On Optimal Performance for Linear Time-Varying Systems
Seddik M. Djouadi and Charalambos D. Charalambous
Abstract— In this paper we consider the optimal disturbance
attenuation problem and robustness for linear time-varying
(LTV) systems. This problem corresponds to the standard
optimal H ∞ problem for LTI systems. The problem is analyzed in the context of nest algebra of causal and bounded
linear operators. In particular, using operator inner-outer
factorization it is shown that the optimal disturbance attenuation problem reduces to a shortest distance minimization
between a certain operator to the nest algebra in question.
Banach space duality theory is then used to characterize
optimal time-varying controllers. Alignment conditions in the
dual are derived under certain conditions, and the optimum
is shown to satisfy an allpass condition for LTV systems,
therefore, generalizing a similar concept known to hold for
LTI systems. The optimum is also shown to be equal to
the norm of a time-varying Hankel operator analogous to
the Hankel operator, which solves the optimal standard H ∞
problem. Duality theory leads to a pair of finite dimensional
convex optimizations which approach the true optimum from
both directions not only producing estimates within desired
tolerances, but also allow the computation of optimal timevarying controllers.
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
•
`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
•
•
•
Zk denotes the k step ahead shift operator for some
integer k.
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
(1)
and stricly causal if it satisfies
Pk+1 APk = Pk+1 A, , ∀k positive integers
(2)
The subscripts “sc ”, “c ” and the superscript “ti ” denote the
restriction of a subspace of operators to its intersection with
S.M. Djouadi is with the Faculty of Electrical & Computer Engineering Department, University of Tennessee, Knoxville, TN 37996-2100
djouadi@ece.utk.edu
C.D. Charalambous is with the Faculty of the Electrical & Computer Engineering Department, University of Cyprus, Nicosia, 1678,
chadcha@ucy.ac.cy
causal, strictly causal (see [12] for the definition), and timeinvariant operators respectively. “? ” 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 [7], [10].
I. I NTRODUCTION
There have been numerous attempts in the literature to
generalize ideas about H ∞ control theory to time-varying
systems. In [15], [16] and more recently [14] the authors
studied the optimal weighted sensitivity minimization
problem, the two-block problem, and the model-matching
problem for LTV systems using inner-outer factorization
for positive operators. They obtained abstract solutions
involving the computation of norms of certain operators,
which is quit difficult since these are infinite dimensional
problems. Moreover, no indication on how to compute
optimal LTV controllers is provided. In [9] both the
sensitivity minimization problem in the presence of plant
uncertainty, and robust stability for LTV systems in the `∞
induced norm is considered. However, their methods could
not be extended to the case of systems operating on finite
energy signals. Analysis of time-varying control strategies
for optimal disturbance rejection for known time-invariant
plants has been studied in [19], [2]. A robust version of
these problems were considered in [18], [8] in different
induced norm topologies. All these references showed that
for time-invariant nominal plants, time-varying control
laws offer no advantage over time-invariant ones.
In this paper, the focus is on optimal disturbance rejection
for time-varying systems. Using inner-outer factorizations
as defined in [3], [13] with respect of the nest algebra of
lower triangular (causal) bounded linear operators defined
on `2 we show that the problem reduces to a distance
minimization between a special operator and the nest
algebra. The inner-outer factorization used here holds
under weaker assumptions than [15], [16], and in fact,
as pointed in [3], is different from the factorization for
positive operators used there. These definitions parallel
well known factorizations of L∞ functions w.r.t. H ∞ as an
algebra on the unit circle. Then the duality structure of the
problem showing existence of optimal LTV controllers, and
predual formulation are provided. Alignment conditions
in the dual of the Banach space B(`2 , `2 ) are derived
under certain conditions. Next, the optimum is shown to
satisfy an allpass condition for LTV systems, therefore,
generalizing a similar concept known to hold for LTI
systems for the optimal standard H ∞ problem [21], [22],
[24]. The optimum is also shown to be equal to the norm
of a time-varying Hankel operator analogous to the Hankel
operator, well known in the optimal standard H ∞ problem.
Duality theory leads to a pair of dual finite dimensional
convex optimizations which approach the real optimal
disturbance rejection performance from both directions
not only producing estimates within desired tolerances,
but also allow the computation of optimal time-varying
controllers.
Our approach is purely input-output and does not
use any state space realization, therefore the results derived
here apply to infinite dimensional LTV systems. Although
the theory is developed for causal stable system, it can be
extended in a straighfoward fashion to the unstable case
using coprime factorization techniques for LTV systems
discussed in [12], [16], [13].
