Proceedings of the European Control Conference 2007
Kos, Greece, July 2-5, 2007
TuD14.1
Time-Varying Optimal Disturbance Rejection in Presence of Plant
Uncertainty
Seddik M. Djouadi, Charalambos D. Charalambous
Abstract— The optimal robust disturbance rejection
problem plays an important role in feedback control
theory. Here its time-varying version is solved explicitly
in terms of duality and operator theory. In particular,
the optimum is shown to satisfy a time-varying allpass
property. Moreover, optimal performance is given in
terms of the norm of a bilinear form. The latter depends
on a lower triangular projection and a multiplication
operator defined on special versions of spaces of compact
operators.
D EFINITIONS AND N OTATION
•
B(E, F ) denotes the space of bounded linear
operators from a Banach space E to a Banach
space F , endowed with the operator norm
kAk :=
sup
kAxk, A ∈ B(E, F )
x∈E, kxk≤1
•
ℓ2 denotes the usual Hilbert space of square
summable sequences with the standard norm
kxk22
:=
∞
X
j=0
•
•
¡
¢
|xj |2 , x := x0 , x1 , x2 , · · · ∈ ℓ2
Pk the usual truncation operator for some integer
k , which sets all outputs after time k to zero.
An operator A ∈ B(E, F ) is said to be causal if
it satisfies the operator equation:
Pk APk = Pk A, ∀k positive integers
The subscript “c ” denotes the restriction of a subspace
of operators to its intersection with causal operators,
that is Bc (E, F ) (see [3], [2] for the definition.)
Bounded and causal linear operators can be represented
by lower triangular “infinite” matrices, with respect
2
to the canonical basis, {ei }∞
1 of ℓ , where the entries
S.M. Djouadi is with the Department of Electrical and Computer Engineering at The University of Tennessee, Knoxville.
djouadi@ece.utk.edu
C.D. Charalambous is with the Electrical & Computer Engineering Department, University of Cyprus, Nicosia, 1678, Cyprus.
chadcha@ucy.ac.cy
ISBN: 978-960-89028-5-5
of {ei } are all zero except that the entry at the i-th
position is 1.
The symbol “⊕” denotes the direct sum of two
spaces. “⋆ ” stands for the adjoint of an operator or
the dual space of a Banach space depending on the
context [10], [14].
I. I NTRODUCTION
The optimal robust disturbance attenuation problem
plays a fundamental role in feedback optimization
[26], [19]. In particular, it has been shown in [26],
for linear time-invariant (LTI) systems, using a
counter example based on a ”two-arc” result, that
approximate solutions employing state space robust
control theory may result in arbitrary poor solutions.
An exact solution based on operator theory and duality
theory for LTI systems has been proposed in [16], [15].
In this paper, we consider the optimal disturbance
rejection problem is considered for time-varying
systems generalizing certain results which hold in
the LTI case. Characterization of the optimal solution
in part by duality theory has been proposed in [17],
albeit for continuous time systems. It was also shown
there that for time-invariant nominal plants and
weighting functions, time-varying control laws offer
no improvement over time-invariant feedback control
laws.
Analysis of time-varying control strategies for optimal
disturbance rejection for known time-invariant plants
has been studied in [24], [5]. A robust version of
these problems were considered in [23], [12], [13] 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 [25], and
considered in [4], [11], [26], [19]. In ORDAP a
stable uncertain linear time-varying plant P is subject
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TuD14.1
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. We consider ORDAP for time-varying systems
subject to time-varying unstructured plant uncertainty,
and therefore generalizing previous results obtained
for LTI systems in [16].
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 ∞ as defined
in expression (1), and the feedback control laws and
weights are allowed to be time-varying. In particular
we show that ORDAP satisfies a time-varying allpass
condition, and that it is given in terms of the norm of
a bilinear form, which depends on a lower triangular
projection and a multiplication operator defined on
special versions of spaces of compacts operators.
