Optimal Fractional Order Proportional—Integral—Differential Controller for Inverted Pendulum with Reduced Order Linear Quadratic Regulator

Abstract

The objective of this chapter is to present an optimal Fractional Order Proportional—Integral-Differential (FOPID) controller based upon Reduced Linear Quadratic Regulator (RLQR) using Particle Swarm Optimization (PSO) algorithm and compared with PID controller. The controllers are applied to Inverted Pendulum (IP) system which is one of the most exciting problems in dynamics and control theory. The FOPID or PID controller with a feed-forward gain is responsible for stabilizing the cart position and the RLQR controller is responsible for swinging up the pendulum angle. FOPID controller is the recent advances improvement controller of a conventional classical PID controller. Fractional-order calculus deals with non-integer order systems. It is the same as the traditional calculus but with a much wider applicability. Fractional Calculus is used widely in the last two decades and applied in different fields in the control area. FOPID controller achieves great success because of its effectiveness on the dynamic of the systems. Designing FOPID controller is more flexible than the standard PID controller because they have five parameters with two parameters over the standard PID controller. The Linear Quadratic Regulator (LQR) is an important approach in the optimal control theory. The optimal LQR needs tedious tuning effort in the context of good results. Moreover, LQR has many coefficients matrices which are designer dependent. These difficulties are talked by introducing RLQR. RLQR has an advantage which allows for the optimization technique to tune fewer parameters than classical LQR controller. Moreover, all coefficients matrices that are designer dependent are reformulated to be included into the optimization process. Tuning the controllers’ gains is one of the most crucial challenges that face FOPID application. Thanks to the Metaheuristic Optimization Techniques (MOTs) which solves this dilemma. PSO technique is one of the most widely used MOTs. PSO is used for the optimal tuning of the FOPID controller and RLQR parameters. The control problem is formulated to attain the combined FOPID controllers’ gains with a feed forward gain and RLQR into a multi-dimensions control problem. The objective function is designed to be multi-objective by considering the minimum settling time, rise time, undershoot and overshoot for both the cart position and the pendulum angle. It is evident from the simulation results, the effectiveness of the proposed design approach. The obtained results are very promising. The design procedures are presented step by step. The robustness of the proposed controllers is tested for internal and external large and fast disturbances.