The document presents a new aggregation procedure for decentralized stabilization of large scale linear interconnected systems. The previous aggregation methods in the literature contained a major gap, as the interaction terms were incorrectly aggregated with the subsystem matrices. The new procedure proposed removes this gap by suggesting a different aggregation approach. It derives inequalities to bound and decouple the interaction terms between subsystems. This allows the subsystems to be stabilized independently through optimal control, ensuring stability of the overall system. The method is illustrated with a numerical example.

1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
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DECENTRALIZED STABILIZATION OF A CLASS OF LARGE SCALE
LINEAR INTERCONNECTED SYSTEM BY OPTIMAL CONTROL
Ranjana Kumari1
, Ramanand Singh2
1
(Department of Electrical Engineering, Bhagalpur College of Engineering, P.O. Sabour,
Bhagalpur-813210, Bihar, India)
2
(Department of Electrical Engineering (Retired Professor), Bhagalpur College of
Engineering, P.O. Sabour, Bhagalpur-813210, Bihar, India)
ABSTRACT
A major gap in literature in the aggregation procedure based on Algebraic Riccati
equations when interaction terms in each subsystem of a linear interconnected system are
aggregated with the co-efficient matrix has been removed by suggesting an alternative simple
aggregation procedure. Optimal controls generated from the solution of the Algebraic Riccati
equations for the resulting decoupled subsystem are the desired decentralized controls which
guarantee the stability of the composite system with nearly optimal response and minimum
cost of control energy. The procedure has been illustrated numerically.
Keywords: Aggregation, decentralized, decoupled, optimal control.
I. INTRODUCTION
Decentralized stabilization of large scale linear, bilinear, non-linear and stochastic
interconnected systems etc. have been studied by various methods. The aim of the present
work is to continue the further study of a computationally simple method in which the
interaction terms of each subsystem is aggregated with the state matrix resulting in complete
decoupling of the subsystems so that the decentralized stabilizing feedback control gain
coefficients can be computed very easily. Two methods of aforesaid aggregation have been
reported in the literature. The first method based on Liapunov function has been studied in [1]
in which the basic methodology has been developed for linear interconnected system and the
same has been extended for non-linear interconnected system in [2] and stochastic
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interconnected system in [10]. But there are two drawbacks in the method. The method
requires the solution of
௡೔ ሺ௡೔ ାଵሻ
ଶ
linear algebraic equations for generating Liapunov function
for the interaction free part of each subsystem of order ݊௜ . Secondly, if the interaction free
parts are unstable, these have to be first stabilized by local controls. After a long gap, these
two drawbacks were removed in [3] where the basic methodology for linear interconnected
system was developed and the same was extended for stochastic bilinear interconnected
system in [4] and stochastic linear interconnected system in [5]. This aggregation method is
based on Algebraic Riccati equation for the interaction free part and requires the solution of
only ሺ݊௜ െ 1ሻ non-linear algebraic equations for the interaction free part of the subsystem of
order ݊௜ by using the simple method in [6]. The method was further improved in [7].
However, after another long gap, the present authors have observed very recently that
in the aggregation procedure of the above authors, there is a major gap affecting the results.
This is illustrated as follows. In the aggregation procedure, the authors have considered the
following inequalities:
In [3]:
‫ܫ‬ ൑
ௌ೔
ఉ೔
ൌ
ீ೔
షభ
ோഢതതതሺீ೔
షభ
ሻ೅
ఉ೔
(1)
In [7]:
‫ܫ‬ ൒
ௌ೔
ఉ೔೘ೌೣ
ൌ
ீ೔
షభ
ோ೔ሺீ೔
షభ
ሻ೅
ఉ೔೘ೌೣ
(2)
where the real symmetric positive definite matrices ܴത௜ in (1) and ܴ௜ in (2) are obtained as the
solution of the algebraic Riccati equation for the interaction free part of the ݅௧௛
subsystem. ‫ܩ‬௜
results from the majorisation of the interaction terms and is a real positive definite diagonal
matrix. It follows that ܵ௜ is a real symmetric positive definite matrix whose all elements are
positive. Hence the Eigen values of ܵ௜ are all real and positive. In (1), ߚ௜ is the minimum
Eigen value of ܵ௜. In (2), ߚ௜௠௔௫ is the maximum Eigen value of ܵ௜. The inequalities (1) and
(2) cannot be true since ܵ௜ is not a diagonal matrix and its elements are all positive.
