Filtering for Linear Systems with Error Variance Constraints

Transcription

1 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUS application of the one-step extrapolation procedure of [3], it is found that the existence of signal z(t) is not valid in the space R( ). II. NONEXISENCE OF z(t) In [3, Sec. VI], under the assumption that there exists a signal z(t) 2 R( ) such that Bz = x (1) the author obtained the following extrapolation equation: x = Bz: (2) Unfortunately, the signal z(t) 2 R( ) assumed above does not exist if the observed signal x is bandlimited. Let p 3 (t) be the characteristic function of subset 3. If a signal z(t) 2 R( ) does exist in (1), let z 0(t) =p 3(t)z(t), and rewrite (2) as 1 01 h(t 0 )z 0( ) d (3) where h(t) = exp(2i!t) d!. After Fourier transformation (F), (3) becomes X(!) =p (!)Z 0(!) (4) where p (!) denotes the characteristic function of subset, and Z 0(!) denotes the F of z 0(t). Obviously, X(!) =0when! =2, and Z 0 (!) =X(!) on. Since z 0 (t) is a time-limited signal, its F Z 0 (!) should be an analytical function. However, X(!) is a function with support in. In most cases, it is not an analytical function, except that it can be analytically expanded to the whole space. his yields a contradiction. A simple counterexample is given below. Let 3=[01; 1], let = [01; 1], and define X(!) = 1 0j!j;! 2 (5) 0; otherwise At the origin, X(0) has no derivative, but X(!) should be differentiable according to (4). his counterexample indicates that we cannot find a signal z(t) 2 R( ), satisfying (2) even for a simple signal sinc 2 (t). he reason for above result is that the convergence of [3, eq. (19)] depends on the condition x 0 0 x 2 B. Actually, we do not now whether [3, eq. (27)] is valid for a signal z(t) 2 R( ). A remedy to remain (2) is to use a specifically defined space instead of L 2. Assume that 3=[0; ], =[0; ], according to [4] for every x(t) 2 R(B) and that it can be expanded as a ' (t) (6) where ' (t) is the prolate spheroidal wave function for the pair (3; ). Letting be the eigenvalue of ' (t) and defining z(t) = we then have the proposed equation 0 a ' (t) (7) h(t 0 )z( ) d (8) where z(t) =2 R( ). On the other hand, for a truncated signal N n=0 a ' (t), (2) is valid. III. CONCLUSION Based on the existence of signal z(t) 2 R( ), a one-step extrapolation procedure was developed in [3]. In L 2 space, the existence is not valid. his can be remedied in a specifically defined space by using the prolate spheroidal wave function. REFERENCES [1] D. Slepian, H. O. Polla, and H. J. Landau, Prolate spheroidal wave functions, Bell Syst. ech. J., vol. 40, pp , [2] A. Papoulis, A new algorithm in spectral analysis and band-limited extrapolation, IEEE rans. Circuits Syst., vol. 22, pp , [3] J. A. Cadzow, An extrapolation procedure for band-limited signals, IEEE rans. Acoust., Speech, Signal Processing, vol. ASSP-27, pp. 4 12, [4] A. Papoulis, Signal Analysis. New Yor: McGraw-Hill, Robust Filtering for Linear Systems with Error Variance Constraints Zidong Wang and Biao Huang Abstract In this correspondence, we consider the robust filtering problem for linear perturbed systems with steady-state error variance constraints. he purpose of this multiobjective problem is to design a linear filter that does not depend on the parameter perturbations such that the following three performance requirements are simultaneously satisfied. 1) he filtering process is asymptotically stable. 2) he steady-state variance of the estimation error of each state is not more than the individual prespecified value. 