Robust reliable control of uncertain 2D discrete switched systems with state delays

Abstract

This paper is concerned with the problem of robust reliable control for a class of uncertain 2D discrete switched systems with state delays and actuator faults represented by a model of Roesser type. The parameter uncertainties are assumed to be norm-bounded. Firstly, based on the average dwell time approach, a delay-dependent sufficient condition for the exponential stability of discrete 2D switched systems with state delays is established in terms of linear matrix inequalities. Then, a reliable state feedback controller is designed to guarantee the exponential stability and reliability for the underlying systems. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Keywords

2D discrete systems average dwell time reliable control state delays switched systems

Introduction

Two-dimensional (2D) systems have received considerable attention over the past few decades due to their wide applications in many areas such as multi-dimensional digital filtering, linear image processing, signal processing and process control (Du and Xie, 2002; Kaczorek, 1985; Lu, 1992). The stability analysis of 2D discrete systems has attracted a great deal of interest and some significant results have been obtained (Anderson et al., 1986; Fornasini and Marchesini, 1986; Lu and Lee, 1985; Ye and Wang, 2011). Furthermore, time delays frequently occur in practical systems and are often the source of instability, and there are many examples containing inherent delays in practical 2D discrete systems; the stability of 2D discrete systems with state delays have also been studied (Chen, 2010a, 2010b, 2010c; Feng et al., 2010; Ye et al., 2008, and the references cited therein).

On the other hand, switched systems have also been attracting considerable attention over the past several decades (Branicky, 1998; Hespanha and Morse, 1999; Hu and Yuan, 2009; Li et al., 2009; Sun XM et al., 2006; Sun YG et al., 2006, 2007; Wang and Zhao, 2007; Xie and Wang, 2005; Zhai et al., 2000). A switched system is a hybrid system, which consists of a finite number of continuous-time or discrete-time subsystems and a switching signal specifying the switch between these subsystems. This class of systems has numerous applications in many fields, such as mechanical systems, the automotive industry, aircraft and air traffic control, and switched power converters. So far, the dwell time approach is an important and effective approach to study switched systems. Recently, the dwell time approach has been applied widely to deal with switched systems (Hespanha and Morse, 1999; Wang and Zhao, 2007; Zhai et al., 2000, and references therein).

However, the switch phenomenon may also occur in 2D discrete systems, and the study of these 2D discrete switched systems will also be significant. There are presently a few reports on 2D discrete switched systems. The stabilization problem of 2D discrete switched systems was investigated in Benzaouia et al. (2011), and sufficient conditions for the existence of a stabilizing controller such that the resulting closed-loop system is asymptotically stable were presented in terms of linear matrix inequalities (LMIs). Recently, Xiang and Huang (2013) investigated the exponential stability of delay-free 2D switched systems via the average dwell time approach, and a switching rule is designed to guarantee the exponential stability of such systems. It should be pointed out that, in the aforementioned results, it has been implicitly assumed that the actuators are perfect. However, the actuators may be subjected to failures in practical operation. When failure occurs, the conventional controller will become conservative and may not satisfy certain control performance indices. Reliable control is a kind of effective control approach to improve system reliability. Several approaches for designing reliable controllers have been proposed, some of which have been used to research the problem of reliable control for switched systems (Wang et al., 2007a, 2007b, 2008; Xiang and Wang, 2009; Zhang and Yu, 2012). To the best of our knowledge, the issue of robust reliable control for 2D discrete switched systems with state delays has not been investigated to date, which motivates our present study.

In this paper, we are interested in designing a reliable stabilizing controller for 2D discrete switched system with state delay and actuator fault such that the resulting closed-loop system is exponentially stable. The average dwell time approach is utilized for the stability analysis and controller design. The main contributions of this paper can be summarized as follows: 1) The stability and reliability of 2D discrete switched systems in the presence of actuator failures are first considered; and 2) a reliable state feedback controller design scheme is proposed for such system, and a controller synthesis condition is formulated in terms of a set of LMIs.

The remainder of the paper is organized as follows. Next, the problem formulation and some necessary lemmas are given. Then, based on the average dwell time approach, the problem of stability analysis for 2D discrete switched systems with state delays are addressed, and a delay-dependent sufficient condition for the existence of a robust reliable controller is derived in terms of a set of LMIs. A numerical example is provided to illustrate the effectiveness of the proposed approach, followed by concluding remarks.

Notations

Throughout this paper, the superscript ‘T’ denotes the transpose, and the notation $X \geq Y$ $(X > Y)$ means that matrix $X - Y$ is positive semi-definite (positive definite, respectively). $‖ \cdot ‖$ denotes the Euclidean norm. $I$ represents identity matrix with appropriate dimension. $I_{h}$ is the identity matrix with $n_{1}$ dimension and $I_{v}$ is the identity matrix with $n_{2}$ dimension. $diag {a_{i}}$ denotes the diagonal matrix with the diagonal elements $a_{i}$ , $i = 1, 2, \dots, n$ . $X^{- 1}$ denotes the inverse of $X$ . The asterisk $*$ in a matrix is used to denote term that is induced by symmetry. The set of all non-negative integers is represented by $Z_{+}$ .

Problem formulation and preliminaries

Consider the following uncertain 2D discrete switched linear systems:

\begin{matrix} [\begin{matrix} x^{h} (i + 1, j) \\ x^{v} (i, j + 1) \end{matrix}] = {\hat{A}}^{σ (i, j)} [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] \\ + {\hat{A}}_{d}^{σ (i, j)} [\begin{matrix} x^{h} (i - d_{h} (i), j) \\ x^{v} (i, j - d_{v} (j)) \end{matrix}] + B^{σ (i, j)} u^{f} (i, j) \end{matrix}

(1)

where $x^{h} (i, j)$ is the horizontal state in $R^{n_{1}}$ , $x^{v} (i, j)$ is the vertical state in $R^{n_{2}}$ and $x (i, j)$ is the whole state in $R^{n}$ with $n = n_{1} + n_{2}$ . $u^{f} (i, j) \in R^{m}$ is the control input of actuator fault. i and j are integers in $Z_{+}$ . $σ (i, j) : Z_{+} \times Z_{+} \to \underline{N} = {1, 2, \dots, N}$ is the switching signal, $N$ is the number of subsystems. $σ (i, j) = k$ , $k \in \underline{N}$ , denotes that the kth subsystem is active. $d_{h} (i)$ and $d_{v} (j)$ are delays along horizontal and vertical directions, respectively. We assume that $d_{h} (i)$ and $d_{v} (j)$ satisfy

d_{hL} \leq d_{h} (i) \leq d_{hH}, d_{vL} \leq d_{v} (j) \leq d_{vH}

(2)

where $d_{hL}$ , $d_{hH}$ and $d_{vL}$ , $d_{vH}$ denote the lower and upper delay bounds along horizontal and vertical directions, respectively.

$B^{k}$ , $k \in \underline{N}$ , are real matrices with appropriate dimensions. ${\hat{A}}^{k}$ and ${\hat{A}}_{d}^{k}$ , $k \in \underline{N}$ , are uncertain real-valued matrices with appropriate dimensions and assumed to be of the form

{\hat{A}}^{k} = A^{k} + H^{k} F^{k} (i, j) E^{k}

{\hat{A}}_{d}^{k} = A_{d}^{k} + H^{k} F^{k} (i, j) E_{d}^{k}

(3)

with

A^{k} = [\begin{matrix} A_{11}^{k} & A_{12}^{k} \\ A_{21}^{k} & A_{22}^{k} \end{matrix}], A_{d}^{k} = [\begin{matrix} A_{d 11}^{k} & A_{d 12}^{k} \\ A_{d 21}^{k} & A_{d 22}^{k} \end{matrix}]

H^{k} = [\begin{matrix} H_{1}^{k} \\ H_{2}^{k} \end{matrix}], E^{k} = [\begin{matrix} E_{1}^{k} E_{2}^{k} \end{matrix}], E_{d}^{k} = [\begin{matrix} E_{d 1}^{k} E_{d 2}^{k} \end{matrix}]

where $A_{11}^{k} \in R^{n_{1} \times n_{1}}$ , $A_{12}^{k} \in R^{n_{1} \times n_{2}}$ , $A_{21}^{k} \in R^{n_{2} \times n_{1}}$ , $A_{22}^{k} \in R^{n_{2} \times n_{2}}$ , $A_{d 11}^{k} \in R^{n_{1} \times n_{1}}$ , $A_{d 12}^{k} \in R^{n_{1} \times n_{2}}$ , $A_{d 21}^{k} \in R^{n_{2} \times n_{1}}$ , $A_{d 22}^{k} \in R^{n_{2} \times n_{2}}$ , $H_{1}^{k} \in R^{n_{1} \times p}$ , $H_{2}^{k} \in R^{n_{2} \times p}$ , $E_{1}^{k} \in R^{q \times n_{1}}$ and $E_{2}^{k} \in R^{q \times n_{2}}$ are constant matrices. $F^{k} (i, j) \in R^{p \times q}$ are unknown matrices representing parameter uncertainties, and they satisfy

F^{k} {(i, j)}^{T} F^{k} (i, j) \leq I

(4)

The control input of actuator fault $u^{f} (i, j)$ can be described as

u^{f} (i, j) = Ω^{σ (i, j)} u (i, j)

(5)

where $u (i, j) = K^{σ (i, j)} x (i, j)$ is the control input to be designed, $Ω^{k}$ , $k \in \underline{N}$ , are the actuator fault matrices with the following form

Ω^{k} = diag {ω_{k 1}, ω_{k 2}, \dots, ω_{kg}, \dots, ω_{km}}

(6)

where $0 \leq ω_{Lkg} \leq ω_{kg} \leq ω_{Hkg}$ .

For simplicity, we denote

Ω_{0}^{k} = diag {{\tilde{ω}}_{k 1}, {\tilde{ω}}_{k 2}, \dots, {\tilde{ω}}_{kg}, \dots, {\tilde{ω}}_{km}}, {\tilde{ω}}_{kg} = \frac{1}{2} (ω_{Lkg} + ω_{Hkg})

(7)

Ξ^{k} = diag {ξ_{k 1}, ξ_{k 2}, \dots, ξ_{kg}, \dots, ξ_{km}}, ξ_{kg} = \frac{ω_{Hkg} - ω_{Lkg}}{ω_{Hkg} + ω_{Lkg}}

(8)

Θ^{k} = diag {Θ_{k 1}, Θ_{k 2}, \dots, Θ_{kg}, \dots, Θ_{km}}, Θ_{kg} = \frac{ω_{kg} - {\tilde{ω}}_{kg}}{{\tilde{ω}}_{kg}}

(9)

It follows that

Ω^{k} = Ω_{0}^{k} (I + Θ^{k}), | Θ^{k} | \leq Ξ^{k} \leq I

(10)

where $| Θ^{k} | = diag {| Θ_{k 1} |, | Θ_{k 2} |, \dots, | Θ_{kg} |, \dots, | Θ_{km} |}$ .

The boundary conditions are given by

\begin{matrix} x^{h} (i, j) = h_{ij}, \forall 0 \leq j \leq z_{1}, - d_{hH} \leq i \leq 0 \\ x^{h} (i, j) = 0, \forall j > z_{1}, - d_{hH} \leq i \leq 0 \\ x^{v} (i, j) = v_{ij}, \forall 0 \leq i \leq z_{2}, - d_{vH} \leq j \leq 0 \\ x^{v} (i, j) = 0, \forall i > z_{2}, - d_{vH} \leq j \leq 0 \end{matrix}

(11)

Where $z_{1} < \infty$ and $z_{2} < \infty$ are positive integers, $h_{ij}$ and $v_{ij}$ are given vectors.

Remark 1. It is assumed that the switch occurs only at each sampling point of i or j in the paper. The switching sequence of the system can be described as

((i_{0}, j_{0}), σ (i_{0}, j_{0})), ((i_{1}, j_{1}), σ (i_{1}, j_{1})), \dots,, ((i_{κ}, j_{κ}), σ (i_{κ}, j_{κ})), \dots

(12)

with $(i_{0}, j_{0}) = (0, 0)$ , $(i_{κ}, j_{κ})$ denotes the $κ$ th switching instant. It should be noted that the value of $σ (i, j)$ only depends on $i + j$ (Benzaouia et al., 2011; Xiang and Huang, 2013).

