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 opera...

Nonlinear Proportional and Rate Feedback Controller Design Synthesis with Experimental Verification

A new systematic procedure for synthesis of nonlinear lead-lag controllers for highly nonlinear unstable plants is developed. The procedure is to stabilize the unstable plant followed by generation of sinusoidal-input describing function (SIDF) models at various operating regimes of interest. The unstable plant frequency domain behavior is extracted, and a set of linear lead-lag controllers are designed that would force the open-loop behavior of the system (comprised of the lead-lag controller and the nonlinear plant) and mimic that of desired. Finally, a describing function inversion is used, and the nonlinear functions describing the values of the lead-lag coefficients as a function of the error signal are obtained. Experimental results are presented for verification, and bounded-input bounded-output stability is demonstrated by successful generation of the describing function models of the closed-loop system comprised of the synthesized nonlinear lead-lag controller and the nonlinear plant.

Nonlinear Lead-Lag Controller Synthesis for Unstable Nonlinear Systems

Numerical Solution to the Inverse Sinusoidal Input Describing Function

In this chapter, a new nonlinear PID controller synthesis approach using describing function inversion technique for unstable systems that a mathematical model may not be available is outlined; the approach is applied to a highly nonlinear inverted pendulum experimental setup. The procedure involves stabilization of the unstable system followed by generation of the describing function models of the stabilized closed-loop system. Then, the corresponding unstable open-loop frequency domain models at various operating regimes are extracted. A controller at nominal conditions is designed, followed by obtaining the corresponding desired open-loop frequency domain model. A set of controllers that force the open-loop behavior of the system mimic that of desired at various operating regimes is designed by optimization. Finally, the controller gains are inverted using a describing function inversion technique followed by experimental verification. The nonlinear PID design is compared with two other alternative designs. The experimental results indicate that the proposed approach is a viable and effective nonlinear controller synthesis technique for use with unstable nonlinear systems.

Experimental Study: Unstable SISO Systems

This paper demonstrates a new non-linear proportional–integral–derivative (PID) controller synthesis approach using a describing function inversion technique for unstable systems where a mathematical model may not be available. The approach is applied to an inverted pendulum experimental set-up whose dynamic behaviour is very sensitive to the amplitude level of excitation. The procedure involves stabilization of the unstable system followed by generation of the describing function models of the stabilized closed-loop system. Then, the corresponding unstable open-loop frequency domain models at various operating regimes are extracted. A controller at nominal conditions is designed, followed by obtaining the corresponding desired open-loop frequency domain model. A set of controllers that force the open-loop behaviour of the system mimic, which is desired at various operating regimes, is designed by optimization. Finally, the controller gains are inverted using a describing function inversion technique foll...

Non-linear proportional–integral–derivative synthesis for unstable non-linear systems using describing function inversion with experimental verification:

S C Li

Papers

Non-linear proportional–integral–derivative synthesis for unstable non-linear systems using describing function inversion with experimental verification: