Proceedings of the 42nd IEEE
Conference on Decision and Control
Maui, Hawii USA,December 2003
FrM04-2
A time-limited balanced reduction method'
Serkan Gugercin
Department of Mathematics,
Virginia Tech,
Blacksburg, VA, USA
[email protected]
Athanasios C. Antoulas
Department of Electrical and Computer
Engineering
Rice University, Houston, TX USA
[email protected]
Abstract-In this paper, we introduce a time-limited balanced
reduction method based on time-domain representations of the
system gramians. The method guarantees stability and yields a
simple 71, ermr bound. A numerical example is illustrated to
examine the efficiency of the proposed method.
to reach are simultaneously difficult to observe. Hence, a
reduced model is obtained by truncating the states which have
this property, i.e. those which correspond to small Hankel
singular values ui.
Let the asymptotically stable and minimal system have
the following balanced realization:
I. INTRODUCTION
In this note, we examine the linear time invariant dynamical systems in state space form:
=
Q
=
diag(C1,Cz)
where
diag(uTII,,,...,ukIm,,)
and
Cz =
diag(ul,+lI",+,,'. . ,uqImq). Then the reduced order
with
where A E WnXn, B E RnXm,C E W p x n , D E !Itpxm;
moreover u(t) E R'" is the input, y(t) E Rp is the output,
and s ( t ) E W" is the state. The transfer function of Z is given
D.The number of states n is
by G(s) := C ( s I - A)-'B
the coniplexify (order) of the system. In many applications,
n is quite large. Then, the problem, we are interested in, is
to find a reduced order system ZT
+
%:
( &(t)
y?(t)
=
=
A,s,(t)+au(t)
C 4 t )+ D u ( t ) '
(1)
where A, E R"',
B, E Rrx" , C, E Rpx', D, E ElPXm,
with T << n such that E, approximates E in some
appropriate sense.
Balanced Model Reduction [B],[17]is one of the most
common and widely used model reduction algorithms. To apply balanced reduction, one needs the contmllahility gramian
P and the observability gramian Q, the solutions to, respectively, the following controllability and the observability
Lyapunov equations:
A P + P A ~+ B B =o,
~
A'Q+ QA+C'C
= 0.
(2)
Under the assumptions that E is asymptotically stable and
minimal, P and Q are unique and symmetric positive definite
matrices. The square roots of the eigenvalues of the product
PQ are the so-called Hankel singular values U@) of the
system E: q ( E ) =
E is called balanced if
m.
C1
P
=
model Zr =
[*]
obtained by the balanced
truncation is asymptotically stable, minimal and satisfies
l l ~ - z ~ l 5l ?2(0$+1
f~
+..'+U,).
(4)
Besides the balancing method mentioned above, other
types of balancing exist such as stochastic balancing [4],[9],
[IO], bounded real balancing [ZO], positive real balancing
[4],LQG balancing [14] and frequency weighted balancing
[51, [211, [22], [251, [SI. For a survey on balancing related
algorithms: see [19], [13].
In this paper we will introduce a balancing method based
on time-limited representation of the system gramians P
and Q. The proposed method is a modification of the
time-limited balancing method of Gawronski and h a n g [SI
where the gramians are computed over a finite time interval
[ t l , t z 1. The impulse response of the resulting reduced
model is expected to match that of E better over [ t l , t z
However, the time-limited approach of [RI does not guarantee
stability and no error bounds exists. With our modification,
we guarantee stability and provide a simple 7-1, error bound.
The rest of the paper is organized as follows. Section II
reviews the time-limited balancing approach of [8]. Then in
Section III, we introduce the proposed method. Section IV
illustrates a numerical example followed by conclusions in
Section V.
1.
(3)
11. GAWRONSKI
A N D JUANG'S TIME-LIMITED BALANCED
where 01 > uz > ... > uq > 0, m,, i = l,...,q are the
multiplicities of ui,and ml
. . . m, = n. The balanced
basis has the property that the states which are difficult
It is well known that in the time domain, the controllability
gramian P and observability gramian Q are given by
P=Q=C=diag(u,l,,,...:a,I,~),
+ +
REDUCTION METHOD
b
m
This work was supp~nedin pan by the NSF through Grants CCR9988393 and DMS-9972591.
0-7803-7924-1103/$17.00
WO03 IEEE
P=
5250
[SI
1
m
eATBBTeATidrand Q =
eAT'CTCeA'dr.
For a finite time interval T = [ t l , t z ] with t z > t l 2 0,
Gawronski and Juang [SI define the time limited gramians
as
PT =
Lr
eATBBTeATTdr,and
and V,(T) are symmetric matrices. Define p := rank(lr,(T))
and @ := rank(Vo(T)).Let
fi
:= Afdiag(1 XI ~ l ~ z , . . ~ , ~ X ~ ~ l ~ z , . ~ ~ , O , . and
.~,O)
C := diag(l61 11/2,...,16g
ll/z,...,O,...,O)NT.
We define the modified time-limited gramians
as the solutions to
Note that both PT and QT are positive semi-definite matrices, Define .5',(t) := eArBBTeArrdr.It simply follows
that
:= eAt
Then the modified time-limited balancing is obtained by
balancing PT and & against each other, i.e.
