Robustness of Discrete Periodically Time-Varying Control under LTI Unstructured Perturbations

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1 1370 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 000 Robustness of Discrete Periodically Time-Varying Control under LTI Unstructured Perturbations Jingxin Zhang and Cishen Zhang Abstract This note investigates robustness of linear periodically time varying (LPTV) control of discrete linear time invariant (LTI) plants subject to LTI unstructured perturbations. The note first derives a necessary and sufficient condition for robust stability of an LPTV system subject to LTI perturbations, which is less conservative than the well known small gain condition. It then presents a quantitative analysis on the robustness of LPTV control under LTI unstructured perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect. Index Terms Robustness, stability margin, time-varying control, unstructured perturbations. I. INTRODUCTION Robustness enhancement is a primary motivation for the investigation of linear periodically time varying (LPTV) control of linear time invariant (LTI) plants. It has been shown by several authors that LPTV control can improve significantly gain/phase margin and can provide strong and simultaneous stabilization. However, it has also been shown that nonlinear time varying (NLTV) control and linear time varying (LTV) control, which include LPTV control as a special case, offer no advantages over LTI control for robust stabilization of LTI plants subject to norm bounded unstructured model perturbations, and that LPTV control systems may be sensitive to the high frequency band unstructured perturbations. For details of the above results, see, e.g., [], [9], [1], [11], [13], [15], [16], [10], [18] and the references therein. These different results appear to be uncorrelated and conflict to a certain extent, and the relevance and tradeoffs between the different results and cases are not clear. For instance, no existing result answers whether or not it is possible to get some benefits, from using LPTV control, at no sacrifice of the robustness for unstructured perturbations. An example case may be that using LPTV control to improve the system gain margin while retaining the same level of the robustness as LTI control for unstructured perturbations. It is therefore important and necessary to gain further understanding of the intrinsic properties of LPTV control systems. This note investigates robustness of LPTV control systems under LTI unstructured perturbations. Using the results of [14] and [19], the note first derives a necessary and sufficient condition for robust stability of a discrete LPTV system subject to LTI perturbations. This condition is expressed in terms of the structured singular values of the lifted system, which is analogous to an condition obtained in [6] characterizing the robustness of sampled-data control systems to LTI perturbations. Since the systems considered here evolve completely in discrete time, our condition is finite dimensional. Compared with the well known small Manuscript received December 1, 1998; revised September 17, 1999 and January 8, 000. Recommended by Associate Editor, A. Varga. This work was supported by Australian Research Council. J. Zhang is with the Cooperative Research Center for Sensor Signal and Information Processing, The Levels Campus, University of South Australia, Mawson Lakes, SA 5095, Australia and the School of Physics and Electronic Systems Engineering, University of South Australia ( jingxin.zhang@cssip.edu.au). C. Zhang is with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Victoria 305, Australia. Publisher Item Identifier S (00) gain condition [15], [16], this condition is less conservative and allows unstable perturbations. Based