$H_\infty$ Control for Lur’e Singular Systems with Time Delays

Abstract

This paper considers the $H_{\infty }$ control problem for Lur’e singular systems with time delays. By using Lyapunov stability theory, sufficient conditions for the system to be exponentially stable and satisfy the performance index of $H_{\infty }$ are obtained; these conditions are based on the linear matrix inequality method. Then the design of a state feedback controller is given, by applying a more clever approach for a nonlinear matrix with a special format to convert it into the sum of several linear matrices, such that the closed-loop system is also exponentially stable. Finally, numerical examples illustrate the effectiveness of the proposed method and its advantages over existing approaches.

1 Introduction

A singular system is a kind of dynamic systems with a more general form that has more describe performance characteristics and has a broader form and wider application background than normal systems [10]. The research content regarding singular systems is diverse, including stability, dissipation and $H_\infty$ control [17]. However, in many practical control problems, there are multiple time delays in the studied systems, whose existence may induce instability, oscillation and poor performance. System stability is a prerequisite for the normal operation of a control system, and $H_\infty$ control can ensure the stability of the system while suppressing the impact of interference on the system performance to a certain extent. Therefore, it is necessary to study stability and $H_\infty$ control problem of a singular time-delay system [1, 10].

A Lur’e system is a type of typical, interval, nonlinear system with widely representative significance. Its nonlinear terms are usually contained in a finite sector interval or an infinite open plane, that is, the linear parts of the system have a fixed matrix and one or more uncertain nonlinear parts are utilized to satisfy sector constraints. Since Lur’e introduced the absolute stability in 1957, many authors have extensively discussed absolute stability of Lur’e control systems based on the Popov frequency domain criteria and the Lyapunov function with a Lur’e form [2, 3].

Reference [10] conducted an in-depth study on the delay-dependent $H_\infty$ control problem of Lur’e singular time-delay systems, reference [7] addressed synchronization methods for a specific Lur’e system, and references [6, 21] studied the absolute stability problem of Lur’e time-delay control systems, that is, the problem of global asymptotic stability. Reference [18] studied system stability and the compactness of the operators describing the solution trajectories. Based on the linear matrix inequality (LMI) method, Wu proposed delay-range-dependent bounded real lemmas and studied sufficient conditions for a system to be exponentially stable and the existence of the linear $H_\infty$ filter [22]. Then, Park studied $H_\infty$ filtering for a class of Markovian jump systems and successfully proved that there are necessary and sufficient conditions of $H_\infty$ filtering for singular Markovian jump systems (SMJSs) whose transfer rates are partially unknown [15]. Based on LMI theory, Kim used a new design method to study the $H_\infty$ control problem for a singular time-delay system and obtained all solutions including the controller gains [5]. In addition, Long designed a dynamic feedback controller to ensure that the developed closed-loop system was impulse-free and stable under the given performance index of $H_\infty$ [11]. By constructing an enhanced Lyapunov-Krasovskii function with triple integral terms, [4] proposed a bounded real lemma to ensure that a singular state-delay system was stable and designed a static output feedback controller. In addition, based on the bounded real lemma, Yang ensured that the singular time-delay system was regular, impulse-free and stable under the conditions of the performance index of $H_\infty$ [24]. Reference [25] studied the exponential $H_\infty$ control problem for a singular system with time-varying delays, and references [19, 23] also studied the control problem for such systems. Furthermore, there are a large number of papers that have studied the stability and $H_\infty$ control problems of discrete-time descriptor systems [9, 13], uncertain systems [12, 26] and SMJSs [8].

This paper mainly studies the $H_\infty$ control problem of Lur’e singular time-delay systems. By using Lyapunov stability theory, a new Lyapunov function is constructed, which applies not only the upper and lower limits of the time delay, but also the time-delay interval. Based on LMI method, sufficient conditions for systems to be exponentially stable and satisfy the performance index of $H_{\infty }$ are given, and a state feedback controller is designed to make the closed-loop system exponentially stable.

Notations: $R^n$ denotes the n-dimensional Euclidean space, and $R^{n\times m}$ is the set of all $m\times n$ real matrices. $||\cdot ||$ stands for the Euclidean norm of a vector and $||f(t)||_d =\sup \limits _{-d\le t\le 0}||f(t)||$. $\alpha _1\bigvee \alpha _2=\max \{\alpha _1,\alpha _2\}$, and $\alpha _1\bigwedge \alpha _2=\min \{\alpha _1,\alpha _2\}$. The superscripts ’T’ and ’$*$’ denote the term that is induced by symmetry, respectively.

2 Problem Formulation

Consider a Lur’e singular time-delay system:

$$\begin{aligned} E{\dot{x}}(t)&=Ax(t)+A_dx(t-d(t))-F\varphi (y(t))+B_ww(t)+B_uu(t) \end{aligned}$$

(1)

$$\begin{aligned} y(t)&=Cx(t)+C_dx(t-d(t))+D_ww(t) \end{aligned}$$

(2)

$$\begin{aligned} z(t)&=Lx(t)+L_dx(t-d(t))+L_ww(t)+L_uu(t) \nonumber \\ x(t)&=\phi (t),t\in [-d_2,0] \end{aligned}$$

(3)

where $x\in R^n$ is the state, $u\in R^m$ is the control input, $w\in R^p$ is the disturbance input that satisfies $\int _0^{+\infty }w^T(t)w(t)<+\infty$, and $z\in R^q$ is the control output of the system. $E, A, A_d, B_w, B_u, C, C_d, D_w, F, L, L_d, L_w, L_u$ are known real constant matrices with appropriate dimensions, and $Rank(E)=r\le n$. $\phi (t):{\mathcal {R}}\rightarrow {{\mathcal {R}}}^n$ is a compatible vector-valued initial function, $y(t)\in R^l$ and the nonlinear function $\varphi (y)\in R^l\rightarrow R^l$ is appropriately smooth and satisfy the sector constraint

$$\begin{aligned} \varphi ^T(y)\varphi (y)\le \varphi ^T(y)My \end{aligned}$$

(4)

where $M\in R^{l\times l}$ is a given positive definite matrix. In addition, d(t) is a time-delay continuous function that is time-differentiable at all times, thereby satisfying

$$\begin{aligned} 0\le d_1\le d(t)\le d_2,{\dot{d}}(t)\le \alpha \end{aligned}$$

(5)

where $d_1$ and $d_2$ represent the upper and lower limits of the time delay, respectively, ${\dot{d}}(t)$ is the corresponding derivative function, and $0\le \alpha <1$.

Throughout this paper, the following definitions and lemmas will be used.

Definition 1

[20] A system is exponentially stable, if there exist scalars $\beta _1>0$ and $\beta _2>0$ such that

$$\begin{aligned} ||x(t)||\le \beta _1e^{-\beta _2t}||x(t)||_{d_2},t>0 \end{aligned}$$

Definition 2

[11] The system (1)–(3) possesses an $H_\infty$ performance of $\gamma$, that is, under zero initial conditions, the system satisfies

$$\begin{aligned} J(t)=\int _0^t[z^T(s)z(s)-\gamma ^2w^T(s)w(s)]\mathrm{d}s<0 \end{aligned}$$

for any nonzero $\omega (t)$ that satisfies $\int _0^{+\infty }w^T(t)w(t)<+\infty$, where $\gamma >0$ is a predefined scalar.

Lemma 1

[20] (Jensen integral inequality) For any positive definite matrix $M=M^T>0$, scalar quantities $\gamma _1$ and $\gamma _2$ and a vector-valued function $v:[\gamma _1,\gamma _2]\rightarrow R^n$, the following inequality holds:

$$\begin{aligned} \left( \int _{\gamma _1}^{\gamma _2}v(s)\mathrm{d}s\right) ^TM\left( \int _{\gamma _1}^{\gamma _2}v(s)\mathrm{d}s\right) \le (\gamma _2-\gamma _1)\int _{\gamma _1}^{\gamma _2}v^T(s)Mv(s)\mathrm{d}s \end{aligned}$$

Lemma 2

[11] Given any real matrices $Q>0$, $W_1$ and $W_2$ of appropriate dimensions and an umber $\lambda >0$, the following inequality holds:

$$\begin{aligned} W_1^TW_2+W_2^TW_1\le \lambda W_1^TQW_1+\lambda ^{-1} W_2^TQ^{-1}W_2 \end{aligned}$$

Lemma 3

[22] Suppose that the positive continuous function f(t) satisfies $f(t)\le \zeta _1\sup \limits _{t-d\le s\le t} f(s) +\zeta _2 e^{-\varepsilon t}$, where $\varepsilon>0,0<\zeta _1<1,0<\zeta _1 e^{\varepsilon d}<1,\zeta _2>0$. For $d>0$, the following inequality holds

$$\begin{aligned} f(t)\le e^{-\varepsilon t} ||f(s)||_d +\frac{\zeta _2 e^{-\varepsilon t}}{1-\zeta _1 e^{-\varepsilon d}} \end{aligned}$$

3 Main Results

First, we consider the exponential stability of the system (1)–(3) when it satisfies the $H_\infty$ performance requirement. Then, we have the following result.

