On feedback capability for a class of semiparametric uncertain systems

The main purpose of this paper is to understand and characterize the maximum capability of the feedback mechanism for a basic class of scalar discrete-time semiparametric minimum-phase control systems, where both parametric and nonparametric uncertainties are included. We will demonstrate that the necessary and sufficient condition to stabilize the class of systems is L<32+2, where LL is the Lipschitz constant describing the “size” of the uncertainty of the nonparametric part. This critical value is the same as that first obtained in Xie and Guo (2000b) for a class of purely nonparametric systems, which shows that the capability of feedback is not influenced by the parameterized uncertainty in the systems, as long as the corresponding parametric nonlinear function has a linear growth rate. While the necessity proof is directly obtainable from Xie and Guo (2000b), the main task of this paper is to prove the sufficiency part.