Event-Triggered ${H}_{\infty}$ Control for Uncertain Markov Jump Systems With Nonlinear Input

This paper investigates the robust event-triggered control scheme for discrete-time Markov jump systems with nonlinear input. An event-based strategy is addressed to reduce the data communication and improve the control efficiency. A Lyaounov function method is utilized to analysis the mean square stability and sufficient conditions are obtained in the form of linear matrix inequalities. Convex combination approach is utilized to handle nonlinear input while parameter uncertainty and external disturbances are both included and the robustness ${H}_{\infty}$ performance is analyzed. Finally, a numerical example is shown to compare the different schemes and a mass spring damping system is given to illustrate the effectiveness of the provided method.

SECTION I. Introduction

Event trigger has become a hot topic in recent year as its outstanding performance in saving communication cost, such as in network control systems [1], network filtering and estimation [2] and consensus analysis of multi-agent systems [3]. From [4], event-based control as an aperiodic technique only works where it is required which is distinguish with the periodic sampling time control. In other words, the event-driven strategy can only transmit the information by satisfying a triggering condition (the error of the control state is greater than that of the threshold value) and then the control scheme is updated. In a network control system, challenges of network-induced delay and data dropping caused by bandwidth limitation is the main obstacle to achieve a good performance of the system. The most important reason is the transmission between sensors and actuators being sample-time-based data. Consequently, the event-based sampling scheme is proposed to reduce the burden of network communication and the computation complexity. With these advantages, the research on event-triggered control mechanism has theoretical value and practical significance.

In published literatures on event-triggered strategy, it has received tons of attention since 1980s [5]–​[7]. Furthermore, event-driven scheme has been used as a new control strategy for several decades. At the initial stage, [8] proposed an event-based sampling scheme to design a PID controller for avoiding the re-computation of the system. In [5], an event-based control strategy is addressed to evaluate the performance and determine the control input in a state-feedback system. For more complex conditions, a switching approach is induced to discover the waiting time in the event-driven strategy [9]. In [10], [11], event-based control problems have been solved for switched linear system by asynchronous control approach, such that leads to an major improvement in event-triggered control theory. By combining event-driven scheme, model predictive control strategy can achieve a satisfactory performance and reduce the computation load, such as in [12], [13]. Recently, adaptive event-triggered controller is designed for uncertain nonlinear systems [14]. Apparently, the event-triggered control scheme has not been fully studied for stochastic systems yet.

Markov jump system (MJS) regarded as a special kind of stochastic systems has been used to model the systems with abrupt changes of the environment and fault occurrence in an industrial process. Since the great practicability of MJLs, it has received lots of attention in the past a few decades, such that in robust control design [15], [16], state estimation [17]–​[19], filtering problem [20], [21] and optimal control scheme [22], etc. As we all know, these results are all in time-triggered manner, which leads to a waste of resources and decreases the efficiency of transmission. Event-triggered control scheme has become widely investigated for MJLs to reduce the communication burden, such as in [23], the measured outputs are transmitted to the estimator only when the triggered condition is satisfied by homogeneous Markov chain is considered. Moreover, event-triggered state estimation problem is also studied for semi-Markov jump systems [24]. In contrast with completely known transition probability, [25] has focused on the ${H}_{\infty}$ filtering problems for MJSs with unknown and uncertain transition probabilities when the transmission from sensor to filter is event-triggered. For more general cases, [26] has considered event-based scheme and network induced delays for discrete Markov jump system. Although these results improve the theoretical research on MJSs a lot, it is still lack of analysis on event-driven control scheme for discrete-time MJSs with parameter uncertainty and external disturbances. For the reason that the deterministic system is nonexist and external disturbance is an essential factor in practical processes, it is necessary to consider robustness for the system [27], [28], which leads to our first motivation in our study.

In another research of line, nonlinearity saturation may cause the unstable and poor performance of the system. In the most practical manufacture processes, actuator saturation is an inevitable issue owing to its limitation in a certain level. This restriction makes the linear input change to the nonlinear one as the value of the input is bounded after some linear operating, which becomes an essential factor should be considered into the systems. Researchers have paid increasing attention on the saturation problems [29], [30]. Among these results, convex combination approach is widely used as its less conservatism, in which a set of linear systems are used to replace the nonlinearity caused by saturation. Based on this method, actuator saturation for MJSs also has achieved much discussion in recent years [31], [32], further for semi-MJSs [33]. However, there are limited results on the event-triggered scheme for MJSs with actuator saturation these years [34], which is another key motivation we focused on in this paper.

Inspired by the reasons above, we consider the event-trigger control scheme instead of the time-triggered control for discrete-time uncertain MJSs with nonlinear input and external disturbance. The diagram is for proposed control strategy is presented in Fig. 1. The main contributions in this paper is twofold: (i) This work, to our knowledge, is the first time to investigate the robust event-triggered control strategy for discrete-time Markov jump systems with uncertain parameters; (ii) The nonlinear saturated input is also considered by convex combination approach when the robust event-triggered control problem is solving for MJSs. Moreover, the largest domain of attraction is presented by solving the optimal control problem in terms of linear matrix inequalities (LMIs).