3) he transfer function from exogenous noise inputs to error state outputs meets the prespecified norm upper bound constraint. We show that in both continuous and discrete-time cases, the addressed filtering problem can effectively be solved in terms of the solutions of a couple of algebraic Riccati-lie equations/inequalities. We present both the existence conditions and the explicit expression of desired robust filters. An illustrative numerical example is provided to demonstrate the flexibility of the proposed design approach. Index erms Algebraic Riccati equation, filtering, Kalman filtering, quadratic matrix inequality, robust filtering. I. INRODUCION In recent years, the study of the so-called cost-guaranteed filters has gained growing interest; see, e.g., [2], [5], [10], and [11]. A common feature of these results is that they have focused on designing a filter that first provides an upper bound on the variance of the estimation error for all admissible parameter perturbations and then minimizes this bound. It is remarable that in this case, the associated upper bound is Manuscript received July 10, 1998; revised March 21, his wor was supported in part by the University of Kaiserslautern and the Alexander von Humboldt Foundation of Germany. he associate editor coordinating the review of this paper and approving it for publication was Prof. Arnab K. Shaw. Z. Wang is with the Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany, on leave from the Department of Automatic Control, Nanjing University of Science and echnology, Nanjing, China ( zwang@mathemati.uni-l.de). B. Huang is with the Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alta., Canada 6G 2G6 ( Biao.Huang@ualberta.ca). Publisher Item Identifier S X(00)05981-X X/00$ IEEE

2 2464 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUS 2000 not specified a priori, and the resulting optimal robust filters are often unique in certain cases. In practical engineering, however, it is often the case that for a large class of filtering problems, the performance objectives are naturally described as the upper bounds on the error variances of estimation; see, e.g., [7] and [12]. Unfortunately, it is usually difficult to utilize traditional methods to deal with this class of constrained variance filtering problems. A novel filtering method, namely, error covariance assignment (ECA) theory (see, e.g., [12]) was recently developed to provide a closed-form solution for directly assigning the specified steady-state estimation error covariance. Subsequently, [8] and [9] extended the ECA theory to the parameter uncertain systems by assigning a prescribed upper bound to the steady-state error variance, but the perturbations were assumed to be time invariant and measurable, and the adopted filter structure depended on the availability of perturbations. his is very restrictive in practical applications. o overcome the drawbac indicated above, this correspondence aims at designing a perturbation-independent filter, where the perturbations are not required to be time-invariant and available, such that we have the following. 1) he filtering process is asymptotically stable. 2) he steady-state variance of the estimation error of each state is not more than the individual prespecified value. 3) he transfer function from exogenous noise inputs to error state outputs meets the prespecified H1 norm upper bound constraint. he results obtained improve those of [8] and [9]. II. PROBLEM FORMULAION: CONINUOUS-IME CASE Consider the following class of linear uncertain continuous-time systems: _(A +1A)x(t) +D 1 w(t) y(t) =(C +1C)x(t)+D 2w(t) (1) where x 2 n ;y 2 p, and A; C; D 1;D 2 are nown constant matrices. w(t) is a zero mean Gaussian white noise process with covariance I > 0. he initial state x(0) has the mean x(0) and covariance P (0) and is uncorrelated with w(t). 