# H ∞ Sampled-Data Control for Singular Neutral System

Authors: Minjie Zheng; Yujie Zhou; Shenhua Yang; Lina Li; Yongfeng Suo
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This study is concerned with theH∞ control problem for singular neutral system based on sampled-data. By input delay approach and a composite state-derivative control law, the singular system is turned into a singular neutral system with time-varying delay. Less conservative result is derived for the resultant system by incorporating the delay decomposition technique, Wirtinger-based integral inequality, and an augmented Lyapunov-Krasovskii functional. Sufficient conditions are derived to guarantee that the resulting system is regular, impulse-free, and asymptotically stable with prescribed H∞ performance. Then, the H∞ sampled-data controller is designed by means of linear matrix inequalities. Finally, two simulation results have shown that the proposed method is effective.


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## Body

## 1. Introduction

In the last decades, modern control system widely used the digital computers to control continuous-time system. In the system, the continuous-time signals are transformed into discrete-time control signals by being sampled and quantized with a digital computer, and they will be transformed into continuous-time signals again by the zero-order holder. Hence, both discrete-time and continuous-time signals occur in one system with a continuous-time framework, which is called sampled-data system. Recently, three main approaches have been adopted to analyze the sampled-data systems. The first one is lifting technology [1, 2], in which the sampled-data system is transformed into a discrete-time system. However, this approach cannot deal with the uncertain sampling intervals problem. The second approach is based on the impulsive modeling of sampled-data systems in which a time-varying periodic Lyapunov function is used [3, 4]. The disadvantage of this approach is that the sampling interval of the process’s output must be constant. Input delay approach, which is proposed by Fridman et al. in [5], is the third methods. It transforms the sampling period into a bounded time-varying delay, and its important advantage is that the sampling distance is not needed to be constant. Besides, the approach can deal with the problem of system with nonuniform uncertain sampling (see [6]), which is difficult for traditional lifting techniques to deal with. In the past years, the approach has considerable successful applications in neural networks system [7–9], fuzzy system [10], chaotic system [11, 12], complex dynamic system [13–15], multiagent system [16, 17], and so on.Singular systems, also referred to as implicit systems or descriptor systems, have gained considerable attentions during the past decades. Compared to regular systems, singular systems can better describe physical systems and have wide applications in various systems such as aerospace systems, power systems, and mechanical systems. In the past years, many results have been reported to analyze the admissibility problem for singular systems [18–23]. For example, the problems of stability analysis and stabilization have been investigated in [19, 20]. The H∞ control problem has been discussed in [21]. The passivity and dissipativity problems of singular systems with time delays have been studied in [22, 23]. In recent year, sampled-data control theory is used for singular system by proposing a control law as [24, 25]. In [24], the dissipative fault-tolerant cascade control synthesis for a class of singular networked cascade control systems (NCCS) with both differentiable and nondifferentiable time-varying delays has been studied. In [25], the problem of the event-triggered stabilization for linear singular systems based on sampled-data is considered. In [24, 25], the state feedback control laws are all considered in the system. However, to the best of our knowledge, no literatures have considered the acceleration feedback for the singular system with sampled-data. The acceleration feedback can improve the system controller’s performance effectively. Moreover, it can suppress varying disturbances, so it has considerable applications in practical systems such as vibration suppression system and mechanical system. Thus, designing a composite control law which includes the state feedback and acceleration feedback for the system is the paper’s aim. Then, the linear sampled-data singular system is turned into a singular neutral system with sampled-data.Up to now, very little interest has been paid for singular neutral systems. In [26], the stability and state feedback stabilization problems of singular neutral systems are considered. Reference [27] studies the stability problems of singular neutral system with mixed