Co-Design of Output-Based Event-Triggered Protocol and Sliding Mode Control for 2D Nonlinear Fornasini-Marchesini Network under Packet Dropouts

Abstract

This paper focuses on the stability and sliding mode issues for the two-dimensional (2D) Fornasini-Marchesini (FMII) networked control system under packet dropouts. Firstly, the output-based 2D event-triggered strategy was constructed to alleviate information transmission pressure caused by limited network resources. Secondly, by considering the impact of packet dropouts, we propose an output-based 2D sliding mode controller and formulate the output-based 2D error-estimation scheme accordingly. Moreover, to get rid of the nonlinear coupling of the conditions (to guarantee the mean-square stability), we established an adaptive intelligence algorithm. Finally, we provide a numerical example to verify the effectiveness and practicability of the proposed algorithm and controller design.

1. Introduction

Two-dimensional (2D) networked control systems realize the modeling of networked control systems under high-dimensional situations and have been applied in many fields, such as network signal processing [1], medical image analysis [2], and linear repetitive process [3]. In recent years, there have been breakthroughs in 2D networked control systems (in both theory and practical applications). For example, Liang et al. [4] solved the complex network’s mean square stability and state estimation issues coupled with the 2D FMII model under multiple random nonlinear disturbances. In [5], a distributed 2D state estimation algorithm was designed to analyze the mean square stability and $H_{\infty }$ performance for the 2D FMII network subjected to nonlinear perturbation.

The 2D networked control system regards the 2D system as the controlled plane [6,7,8] and takes various network constraints as network characteristics [9,10,11]. Among these, the network packet dropout is a notable and unavoidable constraint in the 2D networked control system and has been a concern for many scholars. For instance, in [12], Gao et al. assumed that the 2D random packet losses obeyed the 2D Bernoulli stochastic process and occurred in the information exchange between the controlled system and the filters; they then designed a parameter-dependent discrete 2D filter to formulate the stability and $H_{\infty }$-filtering algorithm for the FMII model with uncertain parameters. In [13], Bu et al. considered the data packet loss that occurred in the information exchange process between the controlled system and the controller and then solved the $H_{\infty }$ control by designing a 2D state feedback controller. Recently, based on an innovative 2D recursive filtering algorithm design, Wang et al. [14] formulated the local optimal performance achieved for the 2D nonlinear FMII model over a random packet loss submitted to the Bernoulli distribution.

The sliding mode control (SMC) is widely accepted for its insensitivity to randomness and uncertainties caused by model errors, parameter changes, and external disturbances [15,16,17]. More explicitly, SMC is regarded as an effective control strategy to guarantee the system’s robustness with structural complexity and randomness. Research on the 2D sliding mode control was developed by Gao et al. [18]. By designing the 2D sliding mode controller, the linear Roesser model’s stability and sliding mode control problems were realized through the model transformation method (MTM) and Choi’s 1997 method, respectively. Yang et al., inspired by this research result, formulated the stability analysis and sliding mode control synthesis schemes for the discrete nonlinear 2D FMII model using MTM and Choi’s 1997 methods, respectively [19,20]. Recently, Yang et al. combined event-triggered protocols with sliding mode technology to discuss the analysis and control synthesis of complex networks. For example, in [21], the authors formulated the state-dependent event-triggered conditions for the switched Fornasini–Marchesini model and Roesser model, respectively, and then the stability and sliding mode control synthesis was realized by formulating the corresponding 2D sliding mode controllers. Innovative communication protocols and control methods are indispensable when dealing with the changeable network environment and the structural complexity of 2D systems. Recently, event-triggered protocols introduced in 1D network systems (achieving satisfactory practical results) have been extended in 2D network control systems to realize stability analysis and control synthesis designs under various network constraints. The 2D event-triggering protocol triggers information updates through specific events that occur; it reduces the number of information transmission times, saves network resources, and improves the efficiency and quality of network transmission. In [22,23], by introducing a state-dependent discrete 2D event-triggering protocol, Ding and his team designed the discrete 2D feedback controller to consider the stabilization and guaranteed cost control for the Roesser model and FMII models, respectively. Yang et al. [24] formulated a 2D event-triggering protocol with a switching form and provided the solutions for the asymptotic stability under arbitrary switching signals and exponential stability under restricted switching signals.

