Geometric Approach to Nonlinear Adaptive Control

In this paper a brief survey is provided on a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years. Since this problem tackling is mainly based on simple geometric and algebraic considerations a brief historical summary is given on the antecedents to exemplify the advantages of geometric way of thinking in Natural Sciences. Following that the most popular branches of the classical and novel, Soft Computing (SC) based robust and adaptive approaches are analyzed with especial emphasis on the supposed need and the consequences of obtaining complete, accurate, and permanent models either for the system to be controlled or to the control situation as a whole. Following that, in comparison to the above mentioned more traditional methods, our novel approach is summarized that has the less ambitious goal of obtaining only partial, incomplete, temporal, and situation-dependent models that require continuous refreshment via observing the behavior of the controlled system in the given actual situation. It will be shown how simple geometric considerations can be used for developing a simple iterative learning control for Single Input - Single Output (SISO) systems the conditions of the convergence of which is easy to satisfy by choosing very simple and primitive initial system models and roughly chosen control parameters. Finally it will be shown how the most attractive mathematical properties of the fundamental symmetry groups of Natural Sciences can be utilized for the generalization of our approach to the control of Multiple Input - Multiple Output (MIMO) systems. The already achieved results are exemplified via simulation, and the possible directions of the future research are briefly outlined, too.