delays. Reference [28] concerns the problem of the delay-dependent robust stability for neutral singular systems with time-varying delays and nonlinear perturbations. In [29], the problem of stability of singular neutral systems with multiple delays is studied. Reference [30] studies the problem of robust stability and stabilization of uncertain neutral singular systems and develops a new stability criterion of the differential operator by the final value theorem for Laplace transform. Reference [31] concerns the problem of output strictly passive control for uncertain singular neutral systems. To analyze the singular neutral systems, several methods have been proposed, among which the more popular approaches are Jensen’s inequality and free weighting matrix approach. Recently Seuret has proposed a new inequality called Wirtinger-based integral inequality in [32], which can provide more accurate estimation than the Jensen inequality. So, if the inequality is employed for investigating the singular neutral systems, we can derive an improved result. Besides, delay decomposition approach, which uses the method to divide the delay interval into N equal-length subintervals, is proposed in [33]. It is worth noticing that the delay decomposition approach can reduce the conservatism. However, to the best of the authors’ knowledge, little literatures have been found to study the sampled-data control problem for singular neutral system combining the delay decomposition approach with Wirtinger-based integral inequality despite its practical importance which motivates our present research work.In the paper, the issue aboutH∞ control for singular neutral system based on sampled-data is discussed. By input delay approach and a composite state-derivative control law, the linear singular system is turned into a singular neutral system with time-varying delay. By adding more information in the integral term, an augmented Lyapunov-Krasovskii functional is constructed. Less conservative results are derived for the resulting system by integrating Wirtinger-based integral inequality with delay decomposition approach. Sufficient conditions are derived to guarantee that the resulting system is regular, impulse-free, and asymptotically stable with prescribed H∞ performance. Then the H∞ sampled-data controller is obtained by means of linear matrix inequalities. Finally, we give two illustrate examples to show that the proposed method is effective.


## 2. Problem Formulation

The linear singular system is considered as follows:(1)Ex˙t=Axt+B1ut+Bwt,t>0,yt=Cxt+Dut,t∈-d,0,xt=ϕt,where xt∈Rn is the state vector, ut∈Rn is the control input and the initial condition, w(t) is the disturbance input vector, and ϕ(t) is the compatible initial function. A, B, and B1 are known constant matrices; matrix E∈Rn×n is assumed to be singular and the rank E=r≤n.In this paper, the state variable of the system is assumed to be measured at the sampling instant0=t0<t1<t2<⋯<tk<⋯; that is, in the interval tk≤t≤tk+1, only x(tk) is available for control purposes. The sampling period follows the assumption that it is bounded by a constant d; that is,(2)tk+1-tk≤d,∀k≥0,d>0.Then, for system (1), we choose the composite state feedback control law as(3)ut=utk=K1xtk+K2x˙tk,tk≤t<tk+1,where tk represents the sampling instant and u(tk) represents the discrete-time control signal. K1 and K2 are the controller gain matrix that will be designed.By substituting (3) into (1), we obtain(4)Ex˙t=Axt+A1xtk+G1x˙tk+Bwt,t>0,yt=Cxt+D1xtk+D2x˙tk,xt=ϕt,t∈-d,0,where(5)A1=B1K1,G1=B1K2,D1=DK1,D2=DK2.Remark 1.
By the composite state-derivative feedback control law (3), system (1) is converted to the singular system in neutral type. The state feedback as well as acceleration feedback is of great significance in actual sampled-data control system. For example, in the dynamic positioning (DP) ship system, by measuring the acceleration and velocity of the DP ship, the acceleration and velocity feedback loop are established, which can improve the DP controller’s performance and suppress varying disturbances like wind, waves, and ocean currents greatly.Remark 2.
Note that (4) involves both discrete and continuous signals, which is more different and practical than continuous-time control approach for singular neutral system. Besides, because the parameter uncertainties exist in the sampled-data control system, the traditional lifting technique is difficult to deal with the problem.Throughout the paper, we introduce the definitions as follows.Definition 3 (see [36]).
(1) The sampled-data control of singular neutral system (4) with w(t)=0(6)Ex˙t=Axt+A1xtk+G1x˙tkis said to be regular and impulse-free if the pair (E,A) is regular and impulse-free.
(2) System (6) is said to be asymptotically admissible, if it is regular, impulse-free, and asymptotically stable.Definition 4.
