H 2 output feedback controller design for discrete-time 2D switched systems

Abstract

This paper investigates the H₂ control problem for a class of discrete-time two-dimensional (2D) switched systems represented by a model of Roesser type. By using a multiple Lyapunov function method, sufficient conditions for evaluation of the H₂ performance for the 2D switched systems are presented in terms of linear matrix inequalities. Based on the obtained results, an H₂ output feedback controller design scheme is developed for the discrete-time 2D switched systems. A numerical example is included to demonstrate the effectiveness of the proposed method.

Keywords

2D systems LMI output feedback switched systems

Introduction

Two-dimensional (2D) systems have attracted considerable research attention in control theory and practice over the past few decades due to their wide applications such as in energy exchange processes, electricity transmission, multi-dimensional digital filtering and linear image processing (Kaczorek, 1985; Kevorkian, 1993; Xu et al., 2004). Two typical examples are the wave equation, which arises in many different fields, such as acoustics, electromagnetics, fluid dynamics, quantum mechanics, general relativity, the heat equation (which describes the temperature in a given region over time) and Poisson’s equation. This stimulates research on 2D discrete systems, which can be represented by different models such as the Roesser model, the Fornasini–Marchesini model and the Attasi model. Some important problems such as stability and stabilization have been investigated in Lam et al. (2004), and many works on the H_∞ control problems and filtering for 2D systems have been provided by many researchers (Du and Xie, 2002; Gao CY et al., 2008; Gao H et al., 2004; Li and Gao, 2012; Paszke, 2004; Xu and Zou, 2008; Xu et al., 2005, 2011).

On the other hand, switched systems have also been intensively studied for their wide applications (Balluchi et al., 1997; Bishop and Spong, 1998; Castillo-Toledo et al., 2008; Sreekumar and Agarwal, 2008; Zhang et al., 2001), such as chemical processes, computer controlled systems, switched circuits, motor engine control, constrained robotics and networked control systems. Many techniques are effective tools for dealing with switched systems, such as the common quadratic Lyapunov function method, multiple Lyapunov function method and average dwell time approach (Branicky, 1998; Lian et al., 2010, 2011; Narendra and Balakrishnan, 1994; Xiang and Chen, 2010). The stability and H_∞ control problems for switched systems have been widely investigated in recent years (Daafouz and Bernussou, 2001; Lee et al., 2000; Song and Zhao, 2007; Li et al., 2008; Wang et al., 2011; Xie et al., 2004; Zhang and Shi, 2009).

In many modelling problems of physical processes, a 2D switching representation is needed, e.g. the transport process with multiple lines (Benzaouia et al., 2011). So 2D switched systems have also attracted considerable research attention. There are a few reports on 2D discrete switched systems; Benzaouia et al. (2011) first considered the problems of stability and stabilization for 2D switched systems with arbitrary switching sequences. Recently, the exponential stability and stabilization of the discrete 2D switched system in the Roesser model was first investigated via the average dwell time approach in Xiang and Huang (2013).

H ₂ optimal control for continuous or discrete-time systems is a classical problem in system theory. Its objective is to minimize the energy of the system output when the system is subjected to a unit impulse input or, equivalently, a white noise input of unit variance. Due to this analytically and practically meaningful specification, the H₂ problem and solution have been well studied and applied for several decades (Du et al., 2006; Feng and Sun, 2002; Mahmoud, 2009; Rotea, 1993). Recently, some results on such a topic for 2D systems have also appeared. For example, by using the linear matrix inequality (LMI) technique, a solution to the mixed H₂/ H_∞ filtering problem of 2D systems was presented in Tuan et al. (2002), where a H₂ norm is defined as the square of the peak amplitude of output when the total energy of the past input is no greater than unity. In Yang et al. (2006), the mixed H₂/H_∞ control problem for 2D discrete systems in the Roesser model was solved, where the H₂ norm of 2D discrete systems was computed with the unit impulse input.

However, because of the complicated behaviour caused by the interaction between the continuous dynamics and discrete switching, the H₂ control problem of 2D switched systems is more difficult to study. To the best of our knowledge, there are no results on such a topic for these systems reported. Moreover, the aforementioned methods cannot be directly applied to 2D switched systems, and this constitutes the main motivation for the present study.

In this paper, we confine our attention to the H₂ control problem of 2D switched discrete systems represented by the Roesser model. The main theoretical contributions are threefold: 1) we further contribute to the development of stabilization for a class of 2D switched systems represented by the Roesser model; 2) the multiple Lyapunov function method is used to derive a condition for evaluation of the H₂ performance of 2D discrete switched systems, which has not been considered in the existing literature; and 3) a mode-dependent dynamic output feedback controller is designed for the underlying systems such that the resulting closed-loop systems are asymptotically stable with H₂ performance under arbitrary switching sequences. The proposed method can also be applied to non-switched 2D discrete linear delay-free systems.

The organization of this paper is as follows. We next formulate the problem and present some preliminary results, then the analysis of asymptotical stability with H₂ performance is given. The H₂ dynamic output feedback controller design scheme is developed, followed by an example to illustrate the effectiveness of the proposed approach. Finally, the paper is concluded.

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. $diag {a_{i}}$ denotes 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 a term that is induced by symmetry. The set of all non-negative integers is represented by $Z^{+}$ . A 2D signal $s$ with $s (i, j) \in R^{n}$ , $i, j \in Z^{+}$ is said to belong to the 2D $l_{2^{-}}$ space if

‖ s ‖_{2} = \sqrt{\sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} s^{T} (i, j) s (i, j)} < \infty

where $‖ \cdot ‖_{2}$ denotes the $l_{2}$ norm of $s$ . A 2D signal in the $l_{2^{-}}$ space is an energy-bounded signal.

Problem formulation and preliminaries

The Roesser model for a 2D switched system $G : u \to y$ is given by the following state equation:

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

(1a)

y (i, j) = C^{σ (i, j)} [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + D^{σ (i, j)} u (i, j)

(1b)

where $x^{h} (i, j) \in R^{n_{1}}$ and $x^{v} (i, j) \in R^{n_{2}}$ are the horizontal state and the vertical state respectively, $x (i, j)$ is the whole state in $R^{n}$ with $n = n_{1} + n_{2}$ , $u (i, j) \in R^{m}$ is the control input, $y (i, j) \in R^{l}$ is the controlled output, $i$ 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. $σ (i, j) = α$ , $α \in \underline{N}$ , denotes that the $α$ th subsystem is activated. $A^{α}$ , $B^{α}$ , $C^{α}$ and $D^{α}$ , $α \in \underline{N}$ , are real matrices with appropriate dimensions. The boundary condition satisfies $‖ X (0) ‖_{2} < \infty$ with $X (0)$ defined as follows:

X (0) = {[x^{hT} (0, 0), x^{hT} (0, 1), x^{hT} (0, 2), \dots, x^{vT} (0, 0), x^{vT} (1, 0), x^{vT} (2, 0), \dots]}^{T}

(2)

Remark 1. In this paper, it is assumed that switching occurs only at each sampling point of i or j. The switching sequence 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

with $(i_{k}, j_{k})$ denoting 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).