on the derived condition, this note then presents a quantitative analysis on the robustness of LPTV control under LTI perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect. For simplicity, only the SISO systems are considered. The obtained results carry easily to MIMO case. II. PRELIMINARIES AND PROBLEM STATEMENT Throughout this note, (1) and (1) denote the spectral radius and maximum singular value of a matrix, respectively; j1jdenotes the magnitude of a complex variable, dim(1) the dimension of a square matrix, X 3 the conjugate transpose of a complex vector X; F (z) and F (e j! ) represent the z transform and Fourier transform of a discrete signal f (t) or the z-transfer function and frequency response of a discrete LTI system F, respectively; for a given transfer function F (z); kf k 1 denotes the 1-norm of F (z), i.e., kf k 1 := sup [F (ej! )]; the term operator refers to the mapping of a discrete (linear or nonlinear) system that maps its input to its output; k1k denotes the -norm induced operator norm which is defined as k(1)k := sup kuk =1 k(1)uk, where 1 kuk := ( t=0 u3 (t)u(t)) 1= ; the boldface letters are used to denote the operators; following [15] and with a little abuse of terminology, the operators with bounded induced norm are called stable operators. Let G be a discrete linear system represented by the following input output equation y(t) = k =0 g(t; )u() (1) where u(t) and y(t) are the input and output, respectively, and g(t; ) is the kernel of G and satisfies g(t + N; + N )=g(t; ): () The following definitions are used in the note:gis a finite order system if G admits a finite dimensional state-space representation. G is a real system if g(t; ) R; 8t; R. G is a stable system if kgk < 1. G is causal if g(t; )=0; 8 > t. G is strictly LPTV if N > 1, and G is LTI if N =1. The above definition on strictly LPTV and LTI systems separates the set of LPTV systems (N 1) into two subsets. This is to distinguish LTI systems which are trivial LPTV systems from strictly LPTV ones and to compare strictly LPTV control with LTI control. This definition coincides with the conventional definition of LPTV systems which includes LTI systems as a special case. Note that if G is LTI, causal and finite order, it can be described by its transfer function G(z), which is a matrix with all elements being proper and finite order rational functions of z, and G is stable if and only if G(z) has no poles outside the open unit disk. Further, if G is stable, kgk = kgk 1 = sup [G(ej! )] < 1. However, this does not apply to unstable G. In such case, if G(z) has no poles on the unit circle, kgk 1 < 1 while kgk = 1. These are standard results, see, e.g., [13], [15], [5] for details. The notions and discussions given in the above are MIMO system oriented, which include the SISO systems to be considered in the following as a special case. Now consider the feedback control system F given in Fig. 1. It is assumed that all the blocks of F are SISO discrete linear systems. In /00$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY the figure, P denotes a real, causal, finite order and stabilizable LTI nominal plant, K is a real, causal and finite order LPTV controller with period N;W is a stable LTI weighting function with stable inverse, the dashed-line enclosed subsystem S is the nominal closed loop system resulting from the interconnection of the nominal plant P, controller K and weighting function W, and 1 is a multiplicative perturbation to P. The following assumptions are made on W; P and 1, which are normal assumptions for the existence of optimal H1 LTI controller and the unstructured LTI perturbations, see, e.g., [5]. A1) W has no zeros and P has no zeros and poles on the unit circle. A) 1 is an LTI, causal, possibly unstable operator with 1(e