Theorem 1

In the system (1)–(3), for given $0\le d_1\le d_2$, $0\le \alpha <1$, if there is a scalar $\lambda >0$, symmetric positive definite matrices $Q_j,R_j,j=1,2,3$ and a matrix P such that

$$\begin{aligned}&E^TP=P^TE\ge 0 \end{aligned}$$

(6)

$$\begin{aligned}&\left[ \begin{array}{ccccccc} \varXi _{11}+L^TL &{}\quad \varXi _{12}+L^TL_d &{}\quad E^TR_1E &{}\quad 0 &{}\quad -A^TWF+C^TM^T\\ *&{}\quad \varXi _{22}+L_d^TL_d &{}\quad E^TR_3E &{}\quad \varXi _{24} &{}\quad -{A_d}^TWF+C_d^TM^T \\ *&{}\quad *&{}\quad \varXi _{33} &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad -Q_2-\varXi _{24} &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \nonumber \\&\left. \begin{array}{ccccccc} L^TL_w+P^TB_w+A^TWB_w &{}\quad P^TF\\ L_d^TL_w+A_d^TWB_w &{}\quad 0\\ 0 &{}\quad 0\\ 0 &{}\quad 0\\ -F^TWB_w+MD_w &{}\quad 0\\ L_w^TL_w+B_wWB_w-\gamma ^2I &{}\quad 0\\ *&{}\quad -\lambda \end{array} \right] <0 \end{aligned}$$

(7)

where $d_{12}=d_2-d_1,W=d_1^2R_1+d_{12}d_2^2R_2+d_{12}^2R_3$, $\varXi _{11}=P^TA+A^TP+\sum _{k=1}^{3}Q_k+A^TWA-E^TR_1E-d_{12}E^TR_2E$, $\varXi _{12}=P^TA_d+A^TWA_d+d_{12}E^TR_2E$,  $\varXi _{22}=-(1-\alpha )Q_3+A_d^TWA_d-E^T((d_{12}+d_2)R_2+2R_3)E$, $\varXi _{24}=E^T(d_2R_2+R_3)E$,  $\varXi _{33}=-Q_1-E^TR_1E-E^TR_3E$, $\varXi _{44}=-Q_2-E^T (d_2R_2+R_3)E$, then for any time-delay function d(t) that satisfies (5), the system is exponentially stable and satisfies the performance index $\gamma$ of $H_\infty$.

Proof

First, we prove that the system (1)–(3) is asymptotically stable ($w(t)=u(t)=0$). We choose the Lyapunov function candidate as

$$\begin{aligned} V(x_t,t)= & {} x^T(t)E^TPx(t)+\sum _{k=1}^{2}\int _{t-d_k}^tx(\alpha )^TQ_kx(\alpha )\mathrm{d}\alpha +\int _{t-d(t)}^tx(\alpha )^TQ_3x(\alpha )\mathrm{d}\alpha \\&+d_1\int _{-d_1}^0\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \\&+d_{12}d_2\int _{-d_2}^0\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \\&+d_{12}\int _{-d_2}^{-d_1}\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \end{aligned}$$

where $x_t=x(t+\theta )$, $-2d_2\le \theta \le 0$. ${\mathcal {A}}$ denotes the derivative of a function, so from (5), for the system (1)–(3), we have

$$\begin{aligned}&{\mathcal {A}}{V}(x_t,t)\\&\quad = 2x^T(t)E^TP{\dot{x}}(t)+\sum _{k=1}^{3}x(t)^TQ_kx(t)-\sum _{k=1}^{2}x(t-d_k)^TQ_kx(t-d_k)\\&\quad \quad -(1-{\dot{d}}(t))x(t-d(t))^TQ_3x(t-d(t))+{\dot{x}}^T(t)E^TWE{\dot{x}}(t)\\&\quad \quad -d_1\int _{t-d_1}^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha -d_{12}\int _{t-d_2}^{t-d_1}{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha \\&\quad \quad -d_{12}d_2\int _{t-d_2}^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha ) \mathrm{d}\alpha \\&\quad \le 2x^T(t)E^TP{\dot{x}}(t)+\sum _{k=1}^{3}x(t)^TQ_kx(t)-\sum _{k=1}^{2}x(t-d_k)^TQ_kx(t-d_k)\\&\quad \quad -(1-\alpha )x(t-d(t))^TQ_3x(t-d(t))+{\dot{x}}^T(t)E^TWE{\dot{x}}(t)\\&\quad \quad -d_1\int _{t-d_1}^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha -d_{12}\int _{t-d_2}^{t-d_1}{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha \\&\quad \quad -d_{12}d_2\int _{t-d_2}^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha ) \mathrm{d}\alpha \end{aligned}$$

Using Lemma 1, the following formula can be obtained:

$$\begin{aligned}&-d_1\int _{t-d_1}^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha - d_{12}d_2\int _{t-d_2}^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha )\mathrm{d}\beta \\&-d_{12}\int _{t-d_2}^{t-d_1}{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha \\&\quad = -d_1\int _{t-d_1}^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha -d_{12}d_2\int _{t-d(t)}^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha )\mathrm{d}\beta \\&-d_{12}\int _{t-d(t)}^{t-d_1}{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha -d_{12} \int _{t-d_2}^{t-d(t)}{\dot{x}}^T(\alpha )E^T(d_2R_2+R_3)E{\dot{x}}(\alpha )\mathrm{d}\alpha \\&\quad \le -\left( \int _{t-d_1}^t{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_1\left( \int _{t-d_1}^tE{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \\&-d_{12}\left( \int _{t-d(t)}^t{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_2\left( \int _{t-d(t)}^tE{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \\&-\left( \int _{t-d(t)}^{t-d_1}{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_3\left( \int _{t-d(t)}^{t-d_1}E{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \\&-\left( \int _{t-d_2}^{t-d(t)}{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) (d_2R_2+R_3)\left( \int _{t-d_2}^{t-d(t)}E{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \end{aligned}$$

The nonlinear function $\varphi (y)$ satisfies inequality (4), and based on Lemma 2, from (1) and (4), we have

$$\begin{aligned}&{\mathcal {A}}{V}(x_t,t)\nonumber \\&\quad \le 2x^T(t)P^T(Ax(t)+A_dx(t-d(t))-F\varphi (y(t)))+\sum _{k=1}^{3}x(t)^TQ_kx(t)\nonumber \\&-\sum _{k=1}^{2}x(t-d_k)^TQ_kx(t-d_k)-(1-\alpha )x(t-d(t))^TQ_3x(t-d(t))\nonumber \\&+{\dot{x}}^T(t)E^TWE{\dot{x}}(t)-\left( \int _{t-d_1}^t{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_1\left( \int _{t-d_1}^tE{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \nonumber \\&-d_{12}\left( \int _{t-d(t)}^t{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_2\left( \int _{t-d(t)}^tE{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \nonumber \\&-\left( \int _{t-d(t)}^{t-d_1}{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) R_3\left( \int _{t-d(t)}^{t-d_1}E{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \nonumber \\&-\left( \int _{t-d_2}^{t-d(t)}{\dot{x}}^T(\alpha )E^T\mathrm{d}\alpha \right) (d_2R_2+R_3)\left( \int _{t-d_2}^{t-d(t)}E{\dot{x}}(\alpha )\mathrm{d}\alpha \right) \nonumber \\&+2(\varphi (y(t))^TMy(t)-\varphi (y(t))^T\varphi (y(t)))\end{aligned}$$

(8)

$$\begin{aligned}&\quad \le \xi ^T\left[ \begin{array}{cccccc} \varXi _{11}+\frac{1}{\lambda }P^TFF^TP &{}\quad \varXi _{12} &{}\quad E^TR_1E &{}\quad 0 &{}\quad -A^TWF+\mu C^TM^T \\ *&{}\quad \varXi _{22} &{}\quad E^TR_3E &{}\quad \varXi _{24} &{}\quad -A_d^TWF+\mu C_d^TM^T \\ *&{}\quad *&{}\quad \varXi _{33} &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad \varXi _{44} &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF\\ \end{array} \right] \xi \nonumber \\ \end{aligned}$$

(9)

where $\xi (t)=\left[ \begin{array}{cccccc} x^T(t)&\quad x^T(t-d(t))&\quad x^T(t-d_1)&\quad x^T(t-d_2)&\quad \varphi ^T(t) \end{array} \right] ^T$.