FIGURE 1.

The block diagram of robust event-triggered control scheme.

Notation: Throughout this paper, an $n$ -dimension Euclidean spaces is given as $R^{n}$ . $\mathbb{N}_{+}$ and $\mathbb{N}$ indicate for the positive and non-negative integers respectively. In addition, $[a,b]$ represents the value taken from the constrained set between $a$ and $b$ . Moreover, the superscript “$T$ ” denotes the transposition while “−1” denotes the matrix inverse. $|\star |$ means the row value of a vector and $\mathbb{E}\{\star \}$ represents the expectation of a function. $\bigcap ^{N}_{i=1}P_{i}$ is the intersection of multiple matrices $P_{i}$ .

SECTION II. Problem Formulation and Preliminaries

The system is described as a Markov linear jump system with nonlinear input as:

$$\begin{align*} x_{k+1}=&\mathcal {A}_{r(k)}x_{k} +B_{r(k)} f(u_{k},r(k))+B^\omega _{r(k)} \omega _{k} \\ z_{k}=&C_{r(k)}x_{k}\tag{1}\end{align*}$$ where $x_{k}\in R^{n}$ is the state variables, $z_{k}\in R^{p}$ represents the measured output. The function $f(u_{k},r(k))$ denotes the nonlinear input dependent on the system mode $r(k)$ . Meanwhile, $r(k)$ is governed by a Markov chain with taking values in a finite set $\varGamma =\{1, 2, \ldots, N\}$ . $\mathcal {A}_{r(k)}=A_{r(k)}+\varDelta A_{r(k)}$ and $\varDelta A_{r(k)}$ indicates the uncertain parameters of the system. Finally, External disturbances $\omega _{k}$ is assumed to be bounded with $\|\omega _{k}\|_{2}^{2} = \sum _{k=0}^{\infty } \omega ^{T}_{k} \omega _{k} \leq \varpi ^{2}$ , where $\varpi >0$ and $B^\omega _{r(k)}$ is the disturbance parameter with appropriate dimension. Moreover, the transition probability between two neighboring modes is given as $\pi _{ij}=Pr\{r(k+1)=j|r(k)=i\}$ with $\pi _{ij} \in [{0,1}]$ and $\sum _{j=1}^{N}\pi _{ij}=1$ .

We recall the following essential definitions and lemmas for our main results in the next section.

Nonlinear saturation here is mainly considered as actuator saturation, which is regarded as a normal phenomenon in physical systems [35], [36] and restricted in a specific constraint when a signal is transmitted from to the actuator. The saturated controller output in a defined boundary may cause the instability and bad performance of the system. Therefore, this kind of nonlinear is replaced by a set of linear systems in a convex hull, by which the controllers are designed as the formation in Lemma 1.

Definition 1[29]:

For the given positive symmetric matrix $P, F$ , an ellipsoid set and a symmetric polyhedron set are defined as follows, respectively:

$$\begin{align*} \Omega (P, 1)=&\{x_{k}\in R^{n}: x^{T}_{k}Px_{k} \leq 1\}. \\ \mathfrak {L}(F)=&\{x_{k}\in R^{n}: | f_{a}x_{k}| \leq 1, a=1,2,\ldots, m\}.\end{align*}$$ where $f_{a}$ denotes the $a$ th row of the matrix $F$ . Then, an ellipsoid $\Omega (P, 1)$ is inside $\mathfrak {L}(F)$ as $\Omega (P, 1)\in \mathfrak {L} (F)$ , if and only if $$\begin{equation*} f_{a}P^{-1}f^{T}_{a}\leq 1.\end{equation*}$$ The actuator saturation in this paper is defined as $\alpha (u_{k})=[\alpha (u_{1}), \alpha (u_{2}), \ldots, \alpha (u_{k})]$ , where $\alpha (u_{i}) = sign(u_{i})\min \{1, |u_{i}|\}$ . Here $\alpha$ denotes both the vector value and scale value of the saturation function.

Lemma 1[32]:

For the given vectors $u_{k}$ and $v_{k}$ , if $v_{k}< 1$ , then, $\alpha (u_{k})=\sum _{l=1}^{2^{m}}\xi _{l}(M_{l}u_{k}+M^{-}_{l}v_{k})$ , where $0\leq \xi _{l}\leq 1$ and $\sum _{l=1}^{2^{m}}\xi _{l}=1$ . Suppose that each element in $m\times m$ matrix $M_{l}$ is either 1 or 0 and $M^{-}_{l}=I{-} M_{l}$ .