Remark 2. It is assumed that system (1) satisfies the following two conditions:

There is no jump in the state x(i, j) at the switching instants;

There is no Zeno behaviour (there is a finite number of switches on every bounded interval of time).

Remark 3. If there is only one subsystem in system (1), it will degenerate to the following 2D Roesser system

[\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}] + Bu (i, j)

y (i, j) = C [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + Du (i, j)

Therefore, the addressed system (1) can be viewed as an extension of discrete 2D Roesser systems to switched systems.

Denoting $X_{r} = sup {‖ x (i, j) ‖ : i + j = r . i, j \in Z^{+}}$ , we first introduce the definition of asymptotic stability for 2D switched system (1).

Definition 1 (Lam et al., 2004). A 2D switched system (1) with $u (i, j) = 0$ is asymptotically stable under the switching signal $σ (i, j)$ if $lim_{r \to \infty} X_{r} = 0$ .

Let $E_{k} \in R^{m}$ , $1 \leq k \leq m$ , denote the $k$ th column of the $m \times m$ identity matrix and $δ (i, j)$ be the 2D discrete time unit impulse signal satisfying

δ (i, j) = {\begin{matrix} 1 if i = 0, j = 0 \\ 0 otherwise \end{matrix}

(3)

Further subject to the zero boundary condition $x^{h} (0, j) = 0$ , $x^{v} (i, 0) = 0$ with $i, j \geq 0$ and the input $u (i, j) = E_{k} δ (i, j)$ , $1 \leq k \leq m$ , let the impulse response of the 2D switched system $G$ be

g_{k} (i, j) = G E_{k} δ (i, j)

If the 2D switched system $G$ is stable, its impulse response $g_{k} (i, j) \in l_{2}$ for $1 \leq k \leq m$ . We can define the H₂ norm of the 2D switched system G as

‖ G ‖_{2} = \sqrt{\sum_{k = 1}^{m} ‖ g_{k} ‖_{2}^{2}} = \sqrt{\sum_{k = 1}^{m} \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} g_{k}^{T} (i, j) g_{k} (i, j)}

(4)

Physically, the square of the H₂ norm of the system represents the amount of the system output energy when it is subject to the unit impulse input or a Gaussian white noise input with unit variance. Now, consider that the 2D switched system G in the Roesser model (1) is subject to the input $u (i, j) = E_{k} δ (i, j)$ . Under such an input, let $x_{k}^{h} (i, j)$ and $x_{k}^{v} (i, j)$ denote the system horizontal state and vertical state, respectively. Furthermore, let

x_{k} (i, j) = [\begin{matrix} x_{k}^{h} (i, j) \\ x_{k}^{v} (i, j) \end{matrix}], x_{k}^{+} (i, j) = [\begin{matrix} x_{k}^{h} (i + 1, j) \\ x_{k}^{v} (i, j + 1) \end{matrix}]

and write the matrices $B^{σ (i, j)} \in R^{n \times m}$ and $D^{σ (i, j)} \in R^{p \times m}$ as

B^{σ (i, j)} = [\begin{matrix} {\tilde{B}}_{1}^{σ (i, j)} & {\tilde{B}}_{2}^{σ (i, j)} & \dots & {\tilde{B}}_{m}^{σ (i, j)} \end{matrix}]

D^{σ (i, j)} = [\begin{matrix} {\tilde{D}}_{1}^{σ (i, j)} & {\tilde{D}}_{2}^{σ (i, j)} & \dots & {\tilde{D}}_{m}^{σ (i, j)} \end{matrix}]

Under the zero boundary condition, we can express the system impulse response $g_{k}$ , $1 \leq k \leq m$ , as

\begin{matrix} x_{k}^{+} (i, j) = A^{σ (i, j)} x_{k} (i, j) + B^{σ (i, j)} E_{k} δ (i, j) \\ = {\begin{matrix} {\tilde{B}}_{k}^{σ (i, j)} \\ A^{σ (i, j)} x_{k} (i, j) \end{matrix} \begin{matrix} if i = 0, j = 0 \\ otherwise \end{matrix} \end{matrix}

(5)

\begin{matrix} g_{k} (i, j) = C^{σ (i, j)} x_{k} (i, j) + D^{σ (i, j)} E_{k} δ (i, j) \\ = {\begin{matrix} {\tilde{D}}_{k}^{σ (i, j)} & if i = 0, j = 0 \\ C^{σ (i, j)} x_{k} (i, j) & otherwise \end{matrix} \end{matrix}

(6)

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:

$S < 0$ ;

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

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

H₂ performance analysis

We now present a sufficient condition for evaluation of the H₂ norm of 2D switched system in the Roesser model (1).

Theorem 1. Given a positive scalar $γ$ , the H₂ norm of 2D system G in form (1) with zero boundary condition is bounded by $γ$ under arbitrary switching sequences, if there exist a block-diagonal matrix $Ω = diag {Ω_{h}, Ω_{v}}$ , symmetric block-diagonal matrices $P^{α} = diag {P_{h}^{α}, P_{v}^{α}}$ with $α \in \underline{N}$ , and matrix $Z$ such that

[\begin{matrix} - P^{α} & {A^{α}}^{T} Ω^{T} & {C^{α}}^{T} \\ Ω A^{α} & P^{β} - (Ω + Ω^{T}) & 0 \\ C^{α} & 0 & - I \end{matrix}] < 0

(7a)

[\begin{matrix} - Z & B^{α T} Ω^{T} & D^{α T} \\ Ω B^{α} & P^{β} - (Ω + Ω^{T}) & 0 \\ D^{α} & 0 & - I \end{matrix}] < 0

(7b)

trace (Z) < γ^{2}, \forall α, β \in \underline{N}

(7c)

Proof. Choose the following Lyapunov functional candidate

V_{k} (i, j) = x_{k}^{T} (i, j) P^{σ (i, j)} x_{k} (i, j)

where

P^{α} = [\begin{matrix} P_{1}^{α} & 0 \\ 0 & P_{2}^{α} \end{matrix}], α \in \underline{N}

