INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int. J. Robust Nonlinear Control 2003; 13:1181–1193 (DOI: 10.1002/rnc.835)
Optimal robust disturbance attenuation for continuous
time-varying systems
Seddik M. Djouadin,y
Department of Systems Engineering, University of Arkansas, 2801 S. University, ETAS 300, Little Rock,
Arkansas 72204, U.S.A.
SUMMARY
In this paper we consider the Optimal robust disturbance attenuation problem (ORDAP) for continuous
time-varying systems subject to time-varying unstructured uncertainty. We show that for causal (possibly
time-varying) continuous systems, ORDAP is equivalent to finding the smallest fixed point of a ‘two-disc’
type optimization problem under time-varying feedback control laws. Duality is applied in the context of
nest algebra of causal stable systems, to prove existence of optimal continuous time-varying controllers.
We also show that for time-invariant nominal plants, time-varying control laws offer no improvement over
time-invariant feedback control laws. Copyright # 2003 John Wiley & Sons, Ltd.
KEY WORDS:
optimization; robust control; time-varying; disturbance attenuation; duality
1. DEFINITIONS AND NOTATION
*
*
*
BðE; F Þ denotes the space of bounded linear operators from a Banach space E to a Banach
space F ; endowed with the operator norm.
L2 ½0; 1Þ the standard Lebesgue space of essentially square integrable functions defined on the
interval ½0; 1Þ:
Dh the family of delay operators defined for h50 by
Dh : L2 ½0; 1Þ ! L2 ½0; 1Þ
f ðtÞ ! f ðt hÞ
*
ð1Þ
and for h50 by
Dh : L2 ½0; 1Þ ! L2 ½0; 1Þ
f ðtÞ ! f ðt hÞw½0;1Þ ðtÞ
ð2Þ
n
y
Correspondence to: Dr. Seddik M. Djouadi, Department of Systems Engineering, University of Arkansas, 2801 S.
University, ETAS 300, Little Rock, Arkansas 72204, U.S.A.
E-mail: msdjouadi@ualr.edu
Published online 26 June 2003
Copyright # 2003 John Wiley & Sons, Ltd.
Received 7 November 2001
Accepted 19 November 2002
1182
S. M. DJOUADI
where w½0;1Þ is the characteristic function of the interval ½0; 1Þ: An operator A 2 BðE; F Þ is
said to be time-invariant if, for all h50; A satisfies the operator equation [1]
ADh ¼ Dh A
*
*
Pt the usual truncation operator, which sets all outputs after time t to zero.
An operator A 2 BðE; F Þ is said to be causal if it satisfies the operator equation:
Pt APt ¼ Pt A;
*
ð3Þ
8t 2 ð0; 1Þ
ð4Þ
h:; :i denote the inner or duality product depending on the context.
The subscripts ‘sc ’, ‘c ’ and the superscript ‘ti ’ denote the restriction of a subspace of
operators to its intersection with causal, strictly causal, and time-invariant operator respectively
(see Reference [2] for the definition.) ‘$ ’ stands for the adjoint of an operator or the dual space
of a Banach space depending on the context. ‘*$ ’ denotes convergence in the weak$
topology [3, 4].
2. INTRODUCTION
Analysis of time-varying control strategies for optimal disturbance rejection for known timeinvariant plants has been studied by Shamma and Dahleh [5], Chapellat and Dahleh [1]. A
robust version of these problems were considered in References [6–8] in different induced norm
topologies. They showed that for time-invariant nominal plants, time-varying control laws offer
no advantage over time-invariant ones.
The optimal robust disturbance attenuation problem (ORDAP) was formulated by Zames [9],
and considered by Bird [10–14]. In ORDAP a stable uncertain linear time-varying plant P is
subject to disturbances at the output (see Figure 1).
The objective is to find a feedback control law which provides the best uniform attenuation of
uncertain output disturbances in spite of uncertainty in the plant model. The ORDAP for
discrete time-varying systems and feedback laws has been studied by Owen [13]. He
characterised the problem in a predual space and proved that in some induced norm topology
the same conclusion as before holds. In this paper we consider the optimal robust disturbance
Figure 1. Feedback control in presence of plant and disturbance uncertainty.