In line with the event-triggered strategy, not all the sampling-time-driven states are applied for the feedback controller. Thus, we take the event driven states as $x_{k}^{E}$ , then, we have the controller in the form of $u_{k}=K_{r(k)}x_{k}^{E}$ . Thus,the event condition is presented below, which means that if the condition is satisfied, the event-triggered controller is used for the system and until the next triggered time.

Definition 2[5]:

Let $t_{k}$ is the current triggered signal and $t_{k+1}$ refers to the next triggered time, then, for $k \in [t_{k}, t_{k+1}]$ , there exist $\gamma _{r(k)}\in [{0, 1}]$ and positive matrix $\varPhi _{r(k)}$ , the event is driven if the following inequality is satisfied.

$$\begin{equation*} e^{T}_{k} \varPhi _{r(k)} e_{k} > \gamma _{r(k)} x^{T}_{k} \varPhi _{r(k)} x_{k}\tag{2}\end{equation*}$$ where $e_{k} = x_{k} - x_{k}^{E}$ is the difference value of sampling-time-triggered state and event-triggered state. The trigger factor $\gamma _{r(k)}\in [{0,1}]$ is a known scalar to determine the trigger instant.

Remark 1:

As event-triggered time in this paper is assumed as $k=lh$ , in which $h$ is the sampling time period and $l$ is the number of sampling period. Thus, the set of succeed transmission can be viewed as the subset of the sampling time instant set. This is the operating how the event-triggered scheme reduce the cost of data communication.

Remark 2:

In this paper, the mode-dependent triggered condition is proposed and the mode is assumed as the same as the system operation mode. When the system jump from $i$ to $j$ , as the time-triggered interval is shorter than that of event-triggered interval, the mode is deemed to be the same.

The robustness control method is applied for the existence of parameter uncertainty and extrinsic disturbance in system (1). Therefore, the following definitions are presented.

Lemma 2[37]:

The parameter uncertainty in system (1) is assumed to only exist in parameter $A$ with

$$\begin{equation*} \Delta A_{r(k)} =H_{r(k)} \Delta _{r(k)} E_{r(k)}\tag{3}\end{equation*}$$ where constant matrices $H_{r(k)}$ and $E_{r(k)}$ are considered with appropriate dimensions while unknown matrix $\Delta _{r(k)}$ is considered with Lebesgue measurable element satisfying $\|\Delta _{r(k)}\|\leq 1$ .

Lemma 3[37]:

Given real matrices of appropriate dimensions $Z$ , $E$ , $F$ and $\Delta$ . For real matrix $\Delta$ satisfying $\Delta ^{\mathrm {T}}\Delta \leq I$ , the following inequality holds if there exist sclar $\alpha > 0$

$$\begin{equation*} Z + E \Delta F+ F^{\mathrm {T}} \Delta ^{\mathrm {T}} E^{\mathrm {T}} \leq Z + \alpha ^{-1} E^{\mathrm {T}} E+ \alpha F^{\mathrm {T}} F\tag{4}\end{equation*}$$

Definition 3[38]:

For given $\rho >0$ and initial state $x_{0}$ , if we have the following inequality holds, then, Markov jump system (1) is satisfactory with a ${H}_{\infty}$ performance index:

$$\begin{equation*} \|T_{z\omega }\|^{2}_\infty = \frac {\sum _{k=0}^{\infty }z^{T}_{k}z_{k}}{\sum _{k=0}^{\infty }\omega ^{T}_{k} \omega _{k}}\leq \rho ^{2}\tag{5}\end{equation*}$$

Definition 4[38]:

System (1) with $u\equiv 0$ and $\omega \equiv 0$ is mean square stable (MSS), for given initial $x_{0}$ and $r_{0}$ , if the following equation holds:

$$\begin{equation*} \lim _{k\rightarrow \infty } \mathbb{E} \{x^{\mathrm {T}}_{k}x_{k}\}|_{x_{0}, r_{0}} \rightarrow 0\tag{6}\end{equation*}$$ The main purpose of this paper is to design a event-triggered controller to stabilizes system (1) and satisfies an ${H}_{\infty}$ performance index with estimating a largest domain of attraction. Based on a Lyapunov function, sufficient conditions are given for the controller design and ${H}_{\infty}$ performance is analyzed by the event-trigger strategy.

SECTION III. Main Results

In this section, an event-based controller is firstly designed with the consideration of actuator saturation in Theorem 1. Then, on the basis of Theorem 1, ${H}_{\infty}$ performance is analyzed and the robust designed controller based on event signal is given in Theorem 2. Finally, the largest domain of attraction is attained in Theorem 3 by solving an optimal problem.