Note that the randomness and uncertainty of the packet dropout occur in a single instant resulting in the coupled state components being unmeasurable at multiple data nodes and preventing the reception of reliable real-time data over the shared network, which causes abnormal operation of the system. In this paper, we consider the system state ’unpredictable’, caused by packet dropouts; in this case, the innovative output-based 2D event-triggered protocol needed to be resorted to reflect the impact of packet dropouts. Moreover, we attempted to formulate the appropriate 2D sliding mode control method and controller design under the output-based event-driven protocol to solve network constraints and structural complexity of the 2D system. To our knowledge, related research studies are rarely involved. Given the aforementioned discussions, the contributions of this paper are as follows.

• We adopted an output-based 2D event-triggered protocol to deal with the information missing from system states caused by packet dropouts.
• We propose an observer-based 2D sliding mode control law to guarantee the mean-square stability of the 2D compensation-estimation system.
• To make the conditions for the mean-square stability applicable, we established an adaptive intelligent algorithm.

The structure of the paper is as follows. In Section 2, we review basic material on the 2D FMII model and formulate the research problem in this paper. Section 3 provides the main results of this paper. Finally, a numerical simulation is provided to show the effectiveness and correctness of the proposed methods and algorithms.

Notations: Let $\mathbb{R}$ and ${\mathbb{R}}^n$ denote the field of real numbers and the space of n-dimensional real vectors, respectively. The superscript $A^T$ stands for the transpose of matrix A. We denote by $P>0$ when the matrix P is real-symmetric and positive-definite. Moreover, $\mathbf{diag}\{·\}$ and $\mathbf{Prob}\{·\}$ refer to block-diagonal matrices and the probability measure, respectively. Finally, $\mathbf{E}\left\{x\right\}$ and $\mathbf{E}\left\{x\right|y\}$ denote the mathematical expectation of x and conditional expectation of x on y, respectively.

2. Preliminaries and Problem Formulation

Consider the discrete nonlinear 2D FMII model given by

$$\begin{array}{cc}\hfill x(h+1,k+1)=& A_{1}x(h+1,k)+A_{2}x(h,k+1)\hfill \\ \hfill & +B_1[u(h+1,k)+f\left(x(h+1,k)\right)]\hfill \\ \hfill & +B_2[u(h,k+1)+f\left(x(h,k+1)\right)]\hfill \\ \hfill y(h,k)=& Cx(h,k),\hfill \end{array}$$

(1)

where $x(h,k)\in {\mathbb{R}}^n$ denotes the system state, $u(h,k)\in {\mathbb{R}}^m$ is the control input, and $y(h,k)\in {\mathbb{R}}^p$ is the measured output with $k,h\in \mathbb{R}$; $f\left(x(h,k)\right)\in {\mathbb{R}}^m$ is an external disturbance satisfying

$$\left|\right|f\left(x\right(h,k\left)\right)\left|\right|\le \gamma$$

(2)

for a given positive constant $\gamma$; $A_1$, $A_2$, $B_1$, $B_2$, and C are system matrices with appropriate dimensions. Next, by introducing the block matrices

$$\begin{array}{c}\hfill \overline{A}=\left[A_1 A_2\right]\mathrm{and}\overline{B}=\left[B_1 B_2\right]\end{array}$$

(3)

the system (1) can be rewritten compactly as

$$\begin{array}{cc}\hfill x(h+1,k+1)=& \overline{A}\overline{x}(h,k)+\overline{B}[\overline{u}(h,k)+\overline{f}(h,k)]\hfill \\ \hfill y(h,k)=& Cx(h,k),\hfill \end{array}$$

(4)

where

$$\overline{x}(h,k)=\left[\begin{array}{c}x(h+1,k)\\ x(h,k+1)\end{array}\right],\overline{u}(h,k)=\left[\begin{array}{c}u(h+1,k)\\ u(h,k+1)\end{array}\right],\mathrm{and}\overline{f}(h,k)=\left[\begin{array}{c}f\left(x\right(h+1,k\left)\right)\\ f\left(x\right(h,k+1\left)\right)\end{array}\right].$$

Throughout this paper, we assume that the following assumption holds

Assumption 1.

The boundary condition of the system (1) is given as

$$\begin{array}{c}\hfill \underset{l\to \infty }{lim}\sum _{k=0}^l(\parallel x(k,0)\parallel +\parallel x(0,k)\parallel )<\infty .\end{array}$$

(5)

To conclude this section, we will formulate the research problem in this paper.

Problem 1.