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
1183
attenuation problem for continuous time-varying systems subject to time-varying unstructured
plant uncertainty, and therefore generalizing previous results obtained for discrete time-varying
systems by Owen [13]. Here the plant uncertainty set is described by a weighted sphere in the
algebra of bounded linear operators from L2 into L2 instead of H 1 as defined in expression (5),
and the feedback control laws are allowed to be time-varying. In particular we show that for
causal continuous (possibly time-varying) systems, ORDAP is equivalent to finding the smallest
fixed point of a ‘two-disc’ type optimization problem under time-varying feedback control laws.
This in turn is expressed as a shortest distance in a special Banach space of bounded linear
operators related to a functional maximization in its predual space. Therefore proving existence
of optimal time-varying control laws. It is also proved that for continuous time-invariant
nominal plants time-varying control laws offer no improvement over time-invariant feedback
control laws and hence settling an open question in Chapter 8 [13]. Finally, note that the
solution of time-varying ORDAP is important for adaptive control in H 1 ; where plant
uncertainty is reduced using identification and the controllers are allowed to be time-varying.
3. PROBLEM FORMULATION
Let Po 2 Bsc ðL2 ½0; 1Þ; L2 ½0; 1ÞÞ be the nominal (possibly time-varying) plant, and denote the set
of plant uncertainty by
CðPo ; V Þ ¼ fP 2 Bc ðL2 ð1; 1Þ; L2 ð1; 1ÞÞ : P ¼ XVPo þ Po ;
X 2 Bc ðL2 ð1; 1Þ; L2 ð1; 1ÞÞ; jjX jj51g
ð5Þ
where V is a causal stable time-invariant weighting function.
The ORDAP can be shown to be equivalent to finding the optimal worst case sensitivity
function with respect to disturbances and plants in CðPo ; V Þ; achievable by a feedback control
law. With reference to Figure 1, mathematically this problem is equivalent to
mo ¼
inf
sup
C stabilizing P 2CðP0 ;V Þ
P 2CðPo ;V Þ
jjW ðI þ PCÞ1 jj
ð6Þ
where W is a causal stable time-invariant weighting function. Expression (6) can be expressed as
mo ¼
inf
sup
Q2Bc ðL2 ;L2 Þ
P 2CðPo ;V Þ
ðIþXVPo QÞ1 2BðL2 ; L2 Þ
jjW ðI Po QÞðI þ XVPo QÞ1 jj
ð7Þ
If a particular controller Q achieves a ‘worst-case’ weighted sensitivity function less than some
r > 0 then
jjW ðI Po QÞðI þ XVPo QÞ1 jj4r
and
ðI þ XVPo QÞ1 2 Bc ðL2 ; L2 Þ;
8X 2 Bc ðL2 ; L2 Þ;
jjX jj51
ð8Þ
Expression (8) is equivalent to
jjW ðI Po QÞðI þ XVPo QÞ1 f jjL2 4rjjf jjL2
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
1184
S. M. DJOUADI
and
ðI þ XVPo QÞ1 2 Bc ðL2 ; L2 Þ;
8X 2 Bc ðL2 ; L2 Þ;
jjX jj51
8f 2 L2 ½0; 1Þ
ð9Þ
which yields
jjW ðI Po QÞF jjL2 4rjjðI þ XVPo QÞF jjL2
8X 2 Bc ðL2 ; L2 Þ;
jjX jj51;
and
8F 2 L2 ½0; 1Þ;
ðI þ XVPo QÞ1 2 Bc ðL2 ; L2 Þ
jjF jjL2 41
ð10Þ
which is in turn implied by (see Chapter 5 of Reference [2])
jjW ðI Po QÞF jjL2 4rjjF jjL2 rjjVPo QF jjL2
8F 2 L2 ;
jjF jjL2 41
ð11Þ
jjW ðI Po QÞF jjL2 þ rjjVPo QF jjL2 4rjjF jjL2
8F 2 L2 ;
jjF jjL2 41
ð12Þ
hence
Let x be the function defined for r 2 ½0; 1 as follows:
xðrÞ ¼
inf
sup
Q2Bc ðL2 ;L2 Þ jjf jj 2 41
L
f 2L2 ½0;1Þ
ðjjW ðI Po QÞf jjL2 þ rjjVPo Qf jjL2 Þ
ð13Þ
then x is a continuous, positive, non-decreasing function of r: We get a similar Theorem to
Theorem 6.1 for discrete time-varying systems in Reference [13], which will be used to establish
the existence of at least one optimal controller. Theorem 1 asserts that for time-invariant
nominal plants subject to time-varying uncertainty and controllers, ORDAP is equivalent to the
non-standard two-disc optimization (13).
