- Email: [email protected]
Optimal robust disturbance attenuation for linear time-varying systems
Systems & Control Letters 46 (2002) 353 – 359
www.elsevier.com/locate/sysconle
Optimal robust disturbance attenuation for linear time-varying systems Avraham Feintuch Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer-Sheva, Israel Received 24 April 2000; received in revised form 2 March 2002
Abstract The time-varying optimal robust disturbance attenuation problem is considered .rst as two-block problem and then as a c 2002 Elsevier Science B.V. All rights reserved. two-disk problem. Keywords: Linear time-varying ordap; Two-disk problem
1. Introduction
general problem reduces to computing
The optimal robust disturbance alternation problem (ORDAP) was formulated for time-invariant systems by George Zames [13] and was studied by him and his students Owen and Djouadi, .rst in their doctoral theses [3,9] and then in a series of papers [4,10,11]. A stable uncertain plant P, lying in a given uncertainty set B(P0 ; V ) is subject to disturbances at the output. The objective is to choose a feedback control law C which provides the best uniform attenuation of uncertain output disturbances in spite of uncertainty in the plant model (Figs. 1 and 2). As opposed to the standard two-block problem of H ∞ design where a genuine worst-case minimum is produced only under the simplifying assumption that the disturbances due to output and sensor noises are mutually orthogonal, ORDAP assumes that this assumption is violated. As was shown in [7], this more
0 = inf∞ {|W (I − PQ)| + |VPQ|∞ };
E-mail address: [email protected] (A. Feintuch).
Q∈H
where P ∈ H ∞ is a given plant and W; V are outer functions. As was shown [3,9], the two-disk problem described by the diagram is a special case of ORDAP. Here the plant P0 is assumed to be known and external disturbances d1 ; d2 are uncertain in the sense that d1 ∈ {Wu: u 6 1}; d2 ∈ {Vu: u 6 1}; the images of the unit discs in H 2 under multiplication by given outer functions W; V . This two-disk problem has been considered in [5,7]. Also, certain types of optimal robust model-matching problems can be put within this framework [3,7]. Because there is no assumption of orthogonality, the standard methods in H ∞ control of reducing the problem to a Nehari distance problem do not apply. Instead classical optimization methods in Banach space, which go back to the classical Hahn–Banach theorem, are used. These require the use of dual spaces and thus the appropriate dual spaces must be identi.ed. In the time-invariant space this was done by Owen [9].
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/02/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 1 4 9 - 4
354
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
to the subspace W W P0 S = P0 T : T ∈ S : V V Of course this distance will depend on the metric considered on this space of operators whose domain is H and range is in f H ×H = : f; g ∈ H : g
Fig. 1.
Djouadi considered in his thesis the time-varying analogue of ORDAP and gave a very elegant operator theoretic formulation of this problem. Let S denote the algebra of stable linear systems (see [6]) acting as operators on a Hilbert space with an appropriate time structure associated with it. The systems W; V; P0 ∈ S are given and ORDAP reduces to a distance problem in the space M2×1 (S) of 2 × 1 matrices with entries in S: Find the distance from W O
Fig. 2.
We consider this problem for two diIerent metrics. The .rst is the operator norm induced by the Hilbert space norms on H and H × H . This was considered by Kwakernaak [8] in the time invariant case. Here the basic idea is standard. By a simple trick we obtain a 2-block Arveson distance problem, which we solve as in [6]. The second is the operator norm induced by considering H × H with the Banach space norm f = f + g; g 1
where · denotes the Hilbert space norm on H . As was pointed by Djouadi [3], this is the time varying analogue of the norm considered by Zames–Owen [10,11] for the two-disk problem. This situation is much more diLcult. In this case the appropriate dual spaces for certain Banach spaces of operators have to be identi.ed. The results used here are classical and were for the most part already
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
355
pointed out by Djouadi [3]. We will list them with the appropriate references. As has been pointed by Francis [7] these norms are in fact topologically equivalent. However, Zames and Owen have already remarked that in problems involving plant uncertainty, a small change in norm can drastically change the solution. More recent work on a related synthesis problem in the time invariant case can be found in [1] and for time varying systems in [12]. In the last part of this paper we will compare the distances obtained and will comment on where diIerences arise.