The solution of time-varying ORDAP is important
for adaptive control in H ∞ , where plant uncertainty
is reduced using identification and the controllers are
allowed to be time-varying.
The paper is organized as follows, section II contains
the formulation of ORDAP in terms of a feedback
optimization. In section III the optimal solution is
characterized in terms of duality theory, where the
annihilator is computed explicitly. This contrasts
with the results of [17] where the annihilator was
characterized implicitly. In section IV, the optimum
is shown to satisfy an allpass condition. Section V
shows that the optimal solution is equal to the operator
induced norm of a bilinear transformation, defined on
particular spaces of compact operators. The bilinear
form is computed explicitly and involves a triangular
projection analogous to the standard Riesz projection
known in the context of Hardy H 2 spaces.
II. P ROBLEM F ORMULATION
Let Po ∈ Bsc (ℓ2 , ℓ2 ) be the nominal (possibly timevarying) plant, and denote the set of plant uncertainty
by
C(Po , V ) = {P ∈ Bc (ℓ2 , ℓ2 ) : P = XV Po + Po ,
2
(1)
u
uncertain Plant
+
-
stable Filter
+
P
output y
C
controller
Fig. 1. Feedback Control in Presence of Plant and Disturbance
Uncertainty
feedback control law. With reference to Figure 1 above,
mathematically this problem is equivalent to
°
°
µo =
inf
sup °W (I + P C)−1 ° (2)
C stabilising P ∈C(P0 ,V )
P ∈ C(Po , V )
where W is a causal stable time-varying weighting
function. Expression (2) can be expressed as
µo =
inf
sup
P ∈C(Po ,V )
Q ∈ Bc (L2 , L2 )
(I + XV Po Q)−1 ∈ B(L2 , L2 )
°
°
°W (I − Po Q)(I + XV Po Q)−1 ° (3)
The optimization (3) is termed as the time-varying optimal robust disturbance attenuation problem, in analogy
with its time-invariant counterpart solved in [15], [16].
(3) is shown in [17] to be equal to the smallest positive
fixed point of the function ξ defined for r ∈ [0, 1] as
follows:
ξ(r) =
inf
Q∈Bc (ℓ2 ,ℓ2 )
sup
kf k2 ≤ 1
f ∈ ℓ2
(kW (I − Po Q)f k2 + r kV Po Qf k2 )
(4)
The function ξ(r) is a continuous, positive, nondecreasing function of r.
Therefore, in principle all that is required to solve
the optimization (3) is to solve the following type of
optimization
2
µo :=
X ∈ Bc (ℓ , ℓ ), kXk < 1}
where V is a causal stable time-varying 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 C(Po , V ), achievable by a
W
inf
Q∈Bc (ℓ2 ,ℓ2 )
sup
kf kL2 ≤ 1
f ∈ ℓ2
(kW (I − Po Q)f k2 + kV Po Qf k2 )
(5)
The rest of the paper characterizes the solution of (5)
in terms of duality and operator theory.
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TuD14.1
III. BANACH S PACE D UALITY T HEORY
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 ⊥ := {f ∈ A⋆ : < f , m > = 0, ∀m ∈ M }
Isometric isomorphism between Banach spaces is denoted by ≃.