Hence in order to remove the aforesaid major gap, the authors have suggested a
different aggregation procedure which is simpler as well as is clear on comparison. The basic
methodology has been developed for large scale linear interconnected system and can be
extended for bilinear, non-linear and stochastic interconnected systems etc. The results have
been illustrated through a numerical example.
II. PROBLEM FORMULATION
As in [1], Large Scale time-invariant systems are considered with linear
interconnection:
‫ݔ‬పሶ ൌ ‫ܣ‬௜௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ ൅ ∑ ‫ܣ‬௜௝‫ݔ‬௝
ே
௝ୀଵ,௝ஷ௜ , ݅ ൌ 1, 2, … , ܰ (3)
In (3), in the interaction-free part of the ݅௧௛
subsystem, ‫ݔ‬௜ is the
݊௜ ܺ 1 state vector, ‫ݑ‬௜ the scalar control, ‫ܣ‬௜௜ is ݊௜ ܺ ݊௜ coefficient matrix and ܾ௜ is the ݊௜ ܺ 1
driving vector. It is assumed that (‫ܣ‬௜௜, ܾ௜ ) is in companion form. In the interaction terms, ‫ݔ‬௝
is the ݊௝ ܺ 1 state vector, ‫ܣ‬௜௝ are ݊௜ ܺ ݊௝ constant real matrices.

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The problem to be studied is the determination of the decentralized (1 x ni) state
feedback control gain vector ࢑࢏ for generating the decentralized control:
‫ݑ‬௜ ൌ ࢑࢏‫ݔ‬௜, ݅ ൌ 1, 2, … , ܰ (4)
for each of the ݅௧௛
subsystem of equation (3) such that the composite system is stabilized with
optimal response and minimum cost of control energy.
III. AGGREGATION-DECOMPOSITION AND DECOUPLED SUBSYSTEMS
It is known that the optimal feedback control ‫ݑ‬௜ for the interaction free part of the ݅௧௛
subsystem of equation (3) which minimizes the quadratic performance criterion:
‫ܫ‬ ൌ ‫׬‬ ሺ‫ݔ‬௜
்∞
଴
ܳ௜‫ݔ‬௜ ൅ ߣ௜‫ݑ‬௜
ଶ
ሻ ݀‫ݐ‬ (5)
is given by:
‫ݑ‬௜ ൌ ݇௜‫ݔ‬௜, ݅ ൌ 1, 2, … , ܰ
with: ݇௜ ൌ െ
ଵ
ఒ೔
ܾ௜
்
ܴ௜ (6)
where ܴ௜ is an ݊௜ ܺ ݊௜ real symmetric positive definite matrix given as the solution of the
Algebraic Riccati equation:
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ ൌ ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ ൅ ܳ௜ (7)
In the equations (5), (6) and (7), ߣ௜ is a positive constant and ܳ௜ is an ݊௜ ܺ ݊௜ real symmetric
positive definite matrix. It follows that:
‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ (8)
Hence for the interconnected subsystems in (3) (proof in Appendix 1):
∑ ‫ݔ‬௜
்
ሺே
௜ୀଵ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ ∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝
ே
௝ୀଵ,௝ஷ௜
ே
௜ୀଵ
ே
௜ୀଵ ‫ݔ‬௝ሻ ൅
൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ (9)
Interaction terms in (9) can be bounded as (proof in Appendix 2):
∑ ∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝
ே
௝ୀଵ,௝ஷ௜
ே
௜ୀଵ ‫ݔ‬௝ሻ ൑ ∑ ሺ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ሻே
௜ୀଵ (10)
where ‫ܯ‬௜ is an ݊௜ ܺ ݊௜ real positive definite diagonal matrix whose elements depend upon the
elements of ܴ௜ and ‫ܣ‬௜௝. The term ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ in the R.H.S. of inequality (10) can be bounded as
(proof in Appendix 3):
‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜
்
ܴ௜‫ݔ‬௜ (11)

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where ݉௜ is a real and positive number given by:
݉௜ ൌ
ఈ೔
ଶఉ೔
(12)
ߙ௜ is the lowest diagonal element of ‫ܯ‬௜ and ߚ௜ is the highest diagonal element of ݊௜ ܺ ݊௜ real
positive definite diagonal matrix ܵ௜ (different from the one in introduction) which is given by:
ܵ௜ ൌ
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬
∑ ‫ݎ‬௜௪ଵ
௡೔
௪ୀଵ ‫ڮ‬ 0
‫ڭ‬ ∑ ‫ݎ‬௜௪ଶ
௡೔
௪ୀଵ
‫ڰ‬
‫ڭ‬
0 ‫ڮ‬ ∑ ‫ݎ‬௜௪௡೔
௡೔
௪ୀଵ ‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
Where ‫,ݓ‬ ‫ݒ‬ ൌ 1, 2, . . . , ݊௜ and ‫ݎ‬௜௪௩ is the element in the ‫ݓ‬௧௛
row and ‫ݒ‬௧௛
column of ܴ௜.
