LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems

This paper studies the problem of the guaranteed cost control via static-state feedback controllers for a class of two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model with norm bounded uncertainties. A convex optimization problem with linear matrix inequality (LMI) constraints is formulated to design the suboptimal guaranteed cost controller which ensures the quadratic stability of the closed-loop system and minimizes the associated closed-loop cost function. Application of the proposed controller design method is illustrated with the help of one example.

Due to assumptions in the modeling process and/or the changing operating conditions of a real world system, it is usually impossible for a mathematical model to describe the real world system exactly.The problem of designing robust controllers for 2-D uncertain systems has drawn the attention of several researchers in recent years [39,40].When controlling a system subject to parameter uncertainty, it is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance.One approach to this problem is the socalled guaranteed cost control approach [45].This approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound.Based on this idea, many significant results have been proposed [42][43][44][45][46][47][48][49][50][51].In [42][43][44], the guaranteed cost control problem for 2-D discrete uncertain systems in FMSLSS setting has been considered and a robust controller design method has been established.The approach of [42] does not provide a true linear matrix inequality (LMI) based result which is not beneficial in terms of numerical complexity.Subsequently, in [43], an LMI based criterion for the existence of robust guaranteed cost controller has been formulated.Robust suboptimal guaranteed cost control for 2-D discrete uncertain systems in FMSLSS setting is an important problem.
In recent years, LMI has emerged as a powerful tool in control design problems [52][53][54][55][56][57][58].The introduction of LMI in control theory has given a new direction in the area of robust control problems.A widely accepted method for solving robust control problems now is to simply reduce them to LMI problems.Since solving LMIs is a convex optimization problem, such formulations offer a numerically efficient means of attacking problems that are difficult to solve analytically.These LMIs can be solved effectively by employing the recently developed Matlab LMI toolbox [53].
This paper, therefore, deals with the suboptimal guaranteed cost control problem for 2-D discrete uncertain systems described by FMSLSS model with norm-bounded uncertainties.The paper is organized as follows.In Section 2, we formulate the problem of robust guaranteed cost control for the uncertain 2-D discrete system described by the FMSLSS model and recall some useful results.An LMI based approach for the design of suboptimal guaranteed cost controller via static-state feedback is presented in Section 3. In Section 4, an application of the presented robust guaranteed cost controller design method is given.Finally, some concluding remarks are given in Section 5.

Problem Formulation and Preliminaries
The following notations are used throughout the paper: R n real vector space of dimension n R nm  set of n  m real matrices 0 null matrix or null vector of appropriate dimension I identity matrix of appropriate dimension G T  transpose of matrix maximum eigenvalue of matrix G.In this paper, we are concerned with the problem of guaranteed cost control for 2-D discrete uncertain systems described by FMSLSS model [14].The system under consideration is given by , 1 where and are the state and control input, respectively.The matrices   an B (k = 1, 2) are real valued matrix functions representing parameter uncertainties in the system model.The parameter uncertainties under consideration are assumed to be norm-bounded and of the form where In the above, L , 1 and 2 can be regarded as known structural matrices of uncertainty and It may be mentioned that the uncertainty of (1c) satisfying (1f) has been widely adopted in robust control literature [38,39,[42][43][44][59][60][61][62].The matrices L and 1 ( 2 ) specify how the elements of the nominal matrices A (B) are affected by the uncertain parameters in can always be restricted as (1f) by appropriately selecting L , 1 and 2 .Therefore, there is no loss of generality in choosing as in (1f).It is assumed that the system (1a) has a finite set of initial conditions [22,34,36,38,43,44] i.e., there exist two positive integers p and q such that and the initial conditions are arbitrary, but belong to the set [42][43][44]  where M is a given matrix.Associated with the uncertain system (1) is the cost function [43,44]: where Suppose the system state is available for feedback, the sign a static-state feedback control law for the system (1) and the cost function (2), such that the closed-loop system (4) is asymptotically stable and the closed-loop cost func Definition 2.1 A control law ( 3) is said to be an optimal lobal asymptotic st 2.1 [44] The 2-D discrete uncertain system (1) is quadratic guaranteed cost control if it ensures the quadratic stability of the closed-loop system (4) and minimizes the closed-loop cost function (5).
As an extension of the result for the g ability condition of 2-D discrete FMSLSS model given in [14, [30][31][32][33], one can easily arrive at the following lemma.

