Robust (Q,S,R)-γ-Dissipative and H₂ Performances for Switched Systems with Mixed Time Delays

Abstract

This paper investigates the switching on switching rule and sampling input to guarantee (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances for switched systems with mixed time delays. Synchronous switching can be used to overcome the difficulty in implementation of real-time switching for signals and sampling. The design of sampling input under arbitrary switching for a switched system is also considered in this paper. A new proposal for a full matrix formulation approach and inequality are applied to achieve the main results. Finally, some numerical examples are illustrated to show the efficiency of the main contribution.

1. Introduction

Switched systems are usually confronted in many practical and physical models, such as automatic engine control systems, chemical processes, constrained robotics, multi--rate control systems, power electronics systems, robot manufacture systems, stepper motors, and water quality control systems [1,2,3]. Switched systems consist of various subsystems and use a switching rule to supervise the switching of dynamics between these subsystems. Switched systems may produce many complicated nonlinear dynamic behaviors, such as chaos and multiple limit cycles, which may be caused due to switching between subsystems [2,3]. On the other hand, delay phenomena are usually encountered in many practical systems such as chemical systems, hydraulic engineering systems, inferred grinding models, the Mackey–Glass equation, network dynamics, reaction-diffusion systems, nuclear reactors, and rolling mills. Time delay may cause bad performance or instability in many dynamic control systems [4,5,6]. It is also well that interval time-varying delay is a more useful formulation for a practical environment regarding signal transmission [1,7,8,9,10]. Hence, some results for stability analysis and performance for switched systems with interval time-varying delay have been proposed and developed in recent years [1,7,8,9,10].

The following two interesting facts can be noticed in switched systems: (1) the stability of a switched system can be reached by selecting a suitable switching method, even when each subsystem is unstable [1,7,8,10], (2) the stability of each subsystem cannot guarantee the overall system under arbitrary switching [9]. Hence, some valuable stability analysis and performance considerations have been proposed for switched systems in recent years. In [11], the average dwell time approach was used to identify the stability of the switched time-delay system under consideration. In [1], the asymptotic analysis of switched systems with state-dependent switching was studied using a Lyapunov-based approach. In [7,8,10,12,13,14,15], some useful design schemes for switching rules are developed to achieve good performance in the switched systems under consideration. The design schemes developed in [7,8,12,13,14] for switching signals are useful and can be studied in discrete and continuous switched systems with time delay.

In the recent years, passivity theory was proposed to guarantee the internal stability of systems. The passive theory has also been investigated in circuit analysis and practical applications [14,16]. On the other hand, $H_{\mathbf{\infty }}$ performance was usually used to improve the effect of the disturbance input on the regulated output of the systems under consideration [7,8,12,13,17]. Hence, it will be interesting to consider (Q,S,R)-$\mathsf{\gamma }$-dissipative performance, which guarantees the integrated property of the system under consideration [18,19,20]. The $H_{\mathbf{\infty }}$ performance and passivity can be seen as two special cases for setting parameters in (Q,S,R)-$\mathsf{\gamma }$-dissipative performance. On the other hand, $H_2$ performance can be used to minimize the dynamics of a system under consideration for some given initial conditions [3,8,18]. For the proposed non-negative inequality approach in [12], more linear matrix inequalities and running times were necessary. In [21], less conservative results were developed by using Wirtinger-based integral inequality instead of the previous Jensen method. The mixed performances of a switched system with time-varying delay were investigated by using real-time switching for rule and feedback control [12]. It is well known that real-time switching is difficult to use in all practical engineering systems. Synchronous switching would be a feasible way to deal with real-time [7,13,14,22]. In [13,14], a proposed aperiodic sampling was considered instead of the periodic system in [23]. Pointwise sampling is relaxed to interval one in this paper. On the other hand, linear fractional perturbation was a more general representation than norm-bounded in our considered systems [7,12,13,14,17,24,25].A major contribution and research motivation is provided as follows:

(1)

Based on many performances and approaches considered in the past, the full matrix formulation approach is developed to achieve (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances of a switched time-delay system with linear fractional perturbations. Linear matrix inequality conditions can be proposed to directly solve our main consideration.

(2)

In practical systems, real-time switching depending on state is difficult to implement. Hence the synchronous switching on rule and sampling input can be used to achieve (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performance of a switched time-delay system, instead of a real-time system. The sampled-data input for arbitrary switching in a switched system can be switched at each sampling instant.

(3)

For more practical results for the considered systems, some upper bounds for sampling intervals can be provided and evaluated using the proposed results. It is interesting to consider the interval time-varying delay, instead of the constant one in [13,14]. The linear matrix inequality results developed in this paper are less conservative and better than other published examples.

(4)

A new inequality in Lemma 2 and Lyapunov-Krasovskii functional, with $X^T\left(t\right)PX\left(t\right),$ is proposed to improve the main results obtained in this paper. Possible information on systems has been included in $X\left(t\right),$ from which the conservativeness of the results obtained can be improved. As in the proposed numerical examples, our approach is efficient and potentially useful.

This paper will be organized as follows. The problem description for our considered system is provided in Section 2. The synchronous switching on rule and sampling input for a switched time-delay system is developed in Section 3. Some examples are illustrated to show the main results in Section 4. A final conclusion is provided in Section 5.

The notation used throughout this paper is provide as follows. For matrix A, we denote a symmetric positive (negative) definite by $A>0$ ($A<0$), the transpose by $A^T$, $Sym\left(A\right)=A+A^T$. $A\le B$ means that matrix $B-A$ is symmetric positive semi-definite, $I$ and 0 denote the identity matrix and zero matrix with appropriate dimensions, respectively, $\Phi$ denotes the empty set, $E_{q,i}=\left[\begin{array}{ccc}{0}_{n\times \left(i-1\right)n}& I& 0_{n\times \left(q-i\right)n}\end{array}\right]\in {\mathfrak{R}}^{n\times qn}$, $i=1,\hspace{0.33em}2,\hspace{0.33em}\cdots ,\hspace{0.33em}q$, $q=2,\hspace{0.33em}3,\cdots ,18$, $0_{n\times \left(q-i\right)n}$ denotes the zero matrix with dimension $n\times \left(q-i\right)n$, and $0_n=\underset{n}{\underbrace{[00\cdots 00]}}$ denotes the matrix composed by n zero matrices with appropriate dimensions,$L_2\left(0,\mathrm{\infty }\right)=\left\{w\in {\mathfrak{R}}^m\left|{\int }_0^{\mathrm{\infty }}{w}^T\left(t\right)w\left(t\right)dt<\mathrm{\infty }\right.\right\}$, $A\backslash B=\left\{x\left|x\in A\hspace{0.33em}and\hspace{0.33em}x\notin B\right.\right\}$, and $\underset{\_}{N}=\left\{1,\hspace{0.33em}2,\hspace{0.33em}\cdots ,\hspace{0.33em}N\right\}$.

2. Problem Description

The following switched system with discrete and distributed delays is considered:

$$\begin{gathered} \dot{x}\left(t\right)={\overline{A}}_{0\sigma }\left(t\right)x\left(t\right)+{\overline{A}}_{1\sigma }\left(t\right)x\left(t-h\left(t\right)\right)+{\overline{A}}_{2\sigma }\left(t\right){\int }_{t-\theta }^{t}x\left(s\right)ds+{\overline{B}}_{u\sigma }\left(t\right)u\left(t\right) \\ +{\overline{B}}_{w\sigma }\left(t\right)w\left(t\right),\hspace{0.33em}t\ge 0, \end{gathered}$$

(1a)

$$\begin{gathered} z\left(t\right)={\overline{A}}_{z0\sigma }\left(t\right)x\left(t\right)+{\overline{A}}_{z1\sigma }\left(t\right)x\left(t-h\left(t\right)\right)+{\overline{A}}_{z2\sigma }\left(t\right){\int }_{t-\theta }^{t}x\left(s\right)ds+{\overline{B}}_{zu\sigma }\left(t\right)u\left(t\right) \\ +{\overline{B}}_{zw\sigma }\left(t\right)w\left(t\right),\hspace{0.33em}t\ge 0, \end{gathered}$$

(1b)

$$x\left(t\right)=\phi \left(t\right),t\in \left[-H,\hspace{0.33em}0\right],$$

(1c)

where ${\overline{A}}_{0\sigma }\left(t\right)=A_{0\sigma }+\Delta A_{0\sigma }\left(t\right)$, ${\overline{A}}_{1\sigma }\left(t\right)=A_{1\sigma }+\Delta A_{1\sigma }\left(t\right)$, ${\overline{A}}_{2\sigma }\left(t\right)=A_{2\sigma }+\Delta A_{2\sigma }\left(t\right)$, ${\overline{B}}_{u\sigma }\left(t\right)=B_{u\sigma }+\Delta B_{u\sigma }\left(t\right)$, ${\overline{B}}_{w\sigma }\left(t\right)=B_{w\sigma }+\Delta B_{w\sigma }\left(t\right)$, ${\overline{A}}_{z0\sigma }\left(t\right)=A_{z0\sigma }+\Delta A_{z0\sigma }\left(t\right)$, ${\overline{A}}_{z1\sigma }\left(t\right)=A_{z1\sigma }+\Delta A_{z1\sigma }\left(t\right)$, ${\overline{A}}_{z2\sigma }\left(t\right)=A_{z2\sigma }+\Delta A_{z2\sigma }\left(t\right)$, ${\overline{B}}_{zu\sigma }\left(t\right)=B_{zu\sigma }+\Delta B_{zu\sigma }\left(t\right)$, ${\overline{B}}_{zw\sigma }\left(t\right)=B_{zw\sigma }+\Delta B_{zw\sigma }\left(t\right)$, $x\left(t\right)\in {\mathfrak{R}}^n$ is the state of system, $u\left(t\right)\in {\mathfrak{R}}^p$ is the input, $w\left(t\right)\in {\mathfrak{R}}^m$ is disturbance, $z\left(t\right)\in {\mathfrak{R}}^q$ is regulated output, and $\sigma$ is the switching rule. The rule $\sigma$ will take its value in the finite set $\underset{\_}{N}$ and is a piecewise constant function. The interval time-varying delay $h\left(t\right)>0$ belongs to $0<h_m\le h\left(t\right)\le h_M$, $\dot{h}\left(t\right)\le h_D<1$, where $h_m$, $h_M$, and $h_D$ are positive constants and can be estimated or given in advance. The distributed delay $\theta >0$ is given, along with the initial vector $\phi \in C_0$, where $C_0$ will be a set of continuous functions from $\left[-H,\hspace{0.33em}0\right]$ to ${\mathfrak{R}}^n$, $H=\mathrm{m}\mathrm{a}\mathrm{x}\{h_M,\theta \}$. Matrices $A_{0i}$, $A_{1i}$, $A_{2i}\in {\mathfrak{R}}^{n\times n}$, $B_{ui}\in {\mathfrak{R}}^{n\times p}$, $B_{wi}\in {\mathfrak{R}}^{n\times m}$, $A_{z0i}$, $A_{z1i}$, $A_{z2i}\in {\mathfrak{R}}^{q\times n}$, $B_{zui}\in {\mathfrak{R}}^{q\times p}$, and $B_{zwi}\in {\mathfrak{R}}^{q\times m}$, $i\in \underset{\_}{N}$, are constant matrices. $\Delta A_{0i}\left(t\right)$, $\Delta A_{1i}\left(t\right)$, $\Delta A_{2i}\left(t\right)$, $\Delta B_{ui}\left(t\right)$, $\Delta B_{wi}\left(t\right)$, $\Delta A_{z0i}\left(t\right)$, $\Delta A_{z1i}\left(t\right)$, $\Delta A_{z2i}\left(t\right)$, $\Delta B_{zui}\left(t\right)$, and $\Delta B_{zwi}\left(t\right)$, $i\in \underset{\_}{N}$, are perturbed matrices. These linear fractional perturbations satisfy:

$$\left[\begin{array}{c}\Delta A_{0i}\left(t\right)\Delta A_{1i}\left(t\right)\Delta A_{2i}\left(t\right)\Delta B_{ui}\left(t\right)\Delta B_{wi}\left(t\right)\end{array}\right]=M_{xi}\cdot {\Delta }_{xi}\left(t\right)\cdot \left[\begin{array}{c}{N}_{x0i}{N}_{x1i}{N}_{x2i}{N_{xui}N}_{xwi}\end{array}\right],$$

(2a)

$$\begin{gathered} \left[\begin{array}{c}\Delta A_{z0i}\left(t\right)\Delta A_{z1i}\left(t\right)\Delta A_{z2i}\left(t\right)\Delta B_{zui}\left(t\right)\Delta B_{zwi}\left(t\right)\end{array}\right] \\ =M_{zi}\cdot {\Delta }_{zi}\left(t\right)\cdot \left[\begin{array}{c}{N}_{z0i}{N}_{z1i}{N_{z2i}N}_{zui}{N}_{zwi}\end{array}\right], \end{gathered}$$

(2b)

$${\Delta }_{xi}\left(t\right)={\left[I-{\Gamma }_{xi}\left(t\right){\Xi }_{xi}\right]}^{-1}{\Gamma }_{xi}\left(t\right),{\Xi }_{xi}{\Xi }_{xi}^T<I,$$

(2c)

$${\Delta }_{zi}\left(t\right)={\left[I-{\Gamma }_{zi}\left(t\right){\Xi }_{zi}\right]}^{-1}{\Gamma }_{zi}\left(t\right),{\Xi }_{zi}{\Xi }_{zi}^T<I,$$

(2d)

where the matrices $M_{xi}$, $M_{zi}$, $N_{x0i}$, $N_{x1i}$, $N_{x2i}$, $N_{xui}$, $N_{xwi}$, $N_{z0i}$, $N_{z1i}$, $N_{z2i}$, $N_{zui}$, $N_{zwi}$, ${\Xi }_{xi}$, and ${\Xi }_{zi}$, $i\in \underset{\_}{N}$ are constant with appropriate dimensions. The matrices ${\Gamma }_{xi}\left(t\right)$ and ${\Gamma }_{zi}\left(t\right)$, $\forall \hspace{0.33em}i\in \underset{\_}{N}$ are unknown and satisfy the following parameter perturbations:

$${\Gamma }_{xi}(t{)}^T\cdot {\Gamma }_{xi}(t)\le I,\forall \hspace{0.33em}i\in \underset{\_}{N},t\ge 0,$$

