Evolutionary Search for Limit Cycle and Controller Design in Multivariable Nonlinear Systems

A feature of many practical control systems is a Multi-Input Multi-Output (MIMO) interactive structure with one or more gross nonlinearities. A primary controller design task in such circumstances is to predict and ensure the avoidance of limit cycling conditions followed by achieving other design objectives. This paper outlines how such a system may be investigated using the Sinusoidal Input Describing Function (SIDF) philosophy quantifying magnitude, frequency and phase of any possible limit cycle operation. While Sinusoidal Input Describing function is a suitable linearization technique in the frequency domain for assessment of stability and limit cycle operation, it can not be employed in time domain. In order to be able to incorporate the time domain requirements in an overall controller design technique, the appropriate linearization technique suggested here is the Exponential Input Describing Function (EIDF). First, an evolutionary search based on a multi-objective formulation is employed for the direct solution of the harmonic balance system matrix equation. The search is based on Multi-Objective Genetic Algorithms (MOGA) and is capable of predicting specified modes of theoretically possible limit cycle operation. Second, the design requirements in time as well as frequency domain are formulated by a set of constraint inequalities. A numerical synthesis procedure also based on Multi-Objective Genetic Algorithm is employed to adjust the initial compensator parameters to meet the imposed constraints. Robust stability and robust performance are investigated with respect to linearization uncertainty within the context of multiobjective formulation. In order to make the Genetic Algorithm (GA) search more amenable to design trade-off between different and often contradictory specifications, a weighted sum of the functions is introduced. This criterion is subsequently optimized subject to the nonlinear system dynamics and a set of design requirements. Examples of use are given to illustrate the effectiveness of the proposed approach.