Table of Contents
ISRN Computational Mathematics
Volume 2012, Article ID 847178, 9 pages
http://dx.doi.org/10.5402/2012/847178
Research Article

Stability Analysis of 2D Discrete Linear System Described by the Fornasini-Marchesini Second Model with Actuator Saturation

1Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India
2Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India

Received 7 November 2011; Accepted 3 December 2011

Academic Editors: J. Buick, P. B. Vasconcelos, and Q.-W. Wang

Copyright © 2012 Richa Negi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper proposes a novel antiwindup controller for 2D discrete linear systems with saturating controls in Fornasini-Marchesini second local state space (FMSLSS) setting. A Lyapunov-based method to design an antiwindup gain of 2D discrete systems with saturating controls is established. Stability conditions allowing the design of antiwindup loops, in both local and global contexts have been derived. Numerical examples are provided to illustrate the applicability of the proposed method.

1. Introduction

An important problem which is always inherent to all dynamical systems is the presence of actuator saturation nonlinearities. Such nonlinearities may lead to performance degradation and even instability for feedback control systems. The stability analysis of the continuous as well as discrete time linear systems with saturating controls has been widely considered for one-dimensional (1D) systems [110]. The commonly used techniques to design controllers taking into account actuator saturation are (i) constrained model predictive control [4, 11], (ii) scheduled controllers [12], and (iii) antiwindup compensators [1318]. Model predictive controllers find applications in chemical industries for the control of systems with saturations. Scheduled controllers also called piecewise linear controller or gain scheduling schemes are often used in aerospace industry. Antiwindup compensators are widely used in practice for the control systems with saturating actuators [14, 15]. Design of antiwindup controllers can be carried out using linear design methods which explain its usefulness and popularity among control engineers. The actuator saturation problem is tackled following the “antiwindup paradigm” which employs a two-step design procedure. The main idea here is to design a linear controller ignoring the saturation nonlinearities and then augment this controller with extra dynamics to minimize the adverse effects of saturation on the closed loop performance. Several results as well as design schemes on the antiwindup problem and compensation gain are formulated and the stability conditions have been mentioned for 1D systems [710, 1418].

In the recent years, two-dimensional (2D) discrete systems have found various applications in many areas such as filtering, image processing, seismographic data processing, thermal processes, gas absorption, and water stream heating [1922]. Mathematically, a 2D discrete system is represented by a set of difference equations with two space coordinates. The stability properties of 2D discrete systems described by Fornasini-Marchesini second local state space (FMSLSS) model [19] has been studied in [2333]. Lyapunov-based sufficient conditions for the global asymptotic stability of linear FMSLSS model have been reported in [2326]. The stabilization problem of 2D continuous time saturated systems by state feedback control has been considered in [34]. The stability analysis of 2D discrete systems with state saturation nonlinearities has been carried out in [24, 2733]. However, to the best of the authors’ knowledge, no previous work has considered the area of “antiwindup paradigm” for 2-D saturated systems.

Inspired by the results [9] for 1D discrete-time systems, this paper investigates the antiwindup problem for 2D discrete systems described by FMSLSS model with saturating controls. The paper aims at providing a technique to compute the antiwindup gain of the 2D dynamic compensator that ensures the stability of the overall closed loop system in local as well as global context. Utilizing the sector description of saturation nonlinearities and 2D quadratic Lyapunov function, linear-matrix-inequality- (LMI-) based stability conditions are obtained.

The paper is organized as follows. In Section 2, the problem considered has been stated. The necessary concepts required in the paper have also been provided. LMI-based criteria for the stability of closed loop systems are developed in Section 3. In Section 4, the applicability of the presented approach has been demonstrated with the help of numerical examples. Section 5 presents the applicability of the designed antiwindup controller.

2. Problem Statement

Consider the 2D discrete system described by FMSLSS model [19]𝐱(𝑖+1,𝑗+1)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐁1𝐮(𝑖+1,𝑗)+𝐁2𝐮(𝑖,𝑗+1),𝐲(𝑖,𝑗)=𝐂𝐱(𝑖,𝑗),(1) where 𝑖𝑍+, 𝑗𝑍+, and 𝑍+ denotes the set of nonnegative integers. The 𝐱(𝑖,𝑗)𝑛 is a state vector, 𝐮(𝑖,𝑗)𝑚 is an input vector, and 𝐲(𝑖,𝑗)𝑝 is an output vector. The matrices 𝐀𝑘𝑛×𝑛, 𝐁𝑘𝑛×𝑚 (𝑘=1,2), and 𝐂𝑝×𝑛 are known constant matrices representing a nominal plant.