(2e)

$${\Gamma }_{zi}(t{)}^T\cdot {\Gamma }_{zi}(t)\le I,\forall \hspace{0.33em}i\in \underset{\_}{N},t\ge 0.$$

(2f)

Some domains of the system by

$${\Omega }_i\left(Z_i\right)=\left\{x\in {\mathfrak{R}}^n{:x}^T{Z}_{i}x\ge 0\right\},i=1,\hspace{0.33em}2,\cdots ,N,$$

(3a)

where matrices $Z_i=Z_i^T$, $i=1,\hspace{0.33em}2,\cdots ,N$ are selected from the proposed results in this paper. Hence the switching domains of system can be selected as follows:

$${\overline{\Omega }}_1={\Omega }_1,{\overline{\Omega }}_2={\Omega }_2\backslash {\overline{\Omega }}_1,{\overline{\Omega }}_3={\Omega }_3\backslash {\overline{\Omega }}_1\backslash {\overline{\Omega }}_2,\hspace{0.33em}\cdots ,\mathrm{and}{\overline{\Omega }}_N={\Omega }_N\backslash {\overline{\Omega }}_1\cdots \cdots \backslash {\overline{\Omega }}_{N-1}$$

(3b)

Defining the sampling instant by $0=T_0<T_1<T_2<\cdots$, we can select the following time-varying function:

$$\tau \left(t\right)=t-T_k,\hspace{0.33em}t\in [\left.T_k,T_{k+1}\right).$$

(4)

Defining ${\tau }_k=T_{k+1}-T_k$, ${\tau }_M={\mathit{max}}_{k=0}^{k=\mathrm{\infty }}{\tau }_k$>0, we have $0\le \tau \left(t\right)\le {\tau }_M$, $t\ge 0$, and $T_k=t-\tau \left(t\right),\hspace{0.33em}t\in [\left.T_k,T_{k+1}\right)$.

From the above definitions, we can provide a simple synchronous sampling for rule sampled-data input as follows:

When

$$x\left(T_k\right)\in {\overline{\Omega }}_i,\sigma \left(t\right)=i,\mathrm{for}\mathrm{all}t\in \left.{[T}_k,T_{k+1}\right),$$

(5a)

when

$$x\left(T_k\right)\in {\overline{\Omega }}_i,u\left(t\right)=-K_{i}x\left(T_k\right),\mathrm{for}\mathrm{all}t\in [\left.T_k,T_{k+1}\right),$$

(5b)

where ${\overline{\Omega }}_i$ is defined in (3b), $K_i\in {\mathfrak{R}}^{p\times n}$, $i\in \underset{\_}{m}$, will be selected from the main proposed results.

The following lemmas will be used to derive the main results in this paper.

Lemma 1.

[22] Let $x\left(t\right)$ be a differentiable function from $\left[t-h_2,\hspace{0.33em}t-h_1\right]\to {\mathfrak{R}}^n$. For a matrix $R>0$ and some matrices $N_1$, $N_2$, $N_3\in {\mathfrak{R}}^{4n\times n}$, and constants $0\le h_1\le h_2$, the following inequality is satisfied:

$$-{\int }_{\hspace{0.33em}t-h_2}^{\hspace{0.33em}t-h_1}{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\le {\delta }^T\left(t\right)\Omega \delta \left(t\right),$$

where

$${\delta }^T\left(t\right)=\left[\begin{array}{cc}{x}^T\left(t-h_1\right)& \begin{array}{ccc}{x}^T\left(t-h_1\right)& \frac{1}{h}{\int }_{\hspace{0.33em}t-h_2}^{\hspace{0.33em}t-h_1}{x}^T\left(s\right)ds& \frac{2}{h^2}{\int }_{t-h_2}^{t-h_1}{\int }_{\hspace{0.33em}t-h_2}^{\hspace{0.33em}s}{x}^T\left(u\right)duds\end{array}\end{array}\right],$$

$$h=h_2-h_1\ge 0,$$

$$\Omega =h\cdot \left[N_1{R}^{-1}{N}_1^T+\frac{1}{3}{N}_2{R}^{-1}{N}_2^T+\frac{1}{5}{N}_3{R}^{-1}{N}_3^T\right]+Sym\left(N_1{\Pi }_1+N_2{\Pi }_2+N_3{\Pi }_3\right),$$

$${\Pi }_1=E_{\mathrm{4,1}}-E_{\mathrm{4,2}},{\Pi }_2=E_{\mathrm{4,1}}+E_{\mathrm{4,2}}-2E_{\mathrm{4,3}},{\Pi }_3=E_{\mathrm{4,1}}-E_{\mathrm{4,2}}-6E_{\mathrm{4,3}}+6E_{\mathrm{4,4}}.$$

Lemma 2.

[18] Let $x\left(t\right)$ be a differentiable function from $\left[t-h_M,\hspace{0.33em}t-h_m\right]\to {\mathfrak{R}}^n$. For matrices $R>0$ and $\mathrm{S}$ and constants $0\le h_m\le h_M$ with

$$\left[\begin{array}{cc}R& S\\ \mathrm{*}& R\end{array}\right]>0,$$

the following inequality is satisfied:

$$-\left(h_M-h_m\right)\cdot {\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}t-h_m}{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds\le$$

$$-{\left[\begin{array}{c}x\left(t-h\left(t\right)\right)-x\left(t-h_M\right)\\ x\left(t-h_m\right)-x\left(t-h\left(t\right)\right)\end{array}\right]}^T\left[\begin{array}{cc}R& S\\ \mathrm{*}& R\end{array}\right]\left[\begin{array}{c}x\left(t-h\left(t\right)\right)-x\left(t-h_M\right)\\ x\left(t-h_m\right)-x\left(t-h\left(t\right)\right)\end{array}\right]={\mathrm{Z}}_2^T\left(t\right){\Omega }_2{Z}_2\left(t\right),$$

where

$$Z_2^T\left(t\right)=\left[\begin{array}{ccc}{x}^T\left(t-h\left(t\right)\right)& x^T\left(t-h_m\right)& x^T\left(t-h_M\right)\end{array}\right],$$

$${\Omega }_2=\left[\begin{array}{ccc}-2R+S+S^T& R-S& R-S^T\\ \mathrm{*}& -R& S^T\\ \mathrm{*}& \mathrm{*}& -R\end{array}\right].$$

Lemma 3.

[21] Let $x\left(t\right)$ be a differentiable function from $\left[t-{\tau }_1,\hspace{0.33em}t-{\tau }_2\right]\to {\mathfrak{R}}^n$. For a matrix $R>0$ and constants $0\le {\tau }_2\le {\tau }_1$, the following inequality is satisfied:

$$\begin{gathered} -\left({\tau }_1-{\tau }_2\right)\cdot {\int }_{\hspace{0.33em}t-{\tau }_1}^{\hspace{0.33em}t-{\tau }_2}{\dot{x}}^T\left(s\right)R\dot{x}\left(s\right)ds \\ \le -{\left[x\left(t-{\tau }_2\right)-x\left(t-{\tau }_1\right)\right]}^{T}R\left[x\left(t-{\tau }_2\right)-x\left(t-{\tau }_1\right)\right]-3{\Psi }^{T}R\Psi =\mathrm{Z}{\left(t\right)}^T\Omega Z\left(t\right), \end{gathered}$$

where

$$\Psi =x\left(t-{\tau }_2\right)+x\left(t-{\tau }_1\right)-\frac{2}{\left({\tau }_1-{\tau }_2\right)}\cdot {\int }_{t-{\tau }_1}^{t-{\tau }_2}x\left(s\right)ds,$$

$$Z^T\left(t\right)=\left[\begin{array}{ccc}{x}^T\left(t-{\tau }_2\right)& x^T\left(t-{\tau }_1\right)& \frac{1}{\tau }{\int }_{t-{\tau }_1}^{t-{\tau }_2}{x}^T\left(s\right)ds\end{array}\right],$$

$$\Omega =\left[\begin{array}{ccc}-4R& -2R& 6R\\ \mathrm{*}& -4R& 6R\\ \mathrm{*}& \mathrm{*}& -12R\end{array}\right],\tau ={\tau }_1-{\tau }_2.$$

Lemma 4.

[12] For constants $0\le {\alpha }_i\le 1$, $i\in \underset{\_}{N}$, ${\sum }_{i=1}^N{\alpha }_i=1$, matrices $Z_i=Z_i^T$, $i\in \underset{\_}{N}$, and

$${\sum }_{i=1}^N{\alpha }_i\cdot Z_i>0,$$

we have

$${{\bigcup }_{i=1}^N\overline{\Omega }}_i={\mathfrak{R}}^n\mathrm{and}{\overline{\Omega }}_i\cap {\overline{\Omega }}_j=\Phi ,\forall \hspace{0.33em}i\ne j,x^T\left(t\right){{\rm Z}}_{i}x\left(t\right)\ge 0\mathrm{and}\sigma \left(x\left(t\right)\right)=i,\forall \hspace{0.33em}x\left(t\right)\in {\overline{\Omega }}_i$$

where ${\overline{\Omega }}_i$ has been defined in definition (3b).

Lemma 5.

[24,25] For the matrix ${\Delta }_{xi}\left(t\right)$ defined in (2a) and satisfied in (2c), matrices $U_i$, $W_i$, $X_i$, and $X_i=X_i^T$, we have the following equivalent conditions:

(I)

$X_i+U_i{\Delta }_{xi}\left(t\right)W_i+W_i^T{\Delta }_{xi}^T\left(t\right)U_i^T<0$.

(II)

With a constant ${\epsilon }_i>0$, we have

$$\left[\begin{array}{ccc}{X}_i& {\epsilon }_i\cdot U_i& W_i^T\\ \mathrm{*}& -{\epsilon }_i\cdot I& {\epsilon }_i\cdot {\Xi }_{xi}^T\\ \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I\end{array}\right]<0,$$

where the matrix ${\Xi }_{xi}$ has been defined in condition (2c).

In this paper, the (Q,S,R)–$\mathsf{\gamma }$ dissipative and $H_2$ performances for our considered switched time-delay system are defined as follows.

Definition 1.

[18,19] Consider the switched system (1) with (2), switching signal in (5a), and sampled-data control in (5b). Assume

(i)

With $w\left(t\right)=0$, the system (1) with (2) is asymptotically stable due to the switching signal in (5a) and sampled-data control in (5b).

(ii)

With $\phi \left(t\right)=0,\hspace{0.33em}-H\le t\le 0$, andgiven constant $\gamma$, matrices $S$, $Q=Q^T\le 0$, and $R=R^T$, the signals $w\left(t\right)$ and $z\left(t\right)$ are constrained in

$$\gamma \cdot {\int }_0^{{\mathcal{l}}_1}{w}^T\left(t\right)w\left(t\right)dt\le {\int }_0^{{\mathcal{l}}_1}{[z}^T\left(t\right)Qz\left(t\right)+2z^T\left(t\right)Sw\left(t\right)+w^T\left(t\right)Rw\left(t\right)]dt,\forall \hspace{0.33em}w\ne 0$$

(6)

for all constants ${\mathcal{l}}_1>0$. With the parameter ${\mathcal{l}}_1$=$\mathrm{\infty }$, $w$should be defined in $L_2\left(0,\mathrm{\infty }\right)$.

(iii)

Under zero disturbance $w\left(t\right)=0$, an upper bound $\alpha >0$ can be found to satisfy the following condition

$${\int }_0^{{\mathcal{l}}_2}{z}^T\left(t\right)z\left(t\right)dt\le \alpha$$

for all constant ${\mathcal{l}}_2>0$.

Then we can state that system (1) with (2) is asymptotically stabilizable by the switching signal in (5a) and sampled-data control in (5b) with strictly (Q,S,R)–$\mathsf{\gamma }$ dissipative performance and $H_2$ measure $\alpha$.

3. Main Results

Now we will provide a delay-dependent linear matrix inequality result to show the asymptotic stability and (Q,S,R)–$\mathsf{\gamma }$ dissipative performance of switched system (1) with (2), using our developed synchronous switching on rule and sampling input in (5).

Theorem 1.

