-
Proceedings of the American Control Conference
San Diego, California June 1999
Optimal Robust Disturbance Attenuation for MIMO
Uncertain Systems in H”
M.S. Djouadi
School of Aerospace Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0150
seddik .djouadi@ae.gatech. edu
Abstract
Banach space, LP(X), (or ILP(X)), 1 5 p 5 03, stands
for the Lebesgue-Bochner space of measurable, and p
In this paper, we consider the optimal robust disturbance th power absolutely integrable X-valued functions on the
attenuation problem (ORDAP) for multi-input multi- unit circle 8 9 under the norm
output (MIMO) uncertain plants. Duality theory is used
<
Ilf1lEP(x)=
ro.zT) I l f ( e ” ) l l ~ d m ,
for
to show existence of optimal feedback laws. Next a key
., ,
multiplication operator acting on particular vector-valued
Ilf(eae)llx, for p = m
(4)
I l f l l L ( X ) - ess 1 3sup
Hardy spaces is introduced. It is then proved that OR-.~ 1 0 . 2 ~ )
DAP-for-MIMO systems is equal to the operator induced
where rn is the normalized Lebesgue measure, f E
norm of a specific operator. The latter is shown to be a
ILp(X), and 11 . I I x denotes the norm on X . HP(X), (or
combination of multiplication and Toeplitz operators. An
W ( X ) ) , 1 5 p 5 CO, is the Hardy space of X-valued an“infinite matrix” representation with respect to a canonialytic functions on the unit disc 9 , viewed as a closed
cal basis is derived, and the norm of the relevant operator
subspace of Lp(X) (or resp. U’(X)).
“Tr(A)” denotes
is approximated by special matrix norms.
the trace of the matrix A, and “ess sup” the essential
supremum.
1
I
I
Not ation
C stands for the field of complex numbers, If z E C then 1 Introduction
Z denotes the complex conjugate of 2. For an n-vector
C E C,,, where C,, denotes the n-dimensional complex The optimal robust disturbance attenuation problem
denotes the space (ORDAP) was initiated by Zames in 1979 (see [3]), and
space, IC1 is the Euclidean norm. G X n
of n x n matrices A with entries in C, and IAl is the largest considered by Bird, Francis, Owen and Zames [4,5, 6, 21.
and q::n
denote the complex In the single-input single-output (SISO) case, Bird and
singular value of A. Gnxn
Francis proposed a two-block H m approximate solution.
A2
A1
A1, Owen and Zames in [6] showed that the approximations
Banach space of 2n x n matrices A , A =
obtained this way can be arbitrary poor. Duality was
A2 E G x nwith
, respectively the following norms
then applied to show existence of optimal solutions, and
IlAll =
(IAlCI + IAZCI)
(1) approximations based on convex programming were proIC1 5 1
posed. In this paper, we consider ORDAP for MIMO
uncertain systems. First, we give the duality structure of
llAllnuc= i n f { x ICkllWklmax, C k E G ,
(2) the problem showing existence of optimal solutions. A key
k
multiplication operator is then introduced, and its norm
W k E eF : AU = C ( < f U ) W k ,
VU E G}
is shown to coincide with the norm involved in ORDAP.
k
Next, we prove that optimal performance for ORDAP in
where 11 . [Inuc is the nuclear norm, and qfl denotes the the MIMO case is equal to the induced norm of a specific
operator. The latter is a Banach space projection of a
Cl,
complex Banach space of 2n-vectors C, C =
multiplication operator. The projection is then explicitly
C2 E C,, with the norm
computed showing that our operator is a combination of
multiplication and Toeplitz operators. This leads natuIClrnax = maX(lClI,lel)
(3)
rally to an infinite matrix representation of the operator
is the dual space of C?n::n
[l, 21, and vise-versa with respect to a canonical basis, and its norm is apsince Gnxnand q ; k n are finite dimensional and there- proximated by special matrix norms. Part of this paper
fore reflexive. If X denotes a finite dimensional complex extends our scalar case results [7, 8, 91.