Theorem 1:

For given initial state $x(0)$ , scalars $\gamma _{i}>0$ and matrices $M_{l}$ , $M^{-1}_{l}$ , if for all $i, j \in \Gamma$ , there exists a set of $G_{i}> 0$ , $Q_{i}> 0$ , $\Xi _{i}>0$ , $Y_{i}>0$ , $Z_{i}>0$ and scalars $\alpha >0$ , such that the following inequalities are satisfied. Then, system (3) is said to be mean square stability in the region $\bigcap ^{N}_{i=1}Q_{i}$ :

$$\begin{align*} & \Biggl [{ \begin{array}{ccc} -G_{i}^{T}-G_{i}+Q_{i}+\gamma _{i}\Xi _{i}^{T} & 0 & \Upsilon _{i}^{T}\\ \ast & - \Xi _{i}^{T} & X_{i}^{T} \\ \ast & \ast & -Q_{j} \\ \ast & \ast & \ast \\ \ast & \ast & \ast \end{array} } \\ & \qquad \qquad \qquad \qquad \qquad \quad { \begin{array}{cc} 0 & G_{i}^{T}H_{i}^{T}\\ 0 & 0\\ \Psi ^{T}_{i} & 0\\ -\alpha I & 0\\ \ast & -\alpha ^{-1} I \end{array} }\Biggr]< 0\tag{7}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \varepsilon (P_{i}, 1)\in \mathfrak {L}(F_{i})\tag{8}\end{align*}$$ where $Z=K_{i}G_{i}$ , $Y=F_{i}G_{i}$ , $\Upsilon _{i}^{T}=[\sqrt {\pi _{i1}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}, \sqrt {\pi _{i2}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}, \cdots, \sqrt {\pi _{iN}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}]^{T}$ , $\Psi ^{T}_{i}=[\sqrt {\pi _{i1}}E_{i}^{T}, \ldots, \sqrt {\pi _{iN}}E^{T}_{i}]^{T}$ , $\Xi _{i}^{T} = G_{i}^{T} \varPhi _{i} G_{i}$ and $X_{i}^{T}=[\sqrt {\pi _{i1}}(B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}, \cdots, \sqrt {\pi _{iN}}(B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}]$ . Then, we have controller $K_{i}=Z_{i} G_{i}^{-1}$ .

Proof:

Let mode-dependent Lyapunov function as: $V(x_{k}, r_{k})=x_{k}^{T} P_{r_{k}} x_{k}$ . Here we show the closed-loop system (1) is mean square stable when external disturbance $\omega _{k} = 0$ . For $r_{k}=i$ and $r_{K+1}=j$ , we have

