To appear in Proceedings of
First IEEE Symposium on Large
Scale Systems, October 1982
LIDS-P-1224
HIERARCHICAL AGGREGATION OF LINEAR SYSTEMS WITH MULTIPLE TIME. SCALES
M. Coderch,* A.S.Willsky,* S.S.Sastry*
and D.A. Castanon*
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
0.
INTRODUCTION
Models of large scale systems typically include weak couplings between subsystems. This results in the evolution of different portions
of the system at different time scales.
Frequently, intuition suggests that subsystems or
groups of states evolve slowly in comparison
with their internal dynamics.
This suggests that
the overall system dynamics analysis may be simplified (or so1metimes made tractable) by separately
analyzing the dynamics at a specific time scale by assuming constancy of variables evolving at
slower time scales and steady state values for
variables at. faster time scales.
If this
intuition were to hold true, it then follows
that system behavior may be approximated by
means of a hierarchy of increasingly simplified
models valid at progressively slower time scales.
The motivation for the present work came from
mathematical models for interconnected power
systems. These models have variations on several
time scales - nearly instantaneous adjustment of
loadbus angles and voltages, dynamics of the swing
equations, voltage regulator and turbine power
generation dynamics, generation scheduling (set
generation o
dynam
shavior
, 'e
point changes) are examples of progressively slower
dynamics.
With this general philosophy in mind, we study
in [8], a very simple class of systems - linear
and time invariant (equation 1.1).
For this class
of systems we model weak coupling by parametric
dependence of (1.1) on C, and obtain tight conditions under which a hierarchy of reduced order
models valid at different time scales may be
constructed. The study of this (deterministic)
eauation is also relevant to the hierarchical
aggregation of finite state Markov processes with
some rare transitions.
The details of the
application of our results in this context are
presented in [3].
1.
PROBLEM FORMULATION
We consider here .linear time-invariant systems
of the form
C
C
x = AO(C)x
x(O)=x 0
(1.1)
0
0
where x C R and the matrix A (E) is an analytic
1
w][4])
Research supported in part by the AFOSR under)
grant
82-02,52. The authors are thankful to
B. Levy and J.C.Willems for the helpful discussions.
function of Et,
of normal rank d except at s
(A)
k
A0)
k
=
0
(1.2)
we analyze the asymptotic behavior of x (t) as
C 4 0 on the time interval [0,=[. In general
Lim sup I X
C+0 t>0
(t)-xO(t)II O
so that asymptotic behavior at several time scales
needs to be considered - x (t) is said to have well
defined behavior at time scale t/g(E) (where g(E)
o
is a monotone increasing C- function on [0,E 3 with
g(0)=0) if there exists a bounded continuous function Yk(t), called the evolution at that time scale,
lim su
EC0 6<t<T
li|x (t/g(c))-y (t)|l=0
6>0, T<-
(1.3)
In this
paper we give tight sufficient conditions under which the multiple time scale bee
of x (t) can be fully described by its evoh
'
lutions at time scales t/E
for integers
k=0,l1,...,m.
(i)
(ii)
These evolutions are used to:
provide a set of reduced order models
valid at different time scales.
provide an asymptotic approximation to
xE(t) valid uniformly on [0,-[.
nxn
Given A e R
, R(A) and N(A) denote the range
and null space of A.
p(A) denotes the resolvent
set of A, i.e. the set of X C C such that the
resolvent, denoted R(X,A): =
(A-AI)
, is well
defined
If X=0 is an eigenvalue of algebraic multiplicity m, then the Laurent series of R(X,A) at
Forrallyv, our results hold even if A (C) is only
assumed to have an asymptotic series (see,for eg.
of the form (1.2), provided that A (c) has
of the form (1.2), provide
constant
rank d for
0
2
A=0 has the form (see 53)
M- 1
(A)
m-k-A)
R
R (XAk
,k
k
-+P=(A0
-
where
where
'ko
So that it
.Akl',1'Ak,+1
only
triangularly as shown in Table 1.
