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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 9, SEPTEMBER 1998
On the Existence of Robust Strictly
Positive Real Rational Functions
Horacio J. Marquez, Member, IEEE, and Panajotis Agathoklis, Senior Member, IEEE
Abstract—A new approach for the analysis of the strict positive
real property of rational functions of the form H (s) = p(s)=q (s)
is proposed. This approach is based on the interlacing properties
of the roots of the even and odd parts of p(s) and q (s) over
the imaginary axis. From the analysis of these properties, an
algorithm to obtain p(s) such that p(s)=q (s) is strictly positive
real (SPR) for a given Hurwitz q (s) is developed. The problem of
finding p(s) when q (s) is an uncertain Hurwitz polynomial is also
considered, using this new approach. An algorithm for obtaining
p(s) such that p(s)=q (s) is SPR, when q (s) has parametric
uncertainites, is presented. This algorithm is easy to use and leads
to p(s) in cases where previously published methods fail.
I. INTRODUCTION
O
NE way to represent uncertainties in linear timeinvariant systems is to let the coefficients of the system
transfer function take values between a lower and an upper
bound. Stability analysis of such systems is important, as well
as other properties such as strict positive realness (SPRness),
which has played a major role in the evolution of several
areas such as nonlinear and adaptive control [1], as well as
adaptive filtering.
Several aspects of the robust SPR problem, i.e., developing
conditions for SPRness for systems with parameter uncertainties, have been considered by a number of authors. Several
important results [2]–[7] have been presented. References
to be
[2]–[7] deal with conditions for the function
and
where
and
SPR for all possible
are two given families of polynomials.
containing parametric
Given a Hurwitz polynomial
uncertainties, the problem of finding a (fixed) polynomial
such that
is SPR was considered in [5]–[7].
The approach in both [5] and [7] is essentially numerical,
with parametric uncertainties,
i.e., given a polynomial
numerical procedures were proposed which, provided that
certain conditions are satisfied, will result in a polynomial
such that
is SPR. A more fundamental question was
raised in [6], namely, given a family of Hurwitz polynomials
, does it always exist a (fixed) polynomial
, of the same
is SPR. For continuous time systems,
order, such that
Manuscript received January 1996; revised March 13, 1998. This work
was supported by the Natural Sciences and Engineering Research Council
of Canada (NSERC). This paper was recommended by Associate Editor H.
Uwe.
H. J. Marquez is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alta., Canada (e-mail: [email protected]).
P. Agathoklis is with the Department of Electrical and Computer Engineering, University of Victoria, Canada (e-mail: [email protected]).
Publisher Item Identifier S 1057-7122(98)06498-8.
no answer was found when the order of the polynomial is
greater than 3 (see also [7]).
In this paper a new approach to investigate the SPR property
of a rational function is proposed and applied to the robust
SPR problem. Our approach is to address the robust SPR
problem in two steps. First, in Section III, we consider
without any uncertainty and study how to
a polynomial
is SPR. Our interest
construct polynomials such that
in this section is to show that SPR rational functions can be
characterized by an interlacing property of the even and odd
parts of the numerator and denominator polynomials following
certain patterns. Second, in Section IV, the results of Section
.
III are extended to a family of Hurwitz polynomials
Here we explicitly show how to use the interlacing property
developed in Section III to construct a single polynomial
that makes
SPR for all members of a given family of
. Our method is shown to work for examples
polynomials
where previous techniques fail.
II. PRELIMINARIES
AND
NOTATION
Let represent the field of real numbers,
the set of real
and
represent the positive
polynomials of th degree.
denotes the
and negative elements of , respectively.
.
degree of the polynomial
be a rational function of the form
Definition 1: Let
, with
and
. Then
is positive real if
.
is said to be strictly positive real, SPR, if
is
.
positive real for some
and assume that
Theorem 1 [4]: Let
. Then
is SPR if and only if the following
conditions are satisfied:
;
(a)
and
are Hurwitz polynomials;
(b)
is Hurwitz for all
.
(c)
. We will denote by
Consider a polynomial
and
, the even and odd parts of
, respectively.
and
such that
Define also polynomials
, as follows:
Property 1 [9], [10]:
is said to satisfy the interlacing
and in addition the roots
property if and only if
and
are real and interlace in the following
of
1057–7122/98$10.00  1998 IEEE
MARQUEZ AND AGATHOKLIS: ROBUST STRICTLY POSITIVE REAL RATIONAL FUNCTIONS
manner:
(1)
, and
are the th real positive roots of
where
and
, respectively.