Theorem 1 (Djouadi [15])
(1) Let Po be time-invariant and mo as above, if there exists an optimal Q 2 Bc ðL2 ; L2 Þ for
each r 2 ½0; 1 in the expression (13), then mo is equal to the smallest fixed point of xðrÞ:
(2) If Po is time-varying then mo is bounded above by the smallest fixed point of xðrÞ:
Proof
The proof follows by slightly modifying the proof of Theorem 6.1 in Reference [13]. Let Q* 2
Bc ðL2 ; L2 Þ be such that
jjW ðI Po Q* ÞðI þ XVPo Q* Þ1 jj4m1
and ðI þ XVPo Q* Þ
1
2
2
2 Bc ðL ; L Þ;
where m1 > mo
8X 2 Bc ðL2 ; L2 Þ;
jjX jj51
ð14Þ
Claim 1
lim
sup jjVPo Q* Dt f jjL2 41
t!1 jjf jj 41
L2
f 2L2
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
Suppose by way of contradiction that there exists e > 0 such that
sup jjVPo Q* Dt f jjL2 > 1 þ e; 8t50
1185
ð15Þ
jjf jjL2 41
f 2L2
Fix d > 0 and consider the following construction of a function F 2 L2 ½0; 1Þ
Step 1: let t0 ¼ 0; and select f1 2 L2 ½0; 1Þ; jjf1 jjL2 41; such that f1 has compact support (i.e.
closed and bounded in the topology of the reals) suppðf1 Þ; where the supp denotes the support
of f1 ; which is defined as the smallest closed subset of ½0; 1Þ outside of which f1 vanishes, and
satisfies (15) for f ¼ f1 and t ¼ 0: Since VPo Q* f1 2 L2 ½0; 1Þ there exists a positive real number
t1 > jsuppðf1 Þj; where jsuppðf1 Þj is the Lebesgue measure of suppðf1 Þ; and such that
jjðI Pt ÞVPo Q* f1 jjL2 5d
ð16Þ
1
2
Step k: select fk 2 L ½0; 1Þ with compact support such that jjfk jjL2 41; Ptk1 fk ¼ 0; fk satisfies
(15) for f ¼ fk and t ¼ 0: Define tk > jsuppðfk Þj and such that
jjðI Ptk ÞVPo Q* fk jjL2 5d
ð17Þ
Next define
F ¼
N
1X
fk
N k¼1
If d is chosen sufficiently small with respect to jjVPo Q* jj and e; we get
e
jjVPo Q* F jjL2 > 1 þ
2
ð18Þ
ð19Þ
Define X 2 Bc ðL2 ; L2 Þ to be the following contractive, causal, finite rank linear operator from
L2 ! L 2
Xg ¼
N 1
X
lk hnk ; gifkþ1
ð20Þ
k¼1
where
lk ¼
1
1
4
*
jjVPo Qfk jjL2 e þ 1
and
nk ¼
X maps ðPtk Ptk1 ÞVPo Q* fk to
ðPtk Ptk1 ÞVPo Q* fk
jjðPtk Ptk1 VPo Q* fk jjL2
jjðPtk Ptk1 ÞVPo Q* fk jjL2
fkþ1
jjVPo Q* fk jjL2
for 14k4N 1; hence
N X
f 1 XVPo Q* fN fk XVPo Q* fk1 *
jjðI XVPo QÞF jjL2 4 þ
2 þ
2
N L2
N
N
L
L
k¼2
4
1 þ jjVPo Q* jj
þ 2d
N
Copyright # 2003 John Wiley & Sons, Ltd.
ð21Þ
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
1186
S. M. DJOUADI
If d was chosen sufficiently small for large enough N (22) contradicts (15) and so claim 1 is
proven.