Next, the following matrix decomposition is executed. Since $Rank(E)=r\le n$, there are nonsingular matrices G, H such that

$$\begin{aligned} GEH=\left[ \begin{array}{cc} I_r &{}\quad 0\\ 0 &{}\quad 0\\ \end{array} \right] \end{aligned}$$

Denoting

$$\begin{aligned} GAH=\left[ \begin{array}{cc} {\hat{A}}_1&{}\quad {\hat{A}}_2\\ {\hat{A}}_3&{}\quad {\hat{A}}_4\\ \end{array} \right] ,G^{-T}PH=\left[ \begin{array}{cc} {\hat{P}}_1&{}\quad {\hat{P}}_2\\ {\hat{P}}_3&{}\quad {\hat{P}}_4\\ \end{array} \right] \end{aligned}$$

we can obtain ${\hat{P}}_2=0$ from (6). We can obtain $\hat{A }_4^T{\hat{P}}_4+{\hat{P}}_4^T{\hat{A}}_4<0$ by pre-multiplying and post-multiplying $\varXi _{11}<0$ by $H^T$ and H, respectively, so ${\hat{A}}_4$ is nonsingular. Setting

$$\begin{aligned} {\hat{G}}=\left[ \begin{array}{cc} I_r &{}\quad -{\hat{A}}_2 {{\hat{A}}_4}^{-1}\\ 0 &{}\quad {{\hat{A}}_4}^{-1}\\ \end{array} \right] G \end{aligned}$$

we have

$$\begin{aligned} {\hat{G}}EH=\left[ \begin{array}{cc} I_r &{}\quad 0\\ 0 &{}\quad 0\\ \end{array} \right] ,{\hat{G}}AH=\left[ \begin{array}{cc} A_1&{}\quad 0\\ A_3&{}\quad I\\ \end{array} \right] ,{\hat{G}}^{-T}PH=\left[ \begin{array}{cc} P_1&{}\quad P_2\\ P_3&{}\quad P_4\\ \end{array} \right] \end{aligned}$$

where $A_1=\hat{A_1}-\hat{A_2} {\hat{A_4}}^{-1}\hat{A_3}, A_3={\hat{A_4}}^{-1} \hat{A_3}.$

Denoting

$$\begin{aligned} {\hat{G}}A_d H=\left[ \begin{array}{cc} A_{d1} &{}\quad A_{d2}\\ A_{d3} &{}\quad A_{d4}\\ \end{array} \right] ,H^T Q_3 H=\left[ \begin{array}{cc} Q_{11}&{}\quad Q_{12}\\ *&{}\quad Q_{22}\\ \end{array} \right] ,{\hat{G}}FH=\left[ \begin{array}{cc} F_1&{}\quad F_2\\ F_3&{}\quad F_4\\ \end{array} \right] \end{aligned}$$

Inequality (6) implies that $P_2=0$ and $P_1\ge 0$. We define $\eta (t)\!=\!\left[ \begin{array}{cccccc} \eta _1(t)&\quad \eta _2(t) \end{array} \right] ^T =H^{-1}x(t),\psi {(y(t))}=\left[ \begin{array}{cccccc} \psi _1{(y(t))}&\quad \psi _2{(y(t))} \end{array} \right] ^T=H^{-1}\varphi {(y(t))}.$ We can obtain

$$\begin{aligned} {\hat{G}}EH {\dot{\eta }}(t)&={\hat{G}}E{\dot{x}}(t)={\hat{G}} [Ax(t)+A_d x{(t-d(t))}-F\varphi {(y(t))}]\\&={\hat{G}}AH\eta (t)+{\hat{G}}A_d H \eta {(t-d(t))}-{\hat{G}}FH\psi {(y(t))} \end{aligned}$$

Then, the system (1)–(3) with $\omega (t)\equiv 0$ is a restricted system that is equivalent to

$$\begin{aligned} \left[ \begin{array}{cc} I_r &{}\quad 0\\ 0 &{}\quad 0\\ \end{array} \right] \left[ \begin{array}{cc} {\dot{\eta }}_1(t)\\ {\dot{\eta }}_2(t)\\ \end{array} \right] =&\left[ \begin{array}{cc} A_1 &{}\quad 0\\ A_3 &{}\quad I\\ \end{array} \right] \left[ \begin{array}{cc} \eta _1(t)\\ \eta _2(t)\\ \end{array} \right] +\left[ \begin{array}{cc} A_{d1} &{}\quad A_{d2}\\ A_{d3} &{}\quad A_{d4}\\ \end{array} \right] \left[ \begin{array}{cc} \eta _1(t-d(t))\\ \eta _2(t-d(t))\\ \end{array} \right] \\&-\left[ \begin{array}{cc} F_1&{}\quad F_2\\ F_3&{}\quad F_4\\ \end{array} \right] \left[ \begin{array}{cccccc} \psi _1{(y(t))} \\ \psi _2{(y(t))} \end{array} \right] \end{aligned}$$

In other words, we have that

$$\begin{aligned}&{\dot{\eta }}_1(t)=A_1\eta _1(t)+A_{d1}\eta _1(t-d(t))\nonumber \\&\quad +A_{d2}\eta _2(t-d(t))-F_1\psi _1(y(t))-F_2\psi _2(y(t)) \end{aligned}$$

(10)

$$\begin{aligned}&0=A_3\eta _1(t)+\eta _2(t)+A_{d3}\eta _1(t-d(t))+A_{d4}\eta _2(t-d(t))\nonumber \\&\quad -F_3\psi _1(y(t))-F_4\psi _2(y(t)) \nonumber \\&\eta (t)=H^{-1}\phi (t),t\in [-d_2,0] \end{aligned}$$

(11)

Defining a new function $W(x_t,t)=e^{\varepsilon t} V(x_t,t)$, we can obtain a scalar $\sigma >0$ such that ${\mathcal {A}}{V}(x_t,t)\le -\sigma ||x(t)||^2$, so ${W}(x_t,t)\le W(x_0,0)+\int _{0}^{t}{e^{\varepsilon s}(\varepsilon V(x_s,s)-\sigma ||x(s)||^2)\mathrm{d}s}$. Due to

$$\begin{aligned} x^T(t)E^TPx(t)&=\eta (t)^TH^TE^T{\hat{G}}^T{\hat{G}}^{-T}PH\eta (t)\\&=\left[ \begin{array}{cc} \eta _1(t)^T&{} \eta _2(t)^T\\ \end{array} \right] \left[ \begin{array}{cc} I_r &{}\quad 0\\ 0 &{}\quad 0\\ \end{array} \right] \left[ \begin{array}{cc} P_1&{}\quad P_2\\ P_3&{}\quad P_4\\ \end{array} \right] \left[ \begin{array}{cc} \eta _1(t)\\ \eta _2(t)\\ \end{array} \right] \\&=\eta _1(t)^T P_1 \eta _1(t) \end{aligned}$$

we can obtain

$$\begin{aligned}&\lambda _{min}(P_1)||\eta _1(t)||^2\le x^T(t)E^TPx(t)\le V(x_t,t)=e^{-\varepsilon t}W(x_t,t)\\&\quad \le e^{-\varepsilon t}\{W(x_0,0)+\int _{0}^{t}e^{\varepsilon s}[\varepsilon V(x_t,t)-\sigma ||x(s)||^2]\mathrm{d}s\} \end{aligned}$$

