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

Tain-Sou Tsay

doi:10.1155/2011/169848

Abstract

A practical method is developed for limit‐cycle predictions in the nonlinear multivariable feedback control systems with large transportation lags. All nonlinear elements considered are linear independent. It needs only to check maximal or minimal frequency points of root loci of equivalent gains for finding a stable limit‐cycle. This reduces the computation effort dramatically. The information for stable limit‐cycle checking can be shown in the parameter plane also. Sinusoidal input describing functions with fundamental components are used to find equivalent gains of nonlinearities. The proposed method is illustrated by a simple numerical example and applied to one 2 × 2 and two 3 × 3 complicated nonlinear multivariable feedback control systems. Considered systems have large transportation lags. Digital simulation verifications give calculated results provide accurate limit cycle predictions of considered systems. Comparisons are made also with other methods in the current literature.

Highlights

For nonlinear multivariable systems the Nyquist, inverse Nyquist and numerical optimization, methods are usually used to predict the existence of limit cycles
Real and imaginary parts of the characteristic equation are used as two simultaneous equations to find the solution of the limit cycle for single-input singleoutput (SISO) nonlinear feedback control systems [11,12,13,14,15,16,17]
Six criteria for finding a stable limit cycle have been developed for nonlinear multivariable feedback control systems with single- and double-valued nonlinearities

Summary

Introduction

For nonlinear multivariable systems the Nyquist, inverse Nyquist and numerical optimization, methods are usually used to predict the existence of limit cycles 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]. This merit of the work rather than the previous work [18] is the six criteria are applied to check maximal frequency (ωmax) or minimal frequency (ωmin) points of root loci only for finding a stable limit cycle It can reduce the computation effort dramatically. Six criteria for finding unique solution are developed; (2) in Section 3, the proposed method are applied to one 2 × 2 and two 3 × 3 complicated nonlinear multivariable feedback control systems. Comparisons are made with other methods in the current literature

The Basic Approach

Methods

Numerical Examples

Discussions

Findings

Conclusions