Claim 2
lim sup ðjjW ðI Po Q* ÞDt f jjL2 þ m1 jjVPo Q* Dt f jjL2 Þ4m1
t!0 jjf jj 2 41
L
f 2L2
ð22Þ
The proof of Claim 2 follows exactly as in the proof of Claim 2 (in the proof of Theorem 6.1
[13]) by making the same modifications to prove Claim 1 above, and therefore will not be
repeated here.
Claim 2 implies that
lim
sup ðjjW ðI Po Dt Q* Dt Þf jjL2 þ m1 jjVPo Dt Q* Dt f jjL2 Þ4m1
t!1 jjf jj 41
L2
f 2L2
ð23Þ
since Po ; W and V are causal time-invariant, and Dt Q* Dt causal, we get
sup ðjjW ðI Po Dt Q* Dt Þf jjL2 þ m1 jjVPo Dt Q* Dt f jjL2 Þ4m1
inf
Q2Bc ðL2 ; L2 Þ jjf jj 2 41
L
f 2L2
ð24Þ
The rest of the proof is similar to the proof of Theorem 2.1 in Reference [13], and therefore
omitted.
Hence the optimization (7) for time-invariant nominal plants reduces to
mo ¼
inf
sup
Q2Bc ðL2 ;L2 Þ jjf jj 2 41
L
f 2L2 ½0;1Þ
ðjjW ðI Po QÞf jjL2 þ rjjVPo Qf jjL2 Þ
ð25Þ
Next we proceed to give the duality structure of the problem, which shows existence of an
optimal Q for mo :
4. DUALITY STRUCTURE AND EXISTENCE OF AN OPTIMAL SOLUTION
Denote by A$ the dual space of any Banach space A: If M is a subspace of A then M ? is the
subspace of A$ which annihilates M; that is
M ? :¼ ff 2 A$ : hf ; mi ¼ 0; 8m 2 mg
Isometric isomorphism between Banach spaces is denoted by ’ :
A$ is said to be the predual space of A if ðA$ Þ$ ’ A; and a subspace ? M of A$ is a
preannihilator of a subspace M of A if, ð? MÞ? ’ M: We shall use the following standard result
of Banach space duality theory, which asserts that when a predual and preannihilator exist, then
for any K 2 A [4]
min jjK mjjA ¼
m2M
sup
jhK; f ij
f 2? M;jjf jjA$ 41
To apply this result we first show that (13) is equivalent to a shortest distance minimisation
problem in a specific Banach space. To this end, let L2 be the Banach space L2 ½0; 1Þ L2 ½0; 1Þ
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
1187
!
F1 ¼ jjF1 jjL2 þ jjF2 jjL2
F2 2
L
ð26Þ
under the norm:
The vector function
W ðI Po QÞ
!
VPo Q
can be viewed as a multiplication operator from L2 ½0; 1Þ into L2 with the operator induced
norm:
! W ðI Po QÞ W ðI Po QÞ
f ¼ sup 2
jjf jjL2 41
VPo Q
VPo Q
L
f 2L2
¼ sup ðjjW ðI Po QÞf jjL2 þ jjVPo Qf jjL2 Þ
ð27Þ
jjf jjL2 41
f 2L2
Therefore the optimization problem (13) can be expressed as a distance problem from the vector
function
!
W
0
2
2
belonging to BðL ; L Þ to the subspace
S¼
W
V
!
Po Bc ðL2 ; L2 Þ
of BðL2 ; L2 Þ: To ensure closedness of S; we assume that
W $ ðioÞW ðioÞ þ V $ ðioÞV ðioÞ > 0;
8o
1
Then there exists an outer spectral factor L1 2 H ; invertible in H 1 such that L$1 ðioÞL1 ðioÞ ¼ W $ ðioÞW ðioÞ þ V $ ðioÞV ðioÞ: Therefore L1 Po as a bounded linear operator in Bc ðL2 ; L2 Þ
has a polar decomposition U1 G; where U1 is a partial isometry and G a positive operator both
defined on L2 (Theorem 4.39 [3]). Next we assume
(A) U1 is unitary, the operator G and its inverse G1 2 Bc ðL2 ; L2 Þ:
In the time-invariant case, (A) is satisfied when, for e.g. the outer factor of the plant is
invertible.
Letting
!
