IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
1607
Operator Theoretic Approach to the Optimal
Two-Disk Problem
Seddik M. Djouadi, Member, IEEE
Abstract—The nonstandard two-disk problem plays a fundamental role in robust feedback optimization. Here, it is shown
via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a
particular 1 -optimization problem. A novel operator theoretic
framework to characterize explicitly its solutions is developed, in
particular, the two-disk optimization is shown to be equal to the
induced norm of a specific operator defined on a projective tensor
product space involving a non-Hilbert version of a vector valued
2 space. Moreover, this operator is shown to be a combination of
multiplication and Toeplitz operators. Under certain conditions,
existence of maximal vectors is established leading to an explicit
formula for the optimal controller. An “infinite matrix” representation with respect to a canonical basis is derived, together with
an algorithm to compute it. The norm of the relevant operator is
approximated by special finite dimensional optimizations whose
solutions lead to solving semi-definite programming problems
involving the computation of a matrix projective tensor norm.
(3)
, is known
where
as the trace-class norm.
denotes the unit disk of the com.
denotes the
plex plane,
. If is a subset
boundary of ,
, then
denotes the complement of in
.
deof
notes the normalized Lebesgue measure on the unit circle
,
.
is the label used for “Lebesgue almost everywhere.” For a matrix or vector-valued function on the unit
is the real-valued function defined on the unit circle
circle,
,
. If
denotes a finite diby
,
, stands
mensional complex Banach space,
for the Lebesgue-Bochner space of th power absolutely inteunder the norm
grable -valued functions on
Index Terms—Feedback systems, functional analysis, optimal
control, optimization methods, robustness, uncertainty.
for
for
NOTATION
stands for the field of complex numbers.
denotes either the inner or duality product depending on the context.
denotes the identity map. If is a Banach space, then
de, where
denotes
notes its dual space. For an -vector
is the Euclidean norm.
the -dimensional complex space,
is the space of
matrices , where
is the largest
,
and
denote the complex
singular value of .
Banach space of
-vectors ,
, and
(4)
(5)
where
,
denotes the norm on
[1], and
is the abbreviation used for the essential supremum. If
,
, the th Fourier coefficient is defined by
, which define the well-known
,
, is the
Fourier series representation of .
Hardy space of -valued analytic functions on the unit disk ,
. In fact these spaces can
viewed as a closed subspace of
be realized as
with, respectively, the norms
if
and
Clearly,
,
of
(1)
is the dual space of
and vise versa.
,
, and
denote the complex Banach space
matrices
,
,
is defined as
such that
.
denotes the space of continuous -valued
Finally,
.
functions defined on
The space
with,
respectively, the following norms:
(2)
Manuscript received June 29, 2001; revised May 13, 2002, November 2, 2002,
and May 26, 2004. Recommended by Associate Editor V. Balakrishnan.
The author is with the Electrical and Computer Engineering Department, University of Tennessee, Knoxville, TN 37996-2100 USA (e-mail:
djouadi@ece.utk.edu).
Digital Object Identifier 10.1109/TAC.2004.835356
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I. INTRODUCTION
I
T IS WELL KNOWN that there are mainly two reasons for
using feedback, the first is to reduce the effect of any unmeasured disturbances acting on the system, the second is to reduce
the effect of plant uncertainty. The objective of this paper is to
investigate the ability of feedback to reduce the effect of unceroptimization known as
tainty by solving the nonstandard
the two-disk problem
0018-9286/04$20.00 © 2004 IEEE
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where
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
is a stable plant and
are outer weighting functions.
It is well known that the problem of worst-case disturbance rejection in a feedback system subject to a pair of uncertain disturbances representing, for example, output and sensor noises,
without the simplifying assumption that these are mutually orthogonal reduces to (7) [2], [3]. More importantly, the two-disk
problem reflects the fundamental tradeoff between sensitivity ,
and complementary sensitivity ,
as outlined in
[2], where it was termed the robust performance problem. Other
fundamental feedback control problems related to (7) are the optimal robust disturbance attenuation problem (ORDAP) [4]–[7],
where an uncertain plant is subject to disturbances at the output,
and where the objective is to find a feedback control law which
provides the best possible uniform attenuation of output disturbances in spite of uncertainty in the plant model; a plant uncertainty reduction or filtering problem posed by Zames [4, p. 316],
where the following question is posed: If a plant lies in some uncertainty set what is the best achievable uncertainty reduction if a
feedback control is to be used?, as well as questions of well posedness in feedback optimization studied by Smith [8] and rank one
-synthesis. The two-disk optimization problem (7) was considered by, among others, Francis, Owen, and Zames [5]–[7]. In particular, they showed that ORDAP reduces to finding the smallest
fixed point of the function shown in (8) at the bottom of the page.
In the single-input–single-output (SISO) case, (8) takes the form
shown in (9) at the bottom of the page, where the right-hand side
(RHS) member is equivalent to the optimization (7).
