TP06 18:OO
Proceedingsof the 37th IEEE
Conference on Decision & Control
Tampa, Florida USA- December 1998
Exact Solution to the Non-Standard Hm Problem
Mohamed Seddik Djouadi
Center for Intelligent Machines, and
Dept. of Electrical Eng., McGill University
Montreal, Quebec, Canada
Corresp. address Dept. of Electrical Eng.
Wayne State University
5050 Anthony Wayne Dr. # 3100
Detroit, Michigan, USA 48202
djouadi@ece.eng.wayne.edu, djouadi@cim.mcgill.ca
functions defined on dD and D with values in the
spaces defined above.
Abstract
The non-standard H w
problem po
=
infQEHm 11 IW(1 - poQ)I + IVpoQI IIw Plays
a fundamental role in robust feedback control. Here we develop a new operator theoretic
framework to characterize explicitly its solution.
We also establish absolute continuity of a dual
extremal measure showing that the optimal performance/controller satisfy an extremal identity,
which provides a test of optimality. Moreover via
Banach space duality theory we show existence of
a maximal vector leading t o an explicit formula
for the optimal controller.
1 Introduction
In this paper we consider the non-standard H w
optimization known as the “two-disk” problem defined as
Notation
C stands for the field of complex numbers, G x n
denotes the space of n x n matrices with entries
in C. q,,, is the space of 2n x n-matrices with
the norm IIAJJ= lAll
+ lAzl where A =
(2)
Al,2 n x n matrices, and IAll is the largest singular
value of Al. C J , ~ ,is the same as GnXn
but with
the norm IlAll = rnax(STr(Al),STr(Az)), where
~ )Cj”=,
oj(Ai),and
STr(Ai) = T r a ~ e ( { A ~ A i }=
aj(Ai) is a singular value of Ai. q, stands for
2n x 1 with the norm IIEllq, = l&l I&I, where
+
( Et )
E U?, and 1&l is the Euclidean
[=
,
norm. Q, is the dual space of q,. D is the
open unit disk, aD the unit circle, and m stands
for the normalized Lebesgue measure. * denotes
either the complex conjugate transpose of a vector
or an operator depending on the context, and “Tr”
is the abbreviation used for trace. We will use freely
the standard spaces ILP and W , 1 5 p 5 00, of
0-7803-4394-8198
$1 0.000 1998 IEEE
2843
where Po E G x nrepresents the plant, W , V are
outer functions in H w ( G x n ) weighting functions,
all defined in the unit disk D (see [l,2, 31). It
is shown in [4, 5, 31 that under certain conditions
the optimal robust disturbance attenuation problem (ORDAP) reduces to (1) as well as problems
involving questions of well posedness of feedback
systems [6].Zames and Owen in [2, 31 used Banach
space duality theory t o show that under specified
conditions there exists a solution t o (l),which satisfies a flatness or “allpass” condition and is unique
in the SISO case. However the operator theoretic
structure of the solution has still not been described
and there has been no closed form solution reported
in the literature essentially because (1) is more difficult than the standard two-block H w problem.
We will exhibit an explicit solution via operator
theory for the optimal performance and the optimal controller. Moreover these are shown t o satisfy
an extremal identity, which can be used as a test of
optimality . Part of this paper extends our scalar
case results [7, 8, 91. Following [3, 21, the following assumptions will be made throughout: (Al)
(W*W+V*V)(eZ‘)> 0, V8 E [0,2n),
and (A2) the
outer part of Po is invertible in H W ( G x , ) . The
optimization (1) has been shown in [l, 2, 31 t o be
where [GIE @.(Gnxn)
is the equivalent class of
G, @,(Gnxn)
denotes the subspace of the Hardy
' (G,,,) given by
space W
equivalent to
c
where U is inner, W and are outer in H w ( G x , ) .
Let IHIw(C&,,)
denote the Banach space of
bounded analytic functions in D with values in
,&a
, with the norm
IJKII=
~ ess.
+ Ilc2(eie)I)
SUP (IKl(eie)I
e~[0,2x)
( 2 ) E W@'(C&,,).
where K =
(3)
Then (2) is
(4)is the distance from
(
K(Gnxn)
denotes
( ) t o the subspace
) H'(@nxn)
of W(C&,,).
