TIME-VARYING OPTIMAL DISTURBANCE MINIMIZATION IN
PRESENCE OF PLANT UNCERTAINTY∗
SEDDIK M. DJOUADI† AND 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.
Key words. Robust, optimal control, disturbance rejection, time-varying.
AMS subject classifications. 49N35, 49N05, 93B52, 93B36
Definitions and Notation.
• 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
• `2 denotes the usual Hilbert space of square summable sequences with the
standard norm
kxk22 :=
∞
X
¡
¢
|xj |2 , x := x0 , x1 , x2 , · · · ∈ `2
j=0
• L2 denotes the Banach space `2 × `2 under the norm
°µ
¶°
° F1 °
°
°
° F2 ° 2 = kF1 k2 + kF2 k2
L
• L2 denotes the Banach space `2 × `2 under the norm:
°µ
¶°
° g1 °
°
°
° g2 ° 2 = max(kg1 k2 , kg2 k2 )
L
(0.1)
(0.2)
• 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
∗ This
work was supported by a Ralph E. Powe Junior Enhancement award.
of Electrical Engineering and Computer Science, The University of Tennessee,
Knoxville TN 37996 (djouadi@eecs.utk.edu).
‡ Department of Electrical & Computer Engineering, University of Cyprus, Nicosia, 1678, Cyprus
(chadcha@ucy.ac.cy).
† Department
1
2
S.M. DJOUADI AND C.D. CHARALAMBOUS
V
Nα denotes the closed linear span and Nα denotes the intersection of a
collection of subspaces {Nα }.
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.)
•
W
Bounded and causal linear operators can be represented by lower triangular “infi2
nite” matrices, with respect to the canonical basis, {ei }∞
1 of ` , where the entries 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, 18].
1. Introduction. The optimal robust disturbance attenuation problem plays a
fundamental role in feedback optimization [30, 23]. In particular, it has been shown
in [30], 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 [20, 19].
In this paper, 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 [21], 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 [28, 5]. A robust version of these problems
were considered in [27, 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 [29], and considered in [4, 11, 30, 23]. In ORDAP a stable uncertain linear
time-varying plant P is subject to disturbances at the output (see Figure 1.1.) 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
[20].
Here the plant uncertainty set is described by a weighted sphere in the algebra of
bounded linear operators from `2 into `2 instead of H ∞ as defined in expression (2.1),
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.
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
3
The paper is organized as follows, section 2 contains the formulation of ORDAP
in terms of a feedback optimization. In section 3 the optimal solution is characterized in terms of duality theory, where the annihilator is computed explicitly. This
contrasts with the results of [21] where the annihilator was characterized implicitly.
In section 4, the optimum is shown to satisfy an allpass condition. Section 5 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.
u
uncertain Plant
W
+
-
stable Filter
+
P
output y
C
controller
Fig. 1.1. Feedback Control in Presence of Plant and Disturbance Uncertainty
2. Problem Formulation. Let Po ∈ Bsc (`2 , `2 ) be the nominal (possibly timevarying) plant, and denote the set of plant uncertainty by
C(Po , V ) = {P ∈ Bc (`2 , `2 ) : P = XV Po + Po , X ∈ Bc (`2 , `2 ), kXk < 1}
(2.1)
where V is a causal stable time-varying weighting function.
The uncertainty set (2.1) corresponds to the common and widespread multiplicative plant uncertainty model [14, 15]. This uncertainty model is equivalent to the
additive uncertainty model [14]. For more general uncertainty models like coprime
factor uncertainty ORDAP remains an open problem. The difficulty is mainly due to
the fact that computing the worst case sensitivity in the right-hand side of (2.2) for
general uncertainty models is a daunting task.
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 feedback control law. With reference to Figure 1.1 above, mathematically this
problem is equivalent to
µo =
inf
C stabilizing
P ∈ C(Po , V )
sup
P ∈C(P0 ,V )
°
°
°W (I + P C)−1 °
(2.2)
4
S.M. DJOUADI AND C.D. CHARALAMBOUS
where W is a causal stable time-varying weighting function. Expression (2.2) can be
expressed as
°
°
°W (I − Po Q)(I + XV Po Q)−1 °
µo =
inf
sup
Q ∈ Bc (`2 , `2 )
kXk ≤ 1
(I + XV Po Q)−1 ∈ B(`2 , `2 ) X ∈ Bc (`2 , `2 )
(2.3)
The optimization (2.3) is termed as the time-varying optimal robust disturbance attenuation problem, in analogy with its time-invariant counterpart solved in [19, 20].
