On Discrete-Time H Problem with a Strictly Proper Controller

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1 On Discrete-Time H Problem with a Strictly Proper Controller Leonid Mirkin Faculty of Mechanical Engineering Technion IIT Haifa 3000, Israel mersglm@txtechnionacil May 8, 1996 Abstract In this paper the standard discrete-time H control problem is considered under the constraint imposed upon the controller to be strictly proper The necessary and sufficient conditions for the existence of a γ-suboptimal controller are derived and the set of all LTI controllers, which solve the problem, is parametrized In contrast with the unconstrained case, the state-space solution obtained is transparent and actually is not more complex then the solution of the continuous-time H problem 1 Introduction This paper deals with a standard discrete-time H control problem: given a positive constant γ, find all stabilizing controllers which make the H norm of the closed-loop system less than γ Such problems have been extensively investigated for the last two decades in both continuous and discrete time, see the books Francis 1987, Stoorvogel 199, Başar & Bernhard 1995, Green & Limebeer 1995, Zhou et al 1995 and the references therein In this paper we are going to consider the H problem under the additional constraint that the controller must be strictly proper The motivation for such an assumption is twofold First, in most cases discrete-time controllers are designed for continuous-time systems, i e, the sampled-data systems interconnection considered, for instance, in Chen & Francis 1995 As recently shown in Mirkin & Rotstein 1997, sampled-data control problems can always be formulated as discrete-time problems with strictly causal controllers Hence, the assumption that the controller is strictly proper, does not lead to a loss of generality Second, although at a conceptual level the solutions to the continuous- and discrete-time H problems are closely connected, technical details of the later are much more involved Moreover, the controller formulae obtained for the discrete-time H problems in Iglesias & Glover 1991, Stoorvogel 199, Green & Limebeer 1995, Stoorvogel et al 1994 lack the transparency of their continuous-time counterparts Doyle et al 1989, Glover & Doyle 1989, thus making computational algorithms more cumbersome This fact has given rise to conclusions such as: International Journal of Control , no 5, pp Supported by The Center for Absorption in Science, Ministry of Immigrant Absorption State of Israel

2 z y P K w u Figure 1: General setup or even: perhaps the simplest way to do discrete-time H -optimal controller design is by converting to a continuous-time problem via bilinear transformation Chen & Francis 1995, Section 7, it is probably more effective to obtain the discrete solution by using a bilinear transformation Zhou et al 1995, Section 17 Also, and perhaps more important, the lack of a transparent solution makes any further analysis much harder; this is especially true when dealing with sampled-data H problems in the lifted domain, where the plant parameters are infinite dimensional Bamieh et al 1991, Mirkin & Palmor 1995 On the other hand, it has been observed that the formulae for the state-space solution of the H optimal control problem simplify considerably when the controller is assumed to be strictly proper Mirkin & Palmor 1995 Then, since the H and H problems are closely connected, it is natural to expect that similar simplification holds also for the H case The goal of this paper is to demonstrate that this is indeed true We consider the regular H problem in a general setting for the system depicted in Fig 1 The only assumptions about the parameters of the generalized plant P are: stabilizability of the plant by means of output feedback and full column and row ranks of the transfer matrices from u to z and from w to y, respectively, on the unit circle No restrictions are imposed on the feedthrough matrices Our approach to the H design for the case of the strictly proper controller is conceptually simple We start from the parametrization, via a free parameter Qz, of all proper suboptimal controllers using the formulae in Stoorvogel 199, Chapter 10 and then find whether there exists Qz such that the controller be strictly proper This yields an additional with respect to the unconstrained case condition for the solvability of the problem which, in turn, makes it possible to simplify the solution considerably In particular, the necessary and sufficient conditions for the existence of a solution of the problem involve a solvability test for two independent discrete algebraic Riccati equations and one coupling condition Thus, the result is in the style of the state-space solution of the continuous-time Doyle et al 1989, Glover & Doyle 1989 and discretetime Iglesias & Glover 1991 H unconstrained problems However the coupling condition in this paper is different from the coupling condition obtained there and involves not only the stabilizing solutions of the Riccati equations but also the parameters of the open-loop transfer matrix from w to z; this has a quite interesting interpretation in terms of a finite horizon H problem with nonzero initial conditions and a terminal state penalty As an additional advantage, the parametrization of all suboptimal strictly proper controllers becomes straightforward and easy to interpret, as in the case of continuous-time H control As far as the author knows, the only previous solution to the discrete-time H output feedback problem with strictly proper controller is the one in Başar & Bernhard 1995, Section 63, where a dynamic game approach is used However the assumptions about the generalized plant