$$\begin{align*} \Delta V=&\mathbb {E}\{V(x_{k+1}, r_{k+1})\}-V(x_{k}, r_{k}) \\=&x_{k+1}^{T} P_{r(k+1)} x_{k+1}-x_{k}^{T} P_{r(k)} x_{k} \\=&[(A_{i}+\Delta A_{i})+B_{i} \theta _{l}(M_{l} K_{i}+M_{l}^{-1}F_{i})u_{k}]^{T} \\&\sum _{j=1}^{N}\pi _{ij}P_{j} [(A_{i}+\Delta A_{i}) \\&+\,B_{i} \theta _{l}(M_{l} K_{i}+M_{l}^{-1}F_{i})u_{k}] - x_{k}^{T} P_{i} x_{k}\tag{9}\end{align*}$$ For the sake of analysis, we have $R_{i}=M_{l} K_{i}+M_{l}^{-1}F_{i}$ and $\theta _{l}=\sum _{l=1}^{2^{m}}\xi _{l}$ . From Definition 1, we obtain that $e_{k}=x_{k}-x_{k}^{E}$ , then, we have $u_{k}=K_{i} x_{k}^{E}=K_{i} (x_{k}-e_{k})$ and substitute to (9). Furthermore, combining the event-trigger condition (2), the following inequality is obtained $$\begin{align*} \Delta V=&[(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}(x_{k}-e_{k})]^{T} \\&\sum _{j=1}^{N}\pi _{ij}P_{j} [(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}(x_{k}-e_{k})] \\&-\, x_{k}^{T} P_{i} x_{k} + e_{k}^{T}\varPhi _{i}e_{k}-e_{k}^{T}\varPhi _{i}e_{k} \\\leqslant&[(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}(x_{k}-e_{k})]^{T} \\&\sum _{j=1}^{N}\pi _{ij}P_{j} [(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}(x_{k}-e_{k})] \\&-\, x_{k}^{T} P_{i} x_{k} + \gamma _{i} x_{k}^{T}\varPhi _{i}x_{k}-e_{k}^{T}\varPhi _{i}e_{k} < 0\tag{10}\end{align*}$$ Then, we have the following inequality satisfied based on inequality (10). $$\begin{align*}&\hspace {-2pc}x_{k}^{T} \{ [(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}]^{T} \mathcal {P}_{j} \\&[(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}]-P_{i}\}x_{k}+\gamma _{i} \varPhi _{i} x_{k} \\&-\,2~x_{k}^{T} { [(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}(x_{k}-e_{k})] }^{T} \mathcal {P}_{j} e_{k} \\&+\,e_{k}^{T}[(B_{i} \theta _{l}R_{i})^{T} \mathcal {P}_{j} (B_{i} \theta _{l}R_{i}) -\varPhi _{i}]e_{k}< 0\tag{11}\end{align*}$$ where $\mathcal {P}_{j}=\sum _{j=1}^{N}\pi _{ij}P_{j}$ . By setting $\eta _{i}^{T}=[x_{k} \quad e_{k}]^{T}$ and define $\tilde {A}_{i}=A_{i}+\Delta A_{i}$ , then we have $\eta _{i}^{T} \Omega \eta _{i}$ . Based on the Schur complement formula, we obtain $$\begin{equation*} \Omega = \left [{ \begin{array}{ccc} -P_{i}+\gamma _{i} \varPhi _{i} & 0 & [\tilde {A}_{i}+B_{i} \theta _{l}R_{i}]^{T} \\ \ast & \varPhi _{i} & (B_{i} \theta _{l}R_{i})^{T} \\ \ast & \ast & - \displaystyle \sum _{j=1}^{N}\pi _{ij}P_{j}^{-1} \end{array} }\right]< 0\tag{12}\end{equation*}$$ To handle the uncertainty in the system by Lemma 3, the following inequality holds. $$\begin{align*}&\hspace{-0.5pc}\Biggl [{ \begin{array}{ccc} -P_{i}+\gamma _{i} \varPhi _{i} & 0 & \displaystyle \sqrt {\pi _{ij}}[A_{i}+B_{i} \theta _{l}R_{i}]^{T}\\ \ast & - \varPhi _{i}^{T}& \displaystyle \sqrt {\pi _{ij}}(B_{i} \theta _{l}R_{i})^{T}\\ \ast & \ast & -P_{j}^{T} \\ \ast & \ast & \ast \\ \ast & \ast & \ast \end{array} } \\& \qquad \qquad \qquad \qquad \quad {{ \begin{array}{cc} 0 & H_{i}^{T}\\ 0 & 0\\ \displaystyle \sqrt {\pi _{ij}}E_{i}^{T} & 0\\ -\alpha I & 0\\ \ast & -\alpha ^{-1} I \end{array} }\Biggr]< 0}\tag{13}\end{align*}$$ As positive matrix $P_{i}^{-1}$ is infeasible by LMIs. Let $P_{i}^{-1}=Q_{i}$ and multiplying diag$\{G_{i}^{T}, G_{i}^{T}, I, I, I\}$ on the left and diag$\{G_{i}, G_{i}, I, I, I\}$ on the right, respectively. For the reason that $G_{i}^{T}Q_{i}^{-1}G_{i} \geqslant G_{i}^{T}-Q_{i}+G_{i}$ is satisfied for any known positive $G_{i}$ . Accordingly, formula (7) is guaranteed.

In this subsection, we derive the sufficient conditions for event triggered ${H}_{\infty}$ control for uncertain system with considering external disturbances. Following gives the conditions.

Theorem 2:

For given initial state $x(0)$ , scalars $\gamma _{i}>0$ and matrices $M_{l}$ , $M^{-1}_{l}$ , If for all $i, j \in \Gamma$ , there exists a set of $G_{i}> 0$ , $Q_{i}> 0$ , $\Xi _{i}>0$ , $Y_{i}>0$ , $Z_{i}>0$ and scalars $\alpha >0$ , such that the following inequalities are satisfied. Then, system (3) is said to be mean square stability in the region $\bigcap ^{N}_{i=1}Q_{i}$ :

$$\begin{align*} & \Biggl [{ \begin{array}{ccc} -G_{i}^{T}-G_{i}+Q_{i}+\gamma _{i} \Xi _{i}^{T} &0 &0\\ \ast &- \Xi _{i}^{T} &0\\ \ast &\ast &-\rho ^{2} I \\ \ast &\ast &\ast \\ \ast &\ast &\ast \\ \ast &\ast &\ast \\ \ast &\ast &\ast \end{array} } \\ & \qquad \qquad \qquad { \begin{array}{cccc} \Upsilon _{i}^{T} &G_{i}^{T}C_{i}^{T} &0 &G_{i}^{T}H_{i}^{T}\\ X_{i}^{T} &0 &0 &0\\ V_{i}^{T} &0 &0 &0\\ -Q_{j} &0 &\Psi ^{T}_{i} &0\\ \ast &-I &0 &0\\ \ast &-\alpha I &0\\ \ast &\ast &\ast &-\alpha ^{-1} I \end{array} }\Biggr]< 0\tag{14}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \varepsilon (P_{i}, 1)\in \mathfrak {L}(F_{i})\tag{15}\end{align*}$$ where $Z=K_{i}G_{i}$ , $Y=F_{i}G_{i}$ , $\Upsilon _{i}=[\sqrt {\pi _{i1}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i})), \sqrt {\pi _{i2}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i})), \cdots, \sqrt {\pi _{iN}}(A_{i}G_{i}+B_{i} \theta _{l} (Z_{i}+Y_{i}))]^{T}$ , $\Xi _{i}^{=}G_{i}^{T} \varPhi _{i} G_{i}$ , $\Psi ^{T}_{i}=[\sqrt {\pi _{i1}}E_{i}^{T}, \ldots, \sqrt {\pi _{iN}}E^{T}_{i}]^{T}$ and $X_{i}^{T}=[\sqrt {\pi _{i1}}(B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}, \sqrt {\pi _{i2}}(B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}, \cdots, \sqrt {\pi _{iN}}(B_{i} \theta _{l} (Z_{i}+Y_{i}))^{T}]$ . Then, we have controller $K_{i}=Z_{i} G_{i}^{-1}$ .

