Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model

doi:10.1016/j.jfranklin.2008.08.003

Journal of the Franklin Institute

Volume 346, Issue 2, March 2009, Pages 159-174

https://doi.org/10.1016/j.jfranklin.2008.08.003 Get rights and content

Abstract

This paper is concerned with the problem of delay-dependent guaranteed cost control for uncertain two-dimensional (2-D) state delay systems described by the Fornasini and Marchesini (FM) second state-space model. Given a scalar $α \in (0, 1)$ , a sufficient condition for the existence of delay-dependent guaranteed cost controllers is given in terms of a linear matrix inequality (LMI) based on a summation inequality for 2-D discrete systems. A convex optimization problem is proposed to design a state feedback controller stabilizing the 2-D state delay system as well as achieving the least guaranteed cost for the resulting closed-loop system. Finally, the simulation example of thermal processes is given to illustrate the effectiveness of the proposed result.

Introduction

Over the past several decades, two-dimensional (2-D) systems have received considerable attention due to their extensive applications of both theoretical and practical interest [1], [2], [3]. The key feature of a 2-D system is that the information is propagated along two independent directions. Many physical processes, such as image processing [4], signal filtering [5], and thermal processes in chemical reactors, head exchangers and pipe furnaces [3], have a clear 2-D structure. The 2-D system theory is frequently used as an analysis tool to solve some problems, e.g., iterative learning control [6], [7] and repetitive process control [8], [9]. However, the analysis and synthesis approaches for 2-D systems cannot simply extend from existing standard (1-D) system techniques because there are many 2-D system phenomena which have no 1-D system counterparts. Thus the study of 2-D systems is an interesting and challenging topic, and a lot of results have published in the literature. Among these results, Hinamoto [10] established a sufficient condition for the asymptotic stability of 2-D systems described by the Fornasini and Marchesini (FM) second model, Du and Xie [11] present a linear matrix inequality (LMI) approach to solve the robust stability problem for uncertain 2-D systems, Guan et al. [12] and Dhawan et al. [13], [14] dealt with the guaranteed cost control for 2-D systems, Paszke et al. [15] address the guaranteed cost control for discrete linear repetitive processes that are a distinct class of 2-D systems.

Delay systems represent a class of infinite dimensional systems largely used to describe propagation and transport phenomena [16]. In some physical processes which are intrinsically 2-D characteristic, delays appear in a natural way since information transmission along time and/or space require the process of spreading, the presence of delays in a system can also simplify the corresponding process model. Therefore, the control problem for 2-D state delay systems has received some attention. Paszke et al. [17] presented a sufficient stability condition and a stabilization method for discrete linear state delay 2-D systems. In order to guaranteed an adequate level of performance when designing a control system, Chang and Peng [18] first introduced the guaranteed cost control approach. Results on the guaranteed cost control for time-delay systems can be divided into two categories, i.e., delay-independent methods [19], [20] and delay-dependent methods [21], [22], [23]. In the latter methods, the designed controller and the achieved guaranteed costs depend on the size of the time-delay. The delay-dependent guaranteed cost control methods can usually give less guaranteed cost than the delay-independent ones especially when the size of the time-delay is small. In order to reduce the conservatism of delay-dependent conditions, an integral inequality was introduced and controller synthesis conditions [24].

In this paper, we first propose a delay-dependent approach to investigate the guaranteed cost control problem for uncertain 2-D discrete state delay systems described by the FM second model. A Lyapunov–Krasovskii functional is introduced for the 2-D state delay discrete systems. Based on a summation inequality for 2-D discrete state delay systems, a sufficient condition for the existence of delay-dependent guaranteed cost controllers is given in terms of an LMI. A delay-dependent optimal state feedback guaranteed cost controller is obtained by solving an LMI optimization problem. Finally, the simulation example of thermal processes is given to illustrate the effectiveness of the proposed result.

