Adjustable parameters in an adaptive controller are usually constrained into a specified convex set from the viewpoint of realizing an adaptive controller and/or guaranteeing the robust stability in adaptive loop. In conventional studies [13, 14, 17, 18, 21], the convex set to be constrained is a hyper-rectangular or a hyper-sphere in order to accomplish a realization of adaptive controller, an efficient estimation and a robust stability of adaptation loop. These methods are suitable for an adaptive control such as dynamic certainty equivalent (DyCE) control or an adaptive backstepping control. In this paper, we propose new two smooth projection high order tuner which can not only achieve the ability of conventional restraint but also improve the deterioration of the transient response of the adaptive control system based on DyCE principle. According to this smooth projection adaptation law, the adjustable parameters can be constrained into the inside of hyper-parallelogram. The validity of the proposed scheme is illustrated by the theoretical analysis and simple numerical simulation results.