1A and 1C are real-valued perturbation matrices satisfying 1A 1C = M 1 M 2 F (t)n (2) where F (t) 2 i2j is a real time-varying uncertain matrix meeting F (t)f (t) I, and M 1 ;M 2 ; and N are nown constant matrices of appropriate dimensions. Assumption 1: he system matrix A is Hurwitz stable, and the matrix D 2 or M 2 is of full row ran. In the continuous-time case, the linear full-order filter under consideration is given by _^G^x(t) +Ky(t) (3) where ^x(t) denotes the state estimation, and G and K are filter parameters to be determined. he estimation error covariance in the steady state is denoted by P := lim t!1 P (t) := lim t!1 E[e(t)e (t)], where e(t) =x(t) 0 ^x(t), if the limit exists. By defining x f (t) := D f := x(t) e(t) ; A f := A 0 A 0 G 0 KC G D 1 D 1 0 KD 2 (4) M 1 M f := ; N f := [ N 0] M 1 0 KM 2 1A f = M f F (t)n f (5) and considering (1) and (3), we obtain the following augmented system: _x f (t) =(A f +1A f )x f (t)+d f w(t): (6) When the system (6) is robustly asymptotically stable, the steadystate covariance defined by X := lim X(t) := lim t!1 t!1 E x f (t)x f (t) := X xx X xe Xxe P exists and satisfies the following Lyapunov matrix equation: (A f +1A f )X + X(A f +1A f ) + D f D f =0: (8) Our objective is to see the filter parameters G and K such that for all admissible parameter perturbations 1A and 1C, the following three requirements are simultaneously satisfied. 1) he augmented system (6) is asymptotically stable. 2) he steady-state error covariance P meets [P ] ii 2 i ;i = 1; 2;...;n, where [P ] ii stands for the ith diagonal element of P. 2 i (i =1; 2;...;n) denotes the steady-state estimation error variance constraint on the ith state, which is not less than the minimal value obtained from the (robust) minimum variance filtering theory. 3) he H1 norm of the transfer function H(s) =C f [si 0 (A f + 1A f )] 01 D f from disturbances w(t) to error state outputs Le(t) (or C f x f (t)) satisfies the constraint H(s)1, where L is the nown error state output matrix, C f := [ 0 L ], H(s)1 = sup!2r max[h(j!)], max[ 1 ] denotes the largest singular value of [ 1 ], and is a given positive constant. III. MAIN RESULS: CONINUOUS-IME CASE Prior to providing the main results, we first mae the following definitions for notational simplicity: ^A := A + "M 1 M 1 + D 1 D 1 P 01 1 (7) ^C := C + "M 2 M 1 + D 2 D 1 P 01 1 (9) R = "M 2 M 2 + D 2 D 2 ~A := ^A 0 "M 1 M 2 + D 1 D 2 R 01 ^C: (10) heorem 1: Let 1 > 0 and 2 > 0 be sufficiently small constants, and let U 2 p2p be an arbitrary orthogonal matrix. If there exist a scalar ">0and a matrix H 2 n2p such that the Riccati equations AP 1 + P 1 A + "M 1 M 1 + " 01 P 1 N NP 1 + D 1D 1 + 1I =0 (11) ~AP 2 + P 2 ~ A 0 P2 ( ^C R 01 ^C 0 02 L L)P2 + "M 1 M 1 + D 1 D 1 + HH 0 "M 1 M 2 + D 1 D 2 2 R 01 "M 1 M 2 + D 1 D I =0 (12) respectively, have positive definite solutions P 1 > 0 and P 2 > 0, then with the parameters determined by K = P 2 ^C + "M1 M 2 + D 1 D 2 R 01 + HUR 01=2 G = ^A 0 K ^C (13)

3 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUS the filter (3) will be such that for all admissible perturbations 1A and 1C, we have the following. 1) he augmented system (6) is asymptotically stable. 2) he steady-state error covariance P exists and meets P<P 2. 3) H(s)1. Proof: 1) For a scalar ">0 and a matrix P f > 0, it is easy to prove that (1A f )P f +P f (1A f ) "M f M f +" 01 P f N f N f P f. Next, by setting P f := Bloc-diag(P 1 ;P 2 ),wehave (A f +1A f )P f + P f (A f +1A f ) + 02 P f C f C f P f + D f D f 9:= where (14) 9 11 = AP 1 + P 1 A + "M 1 M 1 + " 01 P 1 N NP 1 + D 1 D 1 (15) 9 12 = P 1(A 0 G 0 KC) + "M 1(M 1 0 KM 2) + D 1 (D 1 0 KD 2 ) (16) 9 22 = GP 2 + P 2G + "(M 1 0 KM 2)(M 1 0 KM 2) + 02 P 2 L LP 2 +(D 1 0 KD 2 )(D 1 0 KD 2 ) : (17) Equation (11) means that 9 11 = 0 1 I<0, and the expression of G in (13) and (16) implies 9 12 =0. Now, substitute the expression G = ^A0K ^C into (17), and it is not difficult to verify that 9 22 = ^AP 2 + P 2 ^A + "M 1M P 2L LP 2 + D 1D 1 + KR 1=2 0 P 2 ^C + "M 1M 2 + D 1D 2 R 01=2 2 KR 1=2 0 P 2 ^C + "M 1 M 2 + D 1 D 2 R 01=2 0 P 2 ^C + "M 1 M 2 + D 1 D 2 2 R 01 P 2 ^C + "M 1 M 2 + D 1 D 2 : (18) Furthermore, taing into account the expression of K in (13) and noticing the facts that UU = I and KR 1=2 0 (P 2 ^C + "M 1M 2 + D 1D 2 )R 01=2 = HU, we easily obtain from (18) that 9 22 = ~ AP 2 + P 2 ~ A 0 P 2( ^C R 01 ^C 0 02 L L)P 2 + "M 1M 1 + D 1D 1 + HH 0 ("M 1 M 2 + D 1 D 2 )R 01 ("M 1 M 2 + D 1 D 2 ). Finally, it results from (12) that 9 22 = 0 2 I< 0. Now,we have the conclusion that 9 < 0, and thus, (A f +1A f )P f + P f (A f +1A f ) 0( 02 P f C f C f P f + D f D f )+9< 0, which shows from Lyapunov stability theory that the augmented system (6) is asymptotically stable. 