(2.3) is shown in [21] 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 (`2 ,`2 )
(kW (I − Po Q)f k2 + r kV Po Qf k2 )
sup
kf k2 ≤ 1
f ∈ `2
(2.4)
The function ξ(r) is a continuous, positive, non-decreasing function of r.
Therefore, after absorbing r into V , in principle all that is required to solve the optimization (2.3), after absorbing r into V , is to solve the following type of optimization.
µo :=
inf
Q∈Bc (`2 ,`2 )
(kW (I − Po Q)f k2 + kV Po Qf k2 )
sup
kf k2 ≤ 1
f ∈ `2
(2.5)
The rest of the paper characterizes the solution of (2.5) in terms of duality and
operator theory.
3. Banach Space Duality Theory. 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 [18]
min kK − mkA =
m∈M
sup
| < K, f > |
f ∈⊥ M, kf kA? ≤1
To apply this result we first show that (2.4) is equivalent to a shortest distance minimization problem in a specific Banach space. To this end, let L2 be the Banach space
`2 × `2 under the norm
°µ
¶°
° F1 °
°
°
(3.1)
° F2 ° 2 = kF1 k2 + kF2 k2
L
µ
¶
W (I − Po Q)
The vector function
is viewed as a bounded multiplication operator
µV Po Q
¶
W (I − Po Q)
2
2
from ` into L , that is,
∈ Bc (`2 , L2 ) with the operator induced
V Po Q
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
norm
°
° W (I − Po Q)
°
°
V Po Q
=
sup
kf kL2 ≤ 1
f ∈ L2
5
°
°
°=
°
°µ
¶ °
° W (I − Po Q)
°
°
f°
sup
°
° 2
V Po Q
L
kf k2 ≤ 1
f ∈ `2
¡
¢
kW (I − Po Q)f k2 + kV Po Qf k2
Therefore the optimization problem
µ (2.4)
¶ can be expressed as a distance problem
W
from the vector function K :=
belonging to B(`2 , L2 ) to the subspace
0
µ
¶
W
S=
Po Bc (`2 , `2 ) of B(`2 , 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 (`2 , `2 ),
invertible in Bc (`2 , `2 ) such that Λ?1 Λ1 = W ? W + V ? V [1, 2]. By Theorem 14.20 in [6]
Λ1 P as a bounded linear operator in Bc (`2 , `2 ) has an inner-outer factorization U1 G,
where U1 is inner and G an outer operator defined on `2 . Here inner-outer factorization is different from its H ∞ counterpart, in that it is understood in the following way:
Define a nest N as a family of closed subspaces of the Hilbert space `2 containing {0}
and `2 which is closed under intersection and closed span. Let Qn := I − Pn , for n =
−1, 0, 1, · · · , where P−1 := 0 and P∞ := I. Then Qn is a projection, and we can associate to it the following nest N := {Qn `2 , n = −1, 0, 1, · · · }. Since Bc (`2 , `2 ) is the
set of all bounded linear operators T such that T N ⊆ N for every element N in N ,
it is a nest or triangular algebra. That is
Bc (`2 , `2 ) = {A ∈ B(`2 , `2 ) : (I − Qn )AQn = 0, ∀ n}
(3.2)
W 0
V 0
−
0
+
0
Next, let N := {N ∈ N : N ⊂ N } and N := {N ∈ N : N ⊃ N }. The
subspaces N ª N − are called the atoms of N . Since in our case the atoms of N span
`2 , then N is said to be atomic [6]. An operator A in Bc (`2 , `2 ) is called outer if the
range projection P (RA ), RA being the range of A and P the orthogonal projection
onto RA , commutes with N and AN is dense in N ∩ RA for every N ∈ N . A partial
isometry U is called inner in T (N ) if U ? U commutes with N [1, 6, 2]. In our case,
A ∈ Bc (`2 , `2 ) is outer if P commutes with each Qn and AQn `2 is dense in Qn `2 ∩A`2 .
U ∈ Bc (`2 , `2 ) is inner if U is a partial isometry and U ? U commutes with every Qn .
Next, we assume (A1) G is invertible, so U1 is unitary, and the operator G and
its inverse G−1 ∈ Bc (`2 , `2 ). (A1) is satisfied when, for e.g., the outer factor of the
plant is invertible.