We introduce

\begin{matrix} Δ V_{k} (i, j) = [\begin{matrix} x_{k}^{hT} (i + 1, j) & x_{k}^{vT} (i, j + 1) \end{matrix}] \\ [\begin{matrix} P_{h}^{σ (i + 1, j)} & 0 \\ 0 & P_{v}^{σ (i, j + 1)} \end{matrix}] [\begin{matrix} x_{k}^{h} (i + 1, j) \\ x_{k}^{v} (i, j + 1) \end{matrix}] \end{matrix}

- [x_{k}^{hT} (i, j), x_{k}^{vT} (i, j)] [\begin{matrix} P_{h}^{σ (i, j)} & 0 \\ 0 & P_{v}^{σ (i, j)} \end{matrix}] [\begin{matrix} x_{k}^{h} (i, j) \\ x_{k}^{v} (i, j) \end{matrix}]

(8)

Assume that the $α$ th subsystem is activated at $i + j$ , $α \in \underline{N}$ . When there is no switch occurring at $i + j + 1$ , the increment can be computed as follows:

Δ V_{k} (i, j) = x_{k}^{+ T} (i, j) P^{α} x_{k}^{+} (i, j) - x_{k}^{T} (i, j) P^{α} x_{k} (i, j), 1 \leq k \leq m

(9)

It follows from (7a) that the $α$ th subsystem is stable.

When system (1) is switched to the $β$ th subsystem from the $α$ th subsystem at i+j+1, the increment can be obtained as follows

Δ V_{k} (i, j) = x_{k}^{+ T} (i, j) P^{β} x_{k}^{+} (i, j) - x_{k}^{T} (i, j) P^{α} x_{k} (i, j), 1 \leq k \leq m

(10)

From (7a), we have

A^{α T} P^{β} A^{α} - P^{α} < 0

Then the value of Lyapunov function $V_{k} (i, j)$ is decreasing at the switching instants.

Therefore, one can see that the 2D switched system (1) is asymptotically stable.

From (8), one obtains

\begin{matrix} Δ V_{k} (i, j) = x_{k}^{hT} (i + 1, j) P_{h}^{σ (i + 1, j)} x_{k}^{h} (i + 1, j) - x_{k}^{hT} (i, j) P_{h}^{σ (i, j)} x_{k}^{h} (i, j) \\ + x_{k}^{vT} (i, j + 1) P_{v}^{σ (i, j + 1)} x_{k}^{v} (i, j + 1) - x_{k}^{vT} (i, j) P_{v}^{σ (i, j)} x_{k}^{v} (i, j) \end{matrix}

= M_{k} + N_{k}

where

\begin{matrix} M_{k} = x_{k}^{hT} (i + 1, j) P_{h}^{σ (i + 1, j)} x_{k}^{h} (i + 1, j) - x_{k}^{hT} (i, j) P_{h}^{σ (i, j)} x_{k}^{h} (i, j) \\ N_{k} = x_{k}^{vT} (i, j + 1) P_{v}^{σ (i, j + 1)} x_{k}^{v} (i, j + 1) - x_{k}^{vT} (i, j) P_{v}^{σ (i, j)} x_{k}^{v} (i, j) \end{matrix}

Noting that $σ (i + 1, j) = σ (i, j + 1)$ , for any positive scalars $q_{h}$ , $q_{v} \in Z^{+}$ , we have

\begin{matrix} \sum_{i = 0}^{q_{h}} \sum_{j = 0}^{q_{v}} Δ V_{k} (i, j) = \sum_{i = 0}^{q_{h}} \sum_{j = 0}^{q_{v}} M_{k} + \sum_{i = 0}^{q_{h}} \sum_{j = 0}^{q_{v}} N_{k} \\ = \sum_{j = 0}^{q_{v}} (x_{k}^{hT} (q_{h} + 1, j) P_{h}^{σ (q_{h} + 1, 1)} x_{k}^{h} (q_{h} + 1, j) - x_{k}^{hT} (0, j) P_{h}^{σ (0, j)} x_{k}^{h} (0, j)) \\ + \sum_{i = 0}^{q_{h}} (x_{k}^{vT} (i, q_{v} + 1) P_{v}^{σ (i, q_{v} + 1)} x_{k}^{v} (i, q_{v} + 1) - x_{k}^{vT} (i, 0) P_{v}^{σ (i, 0)} x_{k}^{v} (i, 0)) \end{matrix}

(11)

The zero boundary condition together with (11) implies

\sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Δ V_{k} (i, j) = 0

(12)

On the other hand, it can be derived from (5) that

\begin{matrix} \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Δ V_{k} (i, j) = {\tilde{B}}_{k}^{σ (0, 0) T} diag {P_{h}^{σ (1, 0)}, P_{v}^{σ (0, 1)}} {\tilde{B}}_{k}^{σ (0, 0)} \\ + \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} x_{k}^{T} (i, j) (A^{σ (i, j) T} diag {P_{h}^{σ (i + 1, j)}, P_{v}^{σ (i, j + 1)}} A^{σ (i, j)} - P^{σ (i, j)}) x_{k} (i, j) \end{matrix}

(13)

By using (6), (12) and (13), we can express the square of the H₂ norm of the 2D system G as

\begin{matrix} ‖ G ‖_{2}^{2} = \sum_{k = 1}^{m} ‖ g_{k} ‖_{2}^{2} = \sum_{k = 1}^{m} \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} g_{k}^{T} (i, j) g_{k} (i, j) \\ = \sum_{k = 1}^{m} ({\tilde{D}}_{k}^{σ (0, 0) T} {\tilde{D}}_{k}^{σ (0, 0)} + \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} x_{k}^{T} (i, j) C^{σ (i, j) T} C^{σ (i, j)} x_{k} (i, j)) \\ = \sum_{k = 1}^{m} ({\tilde{D}}_{k}^{σ (0, 0) T} {\tilde{D}}_{k}^{σ (0, 0)} + \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} x_{k}^{T} (i, j) C^{σ (i, j) T} C^{σ (i, j)} x_{k} (i, j) + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Δ V_{k} (i, j)) \\ = \sum_{k = 1}^{m} \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} x_{k}^{T} (i, j) (A^{σ (i, j) T} P^{σ (i + 1, j)} A^{σ (i, j)} - P^{σ (i, j)} + C^{σ (i, j) T} C^{σ (i, j)}) x_{k} (i, j) \\ + \sum_{k = 1}^{m} ({\tilde{B}}_{k}^{σ (0, 0) T} P^{σ (1, 0)} {\tilde{B}}_{k}^{σ (0, 0)} + {\tilde{D}}_{k}^{σ (0, 0) T} {\tilde{D}}_{k}^{σ (0, 0)}) \\ = \sum_{k = 1}^{m} \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} x_{k}^{T} (i, j) (A^{σ (i, j) T} P^{σ (i + 1, j)} A^{σ (i, j)} - P^{σ (i, j)} + C^{σ (i, j) T} C^{σ (i, j)}) x_{k} (i, j) \\ + trace (B^{σ (0, 0) T} P^{σ (1, 0)} B^{σ (0, 0)} + D^{σ (0, 0) T} D^{σ (0, 0)}) \end{matrix}

It follows immediately that $‖ G ‖_{2} < γ$ if the following conditions are satisfied.