Definition 1. System (1) is said to be exponentially stable under the switching signal $σ (i, j)$ if for a given $z \geq 0$ , there exist positive constants $c$ and $η$ , such that

\sum_{i + j = D} {‖ x (i, j) ‖}^{2} \leq η e^{- c (D - z)} \sum_{i + j = z} ‖ x (i, j) ‖_{C}^{2}

(13)

holds for all $D \geq z$ , where

\begin{matrix} \sum_{i + j = z} ‖ x (i, j) ‖_{C}^{2} \overset{Δ}{=} sup_{\binom{- d_{hH} \leq θ_{h} \leq 0,}{- d_{vH} \leq θ_{v} \leq 0}} \\ \sum_{i + j = z} {{‖ x^{h} (i - θ_{h}, j) ‖}^{2} + {‖ x^{v} (i, j - θ_{v}) ‖}^{2}, \end{matrix}

{‖ η^{h} (i - θ_{h}, j) ‖}^{2} + {‖ η^{v} (i, j - θ_{v}) ‖}^{2}}

η^{h} (i - θ_{h}, j) = x^{h} (i - θ_{h} + 1, j) - x^{h} (i - θ_{h}, j)

η^{v} (i, j - θ_{v}) = x^{v} (i, j + 1 - θ_{v}) - x^{v} (i, j - θ_{v})

Remark 2. From Definition 1, it is easy to see that when $z$ is given, $\sum_{i + j = z} ‖ x (i, j) ‖_{C}^{2}$ will be bounded and $\sum_{i + j = D} {‖ x (i, j) ‖}^{2}$ will tend to be zero exponentially as $D$ goes to infinity, which also means $‖ x (i, j) ‖$ tends to be zero.

Definition2. (Xiang and Huang, 2013). For any $i + j = D \geq z = i_{z} + j_{z}$ , let $N_{σ} (z, D)$ denote the switching number of $σ (i, j)$ on the interval $[z, D)$ . If

N_{σ} (z, D) \leq N_{0} + \frac{D - z}{τ_{a}}

(14)

holds for given $N_{0} \geq 0$ , $τ_{a} \geq 0$ , then the constant $τ_{a}$ is called the average dwell time and $N_{0}$ is the chatter bound.

Lemma 1 (Boyd et al., 1994). For a given matrix $S = [\begin{matrix} S_{11} & S_{12} \\ S_{12}^{T} & S_{22} \end{matrix}]$ , where $S_{11}$ , $S_{22}$ are square matrices, the following conditions are equivalent.

$S < 0$ ;

$S_{11} < 0, S_{22} - S_{12}^{T} S_{11}^{- 1} S_{12} < 0;$

$S_{22} < 0, S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} < 0 .$

Lemma 2 (Xie, 1996). Let $U$ , $V$ , $W$ and $X$ be real matrices of appropriate dimensions with $X$ satisfying $X = X^{T}$ , then for all $V^{T} V \leq I$ , $X + UVW + W^{T} V^{T} U^{T} < 0$ , if and only if there exists a scalar $ε$ such that $X + ε U U^{T} + ε^{- 1} W^{T} W < 0$ .

Lemma 3 (Petersen, 1987). For matrices $R_{1}$ , $R_{2}$ with appropriate dimensions, there exists a positive scalar $ε$ , such that

R_{1} Σ R_{2} + R_{2}^{T} Σ^{T} R_{1}^{T} \leq ε R_{1} {UR}_{1}^{T} + ε^{- 1} R_{2}^{T} U R_{2}

holds, where $Σ$ is a diagonal matrix and $U$ is known real-value matrix satisfying $| Σ | \leq U$ .

Main results

Stability analysis

In this subsection, we first investigate the problem of stability analysis for the following 2D discrete systems.

[\begin{matrix} x^{h} (i + 1, j) \\ x^{v} (i, j + 1) \end{matrix}] = A [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + A_{d} [\begin{matrix} x^{h} (i - d_{h} (i), j) \\ x^{v} (i, j - d_{v} (j)) \end{matrix}]

(15)

where $A$ and $A_{d}$ are constant matrices with appropriate dimensions; $d_{h} (i)$ and $d_{v} (j)$ satisfy (2); The boundary conditions are given in (11).

Lemma 4. For a given positive scalar $α < 1$ , if there exist positive definite symmetric matrices $P = diag {P_{h}, P_{v}}$ , $Q_{1} = diag {Q_{1 h}, Q_{1 v}}$ , $Q_{2} = diag {Q_{2 h}, Q_{2 v}}$ , $W_{1} = diag {W_{1 h}, W_{1 v}}$ , $W_{2} = diag {W_{2 h}, W_{2 v}}$ , $X_{h} = [\begin{matrix} X_{11 h} & X_{12 h} \\ * & X_{22 h} \end{matrix}]$ , $X_{v} = [\begin{matrix} X_{11 v} & X_{12 v} \\ * & X_{22 v} \end{matrix}]$ , $Y_{h} = [\begin{matrix} Y_{11 h} & Y_{12 h} \\ * & Y_{22 h} \end{matrix}]$ $Y_{v} = [\begin{matrix} Y_{11 v} & Y_{12 v} \\ * & Y_{22 v} \end{matrix}]$ , and any matrices $M_{h} = [\begin{matrix} M_{1}^{h} \\ M_{2}^{h} \end{matrix}]$ , $N_{h} = [\begin{matrix} N_{1}^{h} \\ N_{2}^{h} \end{matrix}]$ , $S_{h} = [\begin{matrix} S_{1}^{h} \\ S_{2}^{h} \end{matrix}]$ , $M_{v} = [\begin{matrix} M_{1}^{v} \\ M_{2}^{v} \end{matrix}],$ $N_{v} = [\begin{matrix} N_{1}^{v} \\ N_{2}^{v} \end{matrix}]$ and $S_{v} = [\begin{matrix} S_{1}^{v} \\ S_{2}^{v} \end{matrix}]$ with appropriate dimensions, such that

Φ = [\begin{matrix} Φ_{1} & Φ_{2}^{T} & Φ_{2}^{T} & Φ_{3}^{T} P \\ * & - Λ_{1}^{- 1} W_{1}^{- 1} & 0 & 0 \\ * & * & - Λ_{2}^{- 1} W_{2}^{- 1} & 0 \\ * & * & * & - P \end{matrix}] < 0

(16)

Ψ = diag {Ψ_{1 h}, Ψ_{2 h}, Ψ_{3 h}, Ψ_{1 v}, Ψ_{2 v}, Ψ_{3 v}} \geq 0

(17)

hold, then along the trajectory of system (15), we have

\sum_{i + j = D} V (i, j) < α^{D - z} \sum_{i + j = z} V (i, j)

(18)

where

Φ_{1} = [\begin{matrix} Φ_{11} & Φ_{12} & Φ_{13} & Φ_{14} \\ * & Φ_{22} & Φ_{23} & Φ_{24} \\ * & * & Φ_{33} & 0 \\ * & * & * & Φ_{44} \end{matrix}]

Λ_{1} = diag {d_{hL} I_{h}, d_{vL} I_{v}}, Λ_{2} = diag {(d_{hH} - d_{hL}) I_{h}, (d_{vH} - d_{vL}) I_{v}}

Λ_{3} = diag {α^{d_{hL}} I_{h}, α^{d_{vL}} I_{v}}, Λ_{4} = diag {α^{d_{hH}} I_{h}, α^{d_{vH}} I_{v}}

\begin{matrix} Φ_{11} = Q_{1} + Q_{2} + Λ_{3} (M_{1} + M_{1}^{T} + Λ_{1} X_{11}) \\ + Λ_{2} Λ_{4} Y_{11} - α P, Φ_{23} = - Λ_{3} M_{2} + Λ_{4} S_{2} \end{matrix}

\begin{matrix} Φ_{12} = Λ_{3} (Λ_{1} X_{12} + M_{2}^{T}) + Λ_{4} (N_{1} - S_{1} + Λ_{2} Y_{12}), \\ Φ_{13} = - Λ_{3} M_{1} + Λ_{4} S_{1}, Φ_{14} = - Λ_{4} N_{1} \end{matrix}

\begin{matrix} Φ_{22} = Λ_{1} Λ_{3} X_{22} + Λ_{4} (N_{2} + N_{2}^{T} - S_{2} - S_{2}^{T} + Λ_{2} Y_{22}), \\ Φ_{24} = - Λ_{4} N_{2}, Φ_{33} = - Λ_{3} Q_{1} \end{matrix}

Φ_{44} = - Λ_{4} Q_{2}, Φ_{2} = [\begin{matrix} A - I & A_{d} & 0 & 0 \end{matrix}], Φ_{3} = [\begin{matrix} A & A_{d} & 0 & 0 \end{matrix}]

X_{11} = diag {X_{11}^{h}, X_{11}^{v}}, X_{12} = diag {X_{12}^{h}, X_{12}^{v}}, X_{22} = diag {X_{22}^{h}, X_{22}^{v}}

Y_{11} = diag {Y_{11}^{h}, Y_{11}^{v}}, Y_{12} = diag {Y_{12}^{h}, Y_{12}^{v}}, Y_{22} = diag {Y_{22}^{h}, Y_{22}^{v}}

M_{1} = diag {M_{1}^{h}, M_{1}^{v}}, M_{2} = diag {M_{2}^{h}, M_{2}^{v}}, N_{1} = diag {N_{1}^{h}, N_{1}^{v}}

N_{2} = diag {N_{2}^{h}, N_{2}^{v}}, S_{1} = diag {S_{1}^{h}, S_{1}^{v}}, S_{2} = diag {S_{2}^{h}, S_{2}^{v}}

Ψ_{1 h} = [\begin{matrix} X_{h} & M_{h} \\ * & W_{1 h} \end{matrix}], Ψ_{2 h} = [\begin{matrix} Y_{h} & N_{h} \\ * & W_{2 h} \end{matrix}], Ψ_{3 h} = [\begin{matrix} Y_{h} & S_{h} \\ * & W_{2 h} \end{matrix}]

Ψ_{1 v} = [\begin{matrix} X_{v} & M_{v} \\ * & W_{1 v} \end{matrix}], Ψ_{2 v} = [\begin{matrix} Y_{v} & N_{v} \\ * & W_{2 v} \end{matrix}], Ψ_{3 v} = [\begin{matrix} Y_{v} & S_{v} \\ * & W_{2 v} \end{matrix}] .

Proof. Consider the following Lyapunov–Krasovskii functional candidate

V (x (i, j)) = V^{h} (x^{h} (i, j)) + V^{v} (x^{v} (i, j))

(19)

where

\begin{matrix} V^{h} (x^{h} (i, j)) = \sum_{g = 1}^{3} V_{g}^{h} (x^{h} (i, j)) \\ V_{1}^{h} (x^{h} (i, j)) = x^{h} {(i, j)}^{T} P_{h} x^{h} (i, j) \\ V_{2}^{h} (x^{h} (i, j)) = \sum_{r = i - d_{hL}}^{i - 1} x^{h} {(r, j)}^{T} Q_{1 h} x^{h} (r, j) α^{i - r - 1} \\ + \sum_{r = i - d_{hH}}^{i - 1} x^{h} {(r, j)}^{T} Q_{2 h} x^{h} (r, j) α^{i - r - 1} \\ V_{3}^{h} (x^{h} (i, j)) = \sum_{s = - d_{hL}}^{- 1} \sum_{r = i + s}^{i - 1} η^{h} {(r, j)}^{T} W_{1 h} η^{h} (r, j) α^{i - r - 1} \\ + \sum_{s = - d_{hH}}^{- d_{hL} - 1} \sum_{r = i + s}^{i - 1} η^{h} {(r, j)}^{T} W_{2 h} η^{h} (r, j) α^{i - r - 1} \\ V^{v} (x^{v} (i, j)) = \sum_{g = 1}^{3} V_{g}^{v} (x^{v} (i, j)) \\ V_{1}^{v} (x^{v} (i, j)) = x^{v} {(i, j)}^{T} P_{v} x^{v} (i, j) \\ V_{2}^{v} (x^{v} (i, j)) = \sum_{t = j - d_{vL}}^{j - 1} x^{v} {(i, t)}^{T} Q_{1 v} x^{v} (i, t) α^{j - t - 1} \\ + \sum_{t = j - d_{vH}}^{j - 1} x^{v} {(i, t)}^{T} Q_{2 v} x^{v} (i, t) α^{j - t - 1} \end{matrix}

\begin{matrix} V_{3}^{v} (x^{v} (i, j)) = \sum_{s = - {d_{v}}_{L}}^{- 1} \sum_{t = j + s}^{j - 1} η^{v} {(i, t)}^{T} W_{1 v} η^{v} (i, t) α^{j - t - 1} \\ + \sum_{s = - {d_{v}}_{H}}^{- {d_{v}}_{L} - 1} \sum_{t = j + s}^{j - 1} η^{v} {(i, t)}^{T} W_{2 v} η^{v} (i, t) α^{j - t - 1} \\ η^{h} (r, j) = x^{h} (r + 1, j) - x^{h} (r, j) \\ η^{v} (i, t) = x^{v} (i, t + 1) - x^{v} (i, t) \end{matrix}