As-
sumptions ( A l ) and (A2) ensure closedness of 8 .
According to [l,2, 31 with a slight modification 8
has an equivalent description given by
JWGraXrl).
It has been shown in [ l l , 1, 2 , 3 ] that Lm(U&,,,)
is isometrically isomorphic t o the dual space of
IL1 (G,,,). Since Woo (q,,,)
may be viewed as
a subspace of lLW(C$,,,), and
is the
preannihilator of WDO (G,,,), then a standard result from Banach space duality theory asserts that
MIm (q,,,)
is isometrically isomorphic t o the dual
and
) we write:
space of Q ( G n x n
W""(G,,,)
Then, every K =
=(@(Gnxn))*
( ) defines a linear bounded
Tr(K;Gl
Let I S be the subspace of
where R*R = I almost everywhere (a.e.).
Existence of solutions is next obtained using duality, and some of the ideas in [2, 31.
2 Duality Structure of the Problem and
Existence of Optimal Solutions
Let IL1 (CA,,,)
denote the Banach space of weakly
Lebesgue measurable [lo], and absolutely inte,,,-valued functions defined on the unit
grable CA,
circle d D under the norm
where G =
max(STr(G1), STr(Gz))(eie)dm
z:
( ) E IL1 (G,,,),
(8)
functional +K defined on H;(@znxn)
by
+K([G]) =
IlGllel =
the space obtained by
E(&,,,)
equivalent to
S =
and
taking complex conjugate of all functions in
ls =
+ KlGz)(e")dm
(9)
(CA,,,) defined by
((I- RR*)IL~(C~,,,)
CB
RE(G,,))/x
(10)
where
X = E ( G n x n )n((I-RR*)lL1
(C~nxn)@RE(Gxn))
(11)
It follows from lemma 1 of [2, 11 modulo minor details that l s is the preannihilator of 8 in
@(&,,,).
Theorem 2 ([12], page 121), implies
the following
Theorem 1 Under assumptions ( A I ) and (Ai?)),
there exists at least one optimal Q, E H w ( G x , )
such that:
(6)
and STr(Gi) =
Tr({G:Gi}*) = Cj"=laj(Gi), i = 1, 2; where
"Tr" denotes the trace. aj(Gi) denotes the singular value of Gi [l, 2, 31. Next,&t ilKf,(C~,~,)
be the quotient space IL1 ( C A ~ ~ ~ ) / E @ ,
un-( C A , ~ ~ )
der the quotient norm
In the next section, we show that if we assume further (A3) W , W ,RI, R2 are continuous, as is the
outer spectral factor of W*W+c*c,and p, > p,,,
where
max (STr(G1
+ gl), STr(G2 + g ~ )( e) i e ) d m
2044
i.e., when the open unit disk analyticity constraint
is the space of G x n is removed, and C!(Gxn)
valued continuous functions on d D , (in the scalar
case ( n = I ) , po0 = II min(lW(eie)l, Iv(eie)l)IIm),
the optimal solution is flat [3, 1,2]),then the supremum in (12) is always achieved.
3 Absolute Continuity and Existence of an
Extrema1 Identity for the Optimal
Performance and Controller
Lemma 1 Under assumptioms ( A l ) , (A2) and
(A3), the following hold:
Let C!(C:nxn)
denote the space of continuous functions on d D with values in Q.,
Iff E
we set
Proof: the first equality follows from Lemma 2
[l,21, the second from Theorem 1 ([12], page 121).
It turns out that the maximum in (22) is actually achieved by a measure U, absolutely continuous with respect to the Lebesgue measure. Then
it will follow by the Radon-Nikodym Theorem that
the supremum in (12) is achieved in 'B.
