A Generalized Newton Raphson Method for Limit Cycle Analysis of Nonlinear Control Systems

A numerical method is presented for the limit cycle analysis of multiloop nonlinear control systems with multiple nonlinearities. Describing functions are used to model the first harmonic gains of the nonlinearities. Existence of a limit cycle is sought by driving the least damped eigenvalue to the imaginary axis. The evolution of the limit cycle is studied next as a function of a critical system-parameter. It is shown that by defining a suitable error function it is possible to use both the eigenvalue as well as the eigenvector sensitivities to formulate a generalized Newton Raphson method to solve simultaneously for the updates of state variable amplitudes in a minimum norm sense. A case study involving a fifth order control system with three nonlinearities is included to illustrate the application of the method.