j! ) satisfying k1k1 r; r R; 0 r<1, and P + 1WP possesses the same number of unstable poles as P. The problems to be considered in the note are the stability margin of F, i.e., the maximum r for which F remains stable, under the perturbations A and the comparison of the stability margin attainable by strictly LPTV K with that attainable by LTI K. For simplicity, only the standard multiplicative perturbation case [5] is considered. However, all the analyses in the sequel carry over to additive perturbation case. III. FREQUENCY DOMAIN LIFTING FOR LPTV SYSTEMS Let G be an LPTV system in the form of (1) with period N. Then g(t + ;) is an N -periodic function of and can be expanded into Fourier series g(t + ;) = g k(t)e jk!, where g k (t) = 1=N g(t =0 + ;)e0jk! and! N := =N. Using this, (1) can be written as y(t) = = t =0 g k (t 0 )e jk! u( ) g k (t) 3 [e jk! u(t)]: (3) Suppose that G is causal, real, finite order and stable and that its input u l, i.e. kuk < 1. Then u(t);g k (t) and y(t) all admit z and Fourier transforms, and (3) can be written in z transform as [14] Y (z) = G k (z)u (ze jk! ) (4) where G k (z) and U (ze jk! ) are z transforms of g k (t) and [e jk! u(t)], respectively. Define ^X(z) := [X(z)X(ze j! ) 111X(ze j()! )] T. Using (4) and the fact that e jl! = e j(l+n)!, it is trivial to show that where ^Y (z) = ^G(z) ^U (z) ^G(z) :=[G k (ze jl! )] k;l=0;1;111; : (5) Letting z = e j! ; ^G(z) can also be written as ^G(z) = [G k (e j(!+l! ) )] k;l=0;1;111; =: ^G(e j! ). ^G(z) and ^G(e j! ) given above are called the lifted frequency transfer function of the LPTV system G, and will be used interchangably. The arguments z and e j! will be suppressed when no confusion arises. Define k ^Xk := ((1=)! ^X 3 (e j! ) ^X(e j! ) d!) 1=. It is trivial 0 to show that k ^Y k = kyk and k ^Uk = kuk. Multiplication by ^G is equivalent to G, via the lifting and Z-transform isomorphisms. Correspondingly, the induced norm of G is equal to the 1-norm of ^G. The following lemma gives the properties of ^G, where a), b), and in d) are proved [19, Secs. 3 and 4], and c) follows trivially from the standard LTI system theory. Fig. 1. LPTV feedback control system F. Lemma 3.1: a) If G is stable, then kgk = k ^Gk 1. b) G is strictly LPTV if and only if G k (z) 6= 0for some k 6= 0, and G is LTI if and only if G k (z) =0; 8k 6= 0, i.e., ^G(z) = diag[g 0 (ze jl! )] l=0;1;111;. c) G is stable if and only if all the poles of ^G(z) belong to the open unit disk. d) Given a stable lifted frequency transfer function ^G(z), the lifted transfer function ^G 0 (z) := diag[ ^G(z)] corresponds a stable LTI system G 0, where for a matrix X = [X ij ] i;j=0;1;111; ; diag(x) ii = X ii and diag(x) ij = 0 for i 6= j. Moreover, k ^Gk 1 [sup G k(e j! )] 1= k^g 0k 1 = kg0k 1. IV. ROBUST STABILITY CONDITIONS OF CLOSED LOOP LPTV SYSTEMS As shown in [15] and [16] the nominal closed loop subsystem S in Fig. 1 can be written in linear operator form S = WP(10PK) 01 K. Assume that S is stabilized by K. Apparently, S is LPTV if K is LPTV. Robust stability of the closed loop system F with NLTV 1 has been studied in [15] and [16]. For a more general case, [15] and [16] have proved that if 1 is an NLTV, causal, stable operator with k1k r and K is NLTV, the closed loop system F is stable if and only if ksk < 1=r. This result includes LPTV K as a special case. In the following it will be shown that if 1 is LTI and K is LPTV, then 1 need not to be stable and the small gain condition ksk < 1=r is not necessary. Denote ^S(z); ^P (z); ^W (z); ^K(z), and ^1(z) the lifted transfer functions of S; P; W; K and 1, respectively. Then F can be represented equivalently by an LTI MIMO system with a nominal closed loop ^S(z) = ^W (z) ^P (z)[i 0 ^P (z) ^K(z)] 01 ^K(z) and a perturbation block ^1(z). From Lemma 3.1-b) and the assumptions