A Model Reference Adaptive Controller Based on Operator-Valued Kernel Functions

This paper extends recent results on model reference adaptive control using reproducing kernel Hilbert space (RKHS) learning techniques for some general cases of multi-input systems. We leverage recent results on error bounds for nonlinear observers in a vector-valued RKHS to design adaptive model reference adaptive controllers (MRAC) that are induced by operator-valued kernels. This paper formulates a model reference adaptive control strategy based on a dead zone robust modification, and derives for this case conditions for the ultimate boundedness of the tracking error. The RKHS setting allows the control designer to influence the ultimate bound by selection and placement of operator-valued kernels. As in the scalar-valued setting, closed-form expressions are obtained for the ultimate upper bound. But in this case the upper bound depends on a generalization of the power function for an operator-valued kernel space. Finally, we provide a detailed illustration our results in practice for the case of attitude control of a streamlined tailed-controlled underwater vehicle.