−u1dMξTtUR1−1UTξt+21−u1ξTtUxt−xt−dt+hMx˙TtR2x˙t+1−u2hMξTtVR2−1VTξt+21−u2ξTtVxt−xt−ht,U=U1TU2T00000000T,V=V1T000V1T00000T.
Similarly,(25)V˙6xt=dtdMxTtR3xt−1−d˙tdM∫t−dttxTsR3xsds+hthMxTtR4xt−1−h˙thM∫t−httxTsR4xsds≤dM2xTtR3xt−1−d˙tdt∫t−dttxTsR3xsds+hM2xTtR4xt−1−h˙tht∫t−httxTsR4xsds.
By using Lemma2, we can get(26)V˙6xt≤dM2xTtR3xt+∫t−dttxsdsT−1−u1R3∫t−dttxsds+hM2xTtR4xt+∫t−httxsdsT−1−u2R4∫t−httxsds,V˙7xt=2∫−dM0du∫u0x˙TtS1x˙t−x˙Tt+vS1x˙t+vdv+2∫−hM0du∫u0x˙TtS2x˙t−x˙Tt+vS2x˙t+vdv=dM2x˙TtS1x˙t−2∫−dM0du∫t+utx˙TsS1x˙sds+hM2x˙TtS2x˙t−2∫−hM0du∫t+utx˙TsS2x˙sds.
Based on the Newton–Leibniz formula, we have(27)2ξTtG1xt−xt−dt−∫t−dttx˙sds=0,2ξTtH1xt−dt−xt−dM−∫t−dMt−dtx˙sds=0,2ξTtL1xt−xt−dM−∫t−dMtx˙sds=0,2ξTtG2xt−xt−ht−∫t−httx˙sds=0,2ξTtH2xt−ht−xt−hM−∫t−hMt−htx˙sds=0,2ξTtL2xt−xt−hM−∫t−hMtx˙sds=0,2ξTtJ12dMxt−∫t−dttxsds−∫t−dMt−dtxsds−∫t−dMtxsds−2∫−dM0du∫t+utx˙sds=0,2ξTtJ22hMxt−∫t−httxsds−∫t−hMt−htxsds−∫t−hMtxsds−2∫−hM0du∫t+utx˙ss=0.
According to Lemmas2 and 3, we can get(28)2ξTtG1∫t−dttx˙sds=2ξTtG1−∫t−dttx˙sds=2dMξTtG1×1dM−∫t−dttx˙sds≤dMξTtG1Z1−1G1Tξt+1dM∫t−dttx˙sdsTZ1∫t−dttx˙sds≤dMξTtG1Z1−1G1Tξt+∫t−dttx˙TsZ1x˙sds.
Then, we can obtain(29)−2ξTtG1∫t−dttx˙sds≤dMξTtG1Z1−1G1Tξt+∫t−dttx˙TsZ1x˙sds,−2ξTtH1∫t−dMt−dtx˙sds≤dMξTtH1Z1−1H1Tξt+∫t−dMt−dtx˙TsZ1x˙sds,−2ξTtL1∫t−dMtx˙sds≤dMξTtL1Z2−1L1Tξt+∫t−dMtx˙TsZ2x˙sds,−2ξTtG2∫t−httx˙sds≤hMξTtG2Z3−1G2Tξt+∫t−httx˙TsZ3x˙sds,−2ξTtH2∫t−hMt−htx˙sds≤hMξTtH2Z3−1H2Tξt+∫t−hMt−htx˙TsZ3x˙sds,−2ξTtL2∫t−hMtx˙sds≤hMξTtL2Z4−1L2Tξt+∫t−hMtx˙TsZ4x˙sds,−2ξTtJ1∫t−dttxsds≤dMξTtJ1Z5−1J1Tξt+∫t−dttxTsZ5xsds,−2ξTtJ1∫t−dMt−dtxsds≤dMξTtJ1Z5−1J1Tξt+∫t−dMt−dtxTsZ5xsds,−2ξTtJ1∫t−dMtxsds≤dMξTtJ1Z6−1J1Tξt+∫t−dMtxTsZ6xsds,−2ξTtJ2∫t−httxsds≤hMξTtJ2Z7−1J2Tξt+∫t−httxTsZ7xsds,−2ξTtJ2∫t−hMt−htxsds≤hMξTtJ2Z7−1J2Tξt+∫t−hMt−htxTsZ7xsds,−2ξTtJ2∫t−hMtxsds≤hMξTtJ2Z8−1J2Tξt+∫t−hMtxTsZ8xsds,−4ξTtJ1∫−dM0du∫t+utx˙sds≤dM2ξTtJ1S1−1J1Tξt+2∫−dM0du∫t+utx˙TsS1x˙sds,−4ξTtJ2∫−hM0du∫t+utx˙sds≤hM2ξTtJ2S2−1J2Tξt+2∫−hM0du∫t+utx˙TsS2x˙sds.