With constants $\eta$, $0\le {\alpha }_i\le 1$, $i\in \underset{\_}{N}$, and ${\sum }_{i=1}^N{\alpha }_i=1$, the following linear matrix inequality conditions:

$${\sum }_{i=1}^N{\alpha }_i\cdot {\widehat{Z}}_i>0,$$

(7a)

$$\left[\begin{array}{cc}{\widehat{R}}_5& {\widehat{S}}_1\\ \mathrm{*}& {\widehat{R}}_5\end{array}\right]>0,$$

(7b)

$$\left[\begin{array}{cc}{\widehat{R}}_6& {\widehat{S}}_2\\ \mathrm{*}& {\widehat{R}}_6\end{array}\right]>0,$$

(7c)

$${\widehat{\Psi }}_i=\left[\begin{array}{ccccccc}{\widehat{\Psi }}_{11i}& {\widehat{\Psi }}_{12i}& {\widehat{\Psi }}_{13i}& {\widehat{\Psi }}_{14i}& {\widehat{\Psi }}_{15i}& {\widehat{\Psi }}_{16i}& {\widehat{\Psi }}_{17}\\ \mathrm{*}& -{(I-Q)}^{-1}& 0& {\epsilon }_i\cdot M_{zi}& 0& 0& 0\\ \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{xi}& 0& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{zi}& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& {\widehat{\Psi }}_{66}\end{array}\right]<0,i\in \underset{\_}{N},$$

(7d)

where

$$\begin{gathered} {\widehat{\Psi }}_{11i}=Sym\left({\Delta }_2^T\widehat{P}{\Delta }_1+\left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right){\widehat{\Gamma }}_{1i}\right)-{Sym(E}_{\mathrm{18,17}}^T{\widehat{\Gamma }}_{2i})+{\widehat{\Omega }}_1+{\widehat{\Omega }}_2+{\widehat{\Omega }}_3{+\widehat{\Omega }}_4, \\ {\widehat{\Psi }}_{12i}={\widehat{\Gamma }}_{2i}^T,{\widehat{\Psi }}_{13i}={\epsilon }_i\cdot \left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right)M_{xi},{{\widehat{\Psi }}_{14i}=-{\epsilon }_i\cdot E}_{\mathrm{18,17}}^T{S^{T}M}_{zi}, \\ {\widehat{\Psi }}_{15i}^T=\left[\begin{array}{c}{N}_{x0i}{\widehat{U}}^T{N}_{x1i}{\widehat{U}}^T{0_2-N_{xui}{\widehat{K}}_i{0}_6\theta ·N_{x2i}{\widehat{U}}^T{0}_{4}N}_{xwi}0\end{array}\right], \\ {\widehat{\Psi }}_{16i}^T=\left[N_{z0i}{\widehat{U}}^T{N}_{z1i}{\widehat{U}}^T{0_2-N_{zui}{\widehat{K}}_i{0}_6\theta ·N_{z2i}{\widehat{U}}^T{0}_{4}N}_{zwi}0\right], \\ {\widehat{\Psi }}_{17}=[\sqrt{h_m}\cdot {\Lambda }_1^T{\widehat{N}}_{11}\sqrt{h_m}\cdot {\Lambda }_1^T{\widehat{N}}_{12}\sqrt{h_m}\cdot {\Lambda }_1^T{\widehat{N}}_{13}\sqrt{h_M}\cdot {\Lambda }_2^T{\widehat{N}}_{21}\sqrt{h_M}\cdot {\Lambda }_2^T{\widehat{N}}_{22} \\ \sqrt{h_M}\cdot {\Lambda }_2^T{\widehat{N}}_{23}\sqrt{{h_M-h}_m}\cdot {\Lambda }_3^T{\widehat{N}}_{31}\sqrt{{h_M-h}_m}\cdot {\Lambda }_3^T{\widehat{N}}_{32}\sqrt{{h_M-h}_m}\cdot {\Lambda }_3^T{\widehat{N}}_{33} \\ \sqrt{{\tau }_M}\cdot {\Lambda }_4^T{\widehat{N}}_{41}\sqrt{{\tau }_M}\cdot {\Lambda }_4^T{\widehat{N}}_{42}\sqrt{{\tau }_M}\cdot {\Lambda }_4^T{\widehat{N}}_{43}] \\ {\widehat{\Psi }}_{66}=diag\left[-{\widehat{R}}_1\begin{array}{c}-3{\widehat{R}}_1-5{\widehat{R}}_1-{\widehat{R}}_2-3{\widehat{R}}_2-5{\widehat{R}}_2-{\widehat{R}}_3-3{\widehat{R}}_3-5{\widehat{R}}_3-{\widehat{R}}_4-3{\widehat{R}}_4-5{\widehat{R}}_4\end{array}\right], \\ {\widehat{\Omega }}_1={\sum }_{i=1}^{3}Sym\left({{\Lambda }_1^T\widehat{N}}_{1i}{\Pi }_i{\Lambda }_1+{\Lambda }_2^T{\widehat{N}}_{2i}{\Pi }_i{\Lambda }_2+{{\Lambda }_3^T\widehat{N}}_{3i}{\Pi }_i{\Lambda }_3+{{\Lambda }_4^T\widehat{N}}_{4i}{\Pi }_i{\Lambda }_4,\right) \\ {{\widehat{\Omega }}_2=\Lambda }_5^T\left[\begin{array}{ccc}-2{\widehat{R}}_5+{\widehat{S}}_1+{\widehat{S}}_1^T& {\widehat{R}}_5-{\widehat{S}}_1& {\widehat{R}}_5-{\widehat{S}}_1^T\\ \mathrm{*}& -{\widehat{R}}_5& {\widehat{S}}_1^T\\ \mathrm{*}& \mathrm{*}& -{\widehat{R}}_5\end{array}\right]{\Lambda }_5 \\ {+\Lambda }_6^T\left[\begin{array}{ccc}-2{\widehat{R}}_6+{\widehat{S}}_2+{\widehat{S}}_2^T& {\widehat{R}}_6-{\widehat{S}}_2& {\widehat{R}}_6-{\widehat{S}}_2^T\\ \mathrm{*}& -{\widehat{R}}_6& {\widehat{S}}_2^T\\ \mathrm{*}& \mathrm{*}& -{\widehat{R}}_6\end{array}\right]{\Lambda }_6, \\ {\widehat{\Omega }}_3=E_{\mathrm{18,1}}^T{{\widehat{Q}}_{0}E}_{\mathrm{18,1}}{-\left(1-h_D\right)·E}_{\mathrm{18,2}}^T{{\widehat{Q}}_{4}E}_{\mathrm{18,2}}-E_{\mathrm{18,3}}^T({{\widehat{Q}}_1-{\widehat{Q}}_3)E}_{\mathrm{18,3}} \\ -E_{\mathrm{18,4}}^T({{\widehat{Q}}_2+{\widehat{Q}}_3)E}_{\mathrm{18,4}}-E_{\mathrm{18,6}}^T{\widehat{Q}}_5{E}_{\mathrm{18,6}}-E_{\mathrm{18,7}}^T{\widehat{Q}}_6{E}_{\mathrm{18,7}}+E_{\mathrm{18,5}}^T{\widehat{{\rm Z}}}_i{E}_{\mathrm{18,5}} \\ -E_{\mathrm{18,17}}^T(R-\gamma \cdot I)E_{\mathrm{18,17}}-\eta \cdot Sym\left(E_{\mathrm{18,1}}^T{{\widehat{U}}^{T}E}_{\mathrm{18,18}}\right)-E_{\mathrm{18,18}}^T\left[\widehat{U}+{\widehat{U}}^T-{\widehat{R}}_0\right]E_{\mathrm{18,18}}, \\ {\widehat{\Omega }}_4={\Lambda }_7^T\left[\begin{array}{ccc}-4{\widehat{R}}_7& -2{\widehat{R}}_7& 6{\widehat{R}}_7\\ \mathrm{*}& -4{\widehat{R}}_7& 6{\widehat{R}}_7\\ \mathrm{*}& \mathrm{*}& -12{\widehat{R}}_7\end{array}\right]{\Lambda }_7, \\ {\widehat{Q}}_0={\widehat{Q}}_1+{\widehat{Q}}_2+{\widehat{Q}}_4+{\widehat{Q}}_5+{\widehat{Q}}_6, \\ {\widehat{R}}_0=h_m\cdot {\widehat{R}}_1+h_M\cdot {\widehat{R}}_2+\left(h_M-h_m\right)\cdot {(\widehat{R}}_3+{\widehat{R}}_5)+{\tau }_M\cdot {(\widehat{R}}_4+{\widehat{R}}_6)+{\theta }^2\cdot {\widehat{R}}_7, \\ {\Pi }_1=E_{\mathrm{4,1}}-E_{\mathrm{4,2}},{\Pi }_2=E_{\mathrm{4,1}}+E_{\mathrm{4,2}}-2E_{\mathrm{4,3}},{\Pi }_3=E_{\mathrm{4,1}}-E_{\mathrm{4,2}}-6E_{\mathrm{4,3}}+6E_{\mathrm{4,4}}, \end{gathered}$$

(7e)

$$\begin{gathered} {\Delta }_1=\left[E_{\mathrm{18,1}}^T{h}_m\cdot E_{\mathrm{18,8}}^T{h}_M\cdot E_{\mathrm{18,9}}^T{(h_M-h}_m\right)\cdot E_{\mathrm{18,10}}^T{\tau }_M\cdot E_{\mathrm{18,11}}^T\theta \cdot E_{\mathrm{18,12}}^T \\ {0.5\cdot h_m^2\cdot E_{\mathrm{18,13}}^{T}0.5\cdot h_M^2\cdot E_{\mathrm{18,14}}^{T}0.5\cdot {\left(h_M-h_m\right)}^2\cdot E_{\mathrm{18,15}}^{T}0.5\cdot {\tau }_M^2\cdot E_{\mathrm{18,16}}^T]}_{}^T, \\ {\Delta }_2=[E_{\mathrm{18,18}}^T{\left(E_{\mathrm{18,1}}-E_{\mathrm{18,3}}\right)}^T{\left(E_{\mathrm{18,1}}-E_{\mathrm{18,4}}\right)}^T{\left(E_{\mathrm{18,3}}-E_{\mathrm{18,4}}\right)}^T \\ {\left(E_{\mathrm{18,1}}-E_{\mathrm{18,6}}\right)}^T{\left(E_{\mathrm{18,1}}-E_{\mathrm{18,7}}\right)}^T{h}_m\cdot {\left(E_{\mathrm{18,8}}-E_{\mathrm{18,3}}\right)}^T{h}_M\cdot {\left(E_{\mathrm{18,9}}-E_{\mathrm{18,4}}\right)}^T \\ {(h_M-h}_m)\cdot {\left(E_{\mathrm{18,10}}-E_{\mathrm{18,4}}\right)}^T{\left.{\tau }_M\cdot {\left(E_{\mathrm{18,11}}-E_{\mathrm{18,6}}\right)}^T\right]}_{}^T, \end{gathered}$$

(7f)

$$\begin{gathered} {\Lambda }_1={\left[\begin{array}{ccc}{E}_{\mathrm{18,1}}^T& E_{\mathrm{18,3}}^T& \begin{array}{cc}{E}_{\mathrm{18,8}}^T& E_{\mathrm{18,13}}^T\end{array}\end{array}\right]}^T,{\Lambda }_2={\left[\begin{array}{ccc}{E}_{\mathrm{18,1}}^T& E_{\mathrm{18,4}}^T& \begin{array}{cc}{E}_{\mathrm{18,9}}^T& E_{\mathrm{18,14}}^T\end{array}\end{array}\right]}^T, \\ {\Lambda }_3={\left[\begin{array}{ccc}{E}_{\mathrm{18,3}}^T& E_{\mathrm{18,4}}^T& \begin{array}{cc}{E}_{\mathrm{18,10}}^T& E_{\mathrm{18,15}}^T\end{array}\end{array}\right]}^T,{\Lambda }_4={\left[\begin{array}{ccc}{E}_{\mathrm{18,1}}^T& E_{\mathrm{18,6}}^T& \begin{array}{cc}{E}_{\mathrm{18,11}}^T& E_{\mathrm{18,16}}^T\end{array}\end{array}\right]}^T, \\ {{\Lambda }_5=\left[\begin{array}{ccc}{E}_{\mathrm{18,2}}^T& E_{\mathrm{18,3}}^T& \begin{array}{c}{E}_{\mathrm{18,4}}^T\end{array}\end{array}\right]}^T,{{\Lambda }_6=\left[\begin{array}{ccc}{E}_{\mathrm{18,5}}^T& E_{\mathrm{18,1}}^T& \begin{array}{c}{E}_{\mathrm{18,6}}^T\end{array}\end{array}\right]}^T, \\ {{\Lambda }_7=\left[\begin{array}{ccc}{E}_{\mathrm{18,1}}^T& E_{\mathrm{18,7}}^T& \begin{array}{c}{E}_{\mathrm{18,12}}^T\end{array}\end{array}\right]}^T, \end{gathered}$$

(7g)

$${\widehat{\Gamma }}_{1i}=\left[A_{0i}{\widehat{U}}^T{A}_{1i}{\widehat{U}}^T{0}_2-B_{ui}{\widehat{K}}_i{0}_6\theta {·A}_{2i}{\widehat{U}}^T{0}_4{B}_{wi}0\right],$$

$${\widehat{\Gamma }}_{2i}=\left[A_{z0i}{\widehat{U}}^T{A}_{z1i}{\widehat{U}}^T{0}_2-B_{zui}{\widehat{K}}_i{0}_6\theta {·A}_{z2i}{\widehat{U}}^T{0}_4{B}_{zwi}0\right],$$

are feasible with constants ${\mathsf{\epsilon }}_{\mathrm{i}}>0$, $\mathrm{i}\in \underset{\_}{\mathrm{N}}$, a nonsingular matrix $\widehat{\mathrm{U}}\in {\mathfrak{R}}^{\mathrm{n}\times \mathrm{n}}$, positive definite symmetric matrices $\widehat{\mathrm{P}}\in {\mathfrak{R}}^{10\mathrm{n}\times 10\mathrm{n}}$, ${\widehat{\mathrm{Q}}}_{\mathrm{i}}$, ${\widehat{\mathrm{R}}}_{\mathrm{j}}\in {\mathfrak{R}}^{\mathrm{n}\times \mathrm{n}}$, $\mathrm{i}\in \underset{\_}{6}$, $\mathrm{j}\in \underset{\_}{7}$, symmetric matrices ${\widehat{\mathrm{Z}}}_{\mathrm{i}}\in {\mathfrak{R}}^{\mathrm{n}\times \mathrm{n}}$, $\mathrm{i}\in \underset{\_}{\mathrm{N}}$, and matrices ${\widehat{\mathrm{S}}}_1,{\widehat{\mathrm{S}}}_2\in {\mathfrak{R}}^{\mathrm{n}\times \mathrm{n}}$, ${\widehat{\mathrm{N}}}_{\mathrm{i}\mathrm{j}}\in {\mathfrak{R}}^{4\mathrm{n}\times \mathrm{n}}$, $\mathrm{i}\in \underset{\_}{4}$,$\mathrm{j}\in \underset{\_}{3}$, ${\widehat{\mathrm{K}}}_{\mathrm{i}}\in {\mathfrak{R}}^{\mathrm{p}\times \mathrm{n}}$, $\mathrm{i}\in \underset{\_}{\mathrm{N}}$. Then the system (1) with (2) is asymptotically stabilizable by the switching rule ${\mathsf{{\rm Z}}}_{\mathrm{i}}={\widehat{\mathrm{U}}}^{-1}{\widehat{\mathsf{{\rm Z}}}}_{\mathrm{i}}{\widehat{\mathrm{U}}}^{-\mathrm{T}}$in (5a) and sampling input ${\mathrm{K}}_{\mathrm{i}}={\widehat{\mathrm{K}}}_{\mathrm{i}}{\widehat{\mathrm{U}}}^{-\mathrm{T}}$in (5b), with strictly (Q,S,R)–$\mathsf{\gamma }$ dissipative performance and ${\mathrm{H}}_2$ measure