( ),
( ;);
0-7803-4990-6199$10.00 0 1999 AACC
1856
2
v
Problem Formulation
In ORDAP a stable uncertain plant is subjected to disturbances at the output. The objective is the synthesis
of a feedback control law which provides the best possible uniform attenuation of output disturbances in spite
of uncertainty in the plant model [3, 4, 5, 61. More precisely, let P be a stable LTI uncertain plant, lying in a
set of uncertainty described by the expression
+
x
B(Po,V) = {(I VX)Po :
E Hrn(Gxn),
IlXllW < 1,Po E H r n ( G x n ) , V*' E H r n ( G x n ) }
(5)
where Pois the nominal plant, V is a fixed stable invertible transfer function, and X is a variable stable transfer
function.
The output disturbances d are assumed to lie in the set
We seek a feedback control law C for the set %(Po,
V)
which minimizes the weighted sensitivity norm IlW1(I
PC)-lIla. This is equivalent to solving the optimization
problem given by
+
po =
inf
C stabilizing
all P E B(Po,V)
IIW1('+PC)-'llm
sup
(7)
~EB(Po~V)
+
=
inf
~ )
IlWi(I - poQ)(I+ VXpoQ)-'Qllm
(8)
The optimization (8) is non-linear and non-convex. However, Owen in [2] showed that (8) is equal to the smallest
fixed-point of the following function in a parameter T ,
which is convex in Q
QEHm(Cnxn)
Then (11) can be rewritten as
(13) is the distance from
( 7 ) to the subspace S
=
(F
' ) H W ( G x nof) W'(Gn,n). According to [2, 61,
the subspace S has an equiva!ent description given by
S RH"(Cxn) where R = ( 1: ) is inner. where
3
SUP
Q E Hw(Gxn) x E H ~ ( G X
IlVpoQll- 5 1
lIXIlm < 1
inf
C E G
=
where W1 is outer in H ' ( G x n ) .
Letting Q = C ( I P,C)-l, (6) can be expressed in the
form [3, 21
Po
Here W , T&',
E H a ( G X n ) are outer functions, and
U E H'(@nxn) is inner.
Following [2, 61, we assume throughout: (Al) (T&'*r?l+
V*8)(eie)> 0, ve E [o, 2 ~ ) .
Define lLa(&.,)
(resp. Hm(Gnxn))
to be the Banach space consisting of 2 n x n essentially bounded (resp.
and analytic) functions defined in the unit disc D, with
values in the space Gnxn,
endowed with the norm for
ess sup
eqo,zr)
max
1 ~ 51 1
C E G
(IWl(1- PoQ)W(e")CI
+ rlPoQV(e'e)CI)
Let lL1(q:;*)
denote the Banach space of absolutely
Lebesgue integrable q:;,-valued
functions defined on
the unit circle aD, under the norm
lzr1 (
llGllL1(c~~Gn)
=
Cnxn)
ess sup
ea0.2~)
(IWl(1- PoQ)W(e")CI
(9)
) (I
nuc
dm
(14)
{ F E IHI'
(q:En) :
/d'"
F(eis)dm = O)
(15)
and $,(q:;n)
denotes the space obtained by
taking complex conjugate of all functions in
@(q::n).
Next, let E$(q:Gn)
be the quotient
max
1
1
15 1
66G
+ IPo&V(e")CI)
GI (eie)
Gz(e'e)
It has been shown in [2] that lLM(Gnxn)
is isometrically isomorphic to the dual space of lL1(qzcn). Define @(q,&) to be the subspace of the Hardy space
Ell1 (@2n,Ukn) given by
Therefore, we consider ORDAP for MIMO systems, as
represented by the performance index (9), specifically
po = QEHiyf
Duality Structure of the Problem
(10)
space L1(q:;n)/$(qz;n),
under the quotient norm
Furthermore, the optimization (10) can be shown to be
equivalent to [2]
1857
II[Glllw =
inf
g€mq:Gn)
IIG + gllL1(q:;n),
(16)
(G,,,)
in L1(
q::,),then W"O (G,X,) is isometrically isomorphic to the dual space of
(@2",":,).