3 3 there are more restrictive than the ones in this paper Moreover, the results in Başar & Bernhard 1995 do not include the parametrization of all suboptimal controllers which is presented here The paper is organized as follows In Section we pose the H suboptimal problem Section 3 contains the main result: the necessary and sufficient conditions for the existence of a γ-suboptimal strictly proper controller and the parametrization of all such controllers The rest of the paper is devoted to the proof of this result In Section 4 we show how the result for unconstrained controller can be adapted to our case In particular, the result of Stoorvogel 199 is summarized in Subsection 41 and modified for the case of strictly proper controller in Subsection 4 In principle, the discussion there contains a solution of the problem, but this solution is far from being transparent For example, the two Riccati equations there are not separate The separation of these equations is the subject of Section 5 Subsection 51 assembles some preliminary results about the connection between the stabilizing solution of the Riccati equation and an extended simplectic matrix pairs In Subsection 5 a new extended simplectic pair is introduced and used for the separation of two H Riccati equations Finally, some tedious but nevertheless necessary simplifications are provided in Section 6 The notation used throughout the paper is fairly standard For a real valued matrix M, M denotes the transpose, ρm denotes the spectral radius if M is square, σm = ρm M implies the maximum singular value The notation M 1/ is adopted for the square root of a matrix M = M Finally, the linear fractional transformation LFT of K on P like that shown in Fig 1 is denoted as F l P, K Problem statement Consider the discrete-time control system setup depicted in Fig 1, where P is a generalized plant, K is a controller, w is an exogenous input, z is a regulated output, y is a measured output, and u is a control input The state-space realization of the plant is taken to be of the form Pz = A B 1 B C 1 D 11 D 1 C D 1 D = We make the following assumptions: A B C 1 D 1 C D A B1 B = C D 1 D A1: The triple C, A, B is stabilizable and detectable; A: A3: A e jθ I B has full column rank θ 0, π; C 1 D 1 A e jθ I B 1 has full row rank θ 0, π C D 1 1 These assumption are the same as those of Stoorvogel 199 and actually imply that the H control problem to be posed is nonsingular It worth stressing that we do not impose any constraints upon the matrices D 1 and D 1 This is in contrast the work of Iglesias & Glover 1991, Başar & Bernhard 1995 and Green & Limebeer 1995, where D 1 and D 1 are assumed to be injective and surjective, respectively The discrete-time H control problem to be dealt with in this paper is the following:

4 4 OP H : Given the discrete-time LTI plant 1 such that A1 3 are satisfied and a number γ > 0 Find if such exists a strictly causal controller K which internally stabilizes P and provides F l P, K < γ The difference from Iglesias & Glover 1991, Stoorvogel 199, Green & Limebeer 1995 is that we will assume that the controller K is strictly proper This is motivated by the fact Mirkin & Rotstein 1997 that any sampled-data problem can be formulated as a discrete-time problem with a strictly causal controller even when the computational delays are negligible For example, consider the simplest feedback interconnection of an LTI continuous-time plant P c with the realization Ac B P c s = c C c 0 and a discrete-time controller K d, connected by the ideal sampler and the zero-order hold with the sampling period h If one is concerned only with the discrete-time performance, such a system can be treated as the pure discrete feedback interconnection of the discretized plant e P d z = A ch h 0 eacτ dτ B c C c 0 and K d However, since P d is inherently strictly proper 1 one can always rearrange the system as the feedback interconnection of the proper plant e A c h h 0 P d z = eacτ dτ B c C c e Ach h = z P C c 0 eacτ d z dτ B c and the strictly proper controller K d z = z 1 K d z 3 Main result In this section we will formulate the solution of OP H To this end we will need the following discrete-time algebraic Riccati equations DTAREs: X = A XA + C 1 C 1 D 1 C 1 + B XA GX 1 D 1 C 1 + B XA a and Y = AYA + B 1 B 1 B 1D 1 + AYC HY 1 B 1 D 1 + AYC, b where GX = D 11 D 11 γ I D 11 D 1 D 1 D 11 D 1 D 1 HY = D11 D 11 γ I D 11 D 1 D 1 D 11 D 1 D 1 + B 1 + B C1 C X B 1 B, 3a Y C 1 C 3b 1 P c has to be strictly proper to provide the boundedness of sampling operations Chen & Francis 1995