In [5] and [6], it was shown that there is a parametric verproblem which provides
sion of the standard two-block
an approximate solution based on the standard two-block
problem. Zames and Owen showed that such an approximation can be arbitrary poor, and is never better than a factor of
, and therefore they stressed the need for an “exact” theory
for such problems. The point is that in situations where the
, a more
fixed-point is sensitive to correct estimation of
exact optimization theory is required than that provided by the
quadratic two-block problem [9], [10], [7]. Using duality theory
they showed that under certain conditions there exists a solution
to the two-disk problem, which satisfies an allpass or flatness
condition, and is unique in the SISO case. By gridding the frequency domain they proposed a numerical algorithm based on
convex programming. However, this algorithm assumes uniform
Lipschitz continuity of the problem data (plant and weights),
which may be restrictive and has the drawback of solely ap-
proximating the optimal solution at a finite number of points.
Moreover, duality theory has not fully been exploited, and there
has been no closed form solution reported in the literature. To
some extent the two-disk problem has also been considered by
Holohan and Safonov in the context of optimizing the stability
and robustness margins, where part of the duality of the problem
was characterized in the SISO case [11]. Recently, Besson and
Shenton proposed an approximate graphical ad-hoc solution to
the suboptimal version of (7) [12]. The nonstandard nature of
the norms involved in the optimizations (7)–(9) make this class
of problems intractable by state-space techniques, simply because these norms (as shown later) are induced on certain Banach (non-Hilbert) spaces for which the Parseval Theorem does
not hold. Here, it is resorted to developing tools from functional
analysis and operator theory parallel to those used during the
control theory in the early eighties (see, e.g.,
emergence of
[6], [13], and [14]). In particular, we develop a Banach space
duality theory which leads to an exact solution in terms of operator theory. First, it is shown that an extremal measure in the
dual is absolutely continuous with respect to the Lebesgue measure, a consequence of this result is that the optimal solution
of the two-disk problem satisfies an extremal identity. It is then
shown that the optimal solution is a dual extremal function for an
space.
approximation problem in a particular vector valued
Next, it is shown, in particular when flatness holds, that the
optimal performance in the two-disk optimization is equal to
the induced norm of a special operator. This operator is a Banach space projection of a multiplication operator. The projection is computed explicitly showing that our operator is in fact a
combination of multiplication and Toeplitz operators defined on
and a subspace of the
the projective tensor product of
non-Hilbertian (Banach) vector valued
space. Under
certain conditions, the existence of maximal vectors is proved
leading to an explicit formula for the optimal controller in the
SISO case. An infinite matrix representation with respect to a
canonical basis is given together with general formulas and a
recursive algorithm to compute its columns. An approximation
of the operator norm by particular convex optimizations subject
to a matrix projective tensor norm is also provided, and allows
the computation of optimal performance as closely as desired.
Finally, it is shown that the computation of the matrix tensor
norms reduce to solving a particular semidefinite programming
(SDP) with linear matrix inequalities. The latter results in a uniform approximation of the optimal solution unlike the pointwise
approximation suggested in [7], [10] under the assumption of
(8)
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DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
uniform Lipschitz continuity of the plant and weights. Although
the results here are developed for stable plants, generalization
to unstable plants can be made without much effort by working
with coprime factorization techniques [15]. In view of the isoon the unit disk and
on
metric isomorphism between
the right-half plane, we will confine ourselves to the unit disk.
The results obtained apply to both discrete and continuous time
systems. Parts of the results here appeared in [15]–[19].
II. EQUIVALENT DISTANCE PROBLEM IN
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III. BANACH SPACE DUALITY STRUCTURE AND EXISTENCE
OF OPTIMAL SOLUTIONS
denotes the space of Lebesgue measurable,
Let
and absolutely integrable functions defined on the unit circle
with values in
under the norm
(13)
where
was defined earlier as the Banach space of (eswith
sentially) bounded analytic functions in the unit disk
, the norm is given by
values in
,
of
,
, 2. Let
, and
the singular values
be the quotient space
, where
(10)
such that
where
. Since is analytic
is a subharmonic function there
in , then
[1], hence, it satisfies the maximum principle and, therefore
(11)
Letting
have inner-outer factorization
, and
inner–outer factorization
,
the outer part of
, then
(7) is equivalent to [10], [9]
(12)
to the
Equation (12) is the shortest distance from
subspace
and are outer in
where
[9] and [10], if we assume that
of
, and
(14)
the space
is obtained by taking complex conjugates. The quotient norm is (15), as shown at the bottom of
denotes the coset associated to . It has
the page, where
is isobeen shown in [21], [22], [9], and [10] that
. Since
metrically isomorphic to the dual space of
may be viewed as a subspace of
,
and
is the preannihilator (or preorthogonal)
in
, then
is isoof
metrically isomorphic to the dual space of
and we write
. Every
defines a continon
given by
uous linear functional
,
(16)
inner. As in
then is a closed subspace of
, and there exists an
, such that
outer spectral factor of
. Assumption (A1) precludes strictly
proper plants, but as in the standard
theory, our theory can be
extended to them by modifying (see, e.g., [20], [9], and [15]).