From system (14), it is easy to get(30)2ξTtWx˙t−Axt−A1xt−dt−A2x˙t−ht−A3∫t−dttxsds−A4∫t−httxsds=0.
Combing formulas (19) to (30), we have(31)V˙xt≤ξTtΩ+dMG1Z1−1G1T+dMH1Z1−1H1T+dML1Z2−1L1T+hMG2Z3−1G2T+hMH2Z3−1H2T+hML2Z4−1L2T+2dMJ1Z5−1J1T+dMJ1Z6−1J1T+2hMJ2Z7−1J2T+hMJ2Z8−1J2T+dM2J1S1−1J1T+hM2J2S2−1J2Tξt.
By Lemma1, we can obtain V˙xt≤ξTtΞξt. Obviously, system (14) is ensured to be asymptotically stable.Remark 3.
In the field of analysis of delay-dependent stability, the main purpose is to find the maximum delay bounds for guaranteeing stability. In the discussed LKFVxt, the three terms ζTtEPζt, ∫−dM0du∫u0dv∫t+vtx˙TsS1x˙sds, and ∫−hM0du∫u0dv∫t+vtx˙TsS2x˙sds play an important role in reducing the conservatism of our results. More specifically, by taking the states ∫t−dMtxsds,∫t−hMtxsds,∫t−dttxsds,∫t−httxsds,∫−dM0du∫t+utx˙sds, ∫−hM0du∫t+utx˙sds,∫−dM0du∫t+utxsds, ∫−hM0du∫t+utxsds as variables, more information can be sufficiently utilized on state variables, which can yield less conservatism. And as mentioned in [37], in order to derive an upper bound of V˙5xt in each intervals by Lemma 4, two different free-weighting matrices U and V are introduced at each interval. In addition, the integral terms in V6xt devote to a less result with Jensen’s inequality presented.Remark 4.
The LMI problem illustrated in Theorem1 is a convex problem in relation to solution variables. And various convex optimization algorithms can solve the problem. In this paper, Matlab LMI Toolbox, as a useful software, for solving convex optimization problem, is used in which the interior-point algorithm was implemented.Remark 5.
The proposed criteria in Theorem1 may give larger stability regions in compassion with some of the existing criteria. This claim is due to the following characteristics of the proposed stability analysis scheme: in [16], the relationship between the augmented LKF and the tightness of inequalities was explored. It has been shown that a tighter inequality does not always lead to a less conservative criterion. Nevertheless, a suitable augmented LKF and a tighter inequality, both of which are indispensable, can effectively reduce conservatism. In this paper, an augmented LKF combined with more cross terms can be well dealt with Lemma 4 technically. In addition, double- and triple-integral terms help to further increase the stability region.Remark 6.
(i) The system modeled in (1) is more representative than the ones in [11, 16, 36, 37]. For example, if ΔAt=ΔA1t=ΔA2t=A3t=A4t=0, the neutral items are linear, then system (1) here can reduce to the neutral model without uncertainties in [11]. Besides, when A2t=A3t=A4t=0, the system becomes the general uncertain delay time model in [16, 37]. If At=A,A1t=A1,A2t=A3t=A4t=0, then the general delay time systems without uncertainties in [16, 36] are got. And if dt=d,ht=h, the varying delay terms are turned into constant delay ones. (ii) In the system of this article, all the coefficient matrices of the states and the derivative are functional matrices with uncertainties, different from the known constant matrices or few uncertain matrices in the existing literature. In the majority of the papers including all the cited papers in this paper, uncertainties are neglected more or less. However, in this article, uncertainties are all considered in the states and the derivative vector. (iii) Most articles about the content of neutral time-delay systems are concerned with the stability problem such as [9, 11–15], and our article not only solves the stability problem but also contains the control strategies.Remark 7.
IfA3=0 and A4=0, we can get a special case of system (14):(32)x˙t−A2x˙t−ht=Axt+A1xt−dt,xt=φt,x˙t=ϕt,t∈−H,0.
By applying Theorem1, solutions of system (32) and the upper delay bounds comparing with other results are shown in Example 1.Theorem 2.