proceeds
of special interest to us in the sequel is
(ii)
since they
,A1 0 A2 0
the structure of A
(2.2)
R(k,A)dk
requires
(i) The computation of Ak+l
Remarks:
(2.1)
where P(A), the eigenprojection; D(A), the eigennilpotent and S(A) are defined by
i|
k 0
__
-ks(A)(A
P(A):=
k
(ki
(ki
nd S
k
(
k -
determine the leading term in the asymptotic expan-XR(X,A)dX
D(A):= '
2iriJ
Y
S(A)-1
(A):-- 2xx ¥X
-R(XA)d
('2.3)
*
AO ()t
. For these matrices, (3.1) can be
sion of e
simplified considerably. (see Remark (ii) after
(2.4)
(2.4)
the Theorem and Corollary).
Theorem
withy a positively oriented closed contour enclosing 0 but no other eigenvalue of A.
Let AO(E)
(1.2) of normal rank d except at E=0.
A is said to have semi-simole null structure.
=
N(A)
(SNS)if D(A)=0. In that case, Rn - (A)
and P(A) is the projection onto N(A) along R(A).
A is said to be semi-stable if it has SNA and all
its non-zero eigenvalues are in the open left
half plane.
At
If A is semistable, then lim e =P(A) and
At
S(A)= -J((e
0O
-P(A))dt= (A+P(A))
Rn = R(A00)
Q
A .
Then
STATEMENT OF MAIN RESULT
below
.
The proof relies
in
[8]
Z
For our development, we need an array of
matrices A.ik, i>O, k>O starting from the AOk of
k=O
(1.2), constructed recursively ((k+l)th row from
kthi row) by the following formula
m
kem
k=0
(kl)
Z+1
k
L
k
)
kl+
+kp+l
(k
(k)
Ak
,
Sk
... Ak V
p-l
(3.2)
m
)
.
(0 N (0
k=0
0
e
=
(gt)
0
(E:,t) is any one of the four expressions
A
k
eA °
O
P
1
-
(3.5)
ml
k
t
(3.6)
iT e
k
(3k7)
t
)A
(3.3)
l
c
)
V.>l, k.>0
1-
.
I-Ph.
lim sup
C+0 t>0
We present here a uniform (in t) asymptotic
A (t)t
A0E) t .bwhere
involving behavior at
approximation of e
; k=0,l ...,m.
time scales t/E
on results in [53 and is given
R(Am
m, be the prolet Pk' for kO,1 . ....
be the projection onto N(Ako0) defined by (2.2) and
-1Further
Since S(A) is a generalized inverse of A
S(A)Ax = x = AS(A)x for all x e R(A)) we denote it
3.
A1
10
+ rank A0
are semistable with rank A
rank A O=d then
where N=
- P(A).
If A
00
further
-
further
be a marrix with SNS of the form
e
1-
With this theorem in hand, -the entire multiple
(1.1) can be read off
time-scale structure of
as follows:
--~~------------------·-~~
~
c
ckt
~'~0 --
`---
-
-
-
3
Corollarv
A20 is the null extension to R
Under the conditions of Theorem 1, x (t) of
(1.1) has the following multiple-time scale
properties:
properties:
A
(i) lim sup
IIxE(t/ck)-
k
e
0
A A
mod R(A00)mod R(A)N(
00
02 010001
i.e., A2 0 is given by
X0 11=0
C+0 0<6<t<T<
of
(A -A A A )P P
(3.8)
(3.8)
10
02
01 00 01
0 1
where P1 is defined as in the theorem Pictorially,
for 6>0, T<- and k=0,l,.....,m-l
A20 is the null extension to R
(ii)
lix C(t/em)- X e
m
sup
0<6<t<
lim
C+0
(ii uIEtE)
Am0t
xO l=0
A02-A01A00 01
(3.9)
for 0>0
k=PoP
..