Consider the 2D FMII network in (4), and assume that the completed state information is not measurable. We will propose the output-based sliding mode control strategy via event-triggered protocols.

3. Main Results

Before developing the blue observer-based sliding mode control strategy, we first introduce the following output-based event trigger condition for the 2D FMII network in (4)

$$\begin{array}{c}\hfill e_y^T\left(l\right)W{e}_y\left(l\right)\le \delta y^T\left(l\right)Wy\left(l\right),\end{array}$$

(6)

in which $W\in {\mathbb{R}}^{p\times p}$ denotes the positive-definite weight matrix, $\delta >0$ the triggered parameter, and $e_y\left(l\right)$ the output error defined as $e_y\left(l\right):=y\left(l\right)-y\left(l_i\right)$, where $l=h+k$ and $l_i$ are the current and last transmission instants, respectively. Consequently, the next transmission instant $l_{i+1}$ will be

$$\begin{array}{c}\hfill l_{i+1}=min\{l=h+k|e_y^T\left(l\right)W{e}_y\left(l\right)>\delta y^T\left(l\right)Wy\left(l\right)\}.\end{array}$$

(7)

Next, we construct the following 2D sliding surface function

$$\begin{array}{c}\hfill s\left(l\right)=Gx\left(l\right)=0,\end{array}$$

(8)

where G is the undetermined sliding mode gain satisfying that $G\overline{B}$ is nonsingular. This implies the following sliding mode control law

$$\begin{array}{cc}\hfill {\overline{u}}_{eq}(h,k)& =-{\left(G\overline{B}\right)}^{-1}G\overline{A}\overline{x}(h,k)-f(h,k).\hfill \end{array}$$

(9)

Note that the above control rate (9) fails to fully consider the unmeasurable status caused by the data packet loss during the information transmission. To deal with this problem, we introduce the following observer-based sliding mode controller

$$\begin{array}{c}\hfill \overline{u}(h,k)=-w(h,k){\left(G\overline{B}\right)}^{-1}G\overline{A}\overline{x}(h,k)-(1-w(h,k))L[\overline{y}\left(l_i\right)-\mathbf{diag}\left(C\right)\overline{x}(h,k)]-f(h,k),\end{array}$$

(10)

where $w(h,k)\in \{0,1\}$ satisfies the Bernoulli stochastic process

$$\begin{array}{cc}\hfill \mathbf{Prob}\left\{w\right(h,k)=1\}& =\mathbf{E}\{w(h,k)=1\}=w_0\hfill \\ \hfill \mathbf{Prob}\left\{w\right(h,k)=0\}& =1-\mathbf{E}\{w(h,k)=1\}=1-w_0\hfill \end{array}$$

(11)

in which $w_0$ denotes the expectation of the variable $w(h,k)$.

Consider the following observer-based estimation system

$$\begin{array}{cc}\hfill x_c(h+1,k+1)& =\overline{A}{\overline{x}}_c(h,k)-\overline{B}L[\overline{y}\left(l_i\right)-\mathbf{diag}\left(C\right){\overline{x}}_c(h,k)]\hfill \end{array}$$

(12)

and then the error system can be described as

$$\begin{array}{cc}\hfill e(h+1,k+1)& =\overline{A}\overline{e}(h,k)+\overline{B}[\overline{u}(h,k)+\overline{f}(h,k)]+\overline{B}L[\overline{y}\left(l_i\right)-\mathbf{diag}\left(C\right){\overline{x}}_c(h,k)],\hfill \end{array}$$

(13)

where $e(h,k)=x(h,k)-x_c(h,k)$. The control law (10) needs to be modified further as

$$\begin{array}{cc}\hfill {\overline{u}}_{eq}(h,k)& =-w(h,k){\left(G\overline{B}\right)}^{-1}G\overline{A}{\overline{x}}_c(h,k)-(1-w(h,k))L[\overline{y}\left(l_i\right)-\mathbf{diag}\left(C\right){\overline{x}}_c(h,k)]-\overline{f}(h,k).\hfill \end{array}$$

(14)

In addition, notice that

$$\begin{array}{c}\hfill y\left(l_i\right)=y\left(l\right)-e_y\left(l\right)=Cx\left(l\right)-e_y\left(l\right),\end{array}$$

(15)

by substituting the modified control law (14) into conditions (12) and (13), we have

$$\begin{array}{cc}\hfill x_c(h+1,k+1)=& \overline{A}{\overline{x}}_c(h,k)+\overline{B}L{\overline{e}}_y(h,k)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill e(h+1,k+1)=& w(h,k)\overline{A}\overline{e}(h,k)-w(h,k){\overline{B}}_G{\overline{x}}_c(h,k)-\overline{B}L{\overline{e}}_y(h,k)\hfill \end{array}$$

(17)

with ${\overline{B}}_G=\overline{B}{\left(G\overline{B}\right)}^{-1}G\overline{A}$.

