IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-27, NO,
3, m 1982
747
process restricted to a subset of the state space. It is well known (e.g., [ I ,
sect. 91) that if the restricted subset consists only of recurrent states, then
therestrjctedprocessis
a well-defined Markovreneyal process.
Let P( a ) denote the transitionprobabilitymatrix
of theembedded
Markov chain.
For a given supervisorstrategy a considerthecostprocess
z,, t =
0,l. . . . defined by
S. Ross, Applied Prohuhdrh. Models wwh 0primi:urion Apphruriom. San Francisco.
CA: Holden-Day. 1970.
P.J. Scheitzer. "Iterative solution of the functional equations of undiscounted Markov
renewal programming." J. " a h . A n d Appl., vol. 34. pp. 495-501, 1971.
V. Borkar and P. Varaiya. "Finite chain approximation for a continuous stochastic
control problem." I€€€ Tram. Aurowr. Conrr.. vol AC-26, Apr. 1981. pp. 466-470.
r-I
-7
7
=
2 k(s,.ah.(s,)).
(5)
r=O
We can associate with any
B in B and c B in
Va the expected cost
Comments on "Decentralization, Stabilization, and
Estimation of Large Scale Linear Systems"
S . K. KATTI
K(B,0~)~-Etia[zT"+,-Z7"(bT.=8]
(6)
Abstruct-It is shown here that the input decentralization scheme for
the large scale linear system presented by Siljak and Vukcevic does not
always result in interconnected subsystems controlled by local (scalar)
inputs.
and the expected transition time
Under the strong ergodicity assumption both expressions (6) and (7)are
well defined. Now consider the mean average cost:
In theabove paper,' Siljak and Vukcevic haveproposed aninput
decentralizationscheme which decomposes a largescale system into s
number of subsystems that are controlled by distinct inputs (cf. equation
7).' This correspondence discusses with counterexample that
the scheme
of siljak and Vukcevic does not always lead to controllable pairs ( A t !b z ) .
€.uump/e: Consider a system with the following state equations:
Io 4 [;;I.
If this expression is defined. it is also equal by ( 5 ) and (6) to
0
f = 2
0
where if n ( t ) is the number of jumps of br up to time t ,
n(r)
Z(r)
2 K( hT#,
0%).
( 10)
1
0
0
0
0
0
4
0
o x + o
The system of equations may besplit
follows:
1
into coupledsubsystems
(1)
as
"=O
It thus resultsthat the supervisorproblemis a very standard semiMarkov decision process for which Theorems 7.6 and 7.7 of [2] apply.
Proposition: <There exists aboundedfunction
h ( B ) , B E B and a
constant g such that
This decomposition satisfies the condition given in Siljakand Vukcevic's
paper' since the following two pairs:
( 1 1)
2) The argument of the minimization in (11) defines astationary
supervisor strategy a* which minimizes Y ( u ) .
3) g = Y( U * ) .
Proof; Apply
Theorem
7.6 and 7.7 of [2].
a
and
are individually controllable.
Using the procedure given in the paper.' (2) may be rewritten as
111. CONCLUSION
A numerical solution of (1 1) is possible by using the successive approximation method of Schweitzer [3] which is not fundamentally different from the algorithm proposed in the paper.' The multilayer control
approach is extremely promising for the solution of very large stochastic
controlproblems. By relatingmoreexplicitlythis
approach to Markov
renewal theory we hope to have prepared the way for further developments, in particulartheextension
of this scheme toa largerclass of
processes than Markov chains or one-dimensional diffusion processes (as
in [4]).
It can be observed from (3) that both subsystems described by
Manuscript receiwd September IO, 1981: rerised November 11. 1981.
The author is with theSchool of Automation. Indian Institute of Science, Bangalore
REFERENCES
560012. India.
[ I ] E. Cinlhar."Markov
M a r . 1975.
renewaltheory:
A survey." Munug. SCI..vol. 21, pp. 727-752.
'D. D. Siljak and M. B. Vukcevic. I E E E Truruw. Auromur. Conrr.. vol. AC-21.pp.
363-3136, June 1976.
0 0 1 8-9286/82/0600-0747$00.75
Cl982 IEEE
748
IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL.
and
AC-27,NO. 3.
JUNE
1982
controllable subsystems. In addition to the computational superiority of
the scheme. the herarchcal structure allows one to ulmosr uhcuvs stabilize
the overall system by stabilizing the individual subsystemsusingdecentralized control.
are not an indibidually controllable pair.
Hence, the decentralized-stabilization scheme proposed in the paper’
does not always work.
REFERENCES
[I]
>I E. Sezcr and D D Siljak. “ O n structural dccomposltien and stabilization of
large-scale >yztemr.” I E E E Troorrs. Arrfonwr. Comr.. m l . AC-26. pp 439-344. 19x1.
ACKNOWLEDGSENT
I am thankful to Dr. P. Sen for useful discussions.
Correction to “Homogeneous Interconnected
Systems: An Example”
Author’s Reply’
M. L. EL-SAYED AND P. S. KRISHXAPRASAD
D. D. SILJAK
We knew all along the more or less obvious fact that our decentralized
scheme might not always succeed. The presented example is beneficial in
making t h s fact explicit. We note that the scheme should be considered
useful to the extent that i t replaces thetransformation of the overall
system. which is required in obtaining the Luenberger canonical form. by
transformations of the loworder subsystems.
Since the publication of our 1976 paper. we have developed an efficient
graph-theoretic scheme [ I ] for decentralization of large control systems.
lvhich results in a hierarchial (lower-triangular) ordering of structurally
’ h l a n u s n p t recci\ed December 1. 1981
T h c author I> uith the S c h d 01 Engineering. Umversir? of Santa Clara. Santa Clara. C A
05053
The following typographical errors should benoted for the above
paper.’
I ) In the second lineafter (3.3). the summation sign. 8. should be
placed right at the beginning of the following line.
2) A line is missing after (1.5). The line reads: “Since f,= f- I . then
letting.. . ’’
3 ) In the equation follolving (11.8). the operator 5 in the right-hand side
of the equation should actually be located before the first parenthesis.
Slanuacnpt rcccned Nwember I I . 1981
ht. L El-Saved 1s with thc Technical Appllsarions ScctiQx-BDC. Rahcock and Wilcox
Cornpan!. Barherton. OH W ? O i
p. S Kn>hnapr&.ad 15 uith the Department o f ElcctncalEngineenng.
U n ~ b e r s t yof
hlapland. College Park. MD 20742
i hl L El-Sa!ed and P S. Kribhnaprasad. I E E E Truns. 4uronwr. C w r r . vol. AC-26. pp.
xY4-901.4Ug
19x1
0018-9286/82/0600-074SSOo.75
01982 E E E