Journal of Signal and Information Processing, 2011, 2, 292300 doi:10.4236/jsip.2011.24042 Published Online November 2011 (http://www.SciRP.org/journal/jsip) Copyright © 2011 SciRes. JSIP 1 LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems Amit Dhawan, Haranath Kar Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad, India. Email: {amit_dhawan2, hnkar1}@rediffmail.com Received September 28th, 2011; revised October 30th, 2011; accepted November 14th, 2011. ABSTRACT This paper studies the problem of the guaranteed cost control via staticstate feedback controllers for a class of twodimensional (2D) discrete systems described by the FornasiniMarchesini second local statespace (FMSLSS) model with norm bound ed uncertain ties. A co nvex optimizatio n problem with linear matrix inequality (LMI) constra ints is formulated to design the suboptimal guaranteed cost controller which ensures the quadratic stability of the closedloop system and minimizes the associated closedloop cost function. Application of the proposed controller de sign method is illustrated with the help of one example. Keywords: Linear Matrix Inequality, Lyapunov Methods, Robust Stability, 2D Discrete Systems, Uncertain System s, FornasiniM ar chesi ni Second Local State S p ace Mo del 1. Introduction In the past few years, due to the rapid increase of a wide variety of applications of twodimensional (2D) discrete systems in many practical application domains such as digital filtering, image and video processing, seismogr aphic data processing, thermal processes, gas absorption, water stream heating, control systems etc. [110], there has emerged a continuously growing interest in the sys tem theoretic problems of 2D discrete systems. Many authors have proposed and analyzed linear statevariable models for 2D discrete systems [1114]. The more popu lar models are Roesser model [11], FornasiniMarchesini first model [13] and FornasiniMarchesini second local statespace (FMSLSS) model [14]. Many publications relating to 2D Lyapunov equation with constant coeffi cients for the Roesser model [11] have appeared [1522]. The stability properties of 2D discrete systems described by the FM first model [13] have been investigated exten sively [2329]. The stability analysis of 2D discrete sys tems described by the FMSLSS model [14] has attracted a great deal of interest and many significant results have been obtained [22,3044]. 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 2D uncertain systems has drawn the attention of several researchers in recent years [39,40]. When controlling a system subject to parameter uncer tainty, 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 so called guaranteed cost control approach [45]. This appr oach 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 signifi cant results have been proposed [4251]. In [4244], the guaranteed cost control problem for 2D 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 ine quality (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 guar anteed cost control for 2D discrete uncertain systems in FMSLSS setting is an important problem. In recent years, LMI has emerged as a powerful tool in control design problems [5258]. The introduction of LMI in control theory has given a new direction in the area of robust control problems. A widely accepted met hod for solving robust control problems now is to simply