Along the trajectory of system (15), one obtains

\begin{matrix} V^{h} (x^{h} (i + 1, j)) - α V^{h} (x^{h} (i, j)) + V^{v} (x^{v} (i, j + 1)) - α V^{v} (x^{v} (i, j)) \\ = \sum_{g = 1}^{3} [V_{g}^{h} (x^{h} (i + 1, j)) - α V_{g}^{h} (x^{h} (i, j))] \\ + \sum_{g = 1}^{3} [V_{g}^{v} (x^{v} (i, j + 1)) - α V_{g}^{v} (x^{v} (i, j))] \end{matrix}

(20)

with

\begin{matrix} V_{1}^{h} (x^{h} (i + 1, j)) - α V_{1}^{h} (x^{h} (i, j)) \\ = x^{h} {(i + 1, j)}^{T} P_{h} x^{h} (i + 1, j) - α x^{h} {(i, j)}^{T} P_{h} x^{h} (i, j) \\ V_{2}^{h} (x^{h} (i + 1, j)) - α V_{2}^{h} (x^{h} (i, j)) \\ = \sum_{r = i + 1 - d_{hL}}^{i} x^{h} {(r, j)}^{T} Q_{1 h} x^{h} (r, j) α^{i - r} \\ + \sum_{r = i + 1 - d_{hH}}^{i} x^{h} {(r, j)}^{T} Q_{2 h} x^{h} (r, j) α^{i - r} \\ - α \sum_{r = i - d_{hL}}^{i - 1} x^{h} {(r, j)}^{T} Q_{1 h} x^{h} (r, j) α^{i - r - 1} - α \\ \sum_{r = i - d_{hH}}^{i - 1} x^{h} {(r, j)}^{T} Q_{2 h} x^{h} (r, j) α^{i - r - 1} \\ = x^{h} {(i, j)}^{T} (Q_{1 h} + Q_{2 h}) x^{h} (i, j) - α^{d_{hL}} x^{h} {(i - d_{hL}, j)}^{T} Q_{1 h} x^{h} (i - d_{hL}, j) \\ - α^{d_{hH}} x^{h} {(i - d_{hH}, j)}^{T} Q_{2 h} x^{h} (i - d_{hH}, j) \\ V_{3}^{h} (x^{h} (i + 1, j)) - α V_{3}^{h} (x^{h} (i, j)) \\ \leq {d_{h}}_{L} η^{h} {(i, j)}^{T} W_{1 h} η^{h} (i, j) + ({d_{h}}_{H} - {d_{h}}_{L}) η^{h} {(i, j)}^{T} W_{2 h} η^{h} (i, j) \\ - α^{d_{hL}} \sum_{r = i - d_{hL}}^{i - 1} η^{h} {(r, j)}^{T} W_{1 h} η^{h} (r, j) - α^{d_{hH}} \sum_{r = i - d_{hH}}^{i - d_{h} (i) - 1} η^{h} {(r, j)}^{T} W_{2 h} η^{h} (r, j) \\ - α^{d_{hH}} \sum_{r = i - d_{h} (i)}^{i - d_{hL} - 1} η^{h} {(r, j)}^{T} W_{2 h} η^{h} (r, j) \\ V_{1}^{v} (x^{v} (i, j + 1)) - α V_{1}^{v} (x^{v} (i, j)) \\ = x^{v} {(i, j + 1)}^{T} P_{v} x^{v} (i, j + 1) - α x^{v} {(i, j)}^{T} P_{v} x^{v} (i, j) \\ V_{2}^{v} (x^{v} (i, j + 1)) - α V_{2}^{v} (x^{v} (i, j)) \\ = \sum_{t = j + 1 - d_{vL}}^{j} x^{v} {(i, t)}^{T} Q_{1 v} x^{v} (i, t) α^{j - t} \\ + \sum_{t = j + 1 - d_{vH}}^{j} x^{v} {(i, t)}^{T} Q_{2 v} x^{v} (i, t) α^{j - t} \end{matrix}

\begin{matrix} - α \sum_{t = j - d_{vL}}^{j - 1} x^{v} {(i, t)}^{T} Q_{1 v} x^{v} (i, t) α^{j - t - 1} \\ - α \sum_{t = j - d_{vH}}^{j - 1} x^{v} {(i, t)}^{T} Q_{2 v} x^{v} (i, t) α^{j - t - 1} \\ = x^{v} {(i, t)}^{T} (Q_{1 v} + Q_{2 v}) x^{v} (i, t) - α^{d_{vL}} x^{v} {(i, j - d_{vL})}^{T} Q_{1 v} x^{v} (i, j - d_{vL}) \\ - α^{d_{vH}} x^{v} {(i, j - d_{vH})}^{T} Q_{2 v} x^{v} (i, j - d_{vH}) \\ V_{3}^{v} (x^{v} (i, j + 1)) - α V_{3}^{v} (x^{v} (i, j)) \\ \leq {d_{v}}_{L} η^{v} {(i, j)}^{T} W_{1 v} η^{v} (i, j) + ({d_{v}}_{H} - {d_{v}}_{L}) η^{v} {(i, j)}^{T} W_{2 v} η^{v} (i, j) \\ - α^{d_{vL}} \sum_{t = j - d_{vL}}^{j - 1} η^{v} {(i, t)}^{T} W_{1 v} η^{v} (i, t) - α^{d_{vH}} \sum_{t = j - d_{vH}}^{j - d_{v} (j) - 1} η^{v} {(i, t)}^{T} W_{2 v} η^{v} (i, t) \\ - α^{d_{vH}} \sum_{t = j - d_{v} (j)}^{j - d_{vL} - 1} η^{v} {(i, t)}^{T} W_{2 v} η^{v} (i, t) \end{matrix}

On one hand, the following equations hold for any matrices $M_{h} = [\begin{matrix} M_{1}^{h} \\ M_{2}^{h} \end{matrix}]$ , $N_{h} = [\begin{matrix} N_{1}^{h} \\ N_{2}^{h} \end{matrix}]$ , $S_{h} = [\begin{matrix} S_{1}^{h} \\ S_{2}^{h} \end{matrix}]$ , $M_{v} = [\begin{matrix} M_{1}^{v} \\ M_{2}^{v} \end{matrix}]$ , $N_{v} = [\begin{matrix} N_{1}^{v} \\ N_{2}^{v} \end{matrix}]$ and $S_{v} = [\begin{matrix} S_{1}^{v} \\ S_{2}^{v} \end{matrix}]$ with appropriate dimensions:

0 = 2 α^{d_{hL}} [x^{h} {(i, j)}^{T} M_{1}^{h} + x^{h} {(i - d_{h} (i), j)}^{T} M_{2}^{h}]

\times [x^{h} (i, j) - x^{h} (i - d_{hL}, j) - \sum_{r = i - d_{hL}}^{i - 1} η^{h} (r, j)]

(21)

0 = 2 α^{d_{hH}} [x^{h} {(i, j)}^{T} N_{1}^{h} + x^{h} {(i - d_{h} (i), j)}^{T} N_{2}^{h}]

\times [x^{h} (i - d_{h} (i), j) - x^{h} (i - d_{hH}, j) - \sum_{r = i - d_{hH}}^{i - 1 - d_{h} (i)} η^{h} (r, j)]

(22)

0 = 2 α^{d_{hH}} [x^{h} {(i, j)}^{T} S_{1}^{h} + x^{h} {(i - d_{h} (i), j)}^{T} S_{2}^{h}]

\times [x^{h} (i - d_{hL}, j) - x^{h} (i - d_{h} (i), j) - \sum_{r = i - d_{h} (i)}^{i - 1 - d_{hL}} η^{h} (r, j)]

(23)

0 = 2 α^{d_{vL}} [x^{v} {(i, j)}^{T} M_{1}^{v} + x^{v} {(i, j - d_{v} (j))}^{T} M_{2}^{v}]

\times [x^{v} (i, j) - x^{v} (i, j - d_{vL}) - \sum_{t = j - d_{vL}}^{j - 1} η^{v} (i, t)]

(24)

0 = 2 α^{d_{vH}} [x^{v} {(i, j)}^{T} N_{1}^{v} + x^{v} {(i, j - d_{v} (j))}^{T} N_{2}^{v}]

\times [x^{v} (i, j - d_{v} (j)) - x^{v} (i, j - d_{vH}) - \sum_{t = j - d_{vH}}^{j - 1 - d_{v} (j)} η^{v} (i, t)]

(25)

0 = 2 α^{d_{vH}} [x^{v} {(i, j)}^{T} S_{1}^{v} + x^{h} {(i, j - d_{v} (j))}^{T} S_{2}^{v}]

\times [x^{v} (i, j - d_{vL}) - x^{v} (i, j - d_{v} (j)) - \sum_{t = j - d_{v} (j)}^{j - 1 - d_{vL}} η^{v} (i, t)]

(26)

On the other hand, for any matrices $X_{h} = [\begin{matrix} X_{11 h} & X_{12 h} \\ * & X_{22 h} \end{matrix}] > 0$ , $Y_{h} = [\begin{matrix} Y_{11 h} & Y_{12 h} \\ * & Y_{22 h} \end{matrix}] > 0$ , $X_{v} = [\begin{matrix} X_{11 v} & X_{12 v} \\ * & X_{22 v} \end{matrix}] > 0$ and $Y_{v} = [\begin{matrix} Y_{11 v} & Y_{12 v} \\ * & Y_{22 v} \end{matrix}] > 0$ , the following equations also hold

0 = α^{d_{hL}} (\sum_{r = i - d_{hL}}^{i - 1} ξ_{1}^{h} {(i, j)}^{T} X_{h} ξ_{1}^{h} (i, j) - \sum_{r = i - d_{hL}}^{i - 1} ξ_{1}^{h} {(i, j)}^{T} X_{h} ξ_{1}^{h} (i, j))

= α^{d_{hL}} (d_{hL} ξ_{1}^{h} {(i, j)}^{T} X_{h} ξ_{1}^{h} (i, j) - \sum_{r = i - d_{hL}}^{i - 1} ξ_{1}^{h} {(i, j)}^{T} X_{h} ξ_{1}^{h} (i, j))

(27)

\begin{matrix} 0 = α^{d_{hH}} (\sum_{r = i - d_{hH}}^{i - 1 - d_{hL}} ξ_{1}^{h} {(i, j)}^{T} Y_{h} ξ_{1}^{h} (i, j) - \sum_{r = i - d_{hH}}^{i - 1 - d_{hL}} ξ_{1}^{h} {(i, j)}^{T} Y_{h} ξ_{1}^{h} (i, j)) \\ = α^{d_{hH}} ((d_{hH} - d_{hL}) ξ_{1}^{h} {(i, j)}^{T} Y_{h} ξ_{1}^{h} (i, j) - \sum_{r = i - d_{h} (i)}^{i - 1 - d_{hL}} ξ_{1}^{h} {(i, j)}^{T} Y_{h} ξ_{1}^{h} (i, j) \\ - \sum_{r = i - d_{hH}}^{i - 1 - d_{h} (i)} ξ_{1}^{h} {(i, j)}^{T} Y_{h} ξ_{1}^{h} (i, j)) \end{matrix}

(28)

0 = α^{d_{vL}} (\sum_{t = j - d_{vL}}^{j - 1} ξ_{1}^{v} {(i, j)}^{T} X_{v} ξ_{1}^{v} (i, j) - \sum_{t = j - d_{vL}}^{j - 1} ξ_{1}^{v} {(i, j)}^{T} X_{v} ξ_{1}^{v} (i, j))

= α^{d_{vL}} (d_{vL} ξ_{1}^{v} {(i, j)}^{T} X_{v} ξ_{1}^{v} (i, j) - \sum_{t = j - d_{vL}}^{j - 1} ξ_{1}^{v} {(i, j)}^{T} X_{v} ξ_{1}^{v} (i, j))

(29)

\begin{matrix} 0 = α^{d_{vH}} (\sum_{t = j - d_{vH}}^{j - 1 - d_{vL}} ξ_{1}^{v} {(i, j)}^{T} Y_{v} ξ_{1}^{v} (i, j) - \sum_{t = j - d_{vH}}^{j - 1 - d_{vL}} ξ_{1}^{v} {(i, j)}^{T} Y_{v} ξ_{1}^{v} (i, j)) \\ = α^{d_{vH}} ((d_{vH} - d_{vL}) ξ_{1}^{v} {(i, j)}^{T} Y_{v} ξ_{1}^{v} (i, j) - \sum_{t = j - d_{v} (j)}^{j - 1 - d_{vL}} ξ_{1}^{v} {(i, j)}^{T} Y_{v} ξ_{1}^{v} (i, j) \\ - \sum_{t = j - d_{vH}}^{j - 1 - d_{v} (j)} ξ_{1}^{v} {(i, j)}^{T} Y_{v} ξ_{1}^{v} (i, j)) \end{matrix}