System (4) is said to have H∞ performance γ if the following inequality is satisfied:(7)yt2≤γwt2for all nonzero w(t)∈L20,∞ under zero initial condition, where γ>0.Using the input delay approach, the state feedback controlleru(t) is rewritten as(8)ut=utk=ut-t-tk=ut-τt,tk≤t<tk+1,τt=t-tk,where the time-varying delay τ(t) is piecewise-linear satisfying(9)0≤τt≤d,τ˙t=1,t≠tk.Thus, the singular neutral system based on sampled-data in (4) can be converted to the singular neutral system with time-varying delay τ(t) as follows:(10)Ex˙t=Axt+A1xt-τt+G1x˙t-τt+Bwt,t>0,yt=Cxt+D1xt-τt+D2x˙t-τt,xt=ϕt,t∈-d,0,whereA1=B1K1, G1=B1K2, D1=DK1, and D2=DK2.For obtaining the main results, we state the lemma as follows.Lemma 5 (see [37]).
For a given matrixR>0 and scalars a and b satisfying a<b, the following inequality holds for all continuously differentiable function ω in [a,b]→Rn(11)∫abw˙TsRw˙sds≥1b-awb-waTRwb-wa+3b-aΩTRΩ,where(12)Ω=wb+wa-2b-a∫abwsds.


## 3. Main Results

TheH∞ sampled-data control problem for singular neutral system is studied in this section. By constructing an augmented Lyapunov-Krasovskii functional, combining the delay decomposition technique with the Wirtinger-based integral inequality, less conservative results can be derived.Theorem 6.
For given scalard>0, the closed-loop system (6) is asymptotically admissible, if there exist symmetric positive-definite matrices P,S,Qi,Zi,i=1,…,N, such that(13)ETP=PTE≥0,(14)Ξ11Ξ210⋯0ΞN+210⋯00Ξ2N+31Ξ2N+41AT∗Ξ22⋱⋱⋯ΞN+22⋱⋱⋮⋮000∗∗⋱ΞNN-100⋱⋱0⋮⋮⋮⋮∗∗∗ΞNNΞN+1N⋮⋱⋱Ξ2N+1N⋮⋮⋮⋮∗∗∗∗ΞN+1N+10⋯0Ξ2N+1N+1⋮⋮⋮⋮∗∗∗∗∗-12ETZiE0⋯⋯⋮⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮⋮∗∗∗∗∗∗∗∗-12ETZNE0⋮⋮⋮∗∗∗∗∗∗∗∗∗Ξ2N+22N+2⋮⋮0∗∗∗∗∗∗∗∗∗∗0⋮A1T∗∗∗∗∗∗∗∗∗∗∗0G1T∗∗∗∗∗∗∗∗∗∗∗∗-δ<0,where(15)Ξ11=PTA+ATP+S1+Q1-4ETZ1E,ΞN+1+ii=dNETRiE+6ETZiE,ΞN+1+ii+1=-dNETRiE+6ETZiE,i=1,…,NΞ2N+31=PA1,Ξ2N+41=PG1,Ξii=-Qi-1+Qi-4ETZi-1E-4ETZiE,i=2,…,NΞi+1i=-2ETZiE,i=1,…,N,ΞN+1N+1=-QN-4ETZNE-S1,Ξ2N+22N+2=-ETS2E,δ=dN2∑i=1NETZiE+ETS2E.Proof.
First of all, we prove system (6) is regular and impulse-free. Since rank E=r≤n, there exist nonsingular matrices M^ and H^ such that(16)E-=M^EH^=Ir000.
Similar to (16), we define(17)A-=M^AH^=A11A12A21A22,P-=M^-TPH^=P11P12P21P22.
From (13) and the expressions (16) and (17), we can obtain that P12=0. Then we premultiply and postmultiply Ξ11<0 by H^T and H^, respectively; the inequality A22TP22+P22TA22<0 can be obtained, which implies A22 is nonsingular and the pair (E,A) is regular and impulse-free. Then, by Definition 3, system (6) is regular and impulse-free.
Next, the asymptotically stability of system (6) will be proved. An augmented Lyapunov- Krasovskii functional is chosen as follows:(18)Vt=∑i=16Vit,t∈tk,tk+1,(19)V1t=xtTETPxt,V2t=∫t-dtxsTS1xsds,V3t=∫t-dtx˙sTETS2Ex˙sds,V4t=∑i=1N∫t-id/Nt-i-1d/NxTsQixsds,V5t=∑i=1N∫t-id/Nt-i-1d/NxTsdsETRiE∫t-id/Nt-i-1d/Nxsds,V6t=hN∑i=1N∫-id/N-i-1d/N∫t+θtx˙TsETZiEx˙sdsdθ.