A^{α T} P^{β} A^{α} - P^{α} + C^{α T} C^{α} < 0

(14a)

trace (B^{α T} P^{β} B^{α} + D^{α T} D^{α}) - γ^{2} < 0, \forall α, β \in \underline{N}

(14b)

Now we introduce a symmetric matrix Z, then the inequalities in (14b) are equivalent to

B^{α T} P^{β} B^{α} + D^{α T} D^{α} < Z

(15a)

trace (Z) < γ^{2}

(15b)

By Lemma 1, we get the following conditions

[\begin{matrix} - P^{α} & A^{α T} & C^{α T} \\ A^{α} & - {(P^{β})}^{- 1} & 0 \\ C^{α} & 0 & - I \end{matrix}] < 0

(16a)

[\begin{matrix} - Z & B^{α T} & D^{α T} \\ B^{α} & - {(P^{β})}^{- 1} & 0 \\ D^{α} & 0 & - I \end{matrix}] < 0

(16b)

trace (Z) < γ^{2}, \forall α, β \in \underline{N}

(16c)

Consider that $Ω$ is invertible. Using $diag {I, Ω, I}$ and $diag {I, Ω^{T}, I}$ to pre- and post-multiply (16a) and (16b), respectively, one obtains

[\begin{matrix} - P^{α} & A^{α T} Ω^{T} & C^{α T} \\ Ω A^{α} & - Ω {(P^{β})}^{- 1} Ω^{T} & 0 \\ C^{α} & 0 & - I \end{matrix}] < 0

(17a)

[\begin{matrix} - Z & B^{α T} Ω^{T} & D^{α T} \\ Ω B^{α} & - Ω {(P^{β})}^{- 1} Ω^{T} & 0 \\ D^{α} & 0 & - I \end{matrix}] < 0

(17b)

The inequality ${(P^{α} - Ω)}^{T} {(P^{α})}^{- 1} (P^{α} - Ω^{T}) \geq 0$ implies that $Ω {(P^{α})}^{- 1} Ω^{T} \geq Ω + Ω^{T} - P^{α}$ , so we can obtain that (17) holds if (7) is satisfied, which implies that 2D switched system (1) is asymptotically stable with H₂ norm bound $γ$ .

This completes the proof.

Remark 4. In Theorem 1, we propose a sufficient condition for the existence of asymptotical stability and H₂ performance for the considered 2D switched system (1). It is worth noting that this condition is obtained by using the multiple Lyapunov function approach, and it also can be applied to the general 2D Roesser models when N=1.

H₂ control problem

Consider the following discrete 2D switched plant in the Roesser model:

[\begin{matrix} x^{h} (i + 1, j) \\ x^{v} (i, j + 1) \end{matrix}] = A^{σ (i, j)} [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + B_{1}^{σ (i, j)} w (i, j) + B_{2}^{σ (i, j)} u (i, j)

(18a)

z (i, j) = C_{1}^{σ (i, j)} [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + D_{11}^{σ (i, j)} w (i, j) + D_{12}^{σ (i, j)} u (i, j)

(18b)

y (i, j) = C_{2}^{σ (i, j)} [\begin{matrix} x^{h} (i, j) \\ x^{v} (i, j) \end{matrix}] + D_{21}^{σ (i, j)} w (i, j) + D_{22}^{σ (i, j)} u (i, j)

(18c)

where $x^{h} (i, j) \in R^{n_{1}}$ , $x^{v} (i, j) \in R^{n_{2}}$ , $w (i, j) \in R^{q}$ , $u (i, j) \in R^{m}$ , $z (i, j) \in R^{p}$ , $y (i, j) \in R^{l}$ are, respectively, the horizontal state, the vertical state, disturbance input, control input, controlled output and measurement output of the plant, $A^{α}$ , $B_{1}^{α}$ , $B_{2}^{α}$ , $C_{1}^{α}$ , $C_{2}^{α}$ , $D_{11}^{α}$ , $D_{12}^{α}$ , $D_{21}^{α}$ , $D_{22}^{α}$ , $α \in \underline{N}$ , are real matrices with appropriate dimensions. We make no assumption on the statistics of the disturbance input signal $w (i, j)$ other than that it is energy bounded, i.e. $‖ w ‖_{2}^{2} < \infty$ . Without loss of generality, we assume $D_{22}^{α} = 0$ for any $α \in \underline{N}$ .

Introduce the following $n_{c}$ th order 2D switched dynamic output feedback controller for the plant

[\begin{matrix} x_{c}^{h} (i + 1, j) \\ x_{c}^{v} (i, j + 1) \end{matrix}] = A_{c}^{σ (i, j)} [\begin{matrix} x_{c}^{h} (i, j) \\ x_{c}^{v} (i, j) \end{matrix}] + B_{c}^{σ (i, j)} y (i, j)

(19a)

u (i, j) = C_{c}^{σ (i, j)} [\begin{matrix} x_{c}^{h} (i, j) \\ x_{c}^{v} (i, j) \end{matrix}] + D_{c}^{σ (i, j)} y (i, j)

(19b)

where $x_{c}^{h} (i, j) \in R^{n_{c 1}}$ , $x_{c}^{v} (i, j) \in R^{n_{c 2}}$ , $A_{c}^{α} \in R^{n_{c} \times n_{c}}$ , $B_{c}^{α} \in R^{n_{c} \times l}$ , $C_{c}^{α} \in R^{m \times n_{c}}$ and $D_{c}^{α} \in R^{m \times l}$ . Note that the controller order $n_{c} = n_{c 1} + n_{c 2}$ is not necessarily the same as the plant order $n$ .