Because $w(t)=u(t)=0$, Lemma 2 implies that we have

$$\begin{aligned}&d_1\int _{-d_1}^0\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \\&\quad = d_1\int _{-d_1}^0\int _{t+\beta }^t[Ax(\alpha )+A_d x{(\alpha -d(\alpha ))}-F\varphi {(y(\alpha ))}]^TR_1[Ax(\alpha ) \\&\quad \quad +A_d x{(\alpha -d(\alpha ))}-F\varphi {(y(\alpha ))}] \mathrm{d}\alpha \mathrm{d}\beta \\&\quad = d_1\int _{-d_1}^0\int _{t+\beta }^t [x^T(\alpha )A^TR_1Ax(\alpha )+ x^T(\alpha -d(\alpha ))A_d^T R_1A_d x(\alpha -d(\alpha ))\\&\quad \quad + \varphi (y(\alpha ))^TF^TR_1F\varphi (y(\alpha )) \\&\quad \quad +x^T(\alpha )A^TR_1A_d x(\alpha -d(\alpha ))+x^T(\alpha -d(\alpha ))A_d^T R_1 Ax(\alpha )\\&\quad \quad -x^T(\alpha -d(\alpha ))A_d^T R_1 F\varphi (y(\alpha ))\\&\quad \quad -\varphi (y(\alpha ))^TF^TR_1A_d x(\alpha -d(\alpha )) - x^T(\alpha )A^TR_1F\varphi (y(\alpha ))\\&\quad \quad -\varphi (y(\alpha ))^TF^TR_1Ax(\alpha ) ]\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le d_1\int _{-d_1}^0\int _{t+\beta }^t [x^T(\alpha )A^TR_1Ax(\alpha )+ x^T(\alpha -d(\alpha ))A_d^T R_1A_d x(\alpha -d(\alpha ))\\&\quad \quad + \varphi (y(\alpha ))^TF^TR_1F\varphi (y(\alpha ))\\&\quad \quad +x^T(\alpha )A^TR_1A_d x(\alpha -d(\alpha ))+x^T(\alpha -d(\alpha ))A_d^T R_1 Ax(\alpha )]\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le d_1\int _{-d_1}^0\int _{t+\beta }^t [x^T(\alpha )A^TR_1Ax(\alpha )+ x^T(\alpha -d(\alpha ))A_d^T R_1A_d x(\alpha -d(\alpha ))\\&\quad \quad + \lambda _{max}(F^TR_1F)\varphi (y(\alpha ))^T\varphi (y(\alpha )) \\&\quad \quad +x^T(\alpha )A^TR_1^2 Ax(\alpha )+x^T(\alpha -d(\alpha ))A_d^TA_d x(\alpha -d(\alpha ))]\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le d_1\int _{-d_1}^0\int _{t+\beta }^t\{[\lambda _{max}(A^TR_1A)+\lambda _{max}(A^TR_1^2 A)]||x(\alpha )||^2\\&\quad \quad +[\lambda _{max}(A_d^TR_1A_d)+\lambda _{max}(A_d^TA_d)]||x(\alpha -d(\alpha ))||^2\\&\quad \quad + \lambda _{max}(F^TR_1F)y(\alpha )^TM^2y(\alpha ) \}\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le d_1\int _{-d_1}^0\int _{t+\beta }^t\{[\lambda _{max}(A^TR_1A)+\lambda _{max}(A^TR_1^2 A)]||x(\alpha )||^2\\&\quad \quad +[\lambda _{max}(A_d^TR_1A_d)+\lambda _{max}(A_d^TA_d)]||x(\alpha -d(\alpha ))||^2\\&\quad \quad + \lambda _{max}(F^TR_1F)\lambda _{max}(M^2)[ Cx(t)\\&\quad \quad +C_dx(t-d(t)) ]^T [Cx(t)+C_dx(t-d(t))] \}\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le d_1\int _{-d_1}^0\int _{t+\beta }^t\{[\lambda _{max}(A^TR_1A)+\lambda _{max}(A^TR_1^2 A)\\&\quad \quad +2\lambda _{max}(F^TR_1F)\lambda _{max}(M^2)\lambda _{max}(C^TC) ]||x(\alpha )||^2_{d_2}\\&\quad \quad +[\lambda _{max}(A_d^TR_1A_d)+\lambda _{max}(A_d^TA_d)\\&\quad \quad +2\lambda _{max}(F^TR_1F)\lambda _{max}(M^2)\lambda _{max}(C_d^TC_d) ]||x(\alpha -d(\alpha ))||^2_{d_2} \}\mathrm{d}\alpha \mathrm{d}\beta \\&\quad \le -\frac{{d_1}^3}{2}K_1 ||x(\alpha )||^2_{d_2} \end{aligned}$$

where $K_i=\lambda _{max}(A^TR_iA)+\lambda _{max}(A^TR_i^2 A)+2\lambda _{max}(F^TR_iF)\lambda _{max}(M^2)\lambda _{max}(C^TC) +\lambda _{max}(A_d^TR_iA_d)+\lambda _{max}(A_d^TA_d) +2\lambda _{max}(F^TR_iF)\lambda _{max}(M^2)\lambda _{max}(C_d^TC_d)$, $i=1,2,3$. By the same way,

$d_{12}d_2\int _{-d_2}^0\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_2E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \le -\frac{{d_{12}}{d_2}^2}{2}K_2 ||x(\alpha )||^2_{d_2}$,

$d_{12}\int _{-d_2}^{-d_1}\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_3E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta \le -\frac{{d_{12}}({d_1}^2-{d_2}^2)}{2}K_3 ||x(\alpha )||^2_{d_2}$. Therefore,

$$\begin{aligned}&\lambda _{min}(P_1)||\eta _1(t)||^2\nonumber \\&\quad \le e^{-\varepsilon t}\{W(x_0,0)+\int _{0}^{t}[\varepsilon e^{\varepsilon s}K_0||x(s)||^2_{d_2} \nonumber \\&\quad + e^{\varepsilon s}\varepsilon K\int _{-d_2}^{0}||x(s+\theta )||^2d\theta -\sigma ||x(s)||^2] \mathrm{d}s\}\nonumber \\&\quad \le e^{-\varepsilon t}[W(x_0,0)+\int _{0}^{t}\varepsilon e^{\varepsilon s}K_0||x(s)||^2_{d_2}\mathrm{d}s + \varepsilon K{\int _{0}^{t}e^{\varepsilon s} \int _{-d_2}^{0}||x(s+\theta )||^2d\theta }\mathrm{d}s ] \nonumber \\ \end{aligned}$$

(12)

where $K_0=\lambda _{max}(E^TP)-\frac{{d_1}^3}{2}K_1-\frac{{d_{12}}{d_2}^2}{2}K_2-\frac{{d_{12}}({d_1}^2-{d_2}^2)}{2}K_3, K=\sum _{k=1}^{3} \lambda _{max}(Q_k)$, and we have

$$\begin{aligned}&{\int _{0}^{t}e^{\varepsilon s} \int _{-d_2}^{0}||x(s+\theta )||^2d\theta }\mathrm{d}s ={\int _{0}^{t}e^{\varepsilon s}\int _{t-d_2}^{t}||x(\theta )||^2d\theta }\mathrm{d}s\nonumber \\&\quad \le {\int _{-d_2}^{t}\left( \int _{\theta \bigvee 0}^{(\theta +d_2)\bigwedge t} e^{\varepsilon s} \mathrm{d}s \right) ||x(\theta )||^2}d\theta \le {\int _{-d_2}^{t} d_2 e^{\varepsilon (s+ d_2)} ||x(s)||^2 }\mathrm{d}s\nonumber \\&\quad \le d_2 e^{\varepsilon d_2} \int _{0}^{t}e^{\varepsilon s} ||x(s)||^2 \mathrm{d}s +d_2 e^{\varepsilon d_2} \int _{-{d_2}}^{0} ||\phi (s)||^2 \mathrm{d}s \end{aligned}$$

(13)

Therefore, from inequalities (12) and (13), if the scalar $\varepsilon$ is sufficiently small, there exists a scalar $k=W(x_0,0)({||\phi (t)||^2}_{d_2})^{-1}+K_0e^{\varepsilon d_2}+\varepsilon K{d_2}^2e^{\varepsilon d_2}>0$, such that

$$\begin{aligned} \lambda _{min}(P_1)||\eta _1(t)||^2 \le k e^{-\varepsilon t} {||\phi (t)||^2}_{d_2} \end{aligned}$$

(14)

Therefore, $\eta _1(t)$ is exponentially stable.

