A general approach for constructing the limit cycle loci of multiple-nonlinearity systems

Abstract

E. I. Jury, Theory and Application of the 2-Transform Method. New York: E. I. Jury and B. W. Lee, "The absolute stability of systems with many Huntington, 1964. A. N. Michel and R. K. Miller, "Stability analysis of discretetime interconnected nonlinearities." Automat. Remote Contr., vol. 26, pp. 943-961, 1965. systems via computer-generated Lyapunov functions uzith application to digital fdters," IEEE Trans. Circuits Syst., vol. CAS-32, pp. 737-753, Aug. 1985. M. S. Mousa, R. K. Miller, and A. N. Michel, "Stability analysis of hybrid composite dynamical systems: Descriptions involving operators and difference equations," . WE1A6stract-This note presents a widely convergent algorithm for finding a limit cycle of systems with multiple nonlinearities. A systematic approach is proposed for constructing the limit cycle loci on the parameter planes. The merits of this approach lie in its simplicity, generality, and ability to provide deeper insight into parameter influences on the limit cycle. Besides, stability of the limit cycle can be predicted with a minimum amount of computations.

Limit Cycle Predictions of Nonlinear Multivariable Feedback Control Systems with Large Transportation Lags

“…These methods are based upon the graphical or numerical solutions of the linearized harmonic-balance equations [1][2][3][4][5][6][7][8][9][10]. It has been shown that for multivariable systems, over arbitrary ranges of amplitudes (A i ), frequency (ω) and phases (θ i ), an infinite number of possible solutions may exist.…”

“…It has been shown that for multivariable systems, over arbitrary ranges of amplitudes (A i ), frequency (ω) and phases (θ i ), an infinite number of possible solutions may exist. Gray has proposed a sequential computational procedure to seek the solutions for only specified ranges of discrete values of A i , ω, and θ i , these specified ranges are determined by use of the Nyquist or inverse Nyquist plots [4,5]. Although the aforementioned methods are powerful, large computational efforts are usually expected.…”

Construction of Central Pattern Generator for Quadruped Locomotion Control

“…Such systems present interesting characteristics like limit cycle stability, synchronization, and the possibility to be entrained and modulated by external signals. Based on this cognition, many methods for constructing the CPG have been proposed [10]- [15]. So far, there are two methods for building the CPG to generating an arbitrary periodic signal.…”

“…So far, there are two methods for building the CPG to generating an arbitrary periodic signal. The first one is dynamic learning based on a kind of dynamical Fourier series representation [10], [11], [14], [15]. Another is that the trajectory is embedded into a canonical oscillator [12], [13].…”

Robust prevention of limit cycle for nonlinear control systems with parametric uncertainties both in the linear plant and nonlinearity

“…However, it requires the plotting of a lot of polar plots and describing function curves depending on the uncertain plant parameters when uncertainties are present in the plant parameters or when the considered system contains two or more inseparable nonlinearities. Moreover, a general approach for constructing the limit cycle loci of multi-nonlinearity systems is proposed in [10]. Further, the analysis of nonlinear control systems using the stability equation method is developed to establish the necessary condition for the occurrence of limit cycle of nonlinear control systems with a known mathematical structure of the linear subsystem in an elegant way.…”