?
2 2
Let R = T2 Λ−1
1 U1 , assumption (A1) implies that the operator R R ∈ B(` , ` ) 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 (`2 , `2 ).
Define R2 = RΛ−1 , then R2? R2 = I, and S has the equivalent representation,
S = R2 Bc (`2 , `2 ). After ”absorbing” Λ into the free parameter Q, the optimization
problem (2.4) is then equivalent to
°µ
°
¶
° W
°
°
µo =
inf 2 2 °
− R2 Q°
(3.3)
°
0
Q∈Bc (` ,` )
6
S.M. DJOUADI AND C.D. CHARALAMBOUS
Let L2 be the Banach space `2 × `2 under the norm:
°µ
¶°
° g1 °
°
°
° g2 ° 2 = max(kg1 k2 , kg2 k2 )
(3.4)
L
The following Lemma characterizing the dual space of L2 follows from [8].
Lemma 3.1. Let L2 and L2 defined as above, then the following hold
• (L2 )? ' L2
• (L2 )? ' L2
Hence all these Banach spaces are reflexive.
Proof. The lemma follows by noticing that `2 is self-reflexive and that the maxnorm is the dual of the `1 -norm.
Introduce the class of compact operators on `2 called the nuclear operators acting
from `2 into L2 , denoted C1 (`2 , L2 ), under the nuclear norm [26],
nX
o
kAkn = inf
kFj k2 · kej kL2
(3.5)
j
where the infimum is taken over all possible representations of A,
X
Af =
< Fj , f > ej , ej ∈ L2 , Fj ∈ `2
(3.6)
j
and
X
kFj k2 · kej kL2 < ∞
(3.7)
j
where < · , · > is the inner product in `2 .
We identify B(`2 , L2 ) with the dual space of C1 (`2 , L2 ), C1? (`2 , L2 ), under trace
duality [25, 26], that is, every operator A in B(`2 , L2 ) induces a continuous linear
functional on C1 (`2 , L2 ) as follows:
ΦA ∈ C1? (`2 , L2 )
is defined by ΦA (T ) = tr(A? T ), and we write
B(`2 , L2 ) ' C1? (`2 , L2 )
Every nuclear operator T in turn induces a bounded linear functional on B(`2 , L2 ),
namely ΦT (A) = tr(T ? A) for all A in B(`2 , L2 ).
The preannihilator of Bc (`2 , `2 ), denoted S, is given by [24]
S := {T ∈ C1 (`2 , `2 ) : (I − Qn )T ? Qn+1 = 0, for all n}
(3.8)
where C1 (`2 , `2 ) is the trace-class for operators acting on `2 into `2 .
Define the following subspace of C1 (`2 , L2 ),
S o := (I − R2 R2? )C1 (`2 , L2 ) ⊕ R2 S
The following Lemma states that S o is the preannihilator of the subspace S.
(3.9)
7
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
Lemma 3.2. If φ ∈ B(`2 , L2 ), then
tr(φ? T ) = 0 for all T ∈ S o ⇐⇒ φ ∈ S
(3.10)
Proof. To show (3.10) it suffices to notice that tr(φ? T ) = 0, ∀T in S o is equivalent
to φ? (I − R2 R2? ) = 0, and φ? R2 = A? for some A ∈ Bc (`2 , `2 ), and so these imply
that φ? = φ? R2 R2? = A? R? . By taking the adjoints we get φ = R2 A ∈ S.
Using Theorem 2, Chapter 5.8 [18], relating the distance from a vector to a
subspace and an extremal functional, we deduce the following Theorem.
Theorem 3.3. Under assumption (A1), there exists at least one optimal Qo ∈
Bc (`2 , `2 ), i.e., a linear time-varying control law such that:
°
° °µ
°µ
¶
¶
°
° ° W
° W
°
°
°
− R2 Qo °
− R2 Q° = °
min
°
0
0
Q∈Bc (`2 ,`2 ) °
¯ µ µ
¶¶¯
¯
¯
¯tr A? W
¯
=
sup
(3.11)
¯
¯
0
kAkn ≤ 1
A ∈ ⊥S
Note that Theorem 3.3 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. However, we show first that the optimum
satisfies a time-varying allpass condition.