Remark 1.

The further modified sliding mode controller (14) includes both the data packet loss and observer-based control rate design. It helps to realize the output compensation for the lost packets and achieve the desired accurate real-time control aim.

In what follows, we will first check the reachability of the 2D sliding surface given by (8) and then analyze the mean-square stability of the 2D state estimation output compensation system (16) and (17) under output-based sliding mode control law (14) and even-triggered conditions (6).

3.1. The Reachability Analysis of 2D Sliding Surface (8)

Theorem 1.

Consider the 2D state estimation output and error system (16) and (17) under even-triggered conditions (6), the state behavior can be driven to the specified sliding surface (8) by the following SMC law:

$$\begin{array}{cc}\hfill \overline{u}(h,k)& =-2{\left(G\overline{B}\right)}^{-1}G\overline{A}{\overline{x}}_c(h,k)-L\left[\overline{y}\left(l_i\right)-diag\left(C\right)\overline{x}(h,k)\right]-\rho {\left(G\overline{B}\right)}^{-1}·sgn\left(s(h,k)\right),\hfill \end{array}$$

(18)

where ρ is a positive constant, and $G={\overline{B}}^{T}P$ such that $G\overline{B}$ is nonsingular.

Proof. 

Define the following energy function

$$\begin{array}{c}\hfill V_s\left(l\right)=\frac{1}{2}{s}^T\left(l\right)s\left(l\right),\end{array}$$

(19)

which implies that

$$\begin{array}{cc}\hfill \mathbf{E}\left(\mathsf{\Delta }{V}_s\left(l\right)\right|\xi \left(l\right))& =\mathbf{E}\left\{V_s(l+1)-V_s\left(l\right)|s\left(l\right)\right\}\hfill \\ \hfill & =\mathbf{E}\left\{\frac{1}{2}{s}^T(l+1)s(l+1)-\frac{1}{2}{s}^T\left(l\right)s\left(l\right)|s\left(l\right)\right\}.\hfill \end{array}$$

(20)

Notice that

$$\begin{array}{cc}\hfill \frac{1}{2}{s}^T(l+1)s(l+1)-\frac{1}{2}{s}^T\left(l\right)s\left(l\right)& =s^T\left(l\right)\mathsf{\Delta }s\left(l\right)+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right)\hfill \\ \hfill & =s^T\left(l\right)(s(l+1)-s\left(l\right))+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right)\hfill \\ \hfill & =s^T\left(l\right)s(l+1)-s^T\left(l\right)s\left(l\right)+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right).\hfill \end{array}$$

(21)

On the other hand, we have

$$\begin{array}{c}\hfill s(l+1)=Gx\left(l\right)=G\left[e(l+1)\right)+x_c(l+1)].\end{array}$$

(22)

Recalling the 2D sliding surface (8) and the sliding mode controller law (10), we have

$$\begin{array}{c}\hfill G{x}_c(l+1)=G\overline{A}{\overline{x}}_c(h,k)+\overline{B}L{\overline{e}}_y\left(l\right),\end{array}$$

(23)

and

$$\begin{array}{cc}\hfill Ge(l+1))=& G\left\{\overline{A}{\overline{x}}_c(h,k)+\overline{B}[\overline{u}(h,k)+\overline{f}(h,k)]+\overline{B}L[\overline{y}\left(l_i\right)-\mathbf{diag}\left(C\right)\overline{x}(h,k)]\right\}\hfill \\ \hfill =& G\overline{A}{\overline{x}}_c(h,k)-2G\overline{B}{\left(G\overline{B}\right)}^{-1}G\overline{A}{\overline{x}}_c(h,k)-\rho G\overline{B}{\left(G\overline{B}\right)}^{-1}sgn\left(s\left(l\right)\right)-\overline{B}L{\overline{e}}_y\left(l\right),\hfill \end{array}$$