Then from the definition of PT in (S), one simply obtains
1
and QT
APT + PTAT + BBT = 0 and
Q T +~A ~ +QCTC
~ = 0.
s,"
&(t)= P - Sc(t)PSc(t)Twhere SJt)
PT
PT = QT = diag(P,,I,,,.--,P?I,~)
m
PT
6',(tz)
- O,(tl) =
eATV,(T)eAr"dr,(6)
where ,Et are the modified singular values, 7%are the multiplicities of /Zt and rl . . . T , = n. The following is the
main result of this paper:
Lenvlra 3.1: Let the asymptotically stable, minimal system
have the following modified-time-limited balanced
realization:
+ +
0
where Vc(T):= eAtlBBTeArfl-eAfZBBTeArfz.
A similar
argument yields
where Vo(T):= eATtlCTCeAt'- eATtzCTCeAfz.
Hence
PT and QT are the solutions to the following Lyapunov
equations:
PT
APT+PTA~+V,(T)= 0 and A T G + Q ~ A + I I , ( T=) 0.
A time-limited balancing is obtained by balancing the timelimited gramians PT and QT, i.e.
PT = QT = d i a g ( f i " ~ I " , , . " , f i ~ ~ I , ~ ) ,
where f i i are the time-limited singular values, ni are the
multiplicities of p, with n.] . . .+n, = n. Then the reduced
model is obtained by truncating in this basis. The impulse
response of the reduced model is expected to match that of
the full order model better in the time interval T = [ t l , t2 1,
see [SI. However, since V,(T)and V,(T) are not guaranteed
to be positive definite, stability of the reduced model is never
guaranteed. Below, we will modify the time limited gramians
and the corresponding model reduction scheme to overcome
this obstacle.
I*[
REDUCTION METHOD
Our modification to the time-limited balancing method of
181 follows a similar approach to Wang's er al. modification
in [22] to Enns' method in [5]. Given the set-up above, let
V,(T) :=AIAAfT=Mdiag(X1,...,X,)AIT
Vo(T)
:= N A N T = Ndiag(&,...,6,)NT
and
. .. ,P T , L k ,P T , , , b + , ,.. . , PTQ17,).
Let C, =
be obtained by the truncation
of the balanced basis. Then G,(s) is modified-time-limited
balanced, stable and minimal. If, in addition,
+
111. T H E MODIRED TIME-LIMITED BALANCED
=eT
= diag(fi,,L,,
rank([ B
rank([ CT
then
1) = rank(@
CT 1)
(8)
and
= rank(CT)
& is asymptotically stable, minimal and satisfies
P
11
- ETllx= 2 211JBlII/JC11
/li
(9)
i=k+l
whereJg :=diag(l XI [-'/',...,I
A, 1-'/2,0,...,0)AITB
and JC := CNdiag(( 61
. . , I se I-'/',O,. .. ,O).
Remark 3.1: (1) We note that [221 makes an assumption
similar to (8) and shows that it holds for almost all cases.
(2) We would like to note that the time-limited balancing
approach allows to match 22 over multiple time intervals
without an increase in the number of Lyapunov equations
to be solved. All one needs to do is to update V,(T)and
Vo(T)to reflect the multiple intervals. For example, for
two intervals [ t l , t z ] and [ t 3 , tq 1, V J T ) simply
becomes V,kT) := eAfiBBTeATt'- eAt2BBTeAT'Z+
be the EVD decompositions of V,(T) and V,(T) with
M A I T = N N T = I,,, I XI 12 '.. I A, 12 0 and I 61 12
. . 1 6, 2 0. Such decompositions exist since both V c ( T ) &BBTeA
5251
13
- eAt< BBTeArh,
IV. EXAMPLES
We consider a randomly generated single-input-singleoutput system C of order n = 10. The only restriction is that
the poles are chosen to have an impulse response with a slow
decay. We reduce the order to r = 2 using Gawronski and
Juang's approach and our modified approach with the time
interval [ t l , tz] = [ 0, 10 ] sec. C A J T B
and X:TB denote
the reduced order models resulting from, respectively, our
modified time-limited approach and Gawronski and Juang's
approach. While E A ~ TisBasymptotically stable as guaranteed by the modified approach, ETBis unstable. The upper
plot in Figure 1 depicts the impulse responses of 2,X A J T B
and ETB.Since ETB is unstable, it diverges after t = 40
sec. On the other hand C,+,:TBis a very good match to E.
To examine the performance over the selected time interval
[ 0, 10 ] sec, in the lower plot of Figure 1, we depict the
error in the impulse responses, i.e. (i) the error between
E and E:YTBand (ii) the error between X and ETB.It
is obvious that in the selected region, X T B outperforms
EAJTB.
This is because of the fact that the modified gramians
are not the exact time-limited gramians, but close to them.
However, by this modification, we guarantee stability and
obtain a smooth behavior even outside the selected interval.
Hence the conclusion is that there is a trade-off between the
performance over the specified region and the guaranteed
stability.
lyrmrere%owMmMarc rodlCadraMs
2
-1
Y
....
O0'
10
m
J)
lo
S1o
50
m
80
90
&
rm
L
ancing approach. However, unlike [XI, the proposed method
' , error bound. A
preserves stability and yields a simple H
numerical example is illustrated to examine the efficiency of
the method.
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J)
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iw
M
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?o
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Fig. 1. Impulse responses for the reduced and error models
V. CONCLUSIONS
In this paper, we have presented a time-limited balanced
reduction method based on Gawronski and Juang's [SI bal-
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