Finite-time $H_{\infty }$ bounded control of networked control systems with mixed delays and stochastic nonlinearities

Abstract

In this study, we investigate the stochastic finite-time $H_{\infty }$ bounded control problem for a class of networked control systems (NCSs) subject to mixed delays, stochastic nonlinearities, and randomly missing measurement. The mixed delays consist of discrete time-varying and distributed delays. The stochastic nonlinearities satisfy statistical means. The missing measurement is modeled by a Bernoulli-distributed random variable. By applying stochastic analysis method we present sufficient conditions guaranteeing the stochastic finite-time boundedness of a closed-loop system with desired $H_{\infty }$ performance level within a finite time interval. Moreover, dynamic output feedback controller can be obtained in terms of a set of matrix inequalities, which can be easily solved by using the cone complementarity linearization method (CCLM). Finally, we provide two numerical examples to illustrate the validity of the proposed design technique.

1 Introduction

Networked control systems (NCSs), which integrate the development of automatic control technology, network communication technology, and microelectronics technology, have become a hot research topic in the international control field in recent years. Different from the traditional control system, it links sensors, controllers, and actuators distributed in different regions of the control system to form a closed-loop feedback control system through a communication network. The transmission of data and information between each node is realized through the network. For many years now, these kinds of systems can be found in a variety of engineering areas such as automobiles, manufacturing plants, and aircrafts. At present the research on networked control systems is mostly concentrated on continuous systems, but discrete-time networked control systems play important roles in complex control systems and remote control systems, for example, pump-controlled motor speed servo control, pendulum control, remote diagnostic, and troubleshooting control [1–6]. NCSs have many advantages in comparison with traditional control systems, but the introduction of communication networks also bring a series of problems. The communication between sensors, controllers, and actuators is accomplished by sharing network platform after the networks are introduced into the control system. However, due to limited spectrum resources, the problems of network connection interruption, data collision, network congestion, or resource competition are inevitable, which will inevitably lead to network delays, packet dropouts, or data missing phenomena in both sensor-to-controller link and controller-to-actuator link. In addition, for the actual control system, nonlinear factors are inevitable and cannot be ignored. Therefore it is necessary to consider the network delays, missing measurement, and nonlinearity simultaneously in the study of NCSs. In this paper, we focus on the problem of stochastic finite-time $H_{\infty }$ control for discrete-time NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement.

Network delays in the NCSs are exposed when the data exchange from the sensor to controller and from the controller to actuator. At present the delays are generally divided into two categories according to the occurrence way, discrete delays and distributed delays. In the recent years, some works have been done on the analysis and synthesis of systems for various types of delays [7–12]. However, most of these studies considered one of the delays mentioned before, whereas few considered two kinds of delays simultaneously, that is, mixed delays (both discrete and distributed). In addition, in the existing literature on mixed delays, continuous systems were considered. Note that infinite distributed delays also exist in discrete case. With the development of applications to digital control systems, it is important to discuss how mixed delays affect the dynamic behavior of discrete NCSs. Some works in this field have been made [13, 14].

On the other hand, the missing measurement is also one of the problems that cause the performance deterioration of NCS indicators. Therefore it is particularly important to consider the impact of missing measurement on the performance of NCSs. The missing measurement is usually described by two forms: the first is to regard the missing measurement as a random variable that satisfies the Bernoulli distribution, and the second is using Markov chains to describe random forms. At present, it is popular describing missing measurement by Bernoulli distribution [15, 16]. Recently, missing measurement has attracted considerable attention, and many results have been obtained; see, for example, [17, 18]. In [17] the quantized recursive filtering problem was further investigated for a class of nonlinear systems. Hu et al. [18] studied event-based filtering problem for a class of nonlinear systems. In addition, it is well known that the nonlinearities, such as delays and missing measurements, also cause instability and poor performance of dynamic systems [19–21]. If the system involves serious nonlinearities, then it is difficult to design a controller with good performance. In recent years, stochastic nonlinearities described by statistical means have became a popular form. Hu et al. [22] investigated the $H_{\infty }$ filtering problem for a class of stochastic nonlinear systems with time-varying delay and multiple missing measurements. Other related results can be found in [23–26].

It should be pointed out that all the mentioned references on the problems of Lyapunov asymptotical stability for NCSs defined over an infinite time interval, whereas in many practical applications the main concern is not only system dynamic behavior on an infinite time interval but also a bound of system trajectories over a fixed finite-time interval. In this sense, Dorato [27] first proposed the concept of finite-time stability (FTS). Subsequently, by using Lyapunov theory and LMI technique some sufficient conditions have been established to ensure FTS and finite-time boundedness in various systems [28–30], particularly, in the NCSs [31–36]. Recently, Ma and Song [37, 38] studied a finite-time dissipative problem for a class of singular discrete-time systems and discrete stochastic systems. However, it is worth noting that most of the mentioned references on problems of finite-time control were considered for discrete-time linear NCSs, whereas in the actual systems, most of the controlled objects or processes exhibit nonlinear features. To the best of authors’ knowledge, so far, the problem of stochastic finite-time $H_{\infty }$ control for discrete-time NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement has not been studied. This motivates our present research. It should be remarked that it is difficult to analyze this problem owing to two reasons: (1) How to characterize a suitable definition of stochastic finite-time boundedness for discrete-time NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement? (2) How to deal with mixed delays, stochastic nonlinearities, and randomly missing measurement in the controller design stage.

Our aim is designing a stochastic finite-time output feedback controller by fully taking into account mixed delays, stochastic nonlinearities, and randomly missing measurement. Through such a output feedback controller, both stochastic finite-time boundedness and desired $H_{\infty }$ performance can be guaranteed. The main contributions of this paper can be itemized as follows: (1) The concept of stochastic finite-time boundedness is extended to more general discrete-time NCSs containing mixed delays, stochastic nonlinearities, and randomly missing measurement. (2) The effects of mixed delays, stochastic nonlinearities, and randomly missing measurement on system performance are considered, which are important in system analysis and synthesis. (3) By constructing a novel Lyapunov–Krasovskii functional and using stochastic analysis method, a more general controller, the output feedback controller, is designed. This controller is more useful than the previous given state feedback controller when the state of the system is not measurable.

Notations

${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m \times n}$ respectively denote the space of n-dimensional real vectors and $m \times n$ real matrices; I and 0 stand for the identity and zero matrices of compatible dimension, respectively; $\operatorname{Prob}\{\cdot \}$ denotes the occurrence probability of the event; and ${\mathbb{E}}\{ \cdot \}$ is the mathematical expectation operator with respect to the given probability measure. $\operatorname{Diag}\{\cdot \}$ stands for the block diagonal matrix. The notation $X\geq Y$ or $X>Y$ means that $X-Y$ is positive semidefinite or positive definite, where X and Y are both symmetric matrices. $\lambda _{\mathrm{max}}$ and $\lambda _{ \mathrm{min}}$ are the maximum and minimum eigenvalues of a matrix, respectively. $M^{T}$ denotes the transpose of matrix M, and ∗ denotes the matrix elements induced by symmetry.

2 Problem formulation

In this paper, we consider the following discrete-time NCS:

$$ \textstyle\begin{cases} x(k + 1) = Ax(k) + A_{d} x(k - d(k)) + A_{l} \sum_{m = 1}^{{+}\infty } {\mu _{m} x(k - m)} \\ \hphantom{x(k + 1) ={}}{} +Bu(k) + f(k) + D\omega (k), \\ z(k) = E_{1} x(k) + E_{2} \omega (k), \\ x(k) = \varphi (k), \quad - \infty < k\leq 0, \end{cases} $$

(1)

where $x(k)\in {\mathbb{R}}^{n}$ is the state vector, $z(k) \in {\mathbb{R}} ^{t}$ is the controlled output, $u(k) \in {\mathbb{R}}^{p}$ is the control input, $\sum_{m = 1}^{+\infty } {\mu _{m} x(k - m)}$ is the distributed delay, and A, $A_{d}$, $A_{l}$, B, D, $E_{1}$, $E_{2}$ are known real constant matrices of appropriate dimensions. For simplicity, we denote $x_{d}=x(k-d(k))$ and $x_{m}=x(k-m)$.

Before proceeding the main results, let us introduce the following assumptions.

Assumption 1

$d(k)$ is time-varying communication delay satisfying

$$ d_{m} \leq d(k)\leq d_{M}, \quad d_{m}>0,d_{M}>0, $$

(2)

where $d_{m}$ and $d_{M}$ are given positive numbers.

Assumption 2

$f(k)$ is stochastic nonlinearities cover $x(k)$, $x(k-d(k))$ and $\sum_{m = 1}^{+\infty } \mu _{m} x(k - m)$ in a statistical sense, which is bounded as follows:

$$ \begin{aligned} &{\mathbb{E}} \bigl[f(k) \bigr] = 0, \\ &{\mathbb{E}} \bigl[f(k)f^{T} (k) \bigr] = \sum _{i = 1}^{q} {\rho _{i} \rho _{i}^{T} \Biggl[ {x^{T} (k)A_{i} x(k) + x^{T}_{d}A_{di} x_{d}+ \Biggl( {\sum_{m = 1}^{ + \infty } {\mu _{m} x_{m}} } \Biggr)^{T} A_{li} \sum_{m = 1}^{ + \infty} {\mu _{m} x_{m}} } \Biggr]}, \end{aligned} $$

(3)

where $A_{i}$, $A_{di}$, and $A_{li}$ are known positive definite matrices of appropriate dimensions, $\rho _{i}$ ($i=1, \ldots, q$) are known as column vectors, and $q>1$ is a known constant. The constants $\mu _{m}$ satisfy the following convergence conditions:

$$ \bar{\mu }= \sum_{m = 1}^{ + \infty } {\mu _{m} }\leq \sum_{m = 1}^{ + \infty } {m\mu _{m} } < + \infty . $$

(4)

Remark 1

At present, delays have attracted the attention of many scholars. However, most of the existing results are concerned with either discrete delays [32] or distributed delays [8]. Different from [32], on the one hand, we introduce stochastic nonlinearities into the study of finite-time bounded control. It is worth noting that deterministic systems and stochastic systems have different properties. We need to handle this separately. On the other hand, authors in [32] considered only discrete delays, whereas we consider both discrete delays and distributed delays. It is shown that the distributed delays have an important effect on the finite-time boundedness of the system.

