Time-Scale Decomposition of a Class of Linear and Nonlinear Cheap Control Problems

Cheap control problems where a small parameter ¿² multiplies the control cost are considered. Due to the cheapness of control, a strong control action in the form of high-gain feedback forces the given system to have slow and fast, low and high amplitude variations. For a class of linear systems (uniform rank systems), a systematic procedure of amplitude scaling and time-scale decomposition which normalizes high and low amplitude variations and which separates slow and fast time-scales is presented. The method permits the explicit characterization of all the limiting properties of the considered cheap control problem as ¿ ¿ 0. Methods of calculating singular controls and how nonuniqueness can arise in them are discussed. Above all, several suboptimal composite control schemes are developed based on the decomposition of the given optimal design into two lower order subsystem designs. Finally all these results are extended to a class of nonlinear systems.