Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 816707, 17 pages
Research Article

Limit Cycle Prediction Based on Evolutionary Multiobjective Formulation

1School of Computing Research, Liverpool Hope University, L16 9JD Liverpool, UK
2Department of Computer Science and Engineering, School of Engineering, Shiraz University, 71348-51154 Shiraz, Iran

Received 19 November 2008; Revised 21 December 2008; Accepted 29 December 2008

Academic Editor: José Roberto Castilho Piqueira

Copyright © 2009 M. Katebi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is concerned with an evolutionary search for limit cycle operation in a class of nonlinear systems. In the first part, single input single output (SISO) systems are investigated and sinusoidal input describing function (SIDF) is extended to those cases where the key assumption in its derivation is violated. Describing function matrix (DMF) is employed to take into account the effects of higher harmonic signals and enhance the accuracy of predicting limit cycle operation. In the second part, SIDF is extended to the class of nonlinear multiinput multioutput (MIMO) systems containing separable nonlinear elements of any general form. In both cases linearized harmonic balance equations are derived and the search for a limit cycle is formulated as a multiobjective problem. Multiobjective genetic algorithm (MOGA) is utilized to search the space of parameters of theoretically possible limit cycle operations. Case studies are presented to demonstrate the effectiveness of the proposed approach.