### Multiobjective Control: Linear Matrix Inequality Techniques and Genetic Algorithms Approach

Abstract

This thesis addresses some of the open problems in multiobjective control.
The aim of this thesis is to compare two emerging techniques in multiobjective
control: evolutionary algorithms (EAs) and convex optimisation over linear
matrix inequalities (LMIs).
In the multiobjective control problem, a trade-off is sought between competing
objectives. In such a problem, no single optimal solution exists, rather a
set of equally valida solutions, known as the Pareto optimal set. It has been
shown that the multiobjective control problem can be tackled with LMI techniques,
due to its ability to include convex constraints such as H_2 performance, H_infty
performance, and pole-placement. The multiobjective control problem is formulated
as a semidefinite programming (SDP) problem, which is a single objective convex
optimisation problem, solved using interior-point methods.
As a novel alternative, multiobjective optimisation using EAs can offer a truly
multiobjective treatment of control systems specifications. This approach is
described herein and compared with numerical results from the counterpart LMI
techniques.
This investigation addresses some of the drawbacks of LMI techniques, such as the
inability to reduce the order of the controller and the LMI technique's inherent
conservatism when tackling multiobjective problems. The truly multiobjective
optimisation using an EA is proposed to overcome these problems. Reduction of the
order of the controller is achieved for problems such as the H_infty controller
design with time-domain specifications. The mixed H_2/H_infty control problem is
treated as a multiobjective H_2/H_infty control problem, and an improvement of
the Pareto optimal set is achieved.
Both methods are applied to a gas turbine engine controller design problem.
Numerical problems with controllers resulting from using the LMI approach are
addressed and a solution, based on the EA, is shown to design more numerically
robust controllers than the LMI approach.
The controller design from this application problem is then extended to a gain
scheduling controller design with the use of linear parametric-varying (LPV)
techniques. The issue of applying LPV modelling and LPV control is addressed,
and the EA-based method is proposed to design multiobjective H_2/h_infty controllers
for two operating points. The results give satisfactory, well-conditioned controllers.