By introducing the elementary matrix $Π$ as

Π = [\begin{matrix} I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & I_{n_{c 1}} & 0 \\ 0 & I_{n_{2}} & 0 & 0 \\ 0 & 0 & 0 & I_{n_{c 2}} \end{matrix}]

and noting that $Π^{T} = Π^{- 1}$ , the closed-loop system of the plant (18) and the controller (19) is of the form

[\begin{matrix} {\bar{x}}^{h} (i + 1, j) \\ {\bar{x}}^{v} (i, j + 1) \end{matrix}] = {\bar{A}}^{σ (i, j)} [\begin{matrix} {\bar{x}}^{h} (i, j) \\ {\bar{x}}^{v} (i, j) \end{matrix}] + {\bar{B}}^{σ (i, j)} w (i, j)

(20a)

z (i, j) = {\bar{C}}^{σ (i, j)} [\begin{matrix} {\bar{x}}^{h} (i, j) \\ {\bar{x}}^{v} (i, j) \end{matrix}] + {\bar{D}}^{σ (i, j)} w (i, j)

(20b)

where

{\bar{x}}^{h} (i + 1, j) = [\begin{matrix} x^{h} (i + 1, j) \\ x_{c}^{h} (i + 1, j) \end{matrix}], {\bar{x}}^{v} (i, j + 1) = [\begin{matrix} x^{v} (i, j + 1) \\ x_{c}^{v} (i, j + 1) \end{matrix}]

{\bar{A}}^{σ (i, j)} = Π [\begin{matrix} A^{σ (i, j)} + B_{2}^{σ (i, j)} D_{c}^{σ (i, j)} C_{2}^{σ (i, j)} & B_{2}^{σ (i, j)} C_{c}^{σ (i, j)} \\ B_{c}^{σ (i, j)} C_{2}^{σ (i, j)} & A_{c}^{σ (i, j)} \end{matrix}] Π^{T}

{\bar{B}}^{σ (i, j)} = Π [\begin{matrix} B_{1}^{σ (i, j)} + B_{2}^{σ (i, j)} D_{c}^{σ (i, j)} D_{21}^{σ (i, j)} \\ B_{c}^{σ (i, j)} D_{21}^{σ (i, j)} \end{matrix}]

{\bar{C}}^{σ (i, j)} = [\begin{matrix} C_{1}^{σ (i, j)} + D_{12}^{σ (i, j)} D_{c}^{σ (i, j)} C_{2}^{σ (i, j)} & D_{12}^{σ (i, j)} C_{c}^{σ (i, j)} \end{matrix}] Π^{T}

{\bar{D}}^{σ (i, j)} = D_{11}^{σ (i, j)} + D_{12}^{σ (i, j)} D_{c}^{σ (i, j)} D_{21}^{σ (i, j)}

Now, we are in a position to present a sufficient condition for the existence of a mode-dependent dynamic output feedback controller for system (18) such that the resulting closed-loop system (20) is asymptotically stable with H₂ norm bound $γ$ based on Theorem 1. The controller design procedure is provided in the following theorem.

Theorem 2. Consider system (18), for a given constant $γ > 0$ , a dynamic output feedback controller (19) can be designed such that the resulting closed-loop system (20) is asymptotically stable with H₂ norm bound $γ$ under arbitrary switching sequences, if there exist block-diagonal matrices $H_{11}^{α} = diag {H_{11 h}^{α}, H_{11 v}^{α}}$ , $H_{12}^{α} = diag {H_{12 h}^{α}, H_{12 v}^{α}}$ , $H_{22}^{α} = diag {H_{22 h}^{α}, H_{22 v}^{α}}$ , $G = diag {G_{h}, G_{v}}$ , $Y = diag {Y_{h}, Y_{v}}$ , $S = diag {S_{h}, S_{v}}$ and matrices $M^{α}$ , $N^{α}$ , $Φ^{α}$ , $D_{c}^{α}$ , $Z$ of appropriate dimensions with $α \in \underline{N}$ , such that

[\begin{matrix} - H_{11}^{α} & * & * & * & * \\ - H_{12}^{α T} & - H_{22}^{α} & * & * & * \\ G A^{α} + M^{α} C_{2}^{α} & Φ^{α} & H_{11}^{β} - (G + G^{T}) & * & * \\ A^{α} + B_{2}^{α} D_{c}^{α} C_{2}^{α} & A^{α} Y^{T} + B_{2}^{α} N^{α} & H_{12}^{β T} - (I + S^{T}) & H_{22}^{β} - (Y + Y^{T}) & * \\ C_{1}^{α} + D_{12}^{α} D_{c}^{α} C_{2}^{α} & C_{1}^{α} Y^{T} + D_{12}^{α} N^{α} & 0 & 0 & - I \end{matrix}] < 0

(21a)

[\begin{matrix} - Z & * & * & * \\ {GB}_{1}^{α} + M^{α} D_{21}^{α} & H_{11}^{β} - (G + G^{T}) & * & * \\ B_{1}^{α} + B_{2}^{α} D_{c}^{α} D_{21}^{α} & H_{12}^{β T} - (I + S^{T}) & H_{22}^{β} - (Y + Y^{T}) & * \\ D_{11}^{α} + D_{12}^{α} D_{c}^{α} D_{21}^{α} & 0 & 0 & - I \end{matrix}] < 0

(21b)

trace (Z) < γ^{2}, \forall α, β \in \underline{N}

(21c)

with

S - G Y^{T} = U V^{T}

(21d)

Moreover, if (21) has a feasible solution, the controller parameters can be constructed as

A_{c}^{α} = U^{- 1} [Φ^{α} - (G A^{α} Y^{T} + {GB}_{2}^{α} D_{c}^{α} C_{2}^{α} Y^{T} + {UB}_{c}^{α} C_{2}^{α} Y^{T} + {GB}_{2}^{α} C_{c}^{α} V^{T})] V^{- T}

(22a)

B_{c}^{α} = U^{- 1} (M^{α} - {GB}_{2}^{α} D_{c}^{α})

(22b)

C_{c}^{α} = (N^{α} - D_{c}^{α} C_{2}^{α} Y^{T}) V^{- T}

(22c)

Proof. By Theorem 1, the closed-loop system (20) is asymptotically stable with H₂ norm bound $γ$ under arbitrary switching sequences, if there exist a matrix $Z > 0$ , block-diagonal matrices $Ω = diag {Ω_{h}, Ω_{v}}$ , $P^{α} = diag {P_{h}^{α}, P_{v}^{α}}$ , $α \in \underline{N}$ , such that

[\begin{matrix} - P^{α} & {\bar{A}}^{α T} Ω^{T} & {\bar{C}}^{α T} \\ Ω {\bar{A}}^{α} & P^{β} - (Ω + Ω^{T}) & 0 \\ {\bar{C}}^{α} & 0 & - I \end{matrix}] < 0

(23a)