Moreover, has the following equivalent representation [10],
, where
,
,
, so that
, Lebesgue almost everywhere
), that is, is inner. This notation should not be confused
(
with that for rational functions in
. Banach space duality theory is a very important tool in solving optimization problems in normed linear spaces, in particular, the so-called duality
product plays a role analogous to the inner product in Hilbert
space. In Section III, we give a similar duality structure to [9],
[10] which will prove to be more useful for our purpose.
where
space of
. Let
be the sub-
defined by
(17)
where denotes the direct sum of two subspaces, is the identity map, and
(18)
The following Lemma characterizes the preannihilator (or preorthogonal) of .
is the preannihilator of in
.
Lemma 1:
Proof: Since
is the preannihilator of
, the Lemma follows from Lemma 1 in [10],
[7], and [3].
From a standard result in Banach space duality theory relating the distance from a vector to a subspace and an extremal
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
functional in the predual [23, Th. 2, p. 121], we deduce the following theorem, which asserts the existence of at least one optimal (controller) .
Theorem 1: Under assumption (A1), that is,
,
, there exist at least one optimal
such that
is given by the space
The dual space of
of
-valued functions of bounded variations on
the norm [21], [9], [10], [3]
under
(22)
, and
denotes the sum of the total
where
of all entries of and . Let
variations on
be the space of
-valued functions which are continuous
on the closure of the unit disk and analytic in .
is known as the disk algebra. The annihilator of
is
completely described by the vector-valued version of the F. and
M. Riesz Theorem [25], which yields
(19)
denotes the coset norm in
.
where
In general, the supremum on the left-hand side of (19) is not
. The best we can say at this point
achieved by any
is that, it follows by the Bishop–Phelps Theorem [24], that the
for which the supremum in
set of functions
. However, if we
(19) is attained is norm dense in
assume
is continuous on the unit circle
and
where
(20)
i.e., when the open unit disk analyticity constraint
),
is removed, (in the scalar case (
), then the optimal solution is flat [9], [10], [7], and the supremum in question is
always achieved. This motivates the following section, where
an alignment condition and an extremal expression are shown
to characterize the optimal solution. Note that assumption (A2)
is weaker than a similar assumption in [10], [7], and [9].
IV. ABSOLUTE CONTINUITY AND EXISTENCE OF AN EXTREMAL
IDENTITY FOR THE OPTIMUM
(23)
The dual space of
norm
is then given by [26],
, under the quotient
.
Next define the subspace
. It folis
lows from Lemma 3 [10], [3] that the annihilator of
given by (24), as shown at the bottom of the page, where
.
In the following lemma, we establish that the distance from the
to the subspace
is the same as
vector function
to .
Lemma 2: Under assumptions (A1) and (A2), the following
holds:
(25)
(26)
A. Alignment in the Dual
Recall that
denote the space of continuous functions on
with values in
. If
we set
(21)
It turns out that under assumptions (A1) and (A2), the maximum
in (26) is achieved by a measure absolutely continuous with
respect to the Lebesgue measure. It will then follow by the F. and
M. Riesz Theorem that the supremum in (19) is in fact achieved
.
in
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DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
Lemma 3: Under assumptions (A1) and (A2), we have
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. The optimal performance
is then a dual extremal function for the
approximation problem
.
,
Proof: “Only if” by assumption
, and
such that
where
(27)
where
denotes the coset norm in
.
Proof: See the Appendix.
Lemma 3 implies that under mild assumptions the optimal
solution is flat or allpass. It follows also from the proof that the
optimal solutions in the dual are aligned.
(31)
but the integrand
B. Dual Extremal Property and Extremal Identity for the
Optimum
-spaces of scalar valued
A well-known fact in the theory of
contains a representative
functions is that every coset in
of the least possible coset norm [27], [28], [26]. More precisely,
, consider the linear functional
on
given a kernel
defined by
. Knowing that
, find a kernel function
, such that
, and
. A dual
extremal function is a function
,
such
that,
[29],
[27], [26]. We will show that these remarkable properties hold
, and are inherited by
. The opfor
can then be intertimal solution
preted as the dual extremal function of an optimization problem
.
in
Lemma 4: [16], [15], [19]
, there exists
a) Given
such that
(28)
b) Given
there exists
such that
(29)
is then an extremal kernel or function for
.
Proof: See the Appendix.
Combining Lemmas 3 and 4, we obtain the following theorem
which looks familiar in the theory of extremal problems [26].