For a prescribed scalarγ>0 and given constants dM,u1,hM,u2, system (5) with ut=0 is robustly asymptotically stable with H∞ performance γ if there exist positive-definite matrices Q1,Q2,Qa,Qb, Zi,i=1,2,3,4,5,6,7,8,Rj,j=1,2,3,4,Sk,k=1,2,P1, and matrices with appropriate dimensions Pll=2,…,9, U¯α,V¯α,G¯α,H¯α,L¯α,J¯α,α=1,2,W¯ satisfying the LMI:(33)Ω¯=Ω¯1+Ω¯2+Ω¯2T,Ω¯1=Ω¯11−Ω¯12Ω¯22−1Ω¯12T,Ω¯11=Ω110∗−γ2I,Ω¯12=Ω120,Ω¯22=Ω22,E¯=CC10⋯0︸8DT,Ω¯2=G¯1+L¯1+G¯2+L¯2+2dMJ¯1+2hMJ¯2H¯1−G¯10−H¯1−L¯1H¯2−G¯20−H¯2−L¯20000+W¯−A−A1−A300−A40I0−A2−Bw,G¯1=G¯01G¯02⋮G¯11,H¯1=H¯01H¯02⋮H¯11,L¯1=L¯01L¯02⋮L¯11,J¯1=J¯01J¯02⋮J¯11,G¯2=G¯11G¯12⋮G¯21,H¯2=H¯11H¯12⋮H¯21,L¯2=L¯11L¯12⋮L¯21,J¯2=J¯11J¯12⋮J¯21,W¯=W¯01W¯02⋮W¯11,Ξ¯=Ω¯dMG¯1dMH¯1dML¯1hMG¯2hMH¯2hML¯2∗−dMZ100000∗∗−dMZ10000∗∗∗−dMZ2000∗∗∗∗−hMZ300∗∗∗∗∗−hMZ30∗∗∗∗∗∗−hMZ4∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗2dMJ¯1dMJ¯12hMJ¯2hMJ¯2dM2J¯1hM2J¯2E¯000000000000000000000000000000000000000000−2dMZ5000000∗−dMZ600000∗∗−2hMZ70000∗∗∗−hMZ8000∗∗∗∗−dM2S100∗∗∗∗∗−hM2S20∗∗∗∗∗∗−I<0,where Ω11,Ω12,Ω22 are the same terms as in Theorem 1.Proof.
Introduce the followingH∞ performance index under the zero initial condition:(34)J∞=∫0∞zTtzt−γ2wTtwtdt.
For any nonzero disturbancewt∈L20,∞, we can see(35)J∞=∫0∞zTtzt−γ2wTtwt+V˙dt−V≤∫0∞zTtzt−γ2wTtwt+V˙dt=∫0∞ξtwtTΞ¯ξtwtdt.
Based on the method which is similar to the one utilized in Theorem1, we can conclude J∞<0, which means zt2≤γwt2. That is to say, system (5) with ut=0 is robustly asymptotically stable with H∞ performance γ.Remark 8.
To overcome the external disturbances in actual operations of engineering models and make them stable artificially, considering the problem of robustH∞ control is a must. The study [18] has shown the artificial applications of robust H∞ control under external disturbances.Remark 9.
As a special case of system (5), the following time-varying system is considered:(36)x˙t=Axt+A1xt−dt+Bwwt,zt=Cxt+C1xt−dt,xt=φt,t∈−dM,0.
LetA2=0,A3=0,A4=0, then the stability analysis can be got for system (36) by Theorem 2, and the conservatism of this method is shown in Example 2.Theorem 3.
For a prescribed scalarγ>0 and given constants dM,u1,hM,u2, system (1) with ut=0 is robustly asymptotically stable with H∞ performance γ if there exist positive-definite matrices Q1,Q2,Qa,Qb, Zii=1,2,3,4,5,6,7,8,Rji=1,2,3,4,Skk=1,2,P1, and matrices with appropriate dimensions Pll=2,…,9,U¯α,V¯α,G¯α,H¯α,L¯α,J¯αα=1,2,W¯, and λ>0 satisfying the following LMI:(37)Ξ˜=Ξ¯ΛmλΛn∗−λI0∗∗−λI<0,where(38)Λm=Λm1M0⋯000⋮00⋯0⋮⋱⋮0⋯00⋮000⋯0︸12M,ΛnT=Λn10⋯000⋮00⋯0⋮⋱⋮0⋯00⋮0Λn20⋯0︸120,Λm1=−W¯+P2P3P4P5P6P7P8P9000T,Λn1=NANA1NA300NA4000NA20,Λn2=NCNC10⋯0︸8ND,with the same terms in Theorem 2.Proof.
