[ z-analysis and Synthesis Toolbox * ( p-tools ) t

THE DEVELOPMENT OF robust control theory and m particular H® optimal control theory dunng the 1980s, following the ploneermg work of Zames (1981), can be considered as a beautiful example where classical control concepts and mathematical theory go hand in hand and lead to new and powerful solutions to control system design This field has now reached a certain maturity that makes It worthwhile for engmeers to study the basic concepts and the now available solutions, and to apply these results in actual practice for robust and muitlvanable control system design An addmonal remarkable development, of great lntnnsic value although its theoretical development still has a number of open questions, is the theory of structured singular value (~t) analysis and synthesis, proposed by Doyle (1982) The theory of H® optimal control allows control problems for linear multwanable systems to be formulated in terms of an optimization problem This optimization problem considers the model of the system to be controlled and a frequency-domam performance function to be minimized It takes a description of the uncertamty involved in the model mto consideration by mmimizlng the performance Index for the worst-case effect of the model uncertainty The theory provides an analytical solution to the problem m terms of a feedback controller, computed on the basis of the solution to a pair of Rlccatl equations The structured singular value or ~-synthesis problem deepens this formulation by allowing for a more structured description of the uncertainty This structure IS taken into consideration in determInmg non-conservatively the effective worst-case disturbance effect within this structured form From an engmeermg point of view it turns out to be a worthwhile exercise to spend considerable effort In precisely formulating the uncertamty m the model parameters of a control design model, and to design a control systems that can cope with the performance deterioration caused by any of these uncertain parameters within the specified range of values In fact the /~-synthesis approach follows in a more structured fashion a design strategy that is quite common in classical control design approaches The trial-and-error steps Involved in classical control are replaced here by a formahzed method effecting the trade-off between performance maximization and the accommodation of uncertamty In contrast to H® optlmizaUon, no analytical solutions exist for/A-synthesis The user has to rely on numerical algorithms (D-K iteration) that turn out to work well in practice although convergence has not been shown theoretically