NO. AC-32, IEEE
VOL.
AUTOMATIC
CONTROL.
TRANSACTIONS
ON
53
1,
1987
JANUARY
order reduction in control theory-An overview." Automatica. vol. 12, pp. 123132,1976.
[6] K. Kendig. ElementaryAlgebraic Geometry. New York:Springer-Verlag.
1977.
satisfies (a). (b) but is not SPR. since it cannot be realized as the driving
point impedance of a dissipative network. Similarly.
1
h(s)=-,
a>o
S+CY
Frequency Domain Conditions for Strictly Positive Real
Functions
PETROS IOANNOU AND GANG TAO
Absfract-Frequency domain conditionsfor strictly positive real (SPR)
functions which appear in literatureareoften only necessary or only
sufficient. This point is raised in [l], [Z], where necessary and sufficient
conditions in thesdomain aregiven for a transfer functionto be SPR. In
this note, thepoints raised in 111, 121 are clarified further by giving
necessary and sufficient conditions in the frequency domain for transfer
functions to be SPR. These frequency-domain conditions areeasier to test
than those given in the s-domain or time domain 111, [2].
I. INTRODUCTION
The definition of positive real (PR) and SPR transfer functions is
motivated from network theory. That is,a PR (SPR) rational transfer
function can be realized asthe driving point impedance of a passive
(dissipative) network. Conversely, a passive (dissipative) network has a
driving point impedance that is rational and PR (SPR).
In [I], the following equivalent definitions have been given for PR
transfer functions by an appeal to network theory.
Definition 1.1 [I]: A rational function h ( s ) of the complex variable s
= u + jw is PRif
1) h ( s ) is real for real s;
2) Re [ h ( s ) ] 2 0 for all Re [ s ] > 0.
Definition 1.2 :]Z[ The rational transfer function h ( s ) is PR if
1) h ( s ) is real for real s;
2) h ( s )is analytic in Re [s] > 0 and poles on thejw-axis are simple
and such that the associated residue is nonnegative;
3) For any real value of w for which Jw is not a pole of h(jw), Re
[h(jo)]5 0.
In most of the literature, h ( s ) is termed to be SPR if
a)
h analytic
( s ) is
in Re [ s ] z O
(1.1)
(1 5 )
is an SPR transfer function but does not satisfy (b'). Furthermore, h ( s )
given by (1.4) does not satisfy the Kalman-Yakubovich lemma [3],[4],
which is fundamental in the stability analysis of adaptive [5]$[6] and other
nonlinear systems [3] via the Lyapunov direct method. Motivated from
network theory. Taylor [2] and Narendra and Taylor [3] proved the
following lemma for SPR transfer functions.
Lemma 1.1: Assume that h ( s ) is not identically zero for all s. Then
h ( s ) is SPR if and only if h ( s - E ) is PR for some E > 0.
In 121. Taylor also showed that if h ( s ) is SPR and strictly proper, then
Re [ h ( j w ) ]can go to zero as I w I
03 not faster than w - * . In the
following section, we give necessary and sufficient conditions in the
frequency domain for proper and improper transfer functions to be SPR.
+
11. FREQUENCY
DOMAINCONDITIOKS FOR SPR FVKCTIONS
Let n* be the relative degree of h ( s ) = n ( s ) / d ( s ) i.e.,
,
n* = degree
of n(s) - degree of d ( s ) . The following theorem establishes necessary
and sufficient conditions for h ( s ) to be SPR.
Theorem 2.1: Assume that h ( s ) is not identically zero for all s. Then
h ( s ) is SPR if and only if
1) h ( s ) is analytic in Re Is] 2 0 ,
2) Re [ h ( j w ) ]> 0, V w E (-a,
m). and
3)
i) Qrn w 2 Re [h(jw)]>O
when n * = l , or
(2.1)
4A-m
ii) lirn Re [h(jw)]>O, lirn h(jw)>O
dz-p
'Y1-m
jw
when n* = - 1.
Proof:
Necessity: If h (s) is SPR, then from Lemma I , ] , h (s- E ) is PR for
some E > 0. Hence, there exists an E* > 0 such that for each E E [0, E * ) ,
h ( s - E )is analytic in Re Is] < 0. Therefore, there exists a real rational
function W ( s )such that [I]
h ( S - t ) + h ( - s + E ) = N'(s-€)W(-s+t)
where W ( s )is analytic and nonzero for all s in Re [s] >
+ jw;then from (2.3) w'e have
(2.3)
-E.