5 5 Define the matrices A F = A + BF and AL = A + LC, where F = GX 1 D 1 C 1 + B F1 XA =, L = B 1 D 1 + AYC HY 1 = L 1 L F We will say that a solution X of a is stabilizing if X = X, the matrix GX is invertible, and the matrix A F is Schur Similarly, a solution Y of b is stabilizing if Y = Y, HY is invertible, and A L is Schur With these definitions we are now in the position to formulate the main result of this paper: Theorem 1 The following statements are equivalent: i There exists a strictly proper controller K which solves OP H ii The DTAREs have stabilizing solutions X 0 and Y 0 such that X 1/ 0 A B1 Y 1/ 0 σ < γ 4 0 I C 1 D 11 0 I Given that the conditions of part ii hold, then the matrix Z = I γ YX is nonsingular and all rational internally stabilizing controllers K which solve OP H are given by K = F l GK, Q K, where G K z = A + BF + Z 1 L C + D F Z 1 L Z 1 B + LD F 0 I C + D F I 0 and Q K RH is arbitrary strictly proper transfer matrix such that F l MP, Q K z < γ, where the matrix M P is defined as follows: X 1/ 0 0 A B 1 B Y 1/ 0 0 M P = 0 I 0 C 1 D 11 D 1 0 I I C D 1 D 0 0 I Remark 31 The solvability test in Theorem 1 is based on two separate H Riccati equations It is in contrast to the solution by Stoorvogel 199, where two Riccati equations are coupled The solution in Iglesias & Glover 1991 is also based on two separate Riccati equations, yet the proof there is essentially based on the assumption that D 1 is injective and D 1 is surjective see also Walker 1990 Remark 3 It is also worth stressing that condition 4 is different from the coupling condition of Doyle et al 1989, Glover & Doyle 1989, Walker 1990, Iglesias & Glover 1991, which is ρxy < γ it will be shown in Section 6 that the latter condition is covered by 4 Condition 4 has an interesting interpretation in terms of finite-horizon H norm of the system P 11 z Denote by x the state vecor of P Then 4 is clearly equivalent to the condition x 1 Xx 1 + z 0 z 0 < γ x 0 Y x 0 + w 0 w 0, w0 R dimw, x 0 Im Y, where Y denotes the pseudoinversion of Y Moreover, it can be shown that the coupling condition for the H problem with k-delyed controller is as follows: x k Xx k 1 k + z i z k 1 i < γ x 0 Y x 0 + w i w i, w l 1k i=0 i=0

6 6 Moreover, for the sampled-data H problem with the sampling period h the condition will be: h h x hxxh + z zdt < γ x 0Y x0 + w wdt, w L 0, h 0 In all these cases the solvability conditions involve not only the bounds on the induced norm of the plant while it is open-loop which is obvious, but also the influence of Y via initial conditions and the influence of X via the terminal state penalty Remark 33 The parametrization of all γ-suboptimal controllers given in Theorem 1 involves the constraint on the free parameter Q K in terms of its LFT However it can easily be transformed to the following affine form: X 1/ 0 A B1 B + Q 0 I C 1 D 11 D K z Y C D 1/ 0 1 < γ 1 0 I by replacing G K z with the transfer matrix G K z D Remark 34 It is of interest to see what happens in the optimal, rather then the suboptimal, case As γ will approach to an optimal level, say γ, the condition 4 will typically be violated, while the DTAREs still have the stabilizing solutions and ρyx < γ Hence in the optimal case the transfer matrix G K given in Theorem 1 is typically well defined This is unlike the unconstrained case when I γ YX is often singular Glover & Doyle 1989 and G K can only be considered in the descriptor form 0 4 Proof: the first step For the sake of simplicity we will prove Theorem 1 for the case γ = 1 It can always be provided by an appropriate scaling of the plant parameters, for example γ 1 C 1, γ 1 D 1, γ X, and γl 1 Also, we will assume throughout this section that D = 0 This assumption also can always be made since the action of a controller K 0 on a plant P 0 is equivalent to the action of the controller K 1 = 0 I Fl I D, K0 on the plant P D Since for a strictly proper K0 the transfer matrix K 1 is well defined and K 1 is strictly proper iff so is K 0, the assumption D = 0 does not lead to loss of generality We will prove Theorem 1 using the solution of the discrete-time H problem with proper controller known in the literature We will start from the solution given in Stoorvogel 199, Chapter 10 and then adjust the γ-suboptimal controller and the conditions of its existence there to handle the case of strictly proper controller 41 Preliminary: unconstraint H problem To start with, denote G11 G GX = 1 G 1 G, where the partitioning is compatible with that in the right-hand side of 3a, assuming that G is nonsingular define X = G11 + G 1 G 1 G 1, That is not necessarily strictly proper

