Robust Model Reference Adaptive Control Based on Reproducing Kernel Hilbert Spaces

ABSTRACT This article introduces native space embedding for robust adaptive control of ordinary differential equations that contain vector‐valued functional uncertainties in a reproducing kernel Hilbert space (RKHS). The proposed approach is based on a two‐phase method for analyzing and designing adaptive controllers. In the first phase, a limiting distributed parameter system (DPS), which describes the ideal closed‐loop system's performance, is introduced. The limiting DPS is not realizable in practice since it evolves in a generally infinite‐dimensional space. In the second phase, consistent finite‐dimensional approximations of the DPS are introduced to determine realizable controllers. Uniform ultimate bounds on the trajectory tracking error dynamics are derived for the functional uncertainty classes contained in the native space. These bounds are derived in terms of the power function of the RKHS or in terms of the fill distance of centers that define the scattered basis for approximations. Two numerical examples demonstrate the applicability of the proposed results.