A⋆ is said to be the predual space of A if (A⋆ )⋆ ≃ A,
and a subspace ⊥ M of A⋆ is a preannihilator of a
subspace M of A if, (⊥ M )⊥ ≃ M . We shall use
the following standard result of Banach space duality
theory asserts that when a predual and preannihilator
exist, then for any K ∈ A [14]
min kK − mkA =
m∈M
sup
| < K, f > |
f ∈⊥ M, kf kA⋆ ≤1
To apply this result we first show that (4) is equivalent
to a shortest distance minimization problem in a specific Banach space. To this end, let L2 be the Banach
space ℓ2 × ℓ2 under the norm
°µ
¶°
° F1 °
°
°
(6)
° F2 ° 2 = kF1 k2 + kF2 k2
L
µ
¶
W (I − Po Q)
The vector function
is viewed as a
V Po Q
2
2
bounded
multiplication
µ
¶ operator from ℓ into L , that
W (I − Po Q)
is,
∈ Bc (ℓ2 , L2 ) with the operator
V Po Q
induced norm
°
°
° W (I − Po Q) °
°
°
(7)
°
°
V Po Q
°µ
¶ °
° W (I − Po Q)
°
°
=
sup
f°
°
° 2
V Po Q
L
kf k2 ≤ 1
2
f ∈ℓ
¡
¢
kW (I − Po Q)f k2 + kV Po Qf k2
=
sup
kf kL2 ≤ 1
f ∈ L2
Therefore the optimization problem (4) can be expressedµ as a distance
problem from the vector function
¶
W
K :=
belonging to B(ℓ2 , L2 ) to the subspace
0
µ
¶
W
S=
Po Bc (ℓ2 , ℓ2 ) of B(ℓ2 , L2 ).
V
To ensure closedness of S , we assume that W ⋆ W +
V ⋆ V > 0, i.e., W ⋆ W + V ⋆ V > 0 is a positive
operator. Then there exists an outer spectral factorization Λ1 ∈ Bc (ℓ2 , ℓ2 ), invertible in Bc (ℓ2 , ℓ2 ) such that
Λ⋆1 Λ1 = W ⋆ W + V ⋆ V [1], [2]. Therefore Λ1 P as a
bounded linear operator in Bc (ℓ2 , ℓ2 ) has an inner-outer
factorization U1 G, where U1 is inner and G an outer
operator defined on ℓ2 [6].
Next, we assume (A1) G is invertible, so U1 is unitary,
and the operator G and its inverse G−1 ∈ Bc (ℓ2 , ℓ2 ).
(A1) is satisfied when, for e.g., the outer factor of the
plant is invertible.
Let R = T2 Λ−1
1 U1 , assumption (A1) implies that the
operator R⋆ R ∈ B(ℓ2 , ℓ2 ) has a bounded inverse,
this ensures closedness of S . According to Arveson
(Corollary 2, [1]), the self-adjoint operator R⋆ R has a
spectral factorization of the form: R⋆ R = Λ⋆ Λ, where
Λ, Λ−1 ∈ Bc (ℓ2 , ℓ2 ).
Define R2 = RΛ−1 , then R2⋆ R2 = I , and S has the
equivalent representation, S = R2 Bc (ℓ2 , ℓ2 ). After ”absorbing” Λ into the free parameter Q, the optimization
problem (4) is then equivalent to
°
°µ
¶
°
° W
°
(8)
µo =
inf
− R2 Q°
°
°
0
Q∈Bc (ℓ2 ,ℓ2 )
Let L2 be the Banach space ℓ2 × ℓ2 under the norm:
°µ
¶°
° g1 °
°
°
(9)
° g2 ° 2 = max(kg1 k2 , kg2 k2 )
L
The following Lemma characterizing the dual space of
L2 follows from [8].
Lemma 1: Let L2 and L2 defined as above, then
the following hold
• (L2 )⋆ ≃ L2
• (L2 )⋆ ≃ L2
Hence all these Banach spaces are reflexive.
Introduce the class of compact operators on ℓ2 called
the nuclear operators acting from ℓ2 into L2 , denoted
C1 (ℓ2 , L2 ), under the nuclear norm [22],
o
nX
kFj k2 · kej kL2
kAkn = inf
j
where the infimum is taken over all possible representations of A,
P
Af =
< Fj , f > ej , ej ∈ L2 , Fj ∈ ℓ2(10)
Pj
(11)
and
j kFj k2 · kej kL2 < ∞
where < · , · > is the inner product in ℓ2 .