The dual space of
is given by the space
M ( @ a n x n ) of Ginxn-functions of bounded variations on d D under the norm [l,2, 31
Lemma 2 Under assumptions (Al),
(A3), the following holds
lo,inl
( zi )
ll4llw =
where
max(STr(Gy,l),S T r ( G y , l ) ) ( e i e ) d w y
E
M ( @ i n x n ) , and wy denote the
sum of the total variations on [0,2n) of all entries
of V I and UZ. Let
be the space of Gnxnvalued functions which are continuous on d D and
is known as the disk alanalytic in D. a(@;nxn)
gebra. The annihilator of a(@;,,,) is completely
described by the vector-valued version of the F. and
M. Resz Theorem [13] which yields
A(Gnxn)l =
{P E
C(Gnx,)
: /I = H m ,
(16)
E $&zxn))
It follows that the dual space
( A ( G n x n ) )=
* M(@znxn)/w:(Gnxn) (17)
Next, define the subspaces 8 , and
by
8, =
([ld],
chap.
IV), we say that
( ",",$' ) is a dual extrema1 function.
Proof by lemma 1 there exists [U,] E S i , II[v,]lllw 5
1 such that (22) holds. If we let wo to be the total
variation of the entries of v,, then a similar argument to the proof of Theorem 2 in [l,21 shows that
w, is such that V E Bore1 set, E C d D , w,(E) = 0
implies m ( E ) = 0. Hence by the Radon-Nikodym
Theorem there exists F E IL' (w,, G2,xn) s.t.
8 ne ( q n X n )
d m = Fdw,
(20)
In the following lemma we establish that the dis-
( ) t o 6, is the same as t o 6.
3F, E IL1(Gdnxn)
(25)
(24) and (25) imply dw, = 0, and therefore U, is absolutely continuous w.r.t. m and hence (23) holds.
A well known fact in the theory of HP spaces of
scalar valued functions is that every coset in L' / H A
contains one representative of the least poscible
coset norm [16]. It is not very hard t o show this
dw, = dw,+dw,
6* = {U E M ( C z n x n ) : dU = ( I - RR*)dP @ RG,
E W @ z n x n ) , G E $AGxnl/Y
(24)
However by the Lebesgue decomposition Theorem
for vector measures [15], there is a measure w, absolutely continuous with respect to m, and a singular measure w, s.t.
(18)
It follows then from lemma 3 [l, 21 that the annihilator of 8, is given by
tance from
Following
V respectively
( ( I - R R * ) M ( @ i n x n )@ R E i ( G x n ) )
P
(A2) and
2845
= F,dm+dw,,
4 Exact Solution via Operator Theory
property holds for @(C2nxn)and is inherited by
‘6. This is summarized in the following lemma
which proof is omitted (see [17]).
Lemma 3
0
4.1 A Key Multiplication Operator
Let L2(C&) stand for the Banach space of C&,valued functions defined on dD under the norm:
Given [F] E @(C,,nxn), there
exists h, E m;(Gnxn) s . t .
where G =
( “,: )
.
.
(q,).
Note that L2(qn)
E L2
is not a Hilbert space.
Let CP =
Given [ F ]E‘ 9, there exists xo E X s.t.
( $:)
E l”(C&xn),
the multiplica-
tion operator associated with CP and mapping the
standard Lebesgue space L2(Cn)into IL2(qn)
is
denoted by M a . More precisely Ma f = CP f for
f E L 2 . The next proposition, is a generalization of
Proposition 2 obtained for the scalar case in [8, 91.
in this case we say that F + x , is an extremal kernel
for PI.
Combining lemma 2 and 3, we obtain the following
Theorem which looks familiar in the theory of extremal problems, and provides a test of optimality.
Theorem 2 Under assumptions ( A l ) , (AZ) and
(A3), Fo =
( 2: )
kernel for [ F ] , and
and only if
E
Qo E
[F] E’ 8 is an extremal
H m ( @ n x n ) i s optimal if
Proof omitted (see [17]).
( 2;
Tr{((W*,O)+ QP*)
)}(~Ze)
I(W - RiQo)(eie)J+ 1R2Qo(eie)I
max(STr(Fol),STr(Fo2))(eie), ax.
Proof
llFoll~1
Next we characterize the dual space of IL2 (C&) and
ndL (q,)where
, nd2(@2*,) to be the closed subspace
of L2(q,) consisting of 11 . Ilazcs;,)
bounded of analytic functions in the unit disk D .
=
“Only if” by assumption 3F0 E’
1, and Qo E H M ( G x n ) s.t.