7 7 and introduce the following LTI system: AX B X C X D X = A X B X C 1,X D 1,X C,X D,X = A + B 1 F 1 B 1 G 1/ F G 1/ C + D 1 F 1 D 1 G 1 1/ X Now, the filtering H ARE associated with this system is Y X = A X Y X A X + B XB X B XD X + A XY X C X H XY X 1 B X D X + A XY X C X, 5 where H X Y X = D1,X D 1,X I D 1,XD,X D,X D 1,X D,X D,X H11,X H = 1,X, H 1,X H,X If H,X is nonsingular then C1,X + Y C X C 1,X C,X,X Y,X = H11,X + H 1,X H 1,X H 1,X Finally, define the matrix L X = BX D X + A XY X C X H XY X 1 = L 1,X L,X The following theorem is essentially from Stoorvogel 199, Chapter 10: Theorem If assumptions A1 3 are satisfied, then the following statements are equivalent: i A causal compensator K exists such that F l P, K < 1 and the closed-loop system is internally stable ii The AREs a and 5 have stabilizing solutions X 0 and Y X 0, respectively, such that the following conditions hold: a G > 0; b X > 0; c H,X > 0; d Y,X > 0 If X and Y X exist satisfying part ii, all controllers satisfying the requirements in part i are given by F l Gα, Q α, where Qα RH is such that Q α < 1 and where G α z = I 0 0 H 1/,X B K = L,X + B + L 1,X G 1/ DK, 1/ D K = G H 1,X H 1,X A F B K C,X B K B + L 1,X G 1/ F D K C,X D K I C,X I 0 I 0 0 G 1/ 1/, Y,X

8 8 4 Parametrization of all strictly proper controllers It is clear that the class of all controllers, which solve OP H, is contained in the controller parametrization given in Theorem Then a possible approach to solve OP H is to extract if such a problem is solvable the set of all strictly proper controllers from F l Gα, Q α In other words, one should find whether there exists a transfer matrix Q α such that Q α < 1 and F l Gα, Q α = 0 To this end note that F l Gα, Q α = D K G 1/ 1/ Y,X Q α H 1/,X Then the controller F l Gα, Q α is strictly proper if and only if Qα = D Q, where D Q = 1/ Y,X G1/ D KH 1/,X = 1/ Y,X H 1,XH 1/,X 6 On the other hand, according to the definition of the H norm for discrete-time systems Zhou et al 1995 the quantity σ Q α is the lower bound for Q α Hence, there exists Q α with the feedthrough term as above such that Q α < 1 if and only if σ 1/ Y,X H 1,XH 1/,X < 1 7 Thus, to adjust Theorem to the case of strictly proper controller one has only to add 7 to the conditions of part ii of Theorem Then the parametrization of all strictly proper controllers can be expressed as F l Gβ, Q β, where G β z = F l Gα z, D Q = I 0 0 H 1/,X A F + L,X C,X L,X B + L 1,X G 1/ F 0 I C,X I 0 I 0 0 G 1/ 1/ Y,X and Q β RH is arbitrary strictly proper transfer matrix such that D Q + Q β z < 1 In principle, 8 together with the constraint on Q β constitutes the complete parametrization of all strictly proper controllers which solve the OP H, while the inequality 7 combined with part ii of Theorem yields necessary and sufficient conditions for the existance of this solution However the condition 7 makes it possible to simplify the formulae considerably 8 5 Three Riccati equations We will start the simplifications with the finding a relationship between the Riccati equations 5 and To this end we will exploit the equivalence between DTARE and certain generalized eigenvalue problems 51 The Riccati operator In this subsection we assemble some results concerning the extended simplectic matrix pairs and their role in the solution of the discrete-time algebraic Riccati equations For more detailed discussion see Van Dooren 1981, Rotstein 199, Ionescu & Weiss 199 All the notation throughout this subsection is independent on the one in the other sections

9 9 Consider the following ordered pair of n + m n + m matrices: A 0 B I 0 0 M 1, M = Q I S, 0 A 0, 9 S 0 R 0 B 0 where A R n n, B R n m, Q = Q R n n, S R n m, and R = R R n n The pair 9 is called the extended simplectic pair ESP since the matrix pencil M 1 λm which is associated with this pair has the following properties: a If λ is a generalized eigenvalue of the pencil M 1 λm of multiplicity r then so is 1 λ ; b If λ = 0 is a generalized eigenvalue of M 1 λm of multiplicity r then λ = is a generalized eigenvalue of M 1 λm of multiplicity r + m An ESP is said to be dichotomic if the associated matrix pencil has no generalized eigenvalues on the unit circle From the properties above it follows that if the ESP 9 is dichotomic then the pencil M 1 λm must have n eigenvalues in D Consider the n-dimensional deflating subspace X M 1, M corresponding to eigenvalues in D It is obvious that X 1 X M 1, M = Im X, X 3 where X 1, X R n n, X 3 R m n, and X 1 X 1 M 1 X = M X M st, ρm st < 1 X 3 X 3 A dichotomic ESP is said to be disconjugate if the matrix X 1 is nonsingular For a disconjugate ESP we can always set X = X X 1 1 Since the matrix X is uniquely determined by M 1, M, we can define the function Ric : M 1, M X; thus X = RicM 1, M We will take the domain of Ric, denoted domric, to consist of all disconjugate ESP such that R + B XB is nonsingular The following lemma is essentially from Rotstein 199, Ionescu & Weiss 199: Lemma 1 Given the ESP 9, then the following two statements are equivalent: i M 1, M domric; ii The DTARE A XA X + Q S + A XBR + B XB 1 B XA + S = 0 10 has a unique stabilizing solution X = RicM 1, M Moreover, given M 1, M domric, then the stabilizing solution X of the DTARE 10 and the matrix F = R + B XB 1 B XA + S satisfy the following equality: I I M 1 X = M X A + BF F F Lemma 1 establishes the strong correspondence between the stabilizing solution of a nonlinear Riccati equation and a linear generalized eigenvalue problem Then it is not a surprise that the latter problem is more tractable