2) Since A f +1A f remains asymptotically stable, the steady-state covariance X exists and meets (8). Define 4 := 9 0 [(A f + 1A f )P f + P f (A f +1A f ) + 02 P f C f C f P f + D f D f ]. Clearly, 4 0, and subsequently (A f +1A f )P f + P f (A f +1A f ) + 02 P f C f C f P f + D f D f 0 9+4=0: (19) Subtract (8) from (19) to obtain (A f +1A f )(P f 0 X)+(P f 0 X)(A f +1A f ) + 02 P f C f C f P f = 0, or equivalently, P f 0 X 1 = exp[(a 0 f +1A f )t]( 02 P f C f C f P f 0 9+4) exp[(a f +1A f ) t] dt > 0, which means that X<P f. Conclusion 2) follows from P =[X] 22;P 2 =[P f ] 22 directly, where [ 1 ] 22 is the 22-sub-bloc of [ 1 ]. 3) Since 09 +4> 0, the proof of H(s)1 can be completed by a standard manipulation of (19) [9]. he proof of heorem 1 is then completed. In view of heorem 1, if the positive definite solutions P 1 and P 2 to (11) and (12) exist, and P 2 > 0 meets [P 2 ] ii 2 i ;i=1; 2;...;n, we will have the following conclusions. 1) he augmented system (6) is asymptotically stable. 2) H(s)1. 3) [P ] ii < [P 2 ] ii 2 i ;i=1; 2;...;n. Hence, with the filter (3), whose parameters K and G are determined by (13), the variance-constrained robust H 2 =H1 filtering gain design tas will be accomplished, and we can see that the ey step in designing the expected filters is to deal with the solvability of the Riccati equations (11) and (12). Lemma 1 [5]: If A is stable and N (si 0 A) 01 M 11 < 1, then there exists a constant > 0 such that for all " 2 (0;), the matrix Riccati equation (11) has a positive definite solution P 1. In addition, since (12) is a parameter-dependent continuous-time Riccati matrix equation, the related numerical algorithms can be found in many papers, such as [5] and [10]. Moreover, we can use the modified quadratic matrix inequalities (QMI s) to restate heorem 1 in a clearer sense and obtain the following results immediately. heorem 2: Let U 2 R p2p be an arbitrary orthogonal matrix. If there exists a positive scalar ">0such that the QMI s AP 1 + P 1 A + "M 1 M 1 + " 01 P 1 N NP 1 + D 1 D 1 < 0 (20) 3:= ~ AP 2 + P 2 ~ A 0 P 2 ( ^C R 01 ^C 0 02 L L)P 2 + "M 1 M 1 + D 1 D 1 0 "M 1 M 2 + D 1 D 2 2 R 01 "M 1M 2 + D 1D 2 < 0 (21) respectively, have positive definite solutions P 1 > 0 and P 2 > 0, then the filter (3) with parameters K = P 2 ^C + "M 1 M 2 + D 1 D 2 R 01 + EUR 01=2 G = ^A 0 K ^C (22) where E 2 n2p (p n) is an arbitrary matrix meeting 3+EE < 0, and 3 is defined in (21) will be such that for all admissible perturbations 1A and 1C, we have the following. 1) he augmented system (6) is asymptotically stable. 2) he steady-state error covariance P exists and meets P<P 2. 3) H(s)1. heorem 3: If there exist positive definite solutions, P 1 > 0 and P 2 > 0, respectively, to Riccati matrix equations (11) and (12) or QMI s (20), (21), and [P 2] ii 2 i (i = 1; 2;...;n), then the filter with parameters determined by (11) or (12) will satisfy the desired robust filtering performance constraints. Remar 1: In practical applications, it is very desirable to directly solve Riccati matrix equations (11) and (12) or QMI s (20) and (21), subject to the constraint [P 2] ii 2 i (i =1; 2;...;n) and then obtain the expected filter parameters readily from (13) or (22). When we deal with the QMI s (20) and (21), the local numerical searching algorithms suggested in [1] are very effective for a relatively low-order model. A related discussion of the solving algorithms for QMI s can also be found in [6].