(24)

which implies that

$$\begin{array}{c}\hfill s(l+1)=G\left[e(l+1)\right)+x_c(l+1)]=-\rho sgn\left(s\left(l\right)\right).\end{array}$$

(25)

Therefore, we obtain

$$\begin{array}{cc}\hfill V_s\left(l\right)& =s^T\left(l\right)[-\rho sgn\left(s\left(l\right)\right)]-s^T\left(l\right)s\left(l\right)+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right)\hfill \\ \hfill & =-\rho \left|\right|s\left(l\right)\left|\right|-s^T\left(l\right)s\left(l\right)+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right)\hfill \\ \hfill & \le -\rho \left|\right|s\left(l\right)\left|\right|+\frac{1}{2}\mathsf{\Delta }{s}^T\left(l\right)\mathsf{\Delta }s\left(l\right).\hfill \end{array}$$

(26)

Consequently, it follows that $\mathbf{E}\left\{\mathsf{\Delta }{V}_s\left(l\right)\right|\xi \left(l\right)\}<0$ can be guaranteed for some scalar $\rho >0$, which can always be selected and, thus, the system state will reach (and be maintained in) some domain of the sliding surface. To summarize, we have completed the proof.    □

3.2. Mean Square Stability Analysis of the Controlled System

Theorem 2.

The estimation-compensation system given by (16) and (17) is mean-square stable under the output-based sliding mode control law (14) and even-triggered conditions (6), if there exist positive-definite matrices $P_1,P_2,Q_1,Q_2$, and L satisfying:

$$\begin{array}{cc}\hfill \mathbf{\Pi }=& {\mathbf{\Pi }}_{\mathbf{1}}+{\mathbf{\Pi }}_{\mathbf{2}}+{\mathbf{\Pi }}_{\mathbf{3}}\hfill \end{array}$$

(27)

with

$$\begin{array}{cc}\hfill {\mathbf{\Pi }}_{\mathbf{1}}& =\left[\begin{array}{ccc}{\overline{A}}^{T}P\overline{A}-P& 0& {\overline{A}}^{T}P\overline{B}L\\ 0& 0& 0\\ {\overline{B}}_L^{T}P\overline{A}& 0& {\overline{B}}_L^{T}P\overline{B}L\end{array}\right]\hfill \end{array}$$

(28)

$$\begin{array}{c}\hfill {\mathbf{\Pi }}_{\mathbf{2}}=\left[\begin{array}{ccc}{W}^2{\overline{B}}_G^{T}Q{\overline{B}}_G& -W{\overline{B}}_G^{T}Q\overline{A}& -W{\overline{B}}_G^{T}Q{\overline{B}}_L\\ 0& {\overline{A}}^{T}Q\overline{A}-\overline{Q}& {\overline{A}}^{T}Q{\overline{B}}_L\\ 0& 0& {\overline{B}}_L^{T}Q\overline{B}L\end{array}\right],\end{array}$$

(29)

$$\begin{array}{c}\hfill {\mathbf{\Pi }}_{\mathbf{3}}=\left[\begin{array}{ccc}\mathbf{diag}\left(C^T\right)\delta I\otimes W\mathbf{diag}\left(C\right)& 0& 0\\ 0& \mathbf{diag}\left(C^T\right)\delta I\otimes W\mathbf{diag}\left(C\right)& 0\\ 0& 0& -I\otimes W\end{array}\right].\end{array}$$

(30)

where $P=P_1+P_2,Q=Q_1+Q_2$, ${\overline{B}}_L=\overline{B}L$, $G={\overline{B}}^{T}P$, and ${\overline{B}}_G=\overline{B}{\left(G\overline{B}\right)}^{-1}G\overline{A}$.

Proof. 

Suppose that there exist matrices denoted by

$$\begin{array}{c}\hfill \overline{P}=\mathbf{diag}\{P_1,P_2\}>0,\overline{Q}=\mathbf{diag}\{Q_1,Q_2\}>0,\overline{R}=\mathbf{diag}\{R_1,R_2\}>0,\end{array}$$

satisfying the condition (27). We then define the energy functions:

$$\begin{array}{c}\hfill V\left(l\right)=V_1\left(l\right)+V_2\left(l\right),\end{array}$$

(31)

and

$$\begin{array}{c}\hfill \mathsf{\Delta }V\left(l\right)=\mathsf{\Delta }{V}_1\left(l\right)+\mathsf{\Delta }{V}_2\left(l\right),\end{array}$$