$$\begin{gathered} \alpha =X^T\left(0\right)PX\left(0\right)+{\int }_{\hspace{0.33em}-h_m}^{\hspace{0.33em}0}{\phi }^T\left(s\right)Q_1\phi \left(s\right)ds+{\int }_{\hspace{0.33em}-h_M}^{\hspace{0.33em}0}{\phi }^T\left(s\right)Q_2\phi \left(s\right)ds \\ +{\int }_{\hspace{0.33em}-h_M}^{\hspace{0.33em}-h_m}{\phi }^T\left(s\right)Q_3\phi \left(s\right)ds+{\int }_{\hspace{0.33em}-h_M}^{\hspace{0.33em}0}{\phi }^T\left(s\right)Q_4\phi \left(s\right)ds+{\int }_{\hspace{0.33em}-{\tau }_M}^{\hspace{0.33em}0}{\phi }^T\left(s\right)Q_5\phi \left(s\right)ds \\ +{\int }_{\hspace{0.33em}-\theta }^{\hspace{0.33em}0}{\phi }^T\left(s\right)Q_6\phi \left(s\right)ds+{\int }_{-h_m}^0{\int }_s^0{\dot{\phi }}^T\left(u\right)R_1\dot{\phi }\left(u\right)duds+{\int }_{-h_M}^0{\int }_s^0{\dot{\phi }}^T\left(u\right)R_2\dot{\phi }\left(u\right)duds \\ +{\int }_{-h_M}^{-h_m}{\int }_s^0{\dot{\phi }}^T\left(u\right)\left(R_3+R_5\right)\dot{\phi }\left(u\right)duds+{\int }_{-{\tau }_M}^0{\int }_s^0{\dot{\phi }}^T\left(u\right){(R}_4+R_6)\dot{\phi }\left(u\right)duds \\ +\theta ·{\int }_{-\theta }^0{\int }_s^0{\dot{\phi }}^T\left(u\right)R_7\dot{\phi }\left(u\right)duds, \end{gathered}$$

(7h)

where some notations are defined as follows:

$$\begin{gathered} P=\widehat{\widehat{\widehat{U}}}\widehat{P}{\widehat{\widehat{\widehat{U}}}}^T>0,\widehat{\widehat{\widehat{U}}}=diag\left[UUUUUUUUUU\right],{U=\widehat{U}}^{-1}, \\ {Q}_i=U{{\widehat{Q}}_{i}U}^T>0,R_j=U{{\widehat{R}}_{j}U}^T>0,i\in \underset{\_}{6},j\in \underset{\_}{7}, \\ X\left(0\right)=[{\phi }^T\left(0\right){\int }_{-h_m}^0{\phi }^T\left(s\right)ds{\int }_{-h_M}^0{\phi }^T\left(s\right)ds{\int }_{-h_M}^{-h_m}{\phi }^T\left(s\right)ds{\int }_{-{\tau }_M}^0{\phi }^T\left(s\right)ds \\ {\int }_{-\theta }^0{\phi }^T\left(s\right)ds{\int }_{-h_m}^0{\int }_{\hspace{0.33em}-h_m}^{\hspace{0.33em}s}{\phi }^T\left(u\right)duds{\int }_{-h_M}^t{\int }_{\hspace{0.33em}-h_M}^{\hspace{0.33em}s}{\phi }^T\left(u\right)duds \\ {\left.{\int }_{-h_M}^{-h_m}{\int }_{\hspace{0.33em}-h_M}^{\hspace{0.33em}s}{\phi }^T\left(u\right)duds{\int }_{-{\tau }_M}^0{\int }_{\hspace{0.33em}-{\tau }_M}^{\hspace{0.33em}s}{\phi }^T\left(u\right)duds\right]}_{}^T \end{gathered}$$

Proof. 

Define the Lyapunov functional

$$\begin{gathered} V\left(x_t\right)=X^T\left(t\right)PX\left(t\right)+{\int }_{\hspace{0.33em}t-h_m}^{\hspace{0.33em}t}{x}^T\left(s\right)Q_{1}x\left(s\right)ds+{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}t}{x}^T\left(s\right)Q_{2}x\left(s\right)ds+{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}t-h_m}{x}^T\left(s\right)Q_{3}x\left(s\right)ds \\ +{\int }_{\hspace{0.33em}t-h\left(t\right)}^{\hspace{0.33em}t}{x}^T\left(s\right)Q_{4}x\left(s\right)ds+{\int }_{\hspace{0.33em}t-{\tau }_M}^{\hspace{0.33em}t}{x}^T\left(s\right)Q_{5}x\left(s\right)ds+{\int }_{\hspace{0.33em}t-\theta }^{\hspace{0.33em}t}{x}^T\left(s\right)Q_{6}x\left(s\right)ds \\ +{\int }_{-h_m}^0{\int }_{t+s}^t{\dot{x}}^T\left(u\right)R_1\dot{x}\left(u\right)duds+{\int }_{-h_M}^0{\int }_{t+s}^t{\dot{x}}^T\left(u\right)R_2\dot{x}\left(u\right)duds \\ +{\int }_{-h_M}^{-h_m}{\int }_{t+s}^t{\dot{x}}^T\left(u\right)\left(R_3+R_5\right)\dot{x}\left(u\right)duds+{\int }_{-{\tau }_M}^0{\int }_{t+s}^t{\dot{x}}^T\left(u\right){(R}_4+R_6)\dot{x}\left(u\right)duds \\ +\theta ·{\int }_{-\theta }^0{\int }_{t+s}^t{\dot{x}}^T\left(u\right)R_7\dot{x}\left(u\right)duds, \end{gathered}$$

(8)

where some notations are defined as follows:

$$P=\widehat{\widehat{\widehat{U}}}\widehat{P}{\widehat{\widehat{\widehat{U}}}}^T>0,\widehat{\widehat{\widehat{U}}}=diag\left[UUUUUUUUUU\right],{U=\widehat{U}}^{-1},$$

$$Q_i=U{{\widehat{Q}}_{i}U}^T>0,R_j=U{{\widehat{R}}_{j}U}^T>0,i\in \underset{\_}{6},j\in \underset{\_}{7},$$

$$\begin{gathered} X\left(t\right)=[x^T\left(t\right){\int }_{t-h_m}^t{x}^T\left(s\right)ds{\int }_{t-h_M}^t{x}^T\left(s\right)ds{\int }_{t-h_M}^{t-h_m}{x}^T\left(s\right)ds{\int }_{t-{\tau }_M}^t{x}^T\left(s\right)ds \\ {\int }_{t-\theta }^t{x}^T\left(s\right)ds{\int }_{t-h_m}^t{\int }_{\hspace{0.33em}t-h_m}^{\hspace{0.33em}s}{x}^T\left(u\right)duds{\int }_{t-h_M}^t{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds \\ {\left.{\int }_{t-h_M}^{t-h_m}{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds{\int }_{t-{\tau }_M}^t{\int }_{\hspace{0.33em}t-{\tau }_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds\right]}_{}^T \end{gathered}$$

The time derivatives of $V\left(x_t\right)$ in (8) along system (1) with (2) will satisfy the following conditions:

$$\begin{gathered} \dot{V}\left(x_t\right)={\dot{X}}^T\left(t\right)PX\left(t\right)+X^T\left(t\right)P\dot{X}\left(t\right)+x^T\left(t\right)Qx\left(t\right) \\ -\left(1-\dot{h}\left(t\right)\right){·x}^T\left(t-h\left(t\right)\right)Q_{4}x\left(t-h\left(t\right)\right) \\ -x^T\left(t-h_m\right)\left(Q_1-Q_3\right)x\left(t-h_m\right)-x^T\left(t-h_M\right){(Q}_2+Q_3)x\left(t-h_M\right) \\ -x^T\left(t-{\tau }_M\right)Q_{5}x\left(t-{\tau }_M\right)-x^T\left(t-\theta \right)Q_{6}x\left(t-\theta \right)+{\dot{x}}^T\left(t\right)R\dot{x}\left(t\right) \\ -[\nabla \left(t\right)+{\int }_{t-h_M}^{t-h_m}{\dot{x}}^T\left(s\right)R_5\dot{x}\left(s\right)ds+{\int }_{t-{\tau }_M}^t{\dot{x}}^T\left(s\right)R_6\dot{x}\left(s\right)ds \\ +\theta ·{\int }_{t-\theta }^t{\dot{x}}^T\left(s\right)R_7\dot{x}\left(s\right)ds], \end{gathered}$$

(9a)

where

$$Q_0=Q_1+Q_2+Q_4+Q_5+Q_6,$$

$$R_0=h_m\cdot R_1+h_M\cdot R_2+\left(h_M-h_m\right)\cdot {(R}_3+R_5)+{\tau }_M\cdot {(R}_4+R_6)+{\theta }^2\cdot R_7,$$

$$\begin{gathered} \nabla \left(t\right)={\int }_{t-h_m}^t{\dot{x}}^T\left(s\right)R_1\dot{x}\left(s\right)ds+{\int }_{t-h_M}^t{\dot{x}}^T\left(s\right)R_2\dot{x}\left(s\right)ds+{\int }_{t-h_M}^{t-h_m}{\dot{x}}^T\left(s\right)R_3\dot{x}\left(s\right)ds \\ +{\int }_{t-{\tau }_M}^t{\dot{x}}^T\left(s\right)R_4\dot{x}\left(s\right)ds. \end{gathered}$$

Define

$$\begin{gathered} Y\left(t\right)=\left[x^T\left(t\right)\right.x^T\left(t-h\left(t\right)\right)x^T\left(t-h_m\right)x^T\left(t-h_M)\right)x^T\left(t-\tau \left(t\right)\right)x^T\left(t-{\tau }_M\right) \\ {x}^T\left(t-\theta \right)\frac{1}{h_m}{\int }_{t-h_m}^t{x}^T\left(s\right)ds\frac{1}{h_M}{\int }_{t-h_M}^t{x}^T\left(s\right)ds\frac{1}{{h_M-h}_m}{\int }_{t-h_M}^{t-h_m}{x}^T\left(s\right)ds \\ \frac{1}{{\tau }_M}{\int }_{t-{\tau }_M}^t{x}^T\left(s\right)ds\frac{1}{\theta }{\int }_{t-\theta }^t{x}^T\left(s\right)ds\begin{array}{c}\begin{array}{c}\frac{2}{h_m^2}\end{array}{\int }_{t-h_m}^t{\int }_{\hspace{0.33em}t-h_m}^{\hspace{0.33em}s}{x}^T\left(u\right)duds\end{array} \\ \begin{array}{c}\frac{2}{h_M^2}\end{array}{\int }_{t-h_M}^t{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds\frac{2}{{\left(h_M-h_m\right)}^2}{\int }_{t-h_M}^{t-h_m}{\int }_{\hspace{0.33em}t-h_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds \\ \frac{2}{{\tau }_M^2}{\int }_{t-{\tau }_M}^t{\int }_{\hspace{0.33em}t-{\tau }_M}^{\hspace{0.33em}s}{x}^T\left(u\right)duds{\left.w^T\left(t\right){\dot{x}}^T\left(t\right)\right]}^T, \\ {\Gamma }_{1i}\left(t\right)=\left[{\overline{A}}_{0i}\left(t\right){\overline{A}}_{1i}\left(t\right)0_2-{\stackrel{-}{B}}_{ui}\left(t\right)K_i{0_6\theta {·\overline{A}}_{2i}\left(t\right)0_4\stackrel{-}{B}}_{wi}\left(t\right)0\right], \\ {\Gamma }_{2i}\left(t\right)=\left[{\overline{A}}_{z0i}\left(t\right){\overline{A}}_{z1i}\left(t\right)0_2-{\stackrel{-}{B}}_{zui}\left(t\right)K_i{0_6\theta {·\overline{A}}_{z2i}\left(t\right)0_4\stackrel{-}{B}}_{wi}\left(t\right)0\right], \end{gathered}$$

we have

$$X\left(t\right)={\Delta }_{1}Y\left(t\right),$$

$$\dot{X}\left(t\right)={\Delta }_{2}Y\left(t\right),$$

where the above matrices ${\Delta }_i,\hspace{0.33em}i\in \underset{\_}{2}$, have been defined in (7f). By using Lemma 1 with the matrix $\nabla \left(t\right)$ defined in (9a), we have

$$-\nabla \left(t\right)\le Y^T\left(t\right){\Omega }_{1}Y\left(t\right),$$

(9b)

where some matrices are defined as follows:

$${\Omega }_1=h_m\cdot {\Lambda }_1^T{\exists }_1{\Lambda }_1+{h_M\cdot \Lambda }_2^T{\exists }_2{\Lambda }_2+{\left(h_M-h_m\right)\cdot \Lambda }_3^T{\exists }_3{\Lambda }_3+{{\tau }_M\cdot \Lambda }_4^T{\exists }_4{\Lambda }_4+{\stackrel{-}{\Omega }}_1,$$

$${\exists }_i=N_{i1}{R}_i^{-1}{N}_{i1}^T+\frac{1}{3}{N}_{i2}{R}_i^{-1}{N}_{i2}^T+\frac{1}{5}{N}_{i3}{R}_i^{-1}{N}_{i3}^T,i\in \underset{\_}{4},$$

$${\stackrel{-}{\Omega }}_1={\sum }_{i=1}^{3}Sym\left({{\Lambda }_1^{T}N}_{1i}{\Pi }_i{\Lambda }_1+{\Lambda }_2^T{N}_{2i}{\Pi }_i{\Lambda }_2+{{\Lambda }_3^{T}N}_{3i}{\Pi }_i{\Lambda }_3{{+\Lambda }_4^{T}N}_{4i}{\Pi }_i{\Lambda }_4,\right)$$