Theorem 2: Under assumptions (A1) and (A2),
,
, is an extremal kernel for
, and
is optimal
if and only if
(30)
integrating implies equality must hold throughout, and (30)
holds. This, combined with flatness, implies
(32)
“If” suppose that (30) holds, integrating it yields
hence, equality must hold throughout and
is then optimal.
Remark 1: Theorem 2 shows that if the convex programming method of [7] and [10] is to be used, the search in the
predual which is convex can be restricted to functions which
,
, (or
satisfy
,
, in the scalar case),
moreover, expression (30) can be used as a test for optimality
. The RHS of (30) shows that the optimal primal and
for
dual solutions are “pointwise” aligned in the dual. Note that
is not constant, (
),
and
are nonzero,
in case
), and under assumptions (A1), (A2),
,
(
,
That is at each
frequency the dual extremal function , (at least in the scalar
. From the
case), is an extreme point of the unit ball of
flatness condition the RHS of expression (30) is equal almost
everywhere to , and points to a tradeoff between the optimal
nominal sensitivity represented by
, and
the optimal nominal complementary sensitivity
at each frequency . Since (30) is equal to , it also shows
how the weighting functions affect the optimal solution at each
frequency.
A novel operator theoretic framework to characterize explicitly the solutions to the two-disk problem is developed next.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
V. OPERATOR THEORETIC SOLUTION TO THE
TWO-DISK PROBLEM
First, introduce a key multiplication operator acting on parand
spaces which are not Hilbert.
ticular vector valued
As pointed out previously, the induced norm of this multiplication operator is equal to the norm involved in the two-disk, and
clearly shows that the non-Hilbertian nature of the vector valued
and
preventing using state–space or time domain techniques to solve this problem. However, these spaces essentially
capture the geometry of the problem.
Proof: See the Appendix.
This proposition shows that these spaces although Banach
(non-Hilbertian) are not only reflexive, but also isomorphic to
and
, respecthe standard Hilbert spaces
tively. Moreover, their norms are equivalent. This will allow us
to bring some Hilbert space structure into the problem. In the
next section, we characterize the optimal solution in terms of
a particular operator.
VI. EXACT OPERATOR THEORETIC SOLUTION
A. Key Multiplication Operator
For the remainder of this paper, we will assume that (A1) and
(A2) hold.
is the complex space of Lebesgue inRecall that
-valued functions defined on the unit circle
,
tegrable
under the norm
A. Optimal Performance
for
(33)
Note that
is not given by any Hilbert space
is not a Hilbert space, but is innorm, so that
, where
deed isomorphic to the Hilbert space
is the standard
-dimensional Euclidean space. Let
, the multiplication operator associated with , and mapping the standard Lebesgue
into
is denoted by
. More precisely
space
It is clear that the identity map from the Hardy space
onto the standard Hilbert space
is isomorphic, since these spaces are topologically identical and their
norms are equivalent.
be the orthogonal projection on the closed
Let
subspace
of
, where
is understood to be the orthogonal
,
.
complement of
Next, define the bounded linear projection as follows:
(34)
In the next proposition, we show that the operator induced norm
is equal to
, and in fact it suffices to work
in the standard Hardy space
.
, then
Proposition 1: Let
1)
(37)
where is the usual composition operation.
It is clear from the definition of , and the properties of the orthogonal projection
that every vector function
can be decomposed uniquely as the sum of two vector functions
as follows:
where
(35)
2)
(36)
Proof: See the Appendix.
is finite dimensional, then the dual space of
Note that
is clearly given by the Banach space
[21].
is reflexive. That is
It follows that
(38)
are orthogonal in the sense that
. Also, by uniqueness, note that
Since the orthogonal projection
gives the best approxiby functions in the subspace
mation of elements of
in the
-norm. The projection
operator plays a similar role as
but for approximations in
-norm by elements of
.
the
Now, define the following key bounded linear operator:
and
,
by
(39)
Define the following continuous bilinear transformation:
Recall that
(
) is the closed sub(
) consisting of
space of
(
)-bounded pairs of analytic functions in the unit
disk. In the next proposition, their dual spaces are characterized.
and
be defined as
Proposition 2: Let
before, then the following statements hold.
1)
.
.
2)
Hence,
and
are reflexive Banach spaces.
(40)
(41)
Then, the norm of
is given by
(42)
DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
The bilinear map may be linearized by invoking a projective
tensor product space [30]. To see this, define a bounded linear operator on the projective tensor product
, by
. The projective norm of
is defined as shown in the first equation at the
, for
bottom of the page. More explicitely, letting
,
, may be written as
, in which case
(43)
A useful property of the projective tensor norm
is that it is
finitely generated [30], that is, the search in (43) can be restricted
and
to finite dimensional subspaces of
. This fact will be exploited in Section VI-D to derive
. The operator plays a central role in
an SDP to compute
finding an explicit solution to our problem through the following
theorem which quantifies optimal performance.