Using (10) and (11), the equation (9) is converted to the following inequality:
∑ ‫ݔ‬௜
்
ሺே
௜ୀଵ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜
ழ
வ
∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜
்
ܴ௜‫ݔ‬௜
ே
௜ୀଵ
ே
௜ୀଵ ൅
൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ (13)
The inequality (13) can be replaced by an equation by replacing ܴ௜ by another ݊௜ ܺ ݊௜ real
symmetric positive definite matrix ࡾ࢏:
∑ ‫ݔ‬௜
்
ሺே
௜ୀଵ
ଵ
ఒ೔
ࡾ࢏ܾ௜ܾ௜
்
ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜
்
ࡾ࢏‫ݔ‬௜
ே
௜ୀଵ
ே
௜ୀଵ ൅
൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ (14)
which is reduced to the following equation (Appendix 4):
ଵ
ఒ೔
ࡾ࢏ܾ௜ܾ௜
்
ࡾ࢏ ൌ ‫ܣ‬௠௜
்
ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௠௜ ൅ ܳ௜ (15)
where: ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜ ൅ ݉௜‫ܫ‬
This is the Algebraic Riccati equation for the decoupled subsystems:
‫ݔ‬పሶ ൌ ‫ܣ‬௠௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ , ݅ ൌ 1, 2, … , ܰ (16)
If (16) is compared with (3), it is observed that the effects of interactions have been
aggregated as ݉௜‫ܫ‬ into the coefficient matrix of the interaction free part of the ݅௧௛
subsystem
so that the N interconnected subsystems have been decomposed into the N decoupled
subsystems of (16). The positive number ݉௜ is, therefore, designated as interaction
coefficient and the procedure is called aggregation-decomposition.
IV. DECENTRALIZED STABILIZATION BY OPTIMAL FEEDBACK CONTROL
‫ܣ‬௠௜ in equation (16) is the modified coefficient matrix of the ݅௧௛
subsystem
incorporating the maximum possible interaction effects. Hence the decentralized stabilization
of interconnected subsystems in (3) implies the stabilization of decoupled subsystems in (16).

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It is noted that ሺ‫ܣ‬௠௜, ܾ௜ሻ in (16) is not in companion form. Hence on applying similarity
transformation [8]:
‫ݔ‬పഥ ൌ ܲ௜‫ݔ‬௜
ܲ௜ being ݊௜ ܺ ݊௜ transformation matrix, subsystems (16) are transformed to:
‫ݔ‬ҧሶ௜ ൌ ‫ܣ‬ҧ௠௜‫ݔ‬ҧ௜ ൅ ܾ௜‫ݑ‬௜, ݅ ൌ 1, 2, … , ܰ (17)
where ሺ‫ܣ‬ҧ௠௜, ܾ௜ሻ is in companion form. Referring to [9], the optimal control function ‫ݑ‬௜, which
minimizes the quadratic performance criteria so that the subsystems (17) and hence (16) are
stabilized with optimal response and minimum cost of control energy, is given by:
‫ݑ‬௜ ൌ െ
ଵ
ఒ೔
ܾ௜
்
ܴത௜ ‫ݔ‬పഥ
ൌ െ
ଵ
ఒ೔
ܾ௜
்
ܴത௜ ܲ௜‫ݔ‬௜
֜ ‫ݑ‬௜ ൌ ࢑࢏‫ݔ‬௜,
‫݁ݎ݄݁ݓ‬ ࢑࢏ ൌ െ
ଵ
ఒ೔
ܾ௜
்
ܴത௜ ܲ௜, ݅ ൌ 1,2, … , ܰ (18)
In equation (18), ܴത௜ is the solution of the Algebraic Riccati equation:
ଵ
ఒ೔
ܴത௜ܾ௜ܾ௜
்
ܴത௜ ൌ ‫ܣ‬ҧ௠௜
்
ܴത௜ ൅ ܴത௜‫ܣ‬ҧ௠௜ ൅ ܳ௜ (19)
Hence the decentralized stabilizing control gain vectors ࢑࢏ to generate the controls ‫ݑ‬௜
as per equation (4), which guarantees the stability of the interconnected subsystems (3), are
optimal control gain vectors for the decoupled subsystems (16) and are given by (18).