$N_{ij}\in {\mathfrak{R}}^{4n\times n}$, $i\in \underset{\_}{4}$,$j\in \underset{\_}{3}$, can be chosen, and ${\Lambda }_i$, ${\Pi }_j$, $i\in \underset{\_}{4}$,$j\in \underset{\_}{3}$, have been defined in (7g) and (7e), respectively. From Lemmas 2–3, we have

$$-{\int }_{t-h_M}^{t-h_m}{\dot{x}}^T\left(s\right)R_5\dot{x}\left(s\right)ds+{\int }_{t-{\tau }_M}^t{\dot{x}}^T\left(s\right)R_6\dot{x}\left(s\right)ds\le Y^T\left(t\right){\Omega }_{2}Y\left(t\right),$$

(9c)

where

$${\Omega }_2{=\Lambda }_5^T\left[\begin{array}{ccc}-2R_5+S_1+S_1^T& R_5-S_1& R_5-S_1^T\\ \mathrm{*}& -R_5& S_1^T\\ \mathrm{*}& \mathrm{*}& -R_5\end{array}\right]{\Lambda }_5{+\Lambda }_6^T\left[\begin{array}{ccc}-2R_6+S_2+S_2^T& R_6-S_2& R_6-S_2^T\\ \mathrm{*}& -R_6& S_2^T\\ \mathrm{*}& \mathrm{*}& -R_6\end{array}\right]{\Lambda }_6,$$

${\Lambda }_5$ and ${\Lambda }_6$ are defined in (7f). Then from system (1) with (2) and (5), we have

$${\left(-\dot{x}\left(t\right)+{\Gamma }_{1i}\left(t\right)Y\left(t\right)\right)}^T{U}^T\left(\eta x\left(t\right)+\dot{x}\left(t\right)\right)+{\left(\eta x\left(t\right)+\dot{x}\left(t\right)\right)}^{T}U\left(-\dot{x}\left(t\right)+{\Gamma }_{1i}\left(t\right)Y\left(t\right)\right)=0.$$

(9d)

By (9a)–(9d), we can obtain the following result:

$$\begin{gathered} \dot{V}\left(x_t\right)+z^T\left(t\right)z\left(t\right)-{[z}^T\left(t\right)Qz\left(t\right)+2z^T\left(t\right)Sw\left(t\right)+w^T\left(t\right)\left[R-\gamma \cdot I\right]w\left(t\right) \\ \le Y^T\left(t\right){\Pi }_i\left(t\right)Y\left(t\right), \end{gathered}$$

(9e)

where

$$\begin{gathered} {\Pi }_i\left(t\right)={\Delta }_2^{T}P{\Delta }_1+{\Delta }_1^{T}P{\Delta }_2{+\Omega }_1+{\Omega }_2+{\Omega }_3+{\Omega }_4+Sym\left(\left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right)U{\Gamma }_{1i}\left(t\right)\right) \\ -{Sym(E}_{\mathrm{18,17}}^T{\Gamma }_{2i}\left(t\right))+{\Gamma }_{2i}^T\left(t\right){(I-Q)\Gamma }_{2i}\left(t\right), \end{gathered}$$

(9f)

$$\begin{gathered} {\Omega }_3=E_{\mathrm{18,1}}^T{Q_{0}E}_{\mathrm{18,1}}{-\left(1-h_D\right)·E}_{\mathrm{18,2}}^T{Q_{4}E}_{\mathrm{18,2}}-E_{\mathrm{18,3}}^T({Q_1-Q_3)E}_{\mathrm{18,3}}-E_{\mathrm{18,4}}^T({Q_2+Q_3)E}_{\mathrm{18,4}} \\ -E_{\mathrm{18,6}}^T{Q}_5{E}_{\mathrm{18,6}}-E_{\mathrm{18,7}}^T{Q}_6{E}_{\mathrm{18,7}}+E_{\mathrm{18,5}}^T{Z}_i{E}_{\mathrm{18,5}}-E_{\mathrm{18,17}}^T\left(R-\gamma \cdot I\right)E_{\mathrm{18,17}} \\ -\eta \cdot Sym\left(E_{\mathrm{18,1}}^T{U^{T}E}_{\mathrm{18,18}}\right)-E_{\mathrm{18,18}}^T\left[U+U^T-R_0\right]E_{\mathrm{18,18}}, \end{gathered}$$

$${\Omega }_4={\Lambda }_7^T\left[\begin{array}{ccc}-4R_7& -2R_7& 6R_7\\ \mathrm{*}& -4R_7& 6R_7\\ \mathrm{*}& \mathrm{*}& -12R_7\end{array}\right]{\Lambda }_7,$$

${\Lambda }_7$ is defined in (7f). Define the following matrices:

$${\Sigma }_i\left(t\right)=\left[\begin{array}{cc}{\stackrel{-}{\Pi }}_i& {\overline{\Gamma }}_{2i}^T\\ \mathrm{*}& -{(I-Q)}^{-1}\end{array}\right]+{\Delta }_{xzi}\left(t\right),$$

(10a)

$$\begin{gathered} {\stackrel{-}{\Pi }}_i={\Delta }_2^{T}P{\Delta }_1+{\Delta }_1^{T}P{\Delta }_2{+\Omega }_1+{\Omega }_2+{\Omega }_3+{\Omega }_4+Sym\left(\left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right)U{\stackrel{-}{\Gamma }}_{1i}\right) \\ -{Sym(E}_{\mathrm{18,17}}^T{\stackrel{-}{\Gamma }}_{2i}), \end{gathered}$$

$${\overline{\Gamma }}_{1i}=\left[A_{0i}{A}_{1i}{\begin{array}{c}\begin{array}{ccc}{0}_2& -B_{ui}{K}_i& \begin{array}{cc}{0}_6& \theta {·A}_{2i}\end{array}\end{array}\end{array}{0}_{4}B}_{wi}0\right],$$

$${\overline{\Gamma }}_{2i}=\left[A_{z0i}{A}_{z1i}{\begin{array}{c}\begin{array}{ccc}{0}_2& -B_{zui}{K}_i& \begin{array}{cc}{0}_6& \theta {·A}_{z2i}\end{array}\end{array}\end{array}{0}_{4}B}_{zwi}0\right],$$

$${\Delta }_{xzi}\left(t\right)=Sym\left(\left[\begin{array}{cc}\left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right)U{M}_{xi}& {-E}_{\mathrm{18,17}}^T{S}^T{M}_{zi}\\ 0& M_{zi}\end{array}\right]\left[\begin{array}{cc}{\Delta }_{xi}\left(t\right)& 0\\ 0& {\Delta }_{zi}\left(t\right)\end{array}\right]N_{xzi}\right),$$

$$N_{xzi}=\left[\begin{array}{c}\begin{array}{c}{N}_{x0i}{N}_{x1i}{0}_2-N_{xui}{K}_i{0}_6\end{array}\theta ·N_{x2i}{0}_4{N}_{xwi}0\\ \begin{array}{c}{N}_{z0i}{N}_{z1i}{0}_2-N_{zui}{K}_i{0}_6\end{array}\theta ·N_{z2i}{0}_4{N}_{zwi}0\end{array}\right].$$

Then the enlarged matrices can be defined as follows:

$${\Psi }_i=\left[\begin{array}{cccccc}{\stackrel{-}{\Pi }}_i& {\overline{\Gamma }}_{2i}^T& {\epsilon }_i\cdot \left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right)U{M}_{xi}& {-{\epsilon }_i\cdot E}_{\mathrm{16,15}}^T{S}^T{M}_{zi}& {\Psi }_{15i}& {\Psi }_{16i}\\ \mathrm{*}& -{\left(I-Q\right)}^{-1}& 0& {\epsilon }_i\cdot M_{yi}& 0& 0\\ \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{xi}& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{yi}\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I\end{array}\right],$$

(10b)

where

$${\Psi }_{15i}^T=\left[\begin{array}{c}{N}_{x0i}{N}_{x1i}{0}_2-N_{xui}{K}_i{0}_6\theta ·N_{x2i}{0}_4\end{array}{N}_{xwi}0\right],$$

$${\Psi }_{16i}^T=\left[\begin{array}{c}{N}_{z0i}{N}_{z1i}{0}_2-N_{zui}{K}_i{0}_6\theta ·N_{z2i}{0}_4\end{array}{N}_{zwi}0\right].$$

By using the switching rule in (5a) for condition (7a) and ${\widehat{{\rm Z}}}_i=\widehat{U}{{\rm Z}}_i{\widehat{U}}^T$, we have

$${\sum }_{i=1}^N{\alpha }_i\cdot {{\rm Z}}_i>0.$$

(10c)

When $x\left(T_k\right)\in {\overline{\Omega }}_i$ with Lemma 4 and (4), we have

$$\begin{gathered} x^T\left(T_k\right){{\rm Z}}_{i}x\left(T_k\right)=x^T\left(t-\tau \left(t\right)\right){{\rm Z}}_{i}x\left(t-\tau \left(t\right)\right)\ge 0, \\ \sigma \left(x\left(t\right)\right)=i,\mathrm{for}\mathrm{all}t\in \left.{[T}_k,T_{k+1}.\right) \end{gathered}$$

(10d)

By using conditions defined in (7c), (10d), and the following definitions

$$\widehat{\widehat{U}}=diag\left[\begin{array}{ccccccccccccc}U& U& U& U& U& U& UUUUUUUUU& UI& U& I& I& I& I\end{array}\right],$$

$$\widehat{U}=U^{-1},\widehat{\widehat{\widehat{U}}}=diag\left[UUUUUUUUUU\right],\widehat{P}={\widehat{\widehat{\widehat{U}}}}^{-1}P{\widehat{\widehat{\widehat{U}}}}^{-T}>0,$$

$${\widehat{S}}_i=U^{-1}{S}_i{U}^{-T},{\widehat{Q}}_i=U^{-1}{Q}_i{U}^{-T}>0,{\widehat{R}}_i=U^{-1}{R}_i{U}^{-T}>0,{\widehat{{\rm Z}}}_i=U^{-1}{{\rm Z}}_i{U}^{-T},$$

we can obtain

$${\widehat{\widehat{U}}}^{-1}{\Psi }_i{\widehat{\widehat{U}}}^{-T}<0.$$

This condition ${\Psi }_i<0$ in (10b) is satisfied. By using Lemma 5 and the Schur complement of [26] with the conditions (10b), we can obtain

$${\Sigma }_i\left(t\right)<0,{\Pi }_i\left(t\right)<0.$$

With $w\left(t\right)=0,$ and from (9e) with ${\Pi }_i\left(t\right)<0$, we have

$$\dot{V}\left(x_t\right)<0,\forall \hspace{0.33em}x\left(t\right)\ne 0,\mathrm{for}\mathrm{all}t\in \left.{[T}_k,T_{k+1}\right).$$

The system (1) with (2)–(3) is asymptotically stable by switching rule (5a) with ${{\rm Z}}_i={\widehat{U}}^{-1}{\widehat{{\rm Z}}}_i{\widehat{U}}^{-T}$ and samping input (5b) with $K_i={\widehat{K}}_i{\widehat{U}}^{-T}$. From the condition (9e) with ${\Pi }_i\left(t\right)<0$, the integrate of the equation in (9e) from 0 to $\mathcal{l}>0$ will yield

$$\begin{gathered} V\left(x_{\mathcal{l}}\right)-V\left(\phi \right)+{\int }_0^{\mathcal{l}}{z}^T\left(t\right)z\left(t\right)dt \\ -{\int }_0^{\mathcal{l}}{[z}^T\left(t\right)Qz\left(t\right)+2z^T\left(t\right)Sw\left(t\right)+w^T\left(t\right)(R-\gamma \cdot I)w\left(t\right)]dt\le 0. \end{gathered}$$

(11)

With $\phi \left(t\right)=0,\hspace{0.33em}-H\le t\le 0$, we can obtain

$$V\left(\phi \right)=0,V\left(x_{{\mathcal{l}}_1}\right)\ge 0.$$

By using the derivations above, we can obtain the following condition from (11) with ${\mathcal{l}=\mathcal{l}}_1$

$$-{\int }_0^{{\mathcal{l}}_1}{[z}^T\left(t\right)Qz\left(t\right)+2z^T\left(t\right)Sw\left(t\right)+w^T\left(t\right)[R-\gamma \cdot I]w\left(t\right)]dt\le 0,\forall \hspace{0.33em}w\ne 0,$$

for all constants ${\mathcal{l}}_1>0$. The condition (ii) of Definition 1 has been guaranteed. From (11), with ${\mathcal{l}=\mathcal{l}}_2$, $w\left(t\right)=0$, and $Q=Q^T\le 0$ in Definition 1, we have

$$V\left(x_{{\mathcal{l}}_2}\right)-V\left(\phi \right)+{\int }_0^{{\mathcal{l}}_2}{z}^T\left(t\right)z\left(t\right)dt\le 0.$$

From (7h) with $V\left(x_{{\mathcal{l}}_2}\right)\ge 0$, we have

$${\int }_0^{{\mathcal{l}}_2}{z}^T\left(t\right)z\left(t\right)dt\le V\left(\phi \right)\le \alpha .$$

The condition (iii) of Definition 1 has been guaranteed. This completes the proof. □

If the switching signal is captured at each sampling instant, the sampling input is selected by

$$\mathrm{when}\mathsf{\sigma }\left(T_k\right)=i,u\left(t\right)=-K_{i}x\left(T_k\right),\mathrm{for}\mathrm{all}t\in [\left.T_k,T_{k+1}\right).$$

(12)

If we consider (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances by the proposed sampling input in (12), the delay-dependent conditions are provided as follows.

Corollary 1.