Theorem 3: Let be the optimal performance index defined
by (7), then under assumptions (A1) and (A2), the following
hold.
i)
is equal to the induced norm of , namely
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operators are defined on the standard Hilbert spaces (or suband
, and where a multiplication by a cerspaces of)
tain unitary operator transform the problem to a one block
problem. The difference here is in the definition of the nonHilbertian norm of the range space of , which precludes any
transformation to one block problem by pre- or postmultiplication by unitary operators. The existence of a maximal vector
for the operator under the previous conditions, implies that
a well-known Hankel–Toeplitz operator, which solves the stanproblem, achieves its norm on its discrete
dard two-block
spectrum for (possibly) infinite-dimensional systems [15]. This
settles an open question in [32] and [33]. The existence and computation of maximal vectors, although difficult, is important and
lead to the exact computation of the optimal controller as shown
in Section VI-B for the SISO case.
be arbitrary but fixed, there
Proof of Theorem 3: Let
,
, and
exists
,
, such that
(46)
and
has the form
However,
It follows then
,
.
, thus
(44)
ii) There exist maximal vectors for , i.e.,
-norm 1, and
,
, or putting
, then
of
such that
(45)
Remark 2: The operator may be viewed as an analog to
the Sarason operator [31], and Young’s operator [13]. In parproblem, where these
ticular, for the standard two-block
since
. Note that
(47)–(49), as shown at the bottom of the page, hold true.
Since is arbitrary, we get (50), as shown at the bottom of the
page. The last equality follows from Theorem 1.
where
and
(47)
therefore
(48)
(49)
(50)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
Now, we would like to prove the reverse inequality. Note that
(51)
, similarly
and has norm 1. The member on
the left-hand side of (53) is equal to (54)–(55), as shown at
is the orthogonal projecthe bottom of the page, where
to
. In fact,
tion from
.
is the left multiplication operator by
and has norm
, acting from
to
By Lemmas 3 and 4 and Theorem 2, there exists
, with
having
, and
denotes the duality
-norm 1, and such
that
product in
(52)
By the proof of Theorem 2,
,
; and there exists
,
,
such that
However,
. Moreover,
then
, since
for some
, i.e.,
,
and
.
; see the first equation shown at
Hence,
,
the bottom of the page. Noting that
.
, since
, so
.
Therefore,
. Likewise,
.
Therefore,
. Now, let
denote the
identity matrix, then
(53)
Exploiting an idea in [34], the subspace
consisting of functions each of whose columns belongs to
can be identified with the tensor
, and will henceforth be denoted so. Likewise
. We have
,
, and
(56)
(57)
. Expressions
Hence,
, and equality in (55)
(50), (52), and (55) imply that
must hold inferring existence of maximal vectors.
The orthogonal projection plays a crucial role in solving
problem,
our problem as well as the standard two-block
e.g., mixed sensitivity problem. Thus, it is important to compute
it explicitly, hence the following theorem.
be the orthogonal projection from
Theorem 4: Let
to
, then
(58)
is the standard Riesz projection, i.e., if
where
has Fourier coefficient
,
, then
and has Fourier coefficients
such that
,
and
,
Remark 3: As noted previously, the projection not only
twosolves our problem, but also the standard quadratic
as acting on the Hilbert space
block problem by viewing
, and without having to rely on unitary transformations to transform the problem to an equivalent one block
problem (see, e.g., [32] and [33].) The same remark applies to
problem (when
) by viewing as
the standard
acting on
instead (see, e.g., [35]).
(54)
(55)
DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
Proof of Theorem 4: For
, let us compute
1615
Hence, equality must hold throughout, in which case it is posso as to satisfy
sible to choose the optimal
,
Remark 4: For SISO systems (
), is a scalar function,
, and by uniqueness of the optimal solution, we
can obtain a formula for the optimum
(64)
(59)
is indeed a projection. Clearly, the adjoint
of
, although defined on different
itself, so that
is
spaces, is equal to
an orthogonal projection. Next, we show that the null space of
,
.
then
Let
, since
, then
and, therefore,
. Hence,
. Conversely, let
, then
so
thus
, so
and, therefore,
It follows from [23, Th. 1, p. 155] that
.
(60)
The following Corollary follows then from Theorems 3 and 4,
and provides an explicit solution to the two-disk problem.
is given exCorollary 1: The optimal performance index
plicitly by the following expression for some
,
, and
,
:
(61)
Next, the optimal controller for the two-disk problem is characterized explicitly.
which determines
through we get
uniquely. Multiplying and dividing
, hence,
,
The optimal con, where
,
troller
achieves the optimal robust performance. For the general
case (MIMO systems) the optimal controller is in fact highly
nonunique, and a more involved procedure is required in order
to compute it. We need sufficiently many independent directions to determine the whole symbol . A good starting point
are [34], [36], and [37].
