IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
41
Generalized KYP Lemma: Unified Frequency
Domain Inequalities With Design Applications
Tetsuya Iwasaki, Senior Member, IEEE, and Shinji Hara, Senior Member, IEEE
Abstract—The celebrated Kalman–Yakubovič–Popov (KYP)
lemma establishes the equivalence between a frequency domain
inequality (FDI) and a linear matrix inequality, and has played
one of the most fundamental roles in systems and control theory.
This paper first develops a necessary and sufficient condition for
an -procedure to be lossless, and uses the result to generalize the
KYP lemma in two aspects—the frequency range and the class
of systems—and to unify various existing versions by a single
theorem. In particular, our result covers FDIs in finite frequency
intervals for both continuous/discrete-time settings as opposed to
the standard infinite frequency range. The class of systems for
which FDIs are considered is no longer constrained to be proper,
and nonproper transfer functions including polynomials can
also be treated. We study implications of this generalization, and
develop a proper interface between the basic result and various
engineering applications. Specifically, it is shown that our result
allows us to solve a certain class of system design problems with
multiple specifications on the gain/phase properties in several
frequency ranges. The method is illustrated by numerical design
examples of digital filters and proportional-integral-derivative
controllers.
Index Terms—Control design, digital filter, frequency domain
inequality, Kalman–Yakubovič–Popov (KYP) lemma, linear matrix inequality (LMI).
I. INTRODUCTION
O
NE OF THE most fundamental results in the field of
dynamical systems analysis, feedback control, and signal
processing, is the Kalman–Yakubovič–Popov (KYP) lemma
[1]–[3]. Various properties of dynamical systems can be characterized by a set of inequality constraints in the frequency
domain. The KYP lemma establishes equivalence between such
frequency domain inequality (FDI) for a transfer function and
a linear matrix inequality (LMI) for its state space realization.
The basic roles of the KYP lemma are two fold: it provides 1)
insights into analytical approaches to systems theory, and 2) a
framework for numerical approaches to systems analysis and
synthesis.
Manuscript received September 4, 2003; revised May 25, 2004 and
September 11, 2004. Recommended by Associate Editor Y. Ohta. This work
was supported in part by the National Science Foundation under Grant 0237708,
by The Ministry of Education, Science, Sport, and Culture, Japan, under Grant
14550439, by CREST of the Japan Science and Technology Agency (JST), and
by the 21st Century COE Program on Information Science and Technology
Strategic Core.
T. Iwasaki is with the Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746 USA (e-mail:
[email protected]).
S. Hara is with the Department of Information Physics and Computing, Graduate School of Information Science and Engineering, The University of Tokyo,
Tokyo 113-8656, Japan (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAC.2004.840475
The KYP lemma [3] states that, given matrices
Hermitian matrix , the FDI
,
, and a
(1)
holds for all
if and only if the LMI
(2)
admits a Hermitian solution . Thus, the infinitely many inequalities (1) parametrized by can be checked by solving the
finite-dimensional convex feasibility problem (2). Appropriate
choices of in (1) allows us to represent various system properties including positive-realness and bounded-realness.
While the KYP lemma has been a major machinery for developing systems theory, it is not completely compatible with
practical requirements. In particular, each design specification
is often given not for the entire frequency range but rather for a
certain frequency range of relevance. For instance, a closed-loop
shaping control design typically requires small sensitivity in a
low frequency range and small complementary sensitivity in a
high frequency range. Thus a set of specifications would generally consists of different requirements in various frequency
ranges. On the other hand, the standard KYP lemma treats FDIs
for the entire frequency range only.
The current state of the art for fixing the incompatibility is to
introduce the so-called weighting functions. A low/band/highpass filter would be added to the system in series as a weight
that emphasizes a particular frequency range and then the design parameters are chosen such that the weighted system norm
is small. The weighting method has proven useful in practice,
but there remains some room for improvement. First, the additional weights tend to increase the system complexity (e.g., controller order), and the amount of increased complexity is positively correlated with the complexity of the weights. Second,
the process of selecting appropriate weights can be time-consuming, especially when the designer has to shoot for a good
tradeoff between the complexity of the weights and the accuracy in capturing desired specifications. We remark that some
of these deficiencies may be addressed by recent developments
for new characterizations of disturbance signals [4], [5].
An alternative approach is to grid the frequency axis. In this
case, infinitely many FDIs are approximated by a finite number
of FDIs at selected frequency points. This approach has a practical significance especially when the system is well damped
and the frequency response (after the design) is expected to be
“smooth” (i.e., no sharp peaks). The resulting computational
0018-9286/$20.00 © 2005 IEEE
42
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
burden often seems moderate. However, it is difficult to determine a priori how fine the grid should be to achieve a certain
performance. The method also suffers from the lack of a rigorous performance guarantee in the design process as the violation of the specifications may occur between grid points.
Another approach that avoids both weighting functions and
frequency gridding is to generalize the fundamental machinery,
the KYP lemma, in such a way that FDIs in finite frequency
ranges can be treated directly. Although the time-consuming design iterations may not be completely eliminated with this approach, it is expected that the ability to directly deal with finite
frequency ranges without weighting functions would provide a
more user-friendly platform for systems design. There are several results in the literature along this line. The finite frequency
,
KYP lemma obtained in [6] states that, given a scalar
matrices , , and a Hermitian matrix , the FDI (1) holds in
the low frequency range
if and only if the LMI
(3)
admits Hermitian solutions
and
. The nonstrict inequality version of this result has been obtained in [7] in the context of integrated design of structure/control systems. Other generalizations of the KYP lemma include those in [8]–[10] where
FDIs for polynomials are considered in the discrete-time setting. Specifically, algebraic conditions are given to characterize
polynomials that are positive on an arc of the unit circle in the
complex plane. Some of these results are developed for the purpose of digital filter design.
The main objective of this paper is to further generalize the
KYP lemma and to unify all the previous extensions previously
mentioned by a single theorem. There are two basic aspects in
our generalization—the frequency range and the system class.
In particular, we consider the frequency intervals characterized
by two quadratic forms of the frequency variable . This
characterization captures curve(s) on the complex plane and
encompasses low/middle/high frequency conditions for both
continuous-time and discrete-time systems. Thus, the result
includes the previous results on the low/middle frequency conditions for continuous-time systems [7] as special cases, and
adds new contributions to the cases involving high frequency
conditions and discrete-time systems. The system class is generalized by, roughly speaking, replacing the term
in (1) with a new term of the form
.
It turns out that this allows us to capture FDIs for descriptor
systems as well as polynomials.
The generalized KYP lemma will be obtained within the
framework of the -procedure [11], [12] that has been extensively used in the systems and control literature (e.g. [13]–[16]).
It converts an inequality condition with multiple constraints
to an unconstrained inequality condition with multipliers. The
-procedure is conservative in general, i.e., the latter condition
implies the former, but the converse is not always true. Fradkov
showed in 1973 that the -procedure with scalar multipliers is
lossless (i.e., nonconservative) if and only if a certain rank-one
property holds for an associated separating hyperplane [17].
This type of rank-one property has been used as a sufficient condition for the -procedure to be exact in a more general setting
of matrix-valued multipliers [3], [6], [18]–[21]. In this paper,
we will prove necessity of the rank-one property, providing a
necessary and sufficient condition for the -procedure to be
lossless in the general setting. Our result on the -procedure
will be used to generalize the KYP lemma, but it will also be of
independent interest as a basis for systems analysis in general.
Another contribution of the paper is to develop a useful interface between the generalized KYP lemma and various engineering applications. We will show that the design specifications encompassed by our result can be summarized, in the case
of single-input–single-output (SISO) transfer function designs,
as follows. The Nyquist plot of the transfer function within
a prescribed frequency interval must lie in the intersection of
prescribed conic sections on the complex plane. It will also
be shown that several classes of important engineering problems can be formulated in terms of such specifications, and can
be solved exactly through LMI optimizations. Finally, design
examples of digital filters and proportional-integral-derivative
(PID) controllers will illustrate the procedures and advantages
of our approach in comparison with existing ones.
This paper is organized as follows. We will first discuss in
Section II system design problems that are naturally described
by FDI specifications in various frequency ranges. These problems motivate our generalization of the KYP lemma to treat finite frequency ranges. Section III develops an extension of the
-procedure and obtains a necessary and sufficient condition
for losslessness. Section IV gives a characterization of general
frequency ranges (curves on the complex plane) in terms of
quadratic forms of the frequency variable. Section V presents
our main results that generalize the standard KYP lemma. We
will then show in Section VI how various FDI specifications
for system design can be described within our framework. Finally, Section VII shows what classes of design problems can be
solved via convex optimizations, where analytical discussions
will be followed by design examples.