Calculating the derivative ofV(t), we can get that (20)V˙1t=2xtTPEx˙t,V˙2t=xtTS1xt-xt-dTS1xt-d,V˙3t=x˙tTETS2Ext-x˙t-dTETS2Ex˙t-d,V˙4t=∑i=1NxTt-i-1dNQixt-i-1dN-xTt-idNQixt-idN,V˙5t=∑i=1NxTt-i-1dN-xTt-idNETRiE∫t-id/Nt-i-1d/Nxsds,V˙6t=∑i=1NdN2x˙TtETZiEx˙t-dN∫t-id/Nt-i-1d/Nx˙TtETZiEx˙tds.
Employing Lemma5, we have(21)-dN∫t-id/Nt-i-1d/Nx˙TtETZiEx˙tds≤xt-i-1dNxt-idNdN∫t-id/Nt-i-1d/NxsdsT-4ETZiE-2ETZiE6ETZiE∗-4ETZiE6ETZiE∗∗-12ETZiExt-i-1dNxt-idNdN∫t-id/Nt-i-1d/Nxsds.Then(22)V˙6t≤∑i=1NdN2x˙TtETZiEx˙t+xt-i-1dNxt-idNdN∫t-id/Nt-i-1d/NxsdsT-4ETZiE-2ETZiE6ETZiE∗-4ETZiE6ETZiE∗∗-12ETZiExt-i-1dNxt-idNdN∫t-id/Nt-i-1d/Nxsds.Substituting inequalities (22) into V˙(t) it can be concluded that(23)V˙t≤2xtTPEx˙t+xtTS1xt-xt-dTS1xt-d+x˙tTETS2Ex˙t-x˙t-dTETS2Ex˙t-d+∑i=1NxTt-i-1dNQixt-i-1dN-xTt-idNQixt-idN+∑i=1NxTt-i-1dN-xTt-idNETRiE∫t-id/Nt-i-1d/Nxsds+∑i=1NdN2x˙TtETZiEx˙t+xt-i-1dNxt-idNdN∫t-id/Nt-i-1d/NxsdsT-4ETZiE-2ETZiE6ETZiE∗-4ETZiE6ETZiE∗∗-12ETZiExt-i-1dNxt-idNdN∫t-id/Nt-i-1d/Nxsds≤ςTtΘ+ψTδψςt,where(24)ςt=xTtxTt-dN⋯xTt-d1N∫t-d/NtxTsds⋯1N∫t-dt-N-1d/NxTsdsx˙Tt-dxTt-τtx˙Tt-τt,ψ=A0⋯0BK10⋯0BK2,Θ=Ξ11Ξ210⋯0ΞN+210⋯00Ξ2N+31Ξ2N+41∗Ξ22⋱⋱⋯ΞN+22⋱⋱⋮⋮00∗∗⋱ΞNN-100⋱⋱0⋮⋮⋮∗∗∗ΞNNΞN+1N⋮⋱⋱Ξ2N+1N⋮⋮⋮∗∗∗∗ΞN+1N+10⋯0Ξ2N+1N+1⋮⋮⋮∗∗∗∗∗-12ETZiE0⋯⋯⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮∗∗∗∗∗∗∗∗-12ETZNE0⋮⋮∗∗∗∗∗∗∗∗∗Ξ2N+22N+2⋮⋮∗∗∗∗∗∗∗∗∗∗0⋮∗∗∗∗∗∗∗∗∗∗∗0.By Schur complement, inequality (14) implies (25)Θ+ψTδψ<0,where Θ is defined in (24); it is clear from (25) that V˙(t)<0; hence, system (6) is asymptotically stable. This completes the proof.Remark 7.
In [38, 39], integral of delay term is always adopted to construct an augmented Lyapunov-Krasovskii functional. Therefore, in order to consider the information of delay term sufficiently in our constructed functional (18), an integral term of(26)∫t-id/Nt-i-1d/NxTsdsETRiE∫t-id/Nt-i-1d/Nxsdsis added in V4(xt).Remark 8.