Response will be slightly deviated from the optimal and cost of control energy slightly higher
than minimum due to majorization.
V. NUMERICAL EXAMPLE
The class of linear interconnected system consisting of three subsystems each of
fourth order as in [3], [7] is considered corresponding to (3) as follows:
‫ݔ‬ሶଵ ൌ ቎
0 1 0 0
0
0
0
0
0
െ6
1
0
െ11
0
1
െ6
቏ ‫ݔ‬ଵ ൅ ቎
0
0
0
1
቏ ‫ݑ‬ଵ ൅ ቎
0 0 0 0
0
0
0
0
0
0.05
0
0
0
0
0
0
቏ ‫ݔ‬ଶ +
቎
0 0 0 0
0
0
0
0
0
0.04
0
0
0
0
0
0
቏ ‫ݔ‬ଷ

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‫ݔ‬ሶଶ ൌ ቎
0 1 0 0
0
0
0
0
0
0
1
0
െ2
0
1
െ3
቏ ‫ݔ‬ଶ ൅ ቎
0
0
0
1
቏ ‫ݑ‬ଶ ൅ ቎
0 0 0 0
0
0
0
0
0
0.02
0
0
0
0
0
0
቏ ‫ݔ‬ଷ +
቎
0 0 0 0
0
0
0
0
0
0.04
0
0
0
0
0
0
቏ ‫ݔ‬ଵ
‫ݔ‬ሶଷ ൌ ቎
0 1 0 0
0
0
0
0
0
6
1
0
െ1
0
1
െ4
቏ ‫ݔ‬ଷ ൅ ቎
0
0
0
1
቏ ‫ݑ‬ଷ ൅ ቎
0 0 0 0
0
0
0
0
0
0.03
0
0
0
0
0
0
቏ ‫ݔ‬ଵ +
቎
0 0 0 0
0
0
0
0
0
0.04
0
0
0
0
0
0
቏ ‫ݔ‬ଶ
With ܳଵ ൌ ܳଶ ൌ ܳଷ ൌ ‫ܫ‬ସ and ߣ ൌ ൅1, on solving the Algebraic Riccati equations
corresponding to (7), values of ܴଵ, ܴଶ, ܴଷ are obtained as:
ܴଵ ൌ ቎
7.8202 12.0776 6.2574 1.0000
12.0776
6.2574
1.0000
22.1915
11.9341
1.8202
11.9341
7.7542
1.0776
1.8202
1.0776
0.2574
቏ , ܴଶ
ൌ ቎
3.5170 5.6847 4.1676 1.0000
5.6847
4.1676
1.0000
15.8253
13.6576
3.5170
13.6576
14.1746
3.6847
3.5170
3.6847
1.1676
቏,
ܴଷ ൌ ቎
7.8202 12.0776 6.2574 1.0000
12.0776
6.2574
1.0000
94.1915
71.9341
13.8202
71.9341
57.7542
11.0776
13.8202
11.0776
2.2574
቏
Solving the inequality (10) ‫ܯ‬ଵ , ‫ܯ‬ଶ , ‫ܯ‬ଷ are computed as:
‫ܯ‬ଵ ൌ ቎
0.09 0 0 0
0
0
0
1.3833
0
0
0
0.0970
0
0
0
0.0232
቏ , ‫ܯ‬ଶ ൌ ቎
0.06 0 0 0
0
0
0
1.545
0
0
0
0.2211
0
0
0
0.0701
቏,
‫ܯ‬ଷ ൌ ቎
0.07 0 0 0
0
0
0
1.3210
0
0
0
0.7754
0
0
0
0.1580
቏

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ܵଵ, ܵଶ, ܵଷ are then computed. Hence, one gets: ߙଵ ൌ 0.0232, ߙଶ ൌ 0.06, ߙଷ ൌ 0.07 and
ߚଵ ൌ 48.0233, ߚଶ ൌ 38.6846, ߚଷ ൌ 192.0233. Then using (12), ݉ଵ ൌ 2.4155‫ܧ‬ െ
4, ݉ଶ ൌ 7.755‫ܧ‬ െ 4, ݉ଷ ൌ 1.8227‫ܧ‬ െ 4 . Hence the three decoupled subsystems
corresponding to (16) are obtained. Then with the transformation matrices:
ܲଵ ൌ ቎
1.0000 0 0 0
0.0002
0
0
1.0000
0.0005
0
0
1.0000
0.0007
0
0
1.0000
቏ , ܲଶ
ൌ ቎
1.0000 0 0 0
0.0008
0
0
1.0000
0.0016
0
0
1.0000
0.0023
0
0
1.0000
቏,
ܲଷ ൌ ቎
1.0000 0 0 0
0.0002
0
0
1.0000
0.0004
0
0
1.0000
0.0005
0
0
1.0000
቏
co-efficient matrices of the transformed decoupled subsystems corresponding to (17) are
obtained as:
‫ܣ‬ҧ௠ଵ ൌ ቎
0 1.0000 0 0
0
0
0.0014
0
0
െ5.9947
1.0000
0
െ10.9957
0
1.0000
െ5.9990
቏,
‫ܣ‬ҧ௠ଶ ൌ ቎
0 1.0000 0 0
0
0
െ0.0000
0
0
0.0031
1.0000
0
െ1.9930
0
1.0000
െ2.9969
቏,
‫ܣ‬ҧ௠ଷ ൌ ቎
0 1.0000 0 0
0
0
െ0.0011
0
0
6.0004
1.0000
0
െ0.9978
0
1.0000
െ3.9993
቏
Hence ܴതଵ, ܴതଶ, ܴതଷ are computed by solving (19). Finally, corresponding to (18), the
desired decentralized stabilizing controller gain vectors are computed as:
