Lyapunov redesign of adaptive controllers for polynomial nonlinear systems

In this paper, we study adaptive control redesign problem of polynomial nonlinear systems with matching parametric uncertainties. By transforming the system into its corresponding error dynamics, we will develop an adaptive control scheme in attenuating the effect of the unknown parameters on the controlled output, which is composed of tracking errors and control efforts. To achieve better controlled performance, the Lyapunov functions will be relaxed from quadratic to higher order and the resulting controller gain is generalized from constant to parameter dependent. The synthesis conditions of adaptive control will be formulated as polynomial matrix inequalities and are solvable by recast the resulting conditions into a Sum of Squares (SOS) optimization problem, from which the adaptive control law as well as the parameter adaptation law are derived with zero tracking and parameter estimation errors. An example is provided to demonstrate effectiveness of the proposed adaptive control redesign approach.