In the Theorem6, delay decomposition approach, which is established by decomposing the delay intervals τ(t) into N equidistant subintervals, is employed for constructing the Lyapunov-Krasovskii functional (18). It is noted that the delay decomposition approach can derive less conservative results. Besides, the Wirtinger-based integral inequality is used for estimating the derivative V˙6(t). It is pointed out that, compared to Jensen inequality in [40–42], the Wirtinger-based integral inequality has tighter upper bounds and less conservatism. Thus, incorporating delay decomposition approach with Wirtinger-based integral inequality, we can derive less conservative result for the system.Next, theH∞ performance will be considered for system (10).Theorem 9.
For given scalard>0 and a prescribed scalar γ>0, the closed-loop system (10) is asymptotically admissible with H∞ performance γ, if there exist symmetric positive-definite matrices P,S,Qi,Zi,i=1,…,N, such that(27)ETP=PTE≥0,(28)Ξ11Ξ210⋯0ΞN+210⋯00Ξ2N+31Ξ2N+41Ξ2N+51ATCT∗Ξ22⋱⋱⋯ΞN+22⋱⋱⋮⋮⋮⋮000∗∗⋱ΞNN-100⋱⋱0⋮⋮⋮⋮⋮⋮∗∗∗ΞNNΞN+1N⋮⋱⋱Ξ2N+1N⋮⋮⋮⋮⋮⋮∗∗∗∗ΞN+1N+10⋯0Ξ2N+1N+1⋮⋮⋮⋮⋮⋮∗∗∗∗∗-12ETZiE0⋯⋯⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗-12ETZNE0⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗∗Ξ2N+22N+2⋮⋮⋮00∗∗∗∗∗∗∗∗∗∗0⋮⋮K1TB1TK1TDT∗∗∗∗∗∗∗∗∗∗∗00K2TB1TK2TDT∗∗∗∗∗∗∗∗∗∗∗∗-γ2IBT0∗∗∗∗∗∗∗∗∗∗∗∗∗-δ0∗∗∗∗∗∗∗∗∗∗∗∗∗∗-I<0,(29)Ξ11=PTA+ATP+S1+Q1-4ETZ1E,ΞN+1+ii=dNETRiE+6ETZiE,ΞN+1+ii+1=-dNETRiE+6ETZiE,i=1,…,N,Ξ2N+31=PB1K1,Ξ2N+41=PB1K2,Ξ2N+51=PB,Ξii=-Qi-1+Qi-4ETZi-1E-4ETZiE,i=2,…,N,Ξi+1i=-2ETZiE,i=1,…,N,ΞN+1N+1=-QN-4ETZNE-S1,Ξ2N+22N+2=-ETS2E,δ=dN2∑i=1NETZiE+ETS2E.Proof.
It is obvious that (14) implies (28) holds which, according to Theorem 6, guarantees the asymptotically admissibility of system (10) when w(t)=0. Then, we will prove that system (10) satisfies the H∞ performance γ.
Choose the same Lyapunov-Krasovskii functional given in (14), and then follow similar lines in above proof; we can get(30)yTtyt-γ2wTtwt+V˙t≤ζTtΩ+φTσφζt,where(31)ζTt=xTtxTt-dN⋯xTt-d1N∫t-d/NtxTsds⋯1N∫t-dt-N-1d/NxTsdsx˙Tt-dxTt-τtx˙Tt-τtwt,φ=φ1φ2,φ1=A0⋯0BK10⋯0BK2E,φ2=C0⋯0D1K10⋯0D1K20,Ω=Ξ11Ξ210⋯0ΞN+210⋯00Ξ2N+31Ξ2N+41Ξ2N+51∗Ξ22⋱⋱⋯ΞN+22⋱⋱⋮⋮000∗∗⋱ΞNN-100⋱⋱0⋮⋮⋮⋮∗∗∗ΞNNΞN+1N⋮⋱⋱Ξ2N+1N⋮⋮⋮⋮∗∗∗∗ΞN+1N+10⋯0Ξ2N+1N+1⋮⋮⋮⋮∗∗∗∗∗-12ETZiE0⋯⋯⋮⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮⋮∗∗∗∗∗∗∗∗-12ETZNE0⋮⋮⋮∗∗∗∗∗∗∗∗∗Ξ2N+22N+2⋮⋮⋮∗∗∗∗∗∗∗∗∗∗0⋮⋮∗∗∗∗∗∗∗∗∗∗∗00∗∗∗∗∗∗∗∗∗∗∗∗-γ2I.