(32)

where

$$\begin{array}{cc}\hfill V_1\left(l\right)& ={\overline{x}}_c^T\left(l\right)\overline{P}{\overline{x}}_c\left(l\right),\hfill \\ \hfill V_2\left(l\right)& ={\overline{e}}^T\left(l\right)\overline{Q}\overline{e}\left(l\right),\hfill \\ \hfill \mathsf{\Delta }{V}_1\left(l\right)& =V_1(l+1)-V_1\left(l\right),\hfill \\ \hfill \mathsf{\Delta }{V}_2\left(l\right)& =V_2(l+1)-V_2\left(l\right).\hfill \end{array}$$

(33)

In what follows, we will show the mathematical expectation $\mathbf{E}\left\{\mathsf{\Delta }V\right(l\left)\right|\xi \left(l\right)\}$ satisfies

$$\begin{array}{c}\hfill \mathbf{E}\left\{\mathsf{\Delta }V\right(l\left)\right|\xi \left(l\right)\}<0,\end{array}$$

(34)

where $\xi \left(l\right)={\left[\begin{array}{ccc}{\overline{x}}_c^T\left(l\right)& {\overline{e}}^T\left(l\right)& {\overline{e}}_y^T\left(l\right)\end{array}\right]}^T$.

Firstly, it follows from (16) that:

$$\mathbf{E}\left(\mathsf{\Delta }{V}_1\left(l\right)\right|\xi \left(l\right))=\left[{\overline{x}}_c^T\left(l\right){\overline{e}}^T\left(l\right){\overline{e}}_y^T\left(l\right)\right]{\mathbf{\Pi }}_1\left[\begin{array}{c}{\overline{x}}_c\left(l\right)\\ \overline{e}\left(l\right)\\ {\overline{y}}_d\left(l\right)\end{array}\right],$$

(35)

where

$$\begin{array}{cc}\hfill {\mathbf{\Pi }}_{\mathbf{1}}& =\left[\begin{array}{ccc}{\overline{A}}^{T}P\overline{A}-\overline{P}& 0& {\overline{A}}^{T}P{\overline{B}}_L\\ 0& 0& 0\\ {\overline{B}}_L^{T}P\overline{A}& 0& {\overline{B}}_L^{T}P{\overline{B}}_L\end{array}\right]\hfill \end{array}$$

(36)

with ${\overline{B}}_L=\overline{B}L.$ Similarly, Equation (17) implies that

$$\begin{gathered} \begin{array}{cc}\hfill \mathbf{E}\left\{\mathsf{\Delta }{V}_2\left(l\right)|\xi \left(l\right)\right\}& =\mathbf{E}\left\{e^T(l+1)Qe(l+1)-{\overline{e}}^T\left(l\right)\overline{Q}\overline{e}\left(l\right)|\xi \left(l\right)\right\}\hfill \end{array} \\ =\left[\begin{array}{ccc}{\overline{x}}_c^T\left(l\right)& {\overline{e}}^T\left(l\right)& {\overline{e}}_y^T\left(l\right)\end{array}\right]{\mathbf{\Pi }}_{\mathbf{2}}\left[\begin{array}{c}{\overline{x}}_c\left(l\right)\\ \overline{e}\left(l\right)\\ {\overline{e}}_y\left(l\right)\end{array}\right] \end{gathered}$$

(37)

in which

$$\begin{array}{c}\hfill {\mathbf{\Pi }}_{\mathbf{2}}=\left[\begin{array}{ccc}{W}^2{\overline{B}}_G^{T}Q{\overline{B}}_G& -W{\overline{B}}_G^{T}Q\overline{A}& -W{\overline{B}}_G^{T}Q\overline{B}L\\ 0& {\overline{A}}^{T}Q\overline{A}-\overline{Q}& {\overline{A}}^{T}Q\overline{B}L\\ 0& 0& {\overline{B}}_L^{T}Q\overline{B}L\end{array}\right].\end{array}$$

(38)

Secondly, the output-based event trigger condition shows that

$$\begin{array}{c}\hfill {\overline{e}}_y^T\left(l\right)(I_2\otimes W){\overline{e}}_y^T\left(l\right)\le {\overline{y}}^T\left(l\right)(\delta I_2\otimes W)\overline{y}\left(l\right)\end{array}$$