Next, defining $e(t)=A_3\eta _1(t)+A_{d3}\eta _1(t-d(t))$, it can be seen from (14) that, if there exists a scalar $m>0$,

$$\begin{aligned} ||e(t)||^2\le m e^{-\varepsilon t} {||\phi (t)||^2}_{d_2} \end{aligned}$$

(15)

We construct the function $L(t)=\eta _2(t)^T Q_{22}\eta _2(t)-\eta _2(t-d(t))^TQ_{22}\eta _2(t-d(t))$. We can obtain following formula by pre-multiplying (11) by $\eta _2(t)^T{P_4}^T$:

$$\begin{aligned} 0=&\eta _2(t)^T{P_4}^T\eta _2(t)+\eta _2(t)^T{P_4}^TA_{d4}\eta _2(t-d(t))+\eta _2(t)^T{P_4}^Te(t)\nonumber \\&+\eta _2(t)^T{P_4}^T[{-F_3\psi _1(y(t))-F_4\psi _2(y(t))}] \end{aligned}$$

(16)

Then, we can obtain

$$\begin{aligned}&L(t)=\eta _2(t)^T({P_4}^T+P_4+Q_{22})\eta _2(t)\nonumber \\&+2\eta _2(t)^T{P_4}^TA_{d4}\eta _2(t-d(t))+2\eta _2(t)^T{P_4}^Te(t)\nonumber \\&+2\eta _2(t)^T{P_4}^T[{-F_3\psi _1(y(t))-F_4\psi _2(y(t))}]-{\eta _2(t-d(t))}^TQ_{22}\eta _2(t-d(t))\nonumber \\ \le&\left[ \begin{array}{cc} \eta _2(t)\\ \eta _2(t-d(t))\\ \psi _1(y(t))\\ \psi _2(y(t))\\ \end{array} \right] ^T \left[ \begin{array}{ccccccc} {P_4}^T+P_4+Q_{22} &{}\quad {P_4}^TA_{d4} &{}\quad -{P_4}^TF_3&{}\quad -{P_4}^TF_4\\ *&{}\quad -Q_{22} &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad 0\\ \end{array} \right] \nonumber \\&\quad \quad \left[ \begin{array}{cc} \eta _2(t)\\ \eta _2(t-d(t))\\ \psi _1(y(t))\\ \psi _2(y(t))\\ \end{array} \right] \nonumber \\&+\varepsilon _1\eta _2(t)^T \eta _2(t)+{\varepsilon _1}^{-1} e(t)^T{P_4}{P_4}^Te(t) \end{aligned}$$

(17)

where $\varepsilon _1$ is an any positive real number.

From (7), we can obtain

$$\begin{aligned} \left[ \begin{array}{ccccccc} \varXi _{11} &{}\quad \varXi _{12} \\ *&{}\quad \varXi _{22} \\ \end{array} \right] <0 \end{aligned}$$

Pre-multiplying and post-multiplying the above formula, respectively, by

$$\begin{aligned} \left[ \begin{array}{ccccccc} H &{}\quad 0 \\ 0 &{}\quad H \\ \end{array} \right] ^T and&\left[ \begin{array}{ccccccc} H &{}\quad 0 \\ 0 &{}\quad H \\ \end{array} \right] \end{aligned}$$

we can derive

$$\begin{aligned} \left[ \begin{array}{ccccccc} {P_4}^T+P_4+Q_{22} &{}\quad {P_4}^TA_{d4} \\ *&{}\quad -Q_{22}\\ \end{array} \right] <0 \end{aligned}$$

Then,

$$\begin{aligned} \varOmega _1=\left[ \begin{array}{ccccccc} {P_4}^T+P_4+Q_{22} &{}\quad {P_4}^TA_{d4} &{}\quad -{P_4}^TF_3&{}\quad -{P_4}^TF_4\\ *&{}\quad -Q_{22} &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad 0\\ \end{array} \right] <0 \end{aligned}$$

so there exists a scalar $\varepsilon _2>0$ such that

$$\begin{aligned} \varOmega _1\le -\left[ \begin{array}{ccccccc} \varepsilon _2I &{}\quad 0 &{}\quad 0&{}\quad 0\\ *&{}\quad 0&{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad 0&{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad 0\\ \end{array} \right] \end{aligned}$$

(18)

Choosing a sufficiently small $\varepsilon _1$, such that $\varepsilon _2-\varepsilon _1>0$. Then we can obtain an $\varepsilon _3>0$ such that

$$\begin{aligned} \varepsilon _3Q_{22}\le Q_{22}-(\varepsilon _1-\varepsilon _2)I \end{aligned}$$

(19)

It follows from (17), (18) and (19) that

$$\begin{aligned} L(t)\le -(\varepsilon _2-\varepsilon _1)\eta _2(t)^T\eta _2(t)+{\varepsilon _1}^{-1} e(t)^TP_4{P_4}^Te(t) \end{aligned}$$

(20)

From (19) and (20), we have that

$$\begin{aligned} \eta _2(t)^T Q_{22}\eta _2(t) \le&{\varepsilon _3}^{-1}\eta _2(t)^T[Q_{22}-(\varepsilon _1-\varepsilon _2)I]\eta _2(t)\nonumber \\ =&{\varepsilon _3}^{-1}[\eta _2(t)^TQ_{22}\eta _2(t)-(\varepsilon _1-\varepsilon _2)\eta _2(t)^T\eta _2(t)]\nonumber \\ =&{\varepsilon _3}^{-1}[\eta _2(t-d(t))^TQ_{22}\eta _2(t-d(t))\nonumber \\&\quad \quad +L(t)-(\varepsilon _1-\varepsilon _2)\eta _2(t)^T\eta _2(t)]\nonumber \\ \le&{\varepsilon _3}^{-1}[\eta _2(t-d(t))^TQ_{22}\eta _2(t-d(t))-(\varepsilon _2-\varepsilon _1)\eta _2(t)^T\eta _2(t)\nonumber \\&+{\varepsilon _1}^{-1} e(t)^TP_4{P_4}^Te(t)-(\varepsilon _1-\varepsilon _2)\eta _2(t)^T\eta _2(t)]\nonumber \\ =&{\varepsilon _3}^{-1}\eta _2(t-d(t))^TQ_{22}\eta _2(t-d(t))+{(\varepsilon _1\varepsilon _3)}^{-1}e(t)^TP_4{P_4}^Te(t)\nonumber \\ \le&{\varepsilon _3}^{-1}\eta _2(t-d(t))^TQ_{22}\eta _2(t-d(t))\nonumber \\&\quad \quad +{(\varepsilon _1\varepsilon _3)}^{-1}me^{-\delta t}||P_4||^2{||\phi (t)||^2}_{d_2} \end{aligned}$$

(21)

Because $0<\delta <\min (\varepsilon ,{d_2}^{-1}\ln {\varepsilon _3})$ and $0<{\varepsilon _3}^{-1}<1$, we can obtain that for $0<{\varepsilon _3}^{-1}e^{-\delta {d_2}}<1$, $\zeta ={(\varepsilon _1\varepsilon _3)}^{-1}m||P_4||^2{||\phi (t)||^2}_{d_2}>0$, where ${d_2}>0$. In addition, Lemma 3 implies that we have

$$\begin{aligned} \eta _2(t)^T Q_{22}\eta _2(t) \le e^{-\delta t}\lambda _{\max }(Q_{22}){||\eta _2(t)||^2}_{d_2}+\frac{\zeta e^{-\delta t}}{1-{\varepsilon _3}^{-1} e^{-\delta d_2}} \end{aligned}$$

$\lambda _{\min }(Q_{22})||\eta _2(t)||^2\le \eta _2(t)^T Q_{22}\eta _2(t)$, so we can obtain

$$\begin{aligned} ||\eta _2(t)||^2\le \lambda _{\min }(Q_{22})^{-1}\lambda _{\max }(Q_{22})e^{-\delta t}{||\eta _2(t)||^2}_{d_2}+\frac{\lambda _{\min }(Q_{22})^{-1}\zeta e^{-\delta t}}{1-{\varepsilon _3}^{-1} e^{-\delta d_2}} \end{aligned}$$

(22)

so $\eta _2(t)$ is exponentially stable.

Because of $||\eta (t)||^2=||\eta _1(t)||^2+||\eta _2(t)||^2$ and $x(t)=H\eta (t)$, we can obtain from (14), (22) and Definition 1 that the system is exponentially stable for any time-delay function d(t) that satisfies (5).

Next, we study the $H_\infty$ performance of the system, from (9) we can obtain

$$\begin{aligned} J(t)&=\int _0^t[z^T(s)z(s)-\gamma ^2w^T(s)w(s)]\mathrm{d}s\nonumber \\&\le \int _0^t[z^T(s)z(s)-\gamma ^2w^T(s)w(s)+{\mathcal {A}}{V}(x_s,s)]\mathrm{d}s\nonumber \\&\le \int _0^t\left[ \begin{array}{cccccc} \xi (t)\\ w(t) \end{array} \right] ^T\varOmega \left[ \begin{array}{cccccc} \xi (t)\\ w(t) \end{array} \right] \end{aligned}$$

(23)

where

$$\begin{aligned} \varOmega&=\left[ \begin{array}{ccccccc} \varXi _{11}+\frac{1}{\lambda }P^TFF^TP+L^TL &{}\quad \varXi _{12}+L^TL_d &{}\quad E^TR_1E &{}\quad 0 &{}\quad -A^TWF+C^TM^T\\ *&{}\quad \varXi _{22}+L_d^TL_d &{}\quad E^TR_3E &{}\quad \varXi _{24} &{}\quad -A_dWF+C_d^TM^T \\ *&{}\quad *&{}\quad \varXi _{33} &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad \varXi _{44} &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \\&\left. \begin{array}{ccccccc} L^TL_w+P^TB_w+A^TWB_w\\ L_d^TL_w+A_d^TWB_w \\ 0\\ 0\\ -F^TWB_w+MD_w \\ L_w^TL_w+B_w^TWB_w-\gamma ^2I \end{array} \right] \end{aligned}$$

By the Schur complements, Formula (7) implies that $\varOmega <0$, so $J(t)<0$ for $\forall t>0$; therefore, $||z(t)||_2\le \gamma ||w(t)||_2,\forall w(t)\in {\mathcal {L}} _2[0,+\infty ]$. $\square$

Remark 1

In Theorem 1, a new Lyapunov–Krasovskii functional is constructed, which not only uses the information of the upper limit $d_2$ of the time-delay; but also uses the information of the lower limit $d_1$ of the time-delay and the time-delay interval $d_{12}$. By using the Jensen Integral inequality and the Schur complements, the system is proven to be exponentially stable and satisfies the performance index $\gamma$ of $H_{\infty }$.

