Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations

Transcription

1 1216 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST 2005 Exanding (A3)gives M 0 8 T +1 T +18 =( T ) F = I (A4) g = 0F 8 T +1 =( T ) f = T T +18 F 8 T +1 =( T ) 2 : (A5) (A6) Now, using the well-nown matrix result [A + BCD] 01 = A 01 0 A 01 B[DA 01 B + C 01 ] 01 DA 01 [10], F in (A4)can be comuted as F = M 0 8 T +1 T +18 =( T ) = M 01 M T +1 T +18 M 01 T T +18 M 01 8 T : (A7) +1 According to the definition of R in (16), it is obvious that Finally, substituting for M that R = I 0 8 M R +1 = I M T +1: (A8) from (A2) and (A5) (A7), and noting 8 T, roduces the required result R +1 = R 0 R +1 T +1R T T +1R +1 R 0 = I; =0; 1;...;n0 1 : (A9) REFERENCES [1] R. Haber and H. Unbehauen, Structure identification of nonlinear dynamic systems A survey on inut/outut aroaches, Automatica, vol. 26, , [2] S. Chen, S. A. Billings, and W. Luo, Orthogonal least squares methods and their alication to nonlinear system identification, Int. J. Control, vol. 50, , [3] S. Chen, C. F. N. Cowan, and P. M. Grant, Orthogonal least squares learning algorithm for radial basis function networs, IEEE Trans. Neural Networs, vol. 2, no. 2, , Mar [4] C. Drioli and D. Rocchesso, Orthogonal least squares algorithm for the aroximation of a ma and its derivatives with a RBF networ, Signal Process., vol. 83, , [5] D. S. Huang and W. Zhao, Determining the centers of radial basis robabilities neural networs by recursive orthogonal least square algorithms, Al. Math. Comut., vol. 162, , [6] S. Chen and J. Wigger, Fast orthogonal least squares algorithm for efficient subset model selection, IEEE Trans. Signal Process., vol. 43, no. 7, , Jul [7] Q. M. Zhu and S. A. Billings, Fast orthogonal identification of nonlinear stochastic models and radial basis function neural networs, Int. J. Control, vol. 64, , [8] K. Z. Mao, Fast orthogonal forward selection algorithm for feature subset selection, IEEE Trans. Neural Networs, vol. 13, no. 5, , Se [9] L. Lawson and R. J. Hanson, Solving Least Squares Problem. Englewood Cliffs, NJ: Prentice-Hall, [10] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, N.J.: Prentice-Hall, B. Proof of Proosition 2 a)equation (18)follows directly from the definition of R. b)setting = j in (19), and using roerty (18), the result is obvious. Now, setting = j +1in (19)and using (17), it follows that R = R j+1 = R j 0 R j j+1 T j+1r T j T j+1 R j j+1 : (A10) If both sides of (A10)are now multilied by R j, then R R j = R j+1r j = R jr j 0 R j j+1 T j+1r T j T j+1 R j j+1 R j (A11) which, from (17)and (18), gives R j+1 R j = R j+1 : (A12) Similarly, it can be roved that R j R j+1 = R j+1. This rocess can be reeated for all i>jto confirm the results in (19). c)in (20), setting = i and substituting from (17)and (18), it follows that R i i = R i01 i 0 R i01 i T i R T i01 i T i R i01 i = R i01 i 0 R i01 i =0: (A13) Reeating this for i +1and using (A13)gives R i+1 i = R i i 0 R i i+1 T i+1r T i i T i+1r i i+1 = 0R i i+1 T i+1r i i T i+1r i i+1 =0: (A14) This rocess can be continued for all >ito confirm that (20)is indeed true. Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations Feng Ding and Tongwen Chen Abstract In this note, we aly a hierarchical identification rincile to study solving the Sylvester and Lyaunov matrix equations. In our aroach, we regard the unnown matrix to be solved as system arameters to be identified, and resent a gradient iterative algorithm for solving the equations by minimizing certain criterion functions. We rove that the iterative solution consistently converges to the true solution for any initial value, and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor (or ste-size) aroriately. Furthermore, the iterative method is extended to solve general linear matrix equations. The algorithms roosed require less storage caacity than the existing numerical ones. Finally, the algorithms are tested on comuter and the results verify the theoretical findings. Index Terms Gauss Seidel iteration, gradient search, hierarchical identification rincile, identification, Jacobi iteration, Lyaunov matrix equation, arameter estimation, Sylvester matrix equation. Manuscrit received July 11, 2003; revised Aril 24, Recommended by Associate Editor S.