Proceedings of the 3 P
Conference on Decision & Control
Phoenix,Arizona USA December 1999
MIMO Robust Disturbance Rejection for Uncertain Plants
M. Seddik Djouadi
School of Aerospace Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0150
seddik .djouadi@ae.gatech. edu
measurable, and p t h power absolutely integrable X valued functions on the unit circle 3 9 under the norm
Abstract
In this paper, we investigate in a unified framework two
basic feedback optimization problems, rejection of output disturbances in presence of plant uncertainty, and
contraction of radius of uncertainty for MIMO plants.
Duality theory is provided showing existence of optimal feedback laws. Optimal, nearly optimal solutions
are characterized, and flatness conditions are derived.
The theory developed here allows previous convex programming based numerical methods to be applied to
this class of problems
Notation
C stands for the field of complex numbers, If x E C
then Z denotes the complex conjugate of x. For an
n-vector E &, where @, denotes the n-dimensional
complex space, I<l is the Euclidean norm. G x ndenotes the space of n x n matrices A with entries in
C, and IAI is the largest singular value of A. A* denotes the complex conjugate transpose of A.
and @2“,“:
denote the complex Banach space of 2n x n
where m is the normalized Lebesgue measure, f E
ILP(X), and 11 Ilx denotes the norm on X . H p ( X ) ,
(or I”(X)),1 5 p 5 CO, is the Hardy space of X valued analytic functions on the unit disc 9,viewed as
a closed subspace of L P ( X ) (or resp. lLP(X)). “Tr(A)”
denotes the trace of the matrix A, and “ess sup” the
essential supremum. If B is a Banach space, then B*
denotes the dual space of B.
<
cnxn
spectively the following norms
where 11 . llnuc is the nuclear norm, and Gnudenotes the complex Banach space of 2n-vectors <, =
( );
<I, <Z
<
E
G with the norm
IClmax = m4lC11, IC211
1.1 Rejection of Output Disturbances in Presence of Plant Uncertainty
In this problem a stable LTI plant P known to belong
to a weighted ball in H w ( G x n )
+
V ) = ((1 V x ) p o : X E H w ( 6 x n ) ,
IIxIlw < 1, po
E Hm(G,xn),
v*’E
(2)
Hm(Gxn)}
(1)
[l, 21, and viseversa since GnXn
y d qz;,, are finite dimensional
and therefore reflexive. If X denotes a finite dimensional complex Banach space, L P ( X ) , (or ILP(X)),
1 5 p 5 CO, stands for the LebesgueBochner space of
o 1999~ E E E
It is well known that there are mainly two reasons for
using feedback, the first is to reduce the effect of any
unmeasured disturbances acting on the system, the
second is to reduce the effect of dynamic uncertainty.
The objective of this paper is to investigate the ability
of feedback to reduce uncertainty by analyzing two optimization problems. The first is the MIMO version of
the optimal robust disturbance attenuation (ORDAP)
[3, 2, 41, and the second is a similar problem which
captures the potential of feedback to contact radius of
uncertainty [3]. It may be worth recalling these two
problems in some details.
B(po>o)
cnxn
is the dual space of qz:n
0-7803-5250-5/99/$tO.00
1 Introduction
4050
is subject to disturbances at the output (see Figure
(1)). The objective is to synthetize a robustly stabilizing feedback control law C for the set B(P,,V),
which minimizes the weighted sensitivity norm IlW(l+
PC)-’Ilm over all P E %(Po,
V ) , where W E Hm(Gxn)
is outer [5]. This problem is equivalent to the opti-
lu
@-
uncertain Plant
\
stable Filter
.