࢑૚ ൌ ሾെ1.0019 െ 1.8236 െ 1.0795 െ 0.2577ሿ, ࢑૛ ൌ ሾെ1.0027 െ 3.5253 െ
3.6925 െ 1.1698ሿ, ࢑૜ ൌ ሾെ1.0014 െ 13.8247 െ 11.0806 െ 2.2580ሿ

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VI. CONCLUSION
A computationally simpler aggregation procedure has been suggested to aggregate the
interaction terms with the co-efficient matrix of each subsystem of a large scale linear
interconnected system. Moreover, the major gap in the aggregation procedure in the
concerned literature has been removed. The desired decentralized feedback co-efficients
generated from the solution of the Algebraic Riccati equation for the resulting decoupled
subsystems will guarantee the stability of the composite system with optimal response and
minimum cost of control energy with slight deviation due to majorization of the interaction
terms.
The results of this paper can be easily extended for bilinear, non-linear and stochastic
interconnected systems and even for time varying, uncertain and robust control
interconnected systems and for the case with output feedback and pole-placement. The results
can also be applied for improvement of dynamic and transient stability of multi-machine
power systems etc. All the above cases will be reported in the literature by the present authors
in due course. The procedure of the paper can be computerized and hence is applicable for
higher order systems.
REFERENCES
Journal Papers
[1] A K Mahalanabis and R Singh, On decentralized feedback stabilization of large-scale
interconnected systems, International Journal of Control, Vol. 32, No. 1, 1980, 115-126.
[2] A K Mahalanabis and R Singh, On the analysis and improvement of the transient
stability of multi-machine power systems, IEEE Transactions on Power Apparatus and
Systems, Vol. PAS-100, No. 4, April 1981, 1574-80.
[3] K Patralekh and R Singh, Stabilization of a class of large scale linear system by
suboptimal decentralized feedback control, Institution of Engineers, Vol. 78, September
1997, 28-33.
[4] K Patralekh and R Singh, Stabilization of a class of stochastic bilinear interconnected
system by suboptimal decentralized feedback controls, Sadhana, Vol. 24, Part 3, June
1999, 245-258.
[5] K Patralekh and R Singh, Stabilization of a class of stochastic linear interconnected
system by suboptimal decentralized feedback controls, Institution of Engineers, Vol. 84,
July 2003, 33-37.
[6] R Singh, Optimal feedback control of Linear Time-Invariant Systems with Quadratic
criterion, Institution of Engineers, Vol. 51, September 1970, 52-55.
[7] B C Jha, K Patralekh and R Singh, Decentralized stabilizing controllers for a class of
large-scale linear systems, Sadhana, Vol. 25, Part 6, December 2000, 619-630.
