Synthèse multimodèle d'un régulateur polynomial robuste
Two new approaches are proposed for the multimodel synthesis of a two-degrees-of-freedom polynomial controller : the multimodel pole placement and the multiobjective optimisation. Using several models enables taking into account the parametric variations of the controlled system without introducing conservativeness, be they due to modelling mismatch or changes in the operating conditions. The resulting controller is robust with respect to stability and performances. Multimodel pole placement consists in specifying the closed-loop pole position corresponding to the whole set of models. It is shown that only an approximate pole placement is possible. The distances between the desired and the actual poles are weighted to penalize more the error on the poles which have the most effect on the stability and the performances, and minimized with the least-squares method. An iterative approach is proposed for improving the performances, and an interactive computer-aided design program is developed that enables the user ﬁnding easily which compromises are possible between performances and robustness. In multiobjective optimisation, one considers a set of performance criteria which should be made as close as possible to goal values, also for several models. This results in a minimax problem. Taking into account the H∞ norm of the sensitivity guarantees robust stability. Many different objectives can be included, such as limits on the time- and frequency-domain responses. The result is a compromise between all the speciﬁcations, which is often the purpose of the design. Depending on the optimisation method used, local minima may prevent from reaching the global optimum ; however, this method can signiﬁcantly improve an existing solution. Both approaches are applied to the design of a controller for a ßexible arm. This system exhibits poorly damped modes and large parametric variations. The speciﬁcations concern robust stability and performances. Multimodel pole placement is used to obtain a controller which satisﬁes all of them. Then multiobjective optimisation enables improving the whole set of performance criteria, or more speciﬁcally reducing the amplitude of the control signal.