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

Abstract

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 followed by experimental verification. The non-linear PID design is compared with two other alternative designs. The experimental results indicate that the proposed approach is a viable and effective non-linear controller synthesis technique for use with unstable non-linear systems.