Asymptotic Stabilization for Uncertain Nonlinear Systems With Input Quantization

Abstract

This paper investigates the problem of asymptotic stabilization for a class of uncertain nonlinear systems involving logarithmic quantization at the system input. Different from the existing results and approaches, a Lyapunov function candidate and an adaptive control law are developed to adaptively estimate the uncertain parameters and to asymptotically stabilize the uncertain nonlinear system, in which the control input also involves uncertain parameters, possibly in the nonlinear form. It is shown that asymptotic stabilization can be achieved under some mild conditions, even though the adaptively estimated parameters do not converge to the true system parameters. A sufficient condition is obtained for the asymptotic stabilizability, in terms of the quantization density and the multiplicative parameter error bound at the control input. More importantly, the proposed adaptive control law is suboptimal for the corresponding LQR control and achieves the $H_{\infty }$-norm to be strictly smaller than $\gamma$, provided that $\gamma >1$, for the uncertain linearized closed-loop system, effectively suppressing energy bounded disturbances. Finally, two simulation examples are worked out to illustrate the effectiveness of the proposed method.