Assumption 3

External disturbance input $\omega (k)$ is time varying and satisfies

$$ \sum_{k = 0}^{N} { \omega ^{T} (k)\omega (k)} \le \bar{\omega }, \quad \bar{\omega } \ge 0, $$

(5)

for any given positive number ω̄.

In this paper, the randomly missing measurement is described by

$$ y(k) = \alpha (k)F_{1} x(k) + F_{2} \omega (k), $$

(6)

where $y(k)$ denotes the measured output vector, $F_{1}$, $F_{2}$ are parameter matrices of appropriate dimensions, and $\alpha (k)$ is a Bernoulli-distributed stochastic variable, that is,

$$ \operatorname{Prob} \bigl\{ {\alpha (k) = 1} \bigr\} = \bar{ \alpha }, \qquad \operatorname{Prob} \bigl\{ { \alpha (k)= 0} \bigr\} = 1 - \bar{ \alpha }, $$

(7)

where $\bar{\alpha }\in [0,1]$ is a constant, and the variance $\tilde{\alpha }^{2}=\bar{\alpha }(1-\bar{\alpha })$.

In this paper, we consider the following dynamic output feedback:

$$ \textstyle\begin{cases} \hat{x}(k + 1) = A_{K} \hat{x}(k) + B_{K} y(k), \\ u(k) = C_{K} \hat{x}(k), \end{cases} $$

(8)

where $\hat{x}(k)\in {\mathbb{R}}^{n}$ is the state estimation of system (1), and $A_{K}$, $B_{K}$, and $C_{K}$ are the controller gain parameters to be determined.

By the combination of (6) and (8) with (1) the resultant closed-loop system can be written as

$$ \textstyle\begin{cases} \eta (k + 1) = \bar{A}\eta (k) + (\alpha (k)-\bar{\alpha })\tilde{A} \eta (k) + \bar{A}_{d} \eta _{d} \\ \hphantom{\eta (k + 1) ={}}{}+ \bar{A}_{l} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{d}} + \bar{D}\omega (k) + \bar{G}f(k), \\ z(k) = \bar{E}_{1} \eta (k) + E_{2} \omega (k), \end{cases} $$

(9)

where

$$\begin{array}{c}\eta (k)=\left[\begin{array}{c}x(k)\\ \hat{x}(k)\end{array}\right],{\eta }_d=\eta (k-d(k)),{\eta }_m=\eta (k-m),\overline{A}=\left[\begin{array}{cc}A& D{C}_K\\ \overline{\alpha }{B}_K{F}_1& A_K\end{array}\right],\hfill \\ \tilde{A}=\left[\begin{array}{cc}0& 0\\ B_K{F}_1& 0\end{array}\right],{\overline{A}}_d=\left[\begin{array}{cc}{A}_d& 0\\ 0& 0\end{array}\right],{\overline{A}}_l=\left[\begin{array}{cc}{A}_l& 0\\ 0& 0\end{array}\right],{\overline{E}}_1={\left[\begin{array}{c}{E}_1\\ 0\end{array}\right]}^T,\hfill \\ \overline{D}=\left[\begin{array}{c}D\\ B_K{F}_2\end{array}\right],\overline{G}=\left[\begin{array}{c}I\\ 0\end{array}\right].\hfill \end{array}$$

The objective of this paper is to deal with stochastic finite-time $H_{\infty }$ control problem for closed-loop dynamic system (9). We introduce the following definitions and lemmas before giving the main results.

Definition 1

For a given symmetric matrix $R>0$ and positive numbers $c_{1}$, $c_{2}$ ($c_{1}< c_{2}$), ω̄, N, system (9) is stochastically finite-time bounded (SFTB) with respect to $(c_{1}, c_{2}, \bar{ \omega }, R, N)$ if

$$ \sup_{ -\infty < l\leq 0 } {\mathbb{E}} \bigl\{ {x (l)^{T} Rx(l)} \bigr\} \le c_{1} \quad \Rightarrow\quad {\mathbb{E}} \bigl\{ {x (k)^{T} Rx (k) } \bigr\} \le c_{2} ,\quad k \in \{ 1, \ldots ,N\}, $$

for all $\omega (k)$ satisfying Assumption 3.

Definition 2

For given symmetric matrix $R>0$, system (9) is stochastically finite-time $H_{\infty }$ bounded (SFT$H_{\infty }$B) with respect to $(c_{1}, c_{2}, \bar{\omega },\gamma , R, N)$ if the following two conditions hold:

• (1) System (9) is SFTB with respect to $(c_{1}, c_{2},\bar{\omega }, R, N)$.

• (2) Under the zero-initial condition, the output $z(k)$ satisfies

  $$ {\mathbb{E}} \Biggl\{ {\sum_{s = 0}^{N} {z^{T} (s) z(s)} } \Biggr\} \le \gamma {\sum _{s = 0}^{N} {\omega ^{T} (s) \omega (s)} } $$

  (10)

  for all $\omega (k)$ satisfying Assumption 3.

Lemma 1

(Schur complement [17])

Let${\mathcal{S}}_{1}$, ${\mathcal{S}}_{2}$, and${\mathcal{S}}_{3}$be constant matrices such that${\mathcal{S}}_{1}= {\mathcal{S}}_{1}^{T}$and${\mathcal{S}}_{2}={\mathcal{S}}_{2}^{T}>0$. Then${\mathcal{S}}_{1}+{\mathcal{S}}_{3}^{T}{\mathcal{S}}_{2}^{-1} {\mathcal{S}}_{3}<0$if and only if

$$ \begin{bmatrix} {\mathcal{S}}_{1} & {\mathcal{S}}_{3}^{T} \\ \ast & -{\mathcal{S}}_{2} \end{bmatrix}< 0 \quad \textit{or}\quad \begin{bmatrix} -{\mathcal{S}}_{2} & {\mathcal{S}}_{3} \\ \ast & {\mathcal{S}}_{1} \end{bmatrix}< 0. $$

Lemma 2

([17])

Let$M\in R^{n\times n}$be a positive semidefinite matrix, let$x_{i} \in R^{n}$, and let$\alpha _{i}$ ($i = 1,2, \ldots$) be constant series. If the series${\alpha _{i} }\geq 1$are convergent, then

$$ \Biggl( {\sum_{m = 1}^{ + \infty } {\alpha _{i} x_{i} } } \Biggr) ^{T} M \Biggl( {\sum_{m = 1}^{ + \infty } {\alpha _{i} x_{i} } } \Biggr) \le \Biggl( {\sum _{m = 1}^{ + \infty } {\alpha _{i} } } \Biggr) \sum_{m = 1}^{ + \infty } {\alpha _{i} x_{i}^{T} Mx_{i}}. $$

(11)

3 Main results

In this section, we first investigate the problem of stochastic finite-time boundedness for dynamic system (9). Furthermore, we design the dynamic gain matrices $A_{K}$, $B_{K}$, and $C_{K}$.

3.1 Stochastic finite-time boundedness analysis

In this subsection, we consider the stochastic finite-time boundedness for discrete-time NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement.

Theorem 1

For given scalar$\delta \ge 1$and symmetric matrix$R>0$, system (9) is SFTB with respect to$(c_{1}, c_{2}, \bar{\omega }, R, N)$if there exist positive scalars$\lambda _{i}$ ($i = 1,2,3,4$) and$\xi _{i}$ ($i = 1,2, \ldots , q$), and symmetric matrices$P>0$, $Q>0$, and$Z>0$such that the following inequalities hold:

$$\left[\begin{array}{cc}-{\xi }_i& {\rho }_i^T{\overline{G}}^T\\ \ast & -P^{-1}\end{array}\right]<0(i=1,2,\dots ,q),$$

(12)

$$\left[\begin{array}{cccccc}\Phi & 0& 0& 0& {\overline{A}}^{T}P& \tilde{\alpha }{\tilde{A}}^{T}P\\ \ast & -{\delta }^{d_m}Q+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{di}& 0& 0& {\overline{A}}_d^{T}P& 0\\ \ast & \ast & -\frac{1}{\overline{\mu }}Z+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{li}& 0& {\overline{A}}_l^{T}P& 0\\ \ast & \ast & \ast & -\frac{\gamma }{{\delta }^N}I& {\overline{D}}^{T}P& 0\\ \ast & \ast & \ast & \ast & -P& 0\\ \ast & \ast & \ast & \ast & \ast & -P\end{array}\right]<0,$$

(13)

$$\begin{aligned}& \lambda _{1} R < P < \lambda _{2} R,\qquad 0 < Q < \lambda _{3} R,\qquad 0 < Z < \lambda _{4} R, \end{aligned}$$

(14)

$$\left[\begin{array}{cccc}\gamma \overline{\omega }-c_2\delta {\lambda }_1& c_1{\delta }^{N+1}{\lambda }_2& {\rho }_1{\lambda }_3& {\rho }_2{\lambda }_4\\ \ast & -c_1{\delta }^{N+1}{\lambda }_2& 0& 0\\ \ast & \ast & -{\rho }_1{\lambda }_3& 0\\ \ast & \ast & \ast & -{\rho }_2{\lambda }_4\end{array}\right]<0,$$

(15)

where

$$\begin{array}{c}\Phi :=(d_M-d_m+1)Q+\overline{\mu }Z-\delta P+\sum _{i=1}^q{\xi }_i{\overline{A}}_i,\hfill \\ {\overline{A}}_i=\left[\begin{array}{cc}{A}_i& 0\\ 0& 0\end{array}\right],{\overline{A}}_{di}=\left[\begin{array}{cc}{A}_{di}& 0\\ 0& 0\end{array}\right],{\overline{A}}_{li}=\left[\begin{array}{cc}{A}_{li}& 0\\ 0& 0\end{array}\right],\hfill \\ {\rho }_1:={\delta }^{N+d_M-1}[d_M\delta +\frac{1}{2}(d_M-d_m)(d_M+d_m-1)],{\rho }_2:={\delta }^{N+1}\sigma ,\hfill \\ \sigma ={\mu }_1+(1+\delta ){\mu }_2+\cdots +(1+\delta +\cdots +{\delta }^{d_M-1}){\mu }_{d_M},\hfill \\ {\tau }_1=d_M-d_m+1,{\tau }_2=d_M+d_m-1.\hfill \end{array}$$