Let a linear 2D dynamic compensator which stabilizes system (1) and meets the desired performance specifications in the absence of actuator saturation be given by𝐱𝑐(𝑖+1,𝑗+1)=𝐀𝑐1𝐱𝑐(𝑖+1,𝑗)+𝐀𝑐2𝐱𝑐(𝑖,𝑗+1)+𝐁𝑐1𝐮𝑐(𝑖+1,𝑗)+𝐁𝑐2𝐮𝑐𝐯(𝑖,𝑗+1),𝑐(𝑖,𝑗)=𝐂𝑐𝐱𝑐(𝑖,𝑗)+𝐃𝑐𝐮𝑐(𝑖,𝑗),(2) where 𝐱𝑐(𝑖,𝑗)𝑛𝑐 is a controller state vector, 𝐮𝑐(𝑖,𝑗)=𝐲(𝑖,𝑗)𝑝 is a controller input vector, and 𝐯𝑐(𝑖,𝑗)𝑚 is a controller output vector. The matrices 𝐀𝑐𝑘𝑛𝑐×𝑛𝑐,𝐁𝑐𝑘𝑛𝑐×𝑝 (𝑘=1,2), 𝐂𝑐𝑚×𝑛𝑐, and 𝐃𝑐𝑚×𝑝 are constant matrices of appropriate dimensions.

Suppose that the input vector 𝐮(𝑖,𝑗) is subject to amplitude limitations defined as𝑢0(𝑙)𝑢(𝑙)(𝑖,𝑗)𝑢0(𝑙),(3) where 𝑢0(𝑙)>0, 𝑙=1,,𝑚, denote the control amplitude bounds. Consequently, the actual control signal injected to the system (1) is a saturated one given by𝐮𝐯(𝑖,𝑗)=sat𝑐𝐂(𝑖,𝑗)=sat𝑐𝐱𝑐(𝑖,𝑗)+𝐃𝑐𝐮𝑐(𝑖,𝑗).(4) The saturation nonlinearities characterized by𝑣sat𝑐(𝑖,𝑗)(𝑙)=𝑢0(𝑙),if𝑣𝑐(𝑙)<𝑢0(𝑙),𝑣𝑐(𝑙),if𝑢0(𝑙)𝑣𝑐(𝑙)𝑢0(𝑙),𝑢0(𝑙),if𝑣𝑐(𝑙)>𝑢0(𝑙),(5) where 𝑙=1,,𝑚, are under consideration.

The actuator saturation causes windup of the controller and to mitigate its effect an antiwindup compensation term is added to the controller. A 2D antiwindup compensator involves adding a correction term of the form 𝐄𝑐[sat(𝐯𝑐(𝑖,𝑗))𝐯𝑐(𝑖,𝑗)]. The modified compensator has the form𝐱(𝑖+1,𝑗+1)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐁1𝐯sat𝑐(𝑖+1,𝑗)+𝐁2𝐯sat𝑐,𝐱(𝑖,𝑗+1)𝐲(𝑖,𝑗)=𝐂𝐱(𝑖,𝑗),𝑐(𝑖+1,𝑗+1)=𝐀𝑐1𝐱𝑐(𝑖+1,𝑗)+𝐀𝑐2𝐱𝑐(𝑖,𝑗+1)+𝐁𝑐1𝐂𝐱(𝑖+1,𝑗)+𝐁𝑐2𝐂𝐱(𝑖,𝑗+1)+𝐄𝑐1𝐯sat𝑐(𝑖+1,𝑗)𝐯𝑐(𝑖+1,𝑗)+𝐄𝑐2𝐯sat𝑐(𝑖,𝑗+1)𝐯𝑐,𝐯(𝑖,𝑗+1)𝑐(𝑖,𝑗)=𝐂𝑐𝐱𝑐(𝑖,𝑗)+𝐃𝑐𝐂𝐱(𝑖,𝑗).(6)