The framework developed here can also be applied
to other performance indexes, such as the two-block mixed
sensitivity problem considered in [16], or even the optimal
robust disturbance attenuation problem considered in [11],
[20], [4], [5], [6].
II. P ROBLEM FORMULATION
In this paper we consider the problem of optimizing
performance for causal linear time varying systems. The
standard block diagram for the optimal disturbance attenuation problem that is considered here is represented in Fig.
1, where u represents the control inputs, y the measured
outputs, z is the controlled output, w the exogenous perturbations. P denotes a causal stable linear time varying plant,
and K denotes a time varying controller.
z
w
P
u
y
K
Fig. 1.
Block Diagram for Disturbance Attenuation
The closed-loop transmission from w to z is denoted
by Tzw . Using the standard Youla parametrization of all
stabilizing controllers [12] the closed loop operator Tzw
can be written as [2], [19]
Tzw = T1 − T2 QT3
(3)
where T1 , T2 and T3 are stable causal operators, that is,
T1 , T2 and T3 ∈ Bc (`2 , `2 ). The Youla parameter Q :=
K(I + P K)−1 is then an operator belonging to Bc (`2 , `2 ),
and is related univoquely to the controller K. Note that Q
is allowed to be time-varying.
The magnitude of the signals w and z is measured in the `2 norm. The problem considered here is disturbance rejection.
In this case the performance index can be written in the
following form
µ :=
=
inf {kTzw k : K being stabilizing
linear time − varying controller}
inf 2 2 kT1 − T2 QT3 k
Q∈Bc (` ,` )
(4)
The performance index (4) will be transformed into a
distance minimization between a certain operator and a
subspace to be specified shortly. To this end, 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, · · · }
(5)
The triangular or nest algebra T (N ) is the set of all
operators T such that T N ⊆ N for every element N in
N . That is
T (N ) = {A ∈ B(`2 , `2 ) : Pn A(I − Pn ) = 0, ∀ n}
= {A ∈ B(`2 , `2 ) : (I − Qn )AQn = 0, ∀ n}
(6)
Note that the Banach space Bc (`2 , `2 ) of causal and
bounded operators may be viewed as a nest algebra since
causal operators leave N invariant, i.e., for all operators
A ∈ Bc (`2 , `2 ), AQn `2 ⊂ Qn `2 for all n. In fact Bc (`2 , `2 )
is exactly T (N ) the algebra of lower triangular operators.
For N belonging to the nest N , N has the form Qn `2 for
some n. Define
_
N− =
{N 0 ∈ N : N 0 < N }
(7)
^
N+ =
{N 0 ∈ N : N 0 > N }
(8)
where N 0 < N means N 0 ⊂ N , and N 0 > N means
N 0 ⊃ 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 [3].
By analogy with the standard H ∞ theory [21], [22], and
following [1], [3], [13] we introduce inner-outer factorizations for the operators in T (N ) as follows:
An operator A in T (N ) 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], [3], [13]. In our case, A ∈ T (N ) = 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 . Applying
these notions to the time-varying operator T2 ∈ Bc (`2 , `2 )
we get T2 = T2i T2o , where T2i and T2o are inner outer
operators in Bc (`2 , `2 ), respectively. Similarly, co-inner-coouter fatorization can be defined, and the operator T3 can
be factored as T3 = T3co T3ci , where T3ci ∈ Bc (`2 , `2 ) is
?
co-inner that is T3ci
is inner, T3co ∈ Bc (`2 , `2 ) is co-outer,
?
that is, T3co is outer. The performance index µ in (4) can
then be written as
µ=
inf
Q∈Bc (`2 ,`2 )
kT1 − T2i T2o QT3co T3ci k
(9)
Following the classical H ∞ control theory [21], [22], [24],
[23], we assume
where < ·, · > denotes the duality product.
Let us apply these results to the problem given in (10) by
letting
X = B(`2 , `2 )
?
?
x = T2i
T1 T3ci
∈ B(`2 , `2 )
2 2
J = Bc (` , ` )
Introduce a class of compact operators on `2 called the
trace-class or Schatten 1-class, denoted C1 , under the traceclass norm [17], [3],
1
kT k1 := tr(T ? T ) 2
where tr denotes the Trace.
We identify B(`2 , `2 ) with the dual space of C1 , C1? , under
the trace duality [17], that is, every A in B(`2 , `2 ) induces
a continuous linear functional on C1 as follows: ΦA in C1?
is defined by ΦA (T ) = tr(AT ), and we write
B(`2 , `2 ) ' C1?