is inner, it is a partial isometry whose initial projections U ∗ U commutes with Pt 0 0 Pt
2. Preliminaries
In this section we consider the time varying analogue of the classical H ∞ problem where it is assumed that disturbances due to output and sensor noises are mutually orthogonal. This is the problem of computing
The input–output space is a Hilbert space H of vector functions L2 [0; ∞; C n ] or vector sequences l2 [0; ∞; C n ]. The truncation projections on these spaces will be denoted by {Pt : t ∈ } where is the appropriate index set. The algebra of stable systems S is the algebra of bounded linear operators {T : Pt T = Pt TPt ; t ∈ }; the algebra of casual operators. This algebra is a nest algebra as is T11 T12 : Tij ∈ S M2×2 (S) = T21 T22 (see [6]). In the discrete time case all operators in S and M2×2 (S) have inner–outer factorizations [6, p. 60 – 62]. In continuous time this is not always so but we will assume such factorizations without getting into conditions required for their existance. We will make use of the fact that for T ∈ M2×2 (S) of the form T1 0 T2 0 its inner–outer factorization is of the special form B 0 U1 0 T= U2 0 0 0 [6, p. 61– 62]. Because U1 0 U= U2 0
for each t [6]. Also, it is standard to assume that the outer operator B from S satis.es that BS = {BQ: Q ∈ S} is dense in S. This holds in particular if B is invertible. All these issues are discussed in detail in [6, Chapter 4].
3. The Hilbert space operator norm
inf {|W (I − PQ)|2 + |VPQ|2 }1=2 ∞
Q∈H ∞
and was studied by Kwakernaak [8]. In the time varying case the H ∞ -norm is replaced by the Hilbert space operator norm and we are required to .nd W W 0 = inf − P0 Q : Q ∈ S : 0 V We begin by noting that W W − P0 Q 0 V is the same as W 0 WP0 Q − 0 0 VP0 Q Let T=
WP0
0
VP0
0
0 0
:
∈ M2×2 (S)
and consider its inner–outer factorization U1 0 B 0 T= : U2 0 0 0
356
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
We can rewrite W W − P0 Q 0 V as W 0
0 0
−
U1 U2
0 0
BQ 0
0 0
:
Now we use a standard trick [6, p. 129]. Since U1 U= U2 is an isometry, so is U∗ : I − UU ∗ Thus, W U1 − BQ 0 U2 U∗ W U1 = − BQ ∗ I − UU 0 U2 ∗ ∗ U1 W − BQ = (I − U1 U1∗ )W : −U2 U1∗ W Under the standard assumption that B is invertible, we obtain the two-block problem: .nd U1∗ W − R 0 = inf (I − U1 U1∗ )W : R ∈ S : −U2 U1∗ W By [6, p. 129] this is just Pt U1∗ W (I − Pt ) ∗ sup (I − U1 U1 )W (I − Pt ) : t −U2 U1∗ W (I − Pt ) Since this is the induced Hilbert space operator norm, we obtain 02 6 sup{Pt U1∗ W (I − Pt )2 } t
2 (I − U U ∗ )W 1 1 + sup ) (I − P : t U2 U1∗ W t