(30)

where

\begin{matrix} ξ_{1}^{h} (i, j) = {[\begin{matrix} x^{h} {(i, j)}^{T} & x^{h} {(i - d_{h} (i), j)}^{T} \end{matrix}]}^{T}, \\ ξ_{1}^{v} (i, j) = {[\begin{matrix} x^{v} {(i, j)}^{T} & x^{v} {(i, j - d_{v} (j))}^{T} \end{matrix}]}^{T} \end{matrix}

Combining (20)–(30) yields

\begin{matrix} V^{h} (x^{h} (i + 1, j)) - α V^{h} (x^{h} (i, j)) + V^{v} (x^{v} (i, j + 1)) - α V^{v} (x^{v} (i, j)) \\ = ξ {(i, j)}^{T} Φ ξ (i, j) - \sum_{r = i - d_{hL}}^{i - 1} ξ_{2}^{h} {(r, j)}^{T} Ψ_{1 h} ξ_{2}^{h} (r, j) \\ - \sum_{r = i - d_{h} (i)}^{i - d_{hL} - 1} ξ_{2}^{h} {(r, j)}^{T} Ψ_{2 h} ξ_{2}^{h} (r, j) - \sum_{r = i - d_{hH}}^{i - d_{h} (i) - 1} ξ_{2}^{h} {(r, j)}^{T} Ψ_{3 h} ξ_{2}^{h} (r, j) \\ - \sum_{t = j - d_{vL}}^{j - 1} ξ_{2}^{v} {(i, t)}^{T} Ψ_{1 v} ξ_{2}^{v} (i, t) - \sum_{t = j - d_{v} (j)}^{j - d_{vL} - 1} ξ_{2}^{v} {(i, t)}^{T} Ψ_{2 v} ξ_{2}^{v} (i, t) \end{matrix}

- \sum_{t = j - d_{vH}}^{j - d_{v} (j) - 1} ξ_{2}^{v} {(i, t)}^{T} Ψ_{3 v} ξ_{2}^{v} (i, t)

(31)

where

ξ_{2}^{h} (r, j) = {[\begin{matrix} ξ_{1}^{h} {(r, j)}^{T} & η^{h} {(r, j)}^{T} \end{matrix}]}^{T}, ξ_{2}^{v} (i, t) = {[\begin{matrix} ξ_{1}^{v} {(i, t)}^{T} & η^{v} {(i, t)}^{T} \end{matrix}]}^{T}

\begin{matrix} ξ (i, j) = {[\begin{matrix} x {(i, j)}^{T} & x_{d} {(i, j)}^{T} & x_{dL} {(i, j)}^{T} & x_{dH} {(i, j)}^{T} \end{matrix}]}^{T} \\ x (i, j) = [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}], x_{d} (i, j) = [\begin{matrix} x^{h} (i - d (i), j) \\ x^{v} (i, j - d (j)) \end{matrix}] \\ x_{dL} (i, j) = [\begin{matrix} x^{h} (i - d_{hL}, j) \\ x^{v} (i, j - d_{vL}) \end{matrix}], x_{dH} (i, j) = [\begin{matrix} x^{h} (i - d_{hH}, j) \\ x^{v} (i, j - d_{vH}) \end{matrix}] \end{matrix}

Then it follows from (16)–(17) that

\begin{matrix} V^{h} (x^{h} (i + 1, j)) - α V^{h} (x^{h} (i, j)) \\ + V^{v} (x^{v} (i, j + 1)) - α V^{v} (x^{v} (i, j)) < 0 \end{matrix}

(32)

For simplicity, we denote

V^{h} (i, j) = V^{h} (x^{h} (i, j)), V^{v} (i, j) = V^{v} (x^{v} (i, j)), V (i, j) = V (x (i, j)) .

Note that, for any non-negative integer $D > z = max (z_{1}, z_{2})$ , $V^{h} (0, D) = V^{v} (D, 0) = 0$ , summing up both sides of (32) from $D - 1$ to 0 with respect to $j$ and 0 to $D - 1$ with respect to $i$ , one gets

\begin{array}{l} \sum_{i + j = D} V (i, j) = V^{h} (0, D) + V^{h} (1, D - 1) \\ + V^{h} (2, D - 2) + \dots + V^{h} (D - 1, 1) + V^{h} (D, 0) \\ + V^{v} (0, D) + V^{v} (1, D - 1) + V^{v} (2, D - 2) \\ + \dots + V^{v} (D - 1, 1) + V^{v} (D, 0) \\ < α (V^{h} (0, D - 1) + V^{v} (0, D - 1) + V^{h} (1, D - 2) + V^{v} (1, D - 2) \end{array}

+ \cdot \cdot \cdot + V^{h} (D - 1, 0) + V^{v} (D - 1, 0))

= α \sum_{i + j = D - 1} V (i, j) < \dots < α^{D - z} \sum_{i + j = z} V (i, j)

(33)

The proof is completed.

Remark 3. Lemma 4 presents an exponential stability criterion for 2D discrete system (15), and it will be essential for the proof in our later development.

Robust reliable controller design

Considering system (1), under the controller $u (i, j) = K^{σ (i, j)} x (i, j)$ , the resulting closed-loop system is given by

\begin{matrix} [\begin{matrix} x^{h} (i + 1, j) \\ x^{v} (i, j + 1) \end{matrix}] = ({\hat{A}}^{σ (i, j)} + B^{σ (i, j)} Ω^{σ (i, j)} K^{σ (i, j)}) [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] \\ + {\hat{A}}_{d}^{σ (i, j)} [\begin{matrix} x^{h} (i - d_{h} (i), j) \\ x^{v} (i, j - d_{v} (j)) \end{matrix}] \end{matrix}

(34)

Theorem 1. Consider system (1), for given positive scalars $α < 1$ , $δ_{k}$ , $ε_{k}$ , if there exist positive definite symmetric matrices ${\bar{P}}^{k} = diag {{\bar{P}}_{h}^{k}, {\bar{P}}_{v}^{k}}$ , ${\bar{Q}}_{1}^{k} = diag {{\bar{Q}}_{1 h}^{k}, {\bar{Q}}_{1 v}^{k}}$ , ${\bar{Q}}_{2}^{k} =$ $diag {{\bar{Q}}_{2 h}^{k}, {\bar{Q}}_{2 v}^{k}}$ , ${\bar{W}}_{1}^{k} = diag {{\bar{W}}_{1 h}^{k}, {\bar{W}}_{1 v}^{k}}$ , ${\bar{W}}_{2}^{k} = diag {{\bar{W}}_{2 h}^{k}, {\bar{W}}_{2 v}^{k}}$ , ${\bar{X}}_{h}^{k} = [\begin{matrix} {\bar{X}}_{11 h}^{k} \\ * \end{matrix} \begin{matrix} {\bar{X}}_{12 h}^{k} \\ {\bar{X}}_{22 h}^{k} \end{matrix}]$ , ${\bar{X}}_{v}^{k} = [\begin{matrix} {\bar{X}}_{11 v}^{k} & {\bar{X}}_{12 v}^{k} \\ * & {\bar{X}}_{22 v}^{k} \end{matrix}]$ , ${\bar{Y}}_{h}^{k} = [\begin{matrix} {\bar{Y}}_{11 h}^{k} & {\bar{Y}}_{12 h}^{k} \\ * & {\bar{Y}}_{22 h}^{k} \end{matrix}]$ , ${\bar{Y}}_{v}^{k} = [\begin{matrix} {\bar{Y}}_{11 v}^{k} & {\bar{Y}}_{12 v}^{k} \\ * & {\bar{Y}}_{22 v}^{k} \end{matrix}]$ , and any matrices ${\bar{M}}_{h}^{k} = [\begin{matrix} {\bar{M}}_{1 h}^{k} \\ {\bar{M}}_{2 h}^{k} \end{matrix}]$ , ${\bar{N}}_{h}^{k} = [\begin{matrix} {\bar{N}}_{1 h}^{k} \\ {\bar{N}}_{2 h}^{k} \end{matrix}]$ , ${\bar{S}}_{h}^{k} = [\begin{matrix} {\bar{S}}_{1 h}^{k} \\ {\bar{S}}_{2 h}^{k} \end{matrix}]$ , ${\bar{M}}_{v}^{k} = [\begin{matrix} {\bar{M}}_{1 v}^{k} \\ {\bar{M}}_{2 v}^{k} \end{matrix}]$ , ${\bar{N}}_{v}^{k} = [\begin{matrix} {\bar{N}}_{1 v}^{k} \\ {\bar{N}}_{2 v}^{k} \end{matrix}]$ , ${\bar{S}}_{v}^{k} = [\begin{matrix} {\bar{S}}_{1 v}^{k} \\ {\bar{S}}_{2 v}^{k} \end{matrix}]$ , $ϒ^{k}$ and $Z =$ $diag {Z_{h}, Z_{v}}$ with appropriate dimensions, $k \in \underline{N}$ , such that

\bar{Φ} = [\begin{matrix} {\bar{Φ}}_{1} & {\bar{Φ}}_{2}^{T} & {\bar{Φ}}_{2}^{T} & {\bar{Φ}}_{3}^{T} & T^{k} & {\bar{Φ}}_{7}^{T} \\ * & {\bar{Φ}}_{4} & {\bar{Φ}}_{6} & {\bar{Φ}}_{6} & 0 & 0 \\ * & * & {\bar{Φ}}_{5} & {\bar{Φ}}_{6} & 0 & 0 \\ * & * & * & Π^{k} & 0 & 0 \\ * & * & * & * & - δ_{k} I & 0 \\ * & * & * & * & * & - ε_{k} Ξ^{k} \end{matrix}] < 0

(35)

{\bar{Ψ}}^{k} = diag {{\bar{Ψ}}_{1 h}^{k}, {\bar{Ψ}}_{2 h}^{k}, {\bar{Ψ}}_{3 h}^{k}, {\bar{Ψ}}_{1 v}^{k}, {\bar{Ψ}}_{2 v}^{k}, {\bar{Ψ}}_{3 v}^{k}} \geq 0

(36)

hold, then, under the reliable controller

u (i, j) = K^{σ (i, j)} x (i, j), K^{k} = ϒ^{k} Z^{- 1}, k \in \underline{N}

(37)

and the following average dwell time scheme

τ_{a} > τ_{a}^{*} = \frac{\ln μ}{- \ln α}

(38)

with $μ \geq 1$ satisfies

{\bar{P}}^{k} \leq μ {\bar{P}}^{l}, {\bar{P}}^{l} \leq μ {\bar{P}}^{k}, {\bar{Q}}_{1}^{k} \leq μ {\bar{Q}}_{1}^{l}, {\bar{Q}}_{1}^{l} \leq μ {\bar{Q}}_{1}^{k}, {\bar{Q}}_{2}^{k} \leq μ {\bar{Q}}_{2}^{l}, {\bar{Q}}_{2}^{l} \leq μ {\bar{Q}}_{2}^{k}

{\bar{W}}_{1}^{k} \leq μ {\bar{W}}_{1}^{l}, {\bar{W}}_{1}^{l} \leq μ {\bar{W}}_{1}^{k}, {\bar{W}}_{2}^{k} \leq μ {\bar{W}}_{2}^{l}, {\bar{W}}_{2}^{l} \leq μ {\bar{W}}_{2}^{k}, \forall k, l \in \underline{N}

(39)

The resulting closed-loop system (34) is exponentially stable, where

Λ_{1} = diag {d_{hL} I_{h}, d_{vL} I_{v}}, Λ_{2} = diag {(d_{hH} - d_{hL}) I_{h}, (d_{vH} - d_{vL}) I_{v}}

Λ_{3} = diag {α^{d_{hL}} I_{h}, α^{d_{vL}} I_{v}}, Λ_{4} = diag {α^{d_{hH}} I_{h}, α^{d_{vH}} I_{v}},