Therefore, the pre-orthogonal of S is given by [lo, 111
Hm
where @ denotes the direct sum of two subspaces, and
t = ((I - RR*)IL'
(G;;:,
e E Z : ( G , ~ ,1)
( 2 ).
( : ) W(cznxn),
then @ may be viewed as
where F =
Let @ =
E
cisely
n-:(GZn)
(18) Clearly, MQ is a bounded linear operator. We show in
The quotient norm in
by
2 is denoted by 11
II [FIIIs = k$IIF+ ~IIL~(C;:;,,),
the next Proposition that
'
@(q::,)
11s and is given the operator induced norm is equal to
V [F]E
3
(19)
Theorem 2 ([12], p.121) implies the following theorem,
which shows existence of optimal feedback control laws
for ORDAP of MIMO systems.
N
11@11-.
(@(G,,,))*
Proposition 1 [lo] Let @ and MQ be defined as
above. Then 1. llMd@(G,";,) N ( @ ( ~ 2 n x n ) ) * ~ ~ =
5
SUP ~ ~ g ~ ~ L z ( c n x n1
) l l ~ * ~ l l L z ( ~ z n x n=
)
ll*Il-,
and 2.
9E~"(cxn)
llgllLz(cnxn) <
- 1 IIM@gIILZ(eznXn)
= ll@ll-.
llM*ll = S U P
g E L2(Cx,)
+
Theorem 1 Under assumption
( A l ) (W*W
and
v * v ) ( e i e ) > 0, V8 E [ 0 , 2 ~ ) . There exists at least The dual space of L'(Gnxn) is given by L2(q::,)
one Q o E H m ( G x , ) (i.e., a n optimal controller CO) vise-versa, hence I
L' (Gnxn)
is reflexive [lo]. In the next
such that
Proposition we characterize the dual space of @ (Gnxn).
=
1(7)
-j j ~ ~ I /
N
and @ (
q::,) defined
(20) Proposition 2 Let I@(Gnxn)
as above. Then 1. @ ( C z n x n ) N (p(q:;,))*,
and 2.
4
(G,,,,)
and @ (
q::,)
Hence
spaces.
IF1 E s
Exact Solution via Operator
Theory
are reflexive Banach
Proof
The proposition follows essentially since
q::n are finite dimensional [lo].
Gnx, and
In this section we give a solution based on operator theory
for the ORDAP of MIMO systems. The key observation 4.2 Operator Theoretic Solution
here is to interpret the norm (12) as an operator induced Let n' be the orthogonal projection on the closed subnorm in special vector-valued Hardy spaces.
of I@(Gnxn),
where
space I@(Gznxn)
e
@
0 f i H 2 ( G x n )is the orthogonal complement
of R H 2 ( G X n )(,f i H 2 ( G x n ) ) l . Here orthogonality is
4.1 A Multiplication Operator
understood to be with respect to the inner product of
Let L z ( G x n )( H 2 ( G x n ) )denote the Banach space of matrices, i.e., A , B E w(GnXn)
are orthogonal if and
Lebesgue square integrable (and analytic) GX
n-valued only if
functions on the unit disc under the norm
(Gnxn)
1
2T
JO
where I . I denotes the largest singular value.
' (Gnxn)
(@
to be the Banach
Likewise define L
space of Gnxn-valued and Lebesgue square integrable
(and analytic) functions on the unit disc endowed with
the norm
(G,,,,))
Tr{A*B}(eie)dm
=0
Then 11' is a bounded linear operator on @
Next, define the following operator
1858
2'
by
:~
(25)
(G,,,).