Let 𝝍𝐯𝑐(𝑖,𝑗)=𝐯𝑐𝐯(𝑖,𝑗)sat𝑐(𝑖,𝑗).(7) Substituting (7) into (6) we obtain𝐱(𝑖+1,𝑗+1)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐁1𝐯𝑐𝐯(𝑖+1,𝑗)𝝍𝑐(𝑖+1,𝑗)+𝐁2𝐯𝑐𝐯(𝑖,𝑗+1)𝝍𝑐(𝑖,𝑗+1)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐁1𝐂𝑐𝐱𝑐(𝑖+1,𝑗)+𝐃𝑐𝐯𝐂𝐱(𝑖+1,𝑗)𝝍𝑐(𝑖+1,𝑗)+𝐁2𝐂𝑐𝐱𝑐(𝑖,𝑗+1)+𝐃𝑐𝐯𝐂𝐱(𝑖,𝑗+1)𝝍𝑐(𝑖,𝑗+1)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐁1𝐂𝑐𝐱𝑐(𝑖+1,𝑗)+𝐃𝑐𝐂𝐱(𝑖+1,𝑗)𝐁1𝝍𝐂𝑐𝐱𝑐(𝑖+1,𝑗)+𝐃𝑐𝐂𝐱(𝑖+1,𝑗)+𝐁2𝐂𝑐𝐱𝑐(𝑖,𝑗+1)+𝐃𝑐𝐂𝐱(𝑖,𝑗+1)𝐁2𝝍𝐂𝑐𝐱𝑐(𝑖,𝑗+1)+𝐃𝑐,𝐱𝐂𝐱(𝑖,𝑗+1)𝑐(𝑖+1,𝑗+1)=𝐀𝑐1𝐱𝑐(𝑖+1,𝑗)+𝐀𝑐2𝐱𝑐(𝑖,𝑗+1)+𝐁𝑐1𝐂𝐱(𝑖+1,𝑗)+𝐁𝑐2𝐂𝐱(𝑖,𝑗+1)+𝐄𝑐1𝐯𝑐𝐯(𝑖+1,𝑗)𝝍𝑐(𝑖+1,𝑗)𝐯𝑐(𝑖+1,𝑗)+𝐄𝑐2𝐯𝑐𝐯(𝑖,𝑗+1)𝝍𝑐(𝑖,𝑗+1)𝐯𝑐(𝑖,𝑗+1)=𝐀𝑐1𝐱𝑐(𝑖+1,𝑗)+𝐀𝑐2𝐱𝑐(𝑖,𝑗+1)+𝐁𝑐1𝐂𝐱(𝑖+1,𝑗)+𝐁𝑐2𝐂𝐱(𝑖,𝑗+1)𝐄𝑐1𝝍𝐂𝑐𝐱𝑐(𝑖+1,𝑗)+𝐃𝑐𝐂𝐱(𝑖+1,𝑗)𝐄𝑐2𝝍𝐂𝑐𝐱𝑐(𝑖,𝑗+1)+𝐃𝑐.𝐂𝐱(𝑖,𝑗+1)(8) Define an extended state vector𝐱𝝃(𝑖,𝑗)=𝐱(𝑖,𝑗)𝑐(𝑖,𝑗)𝑛+𝑛𝑐.(9) Using (8) and (9), the closed loop system can be written as𝝃(𝑖+1,𝑗+1)=𝐀1𝝃(𝑖+1,𝑗)𝐁1+𝐑𝐄𝑐1+𝝍(𝐊𝝃(𝑖+1,𝑗))𝐀2𝝃(𝑖,𝑗+1)𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1)),(10a) where𝐀1=𝐀1+𝐁1𝐃𝑐𝐂𝐁1𝐂𝑐𝐁𝑐1𝐂𝐀𝑐1,𝐀2=𝐀2+𝐁2𝐃𝑐𝐂𝐁2𝐂𝑐𝐁𝑐2𝐂𝐀𝑐2,𝐁1=𝐁1𝟎,𝐁2=𝐁2𝟎,𝟎𝐈𝐑=𝑛𝑐𝐃,𝐊=𝑐𝐂𝐂𝑐(10b)and 𝐈𝑛𝑐 is the identity matrix of order 𝑛𝑐. It is assumed [2734] that the system has a finite set of initial conditions, that is, there exist two positive integers 1 and 2 such that𝝃(𝑖,0)=𝟎,𝑖1;𝝃(0,𝑗)=𝟎,𝑗2.(11)

The aim of this paper is to determine the antiwindup compensator gain matrix [𝐄𝑐1𝐄𝑐2] and an associated region of asymptotic stability of the closed loop system (10a)-(10b) for a given set of admissible initial states.

In the following section, we will establish stability condition for system given by (10a)-(10b) in both local and global contexts.

3. Main Results

Consider a block diagonal matrix 𝐆2𝑚×2(𝑛+𝑛𝑐) such that𝐆𝐆=1𝟎𝟎𝐆2,(12) where 𝐆1𝑚×(𝑛+𝑛𝑐), 𝐆2𝑚×(𝑛+𝑛𝑐) and define a polyhedral set:𝝃(𝑛+𝑛𝑐);𝑢0(𝑙)𝐊(𝑙)𝐆1(𝑙)𝝃(𝑖+1,𝑗)𝑢0(𝑙),𝑢0(𝑙)𝐊(𝑙)𝐆2(𝑙)𝝃(𝑖,𝑗+1)𝑢0(𝑙),𝑙=1,2,,𝑚}.(13)

Now, we have the following lemma.