We use the following notation. For a matrix , its transpose,
complex conjugate transpose, and the Moore–Penrose inverse
,
, and
, respectively. The real and
are denoted by
are denoted by
and
. The
imaginary parts of
stands for the set of
Hermitian matrices. For
symbol
, inequalities
and
a matrix
denote positive (semi)definiteness and negative (semi)definiteis deness, respectively. The set of matrices
means the Kronoted by . For matrices and ,
and
, a function
necker product. For
is defined by
(4)
The convex hull and the interior of a set are denoted by
and
, respectively. Given a positive integer , let
be the
.
set of nonnegative integers up to , i.e.,
Finally, denotes the set of positive integers.
IWASAKI AND HARA: GENERALIZED KYP LEMMA
43
II. MOTIVATION: SYSTEM DESIGN
We will first motivate our research through several examples
of design problems for which our generalized KYP lemma is
naturally suited and can be useful. Later in the paper, some of
these problems will be discussed in more detail with solution
procedures and numerical design examples.
Digital filter design: The problem is to find a stable SISO
transfer function
satisfying a set of frequency domain
specifications. Typical design requirements for band-pass filters
are given by the following Chebyshev approximation setting
(see, e.g., [22]–[25]):
:
:
:
: all
(5)
where
and
is a given function that
. An
has desired gain/phase properties in the pass-band
would have the unity magnitude and the
ideal response
linear phase in the pass-band, i.e.,
. The first three
conditions specify the stop-bands and the pass-band, while the
last condition is added to suppress overshoot in the transition
and
.
bands
Sensitivity-shaping: This is a typical control design with
specifications on the closed-loop transfer functions [26], [27].
and a controller
, the sensitivity and the
For a plant
and
are defined
complementary sensitivity functions
by
Fig. 1.
Open-loop shaping specifications.
the low frequency range for the sensitivity reduction. The servo
bandwidth requirement in the middle frequency range might
be represented by an elliptic constraint as shown in the figure,
where we may maximize the bandwidth. The two straight lines
provide gain/phase
lying to the right of the point
stability margins, and the small circle centered at the origin
relates to the roll-off requirement in the high frequency range
for robust stability.
Structure/control design integration: Control performance
of mechanical structures can be significantly enhanced if the
designs of the structure and the controller are integrated [7],
[28]–[30]. It has been shown that the finite frequency positivereal (FFPR) property in the low frequency range
(6)
The sensitivity-shaping problem typically consists of the following type of requirements:
:
:
: all
: all
where
. The first constraint may represent good
tracking of a reference signal with spectral contents in the
low frequency range, while the second may be interpreted as
a robustness requirement against unmodeled high frequency
dynamics of the plant. The last two constraints tend to avoid
oscillatory time responses.
Open-loop shaping: This is a classical control design
problem of determining the controller parameters to meet
specifications on the open-loop transfer function. Given a
, a set of specifications on
marginally stable SISO plant
is given in terms of the Nyquist plot of
the controller
the open-loop transfer function
. Fig. 1
shows typical design requirements, where the shaded regions
indicate where the Nyquist plot should lie in various frequency
at the lower right
ranges. The half plane constraint on
part of the figure corresponds to the high-gain requirement in
is an important requirement for mechanical structure design
to guarantee the existence of controllers that achieve high
servo-bandwidth [7], [31], where
is the transfer function
of the mechanical system to be designed. These references
demonstrated that structures of practical significance can be
designed to achieve the FFPR property by solving LMI feasibility problem with the aid of an existing version of the finite
frequency KYP lemma [6], [7]. In particular, the method has
been applied to a shape design of a swing arm for hard disk
drives [7] and a smart arm design using piezo-electric film [31].
III.
-PROCEDURE
We will first develop a generalized version of the -procedure that converts a constrained inequality to an unconstrained
inequality with multiplier(s) [12], [17], [32], [33]. The result is
instrumental to the proof of the generalized KYP lemma to be
stated later, and will be of independent interest as a tool for developing systems theory.
A. Classical Form and Its Generalization
The classic version of the -procedure [12], [33] is the following. Given ,
, we have the equivalence
such that
such that
44
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
where the regularity,
version
, is assumed. The strict inequality
such that
such that
also holds, and for this equivalence the regularity assumption is
is negative semidefinite in particular, the
not required. When
statement reduces to the Finsler’s theorem [34]. The -procedure is to replace the condition on the left of “ ” by the one
on the right. It may be difficult to check the former condition
directly because the inequalities impose nonconvex constraints
on in general. On the other hand, the latter condition is easier
to verify as it is a search for the scalar multiplier satisfying
convex constraints.
To generalize the classical -procedures, let us rewrite them
with different notation. First note that the set of matrix-valued
multipliers associated with the classical -procedures can be
defined as
a) admissible if it is a nonempty closed convex cone2 and
;
;
b) regular if
c) rank-one separable if
.
It can readily be verified by a separation theorem (Lemma
is
11 in Appendix II) that a nonempty closed convex cone
admissible if and only if the set is nonempty. Note that is
the set of the hyperplanes that separate
from , motivating
the term “rank-one separable” in c). The definition of regularity
is consistent with the standard notion. For example, the set in
. It will turn out that regularity
(7) is regular if and only if
corresponds to controllability when we consider a special class
that is useful for systems analysis.
of
Some examples of admissible, regular, rank-one separable
in (7) with
and
sets include
(11)
(7)
Also note that a general Hermitian form can be expressed as
where
satisfies
and
. Then alternative expressions for the
aforementioned -procedures are given by1
(8)
(9)
where
is a set specified by
as follows:
rank
(10)
on
in
For brevity, we will omit the dependence of and
the development below.
Clearly, the -procedures are completely specified by the set
through (8)–(10). We consider a generalization of the -procedure to the case where
is an arbitrary subset of Hermitian matrices, rather than the specific set in (7). In this case, the
equivalence in (8) or (9) no longer holds in general. In fact, it
is easy to verify that the implication “ ” always holds, but the
converse “ ” may or may not, depending upon
and . The
-procedure is said to be lossless if not only “ ” but also “ ”
hold.
B. Characterization of Lossless -Procedure
The objective of this section is to show under what condithe associated -procedure is lossless regardless of
tion on
. The previous result in [6] provided a sufficient condition for
the strict version of the -procedure (9) to be lossless. Our result below shows that the previous condition turns out to be also
necessary, and is valid for the nonstrict case (8) as well under
a regularity assumption. To this end, let us introduce the following.
is said to be
Definition 1: A set
tr(2S ) 0
tr(2 ) 0
1Notation
means that
S
holds for all S
type of notation will be used throughout this section.
2S
It is straightforward to verify that these sets are admissible and
regular. The rank-one separability has been proven for (7) in
in [3], [35], and for
in [6]. The set
re[12], for
lates to the entire frequency range in the continuous-time setassociated with
via (10) is
ting. Specifically, the set
with
,
given by matrices of the form
for some
whenever
[3]. On the other hand, the set
relates to a finite frequency range by
and
[6], [19]. The multiplier associated with the ( , ) scaling that
gives a real upper bound [36] has been shown to be rank-one
. Other results in
separable [19], and is closely related to
the literature (e.g. [18] and [37]) may also be interpreted within
the framework of rank-one separability.
The following result provides a necessary and sufficient condition for the S-procedure to be lossless.
Theorem 1 (S-procedure): Let an admissible set
be given and define
by (10). Then, the strict S-procedure is
, if and only if
lossless, i.e., (9) holds, for an arbitrary
is rank-one separable. Moreover, assuming that is regular,
the nonstrict -procedure is lossless, i.e., (8) holds, for any
, if and only if
is rank-one separable.
Proof: We shall prove the result for the nonstrict inequality case. The strict inequality case can be proven similarly.
is rank-one separable.
Let us first prove sufficiency. Suppose
Note that the implication “ ” in (8) is obvious. To show the
. Then, from Lemma
converse, suppose that
guarantees that there exists
10 in Appendix II, regularity of
such that
has no solution
. From
an
a separating hyperplane argument (Lemma 11 in Appendix II),
such that
there exists a nonzero
Since
. This
2A
set
is a convex cone containing the origin, we have
M is a cone if M 2 M implies M 2 M for all 0.
IWASAKI AND HARA: GENERALIZED KYP LEMMA
45
Thus,
trices
. Since
is rank-one separable, there exist masuch that
. Note that
implies that
must be true for some
index . This means that the condition on the left in (8) does not
hold, and we conclude that “ ” holds in (8).
We now prove necessity. The main idea follows a result by
Fradkov [17, Th. 19] that proved necessity for a different case
is of the type given by (7). Suppose
is not rank-one
where
such that
and
separable. Then there exists
because
. Note that the latter condition
is disjoint from
. Since
means that the point
is a closed convex cone and
, the infimum of the disand
is strictly positive. Consequently,
tance between
a separation theorem [38, Th. 11.4] infers that the two sets are
strongly separable by a hyperplane, i.e., there exists a Hermitian
matrix such that
and
Clearly, the second condition implies
. On the
other hand, the first condition implies that
because otherwise we have the following contradiction: For
such that
where we noted that
, and
due to
and
. Thus, we have shown that if
is not rank-one
separable, then there exists for which the left-hand side (LHS)
of (8) holds but the right-hand side (RHS) does not. Hence, for
such , (8) does not hold, proving necessity.