on P; W; K and 1 it is known that ^S(z); ^P (z); ^W (z); ^K(z), and ^1(z) are all N N -dimensional, proper, finite order transfer function matrices, and ^1(z) is diagonal in structure. It follows from the well-known theory for MIMO linear systems that if the number of unstable poles of ^P (z) + ^W (z) ^P (z) ^1(z) equals that of ^P (z), the stability of the system F is equivalent to the nonsingularity of [I 0 ^S(e j! ) ^1(e j! )], see e.g. [3]. Theorem 4.1: Suppose that 1 satisfies A and K is strictly LPTV. Then the closed loop system F is stable if and only if sup [ ^S(e j! )] < 1=r (6)! where [ ^S(e j! )] is the complex structured singular value of ^S(e j! ) [3] defined as [ ^S(e j! )] := 1= minf[ ^1(e j! )] :det[i 0 ^S(e j! ) ^1(e j! )] = 0; ^1(e j! ) Dg (7) D := fdiag[0 ; 1 ; 111; ] : i Cg:

3 137 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 000 Proof: By assumption 1 satisfies A. Hence ^P (z) + ^W (z) ^P (z) ^1(z) and ^P (z) possess the same number of unstable poles. From Lemma 3.1-b), under condition A ^1(e j! ) is diagonal with diagonals 1(e j(!+l! ) );l = 0; 111;N 0 1, which are frequency responses of 1 in different frequency bands. By assumption, k1k1 r. Hence, k ^1k1 =sup! [ ^1(e j! )] r. From the definition (7) for [ ^S], it can be easily seen that if (6) holds, no 1 satisfying A will make I 0 ^S ^1 singular. Now suppose (6) does not hold. Then there exists some! 0 [0;! N ) such that [ ^S(e j! )] 1=r. From the definition (7) it follows that there must exist a constant matrix D =diag[d l ] l=0;1;111; with d l C and jd l jr; l =0; 1; 111;, such that det[i 0 ^S(e j! )D] =0. Since ^1(e j! ) = diag[1(e j(!+l! ) )] l=0;1;111; is given by 1(e j(!+l! ) ), the frequency responses of 1 at different frequency bands, a 1 with 1(e j! ) satisfying k1k1 r and passing through the points (d l ;! 0 + l! N);l = 0; 1; 111;N 0 1, will give rise to the matrix D at! =! 0. From the interpolation theory [4] it is known that a stable and proper transfer function 1 0 (e j! ) with order N can be constructed such that k1 0k1 r and 1 0 (e j! +l! ) = d l for l = 0; 1; 111;N 0 1. This 1 0 (e j! ) represents a causal, finite order and stable LTI 1 0. As the stable 1 is a subclass of the 1 in A, the 1 0 with lifted transfer function ^1 0 (e j! ) = diag[1 0 (e j(!+l! ) )] l=0;1;111; is an admissible perturbation. Thus, if (6) does not hold, there does exist a destabilizing 1 in A. Remarks: As shown in [3], sup! [ ^S(e j! )] k^sk1 and the gap between k ^Sk1 and sup! [ ^S(e j! )] can be large. Hence, Theorem 4.1 actually shows that the small gain condition can be conservative if 1 is LTI. Further, the condition A allows unstable LTI 1, this is different from the condition that 1 must be stable if it is NLTV [15], [16]. That robust stabilization to unstructured LTI perturbations reduces to a structured singular value problem is not new. The first work of this kind appears in [6] where robust stabilization of sampled-data systems to LTI perturbations is reduced to an infinite-dimensional structured singular value problem. As the sampled-data systems are periodic in continuous time and are inherently infinite dimensional in the lifted domain, the result of [6] is infinite dimensional. On the other hand, Theorem 4.1 is a finite dimensional condition since it concerns discrete LPTV systems which are finite-dimensional in the lifted domain. Theorem 4.1 includes LTI K as a special case. To see this, let S LTI be the nominal closed loop system resulting from an LTI controller K LTI, and let S LTI (e j! ) and ^S LTI (e j! ) be the transfer function and lifted transfer function of S LTI, respectively. Since K LTI is LTI, so is S LTI. By Lemma 3.1-b), ^S LTI (e j! ) = diag[s LTI (e j(!+l! and k ^S LTI k1 = sup [ ^S LTI (e j! )]! = sup max js LTI (e j(!