Optimal Controller Design for Inverted Pendulum System: An Experimental Study

Abstract

The Cart Inverted Pendulum System (CIPS) has been considered among the most classical and difficult problem in the field of control engineering. The Inverted Pendulum is considered among the typical representative of a class of under actuated, non-minimal system with non-linear dynamics. The aim of this study is to stabilize the Inverted Pendulum such that position of the cart on 1 meter track is controlled quickly and accurately so that pendulum is always maintained erected in its upright (inverted) position. This thesis begins with the explanation of CIPS together with the hardware setup used for research, its state space dynamics and transfer function models after linearizing it. Since, Inverted Pendulum is inherently unstable i.e. if it is left without a stabilizing controller it will not be able to remain in an upright position when disturbed. So, a systematic iterative method for the state feedback design by choosing weighting matrices key to Linear Quadratic Regulator (LQR) design is presented assuming all the states to be available at the output. After that, Kalman Filter, which is an optimal Observer has been designed to estimate all the four states considering process and measurement noises in the system. Then, a Full State Feedback Controller i.e. Linear Quadratic Gaussian (LQG) compensator has been designed. The compensator aims at providing a proper control input that provides a desired output in terms of the Pendulum Angle and Cart Position. Simulation and Experimental study has been carried out to demonstrate the effectiveness of the proposed approach in meeting the desired specifications. Lastly, Loop Transfer Recovery (LTR) analysis has been performed depending on the trade-off between noise suppression and system robustness for suitably selecting the tuning parameter for Observer design.