-I. Niculescu. This wor was suorted by the Natural Sciences and Engineering Research Council of Canada. F. Ding is with the Control Science and Engineering Research Center, Southern Yangtze University, Wuxi , China, and also with the University of Alberta, Edmonton, AB T6G 2V4, Canada ( fding@sytu.edu.cn; fding@ece.ualberta.ca). T. Chen is with the Deartment of Electrical and Comuter Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada ( tchen@ece.ualberta.ca). Digital Object Identifier /TAC /$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST I. INTRODUCTION Consider linear matrix equations, nown as Sylvester equations, of the form AX + XB = C and AXB + X = C where X 2 m2n is an unnown matrix, A; B, and C are constant (coefficient)matrices of aroriate dimensions. Such equations arise in many alications in systems and control, e.g., when B = A T, the equations reduce to the so-called Lyaunov equations, which are often encountered in continuous- and discrete-time stability analysis. Traditional methods convert such equations into an equivalent one of the form: Ax = b. Here, the vector x has mn unnowns consisting of elements of X; A is an mn 2 mn matrix in the Kronecer roduct form [2]. However, the dimensions of the associated matrix (A) are high when m and n are large, e.g., if m = n = 100, the dimension of A is Such a dimensionality roblem leads to comutational difficulty in that excessive comuter memory is required for comutation and inversion of large matrices. Alternative ways exist which transform the matrix equations into forms for which solutions may be readily comuted. Some numerical methods to solve the matrix equations are based on the reliminary matrix transformations to Jordan canonical form [11], comanion-tye form [3], Hessenberg Schur form [9]. However, these methods require first comuting the desired matrix transformation; moreover, they are not suitable for a general matrix equation of the form A1XB1 + A2XB A XB = C (1) which will also be studied in this note. In the field of linear algebra, iterative algorithms for sets of linear equations, such as Jacobi and Gauss Seidel iterations [10], have received secial attention [4], [8], [10], [16]. For examle, Stare and Niethammer studied an iterative bloc successive overrelaxation (SOR) method for the solution of the Sylvester matrix equation [17]. Recently, Jonsson and Kågström roosed recursive bloc algorithms for solving generalized Sylvester and Lyaunov matrix equations [12], [13]; Styel discussed numerical solution for generalized Lyaunov equations [18]. Also, Benner and Quintana Ortí resented an iterative algorithm for solving stable generalized Lyaunov equations using matrix sign functions; but the main drawbac is that their algorithm requires comuting matrix inversion at each iteration [1]. Recently, the authors derived the iterative least squares solutions of couled matrix equations [5]. To the best of our nowledge, iterative algorithms for Sylvester and Lyaunov matrix equations have not been fully investigated, esecially the iterative solutions of general matrix equations of the form in (1)and the convergence of the iterative solutions involved, which are the focus of this wor. Our methods will generate solutions to the matrix equations which are arbitrarily close to the exact solutions. Although exact solutions are imortant, for many alications such as stability