' ( +
6 w~~ ~
( C)e~T ~~H ~~ ( ~G ~), )
Z' =rI'M(
)
(26)
T h e o r e m 2 Under assumptions ( A l ) , and
(AZ) W , R continuous on the unit circle, if po
where
> poo,
to H m ( G x n ) ,then they all admit power series expansions in the unit disc, for instance
i.e., when the open unit analyticity constraint is removed,
the following hold:
E',
i ) po is equal to the operator induced norm of
namely
Then
k
m
= Il='ll
(27)
ai) There exists a maximal vector f o r
E', i.e., f
k = 0,fl,f2,.. .
E
m
(31)
M
k
Proof
The proof is similar to the proof of theorem 3 [ l l ] . It
should be noted that assumption (A2) implies that the Hence straightforward computations yield the following
optimal solution is allpass, inferring existence of a maxi- recursive algorithm
mal vector (see [lo, 111 for details).
The orthogonal projection n' is also given by proposition 3 [ll],i.e., IT = I - R P + k , where this time I is the
P+ is the positive Ftiesz proidentity map on
into @ (G,,,,,), roughly_speakjection from L2
ing, P+ associates to each vector function in IL2((@2,,,,)
its positive Fourier coefficients. The operator Z' is then
a combination of Toeplitz and multiplication operators,
@(enxn),
(e,,,,)
that is 2 = ( I - R P + k )
5
The operator Z' has then the following infinite matrix
representation with respect to the basis { z k e j }
Computation of the Operator
Norm
In this section we show how it is possible to compute
the norm of the operator Z', and hence the optimal performance index po. First, note that for f E H2(Gxn),
f(z) = CEO
z k f k , for some fk E G,,, the action of 8'
on f is determined by
6
k=O
Approximation by Norms of
Matrices
k=O
In what follows we will show that it is possible to apTherefore with respect to the canonical basis { z k e j } ,k =
proximate the norm of 8' by the norms of finite dimen0 , 1 , - . . , j = 1 , 2 , - . - , n ,where { e j } is a basis for G x n ;
sional operators or matrices. Indeed, if we restrict Z' to
E' has the following representation
finite dimensional subspaces of its domain, then we will
a sequence of finite rank operators which norms
obtain
E' = (E'l,E'z,. . . , 2 z n , . ..)
(30) are "nearly" optimal, in the sense that they approach
The k
l-th column corresponding to the representa- the norm of E' as the dimension of the initial spaces intion of E'zk with respect to the basis { z k e j } ; is then creases. More explicitly, let P, denote the orthogonal
formed by the coefficients of Z'zk in
with re- projection from If2(&,,,) to the subspace spanned by
spect to { z k e j } . To compute these coefficients consider, { z k e j } ,k = 0 , 1 , . . . ,m; and P, the orthogonal projection
@
onto the span of { z k e j } ,k = 0 , 1 , . . . ,m.
Z' =
, since W , R I and R 2 belong from
Clearly, IIP,II
5 1; and likewise it is not very hard to
-RI P+Rr W
+
G,,,,,
(
(G,,,,)
1859
show that IIPmll 5 1. The rank of Pm, rank(P,), defined as the dimension of the image space of P,, is such
that r a n k ( P m ) 5 m2n2. The operators EL = PmZ’Pm,
m = 0 , 1 , 2 , .. .; have rank less than or equal m2n2.Next,
we show that in fact, as m tends to infinity, EL converges to Z‘ in the strong operator topology, i.e., for
f E f f 2 ( G x n ) , I l f IILZ(C,,)
I 1, we have that
mlim
+w
1 1 2 f - Pm6‘Pmf IILZ(Gnxn)
=O
(34)
Indeed, by adding and subtracting Pm2‘f , the triangle
inequality implies
lP’f
- ~ m ~ ’ ~ m f I I L Z ( E z , xI
, ) IP’f - P m w L z ( C , , x , )
References
[l]R. Schatten. N o r m Ideals of Completely Continu-
ous Operators. Springer-Verlag, Berlin, Gottingen,
Heidelberg, 1960.