W
R¼
L1
1 U1
V
assumption (A) implies that the operator R$ R 2 BðL2 ; L2 Þ has a bounded inverse, this ensures
closedness of S: According to Arveson (Corollary 2, [16], see also Reference [2]), the self-adjoint
operator R$ R has a spectral factorization of the form:
R$ R ¼ L$ L;
Copyright # 2003 John Wiley & Sons, Ltd.
where L; L1 2 Bc ðL2 ; L2 Þ
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
1188
S. M. DJOUADI
Defining R2 ¼ RL1 ; then R$2 R2 ¼ I; and S has the equivalent representation, S ¼ R2 Bc ðL2 ; L2 Þ: After absorbing L into the free parameter Q; the optimization problem (13) is then
equivalent to:
!
W
R2 Q ð28Þ
mo ¼
inf Q2Bc ðL2 ;L2 Þ 0
Let L2 be the Banach space L2 L2 under the norm
! g1 ¼ maxðjjg1 jjL2 ; jjg2 jjL2 Þ
g2 2
L
ð29Þ
The following Lemma characterizing the dual space of L2 follows from Reference [17]:
Lemma 1
Let L2 and L2 defined as above, then the following hold:
*
*
ðL2 Þ$ ’ L2
ðL2 Þ$ ’ L2
Hence all these Banach spaces are reflexive.
Define NðB1 ; B2 Þ to be the Banach space of nuclear operators mapping the Banach space B1 to
the Banach space B2 under the nuclear norm. Recall that an operator A : B1 ! B2 is said to be
nuclear if it has the representation [18]
X
Af ¼
hFn$ ; f ien where en 2 B2 ; Fn$ 2 B$1
ð30Þ
n
and
X
jjFn$ jj jjen jj51
ð31Þ
n
where h; i is the duality product and the nuclear norm is defined to be
(
)
X
$
jjAjjnuc ¼ inf
jjFn jj jjen jj
ð32Þ
n
where the infimum is taken over all possible representations of A:
The trace of the nuclear operator A is denoted by trðAÞ and is defined by
X
trðAÞ ¼
hFn$ ; Fn i
ð33Þ
n
This sum is well defined and can be shown to be independent of the representation [18]. The
following Lemma applies to the Banach spaces L2 and L2 :
Lemma 2 (Chapellat and Dahleh [1], Diestel and Uhl [18])
BðL2 ; L2 Þ ’ NðL2 ; L2 Þ$
2
2
ð34Þ
$
If f 2 NðL ; L Þ ; the isometric isomorphism ’ is given by
fðAÞ ¼ hB; Ai ¼ trA$ B ¼ tr B$ A
Copyright # 2003 John Wiley & Sons, Ltd.
where A 2 NðL2 ; L2 Þ; B 2 BðL2 ; L2 Þ
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
1189
Next, we need the following Definition introducing the concept of nest Algebra.
Definition 1 (Davidson [19])
*
*
*
A nest is a chain N of closed subspaces of a Hilbert space H containing f0g and H which is
closed under intersection and closed span.
The triangular algebra or nest algebra TðNÞ is the set of all operators T such that TN N
for every element N in N:
W
Given
V a collection fNa g of subspaces of a Hilbert space, a Na denotes the closed linear span
and a Na denotes intersection of the subspaces Na : For N belonging to a nest N; define
_
N ¼
fN 0 2 N : N0 5Ng
ð35Þ
^
fN 0 2 N : N0 > Ng
ð36Þ
Nþ ¼
where N 0 5N means N 0 N ; and N 0 > N means N 0 *N :
The subspaces N N are called the atoms of N: If there are no atoms, N is called
continuous.
Now let us define for each t 2 ð1; 1Þ; Mt DðI Pt ÞðL2 Þ; i.e. the subspace of L2 consisting of all
%
functions f 2 L2 such that f ¼ 0; a.e. on ð1;
tÞ: Then MDfMt ; 15t51g is clearly a
%
continuous nest since
_
fMt 2 M : t5tg ¼ Mt
ð37Þ
Mt ¼
t
2
2
The space Bc ðL ; L Þ can be viewed as a nest algebra since causal operators leave M invariant,
i.e. for all operators A 2 Bc ðL2 ; L2 Þ; AMt Mt ; for all Mt 2 M:
The Banach space NðL2 ; L2 Þ is nothing but the well-known space of trace-class operators
from L2 to L2 ; and that the nuclear norm in this case reduces to the trace-norm [20]. Applying
Theorem 16.6 [19], we get the following Lemma
Lemma 3
Nc ðL2 ; L2 Þ is the preannihilator of Bc ðL2 ; L2 Þ:
Proof
The preannihilator of Bc ðL2 ; L2 Þ;
?