analysis, it is often not necessary to comute exact solutions and aroximate solutions are sufficient. Also, if the arameters in the system matrices are uncertain, it is not ossible to obtain the exact solution for robust stability results [8], [14] [16]. In this note, the roblem will be tacled in a new way by alying the so-called hierarchical identification rincile [6], [7]. We decomose a system into some subsystems, and then the unnown arameters of each subsystem are identified successively. Here, we regard the unnown matrix X as the arameters (arameter matrix)of a system to be identified, and roose ways of decomosing the system equations, leading to gradient based iterative algorithms for solutions of the matrix equations involved. This note is organized as follows. In Sections II IV, we derive iterative algorithms for the matrix equations, AX+XB = C; AXB+X = C and (1), resectively, and study convergence roerties of the algorithms. In Section V, we resent three examles to illustrate the effectiveness of the algorithms roosed in this wor. Finally, we offer some concluding remars in Section VI. II. EQUATION AX + XB = C In this section, we aly the hierarchical identification rincile to solve the Sylvester matrix equation AX + XB = C (2) where A 2 m2m ;B 2 n2n, and C 2 m2n are given constant matrices, X 2 m2n is the unnown matrix to be solved. For a square matrix M; i[m ] reresents the ith eigenvalue of M. The notation I n is the identity matrix of n 2 n. For two matrices M and N; M N is their Kronecer roduct. For an m 2 n, matrix X =[x1 x2 111 x n ] 2 m2n col[x] is an mn-dimensional vector formed by columns of X, i.e., col[x] =[x T 1 x T 2... x T n ] T. The following result is well nown. Lemma 1: Equation (2)has a unique solution if and only if i[a]+ j [B] 6= 0for any i and j; in this case, the unique solution is given by col[x] =[(I n A) +(B T I m )] 01 col[c] (3) and the corresonding homogeneous equation AX + XB = 0 has a unique solution X =0. In articular, if B = A T, (2)reduces to the continuous-time Lyaunov matrix equation; and the necessary and sufficient condition for the existence of a unique solution is that i [A] + j [A] 6= 0for any i and j. According to the hierarchical identification rincile, the system in (2)is decomosed into two subsystems, and then based on least squares otimization, the arameters of each subsystems are identified, resectively. In this way we obtain the iterative algorithm. The details are as follows. Define two matrices b1 := C 0 XB and b2 := C 0 AX: (4) Then, from (2), we obtain two fictitious subsystems AX = b1 and XB = b2: (5) For these two subsystems in (5), we form two criterion functions J1(X) :=AX 0 b1 2 and J2(X) :=XB 0 b2 2 where the norm of the matrix X is defined by X 2 =tr[xx T ]. Let X1() and X2() be the estimates or iterative solutions of X at iteration, associated with the subsystems in (5), resectively. Minimizing J1(X) and J2(X) or using Lemma 1 in [5] leads to the following recursive equations: X1() =X1( 0 1) + A T [b1 0 AX1( 0 1)] (6) X2() =X2( 0 1) + [b2 0 X2( 0 1)B]B T : (7)

3 1218 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST 2005 Here, is called the iterative ste-size or convergence factor, which can be simly taen to be = f max[aa T ]+ max[b T B]g 01 (8) where max indicates the maximum eigenvalue. Substituting (4)into (6)and (7)gives X 1() =X 1( 0 1) + A T [C 0 AX 1( 0 1) 0 XB] (9) X 2 () =X 2 ( 0 1) + [C 0 AX 0 X 2 ( 0 1)B]B T : (10) Here, a difficulty arises in that the exressions on the right-hand sides of (9)and (10)contain the unnown matrix X; so it is imossible to realize the algorithm in (9)and (10). Our solution is based on the hierarchical identification rincile: The unnown variable X in (9)and (10)is relaced with its estimate at time ( 0 1). Hence, we have X 1 () =X 1 ( 0 1) + A T [C 0 AX 1 ( 0 1) 0 X 1 ( 