Weight Selection in Robust Control: An Optimisation Approach

The primary purpose of this thesis is to present optimisation algorithms which facilitate the selection of weights in robust control techniques. The robust control paradigms investigated in this thesis are μ-synthesis andH∞ loop-shaping, as these techniques offer a systematic framework for synthesising sensible controllers that meet performance objectives and guarantee robustness to model uncertainty and unmeasured disturbances. This dissertation is essentially composed of two parts. In the first part of the thesis, a conceptually new approach to the μ-synthesis robust performance problem is presented, whereby an optimisation problem is proposed which maximises the performance weights in the frequency ranges of interest subject to the existence of an internally stabilising controller that guarantees robust performance with respect to these maximised weights. Thus, performance weights and a robustly stabilising controller are simultaneously synthesised by one algorithm in a systematic way. Two solution algorithms are given for the posed optimisation problem — one being pointwise in frequency and the other using state-space techniques. The latter solution eliminates all of the disadvantages of the pointwise approach and considerably enhances the benefits of this type of optimisation based weight selection. The resulting conceptually novel method for performing μ-synthesis robust performance based design is a valuable alternative to the standard D-K iterative procedure. In the second part of the thesis, several steps of the standard H∞ loop-shaping design procedure are combined into one optimisation problem that maximises the robust stability margin over the loop-shaping weights subject to constraints which ensure that the loop-shape and the singular values/condition numbers of the weights lie in pre-specified regions. In this framework, loop-shaping weights, which can be required to have either a diagonal or a non-diagonal structure, and a robustly stabilising controller are simultaneously synthesised by one algorithm systematically. Correspondingly, the proposed algorithms greatly simplify the design of “good” performance weights and loop-shaping weights, and hence allow the designer to concentrate on more fundamental design issues. These algorithms also give an indication of what performance is achievable, although further research is required in that direction.