4 2466 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUS 2000 IV. MAIN RESULS: DISCREE-IME CASE In this section, we will briefly state the main results for discrete-time systems and only give the necessary setches of the proofs. Consider the following linear uncertain discrete-time stochastic system: x( +1)=(A +1A)x() +D 1w() y() =(C +1C)x()+D 2 w() (23) where x; y; A; C; D 1 ;D 2 ; 1A; and 1C have the similar meanings to the continuous-time case. w() is a zero mean Gaussian white noise sequence with covariance I > 0. he initial state x(0) has the mean x(0) and covariance P (0) and is uncorrelated with w(). Assumption 2: he matrix A is Schur stable and nonsingular, and the matrix D 2 or M 2 is of full row ran. In the discrete-time case, the adopted linear full-order filter is of the following structure: ^x( +1)=G^x() +Ky() (24) where ^x() stands for the state estimation, and G and K are filter parameters to be scheduled. he steady-state estimation error covariance is defined by P := lim!1 P () := lim!1 E[e()e ()]; e(t) =x(t) 0 ^x(t) if the limit exists. Furthermore, define x f () :=[x () e ()] and A f ; D f ;M f ;N f ; and 1A f as in (4) and (5), and we obtain the augmented system x f ( +1)=(A f +1A f )x f ()+D f w(): (25) When the augmented system (25) is robustly asymptotically stable, the steady-state covariance given by X := lim X() := lim!1!1 E x f ()x f () := X xx X xe X xe P (26) exists and meets the discrete-time Lyapunov equation X = (A f + 1A f )X(A f +1A f ) + D f D f. he purpose of this section is to design the filter parameters G and K such that for all admissible perturbations 1A and 1C, we have the following. 1) he augmented system (25) is asymptotically stable. 2) [P ] ii 2 i ; i =1; 2;...;n. 3) he H1 norm of the transfer function H(z) :=C f [zi 0 (A f + 1A f )] 01 D f from disturbances w() to error state outputs Le() (or C f x f ()) satisfies the constraint H(z)1, where L is the nown error state output matrix, C f := [0 L], and H(z)1 =sup 2[0;2] max[h(ej )]. Lemma 2 (see, e.g., [11]): Let a positive scalar ">0and a positive definite matrix Q f > 0 be such that N f Q f N f <"I; then, (A f + 1A f )Q f (A f +1A f ) A f (Q 01 f 0"01 N f N f ) 01 A f +"M f M f holds. For technical convenience, we define the following additional notation: 8:= P " 01 N N 01 A ^A := A + "M 1 M 1 + D 1 D (27) ^C := C + "M 2 M 1 + D 2 D :=8 01 P " 01 N N 01 (8 01 ) (28) ~P 2 := P 2 + P 2L ( 2 I 0 LP 2L ) 01 LP 2 (29) 2:= ^A ~ P 2 ^C + "M 1M 1 + D 1D 1 0 "M 2M 1 +D 2 D 1 "M1 M 2 + D 1 D 2 (30) R := ^C ~ P 2 ^C + "M 2 M 1 + D 2 D 1 0 "M 2 M 1 +D 2 D 1 + "M 2 M 2 + D 2 D 2 : (31) heorem 4: Let 1 > 0 and 2 > 0 be sufficiently small positive constants, and let U 2 p2p be an arbitrary orthogonal matrix. If there exist a scalar ">0and a matrix H 2 n2p such that AP 1 A 0 P 1 + AP 1 N ("I 0 NP 1 N ) 01 NP 1 A + "M 1M 1 + D 1D 1 + 1I =0 (32) ^A ~ P 2 ^A 0 P 2 0 2R "M 1 M 1 + D 1 D "M 1 M 1 + D 1 D 1 + "M 1 M 1 + D 1D 1 + HH + 2I =0 (33) together with the inequality constraints NP 1N "I; LP 2L 2 I (34) respectively, have positive definite solutions P 1 > 0 and P 2 > 0, then the filter (24) with the parameters determined by K =2R 01 + HUR 01=2 ;G= ^A 0 K ^C will be such that for all admissible perturbations 1A and 1C, we have the following. 