0 1)B] (11) X 2 () =X 2 ( 0 1) + [C 0 AX 2 ( 0 1) 0 X 2 ( 0 1)B]B T : (12) In fact, we need only an iterative solution X() rather than two solutions X 1 () and X 2 (). Taing the average of X 1 () and X 2 (),we obtain a gradient iterative (GI)algorithm X() =[X 1 ()+X 2 ()]=2 (13) X 1() =X( 0 1) + A T [C 0 AX( 0 1) 0 X( 0 1)B] (14) X 2 () =X( 0 1) + [C 0 AX( 0 1) 0 X( 0 1)B]B T : (15) To initialize the algorithm, we tae X(0) = 0 or some small real matrix, e.g., X(0) = m2n with 1 m2n being an m 2 n matrix whose elements are 1. If the aroach in [2] is used to solve (2), the number of storage elements required is mn(mn +1), which is much larger than (m + n) 2 +1, the number of storage elements required by the GI algorithm in (13) (15); e.g., when m = n = 100, the two numbers are and 401, resectively. A comarable iterative algorithm is the Benner and Quintana Ortí (B Q)algorithm, which requires comuting matrix inversion at each iteration [1]. However, the GI algorithm does not involve matrix inversion and thus has less comutational burden than the B Q algorithm; see Examle 3. Theorem 1: If the Sylvester equation in (2)has a unique solution X, then the iterative solution X() given by the algorithm in (13) (15) converges to X, i.e., lim!1 X() = X; or, the error X() 0 X converges to zero for any initial value X(0). Proof: Define the error matrices and ~X 1() :=X 1() 0 X X ~ 2() :=X 2() 0 X ~X() :=X() 0 X: (16) () :=A ~ X( 0 1) () := ~ X( 0 1)B: (17) Using (2) and (13) (15), it is not difficult to get ~X 1 () = X( ~ 0 1) + A T [0A X( ~ 0 1) 0 X( ~ 0 1)B] = X( ~ 0 1) + A T [0() 0 ()] (18) ~X 2 () = X( ~ 0 1) + [0A X( ~ 0 1) 0 X( ~ 0 1)B]B T = ~ X( 0 1) + [0() 0 ()]B T : (19) Taing the norm of both sides of (18)and (19), and using (17)give X ~ 1 () 2 = X( ~ 0 1) 2 +2trf X ~ T ( 0 1)A T [0() 0 ()]g + 2 A T [0() 0 ()] 2 X( ~ 0 1) 2 +2trf T ()[0() 0 ()]g + 2 max [AA T ]() +() 2 (20) X ~ 2() 2 X( ~ 0 1) 2 +2trf[0() 0 ()] T ()g + 2 max [B T B]() +() 2 : (21) Hence, from (16)and using (20)and (21), we have X() ~ 2 =[ X ~ 1 ()+ X ~ 2 () 2 ]=4 [ X ~ 1 () 2 + X ~ 2 () 2 ]=2 X( ~ 0 1) 2 0 () +() f max[aa T ] + max [B T B]g() +() 2 = ~ X( 0 1) f2 0 ( max[aa T ] + max [B T B])g() +() 2 ~ X(0) f2 0 ( max[aa T ] + max [B T B])g i=1 If the convergence factor is chosen to satisfy then we have (i) +(i) 2 : 0 <<2f max [AA T ]+ max [B T B]g 01 1 =1 ()+() 2 < 1: This imlies that as!1;() +()! 0, ora ~ X( 0 1) + ~X( 0 1)B! 0. According to Lemma 1, ~ X( 0 1)! 0 as!1. This roves Theorem 1. The algorithm (11)or (12)is nown as the single-side iteration, which can not guarantee that X i () converges to X; and (13) (15) give the balanced iterative algorithm, which can simly be written as X() =X( 0 1) + A T [C 0 AX( 0 1) 0 X( 0 1)B]=2 + [C 0 AX( 0 1) 0 X( 0 1)B]B T =2 (22) = f max [AA T ]+ max [B T B]g 01 or = fa 2 + B 2 g 01 (23) whose convergence is guaranteed. Of course, it is easier to comute the trace than the maximum eigenvalue. Even the convergence factor in (23)may not be the best and may be conservative. In fact, there exists a best such that the fast convergence rate of X() to X may be obtained; see Examle 1. Next, we show that the convergence rate of the iterative algorithm in (22) (23) deends on the condition number of the associated system, just lie the Jacobi and Gauss-Seidel iterations for linear systems of the form Ax = b [10]. From (18), (19), and (16), we may get an error equation col[ X()] ~ = [I mn 0 8]col[ X( ~ 0 1)] 8:=I n (A T A)+B T A T + B A +(BB T ) I m : From here, the closer the eigenvalues of 8 are to 1, the closer the eigenvalues of I mn 0 8 tend to zero, and hence, the faster the error

