Control predictivo de sistemas borrosos mediante teoria de conjuntos invariantes y programación copositiva

Abstract

This thesis studies fuzzy model predictive control, looking for novel contributions in his field. The main idea of the thesis is to adapt the classic linear model predictive control to Takagi-Sugeno fuzzy systems. Although these systems can be dealt with nonlinear model predictive control, there are many advantages when their particular structure is taken into account.The Takagi-Sugeno fuzzy systems are formed by a mixture of linear models that vary their importance weight (membership function) depending on the value of the system state. In the last 15 years, this structure has been used to develop a multitude of new applications and a whole control theory by adapting robust control through linear matrix inequalities (LMIs) to Takagi-Sugeno fuzzy systems. In these contributions the main idea is to design controllers valid for any known membership function. A priori, it seems very restrictive and too generic, but since controllers know the membership functions, the control action will depend on it, being a powerful design solution. This design philosophy has been called shape-independent design.In pre-thesis developments, it was observed that this type of design philosophy for predictive control had not been studied in depth, so a first approach to the fuzzy predictive control was addressed. Unfortunately, the complexity of the problem increased exponentially with the control horizon and it was left for the present thesis to develop the methodologies, notation and theorems necessary to treat the problem in all its complexity, reaching solutions that increase its complexity through adjustable parameters. The obtained controller solves the shape independent fuzzy predictive control problem for a finite horizon, under some complexity assumptions.Pursuing the above goal, interesting intermediate results have been iii iv also developed: the design of a controller that (with adjustable degree of complexity) stabilizes the system in the largest possible region, and a direct adaptation of some iterative nonlinear predictive control techniques to the Takagi-Sugeno fuzzy systems. Considerations about invariant set theory for the Takagi-Sugeno problem have been developed to obtain the relevant feasible/terminal sets.These developments are based, in addition on the linear model predictive control, on copositive programming. Understanding a copositivity problem, the study of whether a polynomial is positive for any positive value of its variables. In the case of fuzzy systems since the membership functions are always positive, they easily enter into this kind of problem with the proviso that there are other variables (the system state) that are not always positive. This problem has been addressed in the thesis by the application of Polya's theorem. ResumenEn esta tesis se estudia el control predictivo para modelos borrosos Takagi-Sugeno, buscando aportaciones novedosas en este campo. La idea principal de la tesis es adaptar el control predictivo lineal clásico a los modelos borrosos Takagi-Sugeno. A pesar de que estos...