ISRN Computational Mathematics

Volume 2012, Article ID 847178, 9 pages

http://dx.doi.org/10.5402/2012/847178

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

^{1}Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India^{2}Department 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 [1–10]. 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 [13–18]. 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 [7–10, 14–18].

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 [19–22]. 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 [23–33]. Lyapunov-based sufficient conditions for the global asymptotic stability of linear FMSLSS model have been reported in [23–26]. 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, 27–33]. 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] where , , and denotes the set of nonnegative integers. The is a state vector, is an input vector, and is an output vector. The matrices , (), and 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 by where is a controller state vector, is a controller input vector, and is a controller output vector. The matrices (), , and are constant matrices of appropriate dimensions.

Suppose that the input vector is subject to amplitude limitations defined as where , , denote the control amplitude bounds. Consequently, the actual control signal injected to the system (1) is a saturated one given by The saturation nonlinearities characterized by where , 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 . The modified compensator has the form

Let Substituting (7) into (6) we obtain Define an extended state vector Using (8) and (9), the closed loop system can be written as whereand is the identity matrix of order . It is assumed [27–34] that the system has a finite set of initial conditions, that is, there exist two positive integers and such that

The aim of this paper is to determine the antiwindup compensator gain matrix 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 such that where , and define a polyhedral set:

Now, we have the following lemma.

Lemma 1. *If then
**
where is positive definite block diagonal matrix and “” denotes the transpose.*

*Proof. *Observe that, (14) can be expressed as
where , are positive definite diagonal matrices and . 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 , a matrix , a matrix , a scalar , and a diagonal positive definite matrix satisfying the following LMIs:
**
where
**Then for the gain matrix the ellipsoid with is a region of asymptotic stability for system (10a)-(10b).*

*Proof. *Consider a 2D quadratic Lyapunov function
where . Now, following [23] (see [24, 25] also), we define as
Using (10a)-(10b), (21) can be rearranged as
Adding to and subtracting from (22) the quantity (see (15)), we obtain Defining
equation (23) can be written as
where is given by (19). From (25) and (15) it is clear that if
Before and after multiplying (26) by with , we obtain
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 is included in the polyhedral set as defined in (13). It can be shown that is equivalent to [35]
Before and after multiplying (28) by we get
By Schur’s complement, (29) together with and leads to (17). Similarly, using (13) with and it can be shown that implies (18). This completes the proof.

*Remark 1. *Using Theorem 1, one can determine antiwindup gain matrix 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 for the system (10a)-(10b) with provided that . Thus, the condition (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, 27–33] deal with the problem of global asymptotic stability of digital filters with state saturation. The nonlinearities considered in [24, 27–33] 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 , a diagonal positive definite matrix , and a matrix such that
**
where is defined in (19). Then, for , the origin of system (10a)-(10b) is globally stable.*

*Proof. *Choosing
one can see that (13) is automatically met for all . Consequently, (14) is also satisfied for all . 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 , . 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 () the matrix inequality (16)–(18) is linear in variables , , , and . 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:
Using MATLAB LMI toolbox [36] and choosing , it is found that (30) is feasible for the following values of unknown parameters:
In this case, the gain matrix is given by
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
Using MATLAB LMI toolbox [36] and choosing and the control bound , it is verified that (16)–(18) are feasible for the following values of unknown parameters
In this example, the gain matrix is obtained as
Therefore, Theorem 1 assures that the system under consideration is asymptotically stable in the region given by the ellipsoid .

#### 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 [37–39]. 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 [37–39] given by with the initial conditions: where is an unknown function at space and time ; is the input function; is the measured output; , , , , , and are real constants.

Define then (38a) can be transformed into an equivalent system of first-order differential equation of the form

It follows from (40) that

Taking , , and applying forward difference quotients for both derivatives in (41), it is easy to verify that (41) can be expressed in the following form: with the initial conditions:

Now, taking

equation (43) can be expressed in the FMSLSS setting: with the initial conditions: In view of (40) and (45), (38b) can be written as where

Now, consider the problem of anti-windup controller for the system represented by (46)–(49) with Consequently, the plant under consideration is described by (1) where For the above plant, the parameters of the stabilizing dynamic compensator (2) is given by Assume that, the control signal injected to the plant is a saturated one characterized by (5) where 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 , it is verified that (16)–(18) are feasible for the present example and the values of unknown parameters are obtained as In this example, the gain matrix Therefore, Theorem 1 assures that the system under consideration is asymptotically stable in the region given by the ellipsoid .

#### 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 [37–39].

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 [23–27] 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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