A design scheme of model reference adaptive control system with using a smooth parameter projection adaptive law

In a practical adaptive control, the adaptive law is often designed so that the adjustable parameter in the adaptive controller is constrained into a specified convex set from the viewpoint of the realization of adaptive controller and the stability of adaptive loop. Such a constraint can be accomplished with an adaptive law with some projection function [1]-[5], [8]-[11]. Particularly, the smooth projection algorithm [8]-[11] is suitable for the adaptive control based on the dynamic certainty equivalent (DyCE) principle, the adaptive backstepping control and so on. Unfortunately, the smooth projection algorithm [8]-[11] cannot be applied to the convex set except for hyper-rectangle one. We have investigated several methods [12], [13] in order to solve the problem. Especially, we indicated that the adaptive law with the desired projection function can be designed and it can be realized as a direct integral law without using intermediate variables when the convex set consists of polyhedron based on hyper-planes whose number is less than the number of adjustable parameters or equal to it. In this paper, we show that new two schemes of the high order tuners in model reference adaptive control system based on DyCE principle which can be constrained to the generality convex set of hyper-planes rather than the hyper-rectangle. The effectiveness of the adaptive law proposed here is verified by the stability analysis and simple numerical simulations.