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems293 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 guaran teed cost control problem for 2D discrete uncertain syst ems described by FMSLSS model with normbounded uncertainties. The paper is organized as follows. In Section 2, we formulate the problem of robust guaranteed cost control for the uncertain 2D discrete system described by the FMSLSS model and recall some useful results. An LMI based approach for the design of suboptimal guaran teed cost controller via staticstate 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. 2. Problem Formulation and Preliminaries The following notations are used throughout the paper: Rn real vector space of dimension n Rnm set of n m real matrices 0 null matrix or null vector of appropriate dimension I identity matrix of appropriate dimension GT transpose of matrix G G > 0 matrix G is positive definite symmetric G < 0 matrix G is negative definite symmetric det (G) determinant of matrix G max (G) maximum eigenvalue of matrix G. In this paper, we are concerned with the problem of guaranteed cost control for 2D discrete uncertain syst ems described by FMSLSS model [14]. The system un der consideration is given by 11 22 11 22 1, 11, ,1 1, ,1 ij ij ij ij ij xAAx AAx BBu BBu , (1a) 12 AA, (1b) where and are the state and control input, respectively. The matrices ,n ij Rx ,m ij Runn kR A and k (k = 1, 2) are known constant matrices repre senting the nominal plant, k Ad k nm R B an (k = 1, 2) are real valued matrix functions representing parameter un certainties in the system model. The parameter uncertain ties under consideration are assumed to be normbounded and of the form 12 ,ij BLF MM, (1c) where 1 2 AA 1 BBB 2 , (1d) 1111 MMM 2 2212 MMM 2 . (1e) In the above, , 1 and 2 can be regarded as known structural matrices of uncertainty and M M ,ijF is an unknown matrix representing parameter uncertainty which satisfies ,ij F1. (1f) It may be mentioned that the uncertainty of (1c) satis fying (1f) has been widely adopted in robust control lit erature [38,39,4244,5962]. The matrices and 1 (2) specify how the elements of the nominal matrices A (B) are affected by the uncertain parameters in M M ,ijF. Note that ,ijF can always be restricted as (1f) by appropriately selecting , 1 and 2. Therefore, there is no loss of generality in choosing M M ,ijF 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 ,0 ,i 0x ; , , (1g) ip 0, j0xjq and the initial conditions are arbitrary, but belong to the set [4244] 1 ,0 ,0,:,0, n Si jRixxx MN 2 0,,1(1, 2) T kk jkxMNNN , (1h) where M is a given matrix. Associated with the uncertain system (1) is the cost function [43, 44]: 1 00 2 1 00 1, 1, ,1,1 T ij T T ij ij ij ij ij ij ij uRu uRu W , (2a) where 1, ,1 ij ij ij x x, (2b) Tm kk R 0RR m (k =1, 2), (2c) 1 1 2 0 0 Q WQ, (2d) Tn kk R 0QQ n objective of this paper is to develop a procedure to de (k =1, 2). (2e) Suppose the system state is available for feedback, the Copyright © 2011 SciRes. JSIP
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems 294 sign a staticstate feedback control law ,,ij ijuKx (3) for the system (1) and the cost function (2), such that the closedloop system 111 1 222 2 1, ,1 ij ij BKABKx ABKA BKx (4) is asymptotically stable and the closedloop cost func 1, 1ij xA tion is minimized where 2 00 T ij ij ij J W, (5a) . (5b) Definition 2.1 A control law (3) is said to be an optimal lobal asymptotic st 2.1 [44] The 2D discrete uncertain system (1) is for all 11 2 22 T T 0 0 QKRK WQKRK quadratic guaranteed cost control if it ensures the quad ratic stability of the closedloop system (4) and mini mizes the closedloop cost function (5). As an extension of the result for the g ability condition of 2D discrete FMSLSS model given in [14,3033], one can easily arrive at the following lemma. Lemma globally asymptotically stable if and only if det zz IALF MALF 2 1111 2120M 2 12 ,, Uzz F, (6) where 2 U= ,1,1zzz . 