Lemma globally asymptotically stable if and only
where W given by (5b) and an n  n positive definite symmet matrix 1 P such that The following lemmas are needed in t m 2,44,51] Let he proof of our ain result.
be give nite n matrices.Then here exis sitive defi for all F satisfying F T F  I, if and only if the Lemma 2.3 [52,63] For real matrices M, L, Q of appro-re exists a scalar   0 such that priate dimensions, where (10) or equivalently (11) Lemma 2.4 [44] Suppose there exists a quadratic guar- anteed cost matrix T   0 P P for the uncertain closedloop system (4) with ditions (1g), (1h) and cost function (5) such that (7) holds.Then, a) the uncertain closed-loop system (4) is quadratically stable and b) the cost function satisfies the bound initial con

Main Result
establish that the problem of deter-T

M
In this section, we mining quadratic guaranteed cost control for system (1) and cost function (2) can be recast to a convex optimization problem.The main result may be stated as follows.Theorem 3.1 Consider system (1) and cost function (2), then there exists a suboptimal static-state feedback con- that solves the addressed robust ed c ol problem if the following optimization problem minimize guarante ost contr has a feasible solution 0 and n n  .The constraint ( 13) is give where In this situation, a suboptimal control law is K = which ensures the minimization of the upper boun for the closed-loop uncertain system.
Premultiplying and postmultiplying (18) by the matrix , which can be rewritten as where S and  19) can be e ed as The equivalence of ( 22) and ( 13) fo Lemma 2.3.Using (20), the bound of t ( 12) becomes (23) fined in (15).Equation ( 22).llows trivially from he cost function Clearly, the upper bound ( 23) is not a convex function

implies the minimization of the guaranteed co
The optimality of the solution of the optimization problem (14) follows from the convexity of the objective function and of the constraints.This completes the proof of Theorem 3.1.
Remark 3.1 It should be pointed out that the optimization problem given by ( 14) is an LMI eigenvalue problem [52,53], which provides a procedure to design subo-ptimal guaranteed cost controller.

Application to the Guaranteed Cost Control of Dynamical Processes Described by the Darboux Equation
In this section, we shall demonstrate our proposed method (Theorem 3.1) in robust guaranteed cost control of processes in the Darboux equation.It is known that some dynamical processes in gas absorption, water stream heating and air drying can be described by the Darboux equation [3,7,8]: with the initial conditions where   then (24) can be transformed into a first-order differential equation of the form: It follows from (26) that and applying the forward difference quotients for both derivatives in (27), it is easy to verify that ( 27) can be expressed in the following form: , , with the initial conditions By setting (30) can be converted into the following FMSLSS model: (33) with the initial conditions Now, consider the problem of suboptimal guaranteed cost control of a system represented by (33) with It is also assumed that the above system is subjected to parameter uncertainties of the form (1c)-( f) with 1 0 1 (37e) Associated with the uncertain system (33)- (37), the cost function is given by ( 2) with 1 0.09 0 0 0.09 2 0.9 0 0 0.9 Applying 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 3.1) to find the suboptimal guaranteed cost controller.It is found using the LMI toolbox in Matlab [53] that the optimization problem ( 14) is feasible for the present example and the optimal solution is given by

Conclusions
In this paper, we have presented a method of designing a suboptimal guaranteed cost controller via static-state feedback for a class of 2-D discrete systems described by the FMSLSS model with norm bounded uncertainties.A suboptimal guaranteed cost controller is obtained through a convex optimization problem which can be solved by using Matlab LMI Toolbox [53].Application of presented controller design method is demonstrated through processes described by a Darboux equation [3,7,8].The presented method can also be applied for the robust guaranteed cost controller design for metal rolling control problem [4,9,10].

1 
in S and  .Hence, finding the minimum of this per bound can not be considered as a convex optimizaturn, implies the constraint (ii) in (14).Thus,