Theorem 1 shows that the rank-one separability is necessary
and sufficient for the -procedure to be lossless regardless of
the choice of . It should be noted that the rank-one separability may not be necessary for a particular choice of . Indeed,
a recent result by Ebihara et al. [39] provides a sufficient condition for losslessness when is negative semidefinite, and their
condition is weaker than the notion of rank-one separability.
A class of rank-one separable sets can be generated by a transformation of a particular rank-one separable set as follows. The
result will be used to derive a generalized KYP lemma in a later
section.
be a rank-one separable set. Then
Lemma 1: Let
the set
is rank-one separable for any matrix
and subset
of positive–semidefinite matrices
containing the origin.
be a nonzero positive–semidefinite
Proof: Let
matrix such that
. Since
, we have
. Since
is rank-one separable, there exist
such that
an integer and vectors
and
. By [34, Th. 2.3.1], the former implies existence
of such that
Then, noting that
hold for all
, it can be verified that
and
IV. FREQUENCY RANGE CHARACTERIZATION
One of the key developments in this paper is a unified characterization of finite frequency ranges for both continuous-time
and discrete-time systems. In general, a frequency range is visualized as a curve (or curves) on the complex plane. For instance, a low frequency range in the continuous-time setting is a
line segment on the imaginary axis containing the origin. A frequency range in the discrete-time setting would be an arc of the
unit circle. The objective of this section is to propose a general
framework for representing various curves (frequency ranges),
which will later be used for characterizing FDIs.
Let us first clarify what we mean by curves on the complex
plane.
Definition 2: A curve on the complex plane is a collection of
continuously parametrized by
infinitely many points
for
where ,
and
.A
is said to represent a curve (or
set of complex numbers
curves) if it is a union of a finite number of curve(s).
We consider the following set of complex numbers that represents a certain class of curves:
(12)
are given matrices and the function
is
where ,
defined in (4). The dependence of on and may be made
when no confusion
implicit and is used in place of
should arise. We will explain what type of curves can be repthrough what choices of and . The
resented by
has been extensively studied in the literaspecial case
ture (e.g. [40]) with the notion of “ -stability”, and the previous
work provides a basis for the analysis in the next paragraph.
is the intersection of
and
Note that the set
. It can readily be verified that the set
represents
. If
, then it is
a curve if and only if
an empty set, the entire complex plane, or a set with a single
, then it is either a circle or a straight line.
element. If
Conversely, every circle or a straight line can be represented by
with such that
. In particular
defines the circle of radius
Without loss of generality, we may assume that
to be zero. Define
lowing for some
be a full rank factorization. Then
. By [34, Th. 2.3.8], there exists such that
. This completes the proof.
with center at
, and
by aland
defines a straight line with the normal vector
. The
set
is a region on the complex plane with the boundary
46
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
whenever
; it is the inside or outside of a
, then it is the entire complex
circle, or a half plane. If
and
, it is
plane. The remaining case is trivial; if
either an empty set or a set with a single element.
represents curve(s) only if
We now see that the set
and either
or
, in which case, it
can be partial or whole segment(s) of a circle or a straight line.
When the necessary condition is satisfied, it is possible that the
and
is empty or
intersection of the two sets
a single point, rendering the condition not sufficient. To state
to represent
a necessary and sufficient condition for
curve(s), let us introduce a simultaneous matrix factorization
result. A proof is given in Appendix I.
Lemma 2: Let ,
be given. Suppose
.
Then there exists a common congruence transformation such
that
(13)
(14)
where , ,
and
. In particular, and can
.
be ordered to satisfy
With the factorizations in (13), the essential part of the set
may be captured by the canonical set
.
Specifically, the former can be generated by the bilinear transformation of the latter. The following lemma formally states
and proves this fact in the general setting where ,
are arbitrary.
and nonsingular
be
Lemma 3: Let ,
by
given. Define scalars , , , and and function
(15)
Then the following holds true:
Conversely, let
that
be an element of the set on the LHS. Note
Hence,
is well defined and it can be verified that
.
Since
then follows from the previous identity,
we see that belongs to the set on the RHS of (16).
with
and
deLet us now examine the set
fined in (14). Note that
holds if and only if
for some
. Moreover, for such , we have
. Hence, the set
is the entire imaginary
if
, and is a partial segment of
if
axis
; otherwise, it is either empty or a single point. Conis conceived as the image of (a segsequently, the set
ment of) the imaginary axis mapped through the bilinear transis an invertible, continuous mapping alformation. Since
most everywhere on ,
represents curve(s) if and only
does so.
if
The following result summarizes the aforementioned development. Its formal proof is omitted for brevity.
be given and define the set
Proposition 1: Let ,
by (12). Then the set
represents curve(s) on
the complex plane if and only if the following two conditions
hold:
;
•
or
;
• either
where and are defined by the factorization in (13). In this
case, there are two possibilities.
I)
is a circle or a line specified by
since
.
is a partial segment (or
II)
segments) of a circle or a line specified by
.
By an appropriate choice of and in (12), the set can be
specialized to define a certain range of the frequency variable .
For the continuous-time setting, we have
(16)
Proof: Let be an element of the set on the RHS of (16).
such that
and
Then there exists
. Note that the following identity holds for any
and
if
:
From this identity and
and
have
, it readily follows that
holds and, hence, we
. Moreover
due to nonsingularity of . Hence, we see that
seton the LHS of (16).
belongs to the
where is a subset of real numbers specified by an additional
choice of , for instance, as follows:
where
, and LF, HF, and MF stand for low,
high, and middle frequency ranges, respectively. Similarly, for
the discrete-time setting, we have
IWASAKI AND HARA: GENERALIZED KYP LEMMA
where
47
is a subset of real numbers specified by
where
,
, and
. Detailed derivations of these tables are omitted here
due to space limitation, but can be found in a conference version
of this paper [41].
The set can also capture a portion of the real axis as
This variation is not fully explored in this paper but will be
useful for robustness analysis with respect to real parametric uncertainties (e.g., [42]). It is worth noting that an interval on the
real axis can also be treated by the unit circle [43] through the
Bliman’s transformation
.
V. GENERALIZED KYP LEMMA
A. Main Theorem
This section derives a generalized KYP lemma using the
-procedure in (9) or (8); an FDI will be specified by the
LHS of each statement, and the RHS then provides an LMI
captures
condition for satisfaction of the FDI. In particular,
the input–output graph of a system in a certain frequency range.
To elaborate on this point, let us first discuss the standard KYP
would be given by
lemma. For the FDI in (1), the set
and to show that the
as in (10) by an appropriate choice of
possesses the properties in Definition 1 so that the
chosen set
-procedure is lossless. We will take these steps in the “canonical coordinates” via the bilinear transformation. In particular,
can be reduced
we first show that the case with general
through the simultaneous factorization in
to that with
(13), and then take the aforementioned two steps for the latter
case.
and a nonsingular matrix
Lemma 4: Let ,
be given and define ,
by (13). Consider
in
in (12), and
in (19). Suppose
(18),
represents curve(s). The following conditions on a given vector
are equivalent.
holds for some
.
i)
holds for some
.
ii)
Proof: When statement i) holds, there are three possible
, b)
,
, or c)
,
,
cases: a)
where the entries of matrix are defined by (15). Hence, in view
is
of Lemma 3, we see that i) holds if and only if: a)
, b)
,
,
unbounded and
and
, or c)
for some
such
. Through some algebraic manipulations and using
that
Lemma 13 in Appendix II, it can be verified that the condition is
,
is unbounded, and
equivalent to: a)
; or
,
, and
, b)
,
is unbounded, and
, or
for some
such that
.
c)
We now see that the condition is equivalent to
for some
.
The previous lemma allows us to convert the -procedure to
a standard form. In particular
for some
is equivalent to
This set can also be characterized (in the spirit of [44]) as
for some
(17)
where
and
(18)
We now generalize this characterization of the FDI with respect
to the frequency range. Recall that the set in (12) captures
finite frequency ranges of our interest. Hence, we consider the
general frequency range in (12) for the definition of in (17)
where is defined as
(if is bounded)
(otherwise).
(19)
Finally, the system description is also generalized by considering a completely arbitrary matrix rather than the one with
the structure as in (18).