Pk
Rn
~ i0~
A 1 cili
0
) C10
(A
00
.4. Rn
0 )
N(A
where 70=l and
P
for
,m.k=l.
N(A00
Equation (3.8) implies the results of [1]
and [2] where the authors analyzed the convergence20
A 0 (E)t/C
for fixed
10
(ii)
It is important to be able to calculate the
AkO for k=0,l,.....,m for the given data A0 0 ,A
(1.2), without having to obtain the
complete matrix of Table 1. This can be
a variety of methods. One approach that
cessful is the formal asymptotics of [7]
the A.,O to Toeplitz matrices constructed
the
{A i
1R(Ao0)
20
s and over compact time
Remarks: (1) The requirement of semi-stability
of the matrices A0 0 ,A
...
,A
to obtain well
0
k
00 10 mO
defined behavior at time scales t/Ek is a tight
sufficient condition. Examples showing the
failure of the theorem without these assumptions,
are given in Section 4.
of
R
A
intervals of the form [O0,T].
A6 2 ,...
of A
A
n
A
is the null extension to R
10 is t:e null extension to
of A01 mod
mod
of A
)
R0(A00)/(A00 , i.e. A!0=PoA01PO, where P0 is
defined in the theorem. Pictorially A1 0 is the null
1
l
10
20
i A0 1
A0 A1
0
00
A0
0
01
00
. Connection is made therein with
of the A k0from the A i proceeds as follows:
10
and so on. The reader may refer to [7] for
complete proof and details as well as the conA
.....and the Toeplitz
nections between AA
matrices between A1 0 1 A 2 0 A
... and the Teplitz
3 0
done by
is sucrelating
with
the Smith McMillan zero of AO(E) at E=O. In par0A00
ticular m is proven to be the order of the Smith
McMillan zero of A (C) at C=O. The construction
00(A
....
0
A00
A03
A2
A0
A02
A01
A00
A01
A00
L A0 0
O
1
O
A0
0
O
-and so on.
(iii)
The reader should observe using the formulae
in remark (ii) above that even if A (E)=A00+A01
t he system can exhibit time scales of order
2
3
t/E_
t/_ and so on.
ciated fact.
This is not a widely appre-
nA
n
extension to R of A10 obtained as below
i
A
n
N (A
,0
Rn
R
N(A00)P
Rn
(iv)
Reduced order models.
It follows from (3.8)
and (3.9) that the evolution of xt(t) at time
scales t/E , k=O,l ....,m is given by
_ P0
Yk.(t) = e
10
ni
R
(here, i
kOx=O,
,.
..
m
Thus, x (t) may be represented asymptotically by
R (AOO)
N(AO)
the followinc expression unifcr-mly valid
m
is inclusion)
Y
x (t) =
k=O
k
t)
(1-
for t>0.
; 7k) + o(1)
k=O
4
From the direct sum decomposition (3.2) of the
m
not implY the semistability of {Ak)k=l
theorem, it is clear that a basis T e Rn
chosen such that
Counterexample 2 (Semistability of A
in
I
A Cmt
m
n
can be
mt
T
seristability of Ako
e
Let
Alt
A () t
-
(1)
O
e
.0
0
T
+A
(E)
eAoCt l
At0
e
I_
where A.....,A
li
(3.10)
are full rank square matrices re-
of A0 (C).
the new basis. (3.10) shows that the system (1.1)
decouples asymptotically into a set of subsystems
evolving at different time scales governed by the
reduced order dynamics of
{A}k=o.
A1A0
The significance of each row of matrices in
Table I is discussed in [81.
0
0
11
1
-2
0+C1o
-1
0
0
-1/2
0
so that A10 is nilpotent.
-2
0
are
semi-stability
In this example
A
li- e
does not exist showing no well!l e
does not exist showing no wellES0
2
defined behavior at time scale t/E
in spite of a
real eigenvalue of order
4.