We design a controller for the Lur’e singular system. By considering the state feedback controller $u(t)=Kx(t)$, we obtain the following closed-loop system:

$$\begin{aligned} E{\dot{x}}(t)&=(A+B_uK)x(t)+A_dx(t-d(t))-F\varphi (y(t))+B_ww(t) \end{aligned}$$

(24)

$$\begin{aligned} y(t)&=Cx(t)+C_dx(t-d(t))+D_ww(t) \end{aligned}$$

(25)

$$\begin{aligned} z(t)&=(L+L_u K)x(t)+L_dx(t-d(t))+L_ww(t)\\ x(t)&=\phi (t),t\in [-d_2,0]\nonumber \end{aligned}$$

(26)

Corollary 1

In the system (24)–(26), for given $0\le d_1\le d_2,0\le \alpha <1$, if there is a positive number $\lambda >0$, symmetric positive definite matrices $N_j,{\bar{R}}_j,j=1,2,3,V_1,V_2$ and matrices X, Y such that

$$\begin{aligned}&XE^T=EX^T\ge 0\end{aligned}$$

(27)

$$\begin{aligned}&\left[ \begin{array}{ccccccccccc} \varXi _{1} &{}\quad A_dX^T &{}\quad EX^T &{}\quad 0 &{}\quad XC^TM^T &{}\quad B_\omega &{}\quad FX^T &{}\quad XL^T+Y{L_u}^T \\ *&{}\quad \varXi _{2} &{}\quad 0 &{}\quad EX^T &{}\quad XC_d^TM^T &{}\quad 0 &{}\quad 0 &{}\quad X{L_d}^T\\ *&{}\quad *&{}\quad V_1+V_2-N_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad V_2-N_2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I &{}\quad MD_\omega &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad -{\gamma }^2I &{}\quad 0 &{}\quad {L_\omega }^T\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad -\lambda I &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad - I\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \nonumber \\&\left. \begin{array}{ccccccccccc} d_1XA^T+d_1Y{B_u}^T &{}\quad d_2XA^T+d_2Y{B_u}^T &{}\quad d_{12}XA^T+d_{12}Y{B_u}^T \\ d_1X{A_d}^T &{}\quad d_2X{A_d}^T &{}\quad d_{12}X{A_d}^T \\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ -d_1F^T &{}\quad -d_2F^T &{}\quad -d_{12}F^T\\ d_1{B_\omega }^T &{}\quad d_2{B_\omega }^T &{}\quad d_{12}{B_\omega }^T\\ 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0\\ -{\bar{R}}_1 &{}\quad 0 &{}\quad 0\\ *&{}\quad -{\bar{R}}_2 &{}\quad 0\\ *&{}\quad *&{}\quad -{\bar{R}}_3 \end{array} \right] <0 \end{aligned}$$

(28)

where $\varXi _{1}=XA^T+AX^T+\sum _{k=1}^{3}N_k+Y{B_u}^T+{B_u}Y^T-EX^T-XE^T+{\bar{R}}_1,\varXi _{2}=-(1-\alpha )N_3-EX^T-XE^T+V_1+{\bar{R}}_2$, then for any time-delay function d(t) that satisfies (5), the closed-loop system is exponentially stable and satisfies the performance index $\gamma$ of $H_\infty$.

Proof

Setting ${\bar{A}}=A+B_u K,{\bar{L}}=L+L_u K$, it can be seen from Theorem 1 that (6) is established and

$$\begin{aligned}&S=\left[ \begin{array}{ccccccc} {\bar{\varXi }}_{11}+{\bar{L}}^T{\bar{L}} &{}\quad {\bar{\varXi }}_{12}+{\bar{L}}^TL_d &{}\quad E^TR_1E &{}\quad 0 &{}\quad -{\bar{A}}^TWF+C^TM^T\\ *&{}\quad \varXi _{22}+L_d^TL_d &{}\quad E^TR_3E &{}\quad \varXi _{24} &{}\quad -{A_d}^TWF+C_d^TM^T \\ *&{}\quad *&{}\quad \varXi _{33} &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad -Q_2-\varXi _{24} &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \\&\left. \begin{array}{ccccccc} {\bar{L}}^TL_w+P^TB_w+{\bar{A}}^TWB_w &{}\quad P^TF\\ L_d^TL_w+A_d^TWB_w &{}\quad 0\\ 0 &{}\quad 0\\ 0 &{}\quad 0\\ -F^TWB_w+MD_w &{}\quad 0\\ L_w^TL_w+B_wWB_w-\gamma ^2I &{}\quad 0\\ *&{}\quad -\lambda \end{array} \right] <0 \end{aligned}$$

where ${\bar{\varXi }}_{11}=P^T{\bar{A}}+{\bar{A}}^TP+\sum _{k=1}^{3}Q_k+{\bar{A}}^TW{\bar{A}}-E^TR_1E-d_{12}E^TR_2E$, ${\bar{\varXi }}_{12}=P^TA_d+{\bar{A}}^TWA_d+d_{12}E^TR_2E$. At this time, the closed-loop system (24)–(26) is stable and satisfies the performance index $\gamma$ of $H_\infty$.

Setting

$$\begin{aligned}&S_1=\left[ \begin{array}{ccccccc} S_{11} &{}\quad S_{12} &{}\quad E^TR_1E &{}\quad 0 &{}\quad -{\bar{A}}^TWF+ C^TM^T\\ *&{}\quad S_{22} &{}\quad 0 &{}\quad d_2 E^TR_2E &{}\quad -{A_d}^TWF+ C_d^TM^T \\ *&{}\quad *&{}\quad -Q_1-E^TR_1E &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad -Q_2-d_2 E^TR_2E &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \\&\left. \begin{array}{ccccccc} {\bar{L}}^TL_w+P^TB_w+{\bar{A}}^TWB_w &{}\quad P^TF\\ L_d^TL_w+A_d^TWB_w &{}\quad 0\\ 0 &{}\quad 0\\ 0 &{}\quad 0\\ -F^TWB_w+MD_w &{}\quad 0\\ L_w^TL_w+B_wWB_w-\gamma ^2I &{}\quad 0\\ *&{}\quad -\lambda \end{array} \right] \\&S_2=\left[ \begin{array}{ccccccc} -d_{12} E^TR_2E &{}\quad d_{12} E^TR_2E &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad -d_{12} E^TR_2E-2E^TR_3E &{}\quad E^TR_3E &{}\quad E^TR_3E &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad -E^TR_3E &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad -E^TR_3E &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad 0 \end{array} \right] \end{aligned}$$

where $S_{11}=P^T{\bar{A}}+{\bar{A}}^TP+\sum _{k=1}^{3}Q_k+{\bar{A}}^TW{\bar{A}}-E^TR_1E+{\bar{L}}^T{\bar{L}}, S_{12}=P^TA_d+{\bar{A}}^TWA_d+{\bar{L}}^TL_d,S_{22}=-(1-\alpha )Q_3+A_d^TWA_d-d_2 E^TR_2E+L_d^TL_d$, it is easy to see that $S=S_1+S_2$ and $S_2\le 0$ are naturally established, so $S_1<0$ is the sufficient condition of $S<0$.