We identify B(ℓ2 , L2 ) with the dual space of
C1 (ℓ2 , L2 ), C1⋆ (ℓ2 , L2 ), under trace duality [21], [22],
that is, every operator A in B(ℓ2 , L2 ) induces a
continuous linear functional on C1 (ℓ2 , L2 ) as follows:
2397
ΦA ∈ C1⋆ (ℓ2 , L2 )
TuD14.1
is defined by ΦA (T ) = tr(A⋆ T ), and we write
However, we show first that the optimum satisfies a
time-varying allpass condition.
B(ℓ2 , L2 ) ≃ C1⋆ (ℓ2 , L2 )
Every nuclear operator T in turn induces a
bounded linear functional on B(ℓ2 , L2 ), namely
ΦT (A) = tr(T ⋆ A) for all A in B(ℓ2 , L2 ).
The preannihilator of Bc (ℓ2 ,
given by [20]
ℓ2 ), denoted S , is
S := {T ∈ C1 (ℓ2 , ℓ2 ) : (I − Qn )T Qn+1 = 0,
for all n}
(12)
where C1 (ℓ2 , ℓ2 ) is the trace-class for operators acting
on ℓ2 into ℓ2 .
Define the following subspace of C1
(ℓ2 ,
L2 ),
S o := (I − R2 R2⋆ )C1 (ℓ2 , L2 ) ⊕ R2 S
(13)
The following Lemma states that S o is the preannihilator of the subspace S .
Lemma 2: If φ ∈ X , then
⋆
o
tr(φ T ) = 0 for all T ∈ S ⇐⇒ φ ∈ S
(14)
Proof. To show (14) it suffices to notice that
tr(φ⋆ T ) = 0, ∀T in S o is equivalent to
φ⋆ (I − R2 R2⋆ ) = 0 and φ⋆ R = A⋆ for some
A ∈ Bc (ℓ2 , ℓ2 ), and so these imply that φ⋆ = A⋆ R⋆ .
By taking the adjoints we get φ = R2 A ∈ S .
Using Theorem 2, Chapter 5.8 [14], relating the
distance from a vector to a subspace and an extremal
functional, we deduce the following Theorem.
Theorem 1: Under assumption (A1), there exists at
least one optimal Qo ∈ Bc (ℓ2 , ℓ2 ), i.e., a linear timevarying control law such that:
°µ
°
¶
° W
°
°
min °
− R2 Q°
°
0
Q∈Bc (ℓ2 ,ℓ2 )
°µ
°
¶
° W
°
− R2 Qo °
=°
° 0
°
¯ µ µ
¶¶¯
¯
¯
W
⋆
¯
¯
=
sup
(15)
¯tr A
¯
0
kAkn ≤ 1
A ∈ ⊥S
Note that Theorem 1 shows only that an optimal
time-varying controller exits, but does not show how to
compute it. We propose to compute such a controller
in the sequel by quantifying µo in terms of operator
theory. The computation of such a controller is important, in particular, in adaptive control where plant
uncertainty is reduced using identification algorithms.
IV. TV A LLPASS P ROPERTY OF THE O PTIMUM
In the standard H ∞ theory the space B(ℓ2 , ℓ2 ) corresponds to L∞ . The dual space of L∞ is given by the
so-called Yosida-Hewitt decomposition L∞ ≃ L1 ⊕C ⊥ ,
where L1 is the standard Lebesgue space of absolutely
integrable functions and C ⊥ is the annihilator of the
space of continuous functions C defined on the unit
circle.