10 10 5 Yet another extended simplectic pair Conventionally, the relationship between DTARE 5 and is derived using similarity of the simplectic matrix pencils associated with 5 and b Walker 1990, Iglesias & Glover 1991 Unfortunately, such an approach appears to be impossible for the ESP-based treatment The difficulty is that the ESP associated with 5 and b have in general different dimensions, since so do the matrices D 11 D 11 and D 11,XD 11,X In this respect we propose to treat the H DTARE using ESP different from 9 Our reasoning will be based on the following two inequalities: I > D 11 D 11 + B 1 XB 1, I > D 1,X D 1,X 11a 11b Inequalities 11 can be extracted from 7 by simple algebra and thus are necessary for the OP H to be solvable Consider the ESP M 1Y, M Y which is associated with 5, where M 1Y = M Y = A B 1 B 1 0 I C 1 B 1 D 11 C B 1 D 1 D 11 B 1 0 D 1 B 1 0 D 11 D 11 I D 1 D 11 D 11D 1 D 1 D 1 I A C C 0 0 Since 11a implies that the matrices Ω = I D 11 D 11 1 and Υ = I D 11 D 11 1 are well defined, one can get that U Y M 1Y P 3,4 = U Y M Y P 3,4 =, A + B 1 ΥD 11 C 1 0 C + D 1 ΥD 11 C 1 0 B 1 ΥB 1 I B 1 ΥD 1 0 D 1 ΥB 1 0 D 1 ΥD 1 0 D 11 B 1 0 D 11 D 1 Ω 1 I C 1 ΩC A + B 1 ΥD 11 C C + D 1 ΥD 11 C C where P 3,4 is the permutation matrix for 3-rd and 4-th block columns and U Y = I 0 C 1 Ω 0 0 I B 1 D 11 Ω D 1 D 11 Ω I 0 0 I 0 Since both U Y and P 3,4 are nonsingular and the pair U Y M 1Y P 3,4, U Y M Y P 3,4 is block lower triangular, it follows that M 1Y, M Y domric iff M 1Y, M Y domric and Ric M 1Y, M Y = RicM 1Y, M Y,,,

11 11 where M 1Y, M Y = where AΥ C Υ = A C A Υ 0 C Υ B 1 ΥB 1 I B 1 ΥD 1, D 1 ΥB 1 0 D 1 ΥD 1 B1 + ΥD 11 D C 1 1 Thus, we can formulate the following lemma: I C 1 ΩC A Υ 0 0 C Υ 0 Lemma Given D 11 D 11 < I then the following two statements are equivalent: i M 1Y, M Y domric; ii The DTARE b has a stabilizing solution Y = Ric M 1Y, M Y Moreover, M 1Y I Y = M Y L I Y A L L and L 1 = A L YC 1 + L D 1 D 11 + B 1D 11 Ω, 1 It is clear that the inequality 11b makes it possible to prove that Y X = Ric M 1Y,X, M Y,X, where the ESP M 1Y,X, M Y,X can be constructed from the equation 5 by analogy with the construction of M 1Y, M Y from b The benefit in using ESP like 1 for the characterizing the stabilizing solution of H DTARE stems from the fact that now the ESP associated with the equations 5 and b have equal dimensions Then we have the following Lemma 3 Define the matrices I A Υ X 0 Ũ X = 0 I B 1 ΥB 1 X 0, 0 D 1 ΥB 1 X I I X 0 Ṽ X = 0 I I Then, if the inequalities 11 are satisfied the matrices ŨX and ṼX are nonsingular and M 1Y = ŨX M Y,X Ṽ X and MY = ŨX M Y,X Ṽ X Proof It is clear that ṼX is nonsingular and ŨX is nonsingular iff so is the matrix I B 1 ΥB 1 X Then the first claim of the lemma follows directly from 11a By simple algebra one can verify that C 1,X Ω XC 1,X = F GXF D 11 C 1 + B 1 XA G 1 11 D 11 C 1 + B 1 XA, AΥ,X A B1 = G 1 11 D 11 C 1 + B 1 XA C Υ,X C D 1