Pre-multiplying $S_1$ by $H_1=diag[P^{-T},P^{-T},P^{-T},P^{-T},I,I,P^{-T}]$ and post-multiplying it by $H_1^{T}$, we can obtain $S_3=H_1S_1H_1^{T}$:

$$\begin{aligned}&S_3=\left[ \begin{array}{ccccccc} {\bar{S}}_{11} &{}\quad {\bar{S}}_{12} &{}\quad P^{-T}E^TR_1EP^{-1} &{}\quad 0 &{}\quad -P^{-T}{\bar{A}}^TWF+P^{-T}C^TM^T\\ *&{}\quad {\bar{S}}_{22} &{}\quad 0 &{}\quad d_2P^{-T} E^TR_2EP^{-1} &{}\quad -P^{-T}{A_d}^TWF+ P^{-T}C_d^TM^T \\ *&{}\quad *&{}\quad S_{33} &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad S_{44} &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I+F^TWF \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \\&\left. \begin{array}{ccccccc} P^{-T}{\bar{L}}^TL_w+B_w+P^{-T}{\bar{A}}^TWB_w &{}\quad FP^{-1}\\ P^{-T}L_d^TL_w+P^{-T}A_d^TWB_w &{}\quad 0\\ 0 &{}\quad 0\\ 0 &{}\quad 0\\ -F^TWB_w+ MD_w &{}\quad 0\\ L_w^TL_w+B_wWB_w-\gamma ^2I &{}\quad 0\\ *&{}\quad -\lambda \end{array} \right] \end{aligned}$$

where ${\bar{S}}_{11}={\bar{A}}P^{-1}+P^{-T}{\bar{A}}^T+\sum _{k=1}^{3}P^{-T}Q_kP^{-1}+P^{-T}{\bar{A}}^TW{\bar{A}}P^{-1} -P^{-T}E^TR_1EP^{-1} +P^{-T}{\bar{L}}^T{\bar{L}}P^{-1}$, ${\bar{S}}_{12}=A_dP^{-1}+P^{-T}{\bar{A}}^TWA_dP^{-1}+P^{-T}{\bar{L}}^TL_dP^{-1}$, ${\bar{S}}_{22}=-(1-\alpha ) P^{-T}Q_3P^{-1}+P^{-T}A_d^TWA_dP^{-1}{-}d_2P^{-T}E^TR_2EP^{-1} {+}P^{{-}T}L_d^TL_dP^{{-}1}$, $S_{33}=-P^{-T}Q_1P^{-1}-P^{-T}E^TR_1EP^{-1}$,$S_{44}=-P^{-T}Q_2P^{-1}-d_2P^{-T} E^TR_2EP^{-1}.$

It can be seen that for a matrix with the following format, there are positive definite matrices $V_1$ and $V_2$ such that

$$\begin{aligned}&\left[ \begin{array}{ccc} -P^{-T}E^TR_1EP^{-1} &{}\quad 0 &{}\quad P^{-T}E^TR_1EP^{-1} \\ *&{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad -P^{-T}E^TR_1EP^{-1} \\ \end{array} \right] \nonumber \\&\quad \le -\left[ \begin{array}{ccc} P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ -P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ \end{array} \right] -\left[ \begin{array}{ccc} EP^{-1} &{}\quad 0 &{}\quad -EP^{-1} \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ \end{array} \right] +\left[ \begin{array}{ccc} {R_1}^{-1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad V_1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad V_2 \\ \end{array} \right] \nonumber \\ \end{aligned}$$

(29)

$$\begin{aligned}&\left[ \begin{array}{ccc} -d_2P^{-T}E^TR_2EP^{-1} &{}\quad 0 &{}\quad d_2P^{-T}E^TR_2EP^{-1} \\ *&{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad -d_2P^{-T}E^TR_2EP^{-1} \\ \end{array} \right] \nonumber \\&\quad \le -\left[ \begin{array}{ccc} P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ -P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ \end{array} \right] -\left[ \begin{array}{ccc} EP^{-1} &{}\quad 0 &{}\quad -EP^{-1} \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ \end{array} \right] +\left[ \begin{array}{ccc} ({d_2R_2})^{-1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad V_1 &{}\quad 0 \\ 0 &{}\quad 0 &{} V_2 \\ \end{array} \right] \nonumber \\ \end{aligned}$$

(30)

Substituting $W=d_1^2R_1+d_{12}d_2^2R_2+d_{12}^2R_3$, Formulas (29) and (30) into matrix $S_3$, by the Schur complements, we can obtain matrix $S_4$:

$$\begin{aligned} S_4=&\left[ \begin{array}{ccccccccccc} \varTheta _{1} &{}\quad A_dP^{-1} &{}\quad EP^{-1} &{}\quad 0 &{}\quad P^{-T}C^TM^T &{}\quad B_\omega \\ *&{}\quad \varTheta _{2} &{}\quad 0 &{}\quad EP^{-1} &{}\quad P^{-T}C_d^TM^T &{}\quad 0 \\ *&{}\quad *&{}\quad V_1+V_2-P^{-T}Q_1P^{-1} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad V_2-P^{-T}Q_2P^{-1} &{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -(2-\lambda )I &{}\quad MD_\omega \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad -{\gamma }^2I \\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad *&{}\quad *\end{array} \right. \nonumber \\&\left. \begin{array}{ccccccccccc} FP^{-1} &{}\quad P^{-T}{\bar{L}}^T &{}\quad d_1P^{-T}{\bar{A}}^{T} &{}\quad d_2P^{-T}{\bar{A}}^{T} &{}\quad d_{12}P^{-T}{\bar{A}}^{T} \\ 0 &{}\quad P^{-T}{L_d}^T &{}\quad d_1P^{-T}{A_d}^T &{}\quad d_2P^{-T}{A_d}^T &{}\quad d_{12}P^{-T}{A_d}^T \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -d_1F^T &{}\quad -d_2F^T &{}\quad -d_{12}F^T\\ 0 &{}\quad {L_\omega }^T &{}\quad d_1{B_\omega }^T &{}\quad d_2{B_\omega }^T &{}\quad d_{12}{B_\omega }^T\\ -\lambda I &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad - I &{}\quad 0 &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad -{R_1}^{-1} &{}\quad 0 &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad -({d_{12}R_2})^{-1} &{}\quad 0\\ *&{}\quad *&{}\quad *&{}\quad *&{}\quad -{R_3}^{-1} \end{array} \right] <0 \end{aligned}$$

(31)

where $\varTheta _{1}={\bar{A}}P^{-1}+P^{-T}{\bar{A}}^T+\sum _{k=1}^{3}P^{-T}Q_kP^{-1}-P^{-T}E^T-EP^{-1}+{R_1}^{-1}, \varTheta _{2}=-(1-\alpha ) P^{-T}Q_3P^{-1}-P^{-T}E^T-EP^{-1}+({d_2R_2})^{-1}+V_1.$

We substitute ${\bar{A}}=A+B_u K$ and ${\bar{L}}=L+L_u K$ into matrix $S_4$, and set $P^{-T}=X,P^{-T}K^T=Y,P^{-T}Q_kP^{-1}=N_k,{R_1}^{-1}={\bar{R}}_1,({d_{12}R_2})^{-1}={\bar{R}}_2$, ${R_3}^{-1}={\bar{R}}_3$, so that $({d_2R_2})^{-1}=\frac{d_{12}}{d_2}{\bar{R}}_2=\frac{d_2-d_1}{d_2}{\bar{R}}_2\le {\bar{R}}_2$. From Formula (28), we can obtain $S_4<0$, so $S_3<0$, which implies that $S_1<0$. $S_2\le 0$, and $S=S_1+S_2<0$ is established; therefore, the closed-loop system is exponentially stable. $\square$

Remark 2

In Corollary 1, we propose a more clever approach to deal with matrices with a special format, where a nonlinear matrix is converted into the sum of several linear matrices. For example, in Formula (29), the nonlinear matrix $\left[ \begin{array}{ccc} -P^{-T}E^TR_1EP^{-1} &{}\quad 0 &{}\quad P^{-T}E^TR_1EP^{-1} \\ *&{}\quad 0 &{}\quad 0 \\ *&{}\quad *&{}\quad -P^{-T}E^TR_1EP^{-1} \\ \end{array} \right]$ is converted into the sum of three linear matrices through scaling: $-\left[ \begin{array}{ccc} P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ -P^{-T}E^T &{}\quad 0 &{}\quad 0 \\ \end{array} \right]$, $-\left[ \begin{array}{ccc} EP^{-1} &{}\quad 0 &{}\quad -EP^{-1} \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ \end{array} \right]$ and $\left[ \begin{array}{ccc} {R_1}^{-1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad V_1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad V_2 \\ \end{array} \right]$. By applying the result of Theorem 1 and repeatedly applying the Schur complements, the closed-loop system is proven to be exponentially stable and to satisfy the performance index $\gamma$ of $H_{\infty }$ .

4 Numerical Examples

In this section, several numerical examples are presented to illustrate the effectiveness of the proposed method, especially regarding the responses of x(t),y(t) and z(t). Through a comparison with existing results,the advantages of new method are demonstrated.

