Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions

Abstract

A novel method in the design of controllers to drive general nonlinear systems to desired periodic or quasi-periodic motions is presented in this paper. The viability of the approach is demonstrated by controlling chaotic systems to desired motions. The proposed control system consists of a combination of a nonlinear feedforward controller and a linear feedback controller. The control gains for the feedback controller are determined by performing the stability analysis of the closed-loop systems that contain periodic or quasi-periodic coefficients. For the case of periodic coefficients, stability is determined using the well-known Floquet theory. Since there is no rigorous mathematical theory for the analysis of quasi-periodic systems, an approximate technique has been applied to determine the stability where the coefficients turned out to be quasi-periodic. In this approach, a quasi-periodic system is replaced by an approximate periodic system with an appropriate large principal period such that Floquet theory can be applied. State transition matrices for these closed-loop systems are computed symbolically in terms of the unknown control gains using shifted Chebyshev polynomials and Picard iterations. Stability diagrams obtained from the symbolic approach are used to select control gains to guarantee asymptotic stability of the feedback systems. Three examples of chaotic systems are studied in order to show applicability to a diverse class of problems. In the first case, the chaotic motion of a forced Duffing oscillator is driven to a two-frequency quasi-periodic square wave and a fixed point. In the control to a fixed point, the system is constrained to follow a logarithmic spiral trajectory. The second case involves the control of the chaotic attractor of a parametrically forced Lorenz system to a periodic orbit whose frequency is irrationally related to the parametric excitation frequency. In the last case, the chaotic behavior of a Mathieu–Duffing oscillator is successfully driven to a two-frequency quasi-periodic motion.