Since for an A ∈ B(H ); A(I − Pt ) is a decreasing function of t, this is just (I − U U ∗ )W 2 1 1 ∗ 2 sup{Pt U1 W (I − Pt ) } + : U2 U1∗ W t Remark. (1) It follows from a standard weak compactness argument (see [6; p. 132]) that the in.mum is in fact attained. Thus; a controller C which gives the best uniform attenuation of the output disturbances in this sense; exists. As is well known; this sort of analysis does not lead to any constructive methods for .nding C. (2) We consider the all-pass case. In our framework this means that U ∗ is also an isometry and therefore that W ∗ W − BQ − UBQ = U 0 0 = U1∗ W − BQ: In this case we obtain 0 = inf {U1∗ W − BQ: Q ∈ S} = sup{Pt U1∗ W (I − Pt )}: t
4. The Banach space operator norm Let L denote the Banach space H × H with norm given by f = f + g; g 1
the sum of the Hilbert space norms of the co-ordinates. We wish to .nd the distance from the given operator W 0 in M2×1 (S) to the closed subspace U1 S U2 of M2×1 (S) in the operator norm induced by the norm of L. The approach used here follows [3,9,10]. We
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
will make use of the following standard facts: 1. L is a reNexive Banach space. Its dual L∗ = X is the same set of vectors H × H with the dual norm f = max{f; g}: g
This is essentially the same as 1, with C1 playing the role of l1 and B(H; L) of l∞ . This analogy is standard [2, Chapter 1]. 5. We make use of a standard consequence of the Hahn–Banach theorem. If B is a Banach space, M is a subspace of B and x∗ ∈ B∗ , then
∞
d(x∗ ; M ⊥ ) = sup{|x∗ (x)|: x ∈ M; x 6 1}:
This is the two-dimensional case of the fact that the dual of l1 is l∞ . The action of the linear functionals is described as follows: for f u ∈ L; ∈ X; g v u f = (u; f) + (v; g): v g The action of linear functionals from L on X is identical. 2. The space of bounded linear operators on H; B(H ), is the dual of the space of trace-class operators C1 [2, p. 11]. A acts on C1 as follows: T → Tr(AT ). The standard norm on C1 will be denoted by T 1 . 3. The pre-annihilator of S as a subspace of B(H ) is a = {A ∈ C1 ∩ S: A is strictly lower triangular} [2, p. 211]. 4. If B(H; L) denotes the bounded linear operators from H to L, then this is the dual space of the space T1 : Ti ∈ C1 : !(H; X ) = T2 The norm on !(H; X ) is given by T 1 = max{T1 1 ; T2 1 } T2 and R=
R1 ∈ B(H; L) R2
= Tr(T1∗ R1
In order to .nd the distance from W U1 to S 0 U2 in B(H; L), we will relate to U1 S U2 as the annihilator M ⊥ of some subspace M ⊂ !(H; X ). Similar results have appeared in [3,9,10]. Lemma 1. Let P = I − UU ∗ ; the orthogonal projection in H ⊕ H (Hilbert space orthogonal direct sum). Then U1 U1 ⊥ S = A: A ∈ a + P!(H; x): U2 U2 Proof. We .rst show that each of two subspaces U1 a and P!(H; x) U2 are in ⊥
If T=
acts on !(H; X ) as a linear functional as T1 R1 = Tr(R∗1 T1 + R∗2 T2 ) R2 T2 +
T2∗ R2 ):
357
U1
U2
S
U1
:
A ;
U2
then for S ∈ S Tr S ∗ [U1∗
U 1 A = Tr(S ∗ A) = 0 U2∗ ] U2
358
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
since S ∈ a⊥ . If A1 A1 T =P with ∈ !(H; X ); A2 A2
By weakstar compactness [2, p. 4] this supremum is in fact a maximum and thus the sought controllers in fact exist.