+l! ) )j! 0l ) )] l=0;1;111; = sup js LTI (e j! )j = ks LTI k1 (8) Using (8), Theorem 4.1 and the definition of the stability margin, the following corollary is immediately obtained. Corollary 4.1: Provided 1 satisfies A, the closed loop system F with K = K LTI is stable if and only if sup! [ ^S LTI(e j! )] < 1=r. The evaluation of [ ^S(e j! )] is generally complicated unless K = K LTI and ^S becomes diagonal. However, in the case N 3, the following simple result holds. Lemma 4.1: If N 3, then there exists a scaling matrix F (!) such that F (!) = diag[f k (!)]; f k (!) R;f k (!) > 0; k =0; 1; 111;N 0 1;! [0;! N ) (9) sup [ ^S(e j! )] = sup [F (!) ^S(e j! )F 01 (!)] = kf ^SF 01 k1 (10) Proof: In [3] it is shown that for a square complex matrix M associated with a block diagonal matrix D from which (M ) is defined, (M ) = inf F (FMF 01 ) if the number of the blocks of D is less than or equal to 3. Where F = diag[f k I X ];f k R;f k > 0;k =0; 1; ;I X = X k -dimensional identity matrix, X k = the dimension of the kth block of D. In the special case considered here, at any fixed! [0;! N ); ^S(e j! ) =M; ^1(e j! ) =D, and X k = 1 for k =0; 1;. Hence F is frequency dependent and is in the form of (9), and the equality (10) follows from the assumption that dim( ^1) = N 3. V. STABILITY MARGINS ATTAINABLE BY STRICTLY LPTV AND LTI CONTROLLERS From Theorem 4.1 and Corollary 4.1 it can be seen that under the perturbations A, the stability margins of F for strictly LPTV K and LTI K LTI are given by 1= sup! [ ^S(e j! )] and 1= sup! [ ^S LTI(e j! )], respectively. In order to compare sup! [ ^S(e j! )] and sup! [ ^S LTI(e j! )] attainable, respectively, by the strictly LPTV controllers and the optimal H 1 LTI controller, a result on the optimal H 1 LTI controller is introduced in the lemma below, its proof can be found in [5, Chs. 9 and 11]. Lemma 5.1: Subject to the assumption A1, there exists an optimal LTI controller K 3 LTI for the plant P such that the nominal closed loop system S 3 LTI resulting from K 3 LTI satisfies ks 3 LTIk = infk ksk and that the transfer function SLTI(e 3 j! ) of S 3 LTI is all pass, i.e., jslti(e 3 j! )j = ksltik1 3 = ks 3 LTIk = const; 8! [0; ). Employing a Youla parametrization of stabilizing controllers [17], there exists a stable ^Q such that ^K =(^Y 0 ^M ^Q)( ^X 0 ^V ^Q) 01 and ^S = ^S (1) 0 ^S () ^Q ^S (3), where ^V; ^M; ^X; ^Y; ^S (1) ; ^S () and ^S (3) are transfer matrices dependent only on the lifted nominal plant ^P [19]. According to Lemma 3.1, if K is strictly LPTV then ^K and hence ^Q must have nonzero off-diagonals, and ^Q is in the form ^Q(e j! ) = [Q k (e j(!0l! ) )] k;l=0;1;111;. Consequently ^S can be written as ^S(e j! ) = [S k (e j(!0l! ) )] k;l=0;1;111; with S k (e j! )=S (1) (e j! ) 0 S () (e j! )Q 0(e j! )S (3) (e j! ) for k =0and S k (e j! )=S () (e j! )Q k (e j! )S (3) (e j! ) for k 6= 0. Let ^Q 0 and ^S 0 be the diagonal matrices consisting of the diagonals of ^Q and ^S, respectively. From Lemma 3.1-d) it is known that ^Q 0 and ^S 0 give rise to two LTI systems Q 0 and S 0, respectively. The following lemma gives an important property of S 0 in relation to Q 0, its proof can be found in [19, Theorem 5.1]. Lemma 5.: Suppose that ^K = (^Y 0 ^M ^Q)( ^X 0 ^V ^Q) 01 is the lifted frequency transfer function of a real, causal, finite order, strictly LPTV controller K, which stabilizes the closed loop system ^S = ^S (1) 0 ^S () ^Q ^S (3). Then ^K 0 = (^Y 0 ^M ^Q 0 )( ^X 0 ^V ^Q 0 ) 01 gives a real, causal, finite order LTI controller K 0. When applied to the plant P; K 0 gives rise to the closed loop system S 0, which is stable and LTI and whose lifted transfer function is ^S 0. Let ^S LTI(e 3 j! ) be the lifted transfer function of SLTI(e 3 j! ). The theorem below presents the comparison between sup [ ^S(e j! )] and sup [ ^S LTI(e 3 j! )].