4. TV Allpass Property of the Optimum. In the standard H ∞ theory the
space B(`2 , `2 ) 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(`2 , `2 ) is given by the space [25],
B(`2 , `2 )? ' C1 ⊕1 K⊥
(4.1)
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
kΦk = kΦo k + kΦT k
(4.2)
(4.3)
where Φo ∈ K⊥ , and ΦT is induced by the operator T ∈ C1 . By the same token, the
dual space of B(`2 , L2 )? is isometrically isomorphic to the Banach space C1 (`2 , L2 )⊕1
K⊥ , i.e.,
B(`2 , L2 )? ' C1 (`2 , L2 ) ⊕1 K⊥
(4.4)
2
2
where in this case K is the space of compact operators acting from ` into L , and
K⊥ its annihilator.
The annihilator S ⊥ of S in C1 (`2 , L2 ) ⊕1 K⊥ is given by
´
³
S ⊥ = (I − R2 R2? ) C1 (`2 , L2 ) ⊕1 K⊥ ⊕R2 S
(4.5)
8
S.M. DJOUADI AND C.D. CHARALAMBOUS
Banach space duality states with the existence of an annihilator that [18]
inf kx − yk =
y∈S
max
Φ∈S ⊥ , kΦk≤1
|Φ(x)|
(4.6)
The maximizing Φopt in the dual space can be written as
Φopt = Φo + ΦTo
kΦopt k = kΦo k + kΦTo kn = 1
(4.7)
(4.8)
where Φo ∈ K⊥ , and ΦTo is induced by the operator To ∈ S o . In others words, the
following result holds
¯
°µ
° ¯ µµ
¶¶
¶
¯
° W
° ¯
W
?
¯
°
°
+ tr ((W , 0)To )¯¯ (4.9)
− R2 Q° = ¯Φo
µo =
min2 2 °
0
0
Q∈Bc (` ,` )
If Qo achieves the minimum in (4.9), then the alignment condition in the dual is given
by
¯ µµ
¯ °µ
°
¶¶
¶
¯
¯ ° W
°¡
¢
W
?
¯Φo
¯
°
°
+
tr
((W
,
0)T
)
=
−
R
Q
o ¯
2 o ° kΦo k + kΦTo kn (4.10)
¯
°
0
0
µ
¶
W
If we further assume that (A2):
as an operator from `2 into L2 is compact,
0
µµ
¶¶
W
then Φo
= 0 maximum in (4.6) is achieved on S o , that is, the supremum
0
in (3.11) becomes a maximum. It is instructive to
µ note¶that in the LTI case assumpW
tion (A2) is the analogue of the assumption that
is the sum of two parts, one
0
∞
part continuous on the unit circle and the other in H , in which case the optimum
is allpass [30, 19]. 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
µ that¶the modulus
R21
of the optimum |(W − R21 Qo )(eiθ )| + |R22 Qo (eiθ )|, where R2 =
, is constant
R22
at almost all 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
(|W − R21 Qo )(eiθ )| + |R22 Qo (eiθ )|
µo
(4.11)
as a multiplication operator on L2 or H 2 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 by first defining
µoo :=
inf
Q∈(`2 ,`2 )
sup
(kW (I − Po Q)f k2 + kV Po Qf k2 )
kf k2 ≤ 1
f ∈ `2
(4.12)
that is, when the causality constraint on Q is removed.
Before showing the allpass property of the optimum we need the following proposition.
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
9
Proposition 4.1. The search for the minimal representation of the nuclear
operators in (3.5) can be restricted to bases in `2 , that is,
nX
o
2
kAkn = ©
inf
kAv
k
(4.13)
j
L
ª
{vj } basis of `2
j
Proof. To show (4.13) note that the infimum in the nuclear norm
P (3.5) can be
taken over all rank-one operators, that is, over expressions of A as i Ai , where Ai
is a rank-one operator for each i, in other words Ai =< ·, xi > yi , for some vectors
xi ∈ `2 , yi ∈ L2 . Therefore
(
)
X
kAkn = inf
kAi k, Ai rank − one operators
(4.14)
i
where kAi k is the norm of Ai as an operator acting from (`2 , k · k2 ) into (L2 , k · kL2 ),
i.e., kAi k = supkxk2 ≤1, x∈`2 kAi xkL2 .