1) he augmented system (25) is asymptotically stable. 2) he steady-state error covariance P exists and meets P<P 2. 3) H(z)1. Setch of the Proof: o start with, we set P f := Bloc-diag(P 1 ;P 2 ); Q f := Bloc-diag(P 1 ; ~ P 2 ), and by means of Lemma 2 and (27) (31), it is easy to verify that Q f = P f + P f C f ( 2 I 0 C f P f C f ) 01 C f P f and (A f +1A f ) P f + P f C f 2 I 0 C f P f C f 2 (A f +1A f ) 0 P f + D f D f 01 Cf P f A f Q 01 f 0 " N f N f A f + "M f M f 0 P f + D f D f := 9: (35) hen, similar derivation as in the proof of heorem 1 shows 9 < 0, and therefore, the augmented system (25) is Schur stable. he proof techniques of second and third conclusions are along the lines of those used in heorem 1 and [4, Lemma 5.1], and thus, the detailed proofs are omitted here. We now briefly discuss the solvability of the Riccati equations (32) and (33). Lemma 3 [3]: If the uncertain system (23) is quadratically stable, then there must exist a scalar ">0and a matrix P 1 > 0 that satisfy NP 1 N <"Iand the discrete-time Riccati equation (32). Next, noting that (33) is actually a generalized parameter-dependent Riccati equation, we can deal with it by using the approach proposed in [6] and [11]. We further restate heorem 1 in terms of two QMI s and then obtain our main results for the discrete-time case. heorem 5: Let U 2 R p2p be an arbitrary orthogonal matrix. If there exists a positive scalar ">0such that the following two QMI s AP 1 A 0 P 1 + AP 1 N ("I 0 NP 1 N ) 01 NP 1 A + "M 1M 1 + D 1D 1 < 0 (36)

5 IEEE RANSACIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUS := ^A ~ P 2 ^A 0 P 2 0 2R "M 1M 1 + D 1D "M 1 M 1 + D 1 D 1 + "M 1 M 1 + D 1 D 1 < 0 (37) together with the inequality constraints (34), respectively, have positive definite solutions P 1 > 0 and P 2 > 0, then the filter (24) with the parameters determined by K =2R 01 + EUR 01=2 ;G= ^A 0 K ^C, where E 2 n2p (p n) is an arbitrary matrix meeting 3+EE < 0 and 3 is defined in (37), will be such that we have the following. 1) he augmented system (25) is asymptotically stable. 2) P < P 2. 3) H(z)1. heorem 6: Subject to the constraints (34), if there exist positive definite solutions P 1 > 0 and P 2 > 0, respectively, to Riccati matrix equations (32) and (33) or QMI s (36), (37), and [P 2 ] ii 2 i (i = 1; 2;...; n), then the filter with parameters determined by heorem 4 or heorem 5 will, respectively, meet the desired robust H 2 =H1 filtering performance requirements. V. NUMERICAL EXAMPLE Consider a linear discrete-time uncertain stochastic system (23) with the following parameters: 0:8 0:05 A = ; C =[1 0] 00:08 00:5 D 1 = 0:1 0 ; D 2 =[0:1 0:08 ] 0 0:1 M 1 = 0:08 ; M2 =0:1 0:06 N =[0:5 0:5]; L =[0:01 0:01 ] : We wish to design a filter (24) such that the augmented system (25) is asymptotically stable, the