A,A1,A2,A3,A4,C,C1,D in Theorem 2 are substituted by A+ΔAt,A1+ΔA1t,A2+ΔA2t,A3+ΔA3t,A4+ΔA4t,C+ΔCt,C1+ΔC1t,D+ΔDt, respectively, and then we can obtain(39)Ξ¯+ΛmF¯tΛnT+ΛnF¯tTΛmT<0,F¯t=Ft0⋯000⋮0Ft⋯0⋮⋱⋮0⋯Ft0⋮000⋯0︸12Ft.
According to Lemma5, there exists λ>0 satisfying(40)Ξ¯+λ−1ΛmΛmT+λΛnΛnT<0.
Applying Lemma1, obviously, (40) is equal to (37) and the theorem is proved.Remark 10.
The uncertain terms are solved in Theorem3, which makes the system less complex to deal with.Remark 11.
The method applied with parameters in Example3 is proved to be less conservative.
In the following theorem, we will put our focus on the synthesis of the controller for system (1) and the goal is to construct a state-feedback control law (6). On the basis of the results of the previous section, closed-loop system (7) is analyzed. The following Theorem 4 gives an LMI-based computational procedure to determine a state-feedback controller.Theorem 4.
For given constantsdM,u1,hM,u2,γ>0 and a small enough scalar ε>0, system (1) is robustly asymptotically stable with performance γ if there exist positive-definite matrices Q1,Q2,Qa,Qb,Zii=1,2,3,4,5,6,7,8, Rjj=1,2,3,4,Skk=1,2,P1, and matrices with appropriate dimensions Plk=2,…,9, U¯α,V¯α,G¯α,H¯α,L¯α,J¯αα=1,2,W¯,K, and λ>0 satisfying the following LMI:(41)Ξ^=Ξ¯ΛmλΛnΛkΛB+εI∗−λI00∗∗−λI0∗∗∗−ΛB+εIT<0,Λk=Λk10⋯0︸130⋮00⋯0⋮⋱⋮0⋯0,ΛB=ΛB10⋯0︸130⋮00⋯0⋮⋱⋮0⋯0,Λk1=−W¯01+P2T−KT−W¯02+P3T0−W¯03+P4T0−W¯04+P5T0−W¯05+P6T0−W¯06+P7T0−W¯07+P8T0−W¯08+P9T0−W¯090−W¯100−W¯110,ΛB1=0−B−BT0,with the same terms in Theorem 3.Proof.
SubstitutingA in Theorem 3 by A+BK, it is easy to get(42)Ξ¯+λ−1ΛmΛmT+λΛnΛnT+ΛkΛBΛkT<0,ΛkΛBΛkT=Λk1ΛB1Λk1T0⋯0︸130⋮00⋯0⋮⋱⋮0⋯0,Λk1ΛB1Λk1T=ψ11ψ12⋯ψ18−KTBTW¯09T−KTBTW¯10T−KTBTW¯11Tψ21⋮ψ810⋯0⋮⋱⋮0⋯00⋮00⋮00⋮0−W¯09BK0⋯0000−W¯10BK0⋯0000−W¯11BK0⋯0000,where(43)ψ11=KTBT−W¯01T+P2+−W¯01+P2TBK,ψ12=KTBT−W¯02T+P3,ψ18=KTBT−W¯08T+P9,ψ21=−W¯02+P3TBK,ψ81=−W¯08+P9TBK.
There must be a small enough scalarε>0 subject to(44)Ξ¯+λ−1ΛmΛmT+λΛnΛnT+ΛkΛBΛkT+εΛkΛkT<0,which equals to(45)Ξ¯+λ−1ΛmΛmT+λΛnΛnT+ΛkΛB+εIΛkT<0.
Then, we can obtain (41). Thus, the theorem is proved.Remark 12.
To get the controller gain in Theorem4, the method utilized in this paper is faster and easier than the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively.Remark 13.
The feasibility of the method mentioned in Theorem4 is shown in Example 4.Remark 14.
In this paper, we focus on the improvement of the robust stability analysis ofH∞ control for delay-dependent uncertain neutral systems with mixed time-varying delays which shows in the following aspects. First, based on a new LKF, robust stabilization criteria for time-varying delays are established. Second, by introducing free-weighting matrices and using the technique of convex combination, sufficient conditions are derived in terms of LIMs. And it is convenient for us to get all the solutions by MATLAB toolbox directly.Remark 15.
In order to guarantee the solutions of the discussed system asymptotically stable, we develop a continuous state-feedback controller scheme with a rather simple structure to make a closed loop of the system. And when the parameters are given, we can know the value of the controller gain whether to be feasible or not by MATLAB toolbox directly. It is easy to know that the larger control gain is, the less conservative of Theorem4 is. However, a larger control gain means higher control cost. Therefore, we need to choose the control gain as small as possible on the basis of satisfying the inequalities in Theorem 4.


## 4. Numerical Examples

In this section, some examples are presented to demonstrate that the proposed methods are feasible and effective.Example 1.
Consider neutral-type system (32) with parameters(46)A=−200−2,A1=00.40.40,A2=c00c.Case I: forc=0.1,u1=0.9,u2=0,dM=1,hM=1, by LMI Toolbox, the main solutions are(47)P1=4.4021−0.0581−0.05814.4021,Q1=0.3829−0.0092−0.00920.3829,Q2=0.3829−0.0092−0.00920.3829,Qa=0.4348−0.00000.2490−0.0475−0.00000.4348−0.04750.24900.2490−0.04750.26240.0080−0.04750.24900.00800.2624,Qb=0.4348−0.00000.08560.0012−0.00000.43480.00120.08560.08560.00120.29200.00570.00120.08560.00570.2920,Z1=0.05080.00140.00140.0508,Z2=0.04200.00160.00160.0420,Z3=0.26950.00560.00560.2695,Z4=0.22680.00590.00590.2268,Z5=0.07090.00330.00330.0709,Z6=0.03420.00250.00250.0342,Z7=0.68120.00560.00560.6812,Z8=0.77190.00610.00610.7719,R1=0.03930.00180.00180.0393,R2=0.15420.00540.00540.1542,R3=0.0507−0.0093−0.00930.0507,R4=0.3829−0.0092−0.00920.3829,S1=0.00490.00020.00020.0049,S2=0.22680.00590.00590.2268,U1=24.6682−1.3527−1.352724.6682,U2=2.6363−0.0556−0.05562.6363,V1=−0.2166−0.0261−0.0261−0.2166,V2=−0.0086−0.0015−0.0015−0.0086.Because of the large number and big dimensions, the solution toPll=2,…,9,Uα,Vα,Gα,Hα,Lα,Jαα=1,2 and W is omitted here.The results of the state response of the neutral system with parameters in case I are, respectively, depicted in Figure1.Case II: by taking parameterc=0.1, the admissible upper bounds of dt for different u2 are listed in Table 1 by Theorem 1. From the comparison, we can see that the methods proposed in this paper are less conservative.
As we can see in Table1, the results of delay bound to guarantee stability with two different u2 were presented in [12, 31, 39] and the best results were in [31]. However, by applying Theorem 1, we can obtain larger values than [31].Table 1
Example1: comparison of maximum allowable delay bound dM.
u200.6[39]1.88421.6904[12]3.70243.3150[31]4.42393.9563Theorem16.03175.5131;Figure 1
State responses with some typical parameters of Example1.Remark 16.
The example above has shown that our obtained methods not only can solve the normal stability problem as a special case proposed in this paper but also can get results which are better than some existing results.Example 2.
Consider time-varying delay system (36) with matrix parameters(48)A=−0.6238−1.01322.0116−0.2106,A1=−0.50110.7871−0.3002−0.5231,Bw=−0.43260.12531.66560.2877,C=0.2134−0.01910.1119−0.1665,C1=0.08160.12900.07120.0669.
By applying the conditions of Theorem2, Table 2 reports the maximum upper bound dM for different γ with u1=0,u2=0.1,hM=0.1, and simulation results are conducted to show the differences between some typical different parameters in Figure 2.
In Table2, comparison of maximum allowable delay bound dM for different values of γ is listed. With the increase of γ, the maximum allowable delay bound becomes larger. And it also shows that Theorem 2 provides improved delay bounds.Table 2
Example2: comparison of maximum allowable delay bound dM for given γ.
γ2.02.53.03.54.04.5[40]0.42030.47790.51460.54010.55890.5733[30]0.47340.52370.55450.57540.59040.6018Theorem20.60130.63020.64250.65620.66350.6700;Figure 2
State responses with some typical parameters of Example2.Example 3.