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY Theorem 5.1: Suppose the condition A1 holds and K is strictly LPTV with 1 < N 3. Then the nominal closed-loop system S resulting from any such K satisfies sup [ ^S(e j! )] > sup [ ^S LTI(e 3 j! )] (11) Proof: By assumption N 3, hence Lemma 4.1 applies to ^S(e j! ). Let F (!) be the scaling matrix as given in Lemma 4.1. Define ^H(e j! ):=F (!) ^S(e j! )F 01 (!) = H k e j(!+l! ) k;l=0;1; H k e j(!+l! ) f k(!) := f S k e j(!+l! ) l (!) It then follows from (10) that (1) : (13) sup [ ^S(e j! )] = sup [ ^H(e j! )] = k ^Hk1 (14) The proof of (11) is now equivalent to proving k ^Hk1 > k ^S 3 LTIk1. To show this, observe (1) and (13). It can be seen that ^H(e j! ) is in the form of (5) with diagonal ^H 0 (e j! )= ^S 0 (e j! ). Thus, by Lemma 3.1-d), k ^Hk1 1= sup jh k j kh 0 k1 = k ^H 0 k1 = k ^S 0 k1: (15) From Lemma 5., it is known that ^S 0(e j! ) represents an LTI closed loop system S 0 resulting from an LTI K 0. Then there are only two possibilities: S 0 6= S 3 LTI or S 0 = S 3 LTI. Now, suppose S 0 6= S 3 LTI. It follows from (15), (8), and the optimality of S 3 LTI, that k ^Hk1 k^s 0k1 > k ^S 3 LTIk1 (16) Conversely, suppose S 0 = S 3 LTI. By assumption, S is strictly LPTV. It then follows from Lemma 3.1-b) and -d) that there must exist k 6= 0 and! [0; ] such that S k (e j! ) 6= 0and hence H k (e j! ) 6= 0. Thus, from (15), H 0 (e j! )=S 0 (e j! ) and Lemma 5.1 khk1 sup H k (e j! ) = + sup Hk(e j! ) > k=1 = k ^S LTI(e 3 j! )k This proves the theorem. VI. EXAMPLE Consider the LTI plant P (z) =k(1 0 z 01 )=(1 0 3z 01 ). This is an example studied by many authors, e.g., [1], [7], [8]. It is shown in [1] that with LTI controller the gain margin := k max =k min =:5 for k>0. Whereas in [7] it is shown that with the LPTV controller (in time delay operator form) K(q 01 )=1+(01) t +[6+6(01) t ]q 01, the gain margin is infinite for k<05 and k>4. Now, take W = 1( ^W (z) = I) and assume k = k 0, a known nominal value. It is routine to show that the lifted transfer function of the nominal closed loop system is given by the equation at the bottom of the page. Let k 0 = 06. It is easy to calculate, e.g., using MATLAB -toolbox, that sup! [ ^S(e j! )] = 48 db. Using bilinear transform and the well-known procedure given in [5], it can readily be shown that the optimal H 1 controller for the plant is K 3 LTI = :5=1:5k 0 and the resulting all-pass and k 0 independent LTI closed loop system is SLTI(z) 3 =0:5(1 0 z 01 )=(1 0 0:5z 01 ) with sup![0;] [ ^S 3 0 ] = 14dB. Thus sup [ ^S] > sup [ S ^3 0 ]. This coincides with the analysis of Theorem 5.1. Indeed, the LPTV control K provides a very large gain margin, but the resultant [ ^S] is very large as well. This makes it very sensitive to LTI unstructured perturbations. When k 0 = 06, a perturbation 1 of the form A with r>0:004 will destabilize the system, while a destabilizing 1 for the optimal LTI K 3 LTI must have r>5. VII. CONCLUSIONS AND DISCUSSIONS The results of [19] have been extended to investigate the robustness of LPTV control system under LTI unstructured perturbations. A necessary and sufficient condition (7) for robust stability of an LPTV control system subject to LTI perturbations has been derived. This condition is less conservative than small gain condition and allows unstable LTI perturbations. It has been shown in [1] that N =is sufficient for strictly LPTV controllers to achieve much larger stability margin for the structured perturbations than that of LTI controllers. Whereas Theorem 5.1 and the presented example show that under such an N, the stability margin for the unstructured perturbations is deteriorated. Theorem 5.1 holds only for N 3 and does not exclude the possibility that for N > 3, strictly LPTV control might attain [ ^S] = [ ^S 3 LTI]. However, the analysis that leads to the theorem does demonstrate the great difficulty in design such kind of strictly LPTV controllers. As shown in Lemma 5. and the proof of Theorem 5.1, the ^S resulting from a strictly LPTV K always contains in its diagonals an LTI closed-loop system ^S 0; in order to achieve [ ^S] =[ ^S 3 LTI], the strictly LPTV K must at least give rise to an ^S, which satisfies [ ^S] =[ ^S 0 ] and ^S 0 is all pass. This amounts to a [ ^S] optimization problem with constraints [ ^S] = [ ^S 0] and ^S 0 being all pass. It is well known that when N > 3, the exact evaluation of [ ^S] becomes extremely difficult, see, e.g., [1, Ch. ], and the existence of the solution and the solvability for such optimization problem is completely unknown. Similar result to this note can also be drawn if 1 is an NLTV, stable operator with k1k r. In this case, the stability margins for strictly LPTV K and LTI K 3 LTI are given respectively by 1=k ^Sk1 and 1=k ^S 3 LTIk1. It has been shown in [19] that under only the assumption A1, k ^Sk1 > k ^S 3 LTIk1 for all N > 1. This result together with that of this note have provided a deeper understanding on the robustness of LPTV control systems under l norm bounded unstructured perturbations. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable suggestions that have improved the presentation. ^S(z) = k 0 (1 0 z 01 )(1 + 3z 01 )(1 + 6z 01 ) (1 0 z 01 )(1 + 3z 01 )(1 0 6z 01 ) 1+k 0 0 9z 0 (1 + z 01 )(1 0 3z 01 )(1 + 6z 01 ) (1 + z 01 )(1 0 3z 01 )(1 0 6z 01 )

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Furuta, Performance analysis of discrete periodically time varying controllers, Automatica, vol. 33, no. 4, pp , On the Optimal Unbiased Functional Filtering M. Darouach Abstract This paper presents a new and simple solution to the optimal unbiased reduced-order (or functional) filtering problem for linear timevarying systems. Necessary and sufficient conditions for the existence of the obtained filter are given. Stability and convergence conditions are developed for the time-invariant systems. Both continuous- and discrete-time cases are considered. Index Terms Convergence, reduced-order filter, Riccati equation, stability, unbiased filter. I. INTRODUCTION The problem of state estimation of a dynamical system is of great importance in the design of optimal control and fault diagnosis. It is well known from the Kalman Bucy filter theory that the order of the optimal filter is the same as that of the system. In many practical situations, however, one may be interested only in a partial state estimation [1]. The reduced-order Kalman filter was developed in [4] and [5]. The case where the measurements are partially noise-free was reported in [] and [3]. In [4] and [5] the problem of partial state estimation is considered where none of the measurements are assumed to be noise-free. The approach used in these works is based on the minimum variance unbiased estimator, it reduces the problem of functional filtering to the optimization under unbiased constraint one. The solution is then given under the condition rank [ C L ]=n. L is the matrix of the linear function of the state to be estimated, C is the measurement matrix, and n is the dimension of the system. In this paper we propose an alternative method to design an optimal unbiased functional filter without any assumption on the rank of matrix [ C L ]. Contrary to the work in reference [5] we do not need any transformation of the initial system. As in [4] the obtained filter has a standard Kalman filter form, a reduced-order innovation process is introduced, and the uniqueness of the filter is proved. Both continuous and discrete-time systems are considered, necessary and sufficient conditions for the existence are given. The convergence and the stability conditions are developed for the time-invariant cases. II. CONTINUOUS-TIME UNBIASED FUNCTIONAL FILTER Consider the linear time-varying system described by _x(t) =A(t)x(t) +w(t) y(t) =C(t)x(t)+v(t) z(t) =L(t)x(t) (1a) (1b) (1c) where x(t) n state vector, y(t) p measurement output, z(t) r vector to be estimated, with r n. The matrices A(t); C(t), and L(t) are real and of appropriate dimensions and without loss of generality, it is assumed that rank C(t) =p and rank L(t) = r. Let the initial state x 0 be a random vector with /00$ IEEE Manuscript received February 3, 1999; revised August 5, 1999 and February 7, 000. Recommended by Associate Editor, T. Parisini. The author is with CRAN, IUT de Longwy, Cosnes et Romain, France. Publisher Item Identifier S (00)