then since
Remark. (1) Owen and Zames consider the all pass case in [10]. As was pointed out in Remark 2 at the end of Section 3; in this framework all-pass means that U ∗ is also an isometry. In this case P = 0; the second direct summand disappears; and we obtain sup{|Tr(W ∗ U1 A)|: A ∈ a; A1 6 1}: Using the argument of [2; Corollary 16.8; p. 211] this reduces to supt Pt U1∗ W (I − Pt ) which is exactly what is obtained in the all-pass case in the two-block problem. The time-invariant case is more complicated. (2) In fact, also in the general case, the distance formula obtained here is closely related to that obtained in the two-block problem. We have seen this for the .rst direct summand. If we consider A1 ∗ sup Tr([W 0])P A2
[U1∗
U2∗ ]P
= [U1∗
U2∗ ]
I−
U1 [U1∗ U2
U2∗ ]
= 0; we have Tr(S ∗ [U1∗ U2∗ ]P) = 0 for all S ∈ S. For the opposite inclusion take B1 ∈ !(H; X ) B2 such that [U1∗
Then U1 U2 and B1 B2
U2∗ ]
B1
= U1∗ B1 + U2∗ B2 = A ∈ a:
B2
[U1∗ =
U2∗ ]
U1 U2
B1 B2
=
A+P
B1 B2
U1
U2
A
:
That the sum is in fact a closed direct sum follows from the fact that the ranges of U1 U2 and P are orthogonal in H ⊕ H . We can now compute W U1 S ˆ0 = d ; U2 0 in the norm of B(H; L). By Lemma 1 this is just sup{|Tr(W ∗ (U1 A + (I − U1 U1∗ )A1 − U1 U2∗ A2 ))|}, where the supremum is taken over the unit ball in the subspace U1 A: A ∈ a + P!(H; x): U2
over the unit ball in the subspace P!(H; X ), this is just [(I − U1 U1∗ ) + U2 U1∗ ]W . Of course, precise inequalities for the relationship between 0 and ˆ0 seem much more diLcult to obtain. (3) We complete this discussion with an estimate for ˆ0 which is easier to compute. As was seen, the supremum is computed over the unit ball U A1 1 A+P 61 ; U2 A2 1
where the norm is the appropriate trace class norm in !(H; X ): T 1 = max{T1 1 ; T2 2 }; T2 1
where Ti 1 is the C1 norm of the trace class operators Ti on H . If A1 U1 A+P ; T= U2 A2 then ∗
∗
T T =A A+
[A∗1
A∗2 ]P
A1 A2
:
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
Now T 1 = (T ∗ T )1=2 1 ¿ T ∗ T 1 : So if we compute the supremum over the larger set {T : A ∈ a; A1 ; A2 ∈ C1 with T ∗ T 1 6 1} we obtain a number ˆ ¿ ˆ0 . But 2 A A1 ; T ∗ T 1 = P A2 2
the square of the Hilbert–Schmidt norm of A A1 : P A2 This is a unit ball in a Hilbert space and the supremum is more transparent, at least in theory. References [1] R. D’Andrea, Generalized ‘2 synthesis, IEEE Trans. Automat. Control 44 (1999) 1486–1497. [2] K. Davidson, Nest Algebras, Pitman Research Notes in Mathematics, Vol. 191, Pitman, London, 1988.
359
[3] M.S. Djouadi, Optimization of highly uncertain feedback systems in H ∞ , Doctoral Dissertation, McGill University, Montreal, 1998. [4] M.S. Djouadi, G. Zames, On optimal robust disturbance attenuation, Systems Control Lett. 46 (2002) 343–351, this issue. [5] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992. [6] A. Feintuch, Robust Control Theory in Hilbert Space, Springer, New York, 1998. [7] B.A. Francis, On disturbance attenuation with plant uncertainty, Workshop on New Prespectives in Industrial Control System Design, 1986. [8] H. Kwakernaak, Minimax frequency domain performance and robustness optimization of linear feedback systems, IEEE Trans. Automat. Control AC-30 (1985) 994–1004. [9] J.G. Owen, Performance optimization of highly uncertain systems in H ∞ , Doctoral Dissertation, McGill University, 1993. [10] J.G. Owen, G. Zames, Robust disturbance minimization by duality, Systems Control Lett. 19 (1992) 255–263. [11] J.G. Owen, G. Zames, Duality theory of robust disturbance attenuation, Automatica 29 (3) (1993) 695–705. [12] C. Pirie, G. Dullerud, Robust controllers synthesis for uncertain time varying systems, SIAM J. Control Optim. 40 (4) (2001) 1312–1331. [13] G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and approximate inverses IEEE Trans. Automat. Control AC-26 (2) (1981) 301–320.
