The control of bifurcations with engineering applications

Abstract

This dissertation develops a general method for the control of the class of local bifurcations of engineering interest, including saddle-node, transcritical, pitchfork and Hopf bifurcations. The method is based on transforming a general affine single input control system into quadratic normal form through coordinate transformations and feedback. (The quadratic normal form includes the quadratic order Poincare normal form of the uncontrolled system as a natural subset.) Then, linear and quadratic state feedback control laws are developed which control the shape of the center manifold of the transformed system. It is shown that control of the center manifold allows the quadratic and cubic order terms of the center dynamics to be influenced to produce nonlinear stability. Specific matrix operations necessary to transform a general affine single input control system into quadratic normal form are provided. Specific control laws to stabilize a general system experiencing a linearly unstable saddle-node, transcritical, pitchfork or Hopf bifurcation are also provided