Proof:

According to Definition 3 before, we make the function of $J_{k}< 0$ shown below,

$$\begin{align*} J_{k}=&\mathbb {E}{\sum _{k=0}^{\infty }(z_{k}^{T}z_{k}-\rho ^{2} \omega _{k}^{T}\omega _{k})} \\=&\mathbb {E}{\sum _{k=0}^{\infty }\{(z_{k}^{T}z_{k}-\rho ^{2} \omega _{k}^{T}\omega _{k})+\Delta V\}}-\mathbb {E}\{\Delta V\} \\\leqslant&\mathbb {E}{\sum _{k=0}^{\infty }(z_{k}^{T}z_{k}-\rho ^{2} \omega _{k}^{T}\omega _{k})+\Delta V}< 0\tag{16}\end{align*}$$ which is to say, if we want $J_{k}< 0$ is satisfied, the former inequality should be guaranteed first. By the similar process, we obtain that $\xi ^{T} \Sigma _{i} \xi$ , where $\xi ^{T}=[\eta _{i}^{T} \quad \omega _{i}^{T}]$ and $$\begin{align*}&\hspace{-0.5pc}\Sigma _{i} = \Biggl [{ \begin{array}{ccc} -P_{i}+\gamma _{i} \varPhi _{i}+C_{i}^{T}C_{i} & 0 & 0 \\ \ast & \varPhi _{i} & 0 \\ \ast & \ast & -\rho ^{2} I \\ \ast & \ast & \ast \\ \end{array} } \\& \qquad \qquad \qquad \qquad {{ \begin{array}{c} [(A_{i}+\Delta A_{i})+B_{i} \theta _{l}R_{i}]^{T} \\ (B_{i} \theta _{l}R_{i})^{T} \\ (B_{i}^\omega)^{T} \\ -\displaystyle \sum _{j=1}^{N}\pi _{ij}P_{j}^{-1} \end{array} }\Biggr]< 0}\tag{17}\end{align*}$$ In a similar way on the basis of Lemma 3 and multiplying diag$\{G_{i}^{T}, G_{i}^{T}, G_{i}^{T}, I, I, I, I\}$ on the left and diag$\{G_{i}, G_{i}, G_{i}, I, I, I, I\}$ on the right. We achieve the follow inequality to be guaranteed. $$\begin{align*}&\hspace{-0.5pc}\Biggl [{ \begin{array}{cccc} -Q_{i}^{-1}+\gamma _{i} \Xi _{i}^{T} & 0 & 0 & \Upsilon _{i}^{T} \\ \ast & - \Xi _{i}^{T} & 0 & X_{i}^{T}\\ \ast & \ast & -\rho ^{2} I & V_{i}^{T} \\ \ast & \ast & \ast & -Q_{j} \\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \end{array} } \\& \qquad \qquad \qquad ~\quad {{ \begin{array}{ccc} G_{i}^{T}C_{i}^{T} & 0 & G_{i}^{T}H_{i}^{T}\\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \Psi _{i} & 0 \\ -I & 0 & 0\\ \ast & -\alpha I & 0\\ \ast & \ast & -\alpha ^{-1} I \end{array} }\Biggr]< 0}\tag{18}\end{align*}$$

In this subsection, for all the feasible solutions, we could obtain the largest domain of the attraction by a reference ellipse within the feasible regions under event-driven control scheme. The following theorem is given for the estimation.