www.elsevier.com/locate/sysconle
Optimal robust disturbance attenuation for linear time-varying systems Avraham Feintuch Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer-Sheva, Israel Received 24 April 2000; received in revised form 2 March 2002
Abstract The time-varying optimal robust disturbance attenuation problem is considered .rst as two-block problem and then as a c 2002 Elsevier Science B.V. All rights reserved. two-disk problem. Keywords: Linear time-varying ordap; Two-disk problem
1. Introduction
general problem reduces to computing
The optimal robust disturbance alternation problem (ORDAP) was formulated for time-invariant systems by George Zames [13] and was studied by him and his students Owen and Djouadi, .rst in their doctoral theses [3,9] and then in a series of papers [4,10,11]. A stable uncertain plant P, lying in a given uncertainty set B(P0 ; V ) is subject to disturbances at the output. The objective is to choose a feedback control law C which provides the best uniform attenuation of uncertain output disturbances in spite of uncertainty in the plant model (Figs. 1 and 2). As opposed to the standard two-block problem of H ∞ design where a genuine worst-case minimum is produced only under the simplifying assumption that the disturbances due to output and sensor noises are mutually orthogonal, ORDAP assumes that this assumption is violated. As was shown in [7], this more
0 = inf∞ {|W (I − PQ)| + |VPQ|∞ };
E-mail address: [email protected] (A. Feintuch).
Q∈H
where P ∈ H ∞ is a given plant and W; V are outer functions. As was shown [3,9], the two-disk problem described by the diagram is a special case of ORDAP. Here the plant P0 is assumed to be known and external disturbances d1 ; d2 are uncertain in the sense that d1 ∈ {Wu: u 6 1}; d2 ∈ {Vu: u 6 1}; the images of the unit discs in H 2 under multiplication by given outer functions W; V . This two-disk problem has been considered in [5,7]. Also, certain types of optimal robust model-matching problems can be put within this framework [3,7]. Because there is no assumption of orthogonality, the standard methods in H ∞ control of reducing the problem to a Nehari distance problem do not apply. Instead classical optimization methods in Banach space, which go back to the classical Hahn–Banach theorem, are used. These require the use of dual spaces and thus the appropriate dual spaces must be identi.ed. In the time-invariant space this was done by Owen [9].
c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/02/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 1 4 9 - 4
354
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
to the subspace W W P0 S = P0 T : T ∈ S : V V Of course this distance will depend on the metric considered on this space of operators whose domain is H and range is in f H ×H = : f; g ∈ H : g
Fig. 1.
Djouadi considered in his thesis the time-varying analogue of ORDAP and gave a very elegant operator theoretic formulation of this problem. Let S denote the algebra of stable linear systems (see [6]) acting as operators on a Hilbert space with an appropriate time structure associated with it. The systems W; V; P0 ∈ S are given and ORDAP reduces to a distance problem in the space M2×1 (S) of 2 × 1 matrices with entries in S: Find the distance from W O
Fig. 2.
We consider this problem for two diIerent metrics. The .rst is the operator norm induced by the Hilbert space norms on H and H × H . This was considered by Kwakernaak [8] in the time invariant case. Here the basic idea is standard. By a simple trick we obtain a 2-block Arveson distance problem, which we solve as in [6]. The second is the operator norm induced by considering H × H with the Banach space norm f = f + g; g 1
where · denotes the Hilbert space norm on H . As was pointed by Djouadi [3], this is the time varying analogue of the norm considered by Zames–Owen [10,11] for the two-disk problem. This situation is much more diLcult. In this case the appropriate dual spaces for certain Banach spaces of operators have to be identi.ed. The results used here are classical and were for the most part already
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
355
pointed out by Djouadi [3]. We will list them with the appropriate references. As has been pointed by Francis [7] these norms are in fact topologically equivalent. However, Zames and Owen have already remarked that in problems involving plant uncertainty, a small change in norm can drastically change the solution. More recent work on a related synthesis problem in the time invariant case can be found in [1] and for time varying systems in [12]. In the last part of this paper we will compare the distances obtained and will comment on where diIerences arise.
