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.September,
1981
'LIDS-P-1141 v
;
:.
HIERARCHICAL AGGRIEGATION OF LINEAR SYSTEMS WITH MULTIPLE TIME SCALES
'~
iM. Coderch,* A.S. Willsky,* S.S.
and D.A. Castanon*
~i~
Sastry,*
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
0.
INTRODUCTIONi
function of £ ,
of normal rank d except at c=O
k
Models of large scale systems typically in-
.
-..
clude weak couplings between some states. This in
turn leads to the evolution of different portions
of the system at' different time scales.
Intuition
suggests that the analysis of phenomena at one time
scale is made tractable (simplified) by assuming
constancy of variables at slower time scales and
steady state values for variables at faster time
scales. It *is this intuition of a hierarchy of
approximations that we make precise in this paper.
Mathematical models of interconnected power systems
have variations on several time scales -nearly instantaneous adjustment of (PV,PQ) load bus angles
(1.2)
A0 (E) =
A
.... k=0 ......
k=O
we analyse the asmptotic behavior of x
as
al
the ampto
b
i
o gea
E +0
on the time interval [0,co[.
In general
and voltages, dynamics of the swing equations,
voltage regulator and generation variation dynamics,
generation set point changes are examples of progressively slower dynamics.
is a monotone increasing C
The presence of uncertainties, load fluctuations, rare catastrophic events in the power system
clearly necessitates stochastic models for large
scale power systems. Model simplification then
consists of identifying all the time scales present
in the given model and presenting approximate models valid (uniformly over time, in a sense made
precise in the paper) at each time scales.
scale, such that
lim
sup I IxE(t)-x°(t)I[x
t>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 ()
ith
o
g(O)=0) if there exists a bounded continuous function Yk(t), called the evolution at that
time
lim s54.p JIx
$%O 6<t<T
(t/g(£))-yk(t) i '=O
k=O,l,...,m.
'
here
We consider
line
We consider tiere
linear tie-invariant
time-invariant
systems
the
.
.
.
.
. form
of thle form
~~~~~~of
x
A (C)x
0
where xE S R
x(O)=x
~~~If
where'*~~~~~
£
and the matrix A (c)
~set
(1.1)
n
is an analyt-ic
6>0, T<-
(1.3)
These evolutions are used to:
(i)
)
provide
order
provide a
a set
set of.reduced
ofreduce
order models
models
valid at different time scales.
(ii)
provide an asymptotic approximation
t)
alid uniformly on
2.
PROBTEM FOPUMULA.TION
function on [O,
in th:is paper we give tight sufficient conditions under which the multiple time scale behavior of xe(t) can be fully escribed by its
evolutions at time scales t/E
for integers
The class of models considered in this paper
is of the linear time invariant forn (1-1).
Study
of this (deterministic) equation is relevant in
the study of hierarchical aggregation of finite
state Markov processes with weak couplings
(symptomatic of multiple time scales), described
by small parameter E. Details of the applications
of our results to this context will be presented
in [3].
1..
>'
1.
RLGiven A e Rnxn
to
NOTATION
R (A) and N (A) denote the range
and null space of A. p(A) denotes the resolvent
of A, i.e. the set of A C C such that the
-l
resolvent, denoted R (,,A):
defined.
plicity
(A-XI)
,is well
X=O is an eigenvalue of algebraic multi-m, then the Laurent series of R (X,A) at
Research supported in part by the DOE under grantET-76-C-01-2295.
The authors are thankful to
our results hold even if A 0 (S) is only
assumed to have an asymptotic series (see, for eg.
[4]) of the form (1.2), provided that A (c) has
B.Levy
constant rank
* Formally,
and J.C.
Willerns for the helpful discussions
for
),
0
2
-~~~~~~~~~~~~~~.
i............
=O has the form (see (5])
xn-l
-PC(A)-
we
)k+
-
X Al
D()
k=1
k
)k
2.)