(39)

for any $l\in [l_i,l_{i+1})$. Recalling the estimation output compensation system (35) and (37), we have:

$$\begin{array}{cc}\hfill \mathbf{E}\left\{\mathsf{\Delta }V\left(l\right)\left|\xi \right(l)\right\}=& \mathbf{E}\left\{\mathsf{\Delta }{V}_1\left(l\right)+\mathsf{\Delta }{V}_2\left(l\right)|\xi \left(l\right)\right\}\hfill \\ \hfill \le & \mathbf{E}\{\mathsf{\Delta }{V}_1\left(l\right)+\mathsf{\Delta }{V}_2\left(l\right)|\xi \left(l\right)\}+{\overline{y}}^T\left(l\right)(\delta I_2\otimes W)\overline{y}\left(l\right)-{\overline{e}}_y^T\left(l\right)(I_2\otimes W){\overline{e}}_y\left(l\right)\hfill \\ \hfill =& \left[\begin{array}{ccc}{\overline{x}}_c^T\left(l\right)& {\overline{e}}^T\left(l\right)& {\overline{e}}_y^T\left(l\right)\end{array}\right]({\mathbf{\Pi }}_{\mathbf{1}}+{\mathbf{\Pi }}_{\mathbf{2}}+{\mathbf{\Pi }}_{\mathbf{3}})\left[\begin{array}{c}{\overline{x}}_c\left(l\right)\\ \overline{e}\left(l\right)\\ {\overline{y}}_d\left(l\right)\end{array}\right]\hfill \end{array}$$

(40)

where

$$\begin{array}{c}\hfill {\mathbf{\Pi }}_{\mathbf{3}}=\left[\begin{array}{ccc}\mathbf{diag}\left(C^T\right)\delta I\otimes W\mathbf{diag}\left(C\right)& 0& 0\\ 0& \mathbf{diag}\left(C^T\right)\delta I\otimes W\mathbf{diag}\left(C\right)& 0\\ 0& 0& -I\otimes W\end{array}\right].\end{array}$$

(41)

To summarize, it follows that

$$\begin{array}{cc}\hfill \mathbf{E}\left\{\mathsf{\Delta }V\left(l\right)\left|\xi \right(l)\right\}& =\mathbf{E}\left\{\mathsf{\Delta }{V}_1\left(l\right)+\mathsf{\Delta }{V}_2\left(l\right)|\xi \left(l\right)\right\}\hfill \\ \hfill & \le \xi {\left(l\right)}^T\left\{{\mathbf{\Pi }}_{\mathbf{1}}+{\mathbf{\Pi }}_{\mathbf{2}}+{\mathbf{\Pi }}_{\mathbf{3}}\right\}\xi \left(l\right)<0.\hfill \end{array}$$

(42)

which implies the 2D networked control system (16) and (17) is mean square stable under the output-based sliding mode control rate (14) and even-triggered conditions (6). We have completed the proof.    □

Note that the existence of coupling among the matrices in Theorem 2 makes the conditions proposed in Theorem 2 infeasible to realize. For instance, the factor ${\left({\overline{B}}_G\right)}^{T}P{\overline{B}}_G$ includes the nonlinear coupling of the controller G and the undetermined gain P, which yields that the feasible solutions are infeasible to be obtained directly from condition (27). Fortunately, it holds that the condition (27) satisfies if and only if the following executable matrix constraint can be realized through the controller design $G={\overline{B}}^{T}P$ and Schur compensation technology

$$\begin{array}{c}\hfill \left[\begin{array}{cccc}-\lambda \overline{P}& A^{T}P& \mathbf{diag}\left(C^T\right)& w_0{A}^T{\left({\overline{B}}^{T}P\right)}^T\\ 0& -P& 0& 0\\ 0& 0& -{(\delta I\otimes W)}^{-1}& 0\\ 0& 0& 0& -w_0^2\left({\overline{B}}^{T}P\overline{B}\right)\end{array}\right]<0.\end{array}$$

(43)

Consequently, we developed an adaptive intelligence algorithm as follows.