Lemma 1. If 𝝃 then 𝛿=𝝍(𝐊𝝃(𝑖+1,𝑗))𝝍(𝐊𝝃(𝑖,𝑗+1))𝑇𝝍𝝃×𝐃(𝐊𝝃(𝑖+1,𝑗))𝝍(𝐊𝝃(𝑖,𝑗+1))𝐆(𝑖+1,𝑗)𝝃(𝑖,𝑗+1)0,(14) where 𝐃 is positive definite block diagonal matrix and “𝑇” denotes the transpose.

Proof. Observe that, (14) can be expressed as 𝛿=𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))×𝐃1𝝍(𝐊𝝃(𝑖+1,𝑗))𝐆1𝝃(𝑖+1,𝑗)+𝝍𝑇(𝐊𝝃(𝑖,𝑗+1))×𝐃2𝝍(𝐊𝝃(𝑖,𝑗+1))𝐆2𝝃(𝑖,𝑗+1)0,(15) where 𝐃1𝑚×𝑚, 𝐃2𝑚×𝑚 are positive definite diagonal matrices and 𝐃=𝐃1𝟎𝟎𝐃2. Following the proof of [9, Lemma 1], it can be shown that both terms of the left hand side of (15) are nonpositive. This completes the proof.

The main result of the paper may be stated as follows.

Theorem 1. Suppose there exists a positive definite symmetric matrix 𝐖(𝑛+𝑛𝑐)×(𝑛+𝑛𝑐), a matrix 𝐙𝑛𝑐×2𝑚, a matrix 𝐘2𝑚×2(𝑛+𝑛𝑐), a scalar 0<𝛼<1, and a diagonal positive definite matrix 𝐒2𝑚×2𝑚 satisfying the following LMIs: 𝐏1𝐘𝑇𝐏1𝐀𝑇𝐘2𝐒𝐁𝐒+𝐑𝐙𝑇𝐀𝐏1𝐖𝛼𝐁𝐒+𝐑𝐙>𝟎,(16)1𝐖𝛼1𝐖𝐊𝑇(𝑙)𝐘𝑇1(𝑙)𝛼1𝐊(𝑙)𝐖𝐘1(𝑙)𝑢20(𝑙)(𝟎,𝑙=1,2,,𝑚,(17)1𝛼)1𝐖(1𝛼)1𝐖𝐊𝑇(𝑙)𝐘𝑇2(𝑙)(1𝛼)1𝐊(𝑙)𝐖𝐘2(𝑙)𝑢20(𝑙)𝟎,(18) where 𝐏1=𝛼1𝐖𝟎𝟎(1𝛼)1𝐖.(19)
Then for the gain matrix 𝐄𝑐=𝐙𝐒1 the ellipsoid 𝜀(𝐏)={𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃min(1/𝛼,1/(1𝛼))} with 𝐏=𝐖1 is a region of asymptotic stability for system (10a)-(10b).

