Modeling, Simulation and Real-Time Control of Underactuated Systems: A Cart-Pendulum Example

Abstract

The present work deals with the modeling, simulation and real-time control of the cart-pendulum system, which is underactuated in nature. An effort is made, to control unstable nature of the cart-pendulum system. The underactuated cart-pendulum system is a typical benchmark problem in the control field, having one control input for two degrees-of-freedom (DOFs) system. It has nonlinear structure, which can be used to validate different nonlinear and linear controllers and has wide range of real-time (realistic) applications like rockets propeller, tank missile launcher, self-balancing robot, stabilization of ships, design of earthquake resistant buildings, etc. As there are broad range of applications, the study of the controllers on the cart-pendulum system should be done on the real-time basis, for significant results. Generally, the cart-pendulum system deals with two type of problems− stabilization of pendulum at unstable equilibrium point and swing-up of the pendulum from hanging position to upward position by moving cart under the restricted track length. The literature study present in this work reveals several types of controllers developed by different researchers to address these problems. The implementation of the many controllers on the real-time basis have not been exploited well in the literature. The present thesis is an attempt to use modern control theory controllers to address the two control problems, swing-up and stabilization in the simulation tasks along with verification on the real-time experimental setup. The simulation tasks are performed in the MATLAB environment and experiments are performed on the Googoltech Linear Inverted Pendulum (GLIP) setup, which works in the SIMULINK environment. To obtain the accurate dynamic model of the system, actuator dynamics is considered. The mathematical model is developed by using Euler-Lagrange approach. In this dissertation, modern control theory is used to address the stabilization problem of the cart-pendulum system. The main aim of this task is to investigate the performance of two different controllers in stabilizing the inverted pendulum at the unstable equilibrium point. This regulation problem is addressed by developing the Pole Placement Controller (PPC) and Linear Quadratic Regulator Controller (LQRC). The analytical results are found in close agreement with the experimental results. Another significant contribution of this work is the swing-up control of the cart-pendulum system under the restricted track length. In this task, the performance of two control strategies is investigated − first to swing-up the pendulum to near unstable equilibrium region and second to stabilize the pendulum at unstable equilibrium point. The swing-up problem is addressed by using energy controller in which cart is accelerated by providing a force to the cart with a AC servo motor with the help of timing pulley arrangement. The initial velocity of the cart is taken into account to confirm swing-up in the restricted track length. The cart displacement in the restricted track length is verified by simulation and experimental test-run. The second control strategy is addressed using Pole Placement Controller (PPC) and LQR Controller (LQRC), which was used in the first stage of this dissertation. The analytically and experimental results show that, the energy controller along with constrained on the linear and angular velocities is able to swing-up the pendulum from hanging position to unstable equilibrium position under the restricted track length. In order to demonstrate the effect of both the stabilizing controllers on the performance of the system, comparison of the experimental results is carried out. It is demonstrated experimentally that LQR controller outperforms the Pole Placement controller, in terms of reduction in the oscillations of the inverted pendulums, as well as the magnitude of maximum control input. Further, robustness of the closed-loop system is investigated by providing external disturbances.