(28)
9,
Proposition 2 Let I L 2 ( Q n ) stand for the Banach
space off&, -valued functions defined on d D under
the n o m :
+
where G =
but the integrand is “5” (IW - RieQol
1 R2QoI) (eie) max(STr( Fol),STr( F o 2 ) ) (e ) a.e.
which is 5 po max(STr(Fol),STr(Fo2))(eie) a.e.
Integrating implies equality must hold throughout.
This combined with “flatness” imply
max(STr(Fol), STr(Foz))(eie)
= 1 a.e.
“If” follows by integrating (as), (see [17] for details).
zl
( ) E !L2(C2n).
Define
l@(C,ln)to
be the analogue to @(C&) but in IL2(C2n).Then
we have the following:
1. L2(CJn)
N (lL2 (U&))*
) (IL2(C2,))*
2. JL2(q,=
3. W C 2 n )21 (W@2*,))*
4.IHf2(qn)
N (nd2(C,2n))*
Hence all these Banach spaces are reflexive.
2846
For the reverse inequality, by lemma 2 and 3 and
Proof 1. and 2. follow from [ll].3. and 4. follow
since Q,, is finite dimensional (see [17]).
In the next section we characterize the optimal solution in terms of an operator which is analogue to
the Sarason operator for the standard HM problem. This generalizes the scalar case results obtained in [7, 8, 91.
Theorem 2, there 3 F =
H2(Cn)
€1 B
ofL1-norm
1 s.t.
4.2 Operator Theoretic Solution
Let II be the orthogonal projection operator on the
closed subspace @(C&) 8 RH2(Cn)of HL(C&).
Then II is a linear bounded operator on w(C&).
Now we define the following key operator [7, 8, 91:
Z:
(2)
+@ (C&) e RH2(Cn)
by
the
proof
of
theorem
2,
= 1 a.e. m, then
max(STr(Fl), STr(FJ))(eie)
there 3h E H 2 s.t. lhI2 = max(STr(Fl), STr(&))
a.e., and llhllHz = 1 [HI. But then
F h E L 2 ( Q n x n ) , and IIF~IIL~(c~,.~)
= 1,
, orthogonal
moreover F h E ( R H 2 ( G x n ) ) Ithe
complement of R H 2 ( G x n )in L 2 ( 2 ~ n x n )Now
.
let I, be the n x n identity matrix, then
Jd
(34)
In the following proposition the form of the orthogonal projection II is computed explicitly.
27T
27T
Tr(hI,(W*, 0 ) F h ) d m
Tr((W*,0)F)dm =
(41)
Exploiting an idea of Young [19], the LHS of (41)
is equal t o
Proposition 3 II = I - RP+R*, where P+ is the
standard Riesz projection, and I the identity map
1171.
I(PM(
7
)hIn,Fh)l
L llPM
The operator E is then a combination of Topelitz
and multiplication operators, more precisely
Z = ( I - RP+R*)
(V
(35)
where P = II@ I,, M
The following Theorem generalizes the results obtained for the scalar case in [7, 8, 91.
Theorem 3 Let p o be the performance index defined by expression (1). Under assumptions ( A l ) ,
(A2) and (A3), p o is equal to the operator induced
norm of E,namely
denotes the tensor product, M
the multi-
plication operator associated to
( 7 ) , and (., .)
denotes the inner product in lL2 (GnX,).
Then,
P o = 11=.11
(36)
Moreover there exist at least a maximal vector f E
H 2 ( C n ) of L2((cCn)-norm1 such that
PM(
=.(
:)@In
(43)
llPM f W ) II = IlEll
(44)
therefore
IlEf l l L z ( q n ) = 11=.11
(37)
( 0 1
Proof: VQ E H w ( G x n ) ,and all g E H 2 ( C n )we
have Q g E H 2 ( C n ) and then IIRQg = 0. Let
f E H2((Cn) with norm at most 1, then the first
inequality follows from (see [7, 91 for the details)
11E.11 =
sup
IIfIILz(,n
min
)<I
1(7)
Expressions (391, (42) and (44) imply that (36) and
(37) hold, for some vector f E L 2 ( C n ) ,I l f l l ~ z p )=
1.
The following corollary follows then from theorem
f - Rg/l (38) (3) and ProPosition (3).