With a constant $\eta$, the following linear matrix inequality conditions apply:

$$\left[\begin{array}{cc}{\widehat{R}}_5& {\widehat{S}}_1\\ \mathrm{*}& {\widehat{R}}_5\end{array}\right]>0,$$

(13a)

$$\left[\begin{array}{cc}{\widehat{R}}_6& {\widehat{S}}_2\\ \mathrm{*}& {\widehat{R}}_6\end{array}\right]>0,$$

(13b)

$${\widehat{\Psi }}_{ij}=\left[\begin{array}{ccccccc}{\widehat{\Psi }}_{11ij}& {\widehat{\Psi }}_{12ij}& {\widehat{\Psi }}_{13i}& {\widehat{\Psi }}_{14i}& {\widehat{\Psi }}_{15ij}& {\widehat{\Psi }}_{16ij}& {\widehat{\Psi }}_{17}\\ \mathrm{*}& -{(I-Q)}^{-1}& 0& {\epsilon }_i\cdot M_{zi}& 0& 0& 0\\ \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{xi}& 0& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& {\epsilon }_i\cdot {\Xi }_{zi}& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& -{\epsilon }_i\cdot I& 0\\ \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& \mathrm{*}& {\widehat{\Psi }}_{77}\end{array}\right]<0,i\in \underset{\_}{N},j\in \underset{\_}{N}$$

(13c)

where

$${\widehat{\Psi }}_{11ij}=Sym\left({\Delta }_2^T\widehat{P}{\Delta }_1+\left(\eta \cdot E_{\mathrm{18,1}}^T+E_{\mathrm{18,18}}^T\right){\widehat{\Gamma }}_{1ij}\right)-{Sym(E}_{\mathrm{18,17}}^T{\widehat{\Gamma }}_{2ij})+{\widehat{\Omega }}_1+{\widehat{\Omega }}_2+{\tilde{\Omega }}_3{+\widehat{\Omega }}_4,$$

$${\widehat{\Psi }}_{12ij}={\widehat{\Gamma }}_{2ij}^T,$$

$${\widehat{\Psi }}_{15ij}^T=\left[\begin{array}{c}{N}_{x0i}{\widehat{U}}^T{N}_{x1i}{\widehat{U}}^T{0_2-N_{xui}{\widehat{K}}_j{0}_6\theta ·N_{x2i}{\widehat{U}}^T{0}_{4}N}_{xwi}0\end{array}\right],$$

$${\widehat{\Psi }}_{16ij}^T=\left[N_{z0i}{\widehat{U}}^T{N}_{z1i}{\widehat{U}}^T{0_2-N_{zui}{\widehat{K}}_j{0}_6\theta ·N_{z2i}{\widehat{U}}^T{0}_{4}N}_{zwi}0\right],$$

$$\begin{gathered} {\tilde{\Omega }}_3=E_{\mathrm{18,1}}^T{{\widehat{Q}}_{0}E}_{\mathrm{18,1}}{-\left(1-h_D\right)·E}_{\mathrm{18,2}}^T{{\widehat{Q}}_{4}E}_{\mathrm{18,2}}-E_{\mathrm{18,3}}^T({{\widehat{Q}}_1-{\widehat{Q}}_3)E}_{\mathrm{18,3}}-E_{\mathrm{18,4}}^T({{\widehat{Q}}_2+{\widehat{Q}}_3)E}_{\mathrm{18,4}} \\ -E_{\mathrm{18,6}}^T{\widehat{Q}}_5{E}_{\mathrm{18,6}}-E_{\mathrm{18,7}}^T{\widehat{Q}}_6{E}_{\mathrm{18,7}}-E_{\mathrm{18,17}}^T(R-\gamma \cdot I)E_{\mathrm{18,17}} \\ -\eta \cdot Sym\left(E_{\mathrm{18,1}}^T{{\widehat{U}}^{T}E}_{\mathrm{18,18}}\right)-E_{\mathrm{18,18}}^T\left[\widehat{U}+{\widehat{U}}^T-{\widehat{R}}_0\right]E_{\mathrm{18,18}}, \end{gathered}$$

$${\widehat{\Gamma }}_{1ij}=\left[A_{0i}{\widehat{U}}^T{A}_{1i}{\widehat{U}}^T{0}_2-B_{ui}{\widehat{K}}_j{0}_6\theta {·A}_{2i}{\widehat{U}}^T{0}_4{B}_{wi}0\right],$$

$${\widehat{\Gamma }}_{2ij}=\left[A_{z0i}{\widehat{U}}^T{A}_{z1i}{\widehat{U}}^T{0}_2-B_{zui}{\widehat{K}}_j{0}_6\theta {·A}_{z2i}{\widehat{U}}^T{0}_4{B}_{zwi}0\right],$$

${\widehat{\Psi }}_{13i}$, ${\widehat{\Psi }}_{14i}$, ${\widehat{\Psi }}_{17}$, ${\widehat{\Psi }}_{77}$, ${\widehat{\Omega }}_1$, ${\widehat{\Omega }}_2$, and ${\widehat{\Omega }}_4$ defined in Theorem 1, are feasible with some constants ${\epsilon }_i>0$, $i\in \underset{\_}{N}$, a nonsingular matrix $\widehat{U}\in {\mathfrak{R}}^{n\times n}$, some positive definite symmetric matrices $\widehat{P}\in {\mathfrak{R}}^{10n\times 10n}$, ${\widehat{Q}}_i$, ${\widehat{R}}_j\in {\mathfrak{R}}^{n\times n}$, $i\in \underset{\_}{6}$, $j\in \underset{\_}{7}$, and some matrices ${\widehat{S}}_1,{\widehat{S}}_2\in {\mathfrak{R}}^{n\times n}$, ${\widehat{N}}_{ij}\in {\mathfrak{R}}^{4n\times n}$, $i\in \underset{\_}{4}$,$j\in \underset{\_}{3}$, ${\widehat{K}}_i\in {\mathfrak{R}}^{p\times n}$, $i\in \underset{\_}{N}$. Then. system (1) with (2) is asymptotically stabilizable by the sampling input $K_i={\widehat{K}}_i{\widehat{U}}^{-T}$in (12), with strictly (Q,S,R)–$\mathsf{\gamma }$ dissipative performance and $H_2$ measure $\alpha$ in (7h).

Remark 1.

(a) With $Q=0$, $S=I$, and $R=2\gamma ·I$, the inequality (6) can be rewritten as

$$-\gamma \cdot {\int }_0^{{\mathcal{l}}_1}{w}^T\left(t\right)w\left(t\right)dt\le 2{\int }_0^{{\mathcal{l}}_1}{z}^T\left(t\right)w\left(t\right)dt,\forall \hspace{0.33em}w\ne 0,$$

System (1) with (2) is asymptotically stabilizable by the switching signal in (5a) and sampling input in (5b) with passivity performance $\gamma$ and$H_2$ measure$\alpha$. (b) For more efficient results in $H_{\infty }$ performance, we can use inequality in (11) with $Q=0$,$S=0$, and $R=2\gamma ·I=2{\stackrel{-}{\gamma }}^2·I$, and the following inequality can be given by 

$${\int }_0^{{\mathcal{l}}_1}{z}^T\left(t\right)z\left(t\right)dt\le {\stackrel{-}{\gamma }}^2\cdot {\int }_0^{{\mathcal{l}}_1}{w}^T\left(t\right)w\left(t\right)dt,\forall \hspace{0.33em}w\ne 0,$$

System (1) with (2) is asymptotically stabilizable by the switching signal in (5a) and sampling input in (5b) with $H_{\infty }$performance $\stackrel{-}{\gamma }$and $H_2$measure $\alpha$[7].

Remark 2.

If $h_D\ge 1$or is unknown, our main results in Theorem 1 and Corollary 1 are still valid by choosing ${\widehat{Q}}_4=0$. With ${\widehat{Q}}_4=0$, the proposed results will be independent, with respect to the parameter $h_D$.

Remark 3.

The rule of switching and sampling input in (5) only happens synchronously at instant $T_k$ of sampling [7]. Hence our proposed approach is easier to implement than the real-time sampling in [12].

Remark 4.

In our past results in [13,14], pointwise sampling was developed to relax the constraint of fixed-time sampling. Interval sampling $\left[0,{\tau }_M\right]$ is proposed to allow sampling period limitation. This will be more useful than some published materials in [13,14].

Remark 5.

In this paper, some free weighting matrices  $\widehat{P}\in {\mathfrak{R}}^{10n\times 10n}$ ,${\widehat{Q}}_i$,${\widehat{R}}_j\in {\mathfrak{R}}^{n\times n}$,$i\in \underset{\_}{6}$,$j\in \underset{\_}{7}$,${\widehat{Z}}_i\in {\mathfrak{R}}^{n\times n}$,$i\in \underset{\_}{N}$,${\widehat{S}}_1,{\widehat{S}}_2\in {\mathfrak{R}}^{n\times n}$,${\widehat{N}}_{ij}\in {\mathfrak{R}}^{4n\times n}$,$i\in \underset{\_}{4}$,$j\in \underset{\_}{3}$, have been provided to improve the proposed results. The proposed results may be conservative for fewer variables in our provided linear matrix inequalities.

4. Some Numerical Examples

Example 1.

The uncertain switched system (1) with (2) and the following parameters are considered:

$$\begin{gathered} N=2,{\tau }_M=0.3,\phi \left(t\right)={\left[-0.10.5\right]}^T,-h_M=-1\le t\le 0, \\ {A}_{01}=\left[\begin{array}{cc}-1.1& 0.1\\ 0.2& -1\end{array}\right],A_{02}=\left[\begin{array}{cc}-1.2& 0\\ -0.1& -1.1\end{array}\right],A_{11}=\left[\begin{array}{cc}-0.9& 0.05\\ -0.2& -1\end{array}\right], \\ {A}_{12}=\left[\begin{array}{cc}-1& 0.1\\ -0.25& -1\end{array}\right],A_{21}=A_{22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],B_{w1}=\left[\begin{array}{c}0.2\\ 0.01\end{array}\right],B_{w2}=\left[\begin{array}{c}0.2\\ 0.02\end{array}\right],B_{u1}=\left[\begin{array}{c}1\\ -0.1\end{array}\right],B_{u2}=\left[\begin{array}{c}1\\ 0.1\end{array}\right],A_{z01}=\left[\begin{array}{cc}1& 0\end{array}\right],A_{z02}=\left[\begin{array}{cc}0.8& -0.1\end{array}\right], \\ {A}_{z11}=\left[\begin{array}{cc}-0.8& 0.6\end{array}\right],A_{z12}=\left[\begin{array}{cc}-0.2& 1\end{array}\right],A_{z21}=A_{z22}=\left[\begin{array}{cc}0& 0\end{array}\right], \\ {B}_{zw1}=B_{zw2}=-0.5,B_{zu1}=B_{zu2}=0.3, \\ {M}_{xi}=\left[\begin{array}{cc}0.01& 0\\ 0& 0.02\end{array}\right],M_{zi}=0.01,N_{x0i}=\left[\begin{array}{cc}0.1& 0\\ 0& 0\end{array}\right],N_{x1i}=\left[\begin{array}{cc}0& 0\\ 0& 0.1\end{array}\right], \\ {N}_{x2i}=N_{xwi}=N_{xui}=0,{N_{z2i}=N}_{zwi}=N_{zui}=0,{\Xi }_{xi}=0.1\cdot I,{\Xi }_{zi}=0.1, \\ {N}_{z0i}=N_{z1i}=\left[\begin{array}{cc}0.1& 0\end{array}\right],i=1,\hspace{0.33em}2. \end{gathered}$$

(14)

(1)

$H_2$ measure and passive switching control. From Theorem 1 and Remarks 1–2 with $T_{k+1}-T_k={\tau }_k\le 0.3={\tau }_M$, $\gamma =1$, $\eta =1$, and ${\alpha }_1={\alpha }_2=0.5$, the proposed linear matrix inequality conditions in (7a)–(7d) with (14) are feasible $h_M=1.4889$. The feasible solution is provided as follows:

$${\widehat{K}}_1=\left[0.0930.0305\right],{\widehat{K}}_2=\left[0.01610.0055\right],\widehat{U}=\left[\begin{array}{cc}0.1345& -0.0275\\ -0.0033& 0.1152\end{array}\right],$$

$${\widehat{{\rm Z}}}_1=\left[\begin{array}{cc}-0.00855& -0.00216\\ -0.00216& -0.001496\end{array}\right],{\widehat{{\rm Z}}}_2=\left[\begin{array}{cc}0.00857& 0.00215\\ 0.00215& 0.0015\end{array}\right].$$

System (1) with (2) and (14) can be stabilizable by the switching rule

$$\sigma \left(x\left(t\right)\right)=\left\{\begin{array}{c}1,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ 2,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left.{[T}_k,T_{k+1}\right),$$

(15a)

where

$${\overline{\Omega }}_1={\Omega }_1,{\overline{\Omega }}_2={\Omega }_2\backslash {\overline{\Omega }}_1={\mathfrak{R}}^2\backslash {\overline{\Omega }}_1,$$

(15b)

${{\rm Z}}_i={\widehat{U}}^{-1}{\widehat{{\rm Z}}}_i{\widehat{U}}^{-T}$, $i=1,\hspace{0.33em}2$, with

$${\Omega }_1=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{1}x=-0.5408x_1^2-0.3582x_1{x}_2-0.1226x_2^2\ge 0\right\},$$

$${\Omega }_2=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{2}x=0.5416\hspace{0.33em}{x}_1^2+0.3572x_1{x}_2+0.1231x_2^2\ge 0\right\}.$$

The switching sampling input is given by

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left.{[T}_k,T_{k+1}\right),$$

(16a)

with

$$K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[0.74970.2861\right],K_2={\widehat{K}}_2{\widehat{U}}^{-T}=\left[0.13050.0514\right].$$

(16b)

Then, system (1) with (2) and (14)–(16) is passive with $\gamma =1$ and $H_2$ measure $\alpha =10.3784$.

(2)

$H_2$ measure and passive control with arbitrary switching. By the same formulation, Corollary 1 and Remarks 1-2 with ${\tau }_k=T_{k+1}-T_k\le {\tau }_M=0.3$, $\gamma =1$, and $\eta =1$ are feasible with $h_M=1.3541$. The switching sampling input is given by

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}\mathsf{\sigma }\left(T_k\right)=1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}\mathsf{\sigma }\left(T_k\right)=2,\end{array}\right.\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}t\in \left.{[T}_k,T_{k+1}\right),$$

(17a)

with

$$\begin{gathered} K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[0.5919-0.2428\right]\times {10}^{-4}, \\ {K}_2={\widehat{K}}_2{\widehat{U}}^{-T}=\left[0.5929-0.2434\right]\times {10}^{-4}. \end{gathered}$$

(17b)

System (1) with (2) and (14) is passive, with $\gamma =1$ and $H_2$ measure $\alpha =10.5365$. Some comparisons are made in

Table 1. Comparisons of switched time-delay system (1) with (2) and (14).