By Schur complement, inequality (28) guarantees(32)Ω+φTσφ<0.
Thus, from (30), we obtain(33)yTtyt-γ2wTtwt+V˙t<0.
Therefore, we havey(t)2≤γw(t)2 for random nonzero wt∈L20,∞ and then establish H∞ performance. This completes the proof.Now, based on Theorem9, the H∞ sampled-data controller (3) will be designed to guarantee the asymptotically admissibility of system (10) in the theorem as follows.Theorem 10.
For given scalarsd>0 and γ>0, the closed-loop system (10) is asymptotically admissible with H∞ performance γ, if there exist symmetric positive-definite matrices P=PT, S=ST, Qi=QiT, and Zi=ZiT,i=1,…,N, such that LMIs hold:(34)ETPT=PE≥0,(35)Ξ-11Ξ-210⋯Ξ-N+11Ξ-N+210⋯0Ξ-2N+21Ξ-2N+31Ξ-2N+41Ξ-2N+51P-ATP-CT∗Ξ-22⋱⋱⋯Ξ-N+22⋱⋱⋮000000∗∗⋱Ξ-NN-100⋱⋱0⋮⋮⋮⋮⋮⋮∗∗∗Ξ-NNΞ-N+1N⋮⋱⋱Ξ-2N+1N⋮⋮⋮⋮⋮⋮∗∗∗∗Ξ-N+1N+10⋯0Ξ-2N+1N+1Ξ-2N+2N+1⋮⋮⋮⋮⋮∗∗∗∗∗-12E-TZiE-0⋯⋯⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗-12E-TZNE-0⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗∗Ξ-2N+22N+2⋮⋮⋮00∗∗∗∗∗∗∗∗∗∗0⋮⋮K-1TB1TK-1TDT∗∗∗∗∗∗∗∗∗∗∗00K-2TB1TK-2TDT∗∗∗∗∗∗∗∗∗∗∗∗-γ2IBT0∗∗∗∗∗∗∗∗∗∗∗∗∗δ--2P-0∗∗∗∗∗∗∗∗∗∗∗∗∗∗-I<0,where(36)Ξ-11=AP-+P-AT+S-1+Q-1-4ETZ-1E,Ξ-N+1+ii=dNETR-iE+6ETZ-iE,Ξ-N+1+ii+1=-dNETR-iE+6ETZ-iE,i=1,…,N,Ξ-2N+31=B1K-1,Ξ-2N+41=B1K-2,Ξ-2N+51=B,Ξ-ii=-Q-i-1+Q-i-4ETZ-i-1E-4ETZ-iE,i=2,…,N,Ξ-i+1i=-2ETZ-iE,i=1,…,N,Ξ-N+1N+1=-Q-N-4ETZ-NE-S-1,Ξ-2N+22N+2=-ETS-2E,δ-=dN2∑i=1NETZ-iE+ETS-2E.
Moreover, a suitable controller withH∞ performance γ in forms of (3) is designed to satisfy the proposed conditions. And the control gain matrices K1 and K2 are given by(37)K1=K-1P--1,K2=K-2P--1.Proof.
By noticing thatδ->0, we have(38)-P-δ-P-≤δ--2P-.
Letς=diag⁡{P-T,P-T,0,…,0,P-T,0,…,0,0,P-T, P-T,I,I,I}. Denoting(39)P-=P-1,K-1=K1P-1,K-2=K2P-1,Q-i=P-TQiP-1,S-=P-TSP-1,R-i=P-TRiP-1,Z-i=P-TZiP-1and premultiplying and postmultiplying (28) by ς and ςT, respectively, the result (35) can be obtained. This completes the proof.Remark 11.
According to Theorem10, sufficient conditions are provided to solve the H∞ control problem for singular neutral system based on sampled-data, and the desired H∞ sampled-data controller is proposed. The conditions take the form of LMI, which can be readily determined by standard numerical software. Moreover, the methods proposed in the work can be easily extended to other system.Remark 12.