(A1) that T2o and T3co are invertible both in Bc (`2 , `2 ).
To compute the preannihilator of Bc (`2 , `2 ) define the
subspace S of C1 by
This assumption guarantees the bijection of the map
Q −→ T2o Bc (`2 , `2 )T3co
S := {T ∈ C1 : (I − Qn )T Qn+1 = 0, for all n}
In the time-invariant case this assumption means essentially
that the outer factor of the plant P is invertible [22], [24].
Under this assumption T2i becomes an isometry and T3ci
?
?
a co-isometry in which case T2i
T2i = I and T3ci T3ci
= I.
By ”absorbing” the operators T2o and T3co into the ”free”
operator Q, expression (9) is then equivalent to
µ=
inf
Q∈Bc (`2 ,`2 )
?
?
− Qk
T1 T3ci
kT2i
(10)
?
?
Expression (9) is the distance from the operator T2i
T1 T3ci
∈
2 2
2 2
B(` , ` ) to the nest algebra Bc (` , ` ).
In the next section we study the distance minimization
probblem (9) in the context of the operator algebra setting
discussed above.
III. E XISTENCE OF A P REDUAL AND AN O PTIMAL
C ONTROLLER
Let X ba a Banach space and X ? its dual space, i.e.,
the space of bounded linear functionals defined on X. For
a subset J of X, the annihilator of J in X ? is denoted J ⊥
and is defined by [10]
J ⊥ := {Φ ∈ X ? : Φ(f ) = 0, f ∈ J}
(11)
Similarly, if K is a subset of X ? then the preannihilator of
K in X is denoted ⊥ K, and is defined by
⊥
K := {x ∈ X : Φ(x) = 0, Φ ∈ K}
(12)
The existence of a preannihilator implies that the following
identity holds [10]
min kx − yk =
y∈K
sup
k∈⊥ K, kkk≤1
| < x, k > |
(14)
(13)
(15)
In the following Lemma we show that S is the preannihilator of Bc (`2 , `2 ).
Lemma 1: The preannihilator of Bc (`2 , `2 ) in C1 ,
⊥
Bc (`2 , `2 ), is isometrically isomorphic to S.
Proof. By Lemma 16.2 in [3] the preannihilator of T (N )
is given by ΦT ∈⊥ Bc (`2 , `2 ) if and only T belongs to the
subspace
{T ∈ C1 : P (N − )⊥ T P (N ) = 0 for all N in ∈ N }
where P (N ) denotes the orthogonal projection on N , likewise for P (N − ). P (N − )⊥ the complementary projection
of P (N − ), that is, P (N − )⊥ = I − P (N − ).
In our case N is atomic, and for any N ∈ N there
exists n such that N = Qn+1 `2 , i.e., P (N ) = Qn+1 . The
immediate predecessor of N , N − , is then given by Qn `2 ,
i.e., N − = Qn `2 . The orthogonal complement of N − is
then `2 ª Qn `2 . So P (N − )⊥ = I − Qn . Therefore,
S = {T ∈ C1 : (I − Qn )T Qn+1 = 0, ∀ n}
is isometrically isomorphic to the preannihilator of
Bc (`2 , `2 ), and the Lemma is proved.
The existence of a predual C1 and a preannihilator S
implies the following Theorem which is a consequence of
Theorem 2 in [10] (Chapter 5.8).
Theorem 1: Under assumption (A1) there exists an optimal Qo in Bc (`2 , `2 ) achieving optimal performance µ in
(10), moreover the following identities hold
µ =
=
=
inf
?
?
kT2i
T1 T3ci
− Qk
sup
− Qo k
?
?
|tr(T T2i
T1 T3ci
)|
Q∈Bc (`2 ,`2 )
?
?
kT2i
T1 T3ci
T ∈S, kT k1 ≤1
(16)
Theorem 1 not only shows the existence of an optimal LTV
controller, but plays an important role in its computation by
reducing the problem to a dual of finite dimensional convex
optimizations. Under a certain condition in the next section
it is shown that the supremum in (16) is achieved. Theorem
1 also leads to a solution based on a Hankel type operator,
which parallel the Hankel operator know in the H ∞ control
theory.
The supremum on the RHS of (16) may be achieved if
the preannihilator S is enlarged. For this let us defined K
as the space of compact operators acting on `2 into `2 . The
dual space of B(`2 , `2 ) is then given by the space C1 ⊕1 K⊥
[17],
(17)
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
(18)
(19)
where Φo ∈ K⊥ , and ΦT is induced by the operator T ∈ C1 .