Let s =
2 Re [ I ~ ( j w ) l = I W ( j w ) ( ~ > O , Vw E (-00, m).
and
b)
. (2.2)
E
(2.4)
Now h ( s ) can be expressed as
Re [h(jw)]>O,
In many cases, (b) is replaced by
b')
Vw E ( -
03, 03).
(1 4
(2.5)
e'),Le.,
Re [ h ( j o ) ] > 6 ,
Vo
E [--03, 031
(1 -3)
where 6 is a positive constant.
As stated by Taylor [2] and Narendra and Taylor [3],(a) and (b) are
only necessary, whereas (a) and (b') are only sufficient for h ( s ) to be
SPR. For example,
I f m = n - l , i . e . , n * = 1,b,-, #O,thenfrom(2.5)itfollowsthatb,-,
> 0 and a n - l b n - l - bn-? > 0 for h ( s - E )to be PRI and
lim e* Re [ h ( j w ) J = a , _ , 6 , ~ l - b , _ ~ r ~ b , _ , > 0 .
<2-=
(2.6)
If m = n + 1. Le., n* = - 1, b,+, # 0, then
Since Re [h(jw - E)] z 0 V o E (--03, 03) and
Manuscript received January 21, 1986: revised September 12. 1986. This
work was
supported by the National Science Foundation under Grant ECS-8312233.
The authors are with the Department of Electrical Engineering-Systems, University
of Southern California. Los Angela, CA 90089.
IEEE Log Number 8611557.
then b,,, > 0 , b, - b,+la,-l2 tb,,, > 0, andtherefore(2.2)follows
directly.
0018-9286/87/0100-00j3f01.00
0 1987 IEEE
54
IEEE TRANSACTIONS ON AUTOMATIC
VOL.
CONTROL,
Sufficiency: Let ( A , b, c, d,fl be a minimal state representation of
h ( s ) . i.e..
(2.8)
h(s)=c(sl-A)-'b+d+fs.
From (2.8) we can write
h ( s - ~ ) = c ( ~ I - - n ) - ' b + d + f s + ~ [ c ( -s d/ --A' ( ~ / - A ) - ' b - f l . (2.9)
Hence,
Re [h(jw-c)]=Re [ h ( j w ) l + ~Re [g(jw--E)]
(2.10)
whereg(jw - E ) = c(jwl - A - d - ' ( j w I - A ) - ' b - f.Thereexists
an E* > 0 such that for all E E [O,e*) and all w E ( - 00, a),(jol- A
- E / ) - ' is analytic. Therefore, for each E E [O?E * ) , I Re [ g ( j w - E ) ] I
<kl<~forallwE(-co,~)andsomekl>O.Ifn*=O,thenRe
[ h ( j w ) ] > k2 > 0 for all w and some k2 > 0. Therefore
Re [h(jw-~)]=Re [ h ( j w ) ] + e Re [ g ( j w - ~ ) ] > k ~ - e k ~ > O(2.11)
for all w E ( - 00, m) and 0 < E < min { E * ,k 2 / k l }Hence,
.
h ( s - E ) is
PR and therefore h ( s ) is SPR.
If n* = 1, then Re [ h ( j w ) ] > k3 > 0 for all I w I < wo and w 2 Re
[ h (j w ) ] > k? > 0 for all I w 1 2 wo, where w,, k3, k, are finite positive
constants. Similarly, I w2 Re [ g ( j w - E ) ] I < k5 and I Re g(jw - E ) 1 <
k6 for all w E (-00, 00) and some finite positive constants k j , kg.
Therefore. Re [ h ( j w - E ) ] > k3 - Ek6 for all I w < wo and w 2 Re
[ h ( j w - E ) ] > k4 - ckj for all I w ( 2 w,. Then, for0 < E c min (kj/
k6, E*.k,/k,} and V u E ( - 00, a),Re [ h ( j w - E ) ] > 0. Hence, h ( 5 E ) is PR and therefore h ( s )is SPR.
If n* = - 1, then d > 0 and therefore
Re [ h ( j w - ~ ) ] > d - ~ k ~ .
Hence, for each E in the interval [0, min {E*, dlk,}), Re [h(j w for all w E ( - 00, 00). Since
(2.12)
E)]
--m
JW
then
h(jw-E )
lim 7 = f >
*-oj
J. H. Taylor. "Strictly positive-real functions and the Lefschetz-KalmanYakubovich (LKY) lemma." IEEE Trans. Circuits Sysf.. pp. 310-311. Mar.
1974.