When the matrixE in (1) is nonsingular, assuming that E is an identity matrix, system (1) reduces to a traditional sampled-data control system.The regular sampled-data control system is considered as follows.(40)x˙t=Axt+B1ut+Bwt,t>0,yt=Cxt+Dut,t∈-d,0,xt=ϕt.Using the input delay approach and a state-derivative control law, the sampled-data control system (1) is turned into a neutral system with time-varying delays.(41)x˙t=Axt+A1xt-τt+G1x˙t-τt+Bwt,t>0yt=Cxt+D1xt-τt+D2x˙t-τt,xt=ϕt,t∈-d,0,where A1=B1K1, G1=B1K2, D1=DK1, and D2=DK2.According to Theorem10, the sampled-data controller is designed for the system (41) such that the system is asymptotically stable.Corollary 13.
For given a scalard>0 and a prescribed scalar γ>0, the closed-loop system (41) is asymptotically stable with H∞ performance γ, if there exist symmetric positive-definite matrices P,S,Qi,Zi,i=1,…,N, such that(42)Ξ-11Ξ-210⋯Ξ-N+11Ξ-N+210⋯0Ξ-2N+21Ξ-2N+31Ξ-2N+41Ξ-2N+51P-ATP-CT∗Ξ-22⋱⋱⋯Ξ-N+22⋱⋱⋮000000∗∗⋱Ξ-NN-100⋱⋱0⋮⋮⋮⋮⋮⋮∗∗∗Ξ-NNΞ-N+1N⋮⋱⋱Ξ-2N+1N⋮⋮⋮⋮⋮⋮∗∗∗∗Ξ-N+1N+10⋯0Ξ-2N+1N+1Ξ-2N+2N+1⋮⋮⋮⋮⋮∗∗∗∗∗-12Z-i0⋯⋯⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗⋱⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗⋱⋱⋮⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗-12Z-N0⋮⋮⋮⋮⋮∗∗∗∗∗∗∗∗∗Ξ-2N+22N+2⋮⋮⋮00∗∗∗∗∗∗∗∗∗∗0⋮⋮K-1TB1TK-1TDT∗∗∗∗∗∗∗∗∗∗∗00K-2TB1TK-2TDT∗∗∗∗∗∗∗∗∗∗∗∗-γ2IBT0∗∗∗∗∗∗∗∗∗∗∗∗∗δ--2P-0∗∗∗∗∗∗∗∗∗∗∗∗∗∗-I<0,where(43)Ξ-11=AP-+P-AT+S-1+Q-1-4Z-1,Ξ-N+1+ii=dNR-i+6Z-i,Ξ-N+1+ii+1=-dNR-i+6Z-i,i=1,…,N,Ξ-2N+31=B1K-1,Ξ-2N+41=B1K-2,Ξ-2N+51=B,Ξ-ii=-Q-i-1+Q-i-4Z-i-1-4Z-i,i=2,…,N,Ξ-i+1i=-2Z-i,i=1,…,N,Ξ-N+1N+1=-Q-N-4Z-N-S-1,Ξ-2N+22N+2=-S-2,δ-=dN2∑i=1NZ-i+S-2.Proof.
Following similar lines in above proof of Theorem10, the proof can be accomplished, so the procedure is omitted.


## 4. Numerical Examples

In this section, two illustrative examples will be provided to demonstrate that the results in this paper are effective and less conservative.Example 1.
Consider the singular system (4) with(44)A=-0.50.10.2-1,B1=-1100.5,C=-1.1100.1,D=0.10.110.1,E=1000.Table1 lists the maximum values of sampling period d which is obtained from different methods.Table 1
Maximum values of sampling periodd.
Methods Maximumd allowed Theorem  3.1 [27] 1 Theorem  1 [28] 1.1547 Theorem  2 [26] 1.1954 Theorem  5 [29] 1.2060 Theorem6 (N=2) 1.2086 Theorem6 (N=3) 1.2457It can be seen from Table1 that the sampling periods obtained from Theorem 6 (N=2) and Theorem 6 (N=3) are larger than those in [26–29], which shows the method in the paper improves the most previous works effectively.When the sampling periodd=0.5 s, the H∞ performance γmin=0.1529, which is achieved by Theorem 9 (N=3). Then, the gain matrix of the sampled-data controller is calculated through LMI toolbox as follows:(45)K1=-3.97550.06462.1741-0.0142,K2=-3.75670.06512.070-0.0128.Then the obtained sampled-data controllers are applied to system (4). With the initial condition x(t)=[-22], the response curve for system (4) is exhibited in Figure 1. From Figure 1, we can see that the state tends to zero; that is, the designed sampled-data controllers can stabilize system (4).Figure 1
State response of system (4).Example 2.