Remarks:
where P(A), the e
oigennrc
ection; D(A), the 2eigen)ilpotnt and S (A) are dfined by
XR(XA)dx
proceeds
A (E)t
(2.3)
(2),0
4 J~lR(XA)dX
X
f
requires
structure of A ,A ,A
since they determine
00'10 20
the leading term in the asymptotic expansion of
For theejesnoctn
e matrices,.(3.1) can be simplified
considerably. (see Femark (ii) after the Theorem
and Corollary).
y
S(A):=
of
i *Z.Z).'
(:'(ii)
of special interest to us in the sequel as the
~
hn~h~a~h
P
__
2(A)
r
whrA)e
(i) The computation
only AkO! A
,l. 7 .IAk-- .l
so that it
triangularly as shwn in Table 1.
27i~i-1
Jf RA)dX
2(ffi
RXA)dX(2.2)
Y
i D(A)P
k
-kk
A
k=O
A 0)
(k. )
S (O)"
(2.4)
Theorem
2r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i~~~~~~~~~~~~~~~~~~~~~~~~~
i
Y
Let A0~
(e) be a matrix with SNS of the form
i~~~~~~~~~~12
ofnralrn-.2xet)teO.I
with y
a positively oriented closed contour enclosing 0 but no other eigenvalue of A.
(1.2) of normal rank d except at
If
A
are semistable with rank A + rank A
A is said to have semi-simple null structure
(SNS) if D(A)=O. In that case, Rn = R(A) 0
N(A)
and P(A) is iss
the
e projection onto N(A) along R(A).
Asai
is to be semi-stable if it has SNS and all
its non-zero eigenvalues are in the open left
half plane.
At
If A is semistable, th.en lim e =P(A) and
-'te
¥t4
....
further
· co
-1f1
rank A ,0d then
S(A)
-
f
0
(e 'P(A))dt
=
(A -P(A)
mO
-P(A).
i
n
p=(
c
A1
Sk
(
~,.l
e
t~-- g)
t
N
+
(3.4)
all
RAP
k
Ak0 C t
-
mI
(3.5)
k=0
kt
(k
Sk p+L
.
(3.6)
k=0
n
~kC
k
t
(3.7)
ek=O
~With
~~
(3.2)
k
AkO
in
e
v.>LI k.>0
-·----~~~~~A
N
A0 ()t
0
(3.3.
limo ssup e
-a
n
A
= t
6~0 t>0
whereP(et) is any one of the four expressions
below
p
V +..+v =k+I
.+v1
p
k +1......4k
=p-lk0
1'
-liasp
e
N(Ako).
IIe
A
mO
I-P
in
(1.2), constructed recursively((k--i)th row from
kth row) by the following formula
Rtt
)
A)ik
a AL,,L,
k
ofAh
( ) (A
:cnFurther
let (, for k= ,L,...,m, be the prok'
onto N(Ak:) defined by (2.2) arnd
~~~~jection
·
k==
For our development, we need an array of
matrices Aik' i>0, k>0 starting from the AOk of
(k
...
...
k=O
.
on resultsin [51 and is outlined in Section 4.
(k
4
Then
We present here a uniform (in t) asymptotic
A (c)t
0~~~~~~~~~~~~
approximation of e
involving behavior at
time scales t/k
k=0,l,...,m. The proof
S relies
'
+ ..
in
[
where N
T .
STATEMENT OF dAIN RSULT
p
100.
10
~~~~~~~incthy'ermn
A 0)
O=R R(A
00
Since S(A) is a genoralized inverse of A
(S(A)Ax=x=AS(A)x for all x e6 R(A)) we denote it .
A'.
3.
00
~
)aypoi
~
~
this theorem in hand, the entire multiple
tine-scale structure of (1.1) can heread off
~as follows:
~
~
I
heeauiom(i
preen
Corollary
~
'
C:orollary
.
......
A20 is
20
Under- the conditions of Theorem 1, x (t)
scale
(1.1) has the following 7multiple-time-i
properties:
(i)
lra
sup - - --
E+O
O<6<t<T<o
t...... ..
of
A -A
02
i.e.,
od R(od
A
A A
-r
R(A
01 0010
A2 0 is given by
N(
)
0
...
00 ol o
(3.8)
)
10
A
(A -A
e
......