Consider now a complex polynomial of the form
963
Hermite–Biehler property satisfied by Hurwitz polynomials.
Our approach to finding polynomials consists of separating
and into their even and odd parts
, and , and
and
so that
then constructing by choosing the roots of
the interlacing property with the corresponding roots of and
is satisfied. Given
, we define
as follows:
In order to state the interlacing property for complex polyno, we define
mials, associated with
(2)
(3)
(4)
(5)
and
(13)
(14)
For simplicity, in the following property, we restrict attention
(the only case needed in this paper).
to the case
(respectively,
) and
(respectively,
) denote
and
in
(respectively,
the th real root of
).
Property 2 [11]: A complex polynomial
where
,
satisfy the interlacing property for complex polynomials if the
and
are real, and satisfy
roots of
, given by (14), we choose the (positive)
To obtain
, and
, of
and
,
roots
respectively, in the following way:
(6)
(16)
(15)
..
.
(17)
(18)
Theorem 2: A real or complex polynomial is Hurwitz if and
only if it satisfies Properties 1 or 2.
. We say that
belongs
Consider a polynomial
of interval polynomials if each coefficient
to the family
. Associated with a family
,
satisfies
we define
(7)
(8)
(9)
(10)
The four Kharitonov polynomials associated with the family
are then defined by:
.
and
so chosen, satisfy the following
The
properties:
(11)
(12)
Theorem 3 [2]–[4]: Let
is SPR for all
Then
the four rational functions
.
if and only if
, are SPR.
III. CONSTRUCTING SPR RATIONAL FUNCTIONS
In this section we consider a given Hurwitz polynomial
, and present a method to find polynomials
such that
is SPR. This method is based on an interlacing property
and
satisfied
between the even and odd parts of
by SPR rational functions, which resembles the well-known
when
when
(19)
(20)
(21)
In the next theorem, it will be shown that if for each
and
are chosen as in (15) and moreover,
is chosen “close enough” to
, and
is “close enough” to
, then there exists a
such that the resulting polynomial
leads to
being SPR. The proof of it is in the
Appendix.
.
Theorem 4: Consider a Hurwitz polynomial
and
such that defining
and
There exist
as in (15)–(20) with
, and
,
, with
as in (21), is SPR
the function
, and
.
and
are given by
if
(22)
where
(23)
(see also Fig. 1).
and
Theorem 4 characterizes the existence of polynomials
such that
is SPR for a given
, in terms of the
proximity of the roots of the even and odd parts of
to those of
. This characterization will be shown to
be fundamental in the case of uncertain polynomials studied
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 9, SEPTEMBER 1998
Fig. 1. Frequency intervals
i ; ij ; i ,
and
i .
in the next section. What is not implied by Theorem 4 is
to be
that this root distribution is necessary for
SPR. It can be shown that, for polynomials with degree less
than or equal to 5, there are four possible ways in which the
, and
can interlace
roots of
with
SPR. For each one of these combinations, it
in
is possible to state and prove a theorem similar to Theorem 4.
, combinations of these patterns are also possible.
If
Theorem 4 also implies a rather simple synthesis procedure
. Two algorithms will now be presented. The first step
for
is common to both of them and consists of choosing the roots
and
. The selection of
to complete the
of
synthesis of will be done in two alternative ways.
Algorithm:
and
that satisfy (15).
(i) Select roots
(ii) Define (see also Fig. 1)
and verify that condition (22) is satisfied. If this is not the case go back to (i) and select
and
such that for each
new roots
is closer to
and
is closer to
.
such that
.
(iii) Select
and
have
Experience has shown that unless the roots
and
, respectively,
is
been placed purposely far from
typically more that two orders of magnitude smaller than . It
substituting
is therefore typically easy to obtain a value for
steps (ii) and (iii) by the following:
Simplified Algorithm:
(
, say).