Lz(qn 1
Corollary 1 The optimal performance index po is
given by the following expression
(39)
2847
[4] J.F Bird and B.A. Francis. On the robust
disturbance attenuation. Proceeding of IEEE Conference on Decision and Control, pages 1804-1809,
1986. .
and there exists f E H'((@"), I l f l l L z p , = 1, such
that
[5] B.A. Francis. On disturbance attenuation
with plant uncertainty. Workshop on New Perspectives in Industrial Control System Design, 1986.
4.3 Optimal Controller
Theorem 1 implies that there exists a vector function Q E Woo ( G n x n
such
) that:
1 1 Q 1 1 ~ = 11Z.11
[6] M.C. Smith. Well posedness of Hw optimal
control problems. SIAM, 28(2):342-358, 1988.
[7] M.S. Djouadi. Banach space optimization of
uncertain systems in Hm. Ph. D. Research Proposal,
Department of Electrical Engineering, McGill University, May 1997.
(47)
and by Theorem 3 there exists f E H 2 ( C n )of norm
1 such that, IlEf 11 = llEllllf l l H a = 11Zl).
By Proposition 1, 4 can be viewed as a multiplication operator n/r, acting from H 2 ( C n ) into
fl(Gn)such that, ~~MQII
= Ilqllw-. Then
[8] M.S. Djouadi and G. Zames. Operator theoretic solution to the optimal robust disturbance attenuation problem. Proceedings of the World Multiconference on Systemics, Cybernetics and Informatics, 3:25-29, July 1997.
Il=llllf l l H 2 = 1
1
3IIw(qn)= llIIQfllEP(qn)
I llQf llw(qn)5 11~11w- Ilf = Il~llllf
11H2
IIHZ
[9] M.S. Djouadi and G. Zames. On optimal robust disturbance minimization. Proceedings of the
the American Control Conference, June 1998.
since the orthogonal projection II acting from
fl(@, ") onto W (C&)0 RH2( G x nhas
) induced
norm less than or equal 1 [7, 91. It follows that,
Qf = Sf, m a.e.
In the scalar case we can divide through, therefore
we have \E =
m a.e., and hence by Theorem
3:
[lo] E. Hille and R.S. Phillips. Functional Analysis and Semi-groups. AMS, Providence, R.I., 1957.
y,
[ll] J . Dieudonnge. Sur le thQorkmede Lebesgue
Nikodym V. Canadian Journal of Mathematics,
3:129-139, 1951.
[12] D.G. Luenberger. Optimization by Vector
Space Methods. John Wiley, 1968.
[13] R. Ryan. The F. and M. Riesz theorem for
vector measures. Indag. Math., 25:408-412, 1963.
[14] J.B. Garnett. Bounded Analytic Functions.
Academic Press, San Diego, New York, Boston,
1981.
[15] J . Diestel and J.J. Uhl. Vector Measures.
AMS, Providence, RI, 1977.
and the optimal Qo is given by the following expression
( Y ) - R Q o = -Sf
f
+
Q~ = R*
( ) - R Zf* ~ m, a.e.
thus the controller CO= Q o V - ' ( I - QoV-lPo)-'
achieves optimal robust performance.
[16] V.P. Havin. Spaces Hm and L1/HL. Journal
of Soviet Mathematics, 39:120-148, 1974.
References
[l] J.G. Owen and G. Zames. Robust disturbance minimization by duality. Systems and Control Letters, 19:255-263, 1992.
[17] M.S. Djouadi. Optimization of Highly Uncertain Feedback Systems in Hw. PhD thesis, Dept.
of Electrical Eng., McGill University, 1998.
[18] K . Hoffman. Bounded Spaces of Analytic
Functions. Dover, New York, 1988.
[19] N.J. Young. The Nevanlinna-Pick problem
for matrix-valued functions. Journal of Operator
Theory, 15:239-265, 1986.
[2] G. Zames and J.G. Owen. Duality theory for
MIMO robust disturbance rejection. IEEE Transactions on Automatic Control, AC-38(5):743-752,
May 1993.
[3] J.G. Owen. Performance Optimization of
Highly Uncertain Systems in Hm. PhD thesis,
Dept. of Electrical Eng., McGill University, 1993.
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