Noninvasive Adaptive Control of a Class of Nonlinear Systems With Unknown Parameters

Abstract

Control-based continuation (CBC) is a general and systematic method to explore the dynamic response of a physical system and perform bifurcation analysis directly during experimental tests. Although CBC has been successfully demonstrated on a wide range of systems, rigorous and general approaches to designing a noninvasive controller underpinning the methodology are still lacking. In this paper, a noninvasive adaptive control strategy for a wide class of nonlinear systems with unknown parameters is proposed. We prove that the proposed adaptive control methodology is such that the states of the dynamical system track a reference signal in a noninvasive manner if and only if the reference is a response of the uncontrolled system to an excitation force. Compared to the existing literature, the proposed method does not require any a priori knowledge of some system parameters, does not require a persistent excitation, and is not restricted to linearly-stable systems, facilitating the application of CBC to a much larger class of systems than before. Rigorous mathematical analyses are provided, and the proposed control method is numerically demonstrated on a range of single- and multi-degree-of-freedom nonlinear systems, including a Duffing oscillator with multiple static equilibria. It is especifically shown that the unstable periodic orbits of the uncontrolled systems can be stabilized and reached, noninvasively, in controlled conditions.