The norm of the operator gives the optimal performance
and controller. It is therefore important to develop an algorithm
to compute it. This motivates the next section.
C. Computation of the Operator Norm
In this section, we show how it is possible to compute the
and hence the optimal performance
norm of the operator
, has a power
index . First, note that every
, for some
.
series expansion
Therefore,
,
;
; form
, where ,
is the usual
a basis for
. To determine a representation of with
canonical basis of
,
;
;
respect to the basis
it suffices to study the action of
on each element of the
for a fixed . We have then for
basis say, for example,
,
, where
is the th column of
. In the representation of , the
th column will be the representation of
with respect
to the basis
,
;
; of
B. Optimal Controller
Theorem 1 implies that there exists a vector function
such that
. Hence, we can interpret
(62)
Moreover by Proposition 1,
may be viewed as a
acting from
into
multiplication operator
such that
. Equivmay be viewed as an operator acting from
alently,
into ,
the projectve tensor product
. We have by
Theorem 4
(63)
, for each ,
as the matrix function whose columns are projections of the
columns of
onto the subspace
.
With this in mind, the action of on is then determined by
. Therefore, with
respect to the canonical basis
,
has the
following representation:
(65)
Each column in (65) is the image of a basis vector after applying
. These columns are determined by studying the action of on
the basis vectors
,
as explained above; the
th column corresponding to the representation of
with
; is then formed by the coeffirespect to the basis
in
with respect to
. To compute
cients of
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these coefficients consider,
,
and
belong to
ries expansion in the unit disk
, since
, they all admit power se-
At the th step, we have
•
Remark 5: The previous formulas show that when the first
column is computed, all the remaining columns follow easily.
However, from a computational point of view, it is more efficient
to compute these columns recursively. Hence, the following recursive algorithm.
Then, we have
, and
Recursive Algorithm:
For
•
(66)
(67)
Where
;
. The
operator has then the following infinite matrix representation
, and for
,
with respect to the basis
,
, the action of on is determined
by
,
where
We get also
.
Now, each column
•
•
,
.
Hence, straightforward computations yield the following.
•
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
(68)
of
By the proof of Theorem 3, there exist
-norm 1,
of
-norm 1, which form a maximal vector for . Next,
we show that is such that
•
(69)
That is
,
,
; we will get
If
, for
(70)
DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
Equality (69) is important since it allows us to bring in some
Hilbert space structure as shown by (70). Indeed alignment in
, we
the dual implies that (71) holds and, for some
have
(71)
1617
over
. Clearly,
; and that
is uniformly bounded. The
defined as the dimension
, is such that
. The
of the image space of
operators
,
, have rank less than
can be represented by a finite
or equal . The operators
matrice formed by the first columns and rows of the infinite
matrix representing in (68). We show that in fact as tends
to infinity
converges to in the strong operator topology
(SOT), i.e., for
(79)
(72)
(73)
Indeed, by adding and subtracting
implies
, the triangle inequality
(74)
Applying Cauchy–Schwarz inequality, we get
(80)
(75)
thus
Since
is finite dimensional, then by the Riesz–Fischer Theorem (cf. [38, Lect. VI]) the two terms on the right-hand side of
(80) tend to zero and, thus, (79) must hold.
has the following
Noting that any in
form
for some
,
to the
similar arguments apply then by restricting
over
. Calling the orthogonal prospan of
,
, the following holds
jection on
as
. Therefore, under assumptions (A1) and (A2) we have
(76)
(77)
Hence, equality must hold throughout, and from (73) and (74)
we deduce
as
and
Note that
the bilinear form
(81)
are finite rank operators. Defining
(82)
(78)
hence
,
, and (69) and (70) must
hold. This fact wil be used in approximating the operator norm
in (84) as discussed in the next section.
D. Approximation by Special Matrix-Norms
In what follows, we will show that it is possible to approximate the norm of by the norms of finite-dimensional
operators or matrices. Indeed, if we restrict to finite dimensional subspaces of its domain, then we will obtain a sequence
of finite rank operators whose norms are “nearly” optimal,
in the sense that they approach the norm of as the dimension of the initial spaces increases. More explicitly, let
denote the orthogonal projection from
to the subspace
,
; and
spanned by
the orthogonal projection from
onto the span of
the corresponding linear operator
is then defined by
. Identity (81) implies
as
in the SOT. Moreover, with
that
, where
respect to the canonical basis
is the standard basis of
,
can
be represented by a matrix obtained by truncating the infinite
.
matrix representation in (68) of , i.e.,
can be computed recursively using the
The columns of
previous recursive algorithm. Similarly,
, can be represented by a matrix which
columns can be computed as in the previous recursive algorithm
, where each entry
developed in Section VI-C,
in
belongs to
,
. Putting
,
.
and
,
are the
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vectors containing the th-first Fourier coefficients of
respectively. Expression (82) of
reduces then to
and
subject to
(83)
(88)
Each can be decomposed as
Call
,
appropriately in such a way that
,
. Partitioning
.