{\bar{Φ}}_{1} = [\begin{matrix} {\bar{Φ}}_{11} & {\bar{Φ}}_{12} & {\bar{Φ}}_{13} & {\bar{Φ}}_{14} \\ * & {\bar{Φ}}_{22} & {\bar{Φ}}_{23} & {\bar{Φ}}_{24} \\ * & * & {\bar{Φ}}_{33} & 0 \\ * & * & * & {\bar{Φ}}_{44} \end{matrix}]

\begin{matrix} {\bar{Φ}}_{11} = {\bar{Q}}_{1}^{k} + {\bar{Q}}_{2}^{k} + Λ_{3} ({\bar{M}}_{1}^{k} + {\bar{M}}_{1}^{kT} + Λ_{1} {\bar{X}}_{11}^{k}) + Λ_{2} Λ_{4} {\bar{Y}}_{11}^{k} \\ - α {\bar{P}}^{k}, {\bar{Φ}}_{13} = - Λ_{3} {\bar{M}}_{1}^{k} + Λ_{4} {\bar{S}}_{1}^{k}, \\ {\bar{Φ}}_{12} = Λ_{3} (Λ_{1} {\bar{X}}_{12}^{k} + {\bar{M}}_{2 k}^{kT}) + Λ_{4} ({\bar{N}}_{1}^{k} - {\bar{S}}_{1}^{k} + Λ_{2} {\bar{Y}}_{12}^{k}), \\ {\bar{Φ}}_{23} = - Λ_{3} {\bar{M}}_{2}^{k} + Λ_{4} {\bar{S}}_{2}^{k}, \\ {\bar{Φ}}_{14} = - Λ_{4} {\bar{N}}_{1}^{k}, {\bar{Φ}}_{22} = Λ_{1} Λ_{3} {\bar{X}}_{22}^{k} \\ + Λ_{4} ({\bar{N}}_{2}^{k} + {\bar{N}}_{2}^{kT} - {\bar{S}}_{2}^{k} - {\bar{S}}_{2}^{kT} + Λ_{2} {\bar{Y}}_{22}^{k}), {\bar{Φ}}_{24} = - Λ_{4} N_{2}^{k} \end{matrix}

\begin{matrix} {\bar{Φ}}_{33} = - Λ_{3} {\bar{Q}}_{1}^{k}, \\ {\bar{Φ}}_{44} = - Λ_{4} {\bar{Q}}_{2}^{k}, {\bar{Φ}}_{2} = [\begin{matrix} (A^{k} - I) Z + B^{k} Ω_{0}^{k} ϒ^{k} & A_{d}^{k} Z & 0 & 0 \end{matrix}] \\ {\bar{Φ}}_{4} = - Λ_{1}^{- 1} {\bar{W}}_{1}^{k} + δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT} \\ {\bar{Φ}}_{3} = [\begin{matrix} A^{k} Z + B^{k} Ω_{0}^{k} ϒ^{k} & A_{d}^{k} Z & 0 & 0 \end{matrix}], \\ {\bar{Φ}}_{6} = δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT} \\ {\bar{Φ}}_{5} = - Λ_{2}^{- 1} {\bar{W}}_{2}^{k} + δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT}, \\ T^{k} = [\begin{matrix} {(E^{k} Z)}^{T} & {(E_{d}^{k} Z)}^{T} & 0 & 0 \end{matrix}] \end{matrix}

\begin{matrix} Π^{k} = - Z - Z^{T} + {\bar{P}}^{k} + δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT}, \\ {\bar{Φ}}_{7} = [\begin{matrix} Ξ^{k} ϒ^{k} & 0 & 0 & 0 \end{matrix}] \end{matrix}

\begin{matrix} {\bar{X}}_{11}^{k} = diag {{\bar{X}}_{11 h}^{k}, {\bar{X}}_{11 v}^{k}}, {\bar{X}}_{12}^{k} = diag {{\bar{X}}_{12 h}^{k}, {\bar{X}}_{12 v}^{k}}, {\bar{X}}_{22}^{k} = diag {{\bar{X}}_{22 h}^{k}, {\bar{X}}_{22 v}^{k}} \\ {\bar{Y}}_{11}^{k} = diag {{\bar{Y}}_{11 h}^{k}, {\bar{Y}}_{11 v}^{k}}, {\bar{Y}}_{12}^{k} = diag {{\bar{Y}}_{12 h}^{k}, {\bar{Y}}_{12 v}^{k}}, {\bar{Y}}_{22}^{k} = diag {{\bar{Y}}_{22 h}^{k}, {\bar{Y}}_{22 v}^{k}} \\ {\bar{M}}_{1}^{k} = diag {{\bar{M}}_{1 h}^{k}, {\bar{M}}_{1 v}^{k}}, {\bar{M}}_{2}^{k} = diag {{\bar{M}}_{2 h}^{k}, {\bar{M}}_{2 v}^{k}}, {\bar{N}}_{1}^{k} = diag {{\bar{N}}_{1 h}^{k}, {\bar{N}}_{1 v}^{k}} \\ {\bar{N}}_{2}^{k} = diag {{\bar{N}}_{2 h}^{k}, {\bar{N}}_{2 v}^{k}}, {\bar{S}}_{1}^{k} = diag {{\bar{S}}_{1 h}^{k}, {\bar{S}}_{1 v}^{k}}, {\bar{S}}_{2}^{k} = diag {{\bar{S}}_{2 h}^{k}, {\bar{S}}_{2 v}^{k}} \\ {\bar{Ψ}}_{1 h}^{k} = [\begin{matrix} {\bar{X}}_{h}^{k} & {\bar{M}}_{h}^{k} \\ * & Z_{h} + Z_{h}^{T} - {\bar{W}}_{1 h}^{k} \end{matrix}], {\bar{Ψ}}_{2 h}^{k} = [\begin{matrix} {\bar{Y}}_{h}^{k} & {\bar{N}}_{h}^{k} \\ * & Z_{h} + Z_{h}^{T} - {\bar{W}}_{2 h}^{k} \end{matrix}], \\ {\bar{Ψ}}_{3 h}^{k} = [\begin{matrix} {\bar{Y}}_{h}^{k} & {\bar{S}}_{h}^{k} \\ * & Z_{h} + Z_{h}^{T} - {\bar{W}}_{2 h}^{k} \end{matrix}] \\ {\bar{Ψ}}_{1 v}^{k} = [\begin{matrix} {\bar{X}}_{v}^{k} & M_{v}^{k} \\ * & Z_{v} + Z_{v}^{T} - {\bar{W}}_{1 v}^{k} \end{matrix}], {\bar{Ψ}}_{2 v}^{k} = [\begin{matrix} {\bar{Y}}_{v}^{k} & {\bar{N}}_{v}^{k} \\ * & Z_{v} + Z_{v}^{T} - {\bar{W}}_{2 v}^{k} \end{matrix}], \\ {\bar{Ψ}}_{3 v}^{k} = [\begin{matrix} {\bar{Y}}_{v}^{k} & {\bar{S}}_{v}^{k} \\ * & Z_{v} + Z_{v}^{T} - {\bar{W}}_{2 v}^{k} \end{matrix}] . \end{matrix}

Proof. Suppose that the kth subsystem is activated during the interval $[m_{κ}, m_{κ + 1})$ . We consider the following Lyapunov function candidate for the kth $(k \in \underline{N})$ subsystem

V_{k} (x (i, j)) = V_{k}^{h} (x^{h} (i, j)) + V_{k}^{v} (x^{v} (i, j))

(40)

where

\begin{array}{l} V_{k}^{h} (x^{h} (i, j)) = \sum_{g = 1}^{3} V_{g k}^{h} (x^{h} (i, j)) \\ V_{1 k}^{h} (x^{h} (i, j)) = x^{h} {(i, j)}^{T} P_{h}^{k} x^{h} (i, j) \\ V_{2 k}^{h} (x^{h} (i, j)) = \sum_{r = i - d_{h L}}^{i - 1} x^{h} {(r, j)}^{T} Q_{1 h}^{k} x^{h} (r, j) α^{i - r - 1} \\ + \sum_{r = i - d_{h H}}^{i - 1} x^{h} {(r, j)}^{T} Q_{2 h}^{k} x^{h} (r, j) α^{i - r - 1} \\ V_{3 k}^{h} (x^{h} (i, j)) = \sum_{s = - d_{h L}}^{- 1} \sum_{r = i + s}^{i - 1} η^{h} {(r, j)}^{T} W_{1 h}^{k} η^{h} (r, j) α^{i - r - 1} \\ + \sum_{s = - d_{h H}}^{- d_{h L} - 1} \sum_{r = i + s}^{i - 1} η^{h} {(r, j)}^{T} W_{2 h}^{k} η^{h} (r, j) α^{i - r - 1} \\ V_{k}^{v} (x^{v} (i, j)) = \sum_{g = 1}^{3} V_{g k}^{v} (x^{v} (i, j)) \\ V_{1 k}^{v} (x^{v} (i, j)) = x^{v} {(i, j)}^{T} P_{v}^{k} x^{v} (i, j) \end{array}

\begin{array}{l} V_{2 k}^{v} (x^{v} (i, j)) = \sum_{t = j - d_{v L}}^{j - 1} x^{v} {(i, t)}^{T} Q_{1 v}^{k} x^{v} (i, t) α^{j - t - 1} \\ + \sum_{t = j - d_{v H}}^{j - 1} x^{v} {(i, t)}^{T} Q_{2 v}^{k} x^{v} (i, t) α^{j - t - 1} \\ V_{3 k}^{v} (x^{v} (i, j)) = \sum_{s = - {d_{v}}_{L}}^{- 1} \sum_{t = j + s}^{j - 1} η^{v} {(i, t)}^{T} W_{1 v}^{k} η^{v} (i, t) α^{j - t - 1} \\ + \sum_{s = - {d_{v}}_{H}}^{- {d_{v}}_{L} - 1} \sum_{t = j + s}^{j - 1} η^{v} {(i, t)}^{T} W_{2 v}^{k} η^{v} (i, t) α^{j - t - 1} . \end{array}

According to Lemma 4, for a given positive scalar $α < 1$ , if there exist positive definite symmetric matrices $P^{k}$ , $Q_{1}^{k}$ , $Q_{2}^{k}$ , $W_{1}^{k}$ , $W_{2}^{k}$ , $X_{h}^{k}$ , $X_{v}^{k}$ , $Y_{h}^{k}$ , $Y_{v}^{k}$ , $M_{h}^{k}$ , $N_{h}^{k}$ , $S_{h}^{k}$ , $M_{v}^{k}$ , $N_{v}^{k}$ and $S_{v}^{k}$ with appropriate dimensions, such that

\hat{Φ} = [\begin{matrix} {\hat{Φ}}_{1} & {\hat{Φ}}_{2}^{T} & {\hat{Φ}}_{2}^{T} & {\hat{Φ}}_{3}^{T} P^{k} \\ * & - Λ_{1}^{- 1} W_{1 k}^{- 1} & 0 & 0 \\ * & * & - Λ_{2}^{- 1} W_{2 k}^{- 1} & 0 \\ * & * & * & - P^{k} \end{matrix}] < 0

(41)

Ψ^{k} = diag {Ψ_{1 h}^{k}, Ψ_{2 h}^{k}, Ψ_{3 h}^{k}, Ψ_{1 v}^{k}, Ψ_{2 v}^{k}, Ψ_{3 v}^{k}} \geq 0

(42)

hold, then the following inequality is satisfied

\sum_{i + j = D} V_{k} (i, j) < α^{D - m_{κ}} \sum_{i + j = m_{κ}} V_{k} (i, j), D \in [m_{κ}, m_{κ + 1})

(43)

where

{\hat{Φ}}_{1} = [\begin{matrix} {\hat{Φ}}_{11} & {\hat{Φ}}_{12} & {\hat{Φ}}_{13} & {\hat{Φ}}_{14} \\ * & {\hat{Φ}}_{22} & {\hat{Φ}}_{23} & {\hat{Φ}}_{24} \\ * & * & {\hat{Φ}}_{33} & 0 \\ * & * & * & {\hat{Φ}}_{44} \end{matrix}]

\begin{matrix} {\hat{Φ}}_{11} = Q_{1}^{k} + Q_{2}^{k} + Λ_{3} (M_{1}^{k} + M_{1}^{kT} + Λ_{1} X_{11}^{k}) \\ + Λ_{2} Λ_{4} Y_{11}^{k} - α P^{k}, {\hat{Φ}}_{13} = - Λ_{3} M_{1}^{k} + Λ_{4} S_{1}^{k}, \\ {\hat{Φ}}_{12} = Λ_{3} (Λ_{1} X_{12}^{k} + M_{2}^{kT}) + Λ_{4} (N_{1}^{k} - S_{1}^{k} + Λ_{2} Y_{12}^{k}), \\ {\hat{Φ}}_{14} = - Λ_{4} N_{1}^{k} \\ {\hat{Φ}}_{22} = Λ_{1} Λ_{3} X_{22}^{k} + Λ_{4} (N_{2}^{k} + N_{2}^{kT} - S_{2}^{k} - S_{2}^{kT} + Λ_{2} Y_{22}^{k}), \\ {\hat{Φ}}_{23} = - Λ_{3} M_{2}^{k} + Λ_{4} S_{2}^{k} \\ {\hat{Φ}}_{24} = - Λ_{4} N_{2}^{k}, {\hat{Φ}}_{33} = - Λ_{3} Q_{1}^{k}, {\hat{Φ}}_{44} = - Λ_{4} Q_{2}^{k} \\ {\hat{Φ}}_{2} = [\begin{matrix} {\hat{A}}^{k} + B^{k} Ω^{k} K^{k} - I & {\hat{A}}_{d}^{k} & 0 & 0 \end{matrix}], \\ {\hat{Φ}}_{3} = [\begin{matrix} {\hat{A}}^{k} + B^{k} Ω^{k} K^{k} & {\hat{A}}_{d}^{k} & 0 & 0 \end{matrix}] \end{matrix}