Books
[8] B C Kuo, Automatic Control Systems, (PHI, 6th
Edition, 1993), 222-225.
[9] D G Schultz and J L Melsa, State Functions and Linear Control Systems, (McGraw Hill
Book Company Inc, 1967).
Proceedings Papers
[10] A K Mahalanabis and R Singh, On the stability of Interconnected Stochastic Systems,
8th
IFAC World Congress, Kyoto, Japan, 1981, No. 248.

9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
164
APPENDIX 1
To derive the equation (9)
Equation (8) is rewritten:
‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
ൌ ‫ݔ‬௜
்
‫ܣ‬௜௜
்
ܴ௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
ൌ ሺ‫ݔ‬௜
்
‫ܣ‬௜௜
்
ܴ௜‫ݔ‬௜ሻ்
൅ ‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ (Since ‫ݔ‬௜
்
‫ܣ‬௜௜
்
ܴ௜‫ݔ‬௜ is a scalar)
ൌ ‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
Hence for the interaction free part of the ݅௧௛
subsystem in (3)
‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ 2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
Thus for the ݅௧௛
linear interconnected subsystem in (3)
‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ 2‫ݔ‬௜
்
ܴ௜ሺ‫ܣ‬௜௜‫ݔ‬௜ ൅ ∑ ‫ܣ‬௜௝
ே
௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
ൌ 2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ 2‫ݔ‬௜
்
ܴ௜ሺ∑ ‫ܣ‬௜௝
ே
௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
֜ ‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ 2‫ݔ‬௜
்
ܴ௜ሺ∑ ‫ܣ‬௜௝
ே
௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜
்
ܳ௜‫ݔ‬௜
For the N linear interconnected subsystems (3)
∑ ‫ݔ‬௜
்
ሺ
ଵ
ఒ೔
ܴ௜ܾ௜ܾ௜
்
ܴ௜ሻ‫ݔ‬௜
ே
௜ୀଵ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜
ே
௜ୀଵ ൅ ∑ ∑ே
௝ୀଵ,௝ஷ௜
ே
௜ୀଵ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൅
൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ
APPENDIX 2
To derive the inequality (10)
It is noted that:
2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝
ൌ 2ሾ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔
ሿ ൦
‫ݎ‬௜ଵଵ ‫ݎ‬௜ଶଵ … ‫ݎ‬௜௡೔ଵ
‫ݎ‬௜ଵଶ ‫ݎ‬௜ଶଶ … ‫ݎ‬௜௡೔ଶ
‫ڭ‬
‫ݎ‬௜ଵ௡೔
‫ݎ‬௜ଶ௡೔
… ‫ݎ‬௜௡೔௡೔
൪
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
ܽ௜௝ଵଵ‫ݔ‬௝ଵ ൅ ܽ௜௝ଵଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝ଵ௡ೕ
‫ݔ‬௝௡ೕ
ܽ௜௝ଶଵ‫ݔ‬௝ଵ ൅ ܽ௜௝ଶଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝ଶ௡ೕ
‫ݔ‬௝௡ೕ
‫ڭ‬
ܽ௜௝௡೔ଵ‫ݔ‬௝ଵ ൅ ܽ௜௝௡೔ଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝௡೔௡௝‫ݔ‬௝௡௝‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
If the multiplications are carried out in the RHS one gets the terms of the form:
2ሾ∑ ሺr୧୵୴a୧୨୵୵′ ሻሿx୧୴ x୨୵′ , i, j ൌ 1,2, … , N; w, v ൌ 1, 2, … … , ݊௜; w’ ൌ 1,2, … , ݊௝.
௡೔
௪ୀଵ
which can be bounded as below :
2ሾ∑ ሺr୧୵୴a୧୨୵୵′ ሻሿx୧୴ x୨୵′ .