?
Bc ðL2 ; L2 Þ is defined by
Bc ðL2 ; L2 Þ ¼ fA 2 NðL2 ; L2 Þ : trðT $ AÞ ¼ 0; 8T 2 Bc ðL2 ; L2 Þg
ð38Þ
Mt
be the closed linear span of fMt 2 M : t5tg; but since M is a continuous nest, then
Let
Mt ¼ Mt : It follows from Theorem 16.6 [19] that ? Bc ðL2 ; L2 Þ ¼ Nc ðL2 ; L2 Þ:
Now let S$ be the subspace of NðL2 ; L2 ð0; 1ÞÞ defined as follows:
S$ ¼ fpðCðI R2 R$2 Þ þ AR$2 ÞjL2 : C 2 NðL2 ; L2 Þ; A 2 Nc ðL2 ; L2 Þg
ð39Þ
where p is the canonical projection of L2 ð1; 1Þ into L2 ½0; 1Þ; and L2 is defined to be the
Banach space L2 ð1; 1Þ L2 ð1; 1Þ endowed with the norm of L2 :
Define the following subspace of NðL2 ; L2 Þ
?
S ¼ fA 2 NðL2 ; L2 Þ : A$ 2 S$ g
Copyright # 2003 John Wiley & Sons, Ltd.
ð40Þ
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
1190
S. M. DJOUADI
Lemma 4
?
S is the preannihilator of S in NðL2 ; L2 Þ:
Proof
Let T 2 BðL2 ½0; 1Þ; L2 Þ; then
hT ; Ai ¼ 0;
8A 2? S , trðpðCðI R2 R$2 ÞjL2 T þ pNR$2 jL2 T Þ ¼ 0
8C 2 NðL2 ; L2 ½0; 1ÞÞ;
N 2 Nc ðL2 ; L2 Þ
, ðI R2 R$2 ÞjL2 Tp ¼ 0 and R$2 jL2 Tp 2 Bc ðL2 ; L2 Þ by Lemma 3
, ðI R2 R$2 ÞjL2 Tp ¼ 0 and R2 R$2 jL2 Tp 2 R2 Bc ðL2 ; L2 Þ
, T 2 R2 Bc ðL2 ; L2 Þ
Using the standard result from Banach space duality theory relating the distance from a vector
to a subspace and an extremal functional in the predual mentioned previously (Theorem 2,
Chapter 5.8 [4]), we deduce the following Theorem
Theorem 2
Under assumption (A) U1 is unitary, G and G1 2 Bc ðL2 ; L2 Þ:
There exists at least one optimal Qo 2 Bc ðL2 ; L2 Þ; i.e. a linear time-varying control law such
that
!
!
W
W
R2 Qo R2 Q ¼ min 0
Q2Bc ðL2 ;L2 Þ 0
!!
W
$
¼ sup tr A
ð41Þ
jjAjjnuc 41
0
A2? S
Moreover by a standard weak$ compactness argument the supremum in (41) is in fact a
maximum.
Note that Theorem 2 shows merely than an optimal time-varying controller exists, but does
not show how to compute it. The computation of such a controller would certainly be
important, in particular, in adaptive control where plant uncertainty is reduced using
identification algorithms.
5. TIME-VARYING VERSUS TIME-INVARIANT CONTROL LAWS
In this section, we assume that the nominal plant Po is time-invariant. It follows also that R2 is
also time-invariant. Define the following performance index when the controllers are
constrained to be linear time-invariant:
!
W
ti
ð42Þ
mo ¼
R
inf
Q
2
ti 2 2 Q2Bc ðL ; L Þ
0
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
1191
In the next Theorem, we show that no advantage is gained in performance if we allow our
controllers to be time-varying, and thus generalizing the same result obtained for discrete time
systems in Reference [13].