is inner, it is a partial isometry whose initial projections U ∗ U commutes with Pt 0 0 Pt
2. Preliminaries
In this section we consider the time varying analogue of the classical H ∞ problem where it is assumed that disturbances due to output and sensor noises are mutually orthogonal. This is the problem of computing
The input–output space is a Hilbert space H of vector functions L2 [0; ∞; C n ] or vector sequences l2 [0; ∞; C n ]. The truncation projections on these spaces will be denoted by {Pt : t ∈ } where is the appropriate index set. The algebra of stable systems S is the algebra of bounded linear operators {T : Pt T = Pt TPt ; t ∈ }; the algebra of casual operators. This algebra is a nest algebra as is T11 T12 : Tij ∈ S M2×2 (S) = T21 T22 (see [6]). In the discrete time case all operators in S and M2×2 (S) have inner–outer factorizations [6, p. 60 – 62]. In continuous time this is not always so but we will assume such factorizations without getting into conditions required for their existance. We will make use of the fact that for T ∈ M2×2 (S) of the form T1 0 T2 0 its inner–outer factorization is of the special form B 0 U1 0 T= U2 0 0 0 [6, p. 61– 62]. Because U1 0 U= U2 0
for each t [6]. Also, it is standard to assume that the outer operator B from S satis.es that BS = {BQ: Q ∈ S} is dense in S. This holds in particular if B is invertible. All these issues are discussed in detail in [6, Chapter 4].
3. The Hilbert space operator norm
inf {|W (I − PQ)|2 + |VPQ|2 }1=2 ∞
Q∈H ∞
and was studied by Kwakernaak [8]. In the time varying case the H ∞ -norm is replaced by the Hilbert space operator norm and we are required to .nd W W 0 = inf − P0 Q : Q ∈ S : 0 V We begin by noting that W W − P0 Q 0 V is the same as W 0 WP0 Q − 0 0 VP0 Q Let T=
WP0
0
VP0
0
0 0
:
∈ M2×2 (S)
and consider its inner–outer factorization U1 0 B 0 T= : U2 0 0 0
356
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
We can rewrite W W − P0 Q 0 V as W 0
0 0
−
U1 U2
0 0
BQ 0
0 0
:
Now we use a standard trick [6, p. 129]. Since U1 U= U2 is an isometry, so is U∗ : I − UU ∗ Thus, W U1 − BQ 0 U2 U∗ W U1 = − BQ ∗ I − UU 0 U2 ∗ ∗ U1 W − BQ = (I − U1 U1∗ )W : −U2 U1∗ W Under the standard assumption that B is invertible, we obtain the two-block problem: .nd U1∗ W − R 0 = inf (I − U1 U1∗ )W : R ∈ S : −U2 U1∗ W By [6, p. 129] this is just Pt U1∗ W (I − Pt ) ∗ sup (I − U1 U1 )W (I − Pt ) : t −U2 U1∗ W (I − Pt ) Since this is the induced Hilbert space operator norm, we obtain 02 6 sup{Pt U1∗ W (I − Pt )2 } t
2 (I − U U ∗ )W 1 1 + sup ) (I − P : t U2 U1∗ W t
