Analysis and Control of Dynamical Systems with Quasi-Periodic Coefficients

Abstract

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. The coefficients of parametric excitation terms are not necessarily small in all cases. A detailed analysis of ‘instability pockets’ appearing in stability diagrams of parametrically excited systems is also discussed. The alterations in ‘instability pockets’ and stability diagrams, in general, due to addition of damping is systematically studied. In particular, the results for some typical cases of Mathieu, Meissner, three-frequency Hill and Quasi-Periodic Hill equations are presented in detail. A methodology to control general nonlinear systems to desired periodic or quasi-periodic motions is also presented. The desired motion could be a periodic orbit, a quasi-periodic motion or a fixed point and does not need to be a solution of the nonlinear system. The applicability of the approach is demonstrated by controlling chaotic systems to desired motions. The controller design is achieved using a combination of a nonlinear feedforward controller and a linear feedback controller.