Theorem 3
If Po is linear, causal, time invariant and assumption (A) holds, then
mo ¼ mtio
ð43Þ
Proof
Since Btic ðL2 ; L2 Þ Bc ðL2 ; L2 Þ; it is obvious that mo 4mtio :
To prove the opposite inequality let Q 2 Bc ðL2 ; L2 Þ
!
W
mo ¼ R2 Q 0
ð44Þ
such a Q exist by Theorem 2. Using an idea of Chapellat and Dahleh [1], let h > 0: Since
jjDu jj41; 8u; we have that 8n > 0
!
! W
ð45Þ
mo 5Dnh
R2 Q Dnh 0
Using time-invariance of W and R2 ; we get
!
W
mo 5
R2 Dnh QDnh 0
ð46Þ
Define for all n50
Qn ðhÞ ¼
Then (46) yields
n
1 X
Dkh QDkh
1 þ n k¼0
ð47Þ
!
W
mo 5
R2 Qn ðhÞ
0
ð48Þ
jjQn ðhÞjj4jjQjj; 8n50 and then jjR2 Qn ðhÞjj is uniformly bounded by jjR2 jjjjQjj: Moreover since the
predual space of BðL2 ; L2 Þ is NðL2 ; L2 Þ; by Alaoglu’s Theorem there exists a subsequence of
fR2 Qnk ðhÞg which converges in the weak$ topology to some vector function R 2 BðL2 ; L2 Þ; and
we write
R2 Qnk ðhÞ *$ R
)
W
0
!
R2 Qnk ðhÞ *
$
W
0
!
R
by (48) and a property of weak$ limits, we have
!
!
W
W
R mo 5 lim inf R2 Qnk ðhÞ5
k!1 0
0
Copyright # 2003 John Wiley & Sons, Ltd.
ð49Þ
ð50Þ
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
1192
S. M. DJOUADI
By definition of the weak$ limit, 8A 2? S; 0 ¼ hR2 Qnk ðhÞ; Ai ! hR; Ai as k ! 1; hence R ¼
R2 Q* ðhÞ; for some Q* ðhÞ 2 Bc ðL2 ; L2 Þ: The adjoint of Dh ; D$h is defined by 8f 2 L2
ðD$h f ÞðtÞ ¼ f ðt þ hÞ ¼ ðDh f ÞðtÞ
Now 8A 2 NðL2 ; L2 Þ we have
hDh R2 Qnk ðhÞ; Ai ¼ tr A$ Dh R2 Qnk ðhÞ ¼ tr ðDh AÞ$ R2 Qnk ðhÞ
¼ hR2 Qnk ðhÞ; Dh Ai !k!1 hR2 Q* ðhÞ; Dh Ai ¼ hDh R2 Q* ðhÞ; Ai
thus Dh R2 Qnk ðhÞ *$ Dh R2 Q* ðhÞ; and likewise R2 Qnk ðhÞDh *$ R2 Q* ðhÞDh : From Reference [1], jj
Dh Qnk ðhÞ Qnk ðhÞDh jj ! 0; as k ! 1; hence jjDh R2 Qnk ðhÞ R2 Qnk ðhÞDh jj ! 0; as k ! 1:
By the same reasoning as previously, 8A 2 NðL2 ; L2 Þ; Dh R2 Qnk ðhÞ *$ R2 Q* ðhÞDh : We
conclude that by uniqueness of the weak$ limit
Dh R2 Q* ðhÞ ¼ R2 Q* ðhÞDh
ð51Þ
jjR2 Q* ðhÞjj4jjR2 jjjjQjj
ð52Þ
!
W
R2 Q* ðhÞ
mo 5 0
ð53Þ
For all h > 0; we showed that
Finally for all n51; define the sequence R2 Q* n ¼ R2 Q* ð1=nÞ: By (52) it is bounded, therefore there
exists Q% such that R2 Q* nk *$ R2 Q% and R2 Q% satisfies (53) as before. Again using properties of
nuclear operators it is easy to see that R2 Q* nk Dh Dh R2 Q* nk *$ R2 Q% Dh R2 Q% : Using exactly the
same procedure as in Reference [1] to prove that R2 Q* nk Dh Dh R2 Q* nk *$ 0 shows that R2 Q% is
time-invariant, but since R2 is time-invariant, Q% must also be time-invariant and this completes
the proof.