Example 1

Consider the linear singular time-delay system of Example 1 (shown in reference [14]), where the coefficient matrices and parameters are

$$\begin{aligned}&E=\left[ \begin{array}{ccccccc} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{array} \right] ,A=\left[ \begin{array}{ccccccc} -0.9 &{}\quad 0.2 \\ -0.1 &{}\quad -0.9 \end{array} \right] ,A_d=\left[ \begin{array}{ccccccc} -1.1 &{}\quad 0.2 \\ -0.1 &{}\quad -1.1 \end{array} \right] \\&F=\left[ \begin{array}{ccccccc} -0.2 &{}\quad 0.1\\ -0.45 &{}\quad -0.3 \end{array} \right] ,d_1=0 \end{aligned}$$

By choosing different values for the derivative upper limits $\alpha$ of the delay function, the maximum allowed time delay $d_2$ of the delay function, which ensures the stability of the system, is obtained. Then, by comparing the obtained results with those of references [16] and [14], we obtain Table 1.

Table 1 The maximum allowed time delay $d_2$ for the different values of $\alpha$

Remark 3

It is clear that the structure of the Lyapunov function in our paper is simpler than that in reference [14]. However, as shown in Table 1, when the derivative $\alpha$ of the delay function takes different values, we derive larger maximum allowed time delays than those in references [16] and [14], which ensures that the system is stable.

Example 2

Consider the linear singular time-delay system of Example 1 (shown in reference [22]), where the coefficient matrices and parameters are

$$\begin{aligned}&E=\left[ \begin{array}{ccccccc} 9 &{}\quad 3 \\ 6 &{}\quad 2 \end{array} \right] , A=\left[ \begin{array}{ccccccc} -13.1 &{}\quad -13.7 \\ -15.4 &{}\quad -23.8 \end{array} \right] , A_d=\left[ \begin{array}{ccccccc} -18.6 &{}\quad -10.4 \\ -25.2 &{}\quad -16.8 \end{array} \right] \\&B_w=\left[ \begin{array}{ccccccc} 1.9 \\ 1.8 \end{array} \right] , L=\left[ \begin{array}{ccccccc} 0.4 \\ -0.8 \end{array} \right] ^T, \alpha =0.2 \end{aligned}$$

(1) Suppose that $w(t)=0$; by comparing the values of the maximum allowed time delay $d_2$ for the different values of $d_1$, we can obtain Table 2.

Table 2 The maximum allowed time delay $d_2$ for the different values of $d_1$

(2) Suppose that $d_1(t)=0$; by comparing the values of the maximum allowed time delay $d_2$ for the different values of $\gamma$, we can obtain Table 3.

Table 3 The maximum allowed time delay $d_2$ for the different values of $\gamma$

It can be seen from Tables 2 and 3 that compared with those in reference [22], the values of the maximum allowed time delay $d_2$ that we obtain are larger, which means that the system is more stable.

Remark 4

In Example 2, we illustrate how we choose the design parameters to affect the control performance. First, in Table 2, through the LMI method, we obtain different values of the maximum allowed time delay $d_2$ by choosing different values of $d_1$ while $w(t)=0$. The values of $d_2$ are larger when that of $d_1$ is chosen to be larger. Then, in Table 3, we choose different values of $\gamma$ when $d_1(t)=0$ and obtain the corresponding values of the maximum allowed time delay $d_2$. It is clear that when we choose larger values of $\gamma$, we can obtain larger values of $d_2$.

Example 3

We design the state feedback controller for the system (1)–(3), and choose scalars as follows:

$$\begin{aligned}&E=\left[ \begin{array}{ccccccc} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{array} \right] ,\Phi =\left[ \begin{array}{ccccccc} -5 \\ 20 \end{array} \right] ,A=\left[ \begin{array}{ccccccc} -32.1 &{}\quad 23.6 \\ -45 &{}\quad 100 \end{array} \right] ,A_d=\left[ \begin{array}{ccccccc} -2 &{}\quad 1 \\ 0 &{}\quad 1 \end{array} \right] , M=\left[ \begin{array}{ccccccc} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{array} \right] \\&B_u=\left[ \begin{array}{ccccccc} -25 &{}\quad 10\\ -0.3&{}\quad 0 \end{array} \right] ,B_w=\left[ \begin{array}{ccccccc} -1 &{}\quad 2\\ -0.2 &{}\quad 0 \end{array} \right] ,F=\left[ \begin{array}{ccccccc} -1.2 &{}\quad -10 \\ 1 &{}\quad -10 \end{array} \right] ,C=\left[ \begin{array}{ccccccc} -24 &{}\quad 0.1 \\ 0 &{}\quad -2 \end{array} \right] \\&C_d=\left[ \begin{array}{ccccccc} -0.5 &{}\quad -16 \\ 3 &{}\quad 0 \end{array} \right] ,D_w=\left[ \begin{array}{ccccccc} -1 &{}\quad 0\\ 0 &{}\quad 0 \end{array} \right] ,L=\left[ \begin{array}{ccccccc} -2 &{}\quad -2.5 \\ 0 &{}\quad -1.4 \end{array} \right] ,L_u=\left[ \begin{array}{ccccccc} -0.1 &{}\quad 0\\ 0 &{}\quad -3 \end{array} \right] \\&L_d=\left[ \begin{array}{ccccccc} -1 &{}\quad -2 \\ 1 &{}\quad -3 \end{array} \right] ,L_w=\left[ \begin{array}{ccccccc} 0 &{}\quad 0\\ 0 &{}\quad -0.3 \end{array} \right] , \varphi {(y(t))}=\omega (t)=\left[ \begin{array}{ccccccc} e^{-t}sin(t)\\ e^{-t}cos(t) \end{array} \right] \\&d(t)=0.1sin(t)+0.1,d_1=0,d_2=0.2,d_{12}=d_2-d_1,\alpha =0.2,\gamma =1,\lambda =0.1 \end{aligned}$$

By using Corollary 1 in this paper and the linear matrix inequality (LMI) method, we can obtain the following results

$$\begin{aligned} X=\left[ \begin{array}{ccccccc} 0.1433 &{}\quad 0.0154 \\ 0.0143 &{}\quad -0.0574 \end{array} \right] ,K=\left[ \begin{array}{ccccccc} 20.3834 &{}\quad 5.0959 \\ 5.4887 &{}\quad -50.9204 \end{array} \right] \end{aligned}$$

which means the state feedback controller we design is

$$\begin{aligned} u(t)=\left[ \begin{array}{ccccccc} 20.3834 &{}\quad 5.0959 \\ 5.4887 &{}\quad -50.9204 \end{array} \right] x(t) \end{aligned}$$

so that the closed-loop system (24)–(26) is exponentially stable and satisfies the performance index $\gamma$ of $H_\infty$.

By applying simulink software, we can obtain the response figures of the state x(t), control input y(t) and control output z(t), as shown in Figs. 1, 2 and 3.

Fig. 1

State responses of x(t) in Example 3

Fig. 2

Input responses of y(t) in Example 3

Fig. 3

Output responses of z(t) in Example 3

In the figures, we can see that, at first, the values of the state x(t), control input y(t) and control output z(t) in the chosen numerical system are not equal to zero. However they rapidly approach zero with the state feedback controller, and after two seconds, they steadily approach zero, which implies that the numerical system is exponentially stable and satisfies the performance index $\gamma$ of $H_\infty$. Therefore, this numerical example illustrates that our method is effective.

5 Conclusion

This paper mainly studies the $H_\infty$ control problem for Lur’e singular time-delay systems. By using Lyapunov stability theory, a new Lyapunov function is constructed. Compared with that developed in a previous paper, our function has one more double integral term $d_1\int _{-d_1}^0\int _{t+\beta }^t{\dot{x}}^T(\alpha )E^TR_1E{\dot{x}}(\alpha )\mathrm{d}\alpha \mathrm{d}\beta$, which makes our obtained result more conservative. Based on the linear matrix inequality (LMI) method, sufficient conditions for the designed system to be exponentially stable and to satisfy the performance index of $H_ {\infty }$ are obtained. During this process, the main difficulty is confirming the exponential stability of the system. Subsequently, a design method for the state feedback controller of the system is given, and by applying a more clever approach for a nonlinear matrix with a special format to convert it into the sum of several linear matrices, the closed-loop system is also exponentially stable and satisfies the $H_ {\infty }$ performance index. Finally, numerical examples illustrate the effectiveness of the proposed method and its advantages over the existing results and the response figures clearly reflect the stability of the system.

In the future, we will consider more normal systems, such as Lur’e singular systems with uncertainties or Markov process, and we will study the problems of stability, $H_\infty$ control and finite-time $H_\infty$ control for these systems.