Robust polynomial controller design

Abstract

The work presented in this thesis was motivated by the desire to establish an alternative approach to the design of robust polynomial controllers. The procedure of pole-placement forms the basis of the design and for polynomial systems this generally involves the solution of a diophantine equation. This equation has many possible solutions which leads directly to the idea of determining the most appropriate solution for improved performance robustness. A thorough review of many of the aspects of the diophantine equation is presented, which helps to gain an understanding of this extremely important equation. A basic investigation into selecting a more robust solution is carried out but it is shown that, in the polynomial framework, it is difficult to relate decisions in the design procedure to the effect on performance robustness. This leads to the approach of using a state space based design and transforming the resulting output feedback controller to polynomial form. The state space design is centred around parametric output feedback which explicitly represents a set of possible feedback controllers in terms of arbitrary free parameters. The aim is then to select these free parameters such that the closed-loop system has improved performance robustness. Two parametric methods are considered and compared, one being well established and the other a recently proposed scheme. Although the well established method performs slightly better for general systems it is shown to fail when applied to this type of problem. For performance robustness, the shape of the transient response in the presence of model uncertainty is of interest. It is well known that the eigenvalues and eigenvectors play an important role in determining the transient behaviour and as such the sensitivities of these factors to model uncertainty forms the basis on which the free parameters are selected. Numerical optimisation is used to select the free parameters such that the sensitivities are at a minimum. It is shown both in a simple example and in a more realistic application that a significant improvement in the transient behaviour in the presence of model uncertainty can be achieved using the proposed design procedure.