Proof. Consider a 2D quadratic Lyapunov function 𝑉(𝑖+𝜂,𝑗+𝜏)=𝝃𝑇(𝑖+𝜂,𝑗+𝜏)𝐏𝝃(𝑖+𝜂,𝑗+𝜏),(20) where 𝐏=𝐏𝑇>𝟎. Now, following [23] (see [24, 25] also), we define Δ𝑉(𝑖,𝑗) as Δ𝑉(𝑖,𝑗)=𝝃𝑇(𝑖+1,𝑗+1)𝐏𝝃(𝑖+1,𝑗+1)𝝃𝑇(𝑖+1,𝑗)𝛼𝐏𝝃(𝑖+1,𝑗)𝝃𝑇(𝑖,𝑗+1)(1𝛼)𝐏𝝃(𝑖,𝑗+1).(21) Using (10a)-(10b), (21) can be rearranged as =Δ𝑉(𝑖,𝑗)𝐀1𝝃(𝑖+1,𝑗)𝐁1+𝐑𝐄𝑐1+𝝍(𝐊𝝃(𝑖+1,𝑗))𝐀2𝝃(𝑖,𝑗+1)𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))𝑇𝐏×𝐀1𝝃(𝑖+1,𝑗)𝐁1+𝐑𝐄𝑐1+𝝍(𝐊𝝃(𝑖+1,𝑗))𝐀2𝝃(𝑖,𝑗+1)𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))𝝃𝑇(𝑖+1,𝑗)𝛼𝐏𝝃(𝑖+1,𝑗)𝝃𝑇(𝑖,𝑗+1)(1𝛼)𝐏𝝃(𝑖,𝑗+1)=𝝃𝑇(𝑖+1,𝑗)𝐀𝑇1𝐏𝐀1𝝃(𝑖+1,𝑗)𝝃𝑇(𝑖+1,𝑗)𝐀𝑇1𝐏𝐁1+𝐑𝐄𝑐1𝝍(𝐊𝝃(𝑖+1,𝑗))𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐀1𝝃(𝑖+1,𝑗)+𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))𝐁1+𝐑𝐄𝑐1𝑇×𝐏𝐁1+𝐑𝐄𝑐1𝝍(𝐊𝝃(𝑖+1,𝑗))+𝝃𝑇(𝑖+1,𝑗)𝐀𝑇1𝐏𝐀2𝝃(𝑖,𝑗+1)𝝃𝑇(𝑖+1,𝑗)𝐀𝑇1𝐏𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐀2𝝃(𝑖,𝑗+1)+𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))𝐁1+𝐑𝐄𝑐1𝑇×𝐏𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))+𝝃𝑇(𝑖,𝑗+1)𝐀𝑇2𝐏𝐀1𝝃(𝑖+1,𝑗)𝝃𝑇(𝑖,𝑗+1)𝐀𝑇2𝐏𝐁1+𝐑𝐄𝑐1𝝍(𝐊𝝃(𝑖+1,𝑗))+𝝃𝑇(𝑖,𝑗+1)𝐀𝑇2𝐏𝐀2𝝃(𝑖,𝑗+1)𝝃𝑇(𝑖,𝑗+1)𝐀𝑇2𝐏𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))𝝍𝑇(𝐊𝝃(𝑖,𝑗+1))𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐀1𝝃(𝑖+1,𝑗)+𝝍𝑇(𝐊𝝃(𝑖,𝑗+1))𝐁2+𝐑𝐄𝑐2𝑇×𝐏𝐁1+𝐑𝐄𝑐1𝝍(𝐊𝝃(𝑖+1,𝑗))𝝍𝑇(𝐊𝝃(𝑖,𝑗+1))𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐀2𝝃(𝑖,𝑗+1)+𝝍𝑇(𝐊𝝃(𝑖,𝑗+1))𝐁2+𝐑𝐄𝑐2𝑇×𝐏𝐁2+𝐑𝐄𝑐2𝝍(𝐊𝝃(𝑖,𝑗+1))𝝃𝑇(𝑖+1,𝑗)𝛼𝐏𝝃(𝑖+1,𝑗)𝝃𝑇(𝑖,𝑗+1)(1𝛼)𝐏𝝃(𝑖,𝑗+1).(22) Adding to and subtracting from (22) the quantity 2𝛿 (see (15)), we obtain𝝃Δ𝑉(𝑖,𝑗)=𝑇(𝑖+1,𝑗)𝝃𝑇(𝑖,𝑗+1)𝝍𝑇(𝐊𝝃(𝑖+1,𝑗))𝝍𝑇×(𝐊𝝃(𝑖,𝑗+1))𝛼𝐏𝐀𝑇1𝐏𝐀1𝐀𝑇2𝐏𝐀1𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐀1𝐃1𝐆1𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐀1𝐀𝑇1𝐏𝐀2(1𝛼)𝐏𝐀𝑇2𝐏𝐀2𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐀2𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐀2𝐃2𝐆2𝐀𝑇1𝐏𝐁1+𝐑𝐄𝑐1𝐆𝑇1𝐃1𝐀𝑇2𝐏𝐁1+𝐑𝐄𝑐12𝐃1𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐁1+𝐑𝐄𝑐1𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐁1+𝐑𝐄𝑐1𝐀𝑇1𝐏𝐁2+𝐑𝐄𝑐2𝐀𝑇2𝐏𝐁2+𝐑𝐄𝑐2𝐆𝑇2𝐃2𝐁1+𝐑𝐄𝑐1𝑇𝐏𝐁2+𝐑𝐄𝑐22𝐃2𝐁2+𝐑𝐄𝑐2𝑇𝐏𝐁2+𝐑𝐄𝑐2×𝝍𝝃(𝑖+1,𝑗)𝝃(𝑖,𝑗+1)𝝍(𝐊𝝃(𝑖+1,𝑗))(𝐊𝝃(𝑖,𝑗+1))+2𝛿.(23) Defining 𝐀=𝐀1𝐀2,𝐁=𝐁1𝐁2,𝐄𝑐=𝐄𝑐1𝐄𝑐2(24)equation (23) can be written as 𝝍Δ𝑉(𝑖,𝑗)=𝝃(𝑖+1,𝑗)𝝃(𝑖,𝑗+1)𝝍(𝐊𝝃(𝑖+1,𝑗))(𝐊𝝃(𝑖,𝑗+1))𝑇𝐗1𝐗2𝐗𝑇2𝐗3×𝐗𝝃(𝑖+1,𝑗)𝝃(𝑖,𝑗+1)𝝍(𝐊𝝃(𝑖+1,𝑗))𝝍(𝐊𝝃(𝑖,𝑗+1))+2𝛿,1=𝐏𝐀𝑇𝐏𝐗𝐀,2=𝐀𝑇𝐏𝐁+𝐑𝐄𝑐𝐆𝑇𝐗𝐃,3=2𝐃𝐁+𝐑𝐄𝑐𝑇𝐏𝐁+𝐑𝐄𝑐,(25) where 𝐏 is given by (19). From (25) and (15) it is clear that Δ𝑉(𝑖,𝑗)0 if 𝐗1𝐗2𝐗𝑇2𝐗3>𝟎.(26) Before and after multiplying (26) by 𝐏1𝟎𝟎𝐃1 with 𝐃=𝐒1, we obtain 𝐏1𝐏1𝐀𝑇𝐏𝐀𝐏1𝐏1𝐀𝑇𝐏𝐁𝐒+𝐑𝐙𝐏1𝐆𝑇𝐁𝐒+𝐑𝐙𝑇𝐏𝐀𝐏1𝐆𝐏12𝐒𝐁𝐒+𝐑𝐙𝑇𝐏𝐁𝐒+𝐑𝐙>0.(27) The equivalence of (27) and (16) trivially follows from Schur’s complement.
Next, we will show that the satisfaction of (17) and (18) implies that the set 𝜀(𝐏)={𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃min(1/𝛼,1/(1𝛼))} is included in the polyhedral set as defined in (13). It can be shown that {𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃1/𝛼} is equivalent to [35] 𝐊𝛼𝐏(𝑙)𝐆1(𝑙)𝑇𝐊(𝑙)𝐆1(𝑙)𝑢02(𝑙)𝟎,𝑙=1,2𝑚.(28) Before and after multiplying (28) by 𝛼1𝐏1 we get 𝛼1𝐏1𝛼1𝐏1𝐊(𝑙)𝐆1(𝑙)𝑇×𝐊(𝑙)𝐆1(𝑙)𝛼1𝐏1𝑢02(𝑙)𝟎.(29) By Schur’s complement, (29) together with 𝐆1=𝛼𝐘1𝐏 and 𝐏=𝐖1 leads to (17). Similarly, using (13) with 𝐆2=(1𝛼)𝐘2𝐏 and 𝐏=𝐖1 it can be shown that {𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃1/(1𝛼)} implies (18). This completes the proof.