+
IIPmE’f - PmE‘PmfllL2(~z,x,)
I
+ ll~mllII=’llllf - P m f 1 1 L 2 ( c n x n ) (35)
IP’f - Pm=’fllLz(cz,,,)
It is therefore possible to approximate as closely as desired
the norm of E‘, and hence pa, by computing the special
matrix norms defined by (38) or (39).
From expression (36), we see that explicit computation of
the maximal vectors of Z L , (or EL*),will shed light on
the form of the maximal vector of Z’, and therefore the
optimal controller Co.
Gn,,
Since
is finite dimensional, by the Riesz-Fischer
Theorem (c.f. Lecture VI, [13]) the two terms on the
right-hand side of (35) tend to zero, and thus (34) must
hold. Therefore (34) implies
IIE’II - mlim
- + mI I P ~ E ’ P ~ ~ I I ~ ~ ~ =
~ ~o , , , )
(36)
It follows then IIPmE’PmII + 112’11 as m + 00, or
equivalently, I IPmz’ Pm f m IIn,zie,,,, + IIE’f II L 2
x. , .
Moreover, as noted earlier with respect to the canonical
basis { z k e j } ,PmE’Pm is represented by a matrix obtained
by truncating the infinite matrix representation (33) of E’,
i.e.,
(e,
[2] J.G. Owen. Performance Optimization of Highly Uncertain Systems in Hw. PhD thesis, McGill University, 1993.
[3] G. Zames. Feedback and optimal sensitivity: Model
reference transformations, multiplicative seminorms,
and approximate inverses. I E E E Transactions on
Automatic Control, AC-26(2):301-320, April 1981.
[4] J.F Bird and B.A. Francis. On the robust disturbance attenuation. Proceeding of I E E E Conference
o n Decision and Control, pages 1804-1809, 1986.
[5] B.A. Francis. On disturbance attenuation with plant
uncertainty. Workshop o n New Perspectives in Industrial Control System Design, 1986.
[6] J.G. Owen and G. Zames. Duality theory of robust
disturbance attenuation. Automatica, 29(3):695-705,
1993.
(37)
The computation of the norm of the matrices EL can be
further simplified, for e.g., in the skalar case (n=l), EL
is a 2(m 1) x ( m l), (each entry ( a k j l ) in (37) is a
complex 2 x 1 vector), the norm of EL can be shown to
reduce to [7] small
+
+
fm
1pm
’ *11=
E Gn
+
max
max(lhlml, lhzmI) = 1
hlm,h2m
I=lm
hlm
+
=;m*h2ml
[ll] M.S. Djouadi. Exact solution to the non-standard
H”’ problem. Proceedings of I E E E Conference on
Decision and Control, Tampa, Florida, 1998.
Clearly, such norms are bounded above and below, by
respectively
lELml and max(lEiml , lZbm1), i.e.
+
7
[lo] M.S. Djouadi. Optimization of Highly Uncertain
Feedback Systems in Hm. PhD thesis, Dept. of Electrical Eng., McGill University, 1998.
E @,
(39)
mm(IEimI
[8] M.S. Djouadi and G. Zames. On optimal robust disturbance minimization. Proceedings of,the American
Control Conference, June 1998.
[9] M.S. Djouadi and G. Zames. On optimal robust disturbance attenuation. Submitted to Systems & Control Letters.
where I . I is the Euclidean norm, and ELrn, k = 1,2; are
(m 1) x (m 1) matrices. Or equivalently, by passing
to the adjoint of E&, a&*,we have
-I
*
+
[7] M.S. Djouadi. Banach space optimization of uncertain systems in Hm. PhD Research Proposal, Department of Electrical Engineering, McGill University,
1997.
I%mI) I II%II I IEimI + IELml
[12] D.G. Luenberger. Optimization by Vector Space
Methods. John Wiley, 1968.
[13] H. Helson. Lectures o n Invariant Subspaces. Academic Press, New York and London, 1964.
(40)
1860