By analogy the dual space of B(ℓ2 , ℓ2 ) is given by the
space [21],
B(ℓ2 , ℓ2 )⋆ ≃ C1 ⊕1 K⊥
(16)
where K⊥ is the annihilator of K and the symbol ⊕1
means that if Φ ∈ C1 ⊕1 K⊥ then Φ has a unique
decomposition as follows
Φ = Φo + ΦT
(17)
kΦk = kΦo k + kΦT k
(18)
where Φo ∈ K⊥ , and ΦT is induced by the operator T ∈ C1 . By the same token, the dual space of
B(ℓ2 , L2 )⋆ is isometrically isomorphic to the Banach
space C1 (ℓ2 , L2 ) ⊕1 K⊥ , i.e.,
B(ℓ2 , L2 )⋆ ≃ C1 (ℓ2 , L2 ) ⊕1 K⊥
(19)
where in this case K is the space of compact operators
acting from ℓ2 into L2 , and K⊥ its annihilator.
The annihilator S ⊥ of S in C1 (ℓ2 , L2 ) ⊕1 K⊥
is given by
´
³
S ⊥ = (I − R2 R2⋆ ) C1 (ℓ2 , L2 ) ⊕1 K⊥ ⊕R2 S (20)
Banach space duality states with the existence of an
annihilator that [14]
inf kx − yk =
y∈S
max
Φ∈S ⊥ , kΦk≤1
|Φ(x)|
(21)
The maximizing Φopt in the dual space can be written
as
Φopt = Φo + ΦTo
kΦopt k = kΦo k + kΦTo kn = 1
(22)
(23)
where Φo ∈ K⊥ , and ΦTo is induced by the operator
To ∈ S o . In others words, the following result holds
°µ
°
¶
° W
°
µo =
min °
− R2 Q°
(24)
°
°
0
Q∈Bc (ℓ2 ,ℓ2 )
¯ µµ
¯
¶¶
¯
¯
W
⋆
¯
(25)
= ¯Φo
+ tr ((W , 0)To )¯¯
0
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TuD14.1
If Qo achieves the minimum in (25), then the alignment
condition in the dual is given by
¯ µµ
¯
¶¶
¯
¯
W
⋆
¯Φo
+ tr ((W , 0)To )¯¯ =
¯
0
°µ
°
¶
°¡
° W
¢
° kΦo k + kΦTo kn (26)
°
−
R
Q
2
o
°
° 0
µ
¶
W
If we further assume that (A2):
as an operator
0 µµ
¶¶
W
2
2
from ℓ into L is compact, then Φo
=0
0
o
maximum in (21) is achieved on S , that is, the
supremum in (15) becomes a maximum. It is
instructive to note that in the LTI case assumption
µ
¶
W
(A2) is the analogue of the assumption that
0
is the sum of two parts, one part continuous on the
unit circle and the other in H ∞ , in which case the
optimum is allpass [26], [15]. By analogy with the
LTI case we would like to find the allpass equivalent
for the optimum in the linear time varying case. This
may be formulated by noting that flatness or allpass
condition in the LTI case means that the modulus
of the optimum µ|(W −¶R21 Qo )(eiθ )| + |R22 Qo (eiθ )|,
R21
, is constant at almost all
where R2 =
R22
frequencies (equal to µo ).
In terms of operator theory, the optimum viewed
as a multiplication operator acting on L2 or H 2 ,
changes the norm of any function in L2 or H 2 by
multiplying it by a constant (=µo ). In other terms
allpass property for the LTI case is equivalent to
A compactness argument (see [18]) shows that the
optimal Qo ∈ µBc (ℓ2¶, ℓ2 ) may be chosen to be
W
compact, when
is.
0
In the next section, we relate our problem to
an LTV bilinear form analogous to the LTI bilinear,
which solves the optimal robust disturbance attenuation
problem in the LTI case [16]. The latter can be realized
by invoking tensor products of operators along the
lines [16], albeit in different spaces of linear causal
compact operators.