Remark 1. Using Theorem 1, one can determine antiwindup gain matrix 𝐄𝑐 in order to ensure the stability for a given region in the state space of 2D discrete systems with saturated inputs.

Remark 2. From (25) it is clear that Δ𝑉(𝑖,𝑗)0 for the system (10a)-(10b) with 𝝍(𝐊𝝃(𝑖+1,𝑗))=𝝍(𝐊𝝃(𝑖,𝑗+1))=𝟎 provided that 𝐏𝐀𝑇𝐏𝐀>𝟎. Thus, the condition 𝐏𝐀𝑇𝐏𝐀>𝟎 (also known as Hinamoto’s condition [23]) is a sufficient condition for the global asymptotic stability of 2D linear FMSLSS model.

Remark 3. It may be mentioned that several previous works [24, 2733] deal with the problem of global asymptotic stability of digital filters with state saturation. The nonlinearities considered in [24, 2733] occur due to the implementation of the system using finite wordlength. In contrast, the present paper tackles the problem of stability of a 2D system in presence of the actuator saturation nonlinearity.

As a direct consequence of Theorem 1, we have the following result for the global exponential stability of system (10a) and (10b).

Corollary 1. Suppose there exist a positive definite symmetric matrix 𝐖(𝑛+𝑛𝑐)×(𝑛+𝑛𝑐), a diagonal positive definite matrix 𝐒2𝑚×2𝑚, and a matrix 𝐙nc×2𝑚 such that 𝐏1𝐏1𝐊𝑇𝐏1𝐀𝑇𝐊𝐏12𝐒𝐁𝐒+𝐑𝐙𝑇𝐀𝐏1𝐖𝐁𝐒+𝐑𝐙>𝟎,(30) where 𝐏1 is defined in (19). Then, for 𝐄𝑐=𝐙𝐒1, the origin of system (10a)-(10b) is globally stable.

Proof. Choosing 𝐆1=𝐆2=𝐊(31) one can see that (13) is automatically met for all 𝝃𝑛+𝑛𝑐. Consequently, (14) is also satisfied for all 𝝃𝑛+𝑛𝑐. Now, substituting (31) into (16) we obtain the global exponential stability condition (30). This completes the proof.

Remark 4. Note that, unlike (16)–(19), (30) is independent of 𝑢0(𝑙), 𝑙=1,2,𝑚. In other words, one is not required to explicitly know the values of control amplitude bounds when dealing with Corollary 1.

Remark 5. It should be observed that for a given 𝛼 (0<𝛼<1) the matrix inequality (16)–(18) is linear in variables 𝐖, 𝐒, 𝐘, and 𝐙. Hence, it can be solved efficiently by employing the MATLAB LMI Toolbox [35, 36].