௡೔
௪ୀଵ ≤ ห∑
௡೔
௪ୀଵ ൫r୧୵୴a୧୨୵୵′ ൯หሺxଶ
୧୴
൅ xଶ
୨୵′ሻ
where r୧୵୴ represents the ‫,ݓ‬ ‫ݒ‬௧௛
element of the matrix ܴ௜ and a୧୨୵୵′ the w, w′୲୦
element of
the matrix ‫ܣ‬௜௝. Using this it is then possible to get the inequality:
2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ ൑ ‫ݔ‬௜
்
‫′ܦ‬௜௜
‫ݔ‬௜ ൅ ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝
where ‫′ܦ‬௜௜ and ‫ܦ‬௜௝ are diagonal matrices whose elements are real non-negative numbers
depending on the elements of the matrices ‫ܣ‬௜௝ and ܴ௜. It then follows that the following
inequality must hold:
∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே
௝ୀଵ,௝ஷ௜ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௜
்
‫′ܦ‬௜௜
‫ݔ‬௜ ൅ ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
ൌ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௜
்
‫′ܦ‬௜௜
‫ݔ‬௜ሻ ൅ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
ൌ ሺܰ െ 1ሻሺ‫ݔ‬௜
்
‫′ܦ‬௜௜
‫ݔ‬௜ሻ ൅ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ

10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
165
ൌ ‫ݔ‬௜
்
ሼሺܰ െ 1ሻ‫′ܦ‬௜௜
ሽ‫ݔ‬௜ ൅ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
ൌ ‫ݔ‬௜
்
‫ܦ‬௜௜‫ݔ‬௜ ൅ ∑ே
௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
where ‫ܦ‬௜௜ ൌ ሺܰ െ 1ሻ‫′ܦ‬௜௜
֜ ∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே
௝ୀଵ,௝ஷ௜ ∑ே
௝ୀଵ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
Hence: ∑ே
௜ୀଵ ∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே
௝ୀଵ,௝ஷ௜ ∑ே
௜ୀଵ ∑ே
௝ୀଵ ሺ‫ݔ‬௝
்
‫ܦ‬௜௝‫ݔ‬௝ሻ
ൌ ∑ே
௜ୀଵ ∑ே
௝ୀଵ ሺ‫ݔ‬௜
்
‫ܦ‬௝௜‫ݔ‬௜ሻ
ൌ ∑ ‫ݔ‬௜
்ே
௜ୀଵ ሺ∑ே
௝ୀଵ ‫ܦ‬௝௜ሻ‫ݔ‬௜
֜ ∑ே
௜ୀଵ ∑ ሺ2‫ݔ‬௜
்
ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே
௝ୀଵ,௝ஷ௜ ∑ே
௜ୀଵ ሺ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ሻ
where ‫ܯ‬௜ ൌ ∑ே
௝ୀଵ ‫ܦ‬௝௜ is in general a real positive definite ݊௜ ܺ ݊௜ diagonal matrix.
APPENDIX 3
In order to prove that ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜
்
ܴ௜‫ݔ‬௜, it is noted that:
‫ݔ‬௜
்
ܴ௜‫ݔ‬௜ ൌ ൣ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔
൧ ൦
‫ݎ‬௜ଵଵ ‫ݎ‬௜ଶଵ … ‫ݎ‬௜௡೔ଵ
‫ݎ‬௜ଵଶ ‫ݎ‬௜ଶଶ … ‫ݎ‬௜௡೔ଶ
‫ڭ‬
‫ݎ‬௜ଵ௡೔
‫ݎ‬௜ଶ௡೔
… ‫ݎ‬௜௡೔௡೔
൪ ൦
‫ݔ‬௜ଵ
‫ݔ‬௜ଶ
‫ڭ‬
‫ݔ‬௜௡೔
൪
ൌ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ‫ݎ‬௜௪௩‫ݔ‬௜௪ ‫ݔ‬௜௩
≤ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ‫ݎ‬௜௪௩ሺ‫ݔ‬ଶ
௜௪൅ ‫ݔ‬ଶ
௜௩ ሻ/2
= ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௪ሻ/2 ൅ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௩
ሻ/2
= ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ‫ݎ‬௜௩௪‫ݔ‬ଶ
௜௩ሻ/2 ൅ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௩