(ii ) Define an arbitrary initial value for
is SPR. If not, go back to
(iii ) Check whether or not
(ii ) and redefine . Typically, a few iterations on this
,
parameters (taking values such as
) will complete the search in a few trials.
or
Example 1: Consider the polynomial
, given by
Here the roots of
and
in
are at the following locations:
Consistent with the proposed algorithm, we define roots
, and
, at the following locations:
We now define
and
as in (16)–(20), and iterate
using values
, as proposed in
on
produces
(ii )–(iii ). We find that after 3 iterations,
such
the polynomial
is SPR. Moreover, an analysis of the intervals
that
and
shows that any
satisfying
will lead to an SPR rational function.
IV. UNCERTAIN POLYNOMIALS
In this section we consider a family of polynomials
of
constant degree . It will be shown that defining functions
and
in a manner analogous to the previous section
of degree , such that
is SPR for
polynomials
, can be found. Let
all possible
MARQUEZ AND AGATHOKLIS: ROBUST STRICTLY POSITIVE REAL RATIONAL FUNCTIONS
Fig. 2. Typical plot of
gi (|!); hi (|!); i
;
= 1 2for a family of Hurwitz polynomials.
be any element of
. Inequalities (11)–(12) show
[respectively,
] lies in the
that the th root of
interval (respectively, ) with end points defined by the th
and
[resp.,
and
],
root of
is Hurwitz by assumption, we have that
see Fig. 2. Since
the intervals and interlace according to Property 1. Thus it
,
is always possible to find real numbers
such that
with
, and if the functions
such that
and
as in (16)–(20), and let
(24)
It is clear that (24) is analogous to (23) for families of
polynomials. The Proof of Theorem 4 was based essentially
on the analysis of the values of (23) in the frequency intervals
and . In this section we will refer to the intervals
and
as a generalization of
and
in the obvious manner,
and
is defined as in Section III, for
,
i.e. each
.
and
of interval polyTheorem 5: Consider a Hurwitz family
such that defining
nomials of degree . If there exists an
and
as in (16)–(20)
, and
are
(25)
is such that
is SPR for all possible
.
The proof is omitted since it is similar to that of Theorem
4. The main difference between Theorems 4 and 5 is that
Theorem 4 was stated and proven under the assumption that
and
can be placed as close as
the roots of
and
, respectively. When
necessary to those of
uncertainties are present, the proximity of the roots of
to the ones of
corresponding
to a polynomial in the family is limited by the size of the
in the
uncertainty. As a consequence, the existence of
presence of uncertainties in is not guaranteed.
It is straightforward to generalize the two algorithms of
Section III to the present case. The application of these
generalizations is shown in the following example.
Example 2: Consider the family of polynomials
defined as follows:
then any
Define
965
with
satisfying
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 9, SEPTEMBER 1998
APPENDIX I
PROOF OF THEOREM 4
For this family, we define the polynomials
as in (7)–(10). The functions
and
have roots at the following locations in
:
, and
satisfies Theorem 1. Conditions a)
We will show that
and b) are satisfied by construction. To show that condition c)
is also satisfied, we define
Thus (see Section II),
Thus, taking account of inequalities (11) and (12), we have
(Fig. 2)
We now define roots
and
functions
and
. We choose
With this selection, we have
for the
, where
An analysis of the intervals
and
reveals that, with
and
, any value of
this root selection for
satisfying
will lead to
SPR, for
.
all
It should be noticed that, in this example, the proposed
algorithm produced the desired polynomial, despite the fact
is exceedingly
that the uncertainty contained in the family
large for most practical purposes. As a comparison, the algorithm developed in [5] cannot be applied since the necessary
conditions for the existence of a solutions required by that
.
method are not satisfied by
V. CONCLUSIONS
AND
FINAL REMARKS
A novel frequency domain approach for the synthesis of
SPR rational functions with a given denominator has been
presented. A major contribution of the paper is the analysis
of the interlacing of the roots of the even and odd parts of
the polynomials and of an SPR rational function over
the imaginary axis, and its application for polynomials with
parametric uncertainties. Algorithms to obtain a polynomial
such that
is SPR were presented for both cases
is a given fixed Hurwitz polynomial or
is an
when
uncertain Hurwitz polynomial.
and we will show that the roots of
and
interlace according to Property 2. Define the frequency intervals in
as shown in Fig. 1(a) and (b): we now
proceed as follows.