4. Numerical Examples

In this section, to check the effectiveness of the proposed criteria and algorithms, we will take the following Darboux equation as an illustrative example,

$$\begin{array}{c}\hfill \frac{{\partial }^{2}s(x,t)}{\partial x\partial t}=a_1\frac{\partial s(x,t)}{\partial t}+a_2\frac{\partial s(x,t)}{\partial x}+a_{0}s(x,t)+bg(x,t).\end{array}$$

(44)

in which $a_0=0.8,a_1=1.2,a_2=0.75,\mathrm{and}b=1.25$. The Darboux equation is always used to describe the practical process, such as air drying, water stream heating, and gas absorption [20]. By applying the similar technique in [20], the system model (44) can be converted into the FMII system in terms of the form given by

$$\begin{array}{cc}\hfill x(h+1,k+1)=& A_{1}x(h+1,k)+A_{2}x(h,k+1)\hfill \\ \hfill & +B_1[u(h+1,k)+f_1\left(x(h+1,k)\right)]\hfill \\ \hfill & +B_2[u(h,k+1)+f_2\left(x(h,k+1)\right)]\hfill \\ \hfill y(h,k)=& Cx(h,k),\hfill \end{array}$$

where $x(h,k)\in {\mathbb{R}}^2$, $f\left(x(h,k)\right)={[x_1^2(h,k)+x_2^2(h,k)]}^{1/2}$, and the coefficient matrices are

$$A_1=\left[\begin{array}{cc}1.08& 0.97\\ 2.01& -2.35\end{array}\right],A_2=\left[\begin{array}{cc}1.2& 1.25\\ 1.05& 0.9\end{array}\right],B_1=\left[\begin{array}{c}-1.02\\ 0.25\end{array}\right],B_2=\left[\begin{array}{c}1.05\\ 0.5\end{array}\right].$$

Take the event-triggered parameters $\delta =0.6$ and $W=I$ and $\mathbf{E}\left\{w\right(h,k)=1\}=0.75.$ It follows from matrix constraints (27)–(30) in Theorem 2 and Algorithm 1 that:

$$P_1=\left[\begin{array}{cc}1.0496& -0.0156\\ -0.0156& 1.2351\end{array}\right];P_2=\left[\begin{array}{cc}1.0306& 0.1132\\ 0.1132& 1.0158\end{array}\right];$$

$$G\overline{B}=\left[\begin{array}{cc}2.4320& -2.0786\\ -2.0063& 2.9189\end{array}\right];$$

$${\overline{L}}^T\left(G\overline{B}\right)=\left[\begin{array}{cc}2.0521& 0.9813\\ 0.2427& 1.0773\\ 1.1251& 0.8852\\ 1.4583& 1.6692\end{array}\right]$$

and then we obtain:

$$G={\overline{B}}^{T}P=\left[\begin{array}{cc}-2.2468& 0.5611\\ 2.2432& 1.1271\end{array}\right],$$

$$\overline{L}=\left[\begin{array}{cccc}2.7176& 0.9799& 1.7278& 2.5970\\ 2.2714& 1.0669& 1.5336& 2.4212\end{array}\right].$$

The state component responses $x_1$ and $x_2$ are shown in Figure 1 and Figure 2, respectively. It is not hard to find that the 2D system (44) with the given coefficient matrices in Example is stable under event triggering conditions $\delta =0.6$, $W=I$ and the sliding mode controller (18).

Algorithm 1 Adaptive intelligence algorithm design for Theorem 2.

Step 1: Set the parameter $\lambda$, $\delta$ and compute the matrix constraints given by (43) with the feasible gain matrix $P>0$. Step 2: if The matrix constraints (43) are satisfied then     Compute the controller gain matrices $G={\overline{B}}^{T}P$ and ${\overline{B}}_G=\overline{B}{\left(G\overline{B}\right)}^{-1}G\overline{A}$ by virtue of P in (43). else Return to Step 1 and resolve the matrices and parameters in (43). end if

The sliding surface function components $s_1$ and $s_2$ are provided as in Figure 3 and Figure 4, respectively. Based on the sliding mode controller design in condition (18), the evolution of the controller components $u_1$ and $u_2$ are plotted as in Figure 5 and Figure 6, respectively.

5. Conclusions

In this paper, we considered the discrete 2D FMII networks subject to packet dropouts. To tackle the unmeasurable status caused by the data packet loss during the information transmission, we introduced one 2D output-based event-triggered protocol and constructed a sliding mode controller satisfying Bernoulli stochastic processes. In addition, we checked the reachability of the operational 2D sliding mode surface, and proved that the obtained estimation system was mean square stable. Finally, to show the effectiveness and practicability of the aforementioned algorithms and controller design, we provided a numerical example.

We conclude this section with some suggestions for future work. Whereas the matrix constraints proposed in this paper are sufficient and conservative, a venue for future research will be to improve the provided criterion further and reduce the conservatism of the criterion.