Section snippets

Problem statement and preliminaries

Consider an uncertain 2-D discrete linear system with state delay described by the following FM second state-space model $x (i + 1, j + 1) = A_{1} x (i, j + 1) + A_{2} x (i + 1, j) + A_{1 d} x (i - d_{1}, j + 1) + A_{2 d} x (i + 1, j - d_{2}) + B_{1} u (i, j + 1) + B_{2} u (i + 1, j),$ where $0 ⩽ i, j \in Z$ are horizontal and vertical coordinates, $x (i, j) \in R^{n}$ is the state vector, $u (i, j) \in R^{m}$ is the input vector, $d_{1}$ and $d_{2}$ are unknown positive integers representing delays along horizontal direction and vertical direction, respectively; $A_{1}, A_{2}, A_{1 d}$ , $A_{2 d}$ , $B_{1}$ and $B_{2}$ are time-varying matrices,

Main results

In this section, we first present a sufficient condition for the existence of a guaranteed cost controller (7) for the 2-D system (1) with any state delays $d_{1}$ and $d_{2}$ satisfying $0 < d_{1} ⩽ d_{1}^{*}$ and $0 < d_{2} ⩽ d_{2}^{*}$ .

Theorem 1

Consider the 2-D system (1) with the initial condition (5). For given integers $d_{1}^{*} > 0$ and $d_{2}^{*} > 0$ , if there exist a positive scale $α < 1$ , matrices $M_{11}, M_{12}, M_{21}, M_{22} \in R^{n \times n}$ , $K \in R^{m \times n}$ , symmetric positive definite matrices $P, S_{1}, S_{2} \in R^{n \times n}$ such that $[\begin{matrix} H_{1} & 0 & Ω_{11}^{T} & Ω_{12}^{T} & 0 & Ω_{14}^{T} \\ * & H_{2} & Ω_{21}^{T} & 0 & Ω_{23}^{T} & Ω_{24}^{T} \\ * & * & - P^{- 1} & 0 & 0 & 0 \\ * & * & * & - W_{1}^{- 1} & 0 & 0 \\ * & * & * & 0 & - W_{2}^{- 1} & 0 \\ * & * & * & * & * & - (d \end{matrix}]$

An illustrative example

This section applies the main results on guaranteed cost control to the thermal processes in chemical reactors, heat exchangers and pipe furnaces, which can be expressed in the partial differential equation with time delays: $\frac{\partial T (x, t)}{\partial x} = - \frac{\partial T (x, t)}{\partial t} - a_{0} (1 + δ) T (x, t) - a_{1} (1 + δ) T (x, t - τ) + b (1 + δ) u (x, t),$ where $T (x, t)$ is usually the temperature at x (space) $\in [0, x_{f}]$ and t (time) $\in [0, \infty]$ , $u (x, t)$ is a given force function, $τ$ is the time delay, $a_{0}, a_{1}, b$ are real coefficients, and $δ$ $(| δ | ⩽ 0.2)$ represents the uncertainty.

Conclusion

This paper has presented a solution to the problem of delay-dependent guaranteed cost control with memoryless state feedback controller for uncertain 2-D state delay systems. In order to reduce the conservatism of delay-dependent conditions, we did not limit the form of the free terms (e.g. $M_{11}$ and $M_{21}$ ) for the summation inequality in deriving delay-dependent results. A controller synthesis condition is obtained in terms of an LMI via the change of variables. A delay-dependent optimal state

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^☆: Supported by the National Natural Science Foundation of China under Grant 60525304.

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Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model☆

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Main results

An illustrative example

Conclusion

Automatica

Signal Process.

Signal Process.

Linear Algebra Appl.

Syst. Control Lett.

Automatica

J. Franklin Inst.

Automatica

Automatica

A discrete state-space model for linear image processing

IEEE Trans. Autom. Control

Doubly indexed dynamical, state-space models and structural properties

Math. System Theory