P^{k} = diag {P_{h}^{k}, P_{v}^{k}}, Q_{1}^{k} = diag {Q_{1 h}^{k}, Q_{1 v}^{k}}, Q_{2}^{k} = diag {Q_{2 h}^{k}, Q_{2 v}^{k}}

W_{1}^{k} = diag {W_{1 h}^{k}, W_{1 v}^{k}}, W_{2}^{k} = diag {W_{2 h}^{k}, W_{2 v}^{k}}

\begin{matrix} X_{11}^{k} = diag {X_{11 h}^{k}, X_{11 v}^{k}}, X_{12}^{k} = diag {X_{12 h}^{k}, X_{12 v}^{k}}, \\ X_{22}^{k} = diag {X_{22 h}^{k}, X_{22 v}^{k}} \end{matrix}

\begin{matrix} Y_{11}^{k} = diag {Y_{11 h}^{k}, Y_{11 v}^{k}}, Y_{12}^{k} = diag {Y_{12 h}^{k}, Y_{12 v}^{k}}, \\ Y_{22}^{k} = diag {Y_{22 h}^{k}, Y_{22 v}^{k}} \end{matrix}

M_{1}^{k} = diag {M_{1 h}^{k}, M_{1 v}^{k}}, M_{2}^{k} = diag {M_{2 h}^{k}, M_{2 v}^{k}}, N_{1}^{k} = diag {N_{1 h}^{k}, N_{1 v}^{k}}

N_{2}^{k} = diag {N_{2 h}^{k}, N_{2 v}^{k}}, S_{1}^{k} = diag {S_{1 h}^{k}, S_{1 v}^{k}}, S_{2}^{k} = diag {S_{2 h}^{k}, S_{2 v}^{k}}

\begin{matrix} X_{h}^{k} = [\begin{matrix} X_{11 h}^{k} & X_{12 h}^{k} \\ * & X_{22 h}^{k} \end{matrix}], X_{v}^{k} = [\begin{matrix} X_{11 v}^{k} & X_{12 v}^{k} \\ * & X_{22 v}^{k} \end{matrix}], Y_{h}^{k} = [\begin{matrix} Y_{11 h}^{k} & Y_{12 h}^{k} \\ * & Y_{22 h}^{k} \end{matrix}], \\ Y_{v}^{k} = [\begin{matrix} Y_{11 v}^{k} & Y_{12 v}^{k} \\ * & Y_{22 v}^{k} \end{matrix}] \end{matrix}

\begin{matrix} M_{h}^{k} = [\begin{matrix} M_{1 h}^{k} \\ M_{2 h}^{k} \end{matrix}], N_{h}^{k} = [\begin{matrix} N_{1 h}^{k} \\ N_{2 h}^{k} \end{matrix}], S_{h}^{k} = [\begin{matrix} S_{1 h}^{k} \\ S_{2 h}^{k} \end{matrix}], M_{v}^{k} = [\begin{matrix} M_{1 v}^{k} \\ M_{2 v}^{k} \end{matrix}], \\ N_{v}^{k} = [\begin{matrix} N_{1 v}^{k} \\ N_{2 v}^{k} \end{matrix}], S_{v}^{k} = [\begin{matrix} S_{1 v}^{k} \\ S_{2 v}^{k} \end{matrix}] \end{matrix}

Ψ_{1 h}^{k} = [\begin{matrix} X_{h}^{k} & M_{h}^{k} \\ * & W_{1 h}^{k} \end{matrix}], Ψ_{2 h}^{k} = [\begin{matrix} Y_{h}^{k} & N_{h}^{k} \\ * & W_{2 h}^{k} \end{matrix}], Ψ_{3 h}^{k} = [\begin{matrix} Y_{h}^{k} & S_{h}^{k} \\ * & W_{2 h}^{k} \end{matrix}]

Ψ_{1 v}^{k} = [\begin{matrix} X_{v}^{k} & M_{v}^{k} \\ * & W_{1 v}^{k} \end{matrix}], Ψ_{2 v}^{k} = [\begin{matrix} Y_{v}^{k} & N_{v}^{k} \\ * & W_{2 v}^{k} \end{matrix}], Ψ_{3 v}^{k} = [\begin{matrix} Y_{v}^{k} & S_{v}^{k} \\ * & W_{2 v}^{k} \end{matrix}] .

Now let $χ = N_{σ} (z, D)$ denote the switching number of $σ (i, j)$ on an interval $[z, D)$ , and let $(i_{κ - χ + 1}, j_{κ - χ + 1})$ , $(i_{κ - χ + 2}, j_{κ - χ + 2})$ , …, $(i_{κ}, j_{κ})$ denote the switching points of $σ (i, j)$ over the interval $[z, D)$ . Denoting $m_{π} = i_{π} + j_{π}$ , $π = κ - χ + 1, \dots, κ$ , one obtains from (39) and (40) that

\sum_{i + j = m_{π}} V_{σ (i_{π}, j_{π})} (i, j) \leq μ \sum_{i + j = m_{π}^{-}} V_{σ (i_{π - 1}, j_{π - 1})} (i, j)

(44)

Combining (43) and (44) yields

\begin{matrix} \sum_{i + j = D} V_{σ (i_{κ}, j_{κ})} (i, j) < α^{D - m_{κ}} \sum_{i + j = m_{κ}} V_{σ (i_{κ}, j_{κ})} (i, j) \\ \leq μ α^{D - m_{κ}} \sum_{i + j = m_{κ}^{-}} V_{σ (i_{κ - 1}, j_{κ - 1})} (i, j) \\ < μ^{2} α^{D - m_{κ}} α^{m_{κ} - m_{κ - 1}} \sum_{i + j = m_{κ - 1}^{-}} V_{σ (i_{κ - 2}, j_{κ - 2})} (i, j) \\ = μ^{2} α^{D - m_{κ - 1}} \sum_{i + j = m_{κ - 1}^{-}} V_{σ (i_{κ - 2}, j_{κ - 2})} (i, j) \\ < \dots \\ < μ^{χ} α^{D - z} \sum_{i + j = z} V_{σ (i_{z}, j_{z})} (i, j) \end{matrix}

(45)

According to Definition 2, we have

χ = N (z, D) \leq N_{0} + \frac{D - z}{τ_{a}}

(46)

Substituting (46) into (45), it can be obtained that

\sum_{i + j = D} V_{σ (i_{κ}, j_{κ})} (i, j) < μ^{χ} α^{D - z} \sum_{i + j = z} V_{σ (i_{z}, j_{z})} (i, j)

= μ^{N_{0}} e^{- (- \frac{\ln μ}{τ_{a}} - \ln α) (D - z)} \sum_{i + j = z} V_{σ (i_{z}, j_{z})} (i, j)

(47)

It follows that

\sum_{i + j = D} {‖ x (i, j) ‖}^{2} \leq \frac{ζ_{1}}{ζ_{2}} μ^{N_{0}} e^{- (- \frac{\ln μ}{τ_{a}} - \ln α) (D - z)} \sum_{i + j = z} ‖ x (i, j) ‖_{C}^{2}

(48)

where

\begin{matrix} ζ_{1} = max_{k, l \in \underline{N}} {λ_{1}, λ_{2}} \\ λ_{1} = λ_{max} (P^{k}) + max (d_{hL}, d_{vL}) λ_{max} (Q_{1}^{k}) \\ + max (d_{hH}, d_{vH}) λ_{max} (Q_{2}^{k}) \\ + max (d_{hL}^{2}, d_{vL}^{2}) λ_{max} (W_{1}^{k}) \\ + max ({(d_{hH} - d_{hL})}^{2}, {(d_{vH} - d_{vL})}^{2}) λ_{max} (W_{2}^{k}), \\ λ_{2} = λ_{max} (P^{l}) + max (d_{hL}, d_{vL}) λ_{max} (Q_{1}^{l}) \\ + max (d_{hH}, d_{vH}) λ_{max} (Q_{2}^{l}) \\ + max (d_{hL}^{2}, d_{vL}^{2}) λ_{max} (W_{1}^{l}) \\ + max ({(d_{hH} - d_{hL})}^{2}, {(d_{vH} - d_{vL})}^{2}) λ_{max} (W_{2}^{l}), \\ ζ_{2} = min_{k, l \in \underline{N}} {λ_{min} (P^{k}), λ_{min} (P^{l})} \end{matrix}

\sum_{i + j = z} ‖ x (i, j) ‖_{C}^{2} \overset{Δ}{=} sup_{\binom{- d_{hH} \leq θ_{h} \leq 0,}{- d_{vH} \leq θ_{v} \leq 0}} \sum_{i + j = z} {{‖ x^{h} (i - θ_{h}, j) ‖}^{2} + {‖ x^{v} (i, j - θ_{v}) ‖}^{2},

{‖ η^{h} (i - θ_{h}, j) ‖}^{2} + {‖ η^{v} (i, j - θ_{v}) ‖}^{2}}

η^{h} (i - θ_{h}, j) = x^{h} (i - θ_{h} + 1, j) - x^{h} (i - θ_{h}, j)

η^{v} (i, j - θ_{v}) = x^{v} (i, j - θ_{v} + 1) - x^{v} (i, j - θ_{v})

It follows readily from (38) that the resulting closed-loop system (34) is exponentially stable.

Note that, for $P^{k}$ , there exits a reversible matrix $G = diag {G_{h}, G_{v}}$ such that $G^{T} P^{- k} G \geq G^{T} + G - P^{k}$ holds. Use $diag {\begin{matrix} I & I & I & G^{T} P^{- k} \end{matrix}}$ and its transpose to pre- and post-multiply the left of (41), respectively, then one can show that (41) holds if the following inequality is satisfied

\hat{Φ} = [\begin{matrix} {\hat{Φ}}_{1} & {\hat{Φ}}_{2}^{T} & {\hat{Φ}}_{2}^{T} & {\hat{Φ}}_{3}^{T} G \\ * & - Λ_{1}^{- 1} W_{1 k}^{- 1} & 0 & 0 \\ * & * & - Λ_{2}^{- 1} W_{2 k}^{- 1} & 0 \\ * & * & * & - G - G^{T} + P^{k} \end{matrix}] < 0

(49)

Denote $Z = G^{- 1}$ , ${\bar{W}}_{1}^{k} = W_{1 k}^{- 1}$ , ${\bar{W}}_{2}^{k} = W_{2 k}^{- 1}$ , and assign the following new matrices

\begin{array}{l} {\bar{P}}^{k} = Z^{T} P^{k} Z, {\bar{Q}}_{1}^{k} = Z^{T} Q_{1}^{k} Z, {\bar{Q}}_{2}^{k} = Z^{T} Q_{2}^{k} Z, {\bar{X}}_{11}^{k} = Z^{T} X_{11}^{k} Z \\ {\bar{X}}_{12}^{k} = Z^{T} X_{12}^{k} Z, {\bar{X}}_{22}^{k} = Z^{T} X_{22}^{k} Z, {\bar{Y}}_{11}^{k} = Z^{T} Y_{11}^{k} Z, {\bar{Y}}_{12}^{k} = Z^{T} Y_{12}^{k} Z \\ {\bar{Y}}_{22}^{k} = Z^{T} Y_{22}^{k} Z, {\bar{M}}_{1}^{k} = Z^{T} M_{1}^{k} Z, {\bar{M}}_{2}^{k} = Z^{T} M_{2}^{k} Z, {\bar{N}}_{1}^{k} = Z^{T} N_{1}^{k} Z \\ {\bar{N}}_{2}^{k} = Z^{T} N_{2}^{k} Z, {\bar{S}}_{1}^{k} = Z^{T} S_{1}^{k} Z, {\bar{S}}_{2}^{k} = Z^{T} S_{2}^{k} Z, ϒ^{k} = K^{k} Z \end{array}

then using $diag {Z^{T}, Z^{T}, Z^{T}, Z^{T}, I, I, Z^{T}}$ and its transpose to pre- and post-multiply the left of (49), respectively, and applying Lemmas 1, 2 and 3, it is easy to get that (35) is equivalent to (49).

