Linear and Nonlinear Multivariable Feedback Control

4Forced oscillation and generalized frequency response characteristics of nonlinear MIMO systems

4.1 INTRODUCTION

The problem of the investigation of forced oscillations in nonlinear systems is very complicated and interesting, owing to the diversity of forms of possible dynamical responses. Thus, if we apply one-frequency sinusoidal signals to the inputs of a limit cycling system, the latter may exhibit a complex motion, which is not strictly periodical and, in many respects, is like the beating mode in a linear oscillatory system driven by a sinusoidal signal. Under some conditions, the phenomenon of capturing (also called entrainment or forced synchronization) can occur in the system, when the external signal enforces its own frequency on the system, suppressing completely the existing limit cycle. Besides, in limit cycling nonlinear systems, the possibility of synchronization at the frequency of one of the subharmonics or higher harmonics is not excluded (Popov 1973; Tsypkin 1974).

Note that so far, in the classical control theory, only the problem of the investigation of one-frequency forced oscillation under the above-mentioned capturing condition has adequately been solved or brought in any way to real engineering applications. At that, if exact methods exist for systems with relay nonlinearities (Thaler and Pastel 1962; Tsypkin 1974), in the case of general nonlinear characteristics, the numerical analysis of systems is mainly based on the describing function method. Consider ...