controller
Figure 2: TwwDegree of F'reedom Feedback Control
Scheme
Figure 1: Feedback Control in Presence of Plant and Dis
turbance Uncertainty
mization [3, 21
po
=
inf
Q E H"(@nxn)
IIVpoQllm 5 1
x
SUP
E~
~
(
IlW(1- P,Q)(I + VXpoQ)-'Qll~
~
~ < ~1 )
l
Following [3, 21, the potential of feedback to
reduce plant uncertainty c m now be formally
stated as the minimization of the "worst-case" Wlweighted closed-loop multiplicative uncertainty radius
~ ~ p p ~ S ( p ,11,Wvl)Amllm over all robustly stabilizing
nominal plant invariant control laws. Mathematically,
l this
~ problem
l l ~can be translated into the following minimization [3]
(3)
In [2] it was shown that p0 is equal to the smallest fixed
point of the function x : IO, 00) e [0,00)defined as a
MIMO version of the "two-disk" problem
po =
inf
Q E H"(Cxn)
IlVpoQllm 5 1
x
SUP
E Hm(Cxn)
l l ~ l<l1~
IlW:(I- poQ)(I+ VXpoQ)-'VXIIm
(7)
It was shown in (Lemma 2.2 [2]) that expression (7)
is equal to
Po
=
inf
Q E Hw(Cxn)
IlVpoQllm I1
SUP
x EH~(C,.~,)
llxllm< 1
+
IlWi(I- K Q ) ( I VXpoQ)-'Vllm
1.2 Attenuation of Plant Uncertainty
In his seminal paper [3], Zames posed the following
question regarding the ability of feedback to reduce
the effect of plant uncertainty: if a plant lies in some
uncertainty set, say %(Po,
V), what is the smallest radius of any set of closed-loop uncertainty that can be
achieved by a single feedback control law ?
To avoid the trivial answer to this question that zero
closed-loop uncertainty can be achieved by disconnecting the system from the input (see Figure (2)), it is
suggested in [3] to adopt the invariant plant scheme,
i.e., the nominal closed-loop system is normalized to
be equal to the open-loop system Po,where Po is the
nominal plant. From Figure (2), we have then
( I + cPo)-'u= I
(5)
The closed-loop map K of the two-degree of freedom
control scheme above belongs to Hw(GXn),
and can
be expressed as
K
- P,
(I + P,Q)(I + AZJQ)-'AP
= AmPo, where Q = C(I + PoC)-' (6)
Expression (8) is then mathematically identical to (3).
We henceforth consider the two-disk optimization (4).
In a similar vein to [6], we develop a duality theory
for (4), showing existence of optimal feedback laws.
We then characterize the optimal and nearly optimal
solutions. We also give a flatness condition for the
optimum. Finally, we demonstrate that the optimal
controller satisfies an extremal identity. The theory
developed here allows the convex programming based
numerical methods discussed in [7, 21 to be applied to
the MIMO extension of ORDAP (4).
2 Duality Results
The optimization problem (4)
. _can be shown to be
equivaient to 121 po =QEHiqf
C X n )
405 1
ess
sup
eE[O,2n)
max
51
C E C
=
where A P = P- Po, and Am E Hw(@nxn)
represents
the closed-loop multiplicative plant uncertainty.
(8)
(IW
- ~ ~ Q ) ( e " ) C+I IvQ(eie)CI)
(9)
where t?l and
E H m ( @ n x n ) are outer functions,
and U E Hw(@n,,) is inner [5].
Hw(Gnx,))
Define I L w ( ~ n x n (resp.
)
to be the Banach space consisting of 2n x n essentially bounded
(resp. analytic) functions defined in the unit disc 9,
with values in the space Gnxn,
endowed with the norm
K=(
2)
Theorem 2 ([8],p.121) implies that there exists at
least one optimal parameter Q, E Ha(&,,) (.i.e. one
optimal controller CO)such that
Then (9) can be rewritten as
Expression (11) is the distance from
subspace S =
('.
) Hm(Gxn) of
( ) to the
W(Gnxn).