, where
,
, are
matrices, the rows of
are the even numbered rows of
, and
’s rows are the odd numbered rows of
. Note
that
, for some (constant)
permutation matrix of appropriate dimension. The optimum
can then be approximated by
If
as
, then the minimization (87) can also be written
(89)
subject to
(90)
..
.
(84)
where
as
. To show that the optimization
in (84) is in fact convex, note first that
is a bilinear form in
and
. To linearize it let
, and invoke again the projective
and
,
tensor product of
, under the projective norm
defined below. Observe that the space of bilinear forms on
is isometrically isomorphic
to the dual space of
[30]. This means that the bilinear form
may be replaced by a bounded linear functional on
. We have then shown that
the optimization (84) can be rewritten as
(85)
is the projective norm of and is defined as shown
where
in (86) at the bottom of the page. The computation (86) has been
shown in [39] to reduce to the following SDP with linear matrix
,
inequality (LMI) constraints, by noting that
and in fact it is a little easier to work with the latter.
(87)
(91)
Therefore, the problem reduces to solving the optimization (85)
subject to the SDP with LMI constraints (89) and (91).
Remark 6: It is therefore possible to approximate uniformly
as closely as desired the norm of , and hence , by computing
the special matrix norms defined by (89). From expression (85),
we see that explicit computation of the maximal vectors of ,
(or ), will shed light on the form of the maximal vector of
(respectively, ) , and therefore the optimal controller
.
As showed this is realized by solving a convex optimization
problem with LMI constraints.
VII. CONCLUSION
This paper was concerned with the so-called two-disk
problem, which arises in many control problems, including the
fundamental tradeoff between sensitivity and complementary
sensitivity functions [2]. The two-disk problem is intractable
by established methods in the standard
theory, and the
duality theory developed in the work of Owen and Zames does
not characterize explicitly exact solutions, in particular when
certain uniform Lipschitz conditions (see [9]) do not hold.
This provided the motivation of the study undertaken here. We
started by a re-examination of the duality theory of the problem
working in a smaller (matrix-valued)
space (rather than a
larger (matrix-valued)
). The two-disk problem was then
interpreted as a distance minimization, allowing predual and
dual representations under a weaker continuity assumption for
the latter. Useful geometric properties were deduced, for example, an extremal identity for optimal performance/controller
which provides a test of optimality; the extremal function in the
predual space satisfies a norm constraint. Then a new operator
theoretic framework was developed based on the introduction of
(86)
DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
two non-Hilbertian versions of the vector valued
space. The
optimal solution was shown, under specific conditions, to be
equal to the norm of a certain operator, which is a combination
of multiplication and Toeplitz operators subject to a projective
tensor norm. Existence of at least one maximal vector was
deduced, which resulted in an explicit formula for the optimal
controller in the SISO case. An infinite matrix representation
of the operator with respect to a canonical basis was provided,
together with approximation by norms of finite rank operators,
whose computations involve solving an SDP problem.
It should be noted here that the results obtained are valid, at
least in theory, for infinite dimensional systems. The theory developed also generalizes to unstable systems using coprime factorization techniques [15]. A similar duality and operator theory,
(although the latter was already developed in the eighties), has
problem in [15],
been developed for the quadratic two-block
which may be useful for infinite-dimensional systems. In particular, a numerical solution based on convex programming along
the lines of [10] and [7] is possible for this class of systems.
1619
The reverse inequality is clear since
follows from [23, Th. 1, p. 121].
. The third equality
B. Proof of Lemma 3
By Lemma 2, there exists a measure
,
, such that (26) holds, i.e., if we let
to be the sum of the total variations of the entries of ,
and
its linear extension to
which satisfies
for all
. Then, by (26), we have (92)–(95),
as shown at the bottom of the page, where
, and
isfies
written as
. The extremal measure
(96)
APPENDIX
A. Proof of Lemma 2
The second equality follows from the same argument used to
prove Lemma 2 [10], [7], [3], where unlike here it is assumed
is continuous on the unit circle. Let
, be
that
. Define
the scaling of the unit disk, and
,
and note that
, and
is
continuous on the unit circle by assumption (A2), then
Note that
satcan be
for some
suppose that
and
and
and
, and
. Next,
is a Borel subset of
, such that
,
. Define
for
,
, for
, then
, which implies
,
on , and from a standard
result from matrix theory
,
on , but since
, then
must be identically the zero matrix.
is restricted
Therefore the maximum in (26) remains , if
, where (97), as shown at the bottom of the page, holds,
to
but
, therefore, by duality
is bounded above by
, since
is subharmonic and satisfies the maximum principle. By continuity
as
. Hence
(98)
contradicting hypothesis (A2) (i.e.,
solutely continuous with respect to
). Hence,
is ab-
. Let
(92)
(93)
(94)
(95)
(97)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 10, OCTOBER 2004
be the Lebesgue decomposition of
part. Since we have that
, where
is its singular
By a), there exists
such that (28) holds.