12 1 Now, using the equality G 1 11 = Υ ΥB 1 XI B 1ΥB 1 X 1 B 1 Υ and rewriting a in the form F GXF = A XA X + C 1 C 1 one can get: C 1,X Ω XC 1,X = C 1 ΩC 1 + A Υ XA Υ,X, A Υ,X = I B 1 ΥB 1 X 1 A Υ, C Υ,X = C Υ + D 1 ΥB 1 XA Υ,X, from which the equality M Y = ŨX M Y,X Ṽ X follows immediately Finally, the equality M 1Y = ŨX M 1Y,X Ṽ X can be obtained from the formulae BX Υ D X B X D,X B1 = G 1,X D 11 B 1 D 1, 1 which can be checked by a direct substitution From Lemma 3 it follows that M 1Y = M Y I Y L I Y L A L M I XY 1Y,X Y L = M I XY Y,X Y A L L Hence, the pair M 1Y,X, M Y,X is disconjugate iff the matrix I XY is nonsingular, and then we have that I I Y X L,X = YI XY 1 L I XY 1 13 Therefore, the relationship between the Riccati equations and 5 is as follows: Lemma 4 Given the DTARE a has the stabilizing solution X 0 such that 11a holds Then the DTARE 5 has the stabilizing solution Y X 0 if and only if the following conditions hold: a The DTARE b has the stabilizing solution Y 0; b ρxy < 1 Moreover, given that the conditions above are satisfied, then Y X = Z 1 Y, L,X = Z 1 L, 14a 14b L 1,X = Z 1 YXB + L 1 D 1 G 1/, 14c where Z = I YX Proof The first part of the lemma together with the formulae for Y X and L,X can be derived from 13 using the arguments like in Doyle et al 1989, Walker 1990, Iglesias & Glover 1991 Thus, we only should prove 14c From Lemma we have: L 1,X = A L,X Y X C 1,X + L,XD,X D 1,X + B XD 1,X Ω X = Z 1 A L YF + L D 1 + ZB 1 1 X G 1G 1 1/ G Ω X = Z 1 A L YF + L D 1 + ZB 1 1 X G 1G 1 G G 1 G 1 11 G 1G 1/ = Z 1 A L YC 1 D 1 + A XB + A L YC 1 D 1/ 11 + L D 1 + B 1 ΥG 1 G,

13 13 where the latter equality is obtained using the relations F G G 1 G 1 11 G 1 = C 1 D 11 + A XB 1 G 1 11 G 1 C 1 D 1 + A XB, 1 X G 1G 1 G G 1 G 1 11 G 1 = G 1 11 G 1, and since A L YA = Y B 1 + LD 1 B 1 A L YC 1 D 11 + A XB 1 + L D 1 + ZB 1 = A L YC 1 D 11 + B 1 + L D 1 ΥG 11 Finally, denoting L α = ZL1,X G 1/ we have: L α = A L YC 1 D 1 + A XB + A L YC 1 D 11 + L D 1 + B 1 ΥG 1 = A L YC 1 D1 ΩD 11 G 1 B 1 XB + B 1 + L D 1 ΥG 1 B 1 XB + YXB = A L YC 1 + B 1 + L D 1 D 11 ΩD1 + YXB = L 1 D 1 + YXB, which completes the proof Thus, Lemma 4 enables to formulate the solution of OP H in terms of two independent Riccati equations and the coupling condition ρxy < 1, like in Walker 1990, Iglesias & Glover 1991 However, in opposite to Walker 1990, Iglesias & Glover 1991 we do not make any restrictive assumptions on the matrices D 1 and D 1 It is worth stressing that in the proof of this result the conditions 11 may be replaced with milder conditions deti D 11 D 11 0, deti B 1 I D 11 D 11 1 B 1 X 0, deti D 1,X D 1,X 0, for which the strict properness of the controller is not necessary Also, the results of Lemma 4 make it possible to simplify the conditions of Theorem and 7 This is the subject of the next section 6 Further simplifications Recall from Subsection 4 that if γ = 1 and D = 0 then all controllers solving OP H can be parametrized as F l Gβ, Q β, where Gβ is as in 8, Q β is an arbitrary strictly proper transfer matrix from RH such that D Q Q β < 1 and D Q is as in 6 On the other hand, it is clear that F l Gβ, Q β = Fl Gγ, Q γ, where G γ z = A F + L,X C,X L,X B + L 1,X G 1/ F 0 I C,X I 0 Q γ z = G 1/ 1/ Y,X Q βz H 1/,X Then the constraint upon Q γ can easily be derived as follows: D Q + C Q Q γ zb Q < 1, 15,