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST col[ ~ X()] or ~ X() converges to zero. In other words, the algorithm in (22)and (23)has a fast convergence rate for small condition numbers of 8; see Examle 2. TABLE I ITERATIVE SOLUTION ( =1 5) III. EQUATION AXB + X = C In this section, we study the solution to the following matrix equation: AXB + X = C (24) where A m2m 2 ;B 2 n2n and C 2 m2n are given constant matrices, X 2 m2n is the unnown matrix to be solved. In articular, if A = B T, (24)is the discrete-time Lyaunov matrix equation. Moreover, if B in (24)is nonsingular, ost-multilying (24)by B 01 yields an equation of the form in (2). Unfortunately, a simle iterative algorithm X() =C 0 AX( 0 1)B cannot guarantee that X() converges to X. Lemma 2: The equation in (24)has a unique solution if and only if i [A] j [B] 6= 01 for any i and j; in this case the unique solution is given by col[x] =[(B T A) +I mn ] 01 col[c] and the corresonding homogeneous equation AXB + X = 0 also has a unique solution X =0. For the equation in (24), we resent the gradient iterative algorithm to comute the solution X X() =[X 1()+X 2()]=2 (25) X 1 () =X( 0 1) + A T [C 0 AX( 0 1)B 0 X( 0 1)]B T (26) X 2 () =X( 0 1) + [C 0 AX( 0 1)B 0 X( 0 1)] (27) = f max [AA T ] max [B T B]+1g 01 or =[A 2 B 2 +1] 01 : (28) The following result discuss the convergence of this algorithm. Theorem 2: If the matrix equation in (24)has a unique solution X, then the iterative solution X() given by the algorithm in (25) (28) converges to the solution X, i.e., lim!1 X() =X. A similar derivation as in Theorem 1 will lead to the result of Theorem 2, hence, it is omitted here. IV. EQUATION A j XB j = C In this section, we will extend the iterative method to solution of a more general matrix equation A 1XB 1 + A 2XB A XB = C (29) where A j 2 m2m ;B j 2 n2n, and C 2 m2n are given constant matrices, X 2 m2n is the unnown matrix to be solved. Lemma 3: The equation in (29)has a unique solution if and only if the matrix (BT j A j ) is nonsingular; in this case the solution is col[x] = B T j A j 01 col[c]; and if C =0, the matrix equation in (29)has a unique solution X =0. For (29), we roose the following gradient iterative algorithm to comute the solution X: X() =[X 1 ()+X 2 ()+111+ X d ()]= (30) X i() =X( 0 1) + A T i C 0 A j X( 0 1)B j Bi T (31) 1 = 1 = max A j A T j max B T j B j or A j 2 Bj 2 : (32) Theorem 3: If the matrix equation in (29)has a unique solution X, then the iterative solution X() given by the algorithm in (30) (32) converges to the solution X. The roof is similar and hence omitted. V. EXAMPLES This section gives three examles to illustrate the erformance of the roosed algorithms. Examle 1: Suose that AX + XB = C, where A = C = : Then, the solution of X from (3)is B = X = x 11 x 12 x 21 x 22 = : Taing X(0) = , we aly the algorithm in (22) (23) to comute X(). The iterative solution of X is shown in Table I with the relative error := X() 0 X=X. From Table I, it is clear that is becoming smaller and goes to zero as increases. This indicates that the roosed algorithm is effective. The effect of changing

5 1220 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST 2005 Fig. 1. Relative error versus (dots and stars). the convergence factor is illustrated in Fig. 1. We see that the larger the convergence factor is, the faster the convergence the algorithm (or the smaller the estimation error). However, if is too large, the algorithm may diverge. How to choose a best convergence factor is still a roject to be studied. Examle 2: Suose that AX + XB = C, where A; B, and C are matrices (m = n =10)and roduced randomly in Matlab, with the simulation rogram given here. Fig. 2. Relative error versus (dots). TABLE II CONDITION NUMBERS OF 8 WITH DIFFERENT ; TABLE III ITERATIVE ERRORS % end This rogram contains a variable. For different values ( = 0:1; 0:5; 1:0; 5:0), the iterative errors versus are shown in Fig. 2 and the corresonding condition numbers of 8 are in Table II, where min[8] reresents the smallest eigenvalue of 