[\begin{matrix} - Z & {\bar{B}}^{α T} Ω^{T} & {\bar{D}}^{α T} \\ Ω {\bar{B}}^{α} & P^{β} - (Ω + Ω^{T}) & 0 \\ {\bar{D}}^{α} & 0 & - I \end{matrix}] < 0

(23b)

trace (Z) < γ^{2}, \forall α, β \in \underline{N}

(23c)

hold, where $P_{h}^{α} \in R^{(n_{1} + n_{c 1}) \times (n_{1} + n_{c 1})}$ and $P_{v}^{α} \in R^{(n_{2} + n_{c 2}) \times (n_{2} + n_{c 2})}$ . Pre- and post- multiplying (23a) by $diag {Π^{T}, Π^{T}, I}$ and $diag {Π, Π, I}$ , and pre- and post- multiplying (23b) by $diag {I, Π^{T}, I}$ and $diag {I, Π, I}$ , one obtains

[\begin{matrix} - {\tilde{P}}^{α} & Π^{T} {\bar{A}}^{α T} Π {\tilde{Ω}}^{T} & Π^{T} {\bar{C}}^{α T} \\ \tilde{Ω} Π^{T} {\bar{A}}^{α} Π & {\tilde{P}}^{β} - (\tilde{Ω} + {\tilde{Ω}}^{T}) & 0 \\ {\bar{C}}^{α} Π & 0 & - I \end{matrix}] < 0

(24a)

[\begin{matrix} - Z & {\bar{B}}^{α T} Π {\tilde{Ω}}^{T} & {\bar{D}}^{α T} \\ {\tilde{Ω}}^{T} Π {\bar{B}}^{α} & {\tilde{P}}^{β} - (\tilde{Ω} + {\tilde{Ω}}^{T}) & 0 \\ {\bar{D}}^{α} & 0 & - I \end{matrix}] < 0

(24b)

with

{\tilde{P}}^{α} = Π^{T} P^{α} Π = Π^{T} diag {P_{h}^{α}, P_{v}^{α}} Π > 0

\tilde{Ω} = Π^{T} Ω Π = Π^{T} diag {Ω_{h}, Ω_{v}} Π > 0

Observe that ${\tilde{Ω}}^{T} + \tilde{Ω} > {\tilde{P}}^{α} > 0$ . Hence, $\tilde{Ω}$ is invertible. Denote

\tilde{Ω} = [\begin{matrix} G & U \\ U_{1} & Ξ \end{matrix}], {\tilde{Ω}}^{- 1} = [\begin{matrix} Y & V \\ V_{1} & Λ \end{matrix}]

and

Θ = [\begin{matrix} I & 0 \\ Y & V \end{matrix}]

In view of $\tilde{Ω} = Π^{T} Ω Π$ with $Ω$ being block-diagonal, it is easy to verify that the matrices $G$ , $U$ , $Y$ , $V$ are also block-diagonal. Pre- and post- multiply (24a) by $diag {Θ, Θ, I}$ and $diag {Θ^{T}, Θ^{T}, I}$ , and pre- and post- multiply (24b) by $diag {I, Θ, I}$ and $diag {I, Θ^{T}, I}$ , we can obtain

[\begin{matrix} - Θ {\tilde{P}}^{α} Θ^{T} & Θ Π^{T} {\bar{A}}^{α T} Π {\tilde{Ω}}^{T} Θ^{T} & Θ Π^{T} {\bar{C}}^{α T} \\ Θ \tilde{Ω} Π^{T} {\bar{A}}^{α} Π Θ^{T} & Θ ({\tilde{P}}^{β} - (\tilde{Ω} + {\tilde{Ω}}^{T})) Θ^{T} & 0 \\ {\bar{C}}^{α} Π Θ^{T} & 0 & - I \end{matrix}] < 0

(25a)

[\begin{matrix} - Z & {\bar{B}}^{α T} Π {\tilde{Ω}}^{T} Θ^{T} & {\bar{D}}^{α T} \\ Θ \tilde{Ω} Π^{T} {\bar{B}}^{α} & Θ ({\tilde{P}}^{β} - (\tilde{Ω} + {\tilde{Ω}}^{T})) Θ^{T} & 0 \\ {\bar{D}}^{α} & 0 & - I \end{matrix}] < 0

(25b)

where

Θ \tilde{Ω} Π^{T} {\bar{A}}^{α} Π Θ^{T} = [\begin{matrix} G A^{α} + M^{α} C_{2}^{α} & Φ^{α} \\ A^{α} + B_{2}^{α} D_{c}^{α} C_{2}^{α} & A^{α} Y^{T} + B_{2}^{α} N^{α} \end{matrix}]

{\bar{C}}^{α} Π Θ^{T} = [\begin{matrix} C_{1}^{α} + D_{12}^{α} D_{c}^{α} C_{2}^{α} & C_{1}^{α} Y^{T} + D_{12}^{α} N^{α} \end{matrix}]

Θ \tilde{Ω} Π^{T} {\bar{B}}^{α} = [\begin{matrix} {GB}_{1}^{α} + M^{α} D_{21}^{α} \\ B_{1}^{α} + B_{2}^{α} D_{c}^{α} D_{21}^{α} \end{matrix}]

{\bar{D}}^{α} = D_{11}^{α} + D_{12}^{α} D_{c}^{α} D_{21}^{α}

(26)

with

Φ^{α} = G A^{α} Y^{T} + {GB}_{2}^{α} D_{c}^{α} C_{2}^{α} Y^{T} + {UB}_{c}^{α} C_{2}^{α} Y^{T} + {GB}_{2}^{α} C_{c}^{α} V^{T} + {UA}_{c}^{α} V^{T}

M^{α} = {GB}_{2}^{α} D_{c}^{α} + {UB}_{c}^{α}

N^{α} = D_{c}^{α} C_{2}^{α} Y^{T} + C_{c}^{α} V^{T}

We can obtain the result of the theorem by letting

H^{α} = [\begin{matrix} H_{11}^{α} & H_{12}^{α} \\ H_{12}^{α T} & H_{22}^{α} \end{matrix}] = Θ {\tilde{P}}^{α} Θ^{T} > 0

Θ \tilde{Ω} Θ^{T} = [\begin{matrix} G & G Y^{T} + U V^{T} \\ I & Y^{T} \end{matrix}]

It can be seen from (21a) that

[\begin{matrix} G + G^{T} & I + S \\ I + S^{T} & Y + Y^{T} \end{matrix}] > [\begin{matrix} H_{11}^{α} & H_{12}^{α} \\ H_{12}^{α T} & H_{22}^{α} \end{matrix}] > 0

(27)

which implies that matrices G and Y are non-singular. Using $[\begin{matrix} - Y & I \end{matrix}]$ and its transpose to pre- and post-multiply (27), one obtains

Y (G Y^{T} - S) + (Y G^{T} - S^{T}) Y^{T} > 0

which implies that $S - G Y^{T}$ is also non-singular, hence there exist non-singular block-diagonal matrices U and V such that

S - G Y^{T} = U V^{T}

Thus, a solution to the controller in the form (19) can be obtained from the solution to the LMI (21). This completes the proof.