Consider the uncertain neutral system(49)x˙t−A2x˙t−ht=A+ΔAxt+A1+ΔA1xt−dt,xt=φt,x˙t=ϕt,t∈−H,0,where(50)A=−200−0.9,A1=−10−1−1,A2=c00c,M=1001,NA=1.6000.05,NA1=0.1000.3,c<1.
Foru1=0.1,u2=0.1 with different values of c and u2=0.1,c=0.1 with different values of u1, Tables 3 and 4 give out the related comparative results, respectively. From messages of the tables, it is easy to get the conclusion that the maximum allowable delay bounds increase along with the decrease of c or u1. Moreover, we can see that the results obtained in Theorem 3 are less conservative than those in [11, 31–33, 39, 42, 43, 45, 46], which illustrates the effectiveness of the methods we proposed.Table 3
Example3: comparison of maximum allowable upper bound dM=hM for u1=0.1,u2=0.1 of different values of c.
c00.10.20.30.40.5[41]0.92420.73860.55990.41230.29350.1914[42]0.97550.78050.60200.45600.31120.1997[11]1.07200.92080.75160.59910.46250.3402[39]1.16600.96210.77810.61640.47270.3467[43]1.17260.96650.78150.61810.47440.3476[44]1.27231.03600.83020.65120.49580.3607[45]1.32091.07770.87010.68270.51230.3596[33]1.34181.08410.85850.66640.52040.3649[31]1.35191.11270.89720.70550.53660.3882[32]2.21651.71871.30220.95790.67640.4488Theorem32.76122.03511.52941.07630.72050.4711;Table 4
Example3: comparison of maximum allowable delay bound dM=hM for u2=0.1,c=0.1 with different values of u1.
u10.10.30.50.70.9[42]0.77120.68950.57800.42190.1704[11]0.91090.85370.79800.74720.7199[39]0.96230.90750.85030.78910.7740[46]0.96680.92020.88760.86600.8537[43]0.96850.95980.95890.95840.9581[45]1.06570.94260.91320.89430.8943[31]1.10041.02660.98580.97080.9701[33]1.10261.00860.97510.97200.9704[32]1.71871.61351.51201.42491.3901Theorem32.10381.89771.66031.53021.4498;Example 4.
Let us consider system (1) with the parameters(51)A=−0.05−0.5−0.01−0.9,A1=1.80.3−1−1,A2=1.50.1−0.1−0.1,A3=−0.1−0.05−0.1−0.1,A4=−0.1−0.01−0.1−0.01,B=−0.1−0.5,Bw=0.01−0.10.1−0.01,C=0.1−0.5−0.01−0.05,C1=0.1−0.5−0.1−0.05,D=−0.30.010.05−0.15,M=0.2000.2,NA=0.1000.15,NA1=0.1000.14,NA2=0.1000.13,NA3=0.1000.12,NA4=0.1000.12,NC=0.1000.11,NC1=0.1000.1,ND=0.1000.09.
Letγ=0.935,dM=3,hM=0.1,u1=0.2,u2=0.3,ε=0.51,Ft=sint, according to the strict LIMs of Theorem 4 by LMI toolbox in MATLAB, we can obtain the state-feedback gain:(52)K=1.81480.7237,and the value γopt=0.3601.


## 5. Conclusion

In this work, the robustH∞ stability and stabilization for the delay-dependent uncertain neutral system with mixed time-varying delays have been studied in detail. Based on an LKF newly constructed and methods of free-weighting matrices and various efficient convex optimization algorithms, several new robust H∞ stability and stabilization criteria are expressed in terms of LMIs. Especially, a desired state-feedback controller gain can be obtained directly by MATLAB, and it can lead to a less conservative result. Finally, numerical examples are shown to illustrate the superiority and feasibility of the proposed method.The weakness of the obtained results is that terms in the constructed Lyapunov function should be handled technically to make the system stable, and then appropriate analysis techniques and methods must be provided. Thus, dealing with a bit too many terms is a challenge. And another weakness is that the nonlinear disturbances have not been contained in our modeled systems. Thus, dealing with nonlinear disturbances in the neutral-type system is in our future work. Besides, the stability problem of neutral-type neural networks with time-varying delay has been more popular recently. Hence, we will also focus on the stability content of different types of delayed neutral-type neural networks and the neural networks with time delays: (i) fixed-time stabilization of neutral-type inertial neural networks with discrete and distributed time-varying delays; (ii) stability criteria Hopfield neural networks of neutral-type with multiple delays.



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