Pn
2
Now letting {vj }∞
1 be any orthonormal basis of ` , then x =
j=1 < x, vj > vj and
Ax =
∞
X
< x, vj > Aaj
(4.15)
j=1
for x ∈ `2 , which shows that kAkn ≤
orthonormal basis it follows that
∞
X
kAkn ≤ inf{
P∞
j=1
kAvj kL2 . Since {vj }∞
1 is an arbitrary
kAvj kL2 : {vj } is an orthonormal basis of L2 }
(4.16)
j=1
P
If A = i Ai , is any representation of A by sums of rank-one operators Ai ’s, then the
reverse inequality can be deduced from
X
X X
XX
X
kAvj kL2 =
k
Ai vj kL2 ≤
kAi vj kL2 ≤
kAi k
(4.17)
j
j
i
j
i
i
since the representation is arbitrary then
X
X
X
kF vj kL2 ≤ inf{
kAi k : F =
Ai , Ai rank − one operators} (4.18)
j
i
i
Hence
kAkn = inf
∞
nX
kAvj kL2 : {vj } is an orthonormal basis of L2
o
(4.19)
j=1
In the following Theorem we show that the optimum is an isometry on a subspace.
Theorem 4.2. Under assumptions (A1) and (A2) there exists at least one optimal linear time varying Qo ∈ Bc (`2 , `2 ) that satisfies if µo > µoo , the allpass
condition
µ W ¶
R2
µo
−
Qo
(4.20)
0
µo
10
S.M. DJOUADI AND C.D. CHARALAMBOUS
is a partial isometry holds. That is, the optimum is an isometry on the range space
of the operator To in (4.10). This is the time-varying counterpart of flatness of the
optimum known to hold in the H ∞ context [20]. The condition µ > µoo is sharp, that
is, if it is not satisfied then there exist W , R2 and Qo such (4.20) is not a partial
isometry.
Proof.
there exists some To ∈ C(`2 , L2 ),
P The dual representation (4.10) implies that
2
To (·) = j < Fj , · > To Fj , for some basis Fj of ` and kTo kn = 1, such that if we
µ
¶
R21
write R2 =
,
R22
¯
¯
¯
¯ ³
´¯ ¯¯X
¯
¯
¯
?
?
?
?
, −Q?o R22
)To ¯ = ¯¯
< Fj , (W ? − Q?o R21
, −Q?o R22
)To Fj >¯¯
µo = ¯tr (W ? − Q?o R21
¯ j
¯
X
?
?
, −Q?o R22
)To Fj >|
≤
|< Fj , (W ? − Q?o R21
j
and by Cauchy-Schwarz inequality this yields
X
?
?
µo ≤
kFj k2 k(W ? − Q?o R21
, −Q?o R22
)To Fj kL2
j
≤
X
j
≤
°µ
¶°
° W − R12 Qo °
° kTo Fj kL2
kFj k2 °
°
°
R22 Qo
X
inf
kTo Fj kL2 µo ≤ µo kTo kn = µo
2
Fj basis in `
j
The last two inequalities follows since they hold for any basis in `2 and therefore by
taking the infimum on the right-hand side, and kFj k2 = 1, ∀j. Moreover, the last
inequality shows that equality must hold throughout yielding
°µ
¶°
° W − R12 Qo °
? ?
?
? ?
° kTo Fj kL2 , ∀j (4.21)
k(W − Qo R21 , −Qo R22 )To Fj kL2 = °
°
°
R22 Qo
°µ
¶°
° W − R12 Qo °
° attains its norm on each To Fj , and
Identity (4.21) implies that °
°
°
R22 Qo
therefore on the range of To , since the latter is given by the span of the {To Fj }’s.
Hence,
µ
¶
1
W − R12 Qo
(4.22)
R22 Qo
µo
is an isometry on the range of To .
If µo = µoo the counter example given in the LTI case in [30] shows that the assertion of the Theorem fails.
2
2
A compactness argument (see
µ [22])¶shows that the optimal Qo ∈ Bc (` , ` ) may
W
be chosen to be 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 [20]. The latter can be realized by invoking tensor products of operators
along the lines [20], albeit in different spaces of linear causal compact operators.
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
11
5. A Solution Based on Operator Theory. Let C2 denote the class of compact operators acting from `2 called the Hilbert-Schmidt or Schatten 2-class [25, 6]
under the norm,
³
´ 12
tr(A? A)
(5.1)
A2 := C2 ∩ Bc (`2 , `2 )
(5.2)
kAk2 :=
Define the space
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 [31], and is analogous
to the standard positive Riesz projection (for functions on the unit circle) for the LTI
case [22].
