Swing-up and Stabilization of a Single Inverted Pendulum: Real-Time Implementation

The single inverted pendulum (SIP) system is a classic example of a nonlinear underactuated system. Despite its simple structure, it is among the most difficult systems to control and is considered as one of the most popular benchmarks of nonlinear control theory. In the past fifty years many nonlinear methods have been proposed for the swing-up and stabilization of a self-erecting inverted pendulum, however, most of these techniques are too complex and impractical for real-time implementation.;In the first part of this dissertation, the successful real-time implementation of a nonlinear controller for the stabilization of the pendulum is discussed. The controller is based on the power series approximation to the Hamilton Jacobi Bellman (HJB) equation. The derivation of the controller is based on work that can be found in the literature, but the controller has not been used for the stabilization of an inverted pendulum before. It performs similarly to the traditional linear quadratic regulator (LQR), but has some important advantages. First, the method can stabilize the pendulum for a wider range of initial starting angle. Additionally, it can also be used with state dependent weighting matrices, Q and R, whereas the LQR problem can only handle constant values for these matrices. The use of state-dependent weighting matrices for the stabilization of an inverted pendulum in real-time has been discussed in the literature before, but only with controls that use a State Dependent Riccati Equation (SDRE) approach. The benefit of the control presented in this thesis over the SDRE controls is that it is computationally less intense and does not require the solution of complicated matrix equations at every time step. However, the control method presented cannot be used to swing-up the pendulum whereas some of the controls using the online solution of the SDRE can.;The second part of the dissertation focuses on the swing-up of the inverted pendulum. The most common and efficient method for the swing-up of the pendulum uses an energy based approach. This method was originally proposed by Astrom and Furuta in 1996, and it was first implemented for the swing-up of a rotary pendulum in 1999. Later, the controller was modified and implemented on a cart pendulum system taking the finite length of the track into account. However, most of the existing swing-up controllers are based on a simplified model for the SIP system, and the e.ects of friction are frequently disregarded. In this thesis, we present a new energy-based swing-up controller that was derived using a more complex dynamical model for the SIP system. We also consider the e.ects of viscous damping, and incorporate physical restrictions like the maximum deliverable voltage by the amplifier, the capacity of the DC motor that drives the cart, and the finite track length.