Since for an A ∈ B(H ); A(I − Pt ) is a decreasing function of t, this is just (I − U U ∗ )W 2 1 1 ∗ 2 sup{Pt U1 W (I − Pt ) } + : U2 U1∗ W t Remark. (1) It follows from a standard weak compactness argument (see [6; p. 132]) that the in.mum is in fact attained. Thus; a controller C which gives the best uniform attenuation of the output disturbances in this sense; exists. As is well known; this sort of analysis does not lead to any constructive methods for .nding C. (2) We consider the all-pass case. In our framework this means that U ∗ is also an isometry and therefore that W ∗ W − BQ − UBQ = U 0 0 = U1∗ W − BQ: In this case we obtain 0 = inf {U1∗ W − BQ: Q ∈ S} = sup{Pt U1∗ W (I − Pt )}: t
4. The Banach space operator norm Let L denote the Banach space H × H with norm given by f = f + g; g 1
the sum of the Hilbert space norms of the co-ordinates. We wish to .nd the distance from the given operator W 0 in M2×1 (S) to the closed subspace U1 S U2 of M2×1 (S) in the operator norm induced by the norm of L. The approach used here follows [3,9,10]. We
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
will make use of the following standard facts: 1. L is a reNexive Banach space. Its dual L∗ = X is the same set of vectors H × H with the dual norm f = max{f; g}: g
This is essentially the same as 1, with C1 playing the role of l1 and B(H; L) of l∞ . This analogy is standard [2, Chapter 1]. 5. We make use of a standard consequence of the Hahn–Banach theorem. If B is a Banach space, M is a subspace of B and x∗ ∈ B∗ , then
∞
d(x∗ ; M ⊥ ) = sup{|x∗ (x)|: x ∈ M; x 6 1}:
This is the two-dimensional case of the fact that the dual of l1 is l∞ . The action of the linear functionals is described as follows: for f u ∈ L; ∈ X; g v u f = (u; f) + (v; g): v g The action of linear functionals from L on X is identical. 2. The space of bounded linear operators on H; B(H ), is the dual of the space of trace-class operators C1 [2, p. 11]. A acts on C1 as follows: T → Tr(AT ). The standard norm on C1 will be denoted by T 1 . 3. The pre-annihilator of S as a subspace of B(H ) is a = {A ∈ C1 ∩ S: A is strictly lower triangular} [2, p. 211]. 4. If B(H; L) denotes the bounded linear operators from H to L, then this is the dual space of the space T1 : Ti ∈ C1 : !(H; X ) = T2 The norm on !(H; X ) is given by T 1 = max{T1 1 ; T2 1 } T2 and R=
R1 ∈ B(H; L) R2
= Tr(T1∗ R1
In order to .nd the distance from W U1 to S 0 U2 in B(H; L), we will relate to U1 S U2 as the annihilator M ⊥ of some subspace M ⊂ !(H; X ). Similar results have appeared in [3,9,10]. Lemma 1. Let P = I − UU ∗ ; the orthogonal projection in H ⊕ H (Hilbert space orthogonal direct sum). Then U1 U1 ⊥ S = A: A ∈ a + P!(H; x): U2 U2 Proof. We .rst show that each of two subspaces U1 a and P!(H; x) U2 are in ⊥
If T=
acts on !(H; X ) as a linear functional as T1 R1 = Tr(R∗1 T1 + R∗2 T2 ) R2 T2 +
T2∗ R2 ):
357
U1
U2
S
U1
:
A ;
U2
then for S ∈ S Tr S ∗ [U1∗
U 1 A = Tr(S ∗ A) = 0 U2∗ ] U2
358
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
since S ∈ a⊥ . If A1 A1 T =P with ∈ !(H; X ); A2 A2
By weakstar compactness [2, p. 4] this supremum is in fact a maximum and thus the sought controllers in fact exist.