Banach space duality asserts that [10]
inf kx − yk =
y∈J
max
Φ∈J ⊥ , kΦk≤1
|Φ(x)k
min
Q∈Bc (`2 , `2 )
?
?
kT2i
T1 T3ci
− Qk =
max
Φo ∈ (K ∩ Bc )⊥
T ∈ S,
kΦo k + kT k1 ≤ 1
?
?
?
?
|Φo (T2i
T1 T3ci
) + tr(T T2i
T1 T3ci
)|
(22)
If
Φopt = Φopt,o + ΦTopt
IV. A LLPASS P ROPERTY OF THE O PTIMUM :
A LIGNMENT IN THE D UAL
B(`2 , `2 )? ' C1 ⊕1 K⊥
Lemma 2: Under assumption (A1) the following holds
(20)
In our case J = Bc (`2 , `2 ). Since B(`2 , `2 )? contains C1
as a subspace, then Bc (`2 , `2 )⊥ contains the preannihilator
S, i.e., the following expression may be deduced
³
´⊥
J ⊥ := Bc (`2 , `2 )⊥ = S ⊕1 K ∩ Bc (`2 , `2 )
(21)
A deep result in [3] asserts that if a linear functional Φ
belongs to the annihilator Bc (`2 , `2 )⊥ , and Φ decomposes
as
Φ = Φo + ΦT
where
³
´⊥
Φo ∈ K ∩ Bc (`2 , `2 )
and ΦT ∈ S. Then
Φo ∈ Bc (`2 , `2 )⊥
and
ΦT ∈ Bc (`2 , `2 )⊥
as well.
Combining Theorem 1, (20) and (21) the following
Lemma is deduced.
achieves the maximum in the RHS of (22), and Qo the
minimum on the LHS, then the alignment condition in the
dual is given by
?
?
?
?
|Φopt,o (T2i
T1 T3ci
) + tr(Topt T2i
T1 T3ci
)| =
?
?
kT2i T1 T3ci − Qo k (kΦopt,o k + kTopt k1 )
(23)
If we further assume that
?
?
T1 T3ci
is a compact operator.
(A2): T2i
?
, T1
This is the case, for example, if anyone of T2i
?
?
?
or T3ci is compact, then Φopt,o (T2i T1 T3ci ) = 0 and the
maximum in (22) is achieved on S, that is the supremum
in (16) becomes a maximum. It is instructive to note that
in the linear time-invariant case assumption (A2) is the
?
?
is continuous on
T1 T3ci
analogue of the assumption that T2i
the unit circle, in which case the optimum is allpass [21],
[24], [20]. In the linear time varying case allpass property
of the optimum is formulated in the following Theorem.
Theorem 2: Under assumptions (A1) and (A2) any optimal linear time varying Qo ∈ Bc (`2 , `2 ) satisfies the
allpass condition
?
?
?
?
kT2i
T1 T3ci
− Qo k = |tr(To T2i
T1 T3ci
)|
?
?
= max k(I − Qn )T2i T1 T3ci Qn k
n
(24)
(25)
where To is some operator in S, and kTo k1 = 1.
Proof. Identity (24) is implied by the previous argument
that the supremum in (16) is achieved by some To in S
with trace-class norm equal to 1. Combining this result with
Corollary 16.8 in [3] (see also [1]), which asserts that
?
?
?
?
kT2i
T1 T3ci
− Qo k = sup k(I − Qn )T2i
T1 T3ci
Qn k
n
shows in fact that the supremum w.r.t. n is achieved proving
that (25) holds. In fact, it can be shown that the operator
?
?
T2i
T1 T3ci
− Qo corresponds to an isometry. Identity (24)
represents the allpass condition in the time-varying case,
since it corresponds exactly to the allpass or flatness
condition in the time-invariant case for the standard optimal
H ∞ problem.
In the next section, we relate our problem to an LTV
operator analogous to the Hankel operator, which is known
to solve the standard optimal H ∞ problem in the LTI case
[21], [23].
V. A S OLUTION BASED ON A H ANKEL O PERATOR
The starting point here is the identity (16)
µ =
inf
Q∈Bc (`2 ,`2 )
=
?
?
kT2i
T1 T3ci
− Qk
sup
A∈S, kT k1 ≤1
?