On the other hand, by uniqueness of the
limit we
have
thus
, and then
It follows by the F. and M. Riesz Theorem that
is absolutely
continuous w.r.t. the Lebesgue measure , therefore likewise
for , and hence (27) holds.
C. Proof of Lemma 4
a) There exists a sequence
such that
(99)
have
Equation (99) implies that the measures
uniformly bounded total variations, by Alaoglu’s Thehas a
cluster
orem [23], the sequence
say. Hence,
point
as
we get
. Putting
,
.
D. Proof of Proposition 1
1) Equality (35) follows from a measure theoretic argument;
is obvious since for
first note that
and
; if
then
we have
(100)
,
However,
therefore
,
and
implying
annihilates
we deduce
. Thus,
, by the F. and M. Riesz Theorem
(101)
and, hence, a) holds.
b) There exists a sequence
where
,
,
such that
(102)
and then by the same argument as (a),
converges in the
topology, we have for
as
hence,
. For the reverse inequality,
, have singular values arranged in decreasing
let ,
,
; and
order
singular value decomposition
,
, 2, where
and
are unitary
-matrix valued
functions, and
. Again by the Riesz Theorem, there exist
such that
Moreover, for each
the columns of
,
, form a complete orthonormal basis of eigenvectors for
each of
,
. The first column of
,
,
, corresponds to the largest singular value of
,
,
. Now let
; then by definition of
of positive Lebesgue
the supremum there exists a set
measure such that
,
Let
be the characteristic function of , and
. Then
L
, and we
define
have
. Therefore we have
and
. Replacing
by their singular value decompositions, yields
(103)
Since
However,
with (103) imply that
, and
together
and
are unitary, we obtain then
DJOUADI: OPERATOR THEORETIC APPROACH TO THE OPTIMAL TWO-DISK PROBLEM
Since
and
correspond to the largest eigenvalues of
and , respectively, then
Since is arbitrary, then
conclude that
2) Clearly by 1)
, and we
.
(104)
be a trigonometric polynomial in
and
. Then
Let
, where
Multiplying by the
th power of
1621
coincide, if and only if the positive Riesz projection
on
is bounded, i.e., if
has
,
, then
Fourier coefficient
and has Fourier coefficients
such
,
and
,
. We set
that
. Then,
is a linear operator on
with
. The negative Riesz projection
is
values in
defined analogously, it associates to its negative Fourier
, where has Fourier coefficoefficients, i.e.,
such that
,
, and
cients
,
.
will be used later to prove a forthcoming
theorem. Since the trigonometric polynomials with coare dense in
, it suffices to
efficients in
,
,
prove that
, trigonometric polynomial
,
,with
.
Hence, we have
, yields
However, since the trigonometric polynomials are dense in
then
and so
is bounded.
2) Follows in a similar fashion.
ACKNOWLEDGMENT
so (36) must hold and we are done.
E. Proof of Proposition 2
1) The dual space of
sociated space
is equal to
sociated norm
can be identified with its as(Theorem 4 [40]), where
, but under the as-
(105)
, and
. Note that
can be regarded
[40]. Define the identity
as a subspace of
, then is conmapping
,
tinuous. By Theorem 5 [40],
i.e., the associated norm
and the norm of
where
This project was initiated while the author was a Ph.D. student under the supervision of the late Prof. G. Zames. The author would like to take this opportunity to extend his sincere
appreciation to Prof. Zames for his guidance and kindness by
dedicating this paper in his memory.
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Seddik M. Djouadi received the B.Sc. degree
(with first class honors) from Ecole Nationale
Polytechnique, Algiers, the M.A.Sc. degree from
Ecole Polytechnique, Montreal, QC, Canada, and
the Ph.D. degree from McGill University, Montreal,
QC, Canada, all in electrical engineering, in 1989,
1992, and 1999, respectively.
He is currently an Assistant Professor in the Electrical and Computer Engineering Department, University of Tennessee, Knoxville. He was an Assistant
Professor with University of Arkansas, Little Rock,
and held postdoctoral positions in the Air Force Research Laboratory and the
Georgia Institute of Technology, Atlanta, where he was also with American Flywheel Systems (AFS), Inc., Medina, WA. His research interests are power control for wireless networks, robust control, active vision, and identification.
Dr. Djouadi received three U.S. Air Force/National Research Council
Summer Faculty Fellowship awards, the Tibbet Award with AFS in 1999, and
a finalist for the American Control Conference Best Student Paper Certificate
(best five in competition) in 1998.