Proof

To show the stochastic finite-time boundedness of system (9), we choose a Lyapunov–Krasovskii functional candidate as follows:

$$ V(k) = V_{1} (k) + V_{2} (k) + V_{3} (k), $$

(16)

where

$$\begin{aligned}& V_{1} (k) = \eta ^{T} (k)P\eta (k), \\& V_{2} (k) = \sum_{s = k - d(k)}^{k - 1} {\delta ^{k - 1 - s} \eta ^{T} (s)Q \eta (s)} + \sum _{s = - d_{M} {+}2}^{ - d_{m} + 1} {\sum_{t = k + s - 1}^{k - 1} {\delta ^{k - 1 - t} \eta ^{T} (t)Q \eta (t)} }, \\& V_{3} (k) = \sum_{m = 1}^{ + \infty } {\mu _{m} } \sum_{l = k - m}^{k - 1} {\delta ^{k - 1 - l} \eta ^{T} (l)Z \eta (l)}. \end{aligned}$$

Let us assume that

$$ {\mathbb{E}} \bigl\{ {\eta (k)^{T} R\eta (k)} \bigr\} \le c_{1}, \quad -\infty < k\leq 0. $$

(17)

The goal is to show that ${\mathbb{E}} \{ {\eta (k)^{T} R\eta (k)} \} \le c_{2}$ for all $k \in \{1, \ldots , N\}$ if conditions (12)–(15) hold.

Firstly, we calculate ${\mathbb{E}} \{ {V_{i} (k + 1) - \delta V_{i} (k)} \}$ ($i=1,2,3$) along the solution of (9):

$$\begin{aligned} &{\mathbb{E}} \bigl\{ {V_{1} (k + 1) - \delta V_{1} (k)} \bigr\} \\ &\quad ={\mathbb{E}} \bigl\{ {\eta ^{T} (k + 1)P\eta (k + 1) - \delta \eta ^{T} (k)P\eta (k)} \bigr\} \\ &\quad ={\mathbb{E}} \Biggl\{ \eta ^{T} (k) \bar{A}^{T} P \bar{A}\eta (k) + 2\eta ^{T} (k)\bar{A}^{T} P \bar{A}_{d} \eta _{d}+2\eta ^{T} (k) \bar{A}^{T} P\bar{A}_{l} \sum _{m = 1} ^{ + \infty } {\mu _{m} \eta _{m}} \\ &\qquad {}+2\eta ^{T} (k)\bar{A}^{T} P \bar{G}f(k)+2\eta ^{T} (k)\bar{A}^{T} P\bar{D}\omega (k) +2 \bigl(\alpha (k) - \bar{\alpha } \bigr)\eta ^{T} (k)\tilde{A}^{T} P \bar{A}_{l} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} \\ &\qquad {}+2 \bigl(\alpha (k) - \bar{ \alpha } \bigr)\eta ^{T} (k) \tilde{A}^{T} P\bar{A}_{d} \eta _{d}+ \bigl( \alpha (k) - \bar{\alpha } \bigr)^{2} \eta ^{T} (k) \tilde{A}^{T} P\tilde{A}\eta (k) \\ &\qquad {}+2 \bigl( \alpha (k) - \bar{\alpha } \bigr)\eta ^{T} (k) \tilde{A}^{T} P\bar{G}f(k)+ 2 \bigl( \alpha (k) - \bar{\alpha } \bigr) \eta ^{T} (k)\tilde{A}^{T} P\bar{D}\omega (k) \\ &\qquad {}+\eta ^{T}_{d}\bar{A}_{d}^{T} P\bar{A}_{d} \eta _{d} + 2\eta ^{T} _{d}\bar{A}_{d}^{T} P\bar{A}_{l} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}}+2\eta ^{T} _{d} \bar{A}_{d}^{T} P\bar{G}f(k) + 2\eta ^{T}_{d} \bar{A}_{d}^{T} P\bar{D}\omega (k) \\ &\qquad {}+ \Biggl( {\sum_{m = 1}^{ + \infty } { \mu _{m} \eta _{m} }} \Biggr)^{T} \bar{A}_{l}^{T} P\bar{A}_{l} \sum _{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} + 2 \Biggl( {\sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} } \Biggr)^{T} \bar{A}_{l}^{T} P\bar{G}f(k) \\ &\qquad {}+2 \Biggl( {\sum_{m = 1}^{ + \infty } { \mu _{m} \eta _{m}} } \Biggr)^{T} \bar{A}_{l}^{T} P\bar{D}\omega (k) + f^{T} (k)\bar{G}^{T} P\bar{G}f(k) + 2f^{T} (k) \bar{G}^{T} P\bar{D}\omega (k) \\ &\qquad {}+\omega ^{T} (k)\bar{D} ^{T} P\bar{D}\omega (k) - \delta \eta ^{T} (k)P\eta (k) \Biggr\} . \end{aligned}$$

By Assumption 2 and inequality (12) we have

$$ \begin{aligned}[b] &{\mathbb{E}} \bigl\{ f^{T} (k) \bar{G}^{T} P\bar{G}f(k) \bigr\} \\ &\quad ={\mathbb{E}} \bigl\{ \operatorname{tr} \bigl[ \bar{G}^{T} P \bar{G}f(k)f^{T} (k) \bigr] \bigr\} \\ &\quad =\sum_{i = 1}^{q} \operatorname{tr} \bigl( \bar{G}^{T} P\bar{G}\rho _{i} \rho _{i}^{T} \bigr) \Biggl[ \eta ^{T} (k) \bar{A}_{i} \eta (k) + \eta ^{T}_{d} \bar{A}_{di} \eta _{d} + \Biggl( {\sum _{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} } \Biggr)^{T} \bar{A}_{li} \sum _{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} \Biggr] \\ &\quad < \sum_{i = 1}^{q} \xi _{i} \Biggl[ \eta ^{T} (k)\bar{A}_{i} \eta (k) + \eta ^{T}_{d}\bar{A}_{di} \eta _{d}+ \Biggl( {\sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} } \Biggr) ^{T} \bar{A}_{li} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} \Biggr]. \end{aligned} $$

(18)

Thus

$$\begin{aligned} &{\mathbb{E}} \bigl\{ {V_{1} (k + 1) - \delta V_{1} (k)} \bigr\} \\ &\quad < \eta ^{T} (k) \Biggl(\bar{A}^{T} P\bar{A} \tilde{ \alpha }^{2} \tilde{A} ^{T} P\tilde{A}+\sum _{i = 1}^{q}{\xi _{i} \bar{A}_{i}}-\delta P \Biggr) \eta (k) + 2\eta ^{T} (k) \bar{A}^{T} P\bar{A}_{d}\eta _{d} \\ &\qquad {}+2\eta ^{T} (k)\bar{A}^{T} P \bar{A}_{l} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} + 2\eta ^{T} (k) \bar{A}^{T} P \bar{D}\omega (k) \\ &\qquad {}+ \eta ^{T}_{d} \Biggl(\bar{A}_{d}^{T} P\bar{A}_{d}+\sum_{i = 1}^{q} {\xi _{i}\bar{A}_{di}} \Biggr)\eta _{d} \\ &\qquad {}+2\eta ^{T}_{d}\bar{A}_{d}^{T} P\bar{A}_{l} \sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} + 2\eta ^{T}_{d} \bar{A}_{d}^{T} P\bar{D} \omega (k) \\ &\qquad {}+ \Biggl( {\sum_{m = 1}^{ + \infty } { \mu _{m} \eta _{m}} } \Biggr) ^{T} \Biggl( \bar{A}_{l}^{T} P\bar{A}_{l}+\sum _{i = 1}^{q}{\xi _{i} \bar{A}_{i}} \Biggr)\sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} \\ &\qquad {}+ 2 \Biggl( {\sum_{m = 1}^{ + \infty } {\mu _{m} \eta _{m}} } \Biggr) ^{T} \bar{A}_{l}^{T} P\bar{D}\omega (k)+\omega ^{T} (k)\bar{D}^{T} P \bar{D}\omega (k), \end{aligned}$$

(19)

$$\begin{aligned} &{\mathbb{E}} \bigl\{ {V_{2} (k + 1) - \delta V_{2} (k)} \bigr\} \\ &\quad =\sum_{s = k + 1 - d(k + 1)}^{k} {\delta ^{k - s} \eta ^{T} (s)Q \eta (s)} - \sum _{s = k - d(k)}^{k - 1} {\delta ^{k - s} } \eta ^{T} (s)Q\eta (s) \\ &\qquad {}+ \sum_{s =- d_{M} {+}2}^{ - d_{m} + 1} {\sum _{t = k + s}^{k} {\delta ^{k - t} \eta ^{T} (t)Q\eta (t)} } - \sum_{s = - d_{M} {+}2}^{ - d_{m} + 1} {\sum_{t = k + s - 1}^{k - 1} {\delta ^{k - t} \eta ^{T} (t)Q\eta (t)} } \\ &\quad \le (d_{M} - d_{m} + 1)\eta ^{T} (k)Q \eta (k) - \delta ^{d_{m} } \eta ^{T} _{d} Q\eta _{d}, \end{aligned}$$

(20)

$$\begin{aligned} &{\mathbb{E}} \bigl\{ {V_{3} (k + 1) - \delta V_{3} (k)} \bigr\} \\ &\quad =\sum_{m = 1}^{ + \infty } {\mu _{m} \Biggl[ {\sum_{l = k + 1 - m}^{k} {\delta ^{k - l} } \eta ^{T} (l)Z\eta (l) - \sum _{l = k - m}^{k - 1} {\delta ^{k - l} } \eta ^{T} (l)Z\eta (l)} \Biggr]} \\ &\quad =\sum_{m = 1}^{ + \infty } {\mu _{m} \bigl[ {\eta ^{T} (k)Z \eta (k) - \delta ^{m} \eta ^{T} _{m}Z\eta _{m}} \bigr]}. \end{aligned}$$