then since
Remark. (1) Owen and Zames consider the all pass case in [10]. As was pointed out in Remark 2 at the end of Section 3; in this framework all-pass means that U ∗ is also an isometry. In this case P = 0; the second direct summand disappears; and we obtain sup{|Tr(W ∗ U1 A)|: A ∈ a; A1 6 1}: Using the argument of [2; Corollary 16.8; p. 211] this reduces to supt Pt U1∗ W (I − Pt ) which is exactly what is obtained in the all-pass case in the two-block problem. The time-invariant case is more complicated. (2) In fact, also in the general case, the distance formula obtained here is closely related to that obtained in the two-block problem. We have seen this for the .rst direct summand. If we consider A1 ∗ sup Tr([W 0])P A2
[U1∗
U2∗ ]P
= [U1∗
U2∗ ]
I−
U1 [U1∗ U2
U2∗ ]
= 0; we have Tr(S ∗ [U1∗ U2∗ ]P) = 0 for all S ∈ S. For the opposite inclusion take B1 ∈ !(H; X ) B2 such that [U1∗
Then U1 U2 and B1 B2
U2∗ ]
B1
= U1∗ B1 + U2∗ B2 = A ∈ a:
B2
[U1∗ =
U2∗ ]
U1 U2
B1 B2
=
A+P
B1 B2
U1
U2
A
:
That the sum is in fact a closed direct sum follows from the fact that the ranges of U1 U2 and P are orthogonal in H ⊕ H . We can now compute W U1 S ˆ0 = d ; U2 0 in the norm of B(H; L). By Lemma 1 this is just sup{|Tr(W ∗ (U1 A + (I − U1 U1∗ )A1 − U1 U2∗ A2 ))|}, where the supremum is taken over the unit ball in the subspace U1 A: A ∈ a + P!(H; x): U2
over the unit ball in the subspace P!(H; X ), this is just [(I − U1 U1∗ ) + U2 U1∗ ]W . Of course, precise inequalities for the relationship between 0 and ˆ0 seem much more diLcult to obtain. (3) We complete this discussion with an estimate for ˆ0 which is easier to compute. As was seen, the supremum is computed over the unit ball U A1 1 A+P 61 ; U2 A2 1
where the norm is the appropriate trace class norm in !(H; X ): T 1 = max{T1 1 ; T2 2 }; T2 1
where Ti 1 is the C1 norm of the trace class operators Ti on H . If A1 U1 A+P ; T= U2 A2 then ∗
∗
T T =A A+
[A∗1
A∗2 ]P
A1 A2
:
A. Feintuch / Systems & Control Letters 46 (2002) 353 – 359
Now T 1 = (T ∗ T )1=2 1 ¿ T ∗ T 1 : So if we compute the supremum over the larger set {T : A ∈ a; A1 ; A2 ∈ C1 with T ∗ T 1 6 1} we obtain a number ˆ ¿ ˆ0 . But 2 A A1 ; T ∗ T 1 = P A2 2
the square of the Hilbert–Schmidt norm of A A1 : P A2 This is a unit ball in a Hilbert space and the supremum is more transparent, at least in theory. References [1] R. D’Andrea, Generalized ‘2 synthesis, IEEE Trans. Automat. Control 44 (1999) 1486–1497. [2] K. Davidson, Nest Algebras, Pitman Research Notes in Mathematics, Vol. 191, Pitman, London, 1988.
359
[3] M.S. Djouadi, Optimization of highly uncertain feedback systems in H ∞ , Doctoral Dissertation, McGill University, Montreal, 1998. [4] M.S. Djouadi, G. Zames, On optimal robust disturbance attenuation, Systems Control Lett. 46 (2002) 343–351, this issue. [5] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992. [6] A. Feintuch, Robust Control Theory in Hilbert Space, Springer, New York, 1998. [7] B.A. Francis, On disturbance attenuation with plant uncertainty, Workshop on New Prespectives in Industrial Control System Design, 1986. [8] H. Kwakernaak, Minimax frequency domain performance and robustness optimization of linear feedback systems, IEEE Trans. Automat. Control AC-30 (1985) 994–1004. [9] J.G. Owen, Performance optimization of highly uncertain systems in H ∞ , Doctoral Dissertation, McGill University, 1993. [10] J.G. Owen, G. Zames, Robust disturbance minimization by duality, Systems Control Lett. 19 (1992) 255–263. [11] J.G. Owen, G. Zames, Duality theory of robust disturbance attenuation, Automatica 29 (3) (1993) 695–705. [12] C. Pirie, G. Dullerud, Robust controllers synthesis for uncertain time varying systems, SIAM J. Control Optim. 40 (4) (2001) 1312–1331. [13] G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and approximate inverses IEEE Trans. Automat. Control AC-26 (2) (1981) 301–320.