It can now be seen that the main technical steps to arrive at
in (17)
the generalized KYP lemma are to express the set
where
, and similarly for the nonstrict inequality case. Note that, if the factorization in (13) is used, the
is (a segment of) the imaginary axis, in which
set
case some prior results already exist [3], [6], [7], [45]. Here, we
provide the aforementioned two steps by two lemmas.
and ,
be given
Lemma 5: Let
by (19)
such that in (12) represents curves. Define and
and (18), respectively. Then, the set
defined in (17) can be
characterized by (10) with
(20)
Proof: We will first prove that the following two conditions are equivalent:
for some
;
i)
for all ,
,
;
ii)
and
are defined in (14). First note that i) holds
where
holds for some
such
if and only if either a)
that
, or b)
is unbounded (i.e.
)
48
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
and
Since
, where ,
are defined by
.
represents curve(s), there are two possibilities:
and
. In the former case, [3, Cor. 4] proves
. It is then straightforward
that i) is equivalent to
to show that this condition is further equivalent to ii). In the latter
case, [6, Lemma 2] can be used to show the equivalence between
i) and ii).
be defined by (17), and
be defined to be
in
Now, let
(10) with (20). Then, for a nonzero vector
for some
for some
for all
where the first and fourth equivalences easily follow from the
definitions, and the second and third hold due to Lemma 4 and
ii), respectively.
the previous equivalence i)
is a closed convex cone. When
, the set
Clearly,
is nonempty and hence
is admissible. From Lemma 2, we
have
is regular and the matrix is a minimal realb) The set
in the sense that
ization of the set
(21)
if and only if
(22)
We are now ready to state and prove a general theorem.
,
Theorem 2 (Generalized KYP): Let matrices
, and ,
be given and define and
by (12) and (19), respectively. Suppose represents curves on
the null space of
where
the complex plane. Denote by
is defined in (18). The following statements are equivalent.
.
i)
ii) There exist ,
such that
and
(23)
Moreover, if the rank condition in (22) is satisfied, then the following statements are equivalent.
.
i)
such that
and
ii) There exist ,
(24)
where
case, defining
or
can be assumed. In the former
the set can be characterized as
with
defined in (11). Since
is rank-one separable, it then follows from Lemma 1 that
is rank-one separable. Similarly,
, the set can be characterized as
in the latter case
for some matrix and for a subset of
positive semidefinite matrices. Again, Lemma 1 infers that is
rank-one separable. These arguments are summarized in statement a) of the following lemma. Statement b) gives a condition
for regularity, and is proven in Appendix III.
Lemma 6: Let matrices ,
and
( ,
) be given such that in (12) represents curves. Define
by (20) and the matrix-valued mapping
by (18).
the set
The following statements hold true.
a) The set
is admissible and rank-one separable.
Proof: We will prove the strict inequality case. The nonstrict inequality case can be shown similarly. Note that (i) holds
if and only if
holds where
is defined in (17).
can be characterized by (10) with
From Lemma 5, the set
in (20). By Lemma 6, the set
is admissible and rank-one separable. Hence, from the generalized -procedure (Theorem 1),
satisfying
condition is equivalent to the existence of
, or equivalently, the existence of ,
such
and (23) hold. Since the inequality in (23) is strict,
that
we can strengthen the positivity of as
without loss of
generality.
In view of the developments in Section IV, the term
in statement ii) of Theorem 2 can be specialized to the
low/middle/high frequency ranges in the continuous/discretetime settings as shown in the equation at the bottom of the page.
When , , , and are all real matrices, and in statement ii) of Theorem 2 can be restricted to be real without loss
of generality. This follows basically from the fact that the real
part of a complex Hermitian positive–definite matrix is positive
definite. Note that, if the frequency region is not symmetric
about the real axis, then one needs to search for complex and
IWASAKI AND HARA: GENERALIZED KYP LEMMA
49
even when and are real. In such case, the LMI in complex variables can be converted to an LMI of larger dimension
in real variables through the following equivalence:
is controllable. Let be the set of eigenvalues
b) the pair
of in . Then, the following statements are equivalent.
, we have
i) For each
ii) There exist
where
and
,
such that
and
are real square matrices.
(27)
B. Specific Extensions of the KYP Lemma
The general result in Theorem 2 can be specialized to several
versions of the KYP lemma. There are two aspects of specialization: the frequency range and the underlying system. The former
has been briefly discussed at the end of the previous section. We
will devote this section to the latter. In particular, we consider
FDIs for descriptor and state space systems as well as for polynomial functions.
Theorem 3 (Descriptor, Strict Inequality Version): Let ma, ,
,
, and ,
trices ,
be given and define and by (12) and (19), respectively. Suppose that: a) represents curves on the complex
plane, b)
for all
, and c) either is
nonsingular or is bounded. Then, the following statements are
equivalent.
i) For
, we have
for all
.
such that
and
ii) There exist ,
(25)
Proof: The result follows from Theorem 2 by choosing
(26)
If is bounded, then the result is immediate by noting that the
is given by
for all
null space of
under the invertibility assumption on
. If is unbounded,
coincides with the null
we need to make sure that the above
at
as well. This is indeed the case if is
space of
nonsingular.
In Theorem 3, statement ii) implies that
for
, provided that the upper left
block of , denoted
all
, is positive semidefinite. To see this, let us assume ii) and
by
for some
. Let
be a vector in the
null space of
, i.e.,
. Denote by
the upper
block of the matrix on the LHS of (25). Then
left
which cannot be true as every terms are nonnegative. By contrafor any
diction, we conclude that ii) implies
. Based on this observation, Theorem 3 can be modified
as follows: Replace supposition b) with
, and add the
for all
to statement i).
condition
Theorem 4: (State–space, nonstrict inequality version) Let
,
,
, and ,
matrices
be given and define and by (12) and (19), respectively.
Suppose that a) represents curves on the complex plane, and
Proof: The result basically follows from the nonstrict inequality case of Theorem 2 by choosing as in (26) with
and
. Note that the rank condition in (22) translates to
and nonsingularity of when
.
controllability of
The only crucial step in this proof is to show that statement i) is
equivalent to
(28)
where
is the null space of
Theorem 3,
can be given by
Hence, i) is equivalent to
. As discussed in the proof of
if
.
By Lemma 12 in Appendix II, this condition is equivalent to
(28) under the controllability assumption.
Theorems 3 and 4 capture certain properties of dynamical
systems expressed in terms of state space, possibly descriptor,
realizations. The results include, as special cases, the standard
version of the KYP lemma [3], its finite frequency extensions
[6], [7], and a unified (continuous/discrete-time) FDI charac) [46]. A
terization for the entire frequency range (i.e.,
and
may also be
special case of Theorem 3 with
obtained through an application of a recent, independent result
by Scherer [21].
The following result characterizes certain properties of polynomials, and is particularly useful for the design of digital filters
as discussed later.
Theorem 5 (Polynomial version): Let matrices
,
, and ,
be given
and define by (12). Suppose that represents curves on the
complex plane. Define
(29)
Then, the following statements are equivalent.
i)
holds for all
.
such that
and
ii) There exist ,
(30)
where
.
,
,
, and
50
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
The equivalence also holds when the three inequalities are
replaced by strict inequalities with the modification that condiis added to statement i) if is unbounded,
tion
where
is the upper left block of .
, define
for
Proof: For a vector
. Then
and
hold where
and
. Noting that
, we see
that, for a fixed
, condition
holds if and
only if
for all
such that
in different frequency ranges
this type, with different sets
where
. We will address such design problems
in the next section. The objective of this section is to show what
classes of can be treated within the framework of our generalized KYP lemma.
Recall that the FDIs arising in various versions of the KYP
lemma (theorems in Section V) are given in terms of quadratic
forms. Thus, a set can be completely captured within our
framework if it can be described by a quadratic form. The idea is
and , a transfer function
to construct, from given
and a Hermitian matrix such that
(31)
If is bounded, the result then directly follows from Theorem
2. When is unbounded, Theorem 2 infers equivalence of i)
and ii) if condition
for all
such that
for each fixed frequency. If
is a polynomial, then the FDI
in a given frequency range can be characterized
by an LMI using Theorem 5. If it is a transfer function
, then the FDI can be expressed as
is added to statement i). This additional condition is equivalent
. We claim that this is already implied by the
to
original statement i) when is unbounded. To see this, note that
This implies that
has a positive eigenvalue for
with a sufficiently large magnitude, whenever
. This completes the proof for the nonstrict inequality case.
Finally, the strict inequality case can be proved in the same
is not immanner. Note, however, that condition
for all
in general and, hence,
plied by
has to be added to statement i).
Versions of positive-real lemma for polynomials have
been obtained in the standard (unrestricted) frequency setting
[47]–[49]. However, results in the restricted frequency setting
are limited. References [8]–[10] gave characterizations of
scalar-valued quasipolynomials that are nonnegative on the
unit circle. Our results extend and unify these prior results to
the case of matrix-valued positive definite polynomials in both
continuous-time and discrete-time settings with both strict and
nonstrict inequality cases. In particular, a convex parametrization of positive quasipolynomials as in [9], [10] can be obtained
from Theorem 5; see [50] for details.