. Notice [Fig. 1(a)] that
Step i): Assume first that
is zero at the lower end of
, and it is positive
is positive everywhere in , except
elsewhere. Also,
at the upper end where it is zero. Since both functions are
, and then
continuous, they must intersect at some
. Therefore, the first real root of
in
lies in
. The same argument shows that
has real roots in
, and that no roots of
can exist in
outside the intervals ’s. A similar
and
for
analysis of the intersections of
shows that there is a root of
in each interval
. We thus conclude that there is one single real
in each frequency interval
in
root of
, and one single real root in each
in
.
Consider now Fig. 1(b). An analysis entirely analogous to
the one above shows that there is one single real root of
in each frequency interval
in
, and one
in
.
single real root in each
are
Notice also that, by construction, the intervals
. Thus, for any
, the roots of
disjoint in
and
in
interlace according to Property 2.
, and consider Figs. 1(a)
Step ii): Assume now that
’s are such that
and (b). The intervals
. As a result, arbitrary values of
may
result in a failure to comply with the interlacing property in
, for some
. Consider now the effect of changing
and
the parameter , and on the root locations of
. By definition,
, and
. It is easy to see from
the roots of
and
Fig. 1 that, for fixed , as
move toward the upper end of the intervals
,
they move in opposite direction. It follows
while when
and
do not satisfy the interlacing property
that
if and only if there exists an interval of real numbers
in
such that if
, then the roots of
and
do not interlace according to Property 2. This can
such that
and
occur if and only if there exists
have a real root at the same frequency . For this to
MARQUEZ AND AGATHOKLIS: ROBUST STRICTLY POSITIVE REAL RATIONAL FUNCTIONS
be the case, we must have
and
(26)
(27)
is SPR if and only if
is selected in
It follows that
such a way that equality (27) is violated at every frequency
. From our analysis, it is clear that (27) can only be
satisfied within the following frequency intervals:
At both end points of each interval , the right-hand side
of (27) equals zero, while for any other value within this
interval it remains positive. Thus, at each frequency interval
, the right-hand side of (27) can be any number within
. Moreover, using a continuity argument
the interval
can be made as small as needed by
it is easy to see that
close to the roots of
choosing the roots of
(i.e. by shrinking the frequency interval ). Similarly, in each
interval , the right-hand side of (27) can be any number
, and moreover,
can be made
within the interval
close to the
as large as needed by choosing the roots of
(i.e., by shrinking the frequency interval ).
roots of
such that, for any
,
Therefore, there exist
, and
ensure
the condition
, and therefore,
that
such that
we have
choosing
and
satisfy the interlacing property for
that
, and the theorem is then
complex polynomials also in
proven.
ACKNOWLEDGMENT
The authors thankfully acknowledge the comments of the
reviewers.
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Horacio J. Marquez (S’88–M’92) received the
M.Sc.E. and Ph.D. degrees in electrical engineering
from the University of New Brunswick, Fredericton,
Canada, in 1990 and 1993.
From 1993 to 1996 he held visiting appointments
at the Royal Roads Military College, and the University of Victoria, Victoria, British Columbia. Since
1996 he has been an Assistant Professor of electrical and computer engineering at the University of
Alberta, Edmonton, Alberta, Canada.
Panajotis Agathoklis (M’81–SM’88) received the
Dipl. Ing. degree in electrical engineering and the
Dr.Sc.Tech. degree from the Swiss Federal Institute
of Technology, Zurich, Switzerland, in 1975 and
1980, respectively.
From 1981 until 1983, he was with the University
of Calgary as a Post-Doctoral Fellow and parttime Instructor. Since 1983, he has been with the
Department of Electrical and Computer Engineering, University of Victoria, B.C., Canada, where he
is currently Professor. He has received a NSERC
University Research Fellowship (1984–1986) and Visiting Fellowships from
the Swiss Federal Institute of Technology (1982, 1984, 1986 and 1993),
from the Australian National University (1987) and the University of Perth,
Australia (1997). He was Associate Editor for the IEEE TRANSACTIONS ON
CIRCUITS AND SYSTEMS in 1990–1993. He has been member of the Technical
Program Committee of many international conferences and has served as the
program Chair of the 1991 IEEE Pacific Rim Conference on Communications,
Computers and Signal Processing. His fields of interest are in digital signal
processing, system theory and stability analysis.