4. Numerical Examples

To illustrate the applicability of the presented results, we now consider the following examples.

Example 1. Consider a closed loop 2D system represented by (10a)-(10b) with the following parameters: 𝐀1=0.050.080.20.3,𝐀2=,𝐁0.10.00.00.11=00.01,𝐁2=0,𝐀0.01𝑐1=[]0.1,𝐀𝑐2=[],𝐁0.1𝑐1=[]0.01,𝐁𝑐2=[],0.01𝐂=10,𝐂𝑐=[],𝐃0.1𝑐=[].10(32) Using MATLAB LMI toolbox [36] and choosing 𝛼=0.5, it is found that (30) is feasible for the following values of unknown parameters: ,,.𝐖=27.61.014.51.0977.73.814.53.81503.1𝐒=4016.8004002.7𝐙=64.839792.7441(33) In this case, the gain matrix 𝐄𝑐=𝐙𝐒1 is given by 𝐄𝑐=0.01610.0232.(34) Therefore, according to Corollary 1, the system under consideration is globally stable.

Example 2. Consider a closed loop system described by (10a) and (10b) with 𝐀1=0.60.080.20.5,𝐀2=,𝐁0.30.00.00.11=00.01,𝐁2=0,𝐀0.01𝑐1=[]0.1,𝐀𝑐2=[],𝐁0.1𝑐1=[]0.01,𝐁𝑐2=[],𝐂0.01𝑐=[],𝐃0.1,𝐂=10𝑐=[].10(35) Using MATLAB LMI toolbox [36] and choosing 𝛼=0.5 and the control bound 𝑢0(1)=1, it is verified that (16)–(18) are feasible for the following values of unknown parameters ,,.𝐖=1.00.0210.01.038213814436𝐒=1511.5001255.2𝐙=4601.43843.7(36) In this example, the gain matrix 𝐄𝑐=𝐙𝐒1 is obtained as 𝐄𝑐=3.04423.0621.(37) Therefore, Theorem 1 assures that the system under consideration is asymptotically stable in the region given by the ellipsoid 𝜀(𝐏)={𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃2}.

5. Application to the Antiwindup Control of Dynamical Processes Described by the Darboux Equation

The design method of antiwindup controller given in Section 3 can be applied to the control of several dynamical processes. It is known that in real world situations, some dynamical processes in water steam heating, gas absorption, and air drying can be described by the Darboux equation [3739]. In this section, we shall illustrate the applicability of our proposed method (Theorem 1) in anti-windup control of processes in a Darboux equation.

Consider the Darboux equation [3739] given by𝜕2𝑠(𝑥,𝑡)𝜕𝑥𝜕𝑡=𝑎1𝜕𝑠(𝑥,𝑡)𝜕𝑡+𝑎2𝜕𝑠(𝑥,𝑡)𝜕𝑥+𝑎0𝑠(𝑥,𝑡)+𝑏𝑓(𝑥,𝑡),(38a)𝑦(𝑥,𝑡)=𝑐1𝜕𝑠(𝑥,𝑡)𝜕𝑡𝑎2𝑠(𝑥,𝑡)+𝑐2𝑠(𝑥,𝑡),(38b) with the initial conditions:𝑠(𝑥,0)=𝑝(𝑥),𝑠(0,𝑡)=𝑞(𝑡),(39) where 𝑠(𝑥,𝑡) is an unknown function at space 𝑥[0,𝑥𝑓] and time 𝑡[0,]; 𝑓(𝑥,𝑡) is the input function; 𝑦(𝑥,𝑡) is the measured output; 𝑎1, 𝑎2, 𝑎0, 𝑏, 𝑐1, and 𝑐2 are real constants.

Define𝑟(𝑥,𝑡)=𝜕𝑠(𝑥,𝑡)𝜕𝑡𝑎2𝑠(𝑥,𝑡)(40) then (38a) can be transformed into an equivalent system of first-order differential equation of the form𝜕𝑟(𝑥,𝑡)𝜕𝑥𝜕𝑠(𝑥,𝑡)=𝑎𝜕𝑡1𝑎1𝑎2+𝑎01𝑎2+𝑏0𝑟(𝑥,𝑡)𝑠(𝑥,𝑡)𝑓(𝑥,𝑡).(41)

It follows from (40) that𝑟(0,𝑡)=𝜕𝑠(𝑥,𝑡)|||𝜕𝑡𝑥=0𝑎2𝑠(0,𝑡)=𝑑𝑞(𝑡)𝑑𝑡𝑎2𝑞(𝑡)=𝑧(𝑡).(42)

Taking 𝑟(𝑖,𝑗)=𝑟(𝑖Δ𝑥,𝑗Δ𝑡), 𝑠(𝑖,𝑗)=𝑠(𝑖Δ𝑥,𝑗Δ𝑡), 𝑓(𝑥,𝑡)=𝑢(𝑖,𝑗) and applying forward difference quotients for both derivatives in (41), it is easy to verify that (41) can be expressed in the following form:=𝑟(𝑖,𝑗)𝑠(𝑖,𝑗)1+𝑎1𝑎Δ𝑥1𝑎2+𝑎0+Δ𝑥00𝑟(𝑖1,𝑗)𝑠(𝑖1,𝑗)00Δ𝑡1+𝑎2+000Δ𝑡𝑟(𝑖,𝑗1)𝑠(𝑖,𝑗1)𝑏Δ𝑥𝑢(𝑖1,𝑗)+𝑢(𝑖,𝑗1)(43) with the initial conditions:𝑠(𝑖,0)=𝑝(𝑖Δ𝑥),𝑟(0,𝑗)=𝑧(𝑗Δ𝑡).(44)

Now, taking 𝐱(𝑖,𝑗)=𝑟(𝑖,𝑗)𝑠(𝑖,𝑗),(45)

equation (43) can be expressed in the FMSLSS setting:𝐱(𝑖+1,𝑗+1)=00Δ𝑡1+𝑎2+Δ𝑡𝐱(𝑖+1,𝑗)1+𝑎1𝑎Δ𝑥1𝑎2+𝑎0+000Δ𝑥00𝐱(𝑖,𝑗+1)𝐮(𝑖+1,𝑗)+𝑏Δ𝑥𝐮(𝑖,𝑗+1)(46) with the initial conditions:𝐱(𝑖,0)=𝑎2𝑝(𝑖Δ𝑥)𝑝(𝑖Δ𝑥),𝐱(0,𝑗)=𝑧(𝑗Δ𝑡)𝑞(𝑗Δ𝑡).(47) In view of (40) and (45), (38b) can be written as𝐲(𝑖,𝑗)=𝐂𝐱(𝑖,𝑗),(48) where𝑐𝐂=1𝑐2.(49)

Now, consider the problem of anti-windup controller for the system represented by (46)–(49) with𝑎0=0.2,𝑎1=1.8,𝑎2𝑐=0.5,𝑏=0.194,Δ𝑥=0.5,Δ𝑡=1,1=1,𝑐2=0.(50) Consequently, the plant under consideration is described by (1) where 𝐀1=0010.5,𝐀2=,𝐁0.10.55001=00,𝐁2=,.0.0970.01𝐂=10(51) For the above plant, the parameters of the stabilizing dynamic compensator (2) is given by𝐀𝑐1=[]0.2,𝐀𝑐2=[],𝐁0.3𝑐1=[]0.01,𝐁𝑐2=[],𝐃0.01𝑐=[]11.12,𝐂𝑐=[].0.1(52) Assume that, the control signal injected to the plant is a saturated one characterized by (5) where10𝑢(𝑙)(𝑖,𝑗)10.(53) The resulting closed loop system is obtained by substituting (51) and (52) in (10a)-(10b). It is understood that the initial conditions of the closed loop system belong to (11). Using MATLAB LMI toolbox [36] and choosing 𝛼=0.5, it is verified that (16)–(18) are feasible for the present example and the values of unknown parameters are obtained as,,.𝐖=0.08100.02466.41520.02460.318114.77506.415214.77509850.1𝐒=21038000010.2344𝐙=114.7141110.7475(54) In this example, the gain matrix 𝐄𝑐=𝐙𝐒1=0.000110.8212.(55) Therefore, Theorem 1 assures that the system under consideration is asymptotically stable in the region given by the ellipsoid 𝜀(𝐏)={𝝃𝑛+𝑛𝑐;𝝃𝑇𝐏𝝃2}.

6. Conclusions

A Lyapunov-based approach to design an antiwindup gain of 2D discrete systems with saturating controls in the FMSLSS setting is established. Stability conditions allowing the design of antiwindup loops, in both local and global contexts have been stated for this system. The proposed criterion is in LMI setting and can be efficiently solved using MATLAB LMI Toolbox [35, 36]. Numerical examples are provided to illustrate the applicability of the presented results. Application of the proposed antiwindup controller design method is demonstrated through processes described by a Darboux equation [3739].

The presented approach utilizes the 2D Lyapunov condition [23] which provides only sufficient condition for the stability and is not necessary. It is known that, unlike its 1D counterpart, the available Lyapunov-based approaches [2327] provide only sufficient condition for the asymptotic stability of 2D linear discrete systems. The problem of determining necessary and sufficient conditions for the stability of 2D discrete saturated systems is very challenging. Further investigation is required to reduce or eliminate the gap between “sufficiency” and the “necessity” for a 2-D system to be stable, which occurs in the proposed approach.

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