In addition, denoting $Z_{h} = G_{h}^{- 1}$ , $Z_{v} = G_{v}^{- 1}$ , ${\bar{W}}_{1 h}^{k} = {(W_{1 h}^{k})}^{- 1}$ , ${\bar{W}}_{1 v}^{k} = {(W_{1 v}^{k})}^{- 1}$ , ${\bar{W}}_{2 h}^{k} = {(W_{2 h}^{k})}^{- 1}$ , ${\bar{W}}_{2 v}^{k} = {(W_{2 v}^{k})}^{- 1}$ , and then using $diag {Z_{h}^{T}, Z_{h}^{T}, Z_{h}^{T}, Z_{h}^{T}, Z_{h}^{T}, Z_{h}^{T}, Z_{v}^{T}, Z_{v}^{T}, Z_{v}^{T}, Z_{v}^{T}, Z_{v}^{T}, Z_{v}^{T}}$ and its transpose to pre- and post-multiply the left of (42), we get

{\hat{Ψ}}^{k} = diag {{\hat{Ψ}}_{1 h}^{k}, {\hat{Ψ}}_{2 h}^{k}, {\hat{Ψ}}_{3 h}^{k}, {\hat{Ψ}}_{1 v}^{k}, {\hat{Ψ}}_{2 v}^{k}, {\hat{Ψ}}_{3 v}^{k}} \geq 0

(50)

where

\begin{array}{l} {\hat{Ψ}}_{1 h}^{k} = [\begin{matrix} {\bar{X}}_{h}^{k} & {\bar{M}}_{h}^{k} \\ * & Z_{h}^{T} W_{1 h}^{k} Z \end{matrix}], {\hat{Ψ}}_{2 h}^{k} = [\begin{matrix} {\bar{Y}}_{h}^{k} & {\bar{N}}_{h}^{k} \\ * & Z_{h}^{T} W_{2 h}^{k} Z_{h} \end{matrix}], \\ {\hat{Ψ}}_{3 h}^{k} = [\begin{matrix} {\bar{Y}}_{h}^{k} & {\bar{S}}_{h}^{k} \\ * & Z_{h}^{T} W_{2 h}^{k} Z_{h} \end{matrix}], {\hat{Ψ}}_{1 v}^{k} = [\begin{matrix} {\bar{X}}_{v}^{k} & {\bar{M}}_{v}^{k} \\ * & Z_{v}^{T} W_{1 v}^{k} Z_{v} \end{matrix}] \\ {\hat{Ψ}}_{2 v}^{k} = [\begin{matrix} {\bar{Y}}_{v}^{k} & {\bar{N}}_{v}^{k} \\ * & Z_{v}^{T} W_{2 v}^{k} Z_{v} \end{matrix}], {\hat{Ψ}}_{3 v}^{k} = [\begin{matrix} {\bar{Y}}_{v}^{k} & {\bar{S}}_{v}^{k} \\ * & Z_{v}^{T} W_{2 v}^{k} Z_{v} \end{matrix}] \end{array}

\begin{matrix} {\bar{X}}_{h}^{k} = Z_{h}^{T} X_{h}^{k} Z_{h}, {\bar{X}}_{v}^{k} = Z_{v}^{T} X_{v}^{k} Z_{v}, {\bar{Y}}_{h}^{k} = Z_{h}^{T} Y_{h}^{k} Z_{h}, \\ {\bar{Y}}_{v}^{k} = Z_{v}^{T} Y_{v}^{k} Z_{v}, {\bar{M}}_{h}^{k} = Z_{h}^{T} M_{h}^{k} Z_{h} \end{matrix}

\begin{matrix} {\bar{M}}_{v}^{k} = Z_{v}^{T} M_{v}^{k} Z_{v}, {\bar{N}}_{h}^{k} = Z_{h}^{T} N_{h}^{k} Z_{h}, {\bar{N}}_{v}^{k} = Z_{v}^{T} N_{v}^{k} Z_{v}, \\ {\bar{S}}_{h}^{k} = Z_{h}^{T} S_{h}^{k} Z_{h}, {\bar{S}}_{v}^{k} = Z_{v}^{T} S_{v}^{k} Z_{v} \end{matrix}

Applying the following relations

Z_{h}^{T} W_{1 h}^{k} Z_{h} \geq Z_{h}^{T} + Z_{h} - {\bar{W}}_{1 h}^{k}, Z_{h}^{T} W_{2 h}^{k} Z_{h} \geq Z_{h}^{T} + Z_{h} - {\bar{W}}_{2 h}^{k}

Z_{v}^{T} W_{1 v}^{k} Z_{v} \geq Z_{v}^{T} + Z_{v} - {\bar{W}}_{1 v}^{k}, Z_{v}^{T} W_{2 v}^{k} Z_{v} \geq Z_{v}^{T} + Z_{v} - {\bar{W}}_{2 v}^{k},

we show that (42) holds if (36) is satisfied. The proof is completed.

Remark 4. It is noticed that (35)–(36) are LMIs; we can firstly solve the LMIs to obtain the solutions of matrices ${\bar{P}}^{k}$ , ${\bar{Q}}_{1}^{k}$ , ${\bar{Q}}_{2}^{k}$ , ${\bar{W}}_{1}^{k}$ , ${\bar{W}}_{2}^{k}$ , ${\bar{X}}_{h}^{k}$ , ${\bar{X}}_{v}^{k}$ , ${\bar{Y}}_{h}^{k}$ , ${\bar{Y}}_{v}^{k}$ , ${\bar{M}}_{h}^{k}$ , ${\bar{N}}_{h}^{k}$ , ${\bar{S}}_{h}^{k}$ , ${\bar{M}}_{v}^{k}$ , ${\bar{N}}_{v}^{k}$ , ${\bar{S}}_{v}^{k}$ , $Z$ and $ϒ^{k}$ . Then $K^{k}$ can be obtained by $K^{k} = ϒ^{k} Z^{- 1}$ .

Remark 5. It is noticed that when $μ = 1$ in (38), (39) turns out to be ${\bar{P}}^{k} = {\bar{P}}^{l}$ , ${\bar{Q}}_{1}^{k} = {\bar{Q}}_{1}^{l}$ , ${\bar{Q}}_{2}^{k} = {\bar{Q}}_{2}^{l}$ , ${\bar{W}}_{1}^{k} = {\bar{W}}_{1}^{l}$ , ${\bar{W}}_{2}^{k} = {\bar{W}}_{2}^{l}$ , $\forall k, l \in \underline{N}$ . In this case, we have $τ_{a} > τ_{a}^{*} = 0$ , which means that the switching signal can be arbitrary.

When the delays in system (1) are constants, i.e. $d_{hH} = d_{hL} = d_{h}$ and $d_{vH} = d_{vL} = d_{v}$ , the resulting closed-loop system can be written as follows

\begin{matrix} [\begin{matrix} x^{h} (i + 1, j) \\ x^{v} (i, j + 1) \end{matrix}] = ({\hat{A}}^{σ (i, j)} + B^{σ (i, j)} Ω^{σ (i, j)} K^{σ (i, j)}) [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] \\ + {\hat{A}}_{d}^{σ (i, j)} [\begin{matrix} x^{h} (i - d_{h}, j) \\ x^{v} (i, j - d_{v}) \end{matrix}] \end{matrix}

(51)

Theorem 2. Consider system (1) with constant delays, for given positive scalars $δ_{k}$ , $ε_{k}$ , $α < 1$ , if there exist positive definite symmetric matrices ${\bar{P}}^{k} = diag {{\bar{P}}_{h}^{k}, {\bar{P}}_{v}^{k}}$ , ${\bar{Q}}_{1}^{k} = diag {{\bar{Q}}_{1 h}^{k}, {\bar{Q}}_{1 v}^{k}}$ , ${\bar{W}}_{1}^{k} = diag {{\bar{W}}_{1 h}^{k}, {\bar{W}}_{1 v}^{k}}$ , ${\bar{X}}_{h}^{k} = [\begin{matrix} {\bar{X}}_{11 h}^{k} & {\bar{X}}_{12 h}^{k} \\ * & {\bar{X}}_{22 h}^{k} \end{matrix}]$ , ${\bar{X}}_{v}^{k} = [\begin{matrix} {\bar{X}}_{11 v}^{k} & {\bar{X}}_{12 v}^{k} \\ * & {\bar{X}}_{22 v}^{k} \end{matrix}]$ , and any matrices $ϒ^{k}$ , ${\bar{M}}_{h}^{k} = [\begin{matrix} {\bar{M}}_{1 h}^{k} \\ {\bar{M}}_{2 h}^{k} \end{matrix}]$ , ${\bar{M}}_{v}^{k} = [\begin{matrix} {\bar{M}}_{1 v}^{k} \\ {\bar{M}}_{2 v}^{k} \end{matrix}]$ , $Z = diag {Z_{h}, Z_{v}}$ with appropriate dimensions, $k \in \underline{N}$ , such that

\tilde{Φ} = [\begin{matrix} {\tilde{Φ}}_{1} & {\tilde{Φ}}_{2} & {\tilde{Φ}}_{3} & T^{k} & ϒ^{kT} Ξ^{k} \\ * & {\tilde{Φ}}_{4} & {\tilde{Φ}}_{5} & 0 & 0 \\ * & * & Π^{k} & 0 & 0 \\ * & * & * & - δ_{k} I & 0 \\ * & * & * & * & - ε_{k} Ξ^{k} \end{matrix}] < 0

(52)

{\bar{Ψ}}_{1 h}^{k} = [\begin{matrix} {\bar{X}}_{h}^{k} & {\bar{M}}_{h}^{k} \\ * & Z_{h} + Z_{h}^{T} - {\bar{W}}_{1 h}^{k} \end{matrix}] \geq 0

(53)

{\bar{Ψ}}_{1 v}^{k} = [\begin{matrix} {\bar{X}}_{v}^{k} & {\bar{M}}_{v}^{k} \\ * & Z_{v} + Z_{v}^{T} - {\bar{W}}_{1 v}^{k} \end{matrix}] \geq 0

(54)

Then, under the reliable controller (37) and the average dwell time scheme (38), the resulting closed-loop system (51) is exponentially stable, where $μ \geq 1$ satisfying

\begin{matrix} {\bar{P}}^{k} \leq μ {\bar{P}}^{l}, {\bar{P}}^{l} \leq μ {\bar{P}}^{k}, {\bar{Q}}_{1}^{k} \leq μ {\bar{Q}}_{1}^{l}, \\ {\bar{Q}}_{1}^{l} \leq μ {\bar{Q}}_{1}^{k}, {\bar{W}}_{1}^{k} \leq μ {\bar{W}}_{1}^{l}, {\bar{W}}_{1}^{l} \leq μ {\bar{W}}_{1}^{k}, \forall k, l \in \underline{N}, \end{matrix}

{\tilde{Λ}}_{1} = diag {d_{h} I_{h}, d_{v} I_{v}}, {\tilde{Λ}}_{3} = diag {α^{d_{h}} I_{h}, α^{d_{v}} I_{v}}, {\tilde{Φ}}_{1} = [\begin{matrix} {\tilde{Φ}}_{11} & {\tilde{Φ}}_{12} \\ * & {\tilde{Φ}}_{22} \end{matrix}]

\begin{matrix} {\tilde{Φ}}_{11} = {\bar{Q}}_{1}^{k} + {\tilde{Λ}}_{3} ({\bar{M}}_{1}^{k} + {\bar{M}}_{1}^{kT} + {\tilde{Λ}}_{1} {\bar{X}}_{11}^{k}) - α {\bar{P}}^{k}, \\ {\tilde{Φ}}_{12} = {\tilde{Λ}}_{3} ({\tilde{Λ}}_{1} {\bar{X}}_{12}^{k} + {\bar{M}}_{2 k}^{kT}) - {\tilde{Λ}}_{3} {\bar{M}}_{1}^{k}, \\ {\tilde{Φ}}_{22} = {\tilde{Λ}}_{1} {\tilde{Λ}}_{3} {\bar{X}}_{22}^{k} - {\tilde{Λ}}_{3} {\bar{M}}_{2}^{k} - {({\tilde{Λ}}_{3} {\bar{M}}_{2}^{k})}^{T} - {\tilde{Λ}}_{3} {\bar{Q}}_{1}^{k} \\ {\tilde{Φ}}_{4} = - {\tilde{Λ}}_{1}^{- 1} {\bar{W}}_{1}^{k} + δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT} \\ {\tilde{Φ}}_{2} = [\begin{matrix} (A^{k} - I) Z + B^{k} Ω_{0}^{k} ϒ^{k} & A_{d}^{k} Z \end{matrix}], \\ {\tilde{Φ}}_{3} = [\begin{matrix} A^{k} Z + B^{k} Ω_{0}^{k} ϒ^{k} & A_{d}^{k} Z \end{matrix}] \\ Π^{k} = - Z - Z^{T} + {\bar{P}}^{k} + δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT} \end{matrix}

T^{k} = [\begin{matrix} {(E^{k} Z)}^{T} & {(E_{d}^{k} Z)}^{T} \end{matrix}], {\tilde{Φ}}_{5} = δ_{k} H^{k} H^{kT} + ε_{k} B^{k} Ω_{0}^{k} Ξ^{k} Ω_{0}^{kT} B^{kT}

\begin{matrix} {\bar{X}}_{11}^{k} = diag {{\bar{X}}_{11 h}^{k}, {\bar{X}}_{11 v}^{k}}, {\bar{X}}_{12}^{k} = diag {{\bar{X}}_{12 h}^{k}, {\bar{X}}_{12 v}^{k}}, \\ {\bar{X}}_{22}^{k} = diag {{\bar{X}}_{22 h}^{k}, {\bar{X}}_{22 v}^{k}} \end{matrix}

{\bar{M}}_{1}^{k} = diag {{\bar{M}}_{1 h}^{k}, {\bar{M}}_{1 v}^{k}}, {\bar{M}}_{2}^{k} = diag {{\bar{M}}_{2 h}^{k}, {\bar{M}}_{2 v}^{k}} .