VI. FREQUENCY RESPONSE SPECIFICATIONS
Consider a transfer function
that depends on the deinclude the opensign parameter vector . Examples of
, the closed-loop sensitivity funcloop transfer function
tions
and
, and the digital filter
as discussed in
Section II. The design problem is to find such that
where is a prescribed subset of the complex matrices. More
generally, may be required to satisfy multiple constraints of
(32)
allows us to treat the FDI
Thus, the appropriate choice of
within the framework of Theorem 3 (or Theorem 4 when
). Hence, the question is how to construct
and
so that
(31) holds. We will answer this for different classes of in the
following sections.
A. Half Plane and Circular Regions
The simplest choice of is
characterized by the FDI in (31) is
. In this case, the set
(33)
This set captures some fundamental properties. For instance, the
small gain condition is given by
and the positive-real condition is given by
To discuss a more general specification on the Nyquist
represents a
plot, let us consider the case where
single-input–single-output (SISO) system. In this case,
for all
means that the segment of the
lies in the
Nyquist plot specified by the frequency range
region on the complex plane. As discussed in Section IV, the
,
region defined by (33) is nontrivial if and only if
in which case, it is given by a half plane or a region inside
or outside of a circle. Clearly, the small gain (disk) and the
positive-real (right-half plane) conditions are characterized by
special cases of such regions.
IWASAKI AND HARA: GENERALIZED KYP LEMMA
51
B. General Conic Sections
TABLE I
10 CASES OF VARIOUS REGIONS
is a scalar (SISO)
Next we consider the case where
transfer function and is a general conic section on the complex
plane.3 Recall that a conic section can be characterized by
(34)
. The following lemma shows that
for some real matrix
this characterization can be converted to another which is compatible with our description of the frequency domain inequality.
be given and consider
Lemma 7: Let a real matrix
the set of complex numbers defined in (34). This set can also be
characterized by
where
is defined by
Proof: Note that
Then, it is straightforward to see that
from which the result follows.
Lemma 7 allows us to establish the equivalence in (31) when
is a conic section by choosing appropriate and . In parbelongs to the conic section
ticular, the transfer function
defined in (34) if and only if is chosen as shown in Lemma
7 and
is specified as
where
is defined by condition
for all
for the continuous-time case
for the discrete-time case. Thus, for
and
, the conic section condition
can
each frequency
with the above augmentation, and
be converted to
the latter can further be converted to an LMI via theorems in
Section V. The conic sections captured by (34) are summarized
later. The proof is omitted for brevity, and can be found in [50].
be given and conProposition 2: Let a real matrix
sider the set in (34). Partition as
, e.g.,
Fig. 2.
Essential regions for 4 .
which is parametrized by a vector
satisfying
In particular, it is obtained from one of the ten regions in the
-plane (see Table I and Fig. 2) through a rotation by and a
.
shift by
VII. APPLICATIONS TO SYSTEM DESIGN
and denote the spectral decomposition of
by
where
and
. Then, the set is nontrivial
and has an interior if and only if is indefinite. In this case,
is given by the set of
such that
This section is devoted to synthesis problems in engineering
applications that can be solved using the LMI characterization
of the FDI in the generalized KYP lemma.
A. Basic Idea for Synthesis
3The developments below can be extended for multiple-input–multiple-output (MIMO) square plants, but the physical meaning is not completely
clear.
From the discussion in the previous section, a general synthesis problem can be formulated as follows. For each
, let transfer function
, Hermitian matrix
, and frequency range
be given where
depends on the
52
parameter vector
for some ,
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
to be designed, and
is specified by (12)
. The problem is to find such that
(35)
for all
. We show how the problem can be solved exactly under certain assumptions. For brevity of exposition, let
, i.e., we have just one specification.
us consider the case
The case with multiple specifications can be handled similarly.
In the sequel, we will drop the subscript “ ” in (35).
A standing assumption is that the design parameter vector
appears affinely in
. If
is a polynomial, then its
is a proper rational
coefficients are affine functions of . If
transfer function, then it has a state–space realization
where
and
are affine functions
of while and are constant matrices. Another assumption
is such that
is that the weighting matrix
(36)
where and are the numbers of inputs and outputs of
.
This condition makes the set of
satisfying
convex. Note in particular that, when
, the design specification (35) defines a convex region in the complex
is desired to lie. This framework capplane in which
tures such standard specifications as the bounded-real and positive-real properties.
Under these assumptions, the condition in (35) is clearly
convex in terms of the design parameter . Furthermore, theorems in Section V can be used to reduce the problem to an LMI.
We will explicitly show how this can be done using Theorem
is a proper transfer function. Other
4 for the case where
is a polynomial or a nonproper (descriptor)
cases where
transfer function can be treated similarly using Theorem 5 or
3, except that the inequality in (35) has to be made strict in the
latter case.
Lemma 8: For a transfer function
, the condition
,
holds if and only if
such that
there exist Hermitian matrices and
(37)
holds, where
provided (36) holds.
Proof: This result follows directly from Theorem 4 with
in (32) and the Schur complement.
When and are affine functions of the design parameter
vector , (37) defines an LMI in terms of the variables , ,
and . Therefore, can be computed via convex programming.
The reduction of the synthesis problem (35) to LMIs is rather
straightforward once the proper assumptions are imposed. Attention must be paid, however, to when the assumptions are satisfied, and how they can be enforced through reparametrization
of the design parameters. We will discuss these points for several engineering problems in the sequel.
Structure/control design integration: The specification is
given by the FFPR condition in (6), which can be converted
to an LMI via Theorem 4. The design parameters would apfor certain problem settings,
pear only in the numerator of
including actuator or sensor placement. The design space may
is affine in the new design
then be reparametrized such that
parameter vector . In this case, the design process is reduced to
an LMI problem, which can be solved via semidefinite programming. This situation occurs in some practical design problems
[7], [31].
Open-loop shaping: The control design via open-loop
shaping discussed in Section II falls into the aforementioned
framework if the poles of the controller are fixed a priori. An
important practical problem that naturally fits in here is the PID
control design. The controller transfer function is described by
(38)
is a small parameter introduced to approximate
where
the differentiator by a proper transfer function. If is fixed, all
the design parameters ( , , and ) appear linearly in the
open-loop transfer function
. Various specifications on the Nyquist plot of may be expressed as confor all
where
straints of the form
is a frequency interval and
is the corresponding convex
region on the complex plane such as a disk, half plane, and a
conic section. The idea described in Section VI can be used to
in terms of quadratic forms of some (possibly
represent
as in (35). The design paramaugmented) transfer function
eters will enter
linearly and, hence, the PID controller can
be designed by solving an LMI as in Lemma 8.
Closed-loop control design: This is a fundamental problem
of designing a controller to satisfy various constraints on the
, including the sensiclosed-loop transfer functions
tivity-shaping problem discussed in Section II as a special case.
With the Youla parametrization of all stabilizing controllers
[51], each closed-loop transfer function depends on the free
in an affine manner. Thus the search for
is
parameter
a convex problem, provided the constraints are convex in terms
of the closed-loop transfer functions [52]. Since the infinite
dimensionality of the parameter space is problematic, one often
approximates the space by a finite dimensional space spanned
by a selected set of basis functions. The problem then becomes
the search for the coefficients of the basis functions, which can
be converted to an LMI problem within the above framework
satisfying
if the specifications are given by (35) with each
(36).
Digital filter design: This is another problem discussed
in Section II. Let us first consider the finite impulse response
(FIR) filter synthesis problem. The denominator of the FIR filter
IWASAKI AND HARA: GENERALIZED KYP LEMMA
is fixed to
53
, and a state–space
realization of
(39)
may be given by the pair
form and
in the controllable canonical
where
are the coefficients of
. Note that
and are fixed matrices and the design parameters
appear
linearly in
and . Moreover, it is not difficult to see that
. Therefore,
the constraints in (5) are convex in terms of
Lemma 8 can be directly applied.
Next, we consider the infinite impulse response (IIR) filter
and
are design variables. The
design where both
problem is fundamentally more difficult than the FIR case
because the above specifications are not convex in terms of
, and the simple-minded approach described previously
does not yield LMI problems due to the dependence of on
, the coefficients of
. However, using the ideas
of [9], [10], [53], and [54], the problem can be made tractable
when the specifications are given by inequality constraints
only. For instance, one can handle the
on the gain
specifications (5) if the third constraint is replaced by
This type of gain specifications can be converted to FDIs in
terms of polynomials based on the Fejer–Riesz result [54, Th.
8.4.5]. Hence, the generalized KYP lemma for polynomials
(Theorem 5) will be useful here. Finally, in certain applications, design objectives may be reasonably well described by
gain constraints only, but the IIR filter design with both gain
and phase constraints still remains as an important practical
problem that needs to be addressed.