ሻ/2
Since ܴ௜ is a real symmetric positive definite matrix: ‫ݎ‬௜௩௪ ൌ ‫ݎ‬௜௪௩, ‫,ݓ‬ ‫ݒ‬ ൌ 1, 2, … , ݊௜
֜ ‫ݔ‬௜
்
ܴ௜‫ݔ‬௜ ൑ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௩ሻ/2 ൅ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௩
ሻ/2
ൌ ∑௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ
௜௩ሻ
ൌ ∑ ሺ
௡೔
௩ୀଵ ∑௡೔
௪ୀଵ ‫ݎ‬௜௪௩ሻ‫ݔ‬ଶ
௜௩
ൌ ∑௡೔
௩ୀଵ ‫݌‬௜௩‫ݔ‬ଶ
௜௩ , (where ‫݌‬௜௩ ൌ ∑௡೔
௪ୀଵ ‫ݎ‬௜௪௩) ൌ
ൣ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔
൧ ൦
‫݌‬௜ଵ 0 … 0
0 ‫݌‬௜ଶ 0
‫ڭ‬
0
‫ڰ‬
…
‫ڭ‬
‫݌‬௜௡௜
൪ ൦
‫ݔ‬௜ଵ
‫ݔ‬௜ଶ
‫ڭ‬
‫ݔ‬௜௡೔
൪
= ‫ݔ‬௜
்
ܵ௜‫ݔ‬௜
where: ܵ௜ ൌ ൦
‫݌‬௜ଵ 0 … 0
0 ‫݌‬௜ଶ 0
‫ڭ‬
0
‫ڰ‬
…
‫ڭ‬
‫݌‬௜௡௜
൪ ൌ
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
∑ r୧୵ଵ
୬୧
୵ୀଵ ‫ڮ‬ 0
‫ڭ‬
‫ڭ‬
∑ r୧୵ଶ
୬୧
୵ୀଵ
‫ڰ‬
‫ڭ‬
‫ڭ‬
0 … ∑ r୧୵୬୧
୬୧
୵ୀଵ ‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
֜ ‫ݔ‬௜
்
ܵ௜‫ݔ‬௜ ൒ ‫ݔ‬௜
்
ܴ௜‫ݔ‬௜
It is noted that Si is a real positive definite ݊௜ ܺ ݊௜ diagonal matrix.
֜ ‫ܫ‬ ൒
ଵ
ఉ೔
ܵ௜
where ߚ௜ is the highest diagonal element of Si. Now referring to Appendix 2, since Mi also is
a real positive definite diagonal matrix,
‫ܯ‬௜ ൒ ߙ௜‫,ܫ‬ where ߙ௜ is the lowest diagonal element of ‫ܯ‬௜

11. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
166
֜ ‫ܯ‬௜ ൒
ఈ೔
ఉ೔
ܵ௜
֜ ‫ܯ‬௜ ൒2݉௜ܵ௜, where ݉௜ ൌ
ఈ೔
ଶఉ೔
is a real and positive number
֜ ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜
்
ܵ௜‫ݔ‬௜
൒ 2݉௜‫ݔ‬௜
்
ܴ௜‫ݔ‬௜
֜ ‫ݔ‬௜
்
‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜
்
ܴ௜‫ݔ‬௜
APPENDIX 4
The equation (14)
∑ ‫ݔ‬௜
்
ሺே
௜ୀଵ
ଵ
ఒ೔
ࡾ࢏ܾ௜ܾ௜
்
ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜
்
ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜
்
ࡾ࢏‫ݔ‬௜
ே
௜ୀଵ
ே
௜ୀଵ ൅
∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௜௜൅݉௜‫ܫ‬ሻ்
ࡾ࢏‫ݔ‬௜
ே
௜ୀଵ ൅ ‫ݔ‬௜
்
ࡾ࢏ሺ‫ܣ‬௜௜൅݉௜‫ܫ‬ሻ‫ݔ‬௜ ൅
∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ
ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௠௜
்
ࡾ࢏
ே
௜ୀଵ ൅ ࡾ࢏‫ܣ‬௠௜ሻ‫ݔ‬௜ ൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ
where ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜൅݉௜‫ܫ‬
֜ ∑ ‫ݔ‬௜
்
ሺே
௜ୀଵ
ଵ
ఒ೔
ࡾ࢏ܾ௜ܾ௜
்
ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜
்
ሺ‫ܣ‬௠௜
்
ࡾ࢏
ே
௜ୀଵ ൅ ࡾ࢏‫ܣ‬௠௜ሻ‫ݔ‬௜ ൅ ∑ ሺ‫ݔ‬௜
்
ܳ௜‫ݔ‬௜ሻே
௜ୀଵ
֜
ଵ
ఒ೔
ࡾ࢏ܾ௜ܾ௜
்
ࡾ࢏ ൌ ‫ܣ‬௠௜
்
ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௠௜ ൅ ܳ௜
which is the Algebraic Riccati equation for the decoupled subsystems:
‫ݔ‬పሶ ൌ ‫ܣ‬௠௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ , ݅ ൌ 1, 2, … , ܰ where ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜ ൅ ݉௜‫ܫ‬