Proof. We consider the following Lyapunov function candidate for the kth $k \in \underline{N}$ subsystem

V_{k} (x (i, j)) = V_{k}^{h} (x^{h} (i, j)) + V_{k}^{v} (x^{v} (i, j))

(55)

where

V_{k}^{h} (x^{h} (i, j)) = \sum_{g = 1}^{3} V_{gk}^{h} (x^{h} (i, j))

V_{1 k}^{h} (x^{h} (i, j)) = x^{h} {(i, j)}^{T} P_{h}^{k} x^{h} (i, j)

V_{2 k}^{h} (x^{h} (i, j)) = \sum_{r = i - d_{h}}^{i - 1} x^{h} {(r, j)}^{T} Q_{1 h}^{k} x^{h} (r, j) α^{i - r - 1}

V_{3 k}^{h} (x^{h} (i, j)) = \sum_{s = - d_{h}}^{- 1} \sum_{r = i + s}^{i - 1} η^{h} {(r, j)}^{T} W_{1 h}^{k} η^{h} (r, j) α^{i - r - 1}

V_{k}^{v} (x^{v} (i, j)) = \sum_{g = 1}^{3} V_{gk}^{v} (x^{v} (i, j))

V_{1 k}^{v} (x^{v} (i, j)) = x^{v} {(i, j)}^{T} P_{v}^{k} x^{v} (i, j)

V_{2 k}^{v} (x^{v} (i, j)) = \sum_{t = j - d_{v}}^{j - 1} x^{v} {(i, t)}^{T} Q_{1 v}^{k} x^{v} (i, t) α^{j - t - 1}

V_{3 k}^{v} (x^{v} (i, j)) = \sum_{s = - d_{v}}^{- 1} \sum_{t = j + s}^{j - 1} η^{v} {(i, t)}^{T} W_{1 v}^{k} η^{v} (i, t) α^{j - t - 1}

The remainder process can be followed by the same lines of the proof of Theorem 1, and we omit the details.

Numerical example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (1) with parameters as follows

\begin{matrix} A_{1} = [\begin{matrix} 1 & 1.5 \\ 0.6 & 0.5 \end{matrix}], A_{d 1} = [\begin{matrix} - 0.2 & 0 \\ - 0.1 & - 0.1 \end{matrix}], \\ B_{1} = [\begin{matrix} - 4.5 & 0 \\ 1 & - 3 \end{matrix}], H_{1} = [\begin{matrix} 0.1 & 0.15 \\ 0.1 & 0.2 \end{matrix}] \\ E_{1} = [\begin{matrix} 0.1 & 0.15 \\ 0.1 & 0 \end{matrix}], E_{d 1} = [\begin{matrix} 0.1 & 0 \\ 0.1 & 0.1 \end{matrix}], \\ F_{1} = [\begin{matrix} \sin (0.5 π (i + j)) & 0 \\ 0 & \sin (0.5 π (i + j)) \end{matrix}] \\ A_{2} = [\begin{matrix} 1.5 & 1.5 \\ 0.5 & 0.5 \end{matrix}], A_{d 2} = [\begin{matrix} - 0.15 & 0.1 \\ - 0.05 & - 0.05 \end{matrix}], \\ B_{2} = [\begin{matrix} - 5 & 1 \\ - 1 & - 3 \end{matrix}], H_{2} = [\begin{matrix} 0.1 & 0.12 \\ 0.1 & 0.1 \end{matrix}] \\ E_{2} = [\begin{matrix} 0.1 & 0.12 \\ 0.12 & 0.1 \end{matrix}], E_{d 2} = [\begin{matrix} 0.05 & 0.1 \\ 0.05 & 0.1 \end{matrix}], \\ F_{2} = [\begin{matrix} \cos (0.5 π (i + j)) & 0 \\ 0 & \cos (0.5 π (i + j)) \end{matrix}] \end{matrix}

d_{h} (i) = 2 + \sin (\frac{π i}{2}), d_{v} (j) = 2 + \sin (\frac{π j}{2})

where state dimensions $n_{1} = 1$ and $n_{2} = 1$ . From this example, it is easy to get that the lower and upper delay bounds along the horizontal and the vertical direction are $d_{hL} = 1$ , $d_{hH} = 3$ , $d_{vL} = 1$ and $d_{vH} = 3$ . The boundary conditions are as follows

\begin{array}{l} x^{h} (i, j) = 0, \forall 0 \leq j \leq 20, - 3 \leq i < 0 \\ x^{h} (i, j) = 4, \forall 0 \leq j \leq 20, i = 0, \\ x^{h} (i, j) = 0, \forall j > 20, - 3 \leq i \leq 0, \\ x^{v} (i, j) = 0, \forall 0 \leq i \leq 20, - 3 \leq j < 0, \\ x^{v} (i, j) = 3, \forall 0 \leq i \leq 20, j = 0, \\ x^{v} (i, j) = 0, \forall i > 20, - 3 \leq j \leq 0 . \end{array}

The fault matrices $Ω^{k} = diag {ω_{k 1}, ω_{k 2}}$ , $k = 1, 2$

0.4 \leq ω_{11} \leq 0.5, 0.5 \leq ω_{12} \leq 0.6, 0.5 \leq ω_{21} \leq 0.6, 0.4 \leq ω_{22} \leq 0.5

Take $α = 0.85$ , $δ_{1} = δ_{2} = 0.2$ , $ε_{1} = ε_{2} = 0.1$ , solving the matrix inequalities in Theorem 1 gives rise to

\begin{array}{l} {\bar{P}}^{1} = [\begin{matrix} 0.5285 & 0 \\ 0 & 0.2755 \end{matrix}], {\bar{Q}}_{1}^{1} = [\begin{matrix} 0.0052 & 0 \\ 0 & 0.0042 \end{matrix}], \\ {\bar{Q}}_{2}^{1} = [\begin{matrix} 0.0477 & 0 \\ 0 & 0.0156 \end{matrix}] \\ {\bar{W}}_{1}^{1} = [\begin{matrix} 0.5470 & 0 \\ 0 & 0.3390 \end{matrix}], {\bar{W}}_{2}^{1} = [\begin{matrix} 0.5893 & 0 \\ 0 & 0.3581 \end{matrix}], \\ ϒ^{1} = [\begin{matrix} 0.0494 & 0.1547 \\ 0.1435 & 0.0262 \end{matrix}] \\ {\bar{X}}_{h}^{1} = [\begin{matrix} 0.1057 & - 0.1047 \\ - 0.1047 & 0.1069 \end{matrix}], {\bar{X}}_{v}^{1} = [\begin{matrix} 0.0398 & - 0.0390 \\ - 0.0390 & 0.0416 \end{matrix}], \\ {\bar{Y}}_{h}^{1} = [\begin{matrix} 0.0137 & - 0.0139 \\ - 0.0139 & 0.0195 \end{matrix}] \\ {\bar{Y}}_{v}^{1} = [\begin{matrix} 0.0055 & - 0.0058 \\ - 0.0058 & 0.0084 \end{matrix}], {\bar{M}}_{h}^{1} = [\begin{matrix} - 0.1690 \\ 0.1671 \end{matrix}], \\ {\bar{M}}_{v}^{1} = [\begin{matrix} - 0.0558 \\ 0.0539 \end{matrix}], {\bar{N}}_{h}^{1} = [\begin{matrix} 0.0001 \\ - 0.0352 \end{matrix}] \\ {\bar{N}}_{v}^{1} = [\begin{matrix} 0.0001 \\ - 0.0115 \end{matrix}], {\bar{S}}_{h}^{1} = [\begin{matrix} - 0.0556 \\ 0.0567 \end{matrix}], \\ {\bar{S}}_{v}^{1} = [\begin{matrix} - 0.0180 \\ 0.0188 \end{matrix}], Z = [\begin{matrix} 0.4141 & 0 \\ 0 & 0.2131 \end{matrix}] \\ {\bar{P}}^{2} = [\begin{matrix} 0.4806 & 0 \\ 0 & 0.2875 \end{matrix}], {\bar{Q}}_{1}^{2} = [\begin{matrix} 0.0078 & 0 \\ 0 & 0.0067 \end{matrix}], \\ {\bar{Q}}_{2}^{2} = [\begin{matrix} 0.0426 & 0 \\ 0 & 0.0220 \end{matrix}] \\ {\bar{W}}_{1}^{2} = [\begin{matrix} 0.6218 & 0 \\ 0 & 0.3429 \end{matrix}], {\bar{W}}_{2}^{2} = [\begin{matrix} 0.6634 & 0 \\ 0 & 0.3479 \end{matrix}], \\ {\bar{X}}_{h}^{2} = [\begin{matrix} 0.0759 & - 0.0764 \\ - 0.0764 & 0.0813 \end{matrix}] \\ {\bar{X}}_{v}^{2} = [\begin{matrix} 0.0555 & - 0.0549 \\ - 0.0549 & 0.0583 \end{matrix}], {\bar{Y}}_{h}^{2} = [\begin{matrix} 0.0103 & - 0.0101 \\ - 0.0101 & 0.0155 \end{matrix}], \\ {\bar{Y}}_{v}^{2} = [\begin{matrix} 0.0070 & - 0.0074 \\ - 0.0074 & 0.0111 \end{matrix}] \\ {\bar{M}}_{h}^{2} = [\begin{matrix} - 0.1219 \\ 0.1247 \end{matrix}], {\bar{M}}_{v}^{2} = [\begin{matrix} - 0.0627 \\ 0.0609 \end{matrix}], {\bar{N}}_{h}^{2} = [\begin{matrix} 0.0022 \\ - 0.0314 \end{matrix}], \\ {\bar{N}}_{v}^{2} = [\begin{matrix} 0.0002 \\ - 0.0153 \end{matrix}] \\ {\bar{S}}_{h}^{2} = [\begin{matrix} - 0.0395 \\ 0.0431 \end{matrix}], {\bar{S}}_{v}^{2} = [\begin{matrix} - 0.0202 \\ 0.0217 \end{matrix}], ϒ^{2} = [\begin{matrix} 0.1352 & 0.1098 \\ 0.0851 & - 0.0553 \end{matrix}] . \end{array}

Then $K^{1}$ and $K^{2}$ can be obtained by $K^{k} = ϒ^{k} Z^{- 1}$ , and the results are as follows

K^{1} = [\begin{matrix} 0.1194 & 0.7262 \\ 0.3466 & 0.1228 \end{matrix}], K^{2} = [\begin{matrix} 0.3265 & 0.5155 \\ 0.2055 & - 0.2597 \end{matrix}]

Furthermore, we can get $μ = 3.2292$ , From (38), it can be obtained that $τ_{a}^{*} = 7.3$ . Choosing $τ_{a} = 8$ , the responses of the states $x^{h} (i, j)$ and $x^{v} (i, j)$ , and the switching signal are shown in Figures 1 and 2.

Figure 1.

The responses of the states $x^{h} (i, j)$ and $x^{v} (i, j)$ : (a) the response of the state $x^{h} (i, j)$ ; (b) the response of the state $x^{v} (i, j)$ .

Figure 2.

The switching signal.

From Figures 1 and 2, one can notice that the reliable controller can guarantee the exponential stability of the closed-loop system. This demonstrates the effectiveness of the proposed method.

Conclusions

This paper has presented a solution to the problem of robust reliable control for a class of uncertain 2D discrete switched systems with state delays and actuator failures. A reliable controller design methodology is proposed for such systems, and a sufficient condition for the existence of the reliable controller is formulated in terms of a set of LMIs. An illustrative example is also given to illustrate the applicability of the proposed approach. We would like to point out that our main results could be extended to other 2D switched systems such as positive 2D switched systems or stochastic 2D switched systems. This will also be one of our future research topics.

Footnotes

Funding

This work was supported by the National Natural Science Foundation of China under grant numbers 60974027 and 61273120.

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