Comparisons about the Sampled-Data Switched Time-Delay System (1) with (2) and (14).
Results	Sampling Interval	$\mathbf{Time}\mathbf{Delay}\mathit{h}\left(\mathbf{t}\right)$	Approaches
Results of [14]	${\tau }_i=0.2$ ${\tau }_j=0.3$ for some i and j pointwise sampling	$h=1$ constant delay	Synchronous switching for: 1. Switching signal 2. Sampled-data input
Results of Theorem 1 in this paper	${0<\tau }_i\le 0.3$	$1\le h\left(\mathrm{t}\right)\le$1.4889 unknown time-varying delay	Synchronous switching for: 1. Switching signal (15) 2. Sampled-data input (16)
Results of Corollary 1 in this paper		$1\le h\left(\mathrm{t}\right)\le$1.3541 unknown time-varying delay	1. Arbitrary switching can be captured at each sampling instant 2. Sampled-data input (17)

to show the improvement of the proposed results. These are less conservative for the conditions in this paper. The proposed switching rule and sampling input can be implemented synchronously.

Example 2.

The uncertain switched time-delay system (1) with (2) and the following parameters can be considered:

$$\begin{gathered} N=2,{\tau }_M=0.5,h_m=4,\phi \left(t\right)={\left[1-1\right]}^T,\hspace{0.33em}-h_M\le t\le 0, \\ {A}_{01}=\left[\begin{array}{cc}-1.1& 0\\ -0.2& 0.25\end{array}\right],A_{02}=\left[\begin{array}{cc}0.35& -0.1\\ 0& -1.1\end{array}\right],A_{11}=\left[\begin{array}{cc}-0.6& -0.1\\ 0.2& -0.2\end{array}\right], \\ {A}_{12}=\left[\begin{array}{cc}-0.3& 0.1\\ 0.1& -0.6\end{array}\right],A_{21}=A_{22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],A_{z01}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.1& 0.1\end{array}\right], \\ {A}_{z02}=\left[\begin{array}{cc}0.1& -0.1\\ -0.1& -\mathrm{0,1}\end{array}\right],A_{z11}=\left[\begin{array}{cc}-0.1& -0.1\\ 0.1& -0.1\end{array}\right], \\ {A}_{z12}=\left[\begin{array}{cc}0.1& 0.1\\ -0.1& -0.1\end{array}\right],A_{z21}=A_{z22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],B_{w1}=B_{w2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right], \\ {B}_{u1}=B_{u2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right],B_{zw1}=B_{zw2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right],M_{xi}=N_{x0i}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right], \\ {N}_{x1i}=\left[\begin{array}{cc}0& 0.1\\ 0.1& 0\end{array}\right],N_{x2i}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],N_{xwi}=\left[\begin{array}{cc}0.1& 0.1\\ 0& 0\end{array}\right],N_{xu1}=N_{xu2}=\left[\begin{array}{cc}0& 0\\ 0.1& 0.1\end{array}\right], \\ {B}_{zui}=M_{zi}=N_{z0i}=N_{z1i}={\Xi }_{zi}=0,{\Xi }_{xi}=0.01,i=1,\hspace{0.33em}2. \end{gathered}$$

(18)

Now, we would like to show the $H_2$measure and passivity of switched system. From Theorem 1 and Remarks 1–2 with $T_{k+1}-T_k={\tau }_k\le {\tau }_M=0.5$, $\eta =1$, $\gamma =5$, and ${\alpha }_1={\alpha }_2=0.5$, the proposed linear matrix inequality conditions in (7a)–(7d) with (18) are feasible, with $h_M=5.4776$. The feasible solution is provided as follows:

$${\widehat{K}}_1=\left[\begin{array}{cc}-0.0521& -0.0106\\ -0.2387& 0.6051\end{array}\right],{\widehat{K}}_2=\left[\begin{array}{cc}0.9112& -0.3739\\ -0.0647& -0.0455\end{array}\right],\widehat{U}=\left[\begin{array}{cc}0.166& -0.0341\\ -0.042& 0.1823\end{array}\right],$$

$${\widehat{{\rm Z}}}_1=\left[\begin{array}{cc}0.019703& -0.002\\ -0.002& -0.012026\end{array}\right],{\widehat{{\rm Z}}}_2=\left[\begin{array}{cc}-0.019702& -0.002\\ -0.002& 0.012032\end{array}\right].$$

System (1) with (2) and (18) can be stabilizable by the switching rule

$$\sigma \left(x\left(t\right)\right)=\left\{\begin{array}{c}1,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ 2,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left.{[T}_k,T_{k+1}\right)$$

(19a)

where

$${\overline{\Omega }}_1={\Omega }_1,{\overline{\Omega }}_2={\Omega }_2\backslash {\overline{\Omega }}_1={\mathfrak{R}}^2\backslash {\overline{\Omega }}_1,$$

(19b)

${{\rm Z}}_i={\widehat{U}}^{-1}{\widehat{{\rm Z}}}_i{\widehat{U}}^{-T}$, $i=1,\hspace{0.33em}2$, with

$${\Omega }_1=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{1}x=0.7414x_1^2+0.0476x_1{x}_2-0.3903x_2^2\ge 0\right\},$$

$${\Omega }_2=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{2}x=-0.7413x_1^2-0.0474x_1{x}_2+0.3905x_2^2\ge 0\right\}.$$

The switching sampling input is given by

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in \left.{[T}_k,T_{k+1}\right),$$

(20a)

with

$$K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[\begin{array}{cc}-0.3418& -0.1368\\ -0.7936& 3.1369\end{array}\right],K_2={\widehat{K}}_2{\widehat{U}}^{-T}=\left[\begin{array}{cc}5.32& -0.8257\\ -0.4628& -0.3561\end{array}\right].$$

(20b)

Then, system (1) with (2) and (18)–(20) is passive, with $\gamma =1$and $H_2$measure $\alpha =2734.8$. Some comparisons are made in

Table 2. Comparisons of the switched time-delay system (1) with (2) and (18).

Results	Sampling Interval	Time Delay	Approach and Delay
Results of [14]	${\tau }_i=0.01$ and ${\tau }_j=0.012$ for some i and j pointwise sampling	$h=4.2136$ Constant delay	Synchronous switching: 1. Switching signal 2. Sampled-data input
The proposed results in Theorem 1	${0<\tau }_i\le 0.5$	$4\le h\left(t\right)\le 5.4776$ unknown time-varying delay	Synchronous switching: 1. Switching signal in (19) 2. Sampled-data input in (20)
The proposed results in Corollary 1	${0<\tau }_i\le 0.2197$	$4\le h\left(t\right)\le 4.8371$ unknown time-varying delay	1. Arbitrary switching can be captured at each sampling instant 2. Sampled-data input

to show the improvement of the proposed results.

With the unperturbed condition and disturbance $w\left(t\right)=\left[\begin{array}{c}2e^{-0.2t}\sin (10t)\\ -2e^{-0.2t}\cos (10t)\end{array}\right]$ illustrated in Figure 1, the output $z\left(t\right)\in {\mathfrak{R}}^2$ for arbitrary switching signal and no sampling input has been illustrated in Figure 2. With our proposed synchronous switching rule in (19), sampling input in (20), and the above disturbance, the output $z\left(t\right)\in {\mathfrak{R}}^2$ with $\phi \left(t\right)=0$ is illustrated in Figure 3. With these results and $\phi \left(t\right)={\left[\begin{array}{cc}1& -1\end{array}\right]}^T$, $-5.4776\le t\le \hspace{0.33em}0$, the output $z\left(t\right)\in {\mathfrak{R}}^2$ and state $x\left(t\right)\in {\mathfrak{R}}^2$ with $\phi \left(t\right)=0$ are illustrated in Figure 4 and Figure 5, respectively. The sampling state $x\left(t\right)=x\left(T_k\right)\in {\mathfrak{R}}^2,\hspace{0.33em}\forall t\in [\left.T_k,T_{k+1}\right)$, $T_{k+1}-T_k\le 0.5,$ and switching rule $\sigma \left(t\right)=\sigma \left(T_k\right)\in \left\{1,\hspace{0.33em}2\right\},\hspace{0.33em}\forall t\in [\left.T_k,T_{k+1}\right)$, with $w\left(t\right)=0$ and $\phi \left(t\right)={\left[\begin{array}{cc}1& -1\end{array}\right]}^T$, $-5.4776\le t\le \hspace{0.33em}0$, are illustrated in Figure 6 and Figure 7, respectively. With the disturbance in Figure 1, the regulated output will be unbounded for an arbitrary switching signal and no control input in Figure 2. The performances of passivity and $H_2$ measure for our proposed synchronous signal in (19) and input in (20) have been shown in Figure 3 and Figure 4, respectively. Some simulation diagrams for state, sampled-data state and switching signal have been provided in Figure 5, Figure 6 and Figure 7. From the simulation, the switching rule in (19) and sampling input in (20) will be effective in achieving the performance of the switched time-delay system.

Example 3.

The unperturbed switched time-delay system (1) with the following parameters is considered:

$N=2$, $h_m=2$, ${\tau }_M=0.3$, $\phi \left(t\right)={\left[0.50.1\right]}^T,\hspace{0.33em}-h_M\le t\le 0$, $h_M$will be evaluated,

$$\begin{gathered} A_{01}=\left[\begin{array}{cc}0.1& -0.001\\ -0.001& 0.15\end{array}\right],\hspace{0.33em}{A}_{02}=\left[\begin{array}{cc}0.15& -0.001\\ -0.001& 0.05\end{array}\right],A_{11}=\left[\begin{array}{cc}-0.7& 0.001\\ 0.001& -0.7\end{array}\right], \\ {A}_{12}=\left[\begin{array}{cc}-0.6& 0.001\\ 0.001& -0.6\end{array}\right],A_{21}=A_{22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],A_{z01}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.1& 0.1\end{array}\right], \\ {A}_{z02}=\left[\begin{array}{cc}0.1& -0.1\\ -0.1& -\mathrm{0,1}\end{array}\right],A_{z11}=A_{z12}=A_{z21}=A_{z22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right], \\ {B}_{w1}=B_{w2}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],B_{u1}=B_{u2}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.2\end{array}\right], \\ {B}_{zw1}=B_{zw2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right],A_{z1i}=B_{zui}=0,i=1,\hspace{0.33em}2. \end{gathered}$$

(21)

We provide some comparisons in

Table 3. Comparing previous results with this paper.

Comparisons of the Switched Time-Delay System (1) with (2) and (22)–(24)
Results	Sampling Interval	Time Delay	Upper Bounds of Time Delay and Performance
[15]	No sampling	$h=1.37$	Fail
[12]	No sampling	$h=1.37$	Real-time state dependent switching signal (No input)
[14]	${\tau }_i=0.025$ and ${\tau }_j=0.03$ for some i and j pointwise sampling	$h=2.8233$	Sampled-data switching on switching signal with passive performance $\gamma =2$ (No input)
		$h=3.4432$	Synchronous switching on switching signal and sampled-data input with passive performance $\gamma =2$
		$h=3.4277$	Switching is arbitrary can be captured anytime for sampled-data input
The proposed results in this paper	${0<\tau }_i\le 0.2$	$2\le h\left(t\right)\le 3.5998$ unknown time-varying delay	Synchronous switching on switching signal and sampled-data input Passive performance $\gamma =2$ $H_2$ measure $\alpha =70.9514$
		$2\le h\left(t\right)\le 3.5783$ unknown time-varying delay	Switching is arbitrary and can be captured only at each sampling instant for sampled-data input Passive performance $\gamma =2$ $H_2$ measure $\alpha =69.5848$

to show the less conservative nature of the results in this paper.

Some observations can be provided in the following from Table 1, Table 2 and Table 3.

With the following matrices $A_{01}$, $A_{02}$, $A_{01}+A_{11}$, and $A_{02}+A_{12}$ not being Hurwitz, the developed approach for arbitrary switching in Corollary 1 has no feasible solution under any sampling input in Example 2. The design of the switching rule is very important in this example.

The sampling input is more efficient than the design of the switching rule in Example 3.

In this paper, the proposed synchronous switching on rule and sampling input is used to achieve the stabilization and performance of an uncertain switched system with mixed time delay. The developed scheme is implemented more easily than a real-time scheme.

A sampling input can be used to achieve the performance of a switched system in arbitrary switching.

Example 4.

