Optimal Decisions for Complex Systems
J. C. Hennet*, S. Caramihai**, M. Marin**, D.Popescu**
*LAAS, Toulouse, France
**University “Politehnica” of Bucharest
Department of Automatic Control and Computer Science
313 Splaiul Independentei, 060041, Bucharest 6, ROMANIA
E-mail: dpopescu@indinf.pub.ro
Abstract-This paper proposes the software package
SISCON, dedicated to the evaluation of optimal
decisions for large scale systems. SISCON firstly
evaluates mathematical models developed from
experimental data using LS methods for linear and
nonlinear systems and after that computes the
optimal decision problems, solving the mathematical
non-linear programming problems. The large scale
systems have generally a complex structure and
global approach computation cannot be carried out.
The authors present a decentralised decision
structure having a well-defined distribution of
supervisory functions. After decomposition of large –
scale problems is carried out, sub problems are solved
using standard optimization techniques. SISCON
offers opportunities for solving non-linear
mathematical programming problems and for
evaluating optimal decisions in large scale systems
control.
I. INTRODUCTION
It is well known that optimization theory and
mathematical nonlinear programming specially, have
been developed by several authors among whom we
mention for instance: Rosen, Fletcher, Reeves, Powell,
Lasdon, Himmelblau. During the 1970’s, many
researchers have considered the optimization theory a
good operating environment for the development of other
theories related to system control (i.e. system
identification). In the early 1980’s, the work in this field
was oriented towards the achievement of optimal
decisions in control and supervising. In our days, this
domain is still one of special interest, as the optimization
techniques are used by specialized software for process
control, systems’ identification, optimal decisions (i.e.
Matlab, Simulink).
Large scale systems are supposed to be represented
by subsystem collections observing some given
arrangements and some given interconnections. Each
subsystem is described through a specific model.
Interconnections between subsystems represent the
constraints of these ones. The management of the global
system is quite complicated and decomposition and
organizing techniques are required. So, the original
problem is transformed into an equivalent one in order to
distribute and reduce the effort made in optimization
computing [Himelbau, 1974], [Ghaoui, 1997].
SISCON software package reduces the computation
complexity, allowing adequate calculus effort per each
subsystem and can be used in many applications of
supervisory architectures.
The computed decision of the considered system is
the solution of a standard linear or non-linear
mathematical programming problem having the form:
1 2 { ( ) ( , ,..., )} N opt F x F x x x = (1)
where
( ) 0, 1,...,
( ) 0, 1,...,
i
j
gx i m
gx j s
= =
= =
(2)
N x x x ,..., , 2 1 sub vectors.
SISCON determines the optimal solution * x using
numerical computation techniques selected by taking into
account the characteristics of the criterion-functions and
of the constraints [Serbanescu, 1999].
II. DECOMPOSITION TECHNIQUES
The following types of decomposition problems can
be used depending on global problem characteristics and
on separation ways [Lasdon, 1975], [Roberts, 2001)]
[Oshuga, 1993]:
a) block-diagonal structure problems associated with
weak coupling systems;
b) additively separable problems depending on
criterion-functions and constraints;
c) relaxation and partitioning techniques;
• The first category includes
The linear problems:
0 ,
min( ) T T
x y
c x c y + (3)
with the coupling constraints:
0 0 Ax D y b + = (4)
Bx Dy b + = (5)
where: 1 2 [ | |...| ] N x x x x = is a set of i n - dimension
vectors xi and y is the coupling vector of the subsystems,
0 , ,..., 1 , 0 = = = y N i xi ; 1 2 [ | |...| ] N c c c c = is the set
of corresponding coefficients, 1 2 [ | | ...| ] N A A A A = is a
.....