12 1 2 , :1,zFF ion 2.2 [4244] Consider the uncertain (1) (7a) where (7b) and 11 (7c) 12 1 Definit system and cost function (2), then the staticstate feedback con troller ,,ij ijuKx is said to define a quadratic guaranteed cost control associated with cost matrix Tnn R 0PP if there exist a 2n 2n positive defi trix 2 W given by (5b) and an n n positive definite symmet matrix 1 P such that 2CL 0ΓW, nite symmetric ma ric 1122 1 112 2 1 T CL 0 0 ΓABKABKP P ABKABKPP 11 11AAAALFM, 22 22AAAALFΜ, (7d) 11 112 BBBLFM, (7e) 22 2222 BBBLFM. (7f) The following lemmas are needed in t m 2,44,51] Let he proof of our ain result. Lemma 2.2 [4 nnnk ,RA,RH ln R and Tn RQQ tts a po matrix P such that T n be give nite n matrices. Then here exissitive defi 0AHFE PAHFEQ (8) for all F satisfying FT F I, if and only if the . (9) Lemma 2.3 [52, 63] For real matrices M, L, Q of appro re exists a scalar 0 such that 1 PH 1 T TT 0 H A AEEQ priate dimensions, where T MM and T 0QQ , then T 0MLQL if an T ML d only if 1 0 LQ (10) or equivalently (11) Lemma 2.4 [44] Suppose there exists a quadratic guar 1 T 0 QL LM . anteed cost matrix T 0PP for the uncertain closed loop system (4) withditions (1g), (1h) and cost function (5) such that (7) holds. Then, a) the uncertain closedloop system (4) is quadratically stable and b) the cost function satisfies the bound initial con max 2Jpq MP (12) 3. Main Result establish that the problem of deter TM In this section, we mining quadratic guaranteed cost control for system (1) and cost function (2) can be recast to a convex optimiza tion problem. The main result may be stated as follows. Theorem 3.1 Consider system (1) and cost function (2), then there exists a suboptimal staticstate feedback con troller ,iju = ,ijKx that solves the addressed robust ed col problem if the following optimization problem minimize guaranteost contr (). 1, (). T i ii 0 IM MS 3 s.t. (14) has a feasible solution 0 , mn R U, nn R 0T SS andnn. The constraint (13) is give 0 n by R T YY Copyright © 2011 SciRes. JSIP
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems Copyright © 2011 SciRes. JSIP 295 12 1/2 1/2 11 111 1/2 1/2 22 212 11 12 /2 1 /2 2 /2 1 /2 2 0 T T SA A A T T T TT T T T T 00 0 0 0 00 00 00 00 00 00 00 0 0 00 00 00 00 0 0000 000 000 00 0 0000 00 0 00 00 L SQU R Y M SQU R ASYM LI MM I I QS I QS I RU I RU , (13) where 11 1 AASBU, (15a) 222 AASBU, (15b) 11 1121 TT MSMUM, T (15c) 12 1222 TT MSMUM . T (15d) In this situation, a suboptimal control law is K = which ensures the minimization of the upper boun for the closedloop uncertain system. Proof: Using (5b) and (7b), matrix Inequality (7a) can be expressed as 112 2 1 112 2 1 11 22 T T T 0 0 R0 0 0R ABKABKP P ABKABKPP QK K QK K 1 US d of (2) , (16) which, in view of (7c)(7f), takes the form (17) Applying Lemma 2.2, (17) can be rearranged as 1 12211211222 1 12211211222 111 12 2 T T T R0 0 0R ABKA BKLFMMKMMK PABKABKLFMMKMMK PQ KK PPQ KK . 11 11 111 1111211121 2 212221121 22 11 211222 12 212221222 . T TT T TT T T T LL Q 0 Q PABK ABKPKRKM MKM MK ABKMMK MMK ABK MMKM MK PPKRKMMK MMK (18) Premultiplying and postmultiplying (18) by the matrix 1/2 1/2 1 1/2 1 00 00 00 I P P , one obtains 1 1 1 11 11 1 1111111 2111 21 11 2 212221121 1 22 11 11 2112 22 1 1 12 212221222 , T TT T TT T T T 0 PLL ABKP PABKPPQKRKM MKMMKP PABKPMMK MMKP ABKP PMMK MMKP PPPQKRKMMKMMKP
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems 296 which can be rewritten as 1 11 111 21211 2 11 12 11 221212 T TT TT T TT 1111 T 0 SLL A AYSQSURUM AMM A MM SYSQSURU MM M (19) where 1 P, (20) S and 1 1 YSP, (21) 1 A, 2 A, 11 M and 12 M xpress are de Equation (19) can be eed 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 1 max 2T Jpq MSM . Clearly, the upper bound (23) is not a convex function 1 in and . Hence, finding the minimum of this per bound can not be considered as a convex optimizaup blemSince tion pro. and are posi may obtainuboost con mi obtain th mume ofteed cost, to 1 max T MS M ptimal guaranteed c 1 max T MS M. To bound of guaran in (23) is changed tive, we troller by e opti the term inim valu a s zing th 1 e upper M max T MS 1 max T MS M1T MS MI which, in turn, implies the constraint (ii) in (14). Thus, the minimization of st in (23). implies the minimization of the guaranteed co The optimality of the solu tion 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 optimi zation problem given by (14) is an LMI eigenvalue prob lem [52,53], which provides a procedure to design subo ptimal guaranteed cost controller. 