B. Design examples
1) PID Controller: We consider the design of PID controllers (38) for a mechanical plant with a lightly damped
flexible mode
The design objective is to have a good tracking performance
with a reasonable stability margin and robustness against unmodeled dynamics. Our approach is to shape the Nyquist plot
. In parof the open-loop transfer function
ticular, we consider the following specifications:
a)
b)
c)
Specification a) with small
ensures robustness against unmodeled dynamics which typically exists in the high frequency
range. Specification b) is meant to guarantee a certain stability
margin. It may be more natural to require the Nyquist plot to be
. Howoutside of a circle with its center at the point
ever, such requirement leads to a nonconvex region on the complex plane and our design method cannot be applied as condition
(36) is not satisfied. On the other hand, the above Spec ification
b) is a half-plane requirement which can be handled within our
framework. The frequency range corresponding to Specification
b) would be the entire imaginary axis, but a small lower bound
has been introduced to exclude
from the range
and to avoid a possible numerical difficulty due to the controller
ensures
pole at the origin. Specification c) with a large
sensitivity reduction in the low frequency range by making
high gain. Again,
would be more direct for the purpose
but does not define a convex constraint on . Specification c) is
a reasonable alternative, given that the phase angle of
apas approaches 0 from above.
proaches
We have designed a PID controller by minimizing subject
to Specifications a)–c) where the other parameters are fixed as
The resulting optimal PID controller is found to be
with the optimal value
. The Nyquist plot, sensitivity functions, and the step response are depicted in Figs. 3–5,
respectively. Also plotted for comparison are the result of a popular heuristic PID tuning rule (Ziegler–Nichols ultimate sensitivity method) [55] where the PID parameters are given by
The stars and the (small) circles in Fig. 3 indicate the points on
the Nyquist plots at frequencies
and , respectively.
We see from Fig. 3 that the constraints in Specifications b) and
c) are active for the optimal PID control, and the optimization
clearly has improved the stability margin and the sensitivity reduction in the low frequency range. The latter effect is transparent in Fig. 4, which shows that the optimization added more
damping to the closed-loop poles than the Ziegler–Nichols design. This can also be seen directly from the oscillatory nature
of the step responses plotted in Fig. 5. Overall, this example
illustrates that our method allows for the tuning of the PID parameters to achieve desired frequency responses in prescribed
frequency ranges.
2) FIR Filter: We consider the design of low-pass FIR filters
to meet the following specifications:
a)
b)
where
is the filter transfer function to be designed. Spec, that the
ification a) requires, within the pass-band
has frequency characteristics similar to the desired
filter
function
having unity gain and the linear phase with group
54
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
Fig. 3. Nyquist plots. (Left) Optimized. (Right) Ziegler-Nichols.
Fig. 4.
Sensitivity functions. (Left) Optimized. (Right) Ziegler-Nichols.
Fig. 5. Step responses. (Solid) Optimized. (Dashed) Ziegler-Nichols.
delay , where is a positive integer.4 Specification b) ensures
has a low gain in the stop-band
. Let be
that
the order of the FIR filter to be designed. For fixed , , , ,
and , we design an FIR filter by minimizing subject to the
above constraints. This is a Chebyshev approximation problem
where the peak value of the frequency response is used as the
design criteria.
Most of the currently available methods for solving the
Chebyshev approximation problem are based on the frequency
4A transfer function H (z ) is said to have a linear phase if it can be represented
as H (z ) = jH (z )jz on the unit circle for some real positive scalar . The
parameter is called the group delay.
gridding [22], [23], [25]. Some recent methods avoid frequency
gridding using KYP lemmas [9], [10], [53], but these methods
can deal only with a subclass of the problem where all the FDIs
. No methods have been
are given in terms of the gain of
known, to our knowledge, to solve the FIR Chebyshev approximation problem with phase constraints as formulated above,
without gridding the frequency axis. Our method naturally
handles such problem.
A standard approach in the literature to achieve the linear
phase for FIR filters is to make the FIR coefficients symis even, it can be
metric [22], [56]. For instance, when
shown that the constraint on the FIR coefficients
in (39) enforces the linear phase with group
delay
. With this symmetry constraint, Specification a) can
be equivalently written as a condition on the gain only
a)
where we used the fact that
holds on the unit
with
. Thus, the
circle for a symmetric FIR filter
design problem is exactly solvable by existing methods [9], [10]
under the symmetry constraint.
A drawback of this approach is that some design freedom is
wasted to enforce the unnecessary constraint of the linear phase
property in the stop-band. By removing the unnecessary constraint, the design freedom may be used to improve the overall
IWASAKI AND HARA: GENERALIZED KYP LEMMA
Fig. 6.
55
FIR responses. (a) n = 30, d = 15. (b) n = 30, d = 10. (c) n = 20, d = 10. (b’) n = 30, d = 10.
performance. We will illustrate this point by numerical examples in the sequel. We fix the frequency ranges and the gain
bound in the stop-band as follows:
advantage of our method that allows us to treat the linear phase
constraint in a finite frequency range, as opposed to the existing
methods that enforce the constraint for all frequencies by symmetry of the FIR coefficients.
VIII. CONCLUSION
and design several FIR filters for different choices of and .
Fig. 6(a) shows the gain response of a typical symmetric FIR
and
, where the frequency is
filter with
normalized by . This filter is obtained as the optimal solution
to the above Chebyshev approximation problem. The optimal
. Now, consider
performance value is found to be
a situation where the group delay of
time steps is too
large and is not acceptable. Such small delay requirement would
be important for certain real-time applications [25], [57]. Supis the desired group delay. Then, the standard
pose that
framework of symmetric FIR filters requires that the filter order
. The gain response of the optimal FIR filter for
be
this case is shown in Fig. 6(c). The optimal performance value
which is significantly larger than the previous
is
design, as seen by the nonflatness of the gain response in the
pass-band. On the other hand, our design method is free from
and, for instance, the group delay of
the constraint
may be (approximately) achieved with a filter of order
equal to the original design, i.e.,
, as shown in Fig. 6(b)
and (b’). We see that, within the pass-band, the linear phase requirement is practically met and the gain response is almost as
good as the original
. This example shows the
We have generalized the KYP lemma to allow for more flexibility in system classes and various frequency ranges. In particular, the system class is described by the null space of a certain
frequency-dependent matrix, capturing descriptor systems and
polynomials in addition to the standard state space systems. The
frequency range is specified by two quadratic forms that define
segment(s) of a circle or a line on the complex plane. In this way,
an FDI condition can be considered in the low/middle/high frequency ranges for both continuous-time and discrete-time settings in a unified manner.
The generalized KYP lemma converts a certain FDI in a finite frequency range to a numerically tractable LMI condition.
When system design specifications are expressed in terms of
such FDIs, the generalized KYP lemma may be used to reduce
the problem to an LMI optimization. We have given some technical conditions under which this reduction is possible, and discussed various engineering applications that fit into the framework, including the design of digital filters, feedback controllers
with fixed poles, and mechanical structures. Further developments have been reported at conferences, addressing such issues as the gain feedback design [58], controller order reduction, a robust generalized KYP lemma [59], and a GKYP design
56
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
toolbox with MATLAB [60]. General control problems with dynamic output feedback, however, still remain open for further
research.
APPENDIX I
SIMULTANEOUS MATRIX FACTORIZATION
In this section, we prove Lemma 2. The following result is
useful for this purpose.
Lemma 9: Let
be given. Then, admits the following factorization:
(40)
where , ,
and
with
In particular, and are the eigenvalues of
Proof: Let the entries of be defined by
, then Lemma 9 guarantees it can
holds. Let
. Since and are the
be decomposed as in (40) with
, they can be ordered so that
.
eigenvalues of
Then, the factorizations of and in (13) are obtained from
(42) and (40) by noting the relation (41) and defining
.
APPENDIX II
TECHNICAL LEMMAS
and
be given. Suppose
Lemma 10: Let
is an admissible, regular set. Then the following statethat
ments are equivalent.
such that
.
i) There exists
, there exists
such that
ii) For each
.
ii) is obvious. To show the
Proof: The implication i)
converse, consider the following.
such that
, where
iii) There exists
.
We will show ii)
iii)
i). Suppose iii) does not hold. Then,
for all
. Since
is closed,
is compact and, hence, there exists
such that
Then
Let
The spectral factorization of
be arbitrary and define
gives
Then, we have
be verified that
where the columns of
are eigenvectors and and are
eigenvalues. Note that , , and are all real. Moreover, since
is
real symmetric, can be chosen to satisfy
. Thus,
belongs to . Now, equating the two ex, we have
pressions for
Finally, it can readily be verified that
(41)
holds for any
. Substituting this expression into the
second term, we obtain the result.