R n of
the
. -null
--- extension to
1
. !.where
P1 is defined as in the theorem Pictorially,
for
6>0,
T<c.
and k=O,l,...,..-l,
.A20 is the null extension to R
A
A0201Ao
(ii)
Ix (t/e )- T e
sup
O<6cS~t<cO
m -.
lim
EJ 0
for
6>0.
where
r0=I and Tk=P
xo
=0
.(o).
(3.9)
(A00)
(-o
--
A
e
.Pk1
for k-=l,..
(E).t/
.
N(Ao )
N~
[OTI.
intervals of the form
N(O )N(AOO)
A 20
(i) The requirement of semi-stability of
the matrices A0 OA 1 0',.
A
to obtain well
defined behavior at time scales t/ £ is a tight
l
sufficient condition. Examples showing the failure
of the theorem without these assumptions, are
given in Section 5.
_00
_0
A
A
01
A0
A01
A00
(1. 9),
without having to obtain the
.1
O
i=O
the {A.}
done by
is suc-oo
relating
with
Connection is made therein with
the Smith McMillan zero of Ao (£)
at C-0.
In
par-
ticular
mn is proven to be the order of the Smith
The construction
McMillan zero of A (C) at e=0.
of the A.0 from the A
of from
the Aothe A
.i 0
A
proceeds as follows:
proceeds as follows:
the null extension to
R of A
mod
whereooisdefied
o A
An
is
RA
enl
A10
thIN(A
R(A00)/N(A 0 0 ),i.e,= A
POA0P
01
1 0 , where P0 is defined
in the theorem. Pictorially A1 0 is the null
to
i
N (A0 0 )
O
n
t
G
A
01
.
n
R
-H.
10
(here, i0
.
is
00
inclusion)
00
.
A
0
O
the system can exhibit tine scales of order
2
t/
, t/£
preciated
3
and so on.
fact.
This is
Reduced order models
not a widely ap-
It follows from (3.8)
and (3.9) that the evolution of xE(t)
k
is given by
oscales t/e , k=0,l,...,m
at time
At
v x
= e
m
k0
kO
C
Thus, x (t) may be represented asymptotically by
the following expression uniformly valid for t>0.
m
~x£(t)
"
A00
(iii) The reader should observe using the formulae
in rcmark (ii) above that even if Ao (E)=A+00+A
k
..
and the Toeplitz
and so on.
y (t)
n
of A10 obtained as below
R
1
A00
O
0
(iv)
is
extension
[R(
A
00
A00
0
A
0
A02
A
of
A
l
(ii) It is important to be able to calculate the
Ako for k=O,l,...m
from the given data A00OA01
...
'A20'A3,...
20
nections between A
matrices
complete matrix of Table 1.
This can be
a variety of methods. One approach that
cessful1is the formal asymptotics of [73
the AkO to Toeplitz matrices constructed
nlIR(A 00
o
and so on.
The reader may refer to [7] for
complete proof and details as well as the con-
.
for fixed s and over compact time
2
Rn
--
20
m..
Equation (.3.8) implies the results of [1]
[2] where the authors analysed the convergence
Remarks:
AtA0001
--
h
and
-A
02
R
of A2 0 ,
~~~~~-,~
W~ =
m
k
y (Fkt)+f(I
-
X
)
k
+ 0(1)
From the direct sum decomposition (3.2) ofn the
theorem, it is clear that a basis T C R x"" can
be chosen si:ch that
IA 0 s)t T.........
............
+o(l:)
__..(..
i
_ ............
...
are full
/
rank square matrices re-
presenting the non zero portions of AOO,...,A
nO
defined.by
()P) (C)
A
.
-
where
If AOO has SNS then Al(E)
'..(iii)
T
Ie=T
has expansion ......
,
. .....
in
A1t(£3)=
(4
lk
are obtained from the AOk by (3.1).
Ok
Proof of theorem
Define
Q0 (c) = I-PO(E).
Then, note that
A (C)t
A () t
A
e
= P0 (c)e
+ Q0 (E)e
(v) Two time scale systems have been the focus of
considerable effort by Kokotovic and coworkers
(see 16], for exaxmple).