Theorem 3:

For given initial state $x(0)$ , scalars $\gamma _{i}>0$ and matrices $M_{l}$ , $M^{-1}_{l}$ , If for all $i, j \in \Gamma$ , there exists a set of $G_{i}> 0$ , $Q_{i}> 0$ , $\Xi _{i}>0$ , $Y_{i}>0$ , $Z_{i}>0$ and scalars $\alpha >0$ , such that the following inequalities are satisfied. Then, system (3) is said to be mean square stability in the region $\bigcap ^{N}_{i=1}Q_{i}$ , such that the following optimization problem is solved. Then, the largest domain of attraction for system (1) is obtained

$$\begin{align*}&\displaystyle \min \limits _{X_{i}, Y_{i}, Z_{i}} \sigma \tag{19}\\&\text {subject to}~\Biggl [{ \begin{array}{ccc} -G_{i}^{T}-G_{i}+Q_{i}+\gamma _{i} \Xi _{i}^{T} & 0 & 0 \\ \ast & - \Xi _{i}^{T} & 0 \\ \ast & \ast & -\rho ^{2} I \\ \ast & \ast & \ast \\ \ast & \ast & \ast \\ \ast & \ast & \ast \\ \ast & \ast & \ast \end{array} } \\&\qquad \qquad { \begin{array}{cccc} \Upsilon _{i}^{T} & G_{i}^{T}C_{i}^{T} & 0 & G_{i}^{T}H_{i}^{T}\\ X_{i}^{T} & 0 & 0 & 0\\ V_{i}^{T} & 0 & 0 & 0 \\ -Q_{j} & 0 & \Psi ^{T}_{i} & 0 \\ \ast & -I & 0 & 0\\ \ast & \ast & -\alpha I & 0 \\ \ast & \ast & \ast & -\alpha ^{-1} I \end{array} }\Biggr]< 0\qquad \quad \tag{20}\\&\hphantom {\text {subject to}~}\left [{ \begin{array}{cc} -\sigma W & I\\ \ast & -Q_{i} \end{array} }\right]< 0\tag{21}\\&\hphantom {\text {subject to}~}\left [{ \begin{array}{cc} -G_{i}^{T}-G_{i}+Q_{i} & G_{i}^{T}f_{i}\\ \ast & -I \end{array} }\right]< 0\tag{22}\end{align*}$$

Proof:

On the ground of Definition 3, the following condition we could obtain,

$$\begin{equation*} \left [{ \begin{array}{cc} -P_{i} & f_{i}^{T}\\ \ast & -I \end{array} }\right]< 0\tag{23}\end{equation*}$$ Then with the same fundamentals of mathematics, we know that (22) is achieved. Then, we choose an ellipsoid reference set and we have $\{x\in W: x^{T}Wx \leq 1\}$ . As $\beta \Omega (W)~\in \mathfrak {L}(F_{i})$ $$\begin{equation*} \left [{ \begin{array}{cc} -1/\beta ^{2} W & I\\ \ast & -P_{i}^{-1} \end{array} }\right]< 0\tag{24}\end{equation*}$$ Let $\sigma =-1/\beta ^{2}$ . (21) is guaranteed. This completes the proof.

Remark 3:

In this part, a reference ellipsoid set $\Omega (W,1)$ is applied to estimate the largest attraction domain of the system (1). On the basis of the feasible domain of the stabilization condition (22), the largest domain of attraction of ellipse $\Omega (W,1)$ is determined by minimizing the parameter $\sigma$ . Then, the estimation of the largest attraction domain problem becomes solving optimization problem above to get the minimum value of $\sigma$ .

SECTION IV. Simulation Examples

In this part, we first give a numerical example to compare the results in Theorem 2 and Theorem 3 for MJSs with uncertainty and bounded disturbance. To verify the effectiveness and feasibility of the mentioned design method for practical system, we give a mass spring damping system in the second part.

Example 1:

The numerical parameters are given as:

$$\begin{align*} A(1) =&\left [{ \begin{array}{cc} 1.3 & -0.45 \\ 0 & 1.1 \\ \end{array} }\right]\quad A(2) = \left [{ \begin{array}{cc} 1 & -0.29 \\ 0.9 & 1.5 \\ \end{array} }\right] \\ A(3) =&\left [{ \begin{array}{cc} 0.8 & -0.5 \\ 1.2 & 0.7 \\ \end{array} }\right]\quad E(1) = \left [{ \begin{array}{c} 0.5 \\ 0.5 \end{array} }\right] E2= \left [{ \begin{array}{c} 0.5 \\ 0.5 \end{array} }\right] \\ B(1) =&\left [{ \begin{array}{c} 0.5 \\ 1.1 \end{array} }\right]\quad B(2) = \left [{ \begin{array}{c} 0.6 \\ 1.4 \end{array} }\right]\quad B(3) = \left [{ \begin{array}{c} 0.8 \\ 1.2 \end{array} }\right] \\ H1=&\left [{ \begin{array}{cc} 0.2 & 0.1\\ \end{array} }\right]\quad H2= \left [{ \begin{array}{cc} 0.1 & 0.2\\ \end{array} }\right]\end{align*}$$

The transition probability of the MJSs is assumed as:

$$\begin{equation*} \pi _{ij} = \left [{ \begin{array}{ccc} 0.2 & 0.5 & 0.3 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.5 & 0.4 \end{array} }\right]\end{equation*}$$

Let the initial state $x_{0} =[0.5 \quad 0.5]^{T}$ , Fig.2 shows the jump mode of the MJSs. Fig.3 present the state responses of the system. From Fig.4 of event trigger signal, we could discover that not every sampling time is triggered when we choose the trigger factor as $\gamma _{1}=0.3, \gamma _{2}=0.3, \gamma _{3}=0.4$ . Furthermore, we set the mode-dependent triggered parameters as in mode 1 (the triggered signal is 2), in mode 2 (the triggered signal is 3) and in mode 3 (the triggered signal is 4). Finally, Fig.5 gives the largest domain attraction (in the intersection part).