0
It may further be verified that
0
(v) Two time scale systems have been the focus of
considerable effort by Kokotovic and coworkers
(see [6], for example).
It is easy to check in
our framework that the assumptions in their
systems guarantee precisely two time scales.
O
0
Note that the eigenvalues of A (E)
0, -2+o(1), -S2+o(E2 ) , showing the
presenting the non zero portions of A00 ..... AmO in
(vi)
)
.
TIGHTNESS OF THE SEMI-STABILITY CONDITION
The requirement of semi-stability for the
matrices AOO'A10
. '
5. CONCLUSIONS
,Am0 for the system (1.1) to
Our theorem in section 3 provides a uniform
aproxiiation over te entire real line [
) to
[O0, to
be
line
approximation over tne entire realcan
tight.Counterexam~les
have well defined behavior at time scales
/Cm
, t
.....
t/E,
t,
be
can Us
is tight.Counterexamples
t, t/C,...,t/Cm.
found for the nonexistence of well defined be-e
havior at different time scales if x0
iS not
x0O
semi-stable for some k:
Counterexamole
1
(A00 does not have SNS)
LetlA
1]
0 A
1
evolution of he system (1.1), thereb extending the results of [1], which are valid only for
inervals
of the form [OT/.
Furthermore,
intervals of the form [0,T/CE.
Furthermore,
the hierarchy of models which result from the
Corollary is an extension to multiple time scales,
of tne aggregation results in [6]. The application
of these approximations to problems of estimation
K ~1 |
(S )~=
be reported in later publications.
1
o
REFERENCES
[13
Then,
cosVs t
-vS
sin¢v
t'
1 0sinvF
.oe.-t
L E sjin¢ t
cos-
t
_
S.L. Campbell and N.J.Rose, "Singular Perturbations of Autonomous Linear Systems", SIAM. J.
Math. Anal., Vol. 1C0,No. 3, p.542,
1979.
eA (Si t
e
[23
Vol. 29, p. 362, 1978.
A (E)t/E
e
does not exist for any t
1Note that l
([3
Finite State Markov Processes," LIDS P-1198,
April 1982, to appear in Stochastics.
Recall that AkO is semistable if it has
SNS a-.ndall non zero eicenvalues have negative
real parts. The necessity of the second condition
is obvious and we will not furnish a counterexample to illustrate it. We would like to note
that the semistability of A0(E) for EE[0,E]
does
------------
M. Coderch, A. Willsky, S.S.Sastrv and
D.A. Castanon, "Hierarchical Aggregation of
.
showing lack of well defined behavior at time
Remark:
S.L. Campbell, "Sincular Perturbations of
Autonomous Linear Systems II," J. Diff. Ec.,
-
-
-
-
-
-
-
-
-
~
[41
W. Eckhaus, "Asymatotic Analysis of Singular
,
No
A
l
or
(Cerators,"
~o
Springer Verlag, Berlin,
1966.
[6)
P.V. Kokotovic, R.E. O'Malley, Jr. and
P. Sannuti, "Singular Perturbations and Order
Reduction in Control Theory-An Overview,"
Automatica, Vol. 12, p. 123, 1976.
[7]
S.S. Sastry and C.A. Desoer, "Asymptotic
Unbounded Root Loci Formulae and Computation,"
College Eng.,. U.Berkeley, Memo UCB/ERL M 81/6
Dec. 1980, to appear in IEEE Trans. (AC),
April 1983.
[8)
M. Coderch, A. Willsky, S.S. Sastry and
D.A. Castanon, "Hierarchical Aggregation of
Linear Systems with Multiple Time Scales,"
to appear in IEEE Trans. on Automatic Control.
A00
A01 .
Ao
A10
A11 ...........
A1,1
Al 1,0
A£-,11
Ai,0
Table 1: The array A
k,Z
is grown triangularly.