8 and cond[8] denotes the condition number of 8. From Fig. 2 and Table II, increasing leads to small iterative errors, i.e., the convergence rate becomes faster as the condition number of 8 is decreasing. Note that too large a condition number imlies an ill-conditioned 8 [10]. Examle 3: Suose that AX + XB = C, where A; B, and C are matrices and generated similarly in Matlab: Here, we set B = A 0 in order to comare with the B-Q algorithm in which B = A 0 is required. This system has a condition number cond[8] = 1: Use the direct Kronecer roduct aroach in Lemma 1 to solve this high-dimensional equation, the comutational time is s on a Pentium 4 (1.8-GHz)comuter; use the B Q algorithm in [1] and the GI algorithm in (13) (15), the comutational times are 0.75 and 0.41 s, resectively, for ten iterations, and the iterative errors are shown in Table III. From here, we can see that the roosed algorithm and the B-Q algorithm have comarable comutational times, but the latter is only suitable for solving AX + XB = C with B = A 0, and also requires comuting matrix inversion at each iteration. From Table III, after ten iterations, the two algorithms give errors samller than 10 06, but the GI algorithm roosed in this note has faster initial convergence than the B-Q algorithm in the beginning eriod. In simulation, the Pentium 4 comuter fails to solve matrix equations of size greater than by using the direct Kronecer method, without introducing secial oerations, but our iterative method succeeds. VI. CONCLUSION The GI algorithms of solving general matrix equations are studied by using the hierarchical identification rincile. The analysis indicates that the algorithms roosed has good convergence roerties for

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST any initial value, and do not require that the coefficient matrices are of the same size, are stable, have common characteristic roots. Although the algorithms are resented for linear matrix equations, the method adoted can be easily extended to study iterative solutions of nonlinear matrix equations, e.g., the Riccati equations. ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their helful comments and suggestions. REFERENCES [1] P. Benner and E. S. Quintana-Ortí, Solving stable generalized Lyaunov equations with the matrix sign function, Numer. Algorithms, vol. 20, no. 1, , [2] R. Bitmead, Exlicit solutions of the discrete-time Lyaunov matrix equation and Kalman Yaubovich equations, IEEE Trans. Autom. Control, vol. AC-26, no. 6, , Dec [3] R. Bitmead and H. Weiss, On the solution of the discrete-time Lyaunov matrix equation in controllable canonical form, IEEE Trans. Autom. Control, vol. AC-24, no. 3, , Jun [4] E. Davison and F. Man, The numerical solution of + =, IEEE Trans. Autom. Control, vol. AC-13, no. 4, , Aug [5] F. Ding and T. Chen, Iterative least squares solutions of couled Sylvester matrix equations, Syst. Control Lett., vol. 54, no. 2, , [6], Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, vol. 41, no. 2, , [7], Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Autom. Control, vol. 50, no. 3, , Mar [8] Y. Fang, K. A. Loaro, and X. Feng, New estimates for solutions of Lyaunov equations, IEEE Trans. Autom. Control, vol. 42, no. 3, , Mar [9] G. H. Golub, S. Nash, and C. F. Van Loan, A Hessenberg-Schur method for the matrix roblem + =, IEEE Trans. Autom. Control, vol. AC-24, no. 6, , Dec [10] G. H. Golub and C. F. Van Loan, Matrix Comutations, 3rd ed. Baltimore, MD: Johns Hoins Univ. Press, [11] J. Heinen, A technique for solving the extended discrete Lyaunov matrix equation, IEEE Trans. Autom. Control, vol. AC-17, no. 1, , Feb [12] I. Jonsson and B. Kågström, Recursive bloced algorithms for solving triangular systems Part I: One-sided and couled Sylvester-tye matrix equations, ACM Trans. Math. Software, vol. 28, no. 