We
po =
ess
inf
QEHOD(Cnxm)
sup
eE[o,zr)
max
1 ~ 51 1
C E G
(IW - kQ)(e")CI + IfizQ(e"kI)
assume throughout: (Al) (W*W + p v ) ( e i e ) > 0, VO E
[O, 27r). Assumption (Al) ensures closedness of S.
According to [2, 71, the subspace S has an equivalent
description given by
where R is inner.
q;:,)denote the Banach space of absolutely
Let lL1 (
Lebesgue integrable q;;,-valued
functions defined on
the unit circle 89,under the norm
Remark 1 This predual description shows that the
convex optimization based techniques discussed in [2, 71
can also be applied to the MIMO extension of ORDAP,
but with diflerent matrix norms. Namely, the matrix
norm
IlAll
=
m=
IiI 5 1
(IAlCI + lAZC1)
(21)
6 E G
It has been shown in [2, 71 that lLm(GnXn)
is isometrically isomorphic to the dual space of lLi (q;;,).
Again, define @
(q;:,) to be the subspace of the
Hardy space Hi (q;;,)
given by
{ F E E' (G:En) :
/a,
F(e")dm = 0)
0
(14)
and @(Cj:;,) denotes the space obtained by
taking complex conjugate of all functions in
l€$,(Q:;,).
Next, let @(q;",) be the quotient space lLi (q;;,)/@,(Cj;;,),
under the quotient
norm
V[GI E L' (GEn)E(GZn)
CB R z i ( G x n ) ) / X
( ii ). Since the norm (21)
is
difler-
entiable on the unit sphere of C,,, it can be computed
using calculus.
3 Allpass Property of the Optimum:
Alignement in the Dual and Extremal Identity
for the Optimum
In this section we assume (A2) W, 21and RZare continuous_onthe p i t circle, as is the outer spectral factor
of +w + v*v.
Let C!(q;;,) denote the Banach space of continuous
&,xn-vdued functions defined on the unit circle under the norm
(15)
may be viewed as a subspace of
Since WO(&,,)
Loo(G,,,), and $ ( ~ ~ ~ , is
, ) the pre-orthogonal of
(G
),,,
in IL1(q;;,)then
, W O ( & x n ) is is*
metrically isomorphic to the dual space of q(q;;,).
Therefore, the pre-orthogonal of S is given by [4, 61
S = ( ( I - RR*)L1(G;:n)
where A =
F=(
2)
It will be shown that the dual space C'(q:;,)*
is isometrically isomorphic to a space M(c:znx,) consisting
(16)
4052
of qn:gn-valued complex bounded measures defined on
the unit circle.
Accordingly, let
U
=
(2 )
Lemma 2 Under assumptions ( A l ) and (A2) thew
ezists at least one optimal Q o E H W ( G x n )such that
E M(cznxn)l and in-
x
troduce the following bilinear form on M(&xn)
e ( q S n1
< v , >=/
~
Tr{K;dvl(o)
+~ , * d v ~ ( o ) } ,
W K )
K I , ZE e(&,,)
(23)
This form has the following equivalent representation:
let w y be the sum of the total variations on [Ole)of all
entries of v1,2. By the Radon-Nikodym Theorem, there
,wU), r = 1, 2,
exists a vector function GY,PE L1(GXn
such that (23) is reduced to
< v,K >=
Next define
poo=
+
Tr{K;GV,i K,*Gv,z}dwv(B) (24)
IWr)
inf
Q € e ( Gx n )
[I (
) - AQll
(33)
N
i.e., when the open unit analyticity constraint is removed. We show that the optimal solution in this case
is flat or allpass, and an approximate flatness condition
holds.
