Multi-Objective Optimal Feedback Controls for Under-Actuated Dynamical System

Abstract

This paper presents a study of optimal control design for a single-inverted pendulum (SIP) system with the multi-objective particle swarm optimization (MOPSO) algorithm. The proportional derivative (PD) control algorithm is utilized to control the system. Since the SIP system is nonlinear and the output (the pendulum angle) cannot be directly controlled (it is under-actuated), the PD control gains are not tuned with classical approaches. In this work, the MOPSO method is used to obtain the best PD gains. The use of multi-objective optimization algorithm allows the control design of the system without the need of linearization, which is not provided by using classical methods. The multi-objective optimal control design of the nonlinear system involves four design parameters (PD gains) and six objective functions (time-domain performance indices). The Hausdorff distances of consecutive Pareto sets, obtained in the MOPSO iterations, are computed to check the convergence of the MOPSO algorithm. The MOPSO algorithm finds the Pareto set and the Pareto front efficiently. Numerical simulations and experiments of the rotary inverted pendulum system are done to verify this design technique. Numerical and experimental results show that the multi-objective optimal controls offer a wide range of choices including the ones that have comparable performances to the linear quadratic regulator (LQR) control.