(21)

Noting that $\bar{\mu }= \sum_{m = 1}^{ + \infty } {\mu _{m} }$ and $\delta \geq 1$, we get

$$ {\mathbb{E}} \bigl\{ {V_{3} (k + 1) - \delta V_{3} (k)} \bigr\} \le \bar{\mu }\eta ^{T} (k)Z\eta (k) - \sum_{m = 1}^{ + \infty } {\mu _{m} } \eta ^{T} _{m} Z\eta _{m}. $$

(22)

Furthermore, from Lemma 2 we obtain

$$ - \sum_{m = 1}^{ + \infty } {\mu _{m} } \eta ^{T}_{m}Z\eta _{m} \le - \frac{1}{{\bar{\mu }}} \Biggl( {\sum_{m = 1}^{ + \infty } {\mu _{m} } \eta _{m}} \Biggr)^{T} Z\sum _{m = 1}^{ + \infty } {\mu _{m} } \eta _{m}. $$

(23)

Then

$$ {\mathbb{E}} \bigl\{ {V_{3} (k + 1) - \delta V_{3} (k)} \bigr\} \le \bar{\mu }\eta ^{T} (k)Z\eta (k) - \frac{1}{{\bar{\mu }}} \Biggl({ \sum_{m = 1}^{ + \infty } {\mu _{m} } \eta _{m}} \Biggr)^{T} Z \sum _{m = 1}^{ + \infty } {\mu _{m} } \eta _{m}. $$

(24)

Thus

$$ {\mathbb{E}} \bigl\{ {V(k + 1) - \delta V(k)} \bigr\} < \xi ^{T} (k) \varPi \xi (k), $$

where

$$\begin{array}{c}{\xi }^T(k)=\left[\begin{array}{cccc}{\eta }^T(k)& {\eta }_d^T& {({\sum }_{m=1}^{+\mathrm{\infty }}{\mu }_m{\eta }_m)}^T& {\omega }^T(k)\end{array}\right],\hfill \\ \Pi =\left[\begin{array}{cccc}{\Pi }_{11}& {\overline{A}}^{T}P{\overline{A}}_d& {\overline{A}}^{T}P{\overline{A}}_l& {\overline{A}}^{T}P\overline{D}\\ \ast & {\Pi }_{22}& {\overline{A}}_d^{T}P{\overline{A}}_l& {\overline{A}}_d^{T}P\overline{D}\\ \ast & \ast & {\Pi }_{33}& {\overline{A}}_l^{T}P\overline{D}\\ \ast & \ast & \ast & {\overline{D}}^{T}P\overline{D}\end{array}\right],\hfill \\ {\Pi }_{11}={\overline{A}}^{T}P\overline{A}+{\tilde{\alpha }}^2{\tilde{A}}^{T}P\tilde{A}+(d_M-d_m+1)Q+\overline{\mu }Z+\sum _{i=1}^q{\xi }_i{\overline{A}}_i-\delta P,\hfill \\ {\Pi }_{22}={\overline{A}}_d^{T}P{\overline{A}}_d+\sum _{i=1}^q{\xi }_i{\overline{A}}_{di}-{\delta }^{d_m}Q,\hfill \\ {\Pi }_{33}={\overline{A}}_l^{T}P{\overline{A}}_l+\sum _{i=1}^q{\xi }_i{\overline{A}}_{li}-\frac{1}{\overline{\mu }}Z.\hfill \end{array}$$

Hence

$$ {\mathbb{E}} \bigl\{ {V(k + 1) - \delta V(k)} \bigr\} \le \xi ^{T} (k)\bar{\varPi } \xi (k) + \frac{\gamma }{{\delta ^{N} }}\omega ^{T} (k) \omega (k), $$

(25)

where

$$\overline{\Pi }=\left[\begin{array}{cccc}{\Pi }_{11}& {\overline{A}}^{T}P{\overline{A}}_d& {\overline{A}}^{T}P{\overline{A}}_l& {\overline{A}}^{T}P\overline{D}\\ \ast & {\Pi }_{22}& {\overline{A}}_d^{T}P{\overline{A}}_l& {\overline{A}}_d^{T}P\overline{D}\\ \ast & \ast & {\Pi }_{33}& {\overline{A}}_l^{T}P\overline{D}\\ \ast & \ast & \ast & {\overline{D}}^{T}P\overline{D}-\frac{\gamma }{{\delta }^N}I\end{array}\right].$$

Then from (13) it follows that $\bar{\varPi }< 0$, and subsequently

$$ {\mathbb{E}} \bigl\{ {V(k + 1) - \delta V(k)} \bigr\} < \frac{ \gamma }{{\delta ^{N} }} \omega ^{T} (k)\omega (k), \quad \forall k \in Z_{+}, $$

(26)

that is,

$$ {\mathbb{E}} \bigl\{ V(k + 1) \bigr\} < \delta E \bigl\{ V(k) \bigr\} + \frac{\gamma }{{\delta ^{N} }}\omega ^{T} (k)\omega (k), \quad \forall k \in Z_{+}. $$

(27)

We can obtain the following inequality from Assumption 3:

$$ \begin{aligned}[b] {\mathbb{E}} \bigl\{ V(k) \bigr\} &< \delta ^{k} {\mathbb{E}} \bigl\{ V(0) \bigr\} + \frac{ \gamma }{{\delta ^{N} }}\sum_{s = 0}^{k - 1} \delta ^{k - 1 - s} \bigl\{ {\omega ^{T} (s) \omega (s)} \bigr\} \\ &< \delta ^{N} {\mathbb{E}} \bigl\{ V(0) \bigr\} + \frac{\gamma }{{\delta ^{N} }}\delta ^{N - 1} \sum_{s = 0}^{N - 1} { \bigl\{ {\omega ^{T} (s)\omega (s)} \bigr\} } \\ &< \delta ^{N} {\mathbb{E}} \bigl\{ V(0) \bigr\} + \frac{\gamma }{\delta }\bar{ \omega }. \end{aligned} $$

(28)

By condition (14) we have

$$\begin{aligned} {\mathbb{E}} \bigl\{ V(0) \bigr\} = & \eta ^{T} (0)P\eta (0) + \sum _{s = - d(0)} ^{ - 1} {\delta ^{ - 1 - s} \eta ^{T} (s)Q\eta (s)} \\ &{}+ \sum_{s = - d_{M} + 2}^{ - d_{m} + 1} {\sum _{t = s - 1}^{ - 1} {\delta ^{ - 1 - t} \eta ^{T} (t)Q\eta (t)} } + \sum_{m = 1}^{ + \infty } {\mu _{m} } \sum_{l = - m} ^{ - 1} {\delta ^{ - 1 - l} \eta ^{T} (l)Z\eta (l)} \\ < & \lambda _{2} \eta ^{T} (0)R\eta (0) + \lambda _{3} \delta ^{d_{M} - 1} \sum_{s = - d_{M} }^{ - 1} {\eta ^{T} (s)R\eta (s)} \\ &{}+ \lambda _{3} \delta ^{d_{M} + 2} \sum _{s = - d_{M} + 2} ^{ - d_{m} + 1} {\sum_{t = - 1 + s}^{ - 1} {\eta ^{T} (t)R \eta (t)} } \\ &{}+ \lambda _{4} \sum_{m = 1}^{ + \infty } {\mu _{m} } \sum_{l = - m}^{ - 1} {\delta ^{ - 1 - l} \eta ^{T} (l)R \eta (l)}. \end{aligned}$$

Since

$$\begin{aligned} &\sum_{m = 1}^{ + \infty } {\mu _{m} } \sum_{l = - m} ^{ - 1} {\delta ^{ - 1 - l} \eta ^{T} (l)R\eta (l)} \\ &\quad = \mu _{1} \eta ^{T} ( - 1)R\eta ( - 1) + \mu _{2} \bigl[\delta \eta ^{T} ( - 2)R\eta ( - 2) + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] \\ &\qquad {}+ \mu _{3} \bigl[\delta ^{2} \eta ^{T} ( - 3)R\eta ( - 3) + \delta \eta ^{T} ( - 2)R\eta ( - 2) + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] \\ &\qquad {}+ \cdots + \mu _{m} \bigl[\delta ^{m - 1} \eta ^{T} ( - m)R\eta ( - m) + \delta ^{m - 2} \eta ^{T} ( - m + 1)R\eta ( - m + 1) \\ &\qquad {}+ \cdots + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] + \cdots \\ &\quad = \mu _{1} \eta ^{T} ( - 1)R\eta ( - 1) + \mu _{2} \bigl[\delta \eta ^{T} ( - 2)R\eta ( - 2) + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] \\ &\qquad {}+ \mu _{3} \bigl[\delta ^{2} \eta ^{T} ( - 3)R\eta ( - 3) + \delta \eta ^{T} ( - 2)R\eta ( - 2) + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] \\ &\qquad {}+\cdots + \mu _{d_{M} } \bigl[\delta ^{d_{M} - 1} \eta ^{T} ( - d_{M} )R \eta ( - d_{M} ) + \delta ^{d_{M} - 2} \eta ^{T} ( - d_{M} + 1)R\eta ( - d_{M} + 1) \\ &\qquad {}+ \cdots + \eta ^{T} ( - 1)R\eta ( - 1) \bigr] \\ &\quad \le c_{1} \bigl[\mu _{1} + (1 + \delta )\mu _{2} + \cdots + \bigl(1 + \delta + \cdots + \delta ^{d_{M} - 1} \bigr)\mu _{d_{M}} \bigr] \\ &\quad \triangleq c_{1} \sigma , \end{aligned}$$

we get

$$ \begin{aligned}[b] {\mathbb{E}} \bigl\{ V(0) \bigr\} &< \biggl[\lambda _{2} + \lambda _{3} d_{M} \delta ^{d_{M} - 1} + \lambda _{3} \frac{{(d_{M} - d_{m} )(d_{M} + d _{m} - 1)}}{2}\delta ^{d_{M} - 2} + \lambda _{4} \sigma \biggr]c_{1} \\ &\triangleq \varsigma _{1}. \end{aligned} $$