Note that Theorem 3 only shows that time-varying controllers offer no advantage over timeinvariant ones, if the nominal plant Po ; the radius of plant uncertainty V and disturbance
weighting W are time-invariant. If either of V or W are reduced, by e.g. identification, timevarying controllers would certainly offer improved performance as in adaptive control.
6. CONCLUSION
In this paper, we formulated the ORDAP for continuous-time varying plants subject to timevarying unstructured uncertainty. We showed that for causal time-varying systems, ORDAP is
equivalent to finding the smallest fixed point of ‘a two-disc’ type optimization problem under
time-varying feedback control laws. The space of causal and stable continuous plants was
viewed as a continuous nest (or triangular) algebra of operators acting on the Hilbert space of
finite energy inputs into finite energy outputs. Duality was then applied to show existence of
optimal continuous time-varying control laws. It was also shown that for continuous timeinvariant nominal plants time-varying control laws offer no improvement over time-invariant
feedback control laws and hence setting an open question in Reference [13].
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193
OPTIMAL ROBUST DISTURBANCE ATTENUATION PROBLEM
1193
REFERENCES
1. Chapellat H, Dahleh M. Analysis of time-varying control strategies for optimal disturbance rejection and
robustness. IEEE Transactions on Automatic Control 1992; 37:1734–1746.
2. Feintuch A, Saeks R. System Theory: A Hilbert Space Approach. Academic Press: NY, 1982.
3. Douglas RG. Banach Algebra Techniques in Operator Theory. Academic Press: NY, 1972.
4. Luenberger DG. Optimization by Vector Space Methods. John-Wiley: NY, 1968.
5. Shamma JS, Dahleh MA. Time-varying versus time-invariant compensation for rejection of persistent bounded
disturbances and robust stabilization. IEEE Transactions on Automatic Control 1991; 36:838–847.
6. Shamma JS. Robust stability with time-varying structured uncertainty. Proceedings of the IEEE Conference on
Decision and Control 1992; 3163–3168.
7. Khammash M, Dahleh M. Time-varying control and the robust performance of systems with structured normbounded perturbations. Proceedings of the IEEE Conference on Decision and Control, Brighton, UK, 1991.
8. Khammash M, Pearson JB. Performance robustness of discrete-time systems with structured uncertainty. IEEE
Transactions on Automatic Control 1991; 36:398–412.
9. Zames G. Feedback and optimal sensitivity: model reference transformation, multiplicative seminorms, and
approximate inverses, IEEE Transactions on Automatic Control 1981; 26:301–320.
10. Bird JF, Francis BA. On the robust disturbance attenuation problem. Proceedings of the IEEE Conference on
Decision and Control 1986; 1804–1809.
11. Francis BA. On disturbance attenuation with plant uncertainty. Workshop on New Perspectives in Industrial Control
System Design, 1986.
12. Zames G, Owen JG. Duality theory for MIMO robust disturbance rejection. IEEE Transactions on Automatic
Control 1993; 38:743–752.
13. Owen JG. Performance optimization of highly uncertain systems in H 1 : PhD Thesis, McGill University, Montreal,
Canada, 1993.
14. Owen JG, Zames G. Robust disturbance minimization by duality. Systems and Control Letters 1992; 29:255–263.
15. Djouadi SM. Optimization of highly uncertain feedback systems in H 1 : PhD Thesis, McGill University, Montreal,
Canada, 1998.
16. Arveson W. Interpolation problems in nest algebras. Journal of Functional Analysis 1975; 4:67–71.
17. Dieudonn!ee J. Sur le Th!eor"eme de Lebesgue Nikodym V. Canadian Journal of Mathematics 1951; 3:129–139.
18. Diestel J, Uhl JJ. Vector measures. Mathematical Surveys, vol. 15. American Mathematical Society: Providence, RI,
1977.
19. Davidson KR. Nest Algebras. Longman Scientific & Technical: UK, 1988.
20. Schatten R. Norm Ideals of Completely Continuous Operators. Springer-Verlag: Berlin, Gottingen, Heidelberg, 1960.
Copyright # 2003 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2003; 13:1181–1193