V. A S OLUTION BASED ON O PERATOR T HEORY
Let C2 denote the class of compact operators acting
from ℓ2 called the Hilbert-Schmidt or Schatten 2-class
[21], [6] under the norm,
³
´1
2
⋆
(29)
kAk2 := tr(A A)
Define the space
A2 := C2 ∩ Bc (ℓ2 , ℓ2 )
then A2 is the space of causal Hilbert-Schmidt
operators. This space plays the role of the standard
Hardy space H 2 in the standard H ∞ theory. Define
the orthogonal projection P of C2 onto A2 . P is the
lower triangular truncation [27], and is analogous to
the standard positive Riesz projection (for functions
on the unit circle) for the LTI case [18].
Any operator A ∈ Bc (ℓ2 , L2 ), can be viewed
as a multiplication operator acting from A2 into the
Banach space A2 := A2 × A2 with the following norm
1
(|W − R21 Qo )(eiθ )| + |R22 Qo (eiθ )|
µo
L2
H2
is a partial isometry holds. That is, the optimum is an
isometry on the range space of the operator To in (26).
This is the time-varying counterpart of the same notion
known to hold in the H ∞ context [16].
1
kBkA := tr(B1⋆ B1 ) 2 + tr(B2⋆ B2 ) 2
¶
µ
B1
B =
B2
(27)
as a multiplication operator on
or
be an isometry. That is, the operator achieves its norm at every
f ∈ L2 of unit L2 -norm. This interpretation is carried
out to the LTV case in the following Theorem,
Theorem 2: Under assumptions (A1) and (A2)
there exists at least one optimal linear time varying
Qo ∈ Bc (ℓ2 , ℓ2 ) that satisfies if µo > 0, the allpass
condition
µ W ¶
R2
µo
−
Qo
(28)
0
µo
(30)
(31)
with the operator induced norm of A, kAk, equal to
the induced norm given by (7).
Let Π be the orthogonal projection on the closed
subspace A2 ⊖ R2 A2 , that is, the orthogonal
complement of R2 A2 in A2 with respect to the
operator inner product tr(·, ·).
Now define the following bounded linear operator
A2 7−→ A2 ⊖ R2 A2
µ
¶
W
by Ξ = Π
0
Ξ
:
(32)
Next, we need the dual space of A2 , which we will
henceforth denote by A2⋆ . The latter may be shown to
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TuD14.1
be given by A2⋆ = A2 × A2 , but with the following
norm
³
´
1
1
kBkA⋆ := max tr(B1⋆ B1 ) 2 , tr(B2⋆ B2 ) 2
¶
µ
B1
(33)
B =
B2
Define the following bilinear form
Γ : A2 × A2⋆ ⊖ R2 A2 7−→ C
¡
¢
Γ(A, B) := tr B ⋆ ΞA ,
2⋆
A ∈ A2 , B ∈ A
(34)
⊖ R2 A2
Then, the norm of Γ is given by
kΓk =
sup
|Γ(A, B)|
kAk2 ≤ 1
A ∈ A2
kBkA⋆ ≤ 1
B ∈ A2⋆ ⊖ R2 A2
(35)
The bilinear form Γ plays a central role in finding in
computing the optimal index µo through the following
theorem, which quantifies optimal performance.
Theorem 3: Let µo be the optimal performance
index defined by (5), then under assumption (A1) the
following holds
µo = kΓk
(36)
The bilinear form Γ depends on the operator Ξ, which
involves the projection Π. The latter is computed explicitly in the following Lemma.
Lemma 3: Let Π be the orthogonal projection from
A2 into A2 ⊖ R2 A2 , then
Π = I − R2 PR2⋆
(37)
where P is the lower triangular projection of C2 onto
A2 .
VI. C ONCLUSION
In this paper, we gave a solution of the timevarying optimal robust disturbance rejection problem.
In particular, the solution is given explicitly in terms
of duality and operator theory. The optimum is shown
to satisfy a time-varying allpass property. Moreover,
optimal performance is shown to be equal to the norm
of a bilinear form. The latter depends on a lower triangular projection and a multiplication operator defined
on special versions of spaces of compact operators.
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