(29)

Taking (28) and (29) into account, we have

$$ {\mathbb{E}} \bigl\{ V(k) \bigr\} < \delta ^{N} \varsigma _{1} + \frac{\gamma }{ \delta }\bar{\omega }, \quad \forall k \in Z_{+}. $$

(30)

On the other hand, we can deduce that

$$ {\mathbb{E}} \bigl\{ V(k) \bigr\} \geqslant \eta ^{T} (k)P\eta (k) > \lambda _{1} \eta ^{T} (k)R\eta (k),\quad \forall k \in Z_{+}. $$

(31)

Consequently, from (30) and (31) it is easy to get that

$$ \eta ^{T} (k)R\eta (k) < \frac{1}{{\lambda _{1} \delta }} \bigl(\delta ^{N + 1} \varsigma _{1} + \gamma \bar{\omega } \bigr). $$

(32)

Note that

$$\begin{array}{rl}& {\delta }^{N+1}{\varsigma }_1+\gamma \overline{\omega }-c_2\delta {\lambda }_1<0\\ & \iff \gamma \overline{\omega }-c_2\delta {\lambda }_1+c_1{\delta }^{N+1}{\lambda }_2+c_1{\rho }_1{\lambda }_3+c_1{\rho }_2{\lambda }_4<0\\ & \iff \gamma \overline{\omega }-c_2\delta {\lambda }_1+\left[\begin{array}{ccc}{c}_1{\delta }^{N+1}{\lambda }_2& {\rho }_1{\lambda }_3& {\rho }_2{\lambda }_4\end{array}\right]\Gamma {\left[\begin{array}{ccc}{c}_1{\delta }^{N+1}{\lambda }_2& {\rho }_1{\lambda }_3& {\rho }_2{\lambda }_4\end{array}\right]}^T<0,\end{array}$$

(33)

where

$$\Gamma ={\left[\begin{array}{ccc}{c}_1{\delta }^{N+1}{\lambda }_2& 0& 0\\ 0& {\rho }_1{\lambda }_3& 0\\ 0& 0& {\rho }_2{\lambda }_4\end{array}\right]}^{-1}.$$

By Lemma 1 inequality (33) is equivalent to LMI (15), so we have

$$ \eta ^{T} (k)R\eta (k) < \frac{1}{{\lambda _{1} \delta }} \bigl(\delta ^{N + 1} \varsigma _{1} + \gamma \bar{\omega } \bigr) < c_{2}. $$

By Definition 1 system (9) is SFTB with respect to $(c_{1}, c_{2}, \bar{ \omega }, R, N)$, which completes the proof of Theorem 1. □

3.2 Stochastic finite-time $H_{\infty }$ boundedness analysis

In this subsection, we provide sufficient conditions guaranteeing the stochastic finite-time $H_{\infty }$ boundedness of system (9).

Theorem 2

For given scalar$\delta \ge 1$and symmetric matrix$R>0$, system (9) is SFT$H_{\infty }$B with respect to$(c_{1}, c_{2}, \bar{\omega }, \gamma , R, N)$if there exist positive scalars$\lambda _{i}$ ($i = 1,2,3,4$) and$\xi _{i}$ ($i = 1,2, \ldots , q$) and symmetric matrices$P>0$, $Q>0$, and$Z>0$such that (16), (17), and the following inequalities hold:

$$\left[\begin{array}{cc}-{\xi }_i& {\rho }_i^T{\overline{G}}^T\\ \ast & -P^{-1}\end{array}\right]<0,i=1,\dots ,q,$$

(34)

$$\begin{aligned}& \begin{bmatrix} \varPhi & 0 & 0 & 0 & {\bar{A}^{T} P} & {\tilde{\alpha }\tilde{A}^{T} P} & {\bar{E}_{1}^{T} } \\ * & { - \delta ^{d_{m} } Q + \sum_{i = 1}^{q} {\xi _{i} \bar{A} _{di} } } & 0 & 0 & {\bar{A}_{d}^{T} P} & 0 & 0 \\ * & * & { - \frac{1}{{\bar{\mu }}}Z + \sum_{i = 1}^{q} {\xi _{i} \bar{A}_{li} } } & 0 & {\bar{A}_{l}^{T} P} & 0 & 0 \\ * & * & * & { - \frac{\gamma }{{\delta ^{N} }}I} & {\bar{D}^{T} P} & 0 & {E_{2}^{T} } \\ * & * & * & * & { - P} & 0 & 0 \\ * & * & * & * & * & { - P} & 0 \\ * & * & * & * & * & * & { - I} \end{bmatrix}< 0, \end{aligned}$$

(35)

with the remaining parameters as in Theorem 1.

Proof

By Lemma 1 and Theorem 1 conditions (34) and (35) can ensure that system (9) is SFTB with respect to $(c_{1}, c_{2}, \bar{ \omega }, R, N)$. In the following, our objective is to prove inequality (10). By using the same Lyapunov–Krasovskii functional as in Theorem 1 from (25) it follows that

$$ \begin{aligned} &{\mathbb{E}} \bigl\{ {V(k + 1)- \delta V(k)} \bigr\} \\ &\quad < \xi ^{T} (k)\tilde{\varPi } \xi (k) + \frac{\gamma }{{\delta ^{N} }} \omega ^{T} (k)\omega (k)+{\mathbb{E}} \bigl\{ {z^{T} (k)z(k)} \bigr\} - {\mathbb{E}} \bigl\{ {z^{T} (k)z(k)} \bigr\} , \quad \forall k \in Z _{+}. \end{aligned} $$

Then

$$ {\mathbb{E}} \bigl\{ {V(k + 1) - \delta V(k)} \bigr\} < \xi ^{T} (k) \tilde{\varPi } \xi (k) + \frac{\gamma }{{\delta ^{N} }}\omega ^{T} (k) \omega (k) - {\mathbb{E}} \bigl\{ {z^{T} (k)z(k)} \bigr\} , \quad \forall k \in Z_{+}, $$

where

$$\tilde{\Pi }=\left[\begin{array}{cccc}{\Pi }_{11}+{\overline{E}}_1^T{\overline{E}}_1& {\overline{A}}^{T}P{\overline{A}}_d& {\overline{A}}^{T}P{\overline{A}}_l& {\overline{A}}^{T}P\overline{D}+{\overline{E}}_1^T{E}_2\\ \ast & {\Pi }_{22}& {\overline{A}}_d^{T}P{\overline{A}}_l& {\overline{A}}_d^{T}P\overline{D}\\ \ast & \ast & {\Pi }_{33}& {\overline{A}}_l^{T}P\overline{D}\\ \ast & \ast & \ast & {\overline{D}}^{T}P\overline{D}+E_2^T{E}_2-\frac{\gamma }{{\delta }^N}I\end{array}\right].$$

From condition (35) we have

$$ {\mathbb{E}} \bigl\{ V(k + 1) \bigr\} < \delta {\mathbb{E}} \bigl\{ V(k) \bigr\} + \frac{\gamma }{{\delta ^{N} }} {\omega ^{T} (k)\omega (k)}-{\mathbb{E}} \bigl\{ {z^{T} (k)z(k)} \bigr\} , \quad \forall k \in Z_{+}. $$

Thus

$$ 0 \le {\mathbb{E}} \bigl\{ V(k) \bigr\} < \delta ^{k} {\mathbb{E}} \bigl\{ V(0) \bigr\} + \sum _{s = 0}^{k - 1} { \delta ^{k - 1 - s} \biggl[ { \frac{ \gamma }{{\delta ^{N} }}\omega ^{T} (s)\omega (s)- {\mathbb{E}} \bigl\{ z^{T} (s) z(s) \bigr\} } \biggr]}. $$

(36)

Therefore, under the zero initial condition, (36) implies

$$ \sum_{s = 0}^{k - 1} {\delta ^{k - 1 - s} {\mathbb{E}} \bigl\{ z^{T} (s)z(s) \bigr\} }< \sum _{s = 0}^{k - 1} \biggl[\delta ^{k - 1 - s} \frac{ \gamma }{{\delta ^{N} }}\omega ^{T} (s)\omega (s) \biggr]. $$

Letting $k = N + 1$, we have

$$ \sum_{s = 0}^{N} { \delta ^{N - s} {\mathbb{E}} \bigl\{ z^{T} (s) z(s) \bigr\} } < \sum_{s = 0}^{N} {\delta ^{N - s} \biggl[\frac{\gamma }{ {\delta ^{N} }}\omega ^{T} (s) \omega (s) \biggr] }. $$

(37)

Note that $1 \le \delta ^{N - s} \le \delta ^{N}$ for $s \in \{ {0,1, \ldots, N} \}$, which, together with (37), yields

$$ \sum_{s = 0}^{N} {{\mathbb{E}} \bigl\{ z^{T} (s)z(s) \bigr\} }< \gamma \sum_{s = 0}^{N} \omega ^{T} (s)\omega (s). $$

This completes the proof. □

3.3 Stochastic finite-time output feedback control

In this subsection, we give the design procedure of the controller gains.