14 14 1/ where B Q = H,X and C Q = 1/ Y,X G1/ Our first goal in this section is to simplify the inequality 15 using the results of the previous section To this end introduce the notation: X MA M B1 M 1/ A X 1/ B B = 1 X 1/ B, C 1 D 11 D 1 M11 = MA Y 1/ M B1 M C C Y 1/ D 1 Then the following lemma can be formulated: Lemma 5 Given any Q γ RH, then the inequality 15 holds if and only if M 11 + M B Q γ z M C < 1 A B Proof It is known Zhou et al 1995, Lemma 115 that given any RH such that C D σd < 1, then A B C D < 1 A + BD I DD 1 C BI D D 1/ I DD 1/ C 0 Therefore, in order to prove the lemma one just needs to prove that σd Q < 1 σm 11 < 1 16a and since I M 11 M 11 1 = I + M 11 I M 11 M 11 1 M 11 Ξ = BQ B Q = + BQ D Q M B M B + C Q To this end introduce the following notation: Ψ = I 0 F 1 Y 1/, I Φ = I Y 1/ XY 1/ 0 0 X Θ = Y 1/ F G 1 G 1 I DQ D Q 1 DQ B Q C Q < 1 M C M B M 11 I M 11 M 11 1 M C M 11 M B 16b, Then rewriting a as X + F GXF = M A M A and noting that according to the definition G 11 F 1 + G 1 F = M B 1 M A we get: Ψ 1 Φ Θ G Θ Ψ 1 I Y = 1/ X + F GXFY 1/ Y 1/ F 1 G 11 + F G 1 G 11 F 1 + G 1 F Y 1/ G 11 = I M 11 M 11, 17

15 15 while taking into account that G 1 F 1 + G F = M B M A : G Θ Ψ 1 = G 1 F 1 + G F Y 1/ G 1 = M B M On the other hand, since Y X = Y 1/ I Y 1/ XY 1/ 1 Y 1/, we have: H 11,X = I + G 1/ Θ Φ 1 ΘG 1/, H 1,X = M C Ψ Φ 1 Θ G 1/, H,X = M C Ψ Φ 1 Ψ M C From 6 it follows that I D Q D Q = 1/ Y,X H 11,X 1/ Y,X Therefore, σd Q < 1 iff H 11,X < 0 The latter inequality is equivalent to G 1 Θ Φ 1 Θ > 0 that in turn holds iff Φ Θ G Θ > 0 Hence, by 17 the relation 16a holds Now consider 16b block by block We have: Ξ 11 = H,X H 1,X H 1 11,X H 1,X = M C Ψ Φ 1 + Φ 1 ΘG 1 Θ Φ 1 Θ 1 Θ Φ 1 Ψ M C = M C ΨΦ Θ G Θ 1 Ψ M C = M C I M 11 M 11 1 M C by 17 Ξ 1 = H 1,X H 1 11,X G1/ = M C ΨΦ 1 Θ G + G Θ Φ Θ G Θ 1 Θ G = M C ΨΦ Θ G Θ 1 Θ G = M C I M 11 M 11 1 M 11 M B by 17 and 18 Ξ = G 1/ H 1 11,X G1/ = G + G Θ Φ Θ G Θ 1 Θ G = M B M B + M B M 11 I M 11 M 11 1 M 11 M B by 17 and 18 This completes the proof Lemma 5 actually provides the controller formulae given in Theorem 1 They should only be adjusted to the case of D 0 as it was discussed in the beginning of Section 4 Moreover, the result of Lemma 5 implies that 7 is equivalent to 4 Now, it is easy to see that condition b of Theorem is guaranteed by 4 Also, 4 together with the fact that X 0 is the stabilizing solution of a provides condition a of Theorem It follows from the proof of Lemma 5 that if ρyx < 1 and X > 0 then 4 is equivalent to the condition H 11,X < 0 and ker H,X = ker M C Hence, the inequality 4 also guarantees that conditions c and d of Theorem hold Thus, in order to complete the proof of Theorem 1 one has only to show that given 4 holds then the inequality ρyx < 1 is redundant Lemma 6 If σm 11 < 1 and M B M B > 0 then ρyx < 1