Any operator A ∈ Bc (`2 , L2 ), can be viewed as a multiplication operator acting
from A2 into the Banach space C2 := A2 × A2 with the following norm
1
1
kBkC := tr(B1? B1 ) 2 + tr(B2? B2 ) 2
µ
¶
B1
B =
B2
(5.3)
with the operator induced norm of A, kAk, equal to the induced norm given by (3.2).
Let Π be the orthogonal projection on the closed subspace C2 ª R2 A2 , that is, the
orthogonal complement of R2 A2 in C2 with respect to the operator inner product
tr(·, ·).
Now define the following bounded linear operator
Ξ : A2 7−→ C2 ª R2 A2
¶
µ
W
by Ξ = Π
0
(5.4)
Next, we need the dual space of C2 , which we will henceforth denote by C2? . The
latter may be shown to be given by C2? = A2 × A2 , but with the following norm
³
´
1
1
kBkC? := max tr(B1? B1 ) 2 , tr(B2? B2 ) 2
µ
¶
B1
B =
(5.5)
B2
The adjoint operator Ξ? of Ξ is then defined as
Ξ? : C2? ª R2 A2 7−→ A2
Ξ? = (W ? , 0)
(5.6)
Next, define the following bilinear form
Γ : A2 × C2? ª R2 A2 7−→ C
¡
¢
Γ(A, B) := tr A? Ξ? B ,
2?
A ∈ A2 , B ∈ C
ª R2 A2
(5.7)
12
S.M. DJOUADI AND C.D. CHARALAMBOUS
Then, the norm of Γ is given by
kΓk =
sup
|Γ(A, B)|
kAk2 ≤ 1
A ∈ A2
kBkC? ≤ 1
B ∈ C2? ª R2 A2
(5.8)
The bilinear form Γ depends on the operator Ξ, which involves the projection Π. The
latter is computed explicitly in the following Lemma.
Lemma 5.1. Let Π be the orthogonal projection from A2 into C2 ª R2 A2 , then
Π = I − R2 PR2?
(5.9)
where P is the lower triangular projection of C2 onto A2 .
Proof. For f ∈ A2 , let us compute
(I − R2 PR2? )2 f = (I − R2 PR2? )(I − R2 PR2? )f
= (I − R2 PR2? − R2 PR2? + R2 PR2? R2 PR2? )f
= (I − 2R2 PR2? + R2 PR2? )f
since R2? R2 = I and P 2 = P
= (I − R2 PR2? )f
(5.10)
so (I − R2 PR2? ) is indeed a projection.
Clearly the adjoint (I −R2 PR2? )? of (I −R2 PR2? ), although defined on different spaces,
is equal to (I − R2 PR2? ) itself, so that (I − R2 PR2? ) is an orthogonal projection.
Next we show that the null space of (I − R2 PR2? ), null(I − R2 PR2? ) = R2 A2 .
Let f ∈ null(I − R2 PR2? ) then
(I − R2 PR2? )f = 0 =⇒ f = R2 PR2? f
then PR2? f ∈ A2 ) and therefore f ∈ R2 A2 . Hence null(I − R2 PR2? ) ⊂ R2 A2 . Conversely, let f ∈ A2 , then
(I − R2 PR2? )R2 f = R2 f − R2 Pf = R2 f − R2 f = 0
thus R2 f ∈ null(I − R2 PR2? ), so R2 A2 ⊂ null(I − R2 PR2? ), and therefore null(I −
R2 PR2? ) = R2 A2 .
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 5.2. Let µo be the optimal performance index defined by (2.5), then
under assumption (A1) the following holds
µo = kΓk
(5.11)
Proof. Note that since the norm of the norm of the bilinear form is given by (5.8),
then for any ² > 0 there exist A² ∈ A2 , kA² k2 ≤ 1, B² ∈ C2? ª R2 A2 , kBkC? ≤ 1, we
Time-Varying Optimal Disturbance Minimization in Presence of Plant Uncertainty
have
13
¯ ¡
¢¯¯
¯
(5.12)
sup
|Γ(A, B)| < ¯tr A?² Ξ? B² ¯+²
kAk2 ≤ 1
A ∈ A2
kBkC2? ≤ 1
B ∈ C2? ª R2 A2
¯ ¡
¯ ¡
¢¯¯
¢¯¯
¯
¯
< ¯tr A?² (W ? , 0)(I − R2 PR2? )B² ¯+² = ¯tr (I − R2 PR2? )B² A?² (W ? , 0) ¯+²
kΓk =
(5.13)
and (I − R2 PR2? )B² ∈ C2? ª R2 A2 , we have that (I − R2 PR2? )B² A?² belongs to
C1 (`2 , L2 ), and note that for all A ∈ Bc (`2 , `2 ), we have
³
´
³
´
tr A? R2? (I − R2 PR2? )B² A?² = tr A?² A? (I − P)R2? B² = 0
This shows that (I−R2 PR2? )B² A?² ∈ S o , and has nuclear norm k(I−R2 PR2? )B² A?² kn ≤
1 since kA?² k2 ≤ 1 and (I − R2 PR2? )B² = B² , so k(I − R2 PR2? )B² k2 ≤ 1 in (5.13).
Therefore, by (3.11) we have
¯ µ µ
¶¶¯
¯
¯
W
?
¯
¯ = µo , ∀² > 0
kΓk − ² <
sup
(5.14)
¯tr D
¯
0
kDkn ≤ 1
D ∈ So
Since (5.14) holds for ² > 0 arbitrary, we have
kΓk ≤ µo
(5.15)
To show the opposite inequality, let Φ ∈ S o as in (3.11) then Φ can be written uniquely
as
Φ = (I − R2 R2? )A + R2 B, ∃A ∈ C1 (`2 , L2 ), ∃B ∈ S
(5.16)
Pre-multiplying by R2? shows that B = R2? Φ, since R2? R2 = I, and A = (I − R2 R2? )Φ.
Now Φ is trace class then it factorizes as Φ = Φ2 Φ1 where Φ1 ∈ C2 and Φ2 ∈ C2 (`2 , L2 )
the space of Hilbert-Schmidt operators from `2 into L2 , kΦ1 k2 = kΦ2 kC? = kΦkn ≤ 1
[25]. Let Φ1 = V Ψ be a polar decomposition of Φ1 . By a Theorem in [32] Ψ factorizes
as Ψ = ΘΘ? , for Θ ∈ A2 , kΘk2 ≤ 1 and Θ outer. Moreover, Φ2 V Θ ∈ C2 (`2 , L2 )
and (I − R2 R2? )Φ2 V Θ + R2 R2? Φ2 V Θ ∈ C2 (`2 , L2 ). Now note that B ? = ΘΘ? V ? Φ?2 R2
is strictly causal, Θ is causal. Since the orthogonal complement of the range of Θ? ,
(Θ? `2 )⊥ is equal to the null space of θ, i.e., R2? Φ2 V Θ = 0 on (Θ? `2 )⊥ , by a Lemma
in [1] Θ? V ? Φ?2 R2 is strictly causal. Hence, for all D ∈ A2 we have
¡
¢
¡
¢
tr D? R2? (I − R2 R2? )Φ2 V Θ + D? R2? R2 R2? Φ2 V Θ = tr D? R2? Φ2 V Θ = 0 (5.17)
implying that (I − R2 R2? )Φ2 V Θ + R2 R2? Φ2 V Θ ∈ C2? ª R2 A2 , and therefore Φ ∈
C2? ª R2 A2 × A?2 , where A?2 := {A? ∈ C2 : A ∈ A2 }. Thus
µo ≤ kΓk
Inequalities (5.14) and (5.14) imply that µo = kΓk.
(5.18)
14
S.M. DJOUADI AND C.D. CHARALAMBOUS
The proof of Theorem 5.2 implies the existence of operators A² ∈ A2 and B² ∈
C2? ª R2 A2 such that an expression for a suboptimal controller Q² can be obtained
by letting
³ £
³
´
¤ ´
(5.19)
tr A?² (W − R12 Q² )? (R22 Q² )? B² = tr A² Ξ? B²
To solve for Q² it suffices to consider solutions to the operator identity
¶
µ
W − R12 Q²
A² = ΞA²
R22 Q²
Note that
°µ
¶ °
° W − R12 Q²
°
°
A² °
°
° ≤ µo + ²
R22 Q²
C
(5.20)
(5.21)
In (5.21) equality holds with ² = 0 when maximizing operators A ∈ A2 and B ∈
C2? ª R2 A2 exist for the bilinear form Γ, that is,
kΓk = |tr(A? Ξ? B)|
(5.22)
and the optimal Qo satisfies (5.20) with A² and B² replaced by A and B, respectively.
6. Conclusion. In this paper, we gave a solution of the time-varying 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 timevarying 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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