It is easy to check in
our framework that the assumptions in their systems
guarantee precisely two time scales.
.
()t..
.
(4.5)
By the assumption that 00 has SNS, Ai(c) has the
expansion (4.4).
Repeating the manipulation
required to yield (4.5) for its first term and
using the SNS of A10, we obtain
(vi) The significance of each row of matrices in
Table 1 is discussed in the comment following the
proof of the theorem in the next section.
A (E)t
2
t
A (e)c t
A2(0)S
e
=P0 ()P1 ($)e
A(Ec)cEt
A(£)
t-
A (EC)t
4.
Sketch of the proof
The proof relies heavily on resultsin [5]
which are summarized in the following lemma:
where P (E) is the total-projection on the zero
1
group of A1 (E ), Q1 (E) = I-P 1 (E) and
P1(c)P (E)A (6)
1
0
0
Lemma
A2 (£)
(i) If X e p(A0 0 ) then
X e P(A0 ())
and
)
for some {R (X)}k O.
~
-R(Qo(c))
R(XRA CE:)) =,,ii
k=O
R
A00
k--0
RA A)+
3
n
R(PO(C)P)=
1
A ()t
e
±+0
R(Q1()
( )
0
R
Under the SNS assumption on A
,A 20,..,A
this
procedure may be repeated m times to yield
Further, the convergence of
R(X,A (E)) to R(X,A O) as
0
00
compact subsets of p(A0 0 ).
2
Note that
satisfies
~~~~~0
0
= -
2
(1.2).
for E small enough
R(X,A 0 (£))
0o~~
(4.6)
(c)e
+Q
Let A0 (e) have the expression
A)
.l,k
.
k=O
where the A
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 '{Ak
'
A
E
A +l()
=P
(
E)P
(E
(
)..
P
m
is uniform on
+
M
+
m+lt
)e
A
k
(00)
t
Qk()e
(4.7)
k=O
(ii)
Let y be a closed contour enclosing only the
zero eigenvalue of A00. The matrix
(
P0
)
JY
Rn=R (QO())
E)
..............
R,A(Ei))dA
p
(4.2)
.
is well defined for £ sufficiently small and
equals the sun of the eigenprojections of all the
eigenvalues of A (£) that converge to 0 (the zerogroup) as
£+0.
P (C) is analytic in e and conmutesMwith Ao (E).
Further, R(PO ())
and R(PO
Franke0
..
are isomorphic.
(Q
1(c))
...
C(D
(C))
R(P (C)..P (C))
:
R
O(
By the part (ii ) of the leamma above
dim R(Qk(£))
further,
= rank AkO,
since
rank A 00 n
± rank A 10 +.+
rank
.
(4.8)
we obtain from (4.8) that'
-
dim R(po(e)
.p(c)=n-ddim N(A
0e]0,....E]!m-
())--for
-
-------- -------------
.......
Sice
0ince
0-e
x eN(Agfe))=>
N(Ak(E)) kkl
' .
and non-zero , it follows that.
N(A (C))=R(P (c) p (s)...p (£)).
Therefore,
0
0
Thus (4.7) simplifies to -Am+ ()=0.
-
e
j
k +PO (E>''' (
(e
(4'.9)-
Pm (E)
C-1tZ~
R(XAk(£)-R(,4O
A
)dh
does not have SNS)
0
1
CoSE t
Note that li'
-
A0(s)t/s
e
sinEt
t
et
does not exist for any t
-k£+
Q· w can choose
tis
showing lack of well defined behavior at time
contor enclosing all no
zero eigenvalues of Ak0
3
By the sem.i-stability of
k to be in .kthe left
.
half
plane
kk~~~~~~~~~~~~kO
uniformly (in t) bounded on rk.
t ti
This establishes
scale t/
Remark:
Recall that A
realparts.
is
to illustrate
wheresernistability
of cE
AOi()
for
imply the
scale
t/
leftsemistability of
O
3 in the
oose
a
plane
i-axis.
Thus, forhalf
each
(3.4).
is established.