4. 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]: the application of 2,sxts a 120 ,, ,, xt sxt aa sxtbfxt xt tx (24) with the initial conditions ,0 xpx, 0, tqt (25) where , xt is an unknown function at space 0, x and time 0,t ,1 a,2 a,0 a eal constants and and are b r , xt is the input function. Let 2 , ,, sxt rxt asxt t (26) then (24) can be transformed into a firstorder differential equation of the form: n equivalent system of 112 0 2 , ,, 1, 0 , rxt aaaa rxtb x xt asxt sxt t . (27) It follows from (26) that 22 0 ,d 0, 0, x sxt qt rtast aqtzt tdt . (28) Taking 12 1/2 1/2 11111 1/2 1/2 21222 1 11 12 /2 1 /2 2 /2 1 2 T T T T T TT T T T 00SA AL 0 00 000 0 0000 0000000 00000 0 0000000 00 00000 00 00000 00000 00 AYMSQ UR ASYM SQUR IL IMM IQS IQS IRU IR/2T 0 U . (22) Copyright © 2011 SciRes. JSIP
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems297 ,,rijri xj t, (29a) ,, ijsixj t, (29b) ,, xtuij (29c) and applying the forward difference quotients for both derivatives in (27), it is easy to verify that (27) can be ex pressed in the following form: , , 1120 2 ,1 1 ,1 00 00 ,1 1,1 0 1,, 1 00 rijri j axaa ax ijsij rij tat sij bxui juij (30) with the initial conditions ,0 ipix, 0,rjzjt. (31) By setting , ,, rij ij ij x, (32) (30) can be converted into the following FMSLSS model: (33) with the initial conditions , (34) Now, consider the problem of suboptimal guaranteed cost control of a system represented by (33) with 2 1120 00 1, 11, 1 1 ,1 00 0 1,,1 00 ij ij tat axaaaxij bx ui juij xx x 2 ,0 apix ipix x 0, zjt jqjt x 0 1 15 a, (35a) 1 3 15 a , (35b) 2 1 3 a , (35c) (35d) (35e) (35f) and the initial conditions (34) satisfy (1g) and (1h) with (36a) 2 b, 0.5x , 0.9t 2pq, 0.01 0.05 0.006 0.001 M. (36b) It is also assumed that the above system is subjected to parameter uncertainties of the form (1c)(f) with 1 0 1 L, (37a) 11 0.0005 0M, (37b) 12 0 0.005M, (37c) (37d) 21 0M, 22 0.007 M. (37e) Associated with the uncertain system (33)(37), the cost function is given by (2) with 1 0.09 0 00.09 Q, (38a) 2 0.9 0 00.9 Q, (38b) 12 0.0025 RR . (38c) 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 5.03810 4.53485 4.53485 7.41531 S, (39a) 1.22942 0.12961 0.12961 1.19500 Y, (39b) 0.35283 1.31762U, (39c) 11.01117 , (39d) 0.00121 . (39e) By Theorem 3.1, the suboptimal guaranteed cost con troller for this system is ,0.19999 0.29999,uijijx, (40) and the least upper bound of the corresponding closed loop cost function is 0.02682J . (41) 5. Conclusions In this paper, we have presented a method of designing a suboptimal guaranteed cost controller via staticstate Copyright © 2011 SciRes. JSIP
LMI Approach to Suboptimal Guaranteed Cost Control for 2D Discrete Uncertain Systems 298 feedback for a class of 2D 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 pres ented 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 con trol problem [4,9,10]. REFERENC [1] N. K. Bose, “Applied Multidim Van Nostrand R, Ne ntice tica, Vol. 37, No. 2, 2001, pp. 205211. doi:10.1016/S00051098(00)001552 ES ensional System Theory,” einholdw York, 1982. [2] R. N. Bracewell, “TwoDimensional Imaging,” Pre Hall, Englewood, 1995, pp. 505537. [3] C. Du, L. Xie and C. 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