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GENETIC SOLUTIONS TO MIXED H
2
/H
∞
PROBLEMS
Limits of Performance
Gustavo S´anchez, Miguel Strefezza
Universidad Sim´on Bol´ıvar, Departamento de Procesos y Sistemas, Caracas, Venezuela
Minaya Villasana
Universidad Sim´on Bolivar, Departamento de C´omputo Cient´ıfico, Caracas, Venezuela
Keywords:
Multi-objective control, Genetic algorithms, LMIs, Pole placement, COMPl
e
ib.
Abstract:
One of the most relevant problems for control engineers is the so-called “mixed H
2
/H
∞
”. To solve it, different
convexifying strategies became popular in the later 1990s, mainly based on Linear Matrix Inequalities (LMIs).
On the other hand, genetic algorithms have also been applied for H
2
/H
∞
synthesis. Indeed, several authors
agree that they are able to find good solutions to this important control problem. However, in most of the
published papers, only low-order SISO models have been considered. In the present paper a LMI-based
algorithm is compared against a genetic algorithm, with respect to three performance indicators: Set Coverage,
Maximum Distance and Efficient Set Spacing. Five open-loop MIMO models extracted from COMPl
e
ib are
studied, for which the degree varies between 5 and 10. Based on numerical results, the genetic algorithm is
not able to improve LMI solutions for problems with more than 42 variables, restricted to a budget of 20.000
function evaluations.
1 INTRODUCTION
One of the most important problems for control engi-
neers is the so-called “mixed H
2
/H
∞
”. Typically, the
H
∞
channel is used to enhance the robustness of the
closed-loop system, whereas the H
2
channel guaran-
tees good performance (Apkarian et al., 2008).
To solve this problem, different convexifyng
strategies became popular in the later 1990s, despite
the inherent conservatism of this approach. For
instance, in (Scherer et al., 1997) controllers are
designed by solving a set of LMIs in tandem with
nonlinear algebraic equalities. In fact, this design
method (enhanced with many improvements over the
years) remains as state of the art for this problem.
On the other hand, Multi-Objective Evolutionary
Algorithms (MOEA) have also been applied for
H
2
/H
∞
synthesis: in (Takahashi et al., 2001) and
(Takahashi et al., 2004), a genetic approach is pro-
posed to obtain H
2
/H
∞
solutions which are consis-
tent with a Pareto set and less conservative compared
to LMI solutions.
After these examples it may seen obvious that, un-
der special circumstances, genetic algorithms are in
fact able to find better solutions than LMI-based al-
gorithms. However, in most of the published works
which have been consulted for this paper, only low-
order and SISO models have been considered, more
appropriate to evaluate low-complexity controllers as
PIDs (Astrom et al., 1998).
In this manner, the question arises as to whether
the genetic algorithm advantage remains true when
the open-loop models are high-order and MIMO
(Multiple Input Multiple Output) as those proposed
in COMPl
e
ib (Leibfritz, 2004).
The rest of this paper is organized as follows. In
section 2, the controller design problem is formulated.
Next, the two design methods to be compared are
described in section 3. In section 4, numerical results
are presented and conclusions are given in section 5.
2 PROBLEM FORMULATION
The closed-loop system is shown in figure 1. Matri-
ces A ∈ R
n×n
, B ∈ R
n×n
u
and C ∈ R
n
y
×n
denote the
corresponding open-loop state matrices.
The open-loop state-space equations are:
282
Sánchez G., Strefezza M. and Villasana M..
GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance.
DOI: 10.5220/0003640702820285
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 282-285
ISBN: 978-989-8425-83-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)