Remark 5. It is worth noting that in Theorems 1 and 2, the results are derived under the assumption that the switching rule is not known a priori but its value is available in each sampling period. In other words, the switching sequence considered here does not include the random switching one.

Remark 6. We would like to stress that the H₂ control problem of 2D discrete switched systems is first considered in the paper. Although some results on H₂ control and filtering problems for 2D systems have been obtained (Tuan et al., 2002; Yang et al., 2006), the existing methods proposed in these papers cannot be directly applied to 2D switched systems. In Theorem 2, a dynamic output feedback controller design scheme is proposed to solve the H₂ control problem for such systems, and the designed controller gains are mode-dependent, which is different from the existing ones.

Remark 7. In the proof of Theorem 2, the technique of change of variables has been used to transform the non-linear matrix inequalities (24a)–(24b) to LMIs (21a)–(21b) such that the H₂ control problem can be solved via the LMI technique.

In what follows, we present an algorithm for the design of dynamic output controller.

Algorithm 1

Step 1. Given a $γ$ , solve the LMIs (21a)–(21c) to obtain matrices $H_{11}^{α}$ , $H_{12}^{α}$ , $H_{22}^{α}$ , $G$ , $Y$ , $S$ , $M^{α}$ , $N^{α}$ , $Φ^{α}$ , $Z$ and the dynamic output feedback controller parameters $D_{c}^{α}$ with $α \in \underline{N}$ .

Step 2. The invertible matrices $U$ and $V$ can be computed by (21d).

Step 3. The rest of the controller parameters can be obtained by (22).

Numerical example

In this section, we shall illustrate the results developed earlier via an example. All simulations are performed with LMI control toolbox (Gahinet et al., 1995). Consider the following 2D switched system of type (18) with two subsystems:

Subsystem 1

A^{1} = [\begin{matrix} 0.2 & 0.1 & ⋮ & 0.018 & 0 \\ 0.6 & 0.3 & ⋮ & 0 & 0 \\ \dots & \dots & ⋮ & \dots & \dots \\ 0.25 & 0 & ⋮ & 0.3 & 0.1 \\ 0 & 0 & ⋮ & - 0.4 & 0.2 \end{matrix}], B_{1}^{1} = [\begin{matrix} 0.1 \\ 0.5 \\ \dots \\ 0.1 \\ 0.6 \end{matrix}], B_{2}^{1} = [\begin{matrix} 0.1 \\ 0.5 \\ \dots \\ 0 \\ 0.1 \end{matrix}]

C_{1}^{1} = [0 0.1 ⋮ 1 0.1], D_{11}^{1} = 0.1, D_{12}^{1} = 0.5

C_{2}^{1} = [0.1 0 ⋮ 0.2 0], D_{21}^{1} = 0.3, D_{22}^{1} = 0

Subsystem 2

A^{2} = [\begin{matrix} 1.1 & 0.2 & ⋮ & 0.06 & 0 \\ 0.3 & 0.3 & ⋮ & 0 & 0 \\ \dots & \dots & ⋮ & \dots & \dots \\ 0.35 & 0.15 & ⋮ & - 0.5 & 0.1 \\ 0 & 0 & ⋮ & 0 & - 0.4 \end{matrix}], B_{1}^{2} = [\begin{matrix} 0.8 \\ 0 \\ \dots \\ 0.3 \\ 0 \end{matrix}], B_{2}^{2} = [\begin{matrix} 0.1 \\ 0 \\ \dots \\ 0 \\ 0.1 \end{matrix}]

C_{1}^{2} = [0 0 ⋮ 0.5 0], D_{11}^{2} = 0.03, D_{12}^{2} = 0

C_{2}^{2} = [0 0.1 ⋮ 0 0], D_{21}^{2} = 0, D_{22}^{2} = 0

Note that the uncontrolled subsystem 2 is not stable and Figure 1 displays one of its state responses.

Figure 1.

Response of state $x_{1}^{h} (i, j)$ for uncontrolled subsystem 2.

Our purpose is to design a 2D switched dynamic output-feedback controller such that the resulting closed-loop system is asymptotically stable with an H₂ norm bound $γ$ .

Step 1. Taking $γ = 1.3$ , solving LMIs (21a)–(21c) yields

\begin{matrix} G = [\begin{matrix} 1.9671 & - 1.7839 & 0 & 0 \\ - 1.6477 & 3.8768 & 0 & 0 \\ 0 & 0 & 3.2304 & - 0.1809 \\ 0 & 0 & - 0.0847 & 5.8885 \end{matrix}], \\ S = [\begin{matrix} 0.6247 & 0.0693 & 0 & 0 \\ 0.1037 & 0.0174 & 0 & 0 \\ 0 & 0 & 0.9302 & - 0.0509 \\ 0 & 0 & - 0.0118 & - 0.2927 \end{matrix}] \end{matrix}

Y = [\begin{matrix} 10.4531 & 1.7337 & 0 & 0 \\ - 0.1633 & 12.8671 & 0 & 0 \\ 0 & 0 & 3.1333 & - 0.6790 \\ 0 & 0 & - 0.0924 & 13.4313 \end{matrix}], Z = 1.2258

H_{11}^{1} = [\begin{matrix} 1.0403 & - 0.4595 & 0 & 0 \\ - 0.4595 & 1.8317 & 0 & 0 \\ 0 & 0 & 3.0431 & - 0.2194 \\ 0 & 0 & - 0.2194 & 3.7912 \end{matrix}]

\begin{matrix} H_{12}^{1} = [\begin{matrix} 0.6155 & 0.1210 & 0 & 0 \\ 0.1790 & 0.0794 & 0 & 0 \\ 0 & 0 & 1.0722 & 0.0190 \\ 0 & 0 & 0.0182 & 0.1348 \end{matrix}], \\ H_{12}^{2} = [\begin{matrix} 0.6549 & 0.0235 & 0 & 0 \\ 0.0308 & 0.1166 & 0 & 0 \\ 0 & 0 & 0.9446 & - 0.0726 \\ 0 & 0 & - 0.0093 & 0.1277 \end{matrix}], \\ H_{22}^{1} = [\begin{matrix} 10.7464 & 0.0153 & 0 & 0 \\ 0.0153 & 11.1297 & 0 & 0 \\ 0 & 0 & 4.0090 & 0.1268 \\ 0 & 0 & 0.1268 & 11.2051 \end{matrix}], \\ H_{22}^{2} = [\begin{matrix} 11.0411 & 0.0742 & 0 & 0 \\ 0.0742 & 11.2357 & 0 & 0 \\ 0 & 0 & 3.9985 & - 0.2753 \\ 0 & 0 & - 0.2753 & 11.3639 \end{matrix}] \end{matrix}