Theorem 3

For given scalar$\delta \ge 1$and symmetric matrix$R>0$, system (1) is SFT$H_{\infty }$B with respect to$(c_{1}, c_{2}, \bar{\omega }, \gamma , R, N)$if there exist positive scalars$\lambda _{i} ^{\prime }$ ($i = 1,2,3,4$) and$\xi _{i}$ ($i = 1,2, \ldots , q$), symmetric matrices$S>0$, $T>0$, $\hat{Q}_{1}>0$, $\hat{Q}_{3}>0$, $\hat{Z}_{1}>0$, and$\hat{Z}_{3}>0$, nonsingular matrices$Y_{12}$and$Y_{22}$, and real matricesÂ, B̂, Ĉ, $\hat{Q}_{2}$, and$\hat{Z}_{2}$such that the following inequalities hold:

$$\left[\begin{array}{ccc}-{\xi }_i& {\rho }_i^T& 0\\ \ast & -S& -Y_{12}\\ \ast & \ast & -Y_{22}\end{array}\right]<0,i=1,\dots ,q,$$

(38)

$$\left[\begin{array}{cc}{\Omega }_{11}& {\Omega }_{12}\\ \ast & {\Omega }_{22}\end{array}\right]<0,$$

(39)

$$\begin{aligned}& \delta ^{N + 1} \varsigma _{2} + \gamma \bar{\omega } - c_{2} \delta \lambda _{1} ^{\prime }< 0, \end{aligned}$$

(40)

$$\begin{aligned}& \lambda _{1} ^{\prime }R_{1} \le S \le \lambda _{2} ^{\prime }R_{1},\qquad \hat{Q}_{3} \le \lambda _{3} ^{\prime }R_{1},\qquad \hat{Z}_{3} \le \lambda _{4} ^{\prime }R_{1}. \end{aligned}$$

(41)

Furthermore, the corresponding gain matrices of the controller can be given by

$$ A_{K} = X_{12}^{ - 1} T \hat{A}S \bigl(X_{12} Y_{12}^{T} \bigr)^{ - 1} X_{12},\qquad B_{K} = X_{12}^{ - 1} T \hat{B},\qquad C_{K} = \hat{C}S \bigl(X_{12} Y_{12}^{T} \bigr)^{ - 1} X_{12}, $$

where

$$\begin{array}{c}{\varsigma }_2=[{\lambda }_2^{\prime }+{\lambda }_3^{\prime }{d}_M{\mu }^{d_M-1}+\frac{1}{2}{\lambda }_3^{\prime }({\tau }_1-1){\tau }_2{\mu }^{d_M-2}+{\lambda }_4^{\prime }\sigma ]c_1,\hfill \\ {\Omega }_{11}=diag\{{\Omega }_{11}^{\prime },{\Omega }_{22}^{\prime },{\Omega }_{33}^{\prime },{\Omega }_{44}^{\prime }\},\hfill \\ {\Omega }_{12}=\left[\begin{array}{ccccc}{A}^T+{\hat{C}}^T{D}^T& A^T+{\hat{C}}^T{D}^T+\overline{\alpha }{F}_1^T{\hat{B}}^T+\hat{A}& 0& \tilde{\alpha }{F}_1^T{\hat{B}}^T& E_1^T\\ A^T& A^T+\overline{\alpha }{F}_1^T{\hat{B}}^T& 0& \tilde{\alpha }{F}_1^T{\hat{B}}^T& E_1^T\\ A_d^T& A_d^T& 0& 0& 0\\ A_d^T& A_d^T& 0& 0& 0\\ A_l^T& A_l^T& 0& 0& 0\\ A_l^T& A_l^T& 0& 0& 0\\ D^T& D^T+F_2^T{\hat{B}}^T& 0& 0& E_2^T\end{array}\right],\hfill \\ {\Omega }_{22}=\left[\begin{array}{ccccc}-S& -T^{-1}& 0& 0& 0\\ \ast & -T^{-1}& 0& 0& 0\\ \ast & \ast & -S& -T^{-1}& 0\\ \ast & \ast & \ast & -T^{-1}& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right],\hfill \\ {\Omega }_{11}^{\prime }=\left[\begin{array}{cc}-\delta S^{-1}+\tau {\hat{Q}}_1+\overline{\mu }{\hat{Z}}_1+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_i& -\delta S^{-1}+\tau {\hat{Q}}_2^T+\overline{\mu }{\hat{Z}}_2^T+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_i\\ \ast & -\delta T+\tau {\hat{Q}}_3+\overline{\mu }{\hat{Z}}_3+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_i\end{array}\right],\hfill \\ {\Omega }_{22}^{\prime }=\left[\begin{array}{cc}-{\delta }^{d_m}{\hat{Q}}_1+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{di}& -{\delta }^{d_m}{\hat{Q}}_2^T+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{di}\\ \ast & -{\delta }^{d_m}{\hat{Q}}_3^T+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{di}\end{array}\right],\hfill \\ {\Omega }_{33}^{\prime }=\left[\begin{array}{cc}-\frac{1}{\overline{\mu }}{\hat{Z}}_1+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{li}& -\frac{1}{\overline{\mu }}{\hat{Z}}_2^T+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{li}\\ \ast & -\frac{1}{\overline{\mu }}{\hat{Z}}_3+{\sum }_{i=1}^q{\xi }_i{\overline{A}}_{li}\end{array}\right],{\Omega }_{44}^{\prime }=-\frac{\gamma }{{\delta }^N}I,\hfill \end{array}$$

$X_{12}$is a nonsingular matrix satisfying$X_{12}Y_{12}^{T}=I-TS$, and the remaining parameters are as in Theorem 1.

Proof

Firstly, we define P, Q, Z, and R in Theorem 2 as follows:

$$\begin{array}{c}P=\left[\begin{array}{cc}T& X_{12}\\ X_{12}^T& X_{22}\end{array}\right],P^{-1}=\left[\begin{array}{cc}S& Y_{12}\\ Y_{12}^T& Y_{22}\end{array}\right],Q=\left[\begin{array}{cc}{Q}_1& Q_2\\ Q_2^T& Q_3\end{array}\right],\hfill \\ Z=\left[\begin{array}{cc}{Z}_1& Z_2\\ Z_2^T& Z_3\end{array}\right],R=\left[\begin{array}{cc}{R}_1& 0\\ 0& R_2\end{array}\right],T_1=\left[\begin{array}{cc}S& I\\ Y_{12}^T& 0\end{array}\right],T_2=\left[\begin{array}{cc}I& T\\ 0& X_{12}^T\end{array}\right],\hfill \end{array}$$

where the partitioning of P, $P^{-1}$, Q, and Z is compatible with that of Ā, Ã, $\bar{A}_{d}$, and $\bar{A}_{l}$, which implies that

$$\begin{aligned}& PT_{1} = T_{2}, \end{aligned}$$

(42)

$$\begin{array}{rl}{T}_1^{T}Q{T}_1& =\left[\begin{array}{cc}S{Q}_{1}S+S{Q}_2{Y}_{12}^T+Y_{12}{Q}_2^{T}S+Y_{12}{Q}_3{Y}_{12}^T& S{Q}_1+Y_{12}{Q}_2^T\\ \ast & Q_1\end{array}\right]\\ & =\left[\begin{array}{cc}S{\hat{Q}}_{1}S& S{\hat{Q}}_2^T\\ \ast & {\hat{Q}}_3\end{array}\right],\end{array}$$

(43)

$$T_1^{T}Z{T}_1=\left[\begin{array}{cc}S{\hat{Z}}_{1}S& S{\hat{Z}}_2^T\\ \ast & {\hat{Z}}_3,\end{array}\right],$$

(44)

where

$$\begin{aligned}& \hat{Q}_{1}= Q_{1}+ Q_{2} Y_{12}^{T} S^{ - 1}+ S^{ - 1} Y_{12} Q_{2}^{T}+ S^{ - 1} Y_{12} Q_{3} Y_{12}^{T} S^{ - 1}, \\& \hat{Q}_{2}= Q_{1}+ Q_{2} Y_{12}^{T} S^{- 1}, \qquad \hat{Q}_{3} = Q_{1}, \\& \hat{Z}_{1} = Z_{1} + Z_{2} Y_{12}^{T} S^{ - 1} + S^{ - 1} Y_{12}Z _{2}^{T} + S^{ - 1} Y_{12} Z_{3} Y_{12}^{T} S^{ - 1}, \\& \hat{Z}_{2} = Z_{1} + Z_{2} Y_{12}^{T} S^{ - 1},\qquad \hat{Z}_{3}= Z_{1}. \end{aligned}$$

It is easy to see that (38) and (34) are equivalent. To obtain (39), pre- and postmultiplying (35) by $\psi ^{T}$ and ψ, we obtain

$$\left[\begin{array}{cc}{\hat{\Pi }}_{11}& {\hat{\Pi }}_{12}\\ \ast & {\hat{\Pi }}_{22}\end{array}\right]<0,$$

(45)

where

$$\begin{array}{c}\psi =diag\{T_1,T_1,T_1,I,T_1,T_1,I\},{\hat{\Pi }}_{11}=diag\{{\hat{\Pi }}_{11}^{\prime },{\hat{\Pi }}_{22}^{\prime },{\hat{\Pi }}_{33}^{\prime },{\hat{\Pi }}_{44}^{\prime }\},\hfill \\ {\hat{\Pi }}_{12}=\left[\begin{array}{ccccc}S{A}^T+Y_{12}{C}_K^T{D}^T& {\mathrm{\Delta }}_1& 0& \tilde{\alpha }S{F}_1^T{B}_K^T{X}_{12}^T& S{E}_1^T\\ A^T& A^{T}T+\overline{\alpha }{F}_1^T{B}_K^T{X}_{12}^T& 0& \tilde{\alpha }{F}_1^T{B}_K^T{X}_{12}^T& E_1^T\\ S{A}_d^T& S{A}_d^{T}T& 0& 0& 0\\ A_d^T& A_d^{T}T& 0& 0& 0\\ S{A}_l^T& S{A}_l^{T}T& 0& 0& 0\\ A_l^T& A_l^{T}T& 0& 0& 0\\ D^T& D^{T}T+F_2^T{B}_K^T{X}_{12}^T& 0& 0& E_2^T\end{array}\right],\hfill \\ {\hat{\Pi }}_{22}=\left[\begin{array}{ccccc}-S& -I& 0& 0& 0\\ \ast & -T& 0& 0& 0\\ \ast & \ast & -S& -I& 0\\ \ast & \ast & \ast & -T& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right],\hfill \\ {\hat{\Pi }}_{11}^{\prime }=\left[\begin{array}{cc}-\delta S+\tau S{\hat{Q}}_{1}S+\overline{\mu }S{\hat{Z}}_{1}S+{\sum }_{i=1}^q{\xi }_{i}S{A}_{i}S& -\delta I+\tau S{\hat{Q}}_2^T+\overline{\mu }S{\hat{Z}}_2^T+{\sum }_{i=1}^q{\xi }_{i}S{A}_i\\ \ast & -\delta T+\tau {\hat{Q}}_3+\overline{\mu }{\hat{Z}}_3+{\sum }_{i=1}^q{\xi }_i{A}_i\end{array}\right],\hfill \\ {\hat{\Pi }}_{22}^{\prime }=\left[\begin{array}{cc}-{\delta }^{d_m}S{\hat{Q}}_{1}S+{\sum }_{i=1}^q{\xi }_{i}S{A}_{di}S& -{\delta }^{d_m}S{\hat{Q}}_2^T+{\sum }_{i=1}^q{\xi }_{i}S{A}_{di}\\ \ast & -{\delta }^{d_m}{\hat{Q}}_3^T+{\sum }_{i=1}^q{\xi }_i{A}_{di}\end{array}\right],\hfill \\ {\hat{\Pi }}_{33}^{\prime }=\left[\begin{array}{cc}-\frac{1}{\overline{\mu }}S{\hat{Z}}_{1}S+{\sum }_{i=1}^q{\xi }_{i}S{A}_{li}S& -\frac{1}{\overline{\mu }}S{\hat{Z}}_2^T+{\sum }_{i=1}^q{\xi }_{i}S{A}_{li}\\ \ast & -\frac{1}{\overline{\mu }}{\hat{Z}}_3+{\sum }_{i=1}^q{\xi }_i{A}_{li}\end{array}\right],\hfill \\ {\hat{\Pi }}_{44}^{\prime }={\Omega }_{44}^{\prime },{\mathrm{\Delta }}_1=S{A}^{T}T+Y_{12}{C}_K^T{D}^{T}T+\overline{\alpha }S{F}_1^T{B}_K^T{X}_{12}^T+Y_{12}{A}_K^T{X}_{12}^T.\hfill \end{array}$$