steady-state error covariance P meets [P ] =0:5; [P ] =1:2, and H(z)1 =0:9. Choose " =0:5, then a positive definite solution to QMI (36), and therefore, 8; ^A; ^C; 0 can be obtained as follows: P 1 = ^A = 0= 0: : :0006 0:0174 ; 8= 0: :0032 0: :0088 1: :3712 ; ^C =[1: :4175 ] 0: : : :0997 : 26: :2372 hen, solve the QMI (37) to give P 2 = 0: : :1484 1:1617 and it is easily seen that [P 2 ] ii < 2 i (i =1; 2), and (34) is satisfied. Next, choose parameter E, which meets 3+EE < 0 [3 is defined in (37)] as E =[0:0800 0:1000 ]. hen, for the two cases of U 1 =1 and U 2 = 01, we obtain the corresponding desired filter parameters, respectively, as K 1 = 0:4812 0:9643 ; G 1 = K 2 = 0:4002 0:8631 ; G 2 = 0:5068 0: : :5144 0:6240 0: : :6579 : It is not difficult to test that the precsribed performance objectives are all realized. VI. CONCLUSION In this correspondence, attention has been focused on designing linear perturbation-independent filters that achieve the multiple prescribed objectives of filtering process: robust stability,; H1 norm; steady-state estimation error variance constraints. he further study will be the development of efficient algorithms with guaranteed convergence. REFERENCES [1] E. Beran and K. Grigoriadis, A combined alternating projections and semidefinite programming algorithm for low-order control design, in Proc. Prep. 13th IFAC World Congr., vol. C, San Francisco, CA, 1996, pp [2] P. Bolzern, P. Colaneri, and G. De Nicolao, Finite escapes and covergence properties of guaranteed-cost robust filters, Automatica, vol. 33, pp , [3] G. Garcia, J. Bernussou, and D. Arzelier, Robust stabilization of discrete-time linear systems with norm bound time-varying uncertainty, Syst. Contr. Lett., vol. 22, pp , [4] W. M. Haddad, D. S. Bernstein, and D. Mustafa, Mixed-norm H =H regulation and estimation: he discrete-time case, Syst. Contr. Lett., vol. 16, pp , [5] I. R. Petersen and D. C. McFarlane, Optimal guaranteed cost control and filtering for uncertain linear systems, IEEE rans. Automat. Contr., vol. 39, pp , [6] A. Saberi, P. Sannuti, and B. M. Chen, H Optimal Contr., ser. Series in Systems and Control Engineering. London, U.K.: Prentice-Hall, [7] R. E. Selton and. Iwasai, Liapunov and covariance controllers, Int. J. Contr., vol. 57, pp , [8] Z. Wang, Z. Guo, and H. Unbehauen, Robust H =H state estimation for discrete-time systems with error variance constraints, IEEE rans. Automat. Contr., vol. 42, pp , [9] Z. Wang and H. Unbehauen, Robust H =H -state estimation for systems with error variance constraints: he continuous-time case, IEEE rans. Automat. Contr., vol. 44, pp , [10] L. Xie and Y. C. Soh, Robust Kalman filtering for uncertain systems, Syst. Contr. Lett., vol. 22, pp , [11] L. Xie, Y. C. Soh, and C. E. de Souza, Robust Kalman filtering for uncertain discrete-time systems, IEEE rans. Automat. Contr., vol. 39, pp , [12] E. Yaz and R. E. Selton, Continuous and discrete state estimation with error covariance assignment, in Proc. IEEE Conf. Decision Contr., Brighton, U.K., 1991, pp