4, , [13], Recursive bloced algorithms for solving triangular systems Part II: Two-sided and generalized Sylvester and Lyaunov matrix equations, ACM Trans. Math. Software, vol. 28, no. 4, , [14] W. H. Kwon, Y. S. Moon, and S. C. Ahn, Bounds in algebraic Riccati and Lyaunov equations: A survey and some new results, Int. J. Control, vol. 64, , [15] T. Mori and A. Derese, A brief summary of the bounds on the solution of the algebraic matrix equations in control theory, Int. J. Control, vol. 39, , [16] H. Muaidani, H. Xu, and K. Mizuami, New iterative algorithm for algebraic Riccati equation related to control roblem of singularly erturbed systems, IEEE Trans. Autom. Control, vol. 46, no. 10, , Oct [17] G. Stare and W. Niethammer, SOR for =, Linear Alg. Al., vol. 154, , [18] T. Styel, Numerical solution and erturbation theory for generalized Lyaunov equations, Linear Alg. Al., vol. 349, no. 1, , An Imroved Lower Matrix Bound of the Solution of the Unified Couled Riccati Equation Chien-Hua Lee Abstract A new lower matrix bound of the solution for the unified couled algebraic Riccati equation (UCARE) is roosed. This bound imroves the drawbac of the results resented in a revious aer. Index Terms Continuous couled algebraic Riccati equation (CCARE), discrete couled algebraic Riccati equation (DCARE), lower matrix bound, unified couled algebraic Riccati equation (UCARE). I. INTRODUCTION It is nown that the so-called couled algebraic Riccati equations arose and layed an imortant role in the jum linear-quadratic control roblem [2], [6], [8], [9] and in many other imortant engineering roblems [3]. Recently, by maing use of the delta oerator develoed by [7], Czorni and Swiernia roosed in [4] a unified couled algebraic Riccati equation (UCARE)and resented several eigenvalue bounds of its solution. By utilizing these eigenvalue bounds, they then derived a lower matrix bound for the UCARE. However, for the roosed matrix bound, its form is very comlicated and a condition must be satisfied. In ractice, matrix bounds of the UCARE can give rough estimates before actually solving it and can chec whether the solution techniques for the UCARE actually yielded valid solutions. Besides, matrix bounds are the most general findings among all the solution bounds of the UCARE. Therefore, we then study the matrix bound of the solution of the UCARE in order to imrove the drawbac of the arallel result of [4]. By extending Lee s method [5], a new matrix bound of the solution of the UCARE is derived. Comarisons show that the obtained result does not need any condition and is more concise. The following symbol conventions are used in this note. A>()B means matrix A 0 B is ositive (semi)definite; i(a) and i (A), resectively, denote the ith eigenvalue and the ith singular value of a symmetric n 2 n matrix A for i = 1; 2;...;n, whereas i(a) and i(a), resectively, are arranged in the nonincreasing order (i.e., 1(A) 2(A) 111 n (A) and 1(A) 2(A) 111 n(a)). Furthermore, scalar function f (a; b; c) is defined as f (a; b; c) (c=a +(a 2 + bc) 1=2 ). II. MAIN RESULTS In recent years, [3] has roosed the UCARE F i A i + A T i F i +1A T i F i A i 0 1A T i + I F i B i I +1B T i F i B i 01 B T i F i (1A i + I) + ij P j = 0Q i (1) j6=i where A i 2< n2n ;B i 2< n2m, matrix Q i 2< n2n is symmetric and ositive semidefinite, P i 2< n2n denotes the ositive semidefinite solution, ij; 1 2 [0; 1);i2 S; S = f1; 2;...;sg is a finite set, and F i = P i +1 j6=i ijpj : (2) Manuscrit received October 10, 2003; revised March 23, 2004, October 11, 2004, and March 31, Recommended by Associate Editor M. Kothare. This wor was suorted by Cheng-Shiu University under Grant CS The author is with the Deartment of Electrical Engineering, Cheng-Shiu University, Kaohsiung 833, Taiwan, R.O.C. ( chienhua@csu.edu.tw). Digital Object Identifier /TAC /$ IEEE