Pre- and postmultiplying (45) by $\operatorname{diag} \{ S^{ - 1},I,S^{ - 1},I,S ^{- 1},I,I,I,T^{ - 1},I,T^{ - 1},I\}$, we can conclude that

$$\left[\begin{array}{cc}{\check{\Pi }}_{11}& {\check{\Pi }}_{12}\\ \ast & {\check{\Pi }}_{22}\end{array}\right]<0,$$

(46)

where

$$\begin{array}{c}{\check{\Pi }}_{11}={\Omega }_{11},\hfill \\ {\check{\Pi }}_{12}=\left[\begin{array}{ccccc}{A}^T+S^{-1}{Y}_{12}{C}_K^T{D}^T& {\mathrm{\Delta }}_2& 0& \tilde{\alpha }{F}_1^T{B}_K^T{X}_{12}^T{T}^{-1}& E_1^T\\ A^T& A^T+\overline{\alpha }{F}_1^T{B}_K^T{X}_{12}^T{T}^{-1}& 0& \tilde{\alpha }{F}_1^T{B}_K^T{X}_{12}^T{T}^{-1}& E_1^T\\ A_d^T& A_d^T& 0& 0& 0\\ A_d^T& A_d^T& 0& 0& 0\\ A_l^T& A_l^T& 0& 0& 0\\ A_l^T& A_l^T& 0& 0& 0\\ D^T& D^T+F_2^T{B}_K^T{X}_{12}^T{T}^{-1}& 0& 0& E_2^T\end{array}\right],\hfill \\ {\check{\Pi }}_{22}={\Omega }_{22},\hfill \\ {\mathrm{\Delta }}_2=A^T+S^{-1}{Y}_{12}{C}_K^T{D}^T+\overline{\alpha }{F}_1^T{B}_K^T{X}_{12}^T{T}^{-1}+S^{-1}{Y}_{12}{A}_K^T{X}_{12}^T{T}^{-1}.\hfill \end{array}$$

Letting $\hat{A} = T^{ - 1} X_{12} A_{K} Y_{12}^{T} S^{ - 1}$, $\hat{B} = T^{ - 1} X_{12} B_{K}$, and $\hat{C} = C_{K} Y_{12}^{T} S ^{ - 1}$, we get inequality (39).

Since P is a positive definite matrix, we have

$$T_1^{T}P{T}_1=\left[\begin{array}{cc}S& I\\ I& T\end{array}\right]>0\text{or}TS-I>0.$$

(47)

From (47) it follows that $I - TS = X_{12} Y_{12}^{T} < 0$. So there exist two nonsingular matrices $X_{12}$ and $Y_{12}$ such that $X_{12} Y_{12}^{T} < 0$. This completes the proof. □

Remark 2

So far, we have investigated the problem of the stochastic finite-time $H_{\infty }$ bounded control for a class of NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement. Also, some sufficient conditions reflecting the impacts from the involved phenomena are given. Some criterion of the error analysis of the proposed theoretical result will be obtained in the near future if the error analysis becomes necessary.

Remark 3

In Theorem 3, sufficient conditions guaranteeing the stochastic finite-time boundedness of the system are obtained. It is easy to see that the obtained results of this paper are influenced by all system parameters, such as the bounds on the delay. Thus the results of this paper are more general and less conservative than the existing results. In addition, we also give sufficient conditions for the existence of the output feedback controller in Theorem 3. However, this condition is not expressed in the form of LMI, and we cannot directly use the standard LMI toolbox to obtain it. Based on the literature [36], we present the following controller design method.

Let us denote $U = S^{ - 1}$ and $V = T^{ - 1}$. Then (41) can be rewritten

$$\left[\begin{array}{cc}{\overline{\Omega }}_{11}& {\overline{\Omega }}_{12}\\ \ast & {\overline{\Omega }}_{22}\end{array}\right]<0,$$

(48)

where

$$\begin{array}{c}{\overline{\Omega }}_{11}=diag\{{\overline{\Omega }}_{11}^{\prime },{\overline{\Omega }}_{22}^{\prime },{\overline{\Omega }}_{33}^{\prime },{\overline{\Omega }}_{44}^{\prime }\},{\overline{\Omega }}_{12}={\Omega }_{12},\hfill \\ {\overline{\Omega }}_{22}=\left[\begin{array}{ccccc}-S& -V& 0& 0& 0\\ \ast & -V& 0& 0& 0\\ \ast & \ast & -S& -V& 0\\ \ast & \ast & \ast & -V& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right],\hfill \\ {\overline{\Omega }}_{11}^{\prime }=\left[\begin{array}{cc}-\delta U+\tau {\hat{Q}}_1+\overline{\mu }{\hat{Z}}_1+{\sum }_{i=1}^q{\xi }_i{A}_i& -\delta U+\tau {\hat{Q}}_2^T+\overline{\mu }{\hat{Z}}_2+{\sum }_{i=1}^q{\xi }_i{A}_i\\ \ast & -\delta T+\tau {\hat{Q}}_3+\overline{\mu }{\hat{Z}}_3+{\sum }_{i=1}^q{\xi }_i{A}_i\end{array}\right],\hfill \\ {\overline{\Omega }}_{22}^{\prime }={\Omega }_{22}^{\prime },{\overline{\Omega }}_{33}^{\prime }={\Omega }_{33}^{\prime },{\overline{\Omega }}_{44}^{\prime }={\Omega }_{44}^{\prime }.\hfill \end{array}$$

Consequently, we can transform the original nonconvex problem into the following minimization problem involving LMI conditions, which can be solved by CCLM:

$$\begin{array}{c}\underset{V>0,S>0,T>0}{min}tr\{US+VT\}\hfill \\ \text{s.t. (48), }\left[\begin{array}{cc}U& I\\ I& S\end{array}\right]>0,\left[\begin{array}{cc}V& I\\ I& T\end{array}\right]>0.\hfill \end{array}$$

Remark 4

For given scalars $\delta \geq 1$, γ, $c_{1}$, $c_{2}$, ω̄, N and symmetric positive definite matrix R, we can get the following algorithm for calculating the output feedback controller by CCLM [36].

• Step 1. Given the maximum iteration N̄ and fixed scalars $\delta \geq 1$, γ, $c_{1}$, ω̄, N.

• Step 2. Determine an initial value of $c_{2}$.

• Step 3. Find a feasible solution $S_{0}$, $U_{0}$, $T_{0}$, $V_{0}$ satisfying (49) and let $S_{k} = S_{0}$, $U_{k} = U_{0}$, $T_{k} = T _{0}$, $V_{k} = V_{0}$. If there are none, then exit. Set $k=0$.

• Step 4. If $\vert {\operatorname{tr} (U_{k + 1} S _{k} + S_{k + 1} U_{k} + \phi _{k + 1} R_{k} + \phi _{k} R_{k + 1} ) - 4n} \vert < c_{2}$, then exit. Otherwise, set $k=k+1$ and go to Step 2.

• Step 5. If $k > \bar{N}$, then stop.

• Step 6. If problem (49) is unfeasible, then the initial value for $c_{2}$ need to be increased. Otherwise, we decrease $c_{2}$ until we get its minimum value.

  $$ \begin{aligned}[b] &\min_{\substack{V > 0,S > 0,T > 0,Y_{12} > 0, \\ Y_{22} > 0,\hat{Z}_{1} ,\hat{Z}_{2} ,\hat{Z}_{3} ,\hat{Q}_{1} ,\hat{Q}_{2} ,\hat{Q}_{3}}} \operatorname{tr} ( US_{k} + ST_{k} + VT_{k} + TV_{k} ) + \gamma \\ &\quad \mbox{s.t. (38), (40), (41), and (48)}. \end{aligned} $$

  (49)

  where γ is the disturbance attenuation, and $c_{2}$ is the prescribed bound of ${\mathbb{E}}\{x^{T}R_{1}x\}$.

4 Numerical examples

In this section, we give two numerical examples to show the validity of the proposed theoretical results for stochastic nonlinear NCSs (1) with mixed delays and randomly missing measurement.