Proof of Lemma 2: Since
, there exists a nonsuch that
singular matrix
(42)
since
is a cone, and it can readily
. Therefore
Since
is arbitrary, we see that ii) does not hold.
Thus, we have shown ii) iii). Now, suppose iii) holds and let
be such that
. If
, then
and regularity implies
, contradicting
. So
, in which case,
is an element of
and
, proving i).
satisfies
Lemma 11: [19], [61] Let be a convex subset of
, and
be an affine function. The following statements
are equivalent.
i) The set
is empty.
ii) There exists nonzero
such that
is nonnegative for all
.
and
,
Lemma 12: Let matrices
and sets of complex numbers and be given such that
and the closure of
coincides with . Denote by
the null space of
where
is defined in (18). Suppose
holds for all
. Then,
holds
rank
for all
if and only if it holds for
.
IWASAKI AND HARA: GENERALIZED KYP LEMMA
57
Proof: The necessity is obvious. To prove the sufficiency,
such that
does
suppose that there exists
such that
not hold. Then, there exists a vector
and
. Let
. Then,
and
holds for all
.
we have
is continuous on due to the rank assumption,
Note that
holds if is sufficiently close to
and hence
. Such can be chosen from
because
is
. Thus, we have shown that if
an element of the closure of
holds for all
, then it must also hold for
.
all
in (12). Suppose it repLemma 13: Consider the set
is
resents curves on the complex plane. Then the set
and
.
unbounded if and only if
and
. Then,
is
Proof: Suppose
a straight line, while
is a half plane or the outside of a
, is clearly unbounded.
circle. Hence, their intersection,
This proves the sufficiency. The necessity can be proved by con, then
is a circle and its subset
tradiction. If
must be bounded. If
, then
for
with sufficiently large magnitude and hence
is bounded.
must be bounded.
Therefore, its subset
APPENDIX III
PROOF OF LEMMA 6
Statement a) has been proven in the paragraph above the
lemma. We will prove statement b). It is tedious but straightforward to verify that (22) holds if and only if
holds where is an arbitrarily fixed nonsingular matrix. Then,
b) is equivalent to the statement obtained by replacing , ,
, and
in b). Hence, without
and by ,
loss of generality, we prove b) for the case where
and
where
and
are given by (14). Note that
is
regular and (21) holds if and only if
(43)
We first show the “only if” part of the proof. Suppose there
exists
such that rank
. Then there exists
. Let
if
a nonzero vector such that
or
and
if
. It can be verified that the
is a counter example to (43). This
choice
proves necessity.
To prove the converse, suppose that the rank condition (22) is
has full row rank and hence there exists a
satisfied. Then
nonsingular matrix that transforms into the following form:
Note that (
,
) is controllable because
Let ,
be such that
and
. We will show that
( , ) is controllable, there exists
has no eigenvalues on
. Then
. Since
such that
If
or
, then it follows from [3], [45] that
implies
under controllability of
.
implies
, eliminating
So we will show that
with
the remaining possibility. For
, we have
Since
represents curve(s),
implies
.
, given by
In this case, the relative interior of
, is nonempty. We then have
for all
. Since
, this implies
has infinitely many roots in
, and hence
that
must vanish identically. It then follows from Theorem 1
due to controllability
(statement 12) in [62, Sec. 34] that
of
.
ACKNOWLEDGMENT
The would like to thank I. Yamada and H. Hasegawa at the
Tokyo Institute of Technology (TIT), Japan, for their helpful
comments on filtering design, and Y. Iwatani at TIT for his help
with the PID design example. They would also like to thank C.
Scherer at Delft University of Technology, The Netherlands, for
providing them with a preprint of [21] accompanied by his insightful comments, to K. Murota at the University of Tokyo for
helpful discussions on the -procedure, and to A. L. Fradkov
at the Institute for Problems of Mechanical Engineering for directing their attention to the connection between [6] and [17],
which has lead to the necessity proof of Theorem 1.
REFERENCES
[1] B. D. O. Anderson, “A system theory criterion for positive real matrices,” SIAM J. Control, vol. 5, pp. 171–182, 1967.
[2] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Autom. Control, vol. AC-16, no. 6,
pp. 621–634, Dec. 1971.
[3] A. Rantzer, “On the Kalman-Yakubovich-Popov lemma,” Syst. Control
Lett., vol. 28, no. 1, pp. 7–10, 1996.
[4] S. Boyarski and U. Shaked, Discrete-Time
and
Control of systems with structured disturbances under probability requirements, 2003.
H
H
58
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005
H
[5] C. Scherer, “Robust
-controller design under structured noise-uncertainty,” presented at the IFAC World Congr., Barcelona, Spain, 2002.
[6] (2000) Mathematical Problems in Engineering [Online]. Available:
http://mpe.hindawi.com/volume-6/S1 024 123X00 001 368.html
[7] T. Iwasaki, S. Hara, and H. Yamauchi, “Dynamical system design from a
control perspective: finite frequency positive-realness approach,” IEEE
Trans. Autom. Control, vol. 48, no. 8, pp. 1337–1354, Aug. 2003.
[8] Y. Nesterov, “Squared functional systems and optimization problems,”
in High Performance Optimization, H. Frenk et al., Eds. Norwell, MA:
Kluwer, 2000, pp. 405–440.
[9] Y. Genin, Y. Hachez, Y. Nesterov, and P. Van Dooren, “Convex optimization over positive polynomials and filter design,” in Proc. UKACC
Int. Conf. Control, 2000.
[10] T. N. Davidson, Z.-Q. Luo, and J. F. Sturm, “Linear matrix inequality
formulation of spectral mask constraints with applications to FIR filter
design,” IEEE Trans. Signal Processing, vol. 50, no. 11, pp. 2702–2715,
Nov. 2002.
[11] F. P. Gantmacher and V. A. Yakubovich, “Absolute stability of nonlinear
control systems,” in Proc. 2nd All-Union Session on Theoretical and
Applied Mechanics, Moscow, Russia, 1966.
[12] V.
A.
Yakubovič,
S-Procedure
in
Nonlinear
Control
Theory. Leningrad, Russia: Vestnik Leningrad Univ., 1971,
vol. 1, pp. 62–77.
[13] Linear Matrix Inequalities in System and Control Theory, 1994.
[14] T. Iwasaki and S. Hara, “Well-posedness of feedback systems: Insights
into exact robustness analysis and approximate computations,” IEEE
Trans. Autom. Control, vol. 43, no. 5, pp. 619–630, May 1998.
[15] A. Megretski and A. Rantzer, “System analysis via integral quadratic
constraints,” IEEE Trans. Autom. Control, vol. 42, no. 6, pp. 819–830,
Jun. 1997.
[16] C. W. Scherer, “LPV control with full block multipliers,” Automatica,
vol. 37, no. 3, pp. 361–375, 2001.
[17] A. L. Fradkov, “Duality theorems for certain nonconvex extremal problems,” Sibirskii Matematicheskii Zhurnal, vol. 14, no. 2, pp. 357–383,
Mar.–Apr. 1973.
[18] A. Packard and J. Doyle, “The complex structured singular value,” Automatica, vol. 29, no. 1, pp. 71–109, 1993.
[19] G. Meinsma, Y. Shrivastava, and M. Fu, “A dual formulation of mixed and on the losslessness of (D; G) scaling,” IEEE Trans. Autom. Contr.,
vol. 42, no. 7, pp. 1032–1036, Jul. 1997.
[20] D. Henrion and G. Meinsma, “Rank-one LMIs and Lyapunov’s inequality,” IEEE Trans. Autom. Control, vol. 46, no. 8, pp. 1285–1288,
Aug. 2001.
[21] C. Scherer, “When are multiplier relaxations exact?,” in Proc. IFAC
Symp. Robust Control, 2003.
[22] T. W. Parks and J. H. McClellan, “Chebyshev approximation for nonrecursive digital filters with linear phase,” IEEE Trans. Circuit Theory,
vol. CT-19, no. 2, pp. 189–194, Feb. 1972.
[23] X. Chen and T. W. Parks, “Design of FIR filters in the complex domain,”
IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, no. 2, pp.
144–153, Feb. 1987.
[24] K. Preuss, “On the design of FIR filters by complex Chebyshev approximation,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 5,
pp. 702–712, May 1989.
[25] L. J. Karam and J. H. McClellan, “Chebyshev digital FIR filter design,”
Signal Process., vol. 76, pp. 17–36, 1999.
[26] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Upper
Saddle River, NJ: Prentice-Hall, 1996.
[27] T. Glad and L. Ljung, Control Theory—Multivariable and Nonlinear
Methods. New York: Taylor Francis, 2000.
[28] A. Messac and K. Malec, “Control-structure integrated design,” AIAA
J., vol. 30, no. 8, pp. 2124–2131, 1992.