The norm on M(Cznxn)is now defined to be
Lemma 1
Proof:
Since GnX,is the dual space of CjE;,, by Lemma 3.3
[2] and Theorem 2.10 191. Gnxnis finite dimensional,
hence reflexive and it follows that G;;,, is the dual
space of-Gnxn. Then by Singer’s Theorem (p.398
Theorem 1 Under assumptions; ( A l ) (W*W +
v * Q ) ( e i e ) > 0, Vo E [0,_2?r); an! W , f i 1 , fiz, and the
outer spectral factor of w*W + V*V are continuous on
the unit circle. If p, > p,, then
i. Any optimal Q, E Hm(@,.,) in (11) satisfies the
flatness or “allpass” condition
[lo]) W G n x n ) = (e(G%n))*.
Assumption (A2) means we are working in the space
of &,xn-valued functions which are continuous on the
closure of the unit disc 3,and analytic inside D. Call
this space A(C:znxn).The F. and M. Riesz Theorem
[ll]characterizes the orthogonal of A(b(c2nxn) as
A(&nxn
ii. If
{Qn}ris any sequence HW(C,,,,)
={A E ~ ( t 2 n x n ): x = Hm,
$(GZn)}
H E
The dual space of A(C,,,,,>
such that
(35)
(27)
is then the quotient space
A(Gnxn)* = W C Z n x n > / N G n x n ) ’
then
(28)
under the quotient norm for v E M(C2nxn)
The annihilator of S, = S n e(G,&) is then given by
& = {U E M(&nxn)
:dv
p E ~ ( G n x n ) ,G E R i ( c x n ) ) / *
where
*= ( ( I -
+
= ( I - RR*)dP RG,
(30)
where 1.i.m is the limit in quadratic mean. The condition po > poo is sharp.
Proof:
i. Let 4, be the extremal functional corresponding to the extremal measure v, which achieves the
maximum in (32), i.e.,
=< U,,. >, where
$,(a)
+RE;(Cxn))n$(C?Zn)(31)
dpo = Gdw(B),G =
E L*(G,%,w).
Let
be
The following Lemma is then a direct consequence of
the continuous linear extension of qj0 to Lw ( G n X n )
Theorem 1 (p. 121 [8]).
4053
RR*)iW(bnxn)
( zt )
4,
Theorem 2 Under assumptions ( A l ) (WW +
v * v ) ( e i e ) > 0, VB E [0,2?r), (A2) W , R I , Rz are continuous on the unit circle, as is the outer spectral factor
which vanishes on S,. Then
fie=
II(
) -fiQeII
( 2; )
of W*W+pv,
and p, > poo,F, =
E [Fl E 9,
II[qjl~=. 1 is an extrema1 kernel for [ F ] , and Q , is
{T+(W - ii1C?,)'G1 + T r ( f i z Q o ) * G z } d w 4 0 )
- Ilom
5
1
optamal af and only if
+
I{Tr(W - i % Q o ) * G ~ Tr(RzQe)'G,}dwu(0)l
[0,2=)
Ld
max
IC1 51
(I(W - filQO)(eie)CI
+ li%Q.(eiB)CI)
T ~ { ( ( w * ,+
oQ
) W)(
2;
)}(eie)
=
C E C
Proof:
The proof is similar to the proof of Theorem 2
[Cl
that
-
4 Properties of the Optimal Behavior of
MIMO OIWDAP
Suppose now that E is a Bore1 subset of 853 such
that $ ( E ) = 0 and rn(E) > 0. Define the following
essentially bounded matrix valued function
R* = Giii*(eis),
=o,
Then
&(k)= 0 =
for
for 0 E E
eE
/ Tr{GiG3(eie)}drn
(40)
In this section we generalize properties of the optimal
behavior of the ORDAP obtained in [2] for SISO systems to MIMO systems. Note that for the MIMO extension of ORDAP, V is scaled by a parameter T , then
assumption (A2) is strengthened to (A2') The outer
spectral factor of W*W r2Q*Qis continuous for any
positive r , and the outer part of Po is invertible in
Hm( G x n ) .