16 16 Proof It is easy to show that the inequality ρyx < 1 is equivalent to I Y 1/ XY 1/ > 0 Hence we will prove the latter inequality To this end rewrite the Riccati equation a as follows: X = M A M A M A MB1 M B GX 1 M B1 M A M B and denote Ω 1 = I M B1 M B1 1 and Υ 1 = I MB1 M B 1 1 Then it is obvious that I GX 1 Ω1 M = B 1 M B Ω1 0 I 0 0 I 0 M B Υ 1 M B 1 M B M B1 Ω 1 I Hence, X = M A Υ 1M A M A Υ 1M B M B Υ 1 M B 1 M B Υ 1 M A This equation guarantees that Note, that I Y 1/ XY 1/ I Y 1/ M A Υ 1M A Y 1/ I Y 1/ M A Υ 1M A Y 1/ > 0 Υ 1/ 1 Υ 1 1 M A YM A Υ1/ 1 > 0 Now the proof is completed noting that Υ 1 1 M A YM A = I M 11M 11 7 Concluding remarks In the paper the discrete-time H problem has been considered under the assumption that the controller is strictly proper The necessary and sufficient conditions for the problem to be solvable have been derived The conditions consist of the solvability condition for two independent discrete-time algebraic Riccati equations and the coupling condition 4 Given that the problem is solvable, the set off all strictly proper stabilizing controllers which provide the desired performance level has been parametrized The formulae obtained are considerable more transparent then ones for the case of proper controller The author believes that the result derived in this paper may be very useful tool in solving various control problems with the H performance measure For example, and it has been the main motivation for studying the problem, the result of Theorem 1 combined with the technique of Mirkin & Palmor 1995 prompts the straightforward way to the H design of sampled-data controllers in the lifted domain Hence, the multistep procedure connected with the convertion of the problem to an equivalent finite dimensional one may be avoided and a closed-form solution in terms of parameters of the continuous-time plant may be obtained Another application of the parametrization obtained in this paper can be for multirate sampled-data control systems The control problems for such systems can be reduced to LTI problems with the so-called causality constraints on the feedthrough matrix of the controller Meyer 1990 Conventionaly, H problems for multirate systems are solved in the frequency domain using the Youla parametrization with the sequel treatment of a constrained modelmatching problem Voulgaris et al 1994, Chen & Qiu 1994 This procedure involves several intermediate steps and no closed form state-space formulae are available An alternative approach may be as follows: start with the solution of the single-rate problem for a least common sampling period, lift both G K and Q K to, say, G K and Q K, and then find whether there exists Q K such that the feedthrough term of the controller satisfies a causality constraint Due to the sampled-data nature of the problem, one can always assume the controller on the first stage above to be strictly proper Hence, the closed form solution can be calculated in an effective manner using the controller formulae of this paper

17 17 Asknowledgement Several discussions with Dr Héctor Rotstein as well as his scepticism are greately appreciated References Bamieh, B, Pearson, J B, Francis, B A & Tannenbaum, A 1991 A lifting technique for linear periodic systems with applications to sampled-data control, Systems & Control Letters 17: Başar, T & Bernhard, P 1995 H -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, nd edn, Birkhäuser, Boston Chen, T & Francis, B A 1995 Optimal Sampled-Data Control Systems, Springer-Verlag, London Chen, T & Qiu, L 1994 H design of general multirate sampled-data control systems, Automatica 307: Doyle, J C, Glover, K, Khargonekar, P P & Francis, B A 1989 State-space solutions to standard H and H control problems, IEEE Transactions on Automatic Control 348: Francis, B A 1987 A Course in H Theory, Vol 88 of Lecture Notes in Control and Information Sciences, Springer-Verlag, NY Glover, K & Doyle, J C 1989 A state-space approach to H optimal control, in H Nijmeijer & J M Schumacher eds, Three Decades of Mathematical Systems Theory: A Collection of Surveys at the Occasion of the 50th Birthday of J C Willems, Vol 135 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, pp Green, M & Limebeer, D J N 1995 Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ Iglesias, P A & Glover, K 1991 State space approach to discrete-time H control, International Journal of Control 545: Ionescu, V & Weiss, M 199 On computing the stabilizing solution of the discrete-time Riccati equation, Linear Algebra and its Applications 174: 9 38 Meyer, D G 1990 A parametrization of stabilizing controllers for multirate sampled-data systems, IEEE Transactions on Automatic Control 35: Mirkin, L & Palmor, Z J 1995 A new representation of lifted systems with applications, Technical Report TME 439, Faculty of Mechanical Engineering, Technion Israel Institute of Technology Mirkin, L & Rotstein, H 1997 On the characterization of sampled-data controllers in the lifted domain, Systems & Control Letters 95: Rotstein, H P 199 Constrained H -Optimization for Discrete-Time Control, PhD thesis, California Institute of Technology, Pasadena, CA Stoorvogel, A A 199 The H Control Problem: A State Space Approach, Prentice-Hall, London Stoorvogel, A A, Saberi, A & Chen, B M 1994 The discrete-time H control problem with measurement feedback, International Journal of Robust and Nonlinear Control 44: Van Dooren, P 1981 A generalized eigenvalue approach for solving Riccati equations, SIAM Journal on Scientific and Statistical Computing : Voulgaris, P G, Dahleh, M A & Valavani, L S 1994 H and H optimal controllers for periodic and multirate systems, Automatica 30: Walker, D J 1990 Relationship between three discrete-time H algebraic Riccati equation solutions, International Journal of Control 54: Zhou, K, Doyle, J C & Glover, K 1995 Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