The switched time-delay system (1) with (2) is considered as follows:

$$\begin{gathered} N=2,h_m=1,h_M=1.2,{\tau }_M=0.3,\phi \left(t\right)={\left[-0.10.5\right]}^T,-h_M\le t\le 0, \\ {A}_{01}=\left[\begin{array}{cc}-1.1& 0.1\\ 0.2& -1\end{array}\right],A_{02}=\left[\begin{array}{cc}-1.2& 0\\ -0.1& -1.1\end{array}\right],A_{11}=\left[\begin{array}{cc}-0.9& 0.05\\ -0.2& -1\end{array}\right], \\ {A}_{12}=\left[\begin{array}{cc}-1& 0.1\\ -0.25& -1\end{array}\right],A_{21}=A_{22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],B_{w1}=\left[\begin{array}{c}0.2\\ 0.01\end{array}\right], \\ {B}_{w2}=\left[\begin{array}{c}0.2\\ 0.02\end{array}\right],B_{u1}=\left[\begin{array}{c}1\\ -0.1\end{array}\right],B_{u2}=\left[\begin{array}{c}1\\ 0.1\end{array}\right], \\ {A}_{z01}=\left[\begin{array}{cc}1& 0\end{array}\right],A_{z02}=\left[\begin{array}{cc}0.8& -0.1\end{array}\right],A_{z11}=\left[\begin{array}{cc}-0.8& 0.6\end{array}\right],A_{z12}=\left[\begin{array}{cc}-0.2& 1\end{array}\right], \\ {A}_{z21}=A_{z22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],B_{zw1}=B_{zw2}=-0.5,B_{zu1}=B_{zu2}=0.3, \\ {M}_{xi}=\left[\begin{array}{cc}0.01& 0\\ 0& 0.02\end{array}\right],M_{zi}=0.01,N_{x0i}=\left[\begin{array}{cc}0.1& 0\\ 0& 0\end{array}\right],N_{x1i}=\left[\begin{array}{cc}0& 0\\ 0& 0.1\end{array}\right], \\ {N}_{x2i}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]N_{xwi}=N_{xui}=0,N_{zwi}=N_{zui}=0,{\Xi }_{xi}=0.1\cdot I,{\Xi }_{zi}=0.1, \\ {N}_{z0i}=N_{z1i}=\left[\begin{array}{cc}0.1& 0\end{array}\right],N_{z2i}=\left[\begin{array}{cc}0& 0\end{array}\right],i=1,\hspace{0.33em}2. \end{gathered}$$

(22)

In this example, we would like to achieve $H_2$ measure and $H_{\mathrm{\infty }}$ switching control. With synchronous switching (5) for rule and input with conditions $1\le h\left(t\right)\le 1.2$ and ${\tau }_k\le 0.3$, $\forall$k, the linear matrix inequality conditions in (7) of Theorem 1 and Remark 1 with $\eta =1$ and ${\alpha }_1={\alpha }_2=0.5$ have a feasible solution with

$$\gamma =0.268,{\widehat{K}}_1=\left[11.92394.6429\right],{\widehat{K}}_2=\left[16.91416.7406\right],$$

$$\widehat{U}=\left[\begin{array}{cc}5.3986& 1.9342\\ 0.6831& 2.3326\end{array}\right],{\widehat{{\rm Z}}}_1=\left[\begin{array}{cc}-8.1649& -3.4668\\ -3.4668& -0.6176\end{array}\right],{\widehat{{\rm Z}}}_2=\left[\begin{array}{cc}12.3651& 5.0374\\ 5.0374& 2.0327\end{array}\right]$$

.

The switched delay system in (1) with (2) and (22) is asymptotically stabilizable by the switching rule given by

$$\sigma \left(x\left(t\right)\right)=\left\{\begin{array}{c}1,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ 2,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\forall t\in \left.{[T}_k,T_{k+1}\right),$$

(23a)

where

$${\overline{\Omega }}_1={\Omega }_1,{\overline{\Omega }}_2={\Omega }_2\backslash {\overline{\Omega }}_1={\mathfrak{R}}^2\backslash {\overline{\Omega }}_1$$

(23b)

$${{\rm Z}}_i={\widehat{U}}^{-1}{\widehat{{\rm Z}}}_i{\widehat{U}}^{-T},i=1,\hspace{0.33em}2,$$

with

$${\Omega }_1=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{1}x=-0.9072x_1^2-0.7704x_1{x}_2-0.0686x_2^2\ge 0\right\},$$

$${\Omega }_2=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{2}x=1.3739x_1^2+1.1194x_2+0.2259x_2^2\ge 0\right\},$$

and sampling input is given by:

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\forall t\in \left.{[T}_k,T_{k+1}\right),$$

(24a)

with

$$K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[3.97461.5476\right],K_2={\widehat{K}}_2{\widehat{U}}^{-T}=\left[5.6382.2469\right].$$

(24b)

In the above analysis, system (1) with (2) and (22) is stabilized, with $H_{\mathrm{\infty }}$ performance $\overline{\gamma }=\sqrt{\gamma }=0.5177$ and $H_2$ measure $\alpha =2.9174$ by the switching rule in (23) and input in (24a), with $K_i={\widehat{K}}_i{\widehat{U}}^{-T}$ in (24b). We make some comparisons in

Table 4. Some comparisons of the sampled-data switched time-delay system.

Comparisons about the Switched Time-Delay System (1) with (2) and (22)–(24).
Results	Time Delay $\mathit{h}\left(\mathbf{t}\right)$	Sampling Period	Disturbance Attenuation
Results of [13] (Synchronous switching for signal and sampled-data input)	$h\left(t\right)=1$ Constant delay	${\tau }_i=0.2$ ${\tau }_j=0.3$ for some i and j pointwise sampling	$\overline{\gamma }=0.632$
Results of Theorem 1 in this paper	$1\le h\left(\mathrm{t}\right)\le 1.2$ $h\left(\mathrm{t}\right)$ unknown	${0<\tau }_i\le 0.3$	$\overline{\gamma }=0.5177$

to show the contribution of the proposed results. The considered switched system with interval time-varying delay $1\le h\left(\mathrm{t}\right)\le 1.2$ can be studied instead of constant delay $h\left(t\right)=h=1$ in [13]. Interval sampling ${0<\tau }_i\le 0.3$ can also be investigated instead of pointwise-sampling ${\tau }_i=0.2$ and ${\tau }_j=0.3$ in [13].

Example 5.

The switched time-delay system (1) with (2) is considered as follows:

$$\begin{gathered} N=2,h_{\mathrm{m}}=3,h_M=3.5,{\tau }_M=0.1,\phi \left(t\right)={\left[1-1\right]}^T,\hspace{0.33em}-h_M\le t\le 0, \\ {A}_{01}=\left[\begin{array}{cc}-1.1& 0\\ -0.2& 0.25\end{array}\right],A_{02}=\left[\begin{array}{cc}0.35& -0.1\\ 0& -1.1\end{array}\right],A_{11}=\left[\begin{array}{cc}-0.6& -0.1\\ 0.2& -0.2\end{array}\right], \\ {A}_{12}=\left[\begin{array}{cc}-0.3& 0.1\\ 0.1& -0.6\end{array}\right],A_{21}=A_{22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],A_{\mathrm{z}01}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.1& 0.1\end{array}\right], \\ {A}_{\mathrm{z}02}=\left[\begin{array}{cc}0.1& -0.1\\ -0.1& -\mathrm{0,1}\end{array}\right],A_{\mathrm{z}11}=\left[\begin{array}{cc}-0.1& -0.1\\ 0.1& -0.1\end{array}\right], \\ {A}_{\mathrm{z}12}=\left[\begin{array}{cc}0.1& 0.1\\ -0.1& -0.1\end{array}\right],A_{z21}=A_{z22}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],B_{xw1}=B_{xw2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right], \\ {B}_{xu1}=B_{xu2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right],B_{zw1}=B_{zw2}=\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right], \\ {M}_{x\mathrm{i}}=N_{x0\mathrm{i}}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],N_{x1\mathrm{i}}=\left[\begin{array}{cc}0& 0.1\\ 0.1& 0\end{array}\right],N_{x2i}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right], \\ {N}_{x\mathrm{w}\mathrm{i}}=\left[\begin{array}{cc}0.1& 0.1\\ 0& 0\end{array}\right],N_{x\mathrm{u}1}=N_{x\mathrm{u}2}=\left[\begin{array}{cc}0& 0\\ 0.1& 0.1\end{array}\right], \\ {B}_{zu\mathrm{i}}=M_{\mathrm{z}\mathrm{i}}=N_{\mathrm{z}0\mathrm{i}}=N_{\mathrm{z}1\mathrm{i}}=N_{\mathrm{z}2\mathrm{i}}={\Xi }_{\mathrm{z}\mathrm{i}}=0,{\Xi }_{x\mathrm{i}}=0.01,i=1,\hspace{0.33em}2. \end{gathered}$$

(25)

Case 1.

$H_2$ measure and $H_{\mathrm{\infty }}$ switching control. With synchronous switching (5) for rule and input, with $3\le h\left(t\right)\le 3.5$ and ${\tau }_k\le 0.1$, $\forall$k, the linear matrix inequality conditions in (7) of Theorem 1 and Remark 1 with $\eta =1$ and ${\alpha }_1={\alpha }_2=0.5$ have a feasible solution with

$$\begin{gathered} \gamma =0.4031,{\widehat{K}}_1=\left[\begin{array}{cc}1.5028& -0.68\\ -0.7354& 2.4712\end{array}\right],{\widehat{K}}_2=\left[\begin{array}{cc}3.9972& 0.2226\\ -0.0169& 0.905\end{array}\right], \\ \widehat{U}=\left[\begin{array}{cc}0.3467& -0.1381\\ 0.1516& 0.6975\end{array}\right],{\widehat{{\rm Z}}}_1=\left[\begin{array}{cc}0.2564& 0.245\\ 0.245& -0.4702\end{array}\right],{\widehat{{\rm Z}}}_2=\left[\begin{array}{cc}-0.2563& -0.245\\ -0.245& 0.4703\end{array}\right]. \end{gathered}$$

The switched delay system in (1) with (2) and (14) is asymptotically stabilizable by the switching rule given by

$$\sigma \left(x\left(t\right)\right)=\left\{\begin{array}{c}1,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ 2,\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\forall t\in \left.{[T}_k,T_{k+1}\right),$$

(26a)

where

$${\overline{\Omega }}_1={\Omega }_1,{\overline{\Omega }}_2={\Omega }_2\backslash {\overline{\Omega }}_1={\mathfrak{R}}^2\backslash {\overline{\Omega }}_1,{{\rm Z}}_i={\widehat{U}}^{-1}{\widehat{{\rm Z}}}_i{\widehat{U}}^{-T},i=1,\hspace{0.33em}2,$$

(26b)

with

$${\Omega }_1=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{1}x=0.0285x_1^2+0.0544x_1{x}_2-0.0522x_2^2\ge 0\right\},$$

$${\Omega }_2=\left\{x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T\in {\mathfrak{R}}^2{:x}^T{{\rm Z}}_{2}x=-0.0285x_1^2-0.0544x_1{x}_2+0.0523x_2^2\ge 0\right\},$$

and switching input is provided in the following:

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}x\left(T_k\right)\in {\overline{\Omega }}_2,\end{array}\right.\forall t\in \left.{[T}_k,T_{k+1}\right),$$

(27a)

with

$$K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[\begin{array}{cc}0.5009& -0.2267\\ -0.2451& 0.8237\end{array}\right],K_2={\widehat{K}}_2{\widehat{U}}^{-T}=\left[\begin{array}{cc}1.3324& 0.0742\\ -0.0056& 0.3017\end{array}\right].$$

(27b)

We conclude that system (1) with (2) and (25) is stabilized with $H_{\mathrm{\infty }}$ performance $\overline{\gamma }=\sqrt{\gamma }=0.6349$ and $H_2$ measure $\alpha =23.4139$ by the switching rule in (26) and sampling input in (27a), with $K_i={\widehat{K}}_i{\widehat{U}}^{-T}$ in (27b).

Case 2.

The switching rule $\sigma \left(t\right)$ is arbitrary and captured at the sampling instant. We would like to consider $H_2$ measure and $H_{\infty }$ sampled-data switching control. With the above statement in Theorem 1 with Corollary 1 and Remark 1, the system (1) with (2) and (25) is stabilizing by switching input given by:

$$u\left(t\right)=\left\{\begin{array}{c}-K_{1}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}\mathsf{\sigma }\left(T_k\right)=1,\\ -K_{2}x\left(T_k\right),\hspace{0.33em}\hspace{0.33em}\mathsf{\sigma }\left(T_k\right)=2,\end{array}\right.\forall t\in \left.{[T}_k,T_{k+1}\right),$$

(28a)

with $\gamma =0.7201$,

$$K_1={\widehat{K}}_1{\widehat{U}}^{-T}=\left[\begin{array}{cc}0.671& 0.0073\\ -0.0796& 0.6107\end{array}\right]$$

(28b)

The system (1) with (2) and (25) is stabilized with $H_{\mathrm{\infty }}$ performance $\overline{\gamma }=\sqrt{\gamma }=0.8486$ and $H_2$ measure $\alpha =27.2064$ by the sampling input in (28a), with $K_i={\widehat{K}}_i{\widehat{U}}^{-T}$ in (28b). We make some comparisons in

Table 5. Comparing the obtained results with published ones.

Results	Sampling Period	Disturbance Attenuation	Approach and Delay
Results of [12]	No sampling	$\overline{\gamma }=1.3342$	1. No control input 2. Real time switching signal design 3. Constant delay $h=3.4$
Results of [13]	No sampling	$\overline{\gamma }=1.2989$
	${\tau }_i=0.03$ and ${\tau }_j=0.05$ for some i and j pointwise sampling	$\overline{\gamma }=0.8355$	Synchronous switching: 1. Novel switching signal 2. Sampled-data input 3. Constant delay $h=3.4$
		$\overline{\gamma }=1.0951$	1. Arbitrary switching 2. Sampled-data input 3. Constant delay $h=3.4$
The proposed results in Theorem 1	${0<\tau }_i\le 0.1$	$\overline{\gamma }=0.6349$	Synchronous switching: 1. Novel switching signal 2. Sampled-data input 3. $3\le \mathrm{h}\left(\mathrm{t}\right)\le 3.5$
The proposed results in Corollary 1		$\overline{\gamma }=0.8486$	1. Arbitrary switching ($\sigma$ can be captured at sampling instant) 2. Sampled-data input 3. $3\le \mathrm{h}\left(\mathrm{t}\right)\le 3.5$

to show the contribution of the developed results. The considered switched system with interval time-varying delay $3\le h\left(\mathrm{t}\right)\le 3.5$ can be studied instead of constant delay $h\left(t\right)=h=3.4$ in [13]. The interval sampling ${0<\tau }_i\le 0.1$ can also be investigated instead of pointwise-sampling ${\tau }_i=0.03$ and ${\tau }_j=0.05$ in [13].

5. Conclusions

In this paper, synchronous switching on signal and sampling input has been investigated to reach robust (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances for a switched delay system. Under the capture of switching rule, the sampling input with arbitrary switching has also been proposed. In this paper, a full matrix formulation approach has been used to show the improvement in the proposed results. Novel inequality and Lyapunov-Krasovskii functional have been investigated to derive the main contribution of this paper.