[29] P. Maghami, S. Joshi, and D. Price, “Integrated controls-structures design methodology for flexible space craft,” J. Spacecrafts Rockets, vol.
32, no. 5, pp. 839–844, 1995.
[30] K. M. Grigoriadis, G. Zhu, and R. E. Skelton, “Optimal redesign of linear
systems,” ASME J. Dyna. Syst. Meas. Control, vol. 118, pp. 596–605,
1996.
[31] S. Hara, T. Iwasaki, and F. Shimizu, “Finite frequency characterization of easily controllable mechanical systems under control effort constraint,” in Proc. IFAC World Congr., 2002.
[32] V. A. Yakubovič, “Minimization of quadratic functionals under
quadratic constraints and the necessity of a frequency condition in the
quadratic criterion for absolute stability of nonlinear control systems,”
Soviet Math. Dokl., vol. 14, no. 2, pp. 593–597, 1973.
[33] A. L. Fradkov and V. A. Jakubovic, The S-Procedure and a Duality Relations in Nonconvex Problems of Quadratic Programming. Leningrad,
Russia: Vestnik Leningrad Univ. Math., 1979, vol. 6, pp. 101–109.
[34] R. E. Skelton, T. Iwasaki, and K. M. Grigoriadis, A Unified Algebraic
Approach to Linear Control Design. New York: Taylor Francis, 1997.
[35] T. Iwasaki, LMI and Control (in Japanese). Tokyo, Japan: Shokodo,
1997.
[36] M. Fan, A. Tits, and J. Doyle, “Robustness in the presence of mixed
parametric uncertainty and unmodeled dynamics,” IEEE Trans. Autom.
Control, vol. 36, no. 1, pp. 25–38, Jan. 1991.
[37] J. C. Doyle, “Analysis of feedback systems with structured uncertainties,” Proc. Inst. Elect. Eng. D, vol. 129, no. 6, pp. 242–250, Nov. 1982.
[38] R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ.
Press, 1970.
[39] Y. Ebihara, K. Maeda, and T. Hagiwara, “Generalized s-procedure for
inequality conditions on one-vector-lossless sets and linear system analysis,” in Proc. IEEE Conf. Decision Control, 2004.
[40] D. Henrion, M. Sebek, and V. Kucera, “Positive polynomials and robust
stabilization with fixed-order controllers,” IEEE Trans. Autom. Control,
vol. 48, no. 7, pp. 1178–1186, Jul. 2003.
[41] T. Iwasaki and S. Hara, “Generalization of Kalman-Yakubovic-Popov
lemma for restricted frequency inequalities,” in Proc. Amer. Control
Conf., 2003.
[42] X. Zhang, P. Tsiotras, and T. Iwasaki, “Parameter-dependent Lyapunov
functions for stability analysis of LTI parameter dependent systems,” in
Proc. IEEE Conf. Decision Control, 2003.
[43] P.-A. Bliman, “A convex approach to robust stability for linear systems
with uncertain scalar parameters,” SIAM J. Control Optim., vol. 42, no.
6, pp. 2016–2042, 2004.
[44] T. Iwasaki and G. Shibata, “LPV system analysis via quadratic separator
for uncertain implicit systems,” IEEE Trans. Autom. Control, vol. 46, no.
8, pp. 1195–1208, Aug. 2001.
[45] V. Balakrishnan and L. Vandenberghe, “Semidefinite programming duality and linear time-invariant systems,” IEEE Trans. Autom. Control,
vol. 48, no. 1, pp. 30–41, Jan. 2003.
[46] A. N. Churilov, On the solvability of some matrix inequalities. Leningrad, Russia: Vestnik Leningrad State Univ., 1980,
vol. 2, pp. 51–55.
[47] Y. Genin, Y. Nesterov, and P. Van Dooren, “Positive transfer functions
and convex optimization,” in Proc. Euro. Control Conf., 1999.
[48] Y. Genin, Y. Nesterov, R. Stefan, P. Van Dooren, and S. Xu, “Positivity
and linear matrix inequalities,” Eur. J. Control, vol. 8, no. 3, pp. 275–298,
2002.
[49] Y. Genin, Y. Hachez, Y. Nesterov, and P. Van Dooren, “Optimization
problems over positive pseudopolynomial matrices,” SIAM J. Matrix
Anal. Appl., vol. 25, no. 1, pp. 57–79, 2003.
[50] T. Iwasaki and S. Hara. (2003, Aug.) Generalized KYP lemma: Unified
characterization of frequency domain inequalities with applications
to system design. Univ. Tokyo, Tokyo, Japan. [Online]. Available:
http://www.keisu.t.u-tokyo.ac.jp/Research/techrep. 0.html
[51] D. C. Youla, H. A. Jabr, and J. J. Bongiorno, “Modern Wiener–Hopf
design of optimal controllers: Part 2,” IEEE Trans. Autom. Control, vol.
AC-21, no. 3, pp. 319–338, Jun. 1976.
[52] S. Boyd and C. H. Barratt, Linear Controller Design: Limits of Performance. Upper Saddle River, NJ: Prentice-Hall, 1991.
[53] J. Tuqan and P. P. Vaidyanathan, “A state–space approach to the design
of globally optimal FIR energy compaction filters,” IEEE Trans. Signal
Process., vol. 48, no. 10, pp. 2822–2838, Oct. 2000.
[54] W. Rudin, Fourier Analysis on Groups. New York: Wiley, 1962.
[55] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems—Theory and Design. New York: Prentice-Hall, 1997.
[56] L. R. Rabiner, “Techniques for designing finite-duration impulse-response digital filters,” IEEE Trans. Commun. Technol., vol. CT-19, no.
2, pp. 188–195, Feb. 1971.
[57] X. Zhang and T. Yoshikawa, “Design of FIR Nyquist filters with
low group delay,” IEEE Trans. Signal Process., vol. 47, no. 5, pp.
1454–1458, May 1999.
[58] T. Iwasaki and S. Hara, “Robust control synthesis with general frequency
domain specifications: Static gain feedback case,” in Proc. Amer. Control
Conf., 2004.
[59] S. Hara, D. Shiokata, and T. Iwasaki, “Fixed order controller design via
generalized KYP lemma,” in Proc. IEEE Conf. Control Applications,
2004.
[60] D. Shiokata, S. Hara, and T. Iwasaki, “From Nyquist/Bode to GKYP
design: design algorithms and cacsd tools,” in Proc. SICE Annu. Conf.,
2004.
[61] D. G. Luenberger, Optimization by Vector Space Methods. New York:
Wiley, 1968.
[62] V. M. Popov, Hyperstability of Control Systems. New York: SpringerVerlag, 1973.
IWASAKI AND HARA: GENERALIZED KYP LEMMA
Tetsuya Iwasaki (S’89–M’94–SM’01) received the
B.E. and M.E. degrees in electrical and electronic
engineering from the Tokyo Institute of Technology,
Tokyo, Japan, in 1987 and 1990, respectively, and
the Ph.D. degree from the School of Aeronautics and
Astronautics, Purdue University, West Lafayette, IN,
in 1993.
He held a postdoctoral position at Purdue University. After holding a faculty position for five years at
the Tokyo Institute of Technology, he moved to the
University of Virginia, Charlottesville, in May 2000,
where he is currently a Professor. His research interests include robust and optimal control, and modeling and control of biological sensing, locomotion, and
oscillation.
Dr. Iwasaki received the 2002 Pioneer Prize from the Society of Instrument and Control Engineers, and the 2003 National Science Foundation
CAREER Award. He is a past Associate Editor of the IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, and currently is on the Editorial Boards of Automatica
and Systems and Control Letters.
59
Shinji Hara (M’87–SM’04) was born in Izumo,
Japan, in 1952. He received the B.S., M.S., and Ph.D.
degrees in engineering from the Tokyo Institute of
Technology, Tokyo, Japan, in 1974, 1976, and 1981,
respectively.
From 1976 to 1980, he was a Research Member
of Nippon Telegraph and Telephone Public Corporation, Japan. He served as a Research Associate at the
Technological University of Nagaoka, Japan, from
1980 to 1984. In 1984, he joined the Tokyo Institute
of Technology as an Associate Professor and has
served as a Full Professor for ten years. Since 2001, he has been a Full Professor
in the Department of Information Physics and Computing, The University
of Tokyo. His current research interests are in robust control, sampled-data
control, hybrid control, learning control, quantum control, and computational
aspects of control system design. He is a Member of SICE and ISCIE.
Dr. Hara was a BoG member of the IEEE Control Systems Society, the General Chair of the CCA04, and an Associate Editor of the IEEE TRANSACTIONS
ON AUTOMATIC CONTROL and Automatica. He received Best Paper Awards from
SICE (the Society of Instrument and Control Engineers, Japan) in 1987, 1991,
1992, 1997, and 1998, from the Japan Society for Simulation Technology in
2001, and from ISCIE in 2002.