FIGURE 2.

The jump mode of the Markov jump systems.

FIGURE 3.

The state response of the colose-loop system.

FIGURE 4.

The event instant of system (1) by event-triggered scheme.

FIGURE 5.

The largest range of attraction with different trigger factors.

In the following table, we present the comparison of different triggered factors in different system modes. As the factors increase, the optimization problems may have no feasible solutions and the largest domain of attraction is also given in the intersection part of the three ellipses by three modes in Fig.4. Meanwhile, with the increasing of trigger factor, the largest feasible solution decrease instead. Thus, the trigger factor is an important parameter for the event-based control mechanism.

TABLE 1 Comparison of Some Parameters of Theorem 2 and Theorem 3

Example 2:

Mass spring damping system [39].

In this part, we give an dynamic model to verify the effectiveness of the proposed method. For reducing the movement energy and improving driving safety, the analysis of a mass spring damping system is necessary. In this subsection, a mass-spring damping system is described as a Markov linear jump system as the spring has a limited reset switch. Then, two modes ($i \in \{1, 2\}$ ) is chosen with different constants $a=0.3\,\,m^{-1}$ and $a=0$ . For the useful of the device, the saturated input is considered here. The dynamic equations of the system are given as, in which $x$ is the position of a reference point and $\dot {x}$ is the speed of the reference point:

$$\begin{equation*} \left [{ \begin{array}{c} \dot {x} \\ \ddot {x} \end{array} }\right]= \left [{ \begin{array}{cc} 0 & 1 \\ \chi & \dfrac {-c}{m} \end{array} }\right] \left [{ \begin{array}{c} x \\ \dot {x} \end{array} }\right]+ \left [{ \begin{array}{c} 0 \\ \frac {1}{m} \end{array} }\right]u+\omega\end{equation*}$$ where $\chi =\frac {-k-4ka^{2}}{m}$ , $m=1\,\,kg$ is the mass of the reference point, $c=2N*m/s$ stands for the coefficient of friction restriction, $k=8\,\,N/m$ represent the spring coefficient. The initial state is $x_{0}=[0.5 \quad 0.5]^{T}$ and the external disturbance $\omega =0.01*cos(0.2k)$ The other parameters are presented as: $$\begin{align*} H1=&\left [{ \begin{array}{cc} 0.2 & 0.1\\ \end{array} }\right]\quad H2= \left [{ \begin{array}{cc} 0.1 & 0.2\\ \end{array} }\right]\\ E(1) =&\left [{ \begin{array}{cc} 0.5 & 0.5\\ \end{array} }\right]^{T}\quad E2= \left [{ \begin{array}{cc} 0.5 & 0.5\\ \end{array} }\right]^{T}\\ C1=&C2= \left [{ \begin{array}{cc} 0.2 & 0\\ 0 & 0.3 \end{array} }\right]^{T}\\ \pi _{ij}=&\left [{ \begin{array}{cc} 0.6 & 0.4 \\ 0.3 & 0.7 \end{array} }\right]\end{align*}$$ According to Theorem 3, we have the controller as $K1=[7.7257 \quad -15.8293]$ and $K2=[5.1098 \quad -11.1935]$ . The mode jump responses and state responses are given in Fig.6 and Fig.7. The event-triggered release instant is shown in Fig.8, it is obviously that the event-driven control strategy save much more communication cost in the system, where “2” means the triggered instant in mode 1 and “3” means the triggered time in mode 2. Finally, the largest domain of attraction is also presented shown as Fig.9 in the intersection part by solving the optimal problem in Theorem 3.

FIGURE 6.

The response of the jump modes.

FIGURE 7.

The response of the system states.

FIGURE 8.

The event instant of the mass spring damping system.

FIGURE 9.

The largest domain of attraction in the mass spring damping system.

SECTION V. Conclusion

The robust event-triggered control problem for Markov jump systems is addressed in this paper with the consideration of uncertain parameters and nonlinear input. The mode-dependent event-triggered condition is given to determine the time to trigger the controller. Meanwhile the controller is designed to stabilize the system and guarantee the ${H}_{\infty}$ performance index. Sufficient conditions are obtained in terms of LMIs and the largest domain of attraction is given by solving a optimal problem. Simulation studies on a numerical and a mass spring damping systems to illustrate the effectiveness of the proposed method and achieve a range feasible solutions.