K. S. Narendra and J. H. Taylor. Frequency Domain Crireria f o r Absolute
Stability. New York: Academic. 1973.
S. Lefschetz. Stability of Nonlinear Control Sysrems. New York: Academic.
1963.
K. S.Narendra and L. Valavani. "A comparison of Lyapunov and hyperstability
approaches to adaptive control of continuous systems," IEEE Trans. Aufornar.
Conrr.. vol. AC-25, pp. 243-247, 1980.
I.D. Landau. AdaptiveControl-TheModel
Reference Approach. NeuYork: Dekker. 1979.
W . H. Kim and H. E. Meadows, ModernNetworkAnalysis.
New York:
Wiley, 1971.
On the Relationship Between the Model Order Reduction
Problem and the Simultaneous Stabilization Problem
PHILIP D.OLIVER
Abstract-This note points out the fact that, under certain conditions,
the model order reduction problem is very closely tied to the simultaneous
stabilization problem. Theseconditions occur when the reduced order
model is to be used to design a suboptimal controller for a high-order
plant. Under these conditions the controller must not only stabilize the
low-order model, but must also stabilize the high-order plant. The result
concerning the existence of a controller that will simultaneously stabilize
two plants is used to conclude that the approximate inclusion of any
unstable real modes of the high-order plant in the low-order model will
guarantee the existence of such a controller.
>0
lim h ( j o ) = f > O ,
O
JW
and therefore, all the conditions of Definition 2.2 are satisfied by h ( s E ) ; hence, h ( 5 - e) is PR, Le., h ( 5 ) is SPR and the sufficiency proof is
complete.
Remark I : The condition which is missing from the definition of SPR
transfer functions given by (1.1) and (1.2) is the limiting condition at w =
00 when n* = 1 or - 1 and is given by Theorem 2.1. Conditions 1-3
of Theorem 2.1 are much easier to apply than those given by Lemma I . 1
in the sdomain, and are therefore useful.
Remark 2: It should be noted that when n* = 0, conditions (1.1) and
(1.2) or (1.1) and (1.3) are both necessary and sufficient for h ( s ) to be
SPR.
Remark 3: In [A it was shown that h ( s ) is PR iff I / h ( s ) is PR. A
straightforward extension of this result is that h ( s ) is SPR iff l/h(s) is
SPR. In view of this result, Theorem 2.1 may be used to show that if h ( s )
is SPR, then the zeros and poles of h ( s ) lie in Re [s] < 0.
rrr. CONCLUSlOh'
Necessary and sufficient conditions in the frequency domain are given
for a transfer function to be SPR. These conditions extend earlier results
in the s-domain and further clarify the only necessary or only sufficient
frequency domain conditions which often appear in literature.
AC-32. NO. 1, JANUARY 1987
IhTRODUCrlON
In a very real sense, almost all control system design problems are
simultaneous stabilization problems. One uses a mathematical model to
design a controller that produces a stable feedback system when either the
model or the physical reality is in the loop.
Model order reduction refers to any of many procedures that replace a
high-order model with a lower order one. The two problems of physical
modeling and model order reduction are both concerned with obtaining a
simpler mathematical approximation that retains the key features of the
more accurate original.
Precisely which features are important to retain in order to guarantee
stability of the loop has not been determined. In fact. this issue has
received very little attention in the model order reduction literature. This
paper can be viewed as a beginning of this discussion. The answer. or
more likely answers, are very likely to depend on the precise topology of
the control system.
SIMULTANEOUS STABILIZATION AND MODEL
REDUCTION
The simultaneous stabilization problem can be stated as follows: given
two plants G and 6, find a single controller H such that both closed loop
systems in Fig. 1 are stable.
If G is viewed as the transfer function of a high-order plant, G as the
transfer function of a low-order model, and H a s a compensator that is to
be designed based on C yet placed in the loop with 6,then it is easy to see
that the simultaneous stabilization problem is intimately tied to the model
order reduction problem.
Before discussing simultaneous stabilization, it is important to emphasize that Fig. 1 actually represents two situations. First. with a minor
abuse of rotation. G(s)represents the physical reality. and
represents
REFERENCES
[I] B. D. 0 . Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A
ModernSystems Theory Approach. Englewood Cliffs. XI: F'rentice-Hall,
1973.
Manuscript received May 30. 1985; revised October 30. 1985 and July 7. 1986.
The author is with the Division of Engineering, UniversiCy of Texas at San Antonio.
San Antonio, TX 78285.
IEEE Log Number 8611553.
0018-9286/87/0100-0054$01.00
0 1987 IEEE