Nonlinear Proportional and Rate Feedback Controller Design Synthesis with Experimental Verification

A new procedure for synthesis of nonlinear proportional and nonlinear rate-feedback controllers for use with unstable nonlinear systems with application to a direct drive inverted pendulum is presented. The approach is to stabilize the nonlinear system followed by generation of the corresponding describing function models at various operating regimes of interest. With the known stabilizing controller and the stabilized frequency domain models, the frequency domain models of the unstable plant are algebraically extracted. A computer-aided design technique is used, and a set of proportional plus rate feedback controllers for the set of obtained open-loop frequency domain models is designed. The table of rate feedback gains as a function of rate feedback block input amplitudes is treated as the describing function model of the unknown nonlinear rate feedback gain; describing function inversion is used to obtain the nonlinear rate feedback gain. One linear proportional controller at an arbitrary operating regime of interest is designed to force the closed-loop system comprising the linear proportional gain and the nonlinear rate feedback to mimic the desired behavior. In continuation, the describing function model of the closed-loop system is generated, and the desired open-loop frequency domain behavior is extracted. A set of proportional gains at various operating regimes is designed, followed by describing inversion to determine the nonlinear proportional gain. Finally, the closed-loop system comprising the nonlinear proportional gain and the nonlinear plant with the nonlinear rate feedback gain is tested for stability and robustness performance. The procedure is experimentally verified using a direct drive inverted pendulum, and it is demonstrated that the synthesized controller performs favorably to a linear proportional-integral-derivative (PID) controller, and its performance is competitive with two other nonlinear controllers. The primary differences between this research and prior work are that the procedure is developed for unstable systems and the developed procedure is experimentally verified.