Positive semi-definite Lyapunov function-based adaptive control for nonlinear discrete-time systems with application to chaotic Duffing oscillator

Abstract

One of several system classes that are frequently encountered in nonlinear and chaos control theory and its applications is the strict feedback form. The dynamics of many nonlinear and chaotic systems can be transformed into the strict feedback form via a change of variables as well. In this paper, a new discrete-time adaptive nonlinear controller is examined for a class of discrete-time high-order nonlinear systems in strict-feedback form. The proposed controller structure consists of a discrete-time controller built on a Lyapunov function, including a positive semi-definite function, and an adaptation mechanism that estimates unknown time-invariant parameters. The contributions of the proposed discrete-time method are twofold. One is that the discrete-time strict-feedback nonlinear systems are re-evaluated by a transformation, which enables us to design a flexible discrete-time controller. An elegant adaptive nonlinear controller is constructed without the complicated design steps, which is one of the main drawbacks encountered in controller design for general strict-feedback nonlinear systems. Owing to the mentioned transformation, computational complexity associated with other examples in the literature is effectively tackled and reduced. Another is that conventional least squares estimation is employed, demonstrating that the closed-loop error dynamics converge to the origin in an uncertain discrete-time internal model. Simulation results on two discrete-time nonlinear systems and a chaotic system verify the contributions of the proposed method.