Example 1

To demonstrate that our methods are better than some existing results, we consider the same example as in [29] with the following system parameters:

$$\begin{array}{c}A=\left[\begin{array}{cc}0.4& 0.1\\ 0.3& 0.5\end{array}\right],A_d=\left[\begin{array}{cc}0.2& -0.15\\ 0.15& 0.1\end{array}\right],B=\left[\begin{array}{c}0.1\\ 0.2\end{array}\right],D=\left[\begin{array}{c}0.25\\ 0.3\end{array}\right],\hfill \\ R_1=\left[\begin{array}{cc}1.2& 0\\ 0& 1.3\end{array}\right],E_1=\left[\begin{array}{c}0.2\\ 0.3\end{array}\right],E_2=0.1,F_1=\left[\begin{array}{c}2.738\\ 2.287\end{array}\right],F_2=0.2.\hfill \end{array}$$

Without loss of generality, we assume that the noise $\omega (k)$ in system (1) is selected as $\omega (k)=\frac{1}{0.1+k^{2}}$, the time-varying communication delay satisfies $1 \le d(k) \le 12$, the initial states of NCSs and the estimators are $x(0) = [ {0.5}\ { - 0.2}]^{T}$ and $\hat{x}(0) = [0 \ 0 ]^{T}$, respectively, $N=40$, $\bar{\omega }=2$, $c_{1}=1$, $\delta =1.00027$, and $f(k)=0$. We easily check that the initial condition ${\mathbb{E}}\{x^{T}(0)R_{1}x(0)\}\leq c_{1}$ is satisfied.

Firstly, we let $A_{l}=0$. By applying the method in this paper and the method in [29], we can obtain the minimum values of γ and $c_{2}$ for different values of $d_{M}$ in Table 1. From Table 1 we can obviously observe that the obtained minimum values of this paper are smaller than that in [29]. Since $c_{2}$ is the prescribed bound of ${\mathbb{E}} \{x^{T}R_{1}x\}$, the smaller $c_{2}$ is, the better the state trajectory of the system converges. Since γ is the level of disturbance attenuation, the smaller the γ, the better the system performance. This shows that the obtained results of this paper are better than that in [29].

Table 1 Optimal values for different time-delay upper bound $d_{M}$ when $\bar{\mu }=0$

In the following, we choose $A_l=\left[\begin{array}{cc}0.2& 0\\ 0.3& 0.4\end{array}\right]$. The constant sequence $\{\mu _{m}\}\in [0,+\infty )$ is chosen as $\mu _{m}=3^{-(3+m)}$, and the stochastic nonlinear function is selected as follows:

$$\begin{array}{rcl}f(k)& =& \left[\begin{array}{c}0.3\\ 0.3\end{array}\right][0.2x_1(k){\nu }_1(k)+0.2x_2(k){\nu }_2(k)\\ & & +0.1x_1(k-d(k)){\xi }_1(k)+0.2x_2(k-d(k)){\xi }_2(k)],\end{array}$$

where $x_{i} (k)$ ($i=1,2$) stand for the ith element of the system state, $\nu _{i} (k)$ and $\xi _{i} (k)$ ($i = 1,2$) represent the mutually uncorrelated Gaussian white noise sequences with unity variances:

$$ {\mathbb{E}} \bigl\{ {\nu _{i} (k)} \bigr\} = 0, \qquad { \mathbb{E}} \bigl\{ {\nu _{i} (k)}^{2} \bigr\} = 1, \qquad {\mathbb{E}} \bigl\{ {\xi _{i} (k)} \bigr\} = 0,\qquad { \mathbb{E}} \bigl\{ {\xi _{i} (k)} ^{2} \bigr\} = 1. $$

For different upper bounds of discrete delay and distributed delays, the optimal values of γ and $c_{2}$ are listed in Table 2. From Table 2 we can easily find that as upper bounds of discrete delay and distributed delays increase, the optimal values of γ and $c_{2}$ get bigger. Thus, when upper bounds of discrete delay and distributed delays increase, the system performance becomes worse, which means that the mixed delays have important influence on the system performance. Thus the results proposed in this paper are less conservative than those obtained in [29].

Table 2 Optimal values for different upper bound $d_{M}$ and μ̄

Example 2

In the following, we consider system (1) with following system parameters:

$$\begin{array}{c}A=\left[\begin{array}{ccc}-0.9226& -0.4330& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],A_d=\left[\begin{array}{ccc}0.2& 0& 0\\ 0& -0.3& 0.4\\ 0& 0& 0.4\end{array}\right],\hfill \\ A_l=\left[\begin{array}{ccc}-0.1& 0.2& 0\\ 0& 0.3& 0\\ 0.2& 0.1& 0.4\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],D=\left[\begin{array}{c}0.5\\ 0\\ -1.01\end{array}\right],\hfill \\ R_1=\left[\begin{array}{ccc}1.2& 0& 0\\ 0& 1.3& 0\\ 0& 0& 1.4\end{array}\right],E_1={\left[\begin{array}{c}0\\ 0.2\\ 0.1\end{array}\right]}^T,E_2={\left[\begin{array}{c}2.738\\ 2.287\\ 0\end{array}\right]}^T,F_2=0.1,\hfill \\ F_2=0.2,N=60,d(k)=1+sin{(0.5k\pi )}^2,c_1=1,\delta =1.03.\hfill \end{array}$$

We consider the stochastic nonlinear function

$$\begin{array}{rcl}f(k)& =& \left[\begin{array}{c}0.3\\ 0.3\\ 0.3\end{array}\right][0.2x_1(k){\nu }_1(k)+0.2x_2(k){\nu }_2(k)+0.2x_3(k){\nu }_3(k)+0.1x_1(k-d(k)){\xi }_1(k)\\ & & +0.2x_2(k-d(k)){\xi }_2(k)+0.1x_3(k-d(k)){\xi }_3(k)].\end{array}$$

We easily obtain that

$$\begin{array}{rcl}\mathbb{E}[f(k)f^T(k)]& =& \left[\begin{array}{c}0.3\\ 0.3\\ 0.3\end{array}\right]{\left[\begin{array}{c}0.3\\ 0.3\\ 0.3\end{array}\right]}^T[x^T(k)\left[\begin{array}{ccc}0.04& 0& 0\\ 0& 0.04& 0\\ 0& 0& 0.04\end{array}\right]x(k)\\ & & +x^T(k-d(k))\left[\begin{array}{ccc}0.01& 0& 0\\ 0& 0.04& 0\\ 0& 0& 0.01\end{array}\right]x(k-d(k))].\end{array}$$

The initial conditions are supposed to be $x(0)= [ -0.2 \ 0.5 \ 0.2]^{T}$, $\hat{x}(0) = [ 0 \ 0 \ 0]^{T}$. The external disturbance is chosen as $\omega (k)=\frac{1}{k ^{2}}$. We easily check that the initial condition ${\mathbb{E}}\{x ^{T}(0)R_{1}x(0)\}\leq c_{1}$ is satisfied. The corresponding lower and upper bounds of discrete delay are $d_{m}=1$ and $d_{M}=2$, respectively. Other parameters are chosen as those in Example 1.

Solving the minimization problem given in Remark 4 by using Matlab LMI toolbox, it follows that the optimal values of $\gamma _{\min }=3.0314$ and $c_{2\min }=1.5660$ can be obtained with the desired controller as

$$\begin{array}{c}{A}_K=\left[\begin{array}{ccc}-0.6051& 0.0572& 0.0045\\ 2.1737& -0.0815& -0.0052\\ -0.4089& -0.1916& -0.0020\end{array}\right],B_K=\left[\begin{array}{c}0.0947\\ -0.1708\\ -1.0978\end{array}\right],\hfill \\ C_K={\left[\begin{array}{c}-0.4020\\ -0.2370\\ -0.0035\end{array}\right]}^T.\hfill \end{array}$$

In addition, the simulation results of $x^{T}(k)R_{1}x(k)$ are shown in Fig. 1. Figure 1 depicts the trajectories of $x^{T}(k)R_{1}x(k)$ along nine individual experiments. From Fig. 1 we can conclude that the state trajectories of ${\mathbb{E}}\{x^{T}(k)R_{1}x(k)\}$ remain within the obtained optimal value $c_{2\min }=1.5660$ over the fixed interval despite facing mixed delays, stochastic nonlinearities, and randomly missing measurement, which means that the systems are both SFTB under nine individual experiments. Furthermore, by applying the method in [36] we define the following function $\gamma (k)$ to show the effect on the output energy $z(k)$ from the disturbance input energy $\omega (k)$:

$$ \gamma (k)=\frac{{\sum_{s = 0}^{k}{{\mathbb{E}}\{z^{T} (s)z(s) \}}}}{\sum_{s = 0}^{k} {\omega ^{T} (s) \omega (s)}},\quad k=1,\ldots ,N. $$

The corresponding curves of $\gamma (k)$ are plotted in Fig. 2. We can obtain that ${\mathbb{E}}\{\gamma (60)\}\approx \frac{1}{9} \sum_{i =1}^{9}[\gamma (60)]=0.4693<\gamma _{\min }=3.0314$. Thus the developed output feedback controller is a finite-time $H_{\infty }$ controller for the discrete-time NCSs with mixed delays, stochastic nonlinearities, and randomly missing measurement (1) according to Definition 2.

Figure 1

The responses of $x^{T}(k)R_{1}x(k)$

Figure 2

The responses of the function $\gamma (k)$ during the simulation

Remark 5

In [29, 32, 34] the problem of stochastic finite-time $H_{\infty }$ control for discrete time NCSs has been investigated. However, they did not consider mixed delays, stochastic nonlinearity, and randomly missing measurement. Thus our proposed results are more general than those given in [29, 32, 34].

5 Conclusions

In this paper, we investigated the problem of stochastic finite-time $H_{\infty }$ control for stochastic nonlinear NCSs with mixed delays and randomly missing measurement. Based on a novel Lyapunov–Krasovskii functional and stochastic analysis method, we provided the sufficient conditions for stochastic finite-time boundedness of discrete-time NCSs. Then we designed an $H_{\infty }$ output feedback controller ensuring that discrete-time NCS (1) is SFTB. Although the derived results are for a nonconvex feasibility problem, we can turn it into LMI feasibility problem by CCLM. In addition, we analyzed effects of mixed delays on dynamic performance of NCSs and the effective of the developed controller by two numerical examples. Note that the main results proposed in this paper can be extended to the stochastic finite-time filtering design problem. In addition, other factors can be considered in the finite-time $H_{\infty }$ control issues for discrete-time NCSs with mixed delays in the future. For example, we can consider actuator saturation [37], stochastic fading measurement [38], and event-triggered scheme [39].