According to Corollary13, when E is an identity matrix, system (1) is turned into the traditional linear sampled-data control system. To illustrate the behavior of presented state-derivative feedback control law, we use the model of a linear dynamic positioning (DP) ship. The main parameters are referenced to the Ship Handling Simulator designed by the Institute of Navigation Jimei University (length = 175 m, beam = 25.4 m, tonnage = 2.4 × 107 kg, and draft = 9.5 m).Consider the following linearized equations of DP ship.(46)x˙t=Axt+B1ut+Bwt,yt=Cxt+Dut,where(47)A=0I0-M-1D,B1=0M-1,B=0M-1,C=I0,M=2.6415×1070003.345×1071.492×10701.492×1076.52×109,D=2.22×1040002.22×105-1.774×1060.2-1.774×1067.151×108,where x(t)=[xyψpvr] represents the position, heading, and velocities of ship; w(t) represents the external disturbance input of the system like waves, wind, and ocean currents; u(t) is control input vector of forces and moments; y(t) is the controlled system output. The initial position and heading of the ship are ηi=[0m0m2.5°], and body-fixed velocities vi=[2m/s3m/s0.2°/s]. The final desired state is ηf=[10100°] and vf=[0m/s0m/s0°/s]. When the γ = 0.5, the sampling interval d=3.15 s, which is achieved by Theorem 9 (N=3). Then, the control gain matrix is(48)K1=-0.092800-1.37150000.6782-0.18601.2451.58210-0.1860.7530-0.0460.4923,K2=-0.1025000.01030000.0329-0.541200.01030.2350-1.37512.167201.421-3.392.Comparing the result obtained from Corollary13 with [34, 35] adopting the same state-derivative feedback, the calculated result of upper bound delay d for different values of γ is shown in Table 2. It can be seen that the proposed H∞ sampled-data controller in the paper gives larger delay bound than those in [34, 35], which illustrates that the technique proposed in the paper is more effective and the stability criteria are less conservative.Table 2
The upper bounds ofd for various of γ.
Method 0.5 1.0 10.0 Lu et al. [34] 1.56 3.61 5.65 Zhao et al. [35], Theorem 9 (N=2) 2.86 6.16 10.16 Zhao et al. [35], Theorem 9 (N=3) 3.07 6.27 10.28 Theorem9 (N=2) 2.93 6.23 10.27 Theorem9 (N=3) 3.21 6.39 10.41Then the obtained sampled-data controllers are applied to system (46). From Figures 2–9, we can see that the proposed sampled-data controller guaranteed H∞ control performance and the ships can maintain desired position, heading, velocity, and acceleration under the external disturbance like waves, wind, and ocean currents.Figure 2
Position of thex direction of the ship.Figure 3
Position of they direction of the ship.Figure 4
Headingψ of the ship.Figure 5
Surge velocityu of the ship.Figure 6
Sway velocityv of the ship.Figure 7
Yaw rater of the ship.Figure 8
Acceleration of thex direction of the ship.Figure 9
Acceleration of they direction of the ship.


## 5. Conclusion

TheH∞ control problem of singular neutral system with sampled-data has been studied in the paper. By input delay approach and the state-derivative control law, the singular system has been converted to a singular neutral system with time-varying delay. Less conservative results are derived for the resultant system by incorporating the delay decomposition technique, Wirtinger-based integral inequality, and an augmented Lyapunov-Krasovskii functional. The H∞ sampled-data controller is designed to guarantee the asymptotically admissibility of the resultant system. Finally, two simulation examples have illustrated the improvement of the proposed method. Our future research topic is to investigate the fuzzy-model-based sampled-data control for nonlinear singular neutral systems.



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