Recall that in the MIMO case ORDAP assumes the
following form
+
(41)
E
which implies that G3(eie) = 0 rn a.e. in E , but
since m ( E ) > 0, G3 must be identically zero. Therefore the maximum in (32) is achieved on the set
{U E M(c2nxn) : u(0) =. J[i,e)(I- RR*)du'(B), U' E
~ ( ~ 2 n x n ) }which
/ ~ is the annihilator of &'(@nxn)
implying by duality po = poo and then contradicting
our assumption,
. and thus i. must hold.
ii. Follows from the same argument used to prove ii.
Theorem 2 [7].
(43)
Theorem 3 1. Under assumptions (Al), @*I@
+
Q*v)(ei6e)
> " E [o,2.rr)), and
above. If
p, > 0, then there exists an optimal feedback control
V ) such that
law stabilazing every system in %?(Po,
lXloO
Remark 2 The argument used in the proof of i. above
shows that the extrema1 measure U, is absolutely continuous with respect to the Lebesgue measure. Hence
the supremum in (20) is achieved at some function F E
( )
$, II[F]llg = 1. Therefore the optimum
-RQo
satisfies the following Theorem analogoui to Theorem 2
[6], and which provides a test of optimality for feedback
controllers.
4054
IlWl(I - P o Q ) ( I + XVJ'oQ)-'llm
SUP
X E Hw(C,n)
1
=po
(44)
2. If in addition po > boo,then there exasts at least
one optimal controller which stabilizes evew system in
%?(Po:
V ) and achieves (43). I n this case, t h e magnitude of the weighted sensitivity function for the nominal plant Po satisfies
+
I W ( I PoCO)-'(eaB)I
= po - p,1vp,~,(1
(eie)l, rn a.e.
+ poco)-'
(45)
application of Theorem 1 are met. Thus, Q , satisfies
max
IC1 I 1
(I(W - &Qo>(eis)CI + lhQo(eie)CI)=PO
CEG
V8 E [0,2n)
proving that (44) holds, and we are done.
References
[l] R. Schatten. Norm Ideals of Completely Continuous Operators. Springer-Verlag, Berlin, Gottingen,
Heidelberg, 1960.
[2] J.G. Owen. Performance Optimization of Highly
Uncertain Systems in HO". PhD thesis, Dept. of Electrical Eng., McGill University, 1993.
[3] G. Zames. Feedback and optimal sensitivity:
Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE l'hnsactions
on Automatic Control, AC-26(2):301-320, April 1981.
[4] M.S. Djouadi. Optimization of Highly Uncertain
Feedback Systems in Hm. PhD thesis, Dept. of Electrical Eng., McGill University, 1998.
[5] H. Helson. Lectures on Invariant Subspaces. Academic Press, New York and London, 1964.
[6] M.S. Djouadi. Exact solution to the nonstandard Hw problem. Proceedings of the IEEE CDC,
December 1998.
[7] 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.
[SI D.G. Luenberger. Optimization bv Vector Space
Methods. John Wiley, 1968.
191 H. Chapellat and M. Dahleh. Analysis of time
varying control strategies for optimal disturbance rejection and robustness. IEEE lhnsactions on Automatic Control, 37(11):1734-1746,1992.
2. By 1. there exists at least one optimal controller
which stabilizes every system in B(P,, V ) . Therefore
x ( p o )= p o =
inf
QEHm(Gxn)
ess
sup
max
ee[o,zn) IcI 5 1
C E G
(IWI- PoQ)(e")CI + polVPoQ(eie)CI)
implies
x(po) = p, = ess
SUP
max
worn 161 5 1
(IW(I- PoQo)(eie)CI
C E G
+ poIVPoQO(e")Cl I
Assumption (A27 implies that (A2) holds for r?l and
r v for each positive r . Hence all conditions for the
4055
[lo] I. Singer. Sur les applications lineaires integrales
des espaces de fonctions continues. Revue Roumaine
de Math. puws et appl., 4:391-401, 1959.
[ll] R. Ryan. The F. and M. Riesz theorem for vector
measures. Indag. Math., 25:408412, 1963.