Adaptive control with lower-order dynamics for nonlinear systems with unknown control coefficients

While adaptive control has been extensively studied for a variety of uncertain nonlinear systems, overparametrization still exists in many works. By overparametrization, it means that the order of the adaptive controller is (much) higher than the number of unknown parameters of the system, which is a defect from a theoretical perspective and indicates high cost in controller's implementation. This paper investigates adaptive control with lower-order dynamics for nonlinear systems with unknown control coefficients, managing to significantly reduce or eliminate overparametrization and showing the differences in controllers' dynamic orders of different control strategies. The uncertain nonlinear system in question has n unknown control coefficients and a p-dimension unknown vector in the system nonlinearities. We consider two cases of unknown control coefficients: case (a): their signs are known; case (b): their signs are unknown, i.e. unknown control directions. For case (a), first by compensating for unknown control coefficients and their reciprocals, as well as the unknown vector, we design an adaptive controller with $ (2n+p-1) $ (2n+p-1)-order dynamics, where no inequality estimate is used to avoid conservatism of the controller. Second, for case (a), via bounding the system uncertainties by a single unknown constant, we design an adaptive controller with only 1-order dynamics, where inequality estimates are used and hence the controller suffers conservatism. For case (b), by adopting multiple Nussbaum functions, we design a $ (2n+p-1) $ (2n+p-1)-order adaptive controller, where n-order dynamic compensation is employed to overcome n non-identical unknown control directions. As in case (a), because no inequality estimate is used, conservatism of the controller is avoided. Regardless of conservatism, for case (b), by bounding the uncertainties in the destabilizing terms, we design n-order adaptive controller, where, of course, inequality estimates have to be used.