Bifurcation characteristics of nonlinear systems under conventional pid control

Abstract

A bifurcation approach to nonlinear systems stabilized by a conventional PID controller is presented to extend the linear control theories and complement the state-space numerical methods. The types of bifurcation and the nature of the new solutions that are created by individual modes in the closed-loop system are revealed by several theorems. Numerical routines initiated by the local bifurcation analyses are presented to trace all solution branches in the gain space and to locate the region in which the set-point is globally stable. A substrate inhibition model of bioreactors, used to display the techniques, exemplifies the richness of dynamic behaviors that can be instigated by a simple controller by exhibiting multiple equilibrium points, limit-cycles, tori, and chaotic strange attractors in the closed-loop system.