Feedback Stabilization of Inverted Pendulum Models

Many mechanical systems exhibit nonlinear movement and are subject to perturbations from a desired equilibrium state. These perturbations can greatly reduce the efficiency of the systems. It is therefore desirous to analyze the asymptotic stabilizability of an equilibrium solution of nonlinear systems; an excellent method of performing these analyses is through study of Jacobian linearization’s and their properties. Two enlightening examples of nonlinear mechanical systems are the Simple Inverted Pendulum and the Inverted Pendulum on a Cart (PoC). These examples provide insight into both the feasibility and usability of Jacobian linearizations of nonlinear systems, