Robust Suboptimal Guaranteed Cost Control for 2-D Discrete Systems Described by Fornasini-Marchesini First Model

This paper considers the guaranteed cost control problem for a class of two-dimensional (2-D) uncertain discrete systems described by the Fornasini-Marchesini (FM) first model with norm-bounded uncertainties. New linear matrix inequality (LMI) based characterizations are presented for the existence of static-state feedback guaranteed cost controller which guarantees not only the asymptotic stability of closed loop systems, but also an adequate performance bound over all the admissible parameter uncertainties. Moreover, a convex optimization problem is formulated to select the suboptimal guaranteed cost controller which minimizes the upper bound of the closed-loop cost function.


Introduction
In recent years, due to theoretical as well as application importance in the fields such as digital filtering, image processing, seismographic data processing, thermal processes, gas absorption, water stream heating etc. [1][2][3][4][5][6][7][8], the two-dimensional (2-D) systems have received considerable attention.The stability analysis of 2-D discrete systems described the Fornasini-Marchesini (FM) first model [9] have been investigated extensively [10][11][12][13][14][15][16][17][18][19][20].In [10][11][12]20], the method of nonnegative matrix theory has been proposed for the investigation of stability of 2-D systems described by the FM first model.Sufficient conditions for the asymptotic stability of 2-D systems described by the FM first model have been presented in [11][12][13].In [12], it has been shown with the help of a counterexample that condition for the asymptotic stability given in [11] is incorrect and its corrected version is proposed.An improved Lyapunov based sufficient condition for the stability of 2-D linear systems described by the FM first model has been proposed in [14], and it is shown that the criterion given in [14] is less restrictive than those reported in [11,12].A computationally attractive necessary and sufficient condition for the asymptotic stability of the FM first model has been derived in [15].On the basis of the results given in [15], a connection has been established among structured singular value of a constant matrix and the stability, as well as the stability margin of the FM first model.Based on this connection, a novel sufficient condition for the asymptotic stability of the FM first model is obtained and it is shown by numerical simulations that the condition given in [15] is usually less conservative than that of [14].In [16], the stability of 2-D periodically shift variant system represented by the FM first model has been studied.In [19], based on the sum-of-squares polynomials with matrix coefficients, an LMI based sufficient condition for asymptotic stability of 2-D systems described by the FM first model has been derived.Furthermore, it has been shown that the criterion given in [19] is more relaxed than those presented in [15,18].In [20], necessary and sufficient conditions for the asymptotic stability of the FM first model using non-negative matrix theory have been proposed, and it has been shown that the conditions are sharper than those reported in [10,11,17].Further, a survey of the existing literature on the stability of the 2-D discrete systems described by FM first model has been presented in [21].
The guaranteed cost control of uncertain 2-D discrete systems aims to design a robust controller to stabilize the closed loop system and mean while guarantee a specified level of performance index for all the admissible uncertainties.This issue has been discussed for uncertain 2-D discrete systems described by the FM second model [22][23][24][25][26] and Roesser model [27].The guaranteed cost con- This paper, therefore, deals with the guaranteed cost control problem for uncertain 2-D discrete systems described by the FM first model with norm bounded uncertainties.The paper is organized as follows.The problem of robust guaranteed cost control for 2-D discrete uncertain systems described by the FM first model is formulated in Section 2. Some useful related results are also recalled in this section.In Section 3, we relate the notion of cost matrix to the quadratic stability and an upper bound on the closed-loop cost function.Sufficient conditions for the existence of static-state feedback guaranteed cost controllers are derived based on LMI approach.The static-state feedback guaranteed cost controllers are characterized by the feasible solution to a certain LMI.Further, a convex optimization problem is introduced to select the suboptimal guaranteed cost controller which minimizes the upper bound of the closed-loop cost function.An illustrative example showing the potential of the proposed technique is given in Section 4.

Problem Formulation and Preliminaries
Throughout the paper the following notations are used: n R n m  real vector space of dimension n; R set of n  m real matrices; 0 null matrix or null vector of appropriate dimension; I identity matrix of appropriate dimension; maximum eigenvalue of matrix G; block diagonal matrix.This paper deals with the problem of guaranteed cost control for a class of 2-D uncertain discrete systems described by the FM first model [9].Specifically, the system under consideration is given by where is an state vector, 1   represents parameter uncertainties which are assumed to be of the form where   In the above, can be regarded as known structural matrices of uncertainty and is an unknown matrix representing parameter uncertainty which satisfies or equivalently, , 1 .
It is assumed that the system (1a) has a finite set of initial conditions [24,25] i.e., there exist two positive integers r 1 and r 2 such that and the initial conditions are arbitrary, but belong to the set [22][23][24][25][26]  ,0 , 0, : ,0 , 0 where is a given matrix.Note that the vector can always be restricted as 1  by appropriately choosing .In other words, there is no loss of generality by choosing initial conditions as in (1h).

M
Associated with the uncertain system (1a) is the cost function: where ( 1,2,3 We are interested in designing a static-state feedback control law for the system (1) and the cost function (2), such that for all satisfying (1f), the resulting closed-loop system , , is asymptotically stable and the closed loop value of the cost function where satisfies J J   , where J  is some specified constant.Definition 2.1.Consider the system (1) and cost function (2), if there exist a control law and a positive scalar ( , ) i j  u J  such that for all admissible uncertainties, the closed-loop system (4) is asymptotically stable and the closed-loop value of the cost function (5) satisfies J J   , then J  is said to be a guaranteed cost and is said to be a guaranteed cost control law for the uncertain system (1).
where 6) is identified as global asymptotic stability condition ( [14], Equation (3)) of the nominal system described by FM first model.Now, as an extension of the result for the global asymptotic stability condition of the 2-D discrete FM first model given in [14], one can easily arrive at the following lemma.
Lemma 2.2.[14].The uncertain system (4) is globally asymptotically stable, provided there exist n n  positive definite symmetric matrices , and such that where is said to define a quadratic guaranteed cost control associated with cost matrix for the system (4) and cost function (5) if there exist a 3n  3n positive definite symmetric matrix 2 given by (5b) and n  n positive definite symmetric matrices P 1 and P 2 such that The following well-known lemma is needed in the proof of our main results.
Lemma 2.3.[22,25,27] Let and be given matrices.Then there exists a positive definite matrix P such that for all F satisfying T  F F I , if and only if there exists a scalar 0

Main Results
In the following, we aim to relate the notion of cost matrix to the quadratic stability and an upper bound on the closed-loop cost function.Lemma 3.1.Suppose there exists a cost matrix.
T n R n     P P 0 for the system (4) with initial conditions (1g), (1h) and cost function (5) such that (8) holds.Then, (i) system ( 4) is quadratically stable and (ii) the cost function satisfies the bound ) for all admissible parameter uncertainties.
Proof.Proof of (i) directly follows from Lemma 2.2 and Definition 2.2.

 0 x
The following theorem establishes that the problem of determining guaranteed cost control for system (4) and the cost function (5) can be recast to an LMI feasibility problem.
1 2 0    P P P which implies that Theorem 3.1.Consider system (4) with initial conditions (1g), (1h) and cost function (5), then there exists a static-state feedback controller that solves the addressed robust guaranteed cost control problem if there exist a positive scalar Therefore, the upper bound in (11) can be obtained by applying (18) in (17).This completes the proof of the Lemma 3.1.
In this situation, the suitable control laws are given by Moreover, the closed-loop cost function satisfies the following bound Proof.Using 1(c)-1(f), 5(b) and Lemma 2.3, (8) can be rearranged as .
Pre-and post-multiplying ( 22) by the   where and The equivalence of ( 23) and (19) follows trivially from the schur complements.Using (24), the bound of cost function can be easily obtained from (11).This completes the proof of the Theorem 3.1.
Remark 3.1.Note that the matrix inequality ( 19) is linear in variables , , , , ,  U U U V R and 2 which can be easily solved using Matlab LMI Toolbox [28,29].

R
Remark 3.2.It is clear that the upper bound on the closed-loop cost function is dependent on the choice of the guaranteed cost controllers.In particular, the guaran-teed cost controller that renders the corresponding guaranteed cost upper bound as small as possible is more interesting; such a controller is said to be an optimal guaranteed cost controller.Apparently, the upper bound (21) is not a convex function in V and  .Hence, finding the minimum of this upper bound cannot be considered as a convex optimization problem.Since  and Theorem 3.2.Consider system (4) with initial conditions (1g), (1h) and cost function (5), then there exists a suboptimal static-state feedback controller that solves the addressed robust guaranteed cost control problem if the following optimization problem minimize ( ) (i). ( 19), (ii)., which ensures the minimization of the guaranteed cost in (21).
Proof.By Theorem 3.1, the control laws (20) constructed in terms of any feasible solution  , , 1 , 2 , 1 , 2 and 3 U are the guaranteed cost controllers of system (4).To obtain the optimum value of the upper bound of guaranteed cost, the term which, in turn, implies the constraint (ii) in (26).Thus, the minimization of    implies the minimization of the guaranteed cost in (21).This completes the proof of Theorem 3.2.

Illustrative Example
In this section, we will give a specific example to demonstrate the effectiveness of Theorem 3.2.Consider a 2-D discrete uncertain system given by ( 1) and (2) with , diag , 0.66, 0.66 0.044, 0.044 0.0100 .0.0800 Using Lemma 2.1, it is easy to verify that the above system is unstable.We wish to construct a suitable guaranteed cost controller for this system, such that the corresponding cost bound is minimized.To this end, we apply our proposed method (Theorem 3.2) to find the suboptimal guaranteed cost controller.It is found using the Matlab LMI toolbox [28,29] that the optimization problem ( 26) is feasible for the present example and the optimal solution is given by     

Conclusion
In this paper, we have presented a solution to the guaranteed cost control problem for a class of uncertain 2-D discrete systems described by the FM first model in a LMI framework.The existence condition for the staticstate feedback guaranteed cost controller has been derived in terms of a certain LMI.The parameterized representation of a set of guaranteed cost controllers (if they exist) has been presented in terms of the feasible solutions to the LMI.Finally, a convex optimization problem has been introduced to design the suboptimal guaranteed cost controller that minimizes the upper bound of the closed-loop cost function.
are positive, we may obtain a suboptimal guaranteed cost controller by minimizing Theorem 3.1, the design problem of such a suboptimal guaranteed cost controller can be formulated as an optimization problem.