?
|tr(AT2i
T1 T3ci
)|
We need now to introduce the class of compact operators on
`2 called the Hilbert-Schmidt or Schatten 2-class, denoted
C2 , under the Hilbert-Schmidt norm [17], [3],
³
´ 12
kAk2 := tr(A? A)
(26)
In [27], [26] it is shown that any operator A in T (N ) ∩ C1
admits a Riesz factorization, that is, there exist operators
A1 and A2 in T (N ) ∩ C2 such that A factorizes as
A = A1 A2
and kAk1 = kA1 k2 kA2 k2
(27)
(28)
A Hankel form [·, ·]B associated to a bounded linear
operator B ∈ B(`2 , `2 ) is defined by [27], [26]
[A1 , A2 ]B = tr(A1 BA2 )
(29)
Since any operator in the preannihilator S belongs also to
T (N ) ∩ C1 , then any A ∈ S factorizes as in (27). And if
kAk1 ≤ 1, as on the LHS of (16), A ∈ S, the operators
A1 and A2 both in T (N ) ∩ C2 can be chosen such that
kA1 k2 ≤ 1, kA2 k2 ≤ 1, and for all atoms ∆n := Qn+1 −
Qn , ∆n A1 ∆n = 0, n = 0, 1, 2, · · · [26].
Define the orthogonal projection P of C2 onto Bc ∩ C2 ,
and write P+ for the orthogonal projection with range the
subspace of operators T in T (N )∩C2 such that ∆n T ∆n =
0, n = 0, 1, 2, · · · .
Introducing the notation (B1 , B2 ) = tr(B2? B1 ), and the
?
?
Hankel form associated to B := T2i
T1 T3ci
, we have by a
result in [26]
[A1 , A2 ]B
=
=
=
=
=
tr(BA2 )
¡
¢
?
?
A1 , (T2i
T1 T3ci
A2 )?
¡
¢
P+ A1 , (BA2 )?
¡
¢
A1 , P+ (BA2 )?
¡
¢
?
A1 , (HB
A2 )
(30)
(31)
(32)
(33)
(34)
where HB is the Hankel operator (I − P)BP, and we
can see from above that its operator norm coincides with
the norm of the Hankel form computed on A1 and A2 of
unit Schatten 2-norm and satisfying the conditions above.
Factorizing the operator T in (16) as T = t1 t2 , where
kT k1 = kt1 k2 kt2 k2 = 1, t1 , t2 ∈ T (N ) ∩ C2 , shows that
following Theorem must hold.
Theorem 3:
? k
µ = kHT2i? T1 T3ci
(35)
?
?
= k(I − P)T2i T1 T3ci Pk
(36)
Finally in the next section we show that computing µ, within
desired tolerances, reduces to two finite dimensional convex
optimizations in the entries of Q.
VI. A PPROXIMATION BY F INITE D IMENSIONAL
C ONVEX O PTIMIZATIONS
If {en : n = 0, 1, 2, · · · } is the standard orthonormal
basis in `2 , then Qn `2 is the linear span of {ek : k =
n + 1, n + 2, · · · }. The matrix representation of A ∈ T (N )
w.r.t. this basis is lower triangular.
Since Pn = I − Qn −→ I as n −→ ∞ in the strong
operator topology (SOT), if we restrict the miminimization
in (16) over Q ∈ Bc = T (N ) to the span of {en :
n = 0, 1, 2, · · · , N }, that is, PN `2 =: `2N we get a
finite dimensional convex optimization problem in lower
triangular matrices QN of dimension N , that is,
µN :=
inf
QN ∈Bc (PN `2 ,PN `2 )
?
?
k(T2i
T1 T3ci
)|N − QN k
which overestimates µ and result in upper bounds. Solving
such problems are then applications of convex programming
techniques. However, this is not of much use unless we
know how far from the optimum µ this approximation has
caused us to go. Applying the same argument to the dual
optimization on the RHS of (16), that is,
µ0N :=
sup
TN ∈SN , kTN k1 ≤1
?
?
|tr(TN (T2i
T1 T3ci
)|N |
where
SN :=
{TN ∈ C1 (PN `2 , PN `2 ) :
(I − Qn )TN Qn+1 = 0, for all n}
yields other finite dimensional convex optimizations in TN
as the restrictions of T to `2N . We get lower bounds for µ
since the dual optimization involves a supremum rather than
an infimum. Since PN −→ I as N −→ ∞ in the SOT, it
can be shown that these upper and lower bounds converge to
the optimum µ as N −→ ∞. These optimizations estimate
µ within known tolerance and compute the corresponding
LTV operators QN , which in turn allow the computation of
LTV controllers K through the Youla parametrization.
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