Since,
R(Qk())R(Ak)
it.
Counterexample 2
From (3.2) and (3.4), it
(3.5),(3.6)
semistable- if
it
has
The necessity of the second condition
kk0
zero 0wecanchoe
eigenvalues
k A e
limbounded
supt
e
follows that
A0 (0]
(e)t
Ik
(3.2)
Let
0L
.A
where
k is any coact
>0,e
have well defined behavior at time scales
'
(A;
t, fr
t/Ek ... t/6m is tight-counterexamples can be
found for the nonexistence of well defined behavior at different time scales if A
is not
:
semi-stable for some k:
Then,
Then,+
~
i4 ·
e
- The requirement of semi-stability for the
matrices A00,A10, .,Am
for the system (1.1) to
-)
.
We now prove that Qk (e), Ck
(c) ,Ak (e) can be replaced
k
Ak
by Qk=I-Pk, Pk and Ako respectively. To estimate
the difference, we have
TIGHTNESS OF THE SEMI-STABILITY CONDITION
- Counterexample 1 (A
.'
e
5.
he would like to
note that the
0...
[0,ccn3 does not
(Semistability
of
ndondition
semistability of Ako)
e
and (3.7) are also valid.
above, it is clear that
A (E)
t
m
Ak (£3skt
Co
'ko
e
Note that
(E) . ..P (E)
(4.1abov2),e,~defined
(r t
dwherethle Ak-~=0----·------·~---are as given in Table
it
(4. 11)
;0,
.
matriee s
iii)the
of
lea
(1) , -E+o (
t1/2
),
showing the semi-stability
~A
]
t/
in spite of a
-- relegnauofrd behavior at time scale
2 .
Note
that
igenvalue
the
el of order
) are
------------A1/20.
A
~
so
Let that A0 is nilpotent
only the leading
aneed be retained.
oof0
''
the eigenvalues
of A (£6)are
0
0
~~~~-k~
Further,
is clear
for the uniform
apfollows
that that
(3.5),(3.6)
and (3.7)asymptotic
are also valid.
proximtionwsof
From part
(
-2+
(4.12)
AQk£
C~+Ot>O Qk k
im(E)
sup
terms in
the
lim
~
does not exist
real eigenvalue of order S
In this example
showing no well-2
:
:'ti';:
'
6.
6 '
;.2-r:
';:.. " ...
-''
.: REFEPENCES
S.L. Campbell and N.J. Rose, "Singular Perturbations of Autonomous Linear Systems,"
SIAM J. Math. Anal., Vol. 10, No. 3,
p. 542, 1979.
[2
-S.L. Campbell, "Singular Perturbations of
Autonomous Linear Systems II," J. Diff. Eq.,
Vol. 29, p. 362, 1978.
[31
M. Coderch and A.S. Willsky, "Hierarchical
Aggregation of Finite State Markov Processes,"
in preparation.
[43
W. Eckhaus, "Asymptotic Analysis of Singular
Perturbation," North-Holland, Amsterdam,
1979.
[5)
T. Kato, "Perturbation Theory for Linear
Operators, 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.
[71
S.S. Sastry and C.A. Desoer, "Asymptotic
Unbounded Root Loci Fornulae and Computation,"
College Eng., U. Berkely, Memo UCB/ERL M81/6
Dec. 1980.
A01 ......... A0,
A10
All ...... A1, -1
, :
:
,.--
[11
A00
~~~~~~~~~~~~A
A~~~~~~~.
AQ-1,
0
TABLE ]:
-l,1
The array Ag
-
.
CONCLUSION
Our theorem in section 3 provides a uniform
approximation over the entire real line [0,oo) to
the evolution of the system (1.1), thereby exten
,ding the 'results of [1], which are valid only for,
Furthermore, the
'intervals of the form [0-T/E].
hierarchy of models which results from the
Corollary is an extension to multiple time scales,- :
of the aggregation results in [6].
The application
of these approximations to problems in estimation
and control, are currently under study, and will
be reported in later publications. .
-
;
is grown triangularly.
..
....
~----------
...