\begin{matrix} Φ^{1} = [\begin{matrix} 0.0805 & - 0.0261 & - 0.1528 & - 0.0756 \\ 0.4885 & 0.2678 & - 0.1113 & - 0.1358 \\ 1.1446 & - 0.0401 & 0.3186 & 0.4943 \\ - 0.2283 & - 0.0108 & - 0.3086 & 0.2789 \end{matrix}], \\ Φ^{2} = [\begin{matrix} 0.6799 & - 0.0272 & 0.0561 & - 0.0511 \\ 0.2167 & 0.2203 & 0.0284 & 0.0197 \\ - 0.2012 & - 0.0627 & - 0.2243 & 0.1483 \\ - 0.0129 & - 0.0007 & 0.0025 & - 0.1889 \end{matrix}] \end{matrix}

\begin{matrix} N^{1} = [\begin{matrix} - 1.2218 & - 2.5216 & - 4.6176 & - 2.4972 \end{matrix}], \\ N^{2} = [\begin{matrix} - 51.9928 & - 19.0783 & 12.6451 & - 12.6451 \end{matrix}] \end{matrix}

\begin{matrix} M^{1} = {[\begin{matrix} 2.3912 & - 6.6439 & - 2.3902 & - 4.4202 \end{matrix}]}^{T}, \\ M^{2} = {[\begin{matrix} - 14.3474 & - 2.6334 & - 17.0343 & 0.3249 \end{matrix}]}^{T} \end{matrix}

D_{c}^{1} = - 2.6090, D_{c}^{2} = - 61.6731

Step 2. The invertible matrices $U$ and $V$ can be computed by (21d).

\begin{matrix} U = [\begin{matrix} 24.3826 & 0 & 49.3662 & 0 \\ 60.7817 & 0 & - 19.8033 & 0 \\ 0 & - 1.0778 & 0 & 29.9828 \\ 0 & 62.9415 & 0 & 0.5134 \end{matrix}], \\ V = [\begin{matrix} - 0.5097 & - 0.8604 & 0 & 0 \\ 0 & 0 & 0.0262 & - 0.9997 \\ - 0.8604 & 0.5097 & 0 & 0 \\ 0 & 0 & - 0.9997 & - 0.0262 \end{matrix}] \end{matrix}

Step 3. The rest of the controller parameters $A_{c}^{α}$ , $B_{c}^{α}$ and $C_{c}^{α}$ , $α = 1, 2$ , can be obtained by (22).

\begin{matrix} A_{c}^{1} = [\begin{matrix} - 0.2525 & - 0.2371 & 0.0433 & 0.0314 \\ - 0.1830 & 0.1217 & - 0.0657 & 0.3012 \\ - 0.2006 & 0.0153 & - 0.1302 & 0.3134 \\ 0.0766 & 0.0049 & 0.3874 & 0.5346 \end{matrix}], \\ A_{c}^{2} = [\begin{matrix} 0.0215 & 0.0015 & - 0.3961 & - 0.0202 \\ - 0.0658 & 0.0102 & 0.0393 & - 0.0305 \\ 0.3812 & - 0.7867 & - 0.1117 & 0.4993 \\ - 0.2188 & 0.3499 & 0.1376 & - 0.3673 \end{matrix}] \end{matrix}

\begin{matrix} B_{c}^{1} = {[\begin{matrix} 0.1002 & - 0.0676 & 0.0296 & - 0.2675 \end{matrix}]}^{T}, \\ B_{c}^{2} = {[\begin{matrix} - 0.0039 & - 0.1769 & - 0.7749 & 0.3146 \end{matrix}]}^{T} \end{matrix}

\begin{matrix} C_{c}^{1} = [\begin{matrix} 2.0795 & 3.3388 & 2.6286 & - 1.3549 \end{matrix}], \\ C_{c}^{2} = [\begin{matrix} 5.9564 & - 61.3010 & 6.7967 & 41.7661 \end{matrix}] \end{matrix}

The trajectories of states $x_{1}^{h} (i, j)$ , $x_{2}^{h} (i, j)$ , $x_{1}^{ν} (i, j)$ , $x_{2}^{ν} (i, j)$ are shown in Figures 2 –5, where the external signal is $w (i, j) = \frac{1}{i + j}$ , and the boundary condition of the system is

Figure 2.

Response of state $x_{1}^{h} (i, j)$ .

Figure 3.

Response of state $x_{2}^{h} (i, j)$ .

Figure 4.

Response of state $x_{1}^{ν} (i, j)$ .

Figure 5.

Response of state $x_{2}^{ν} (i, j)$ .

x_{h}^{p} (i, j) = \frac{1}{0.5 (j + 1)}, \forall 0 \leq j \leq 60, i = 0, p = 1, 2

x_{v}^{p} (i, j) = \frac{1}{0.5 (i + 1)}, \forall 0 \leq i \leq 60, j = 0, p = 1, 2

The control output sequence $z (i, j)$ is displayed in Figure 6 and the switching sequence is plotted in Figure 7. It is quite evident that the simulation results are satisfactory, which demonstrates the effectiveness of the proposed method.

Figure 6.

Controlled output trajectory.

Figure 7.

Switching signal.

Conclusions

The H₂ output feedback control problem for discrete-time 2D switched systems has been investigated. A sufficient condition based on the multiple Lyapunov function method has been established to guarantee the asymptotic stability and H₂ performance. An H₂ dynamic output feedback controller design scheme for such systems has been developed based on the condition. A numerical example is given to demonstrate the applicability of the proposed approach. The future work will be associated with the following directions: 1) dealing with the same problem for more general systems within the similar framework; and 2) considering H₂ output feedback control problem for discrete-time 2D switched systems with and without state delays.

Footnotes

Funding

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

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H 2 output feedback controller design for discrete-time 2D switched systems

Abstract

Keywords

Introduction

Notations

Problem formulation and preliminaries

H2 performance analysis

H2 control problem

Numerical example

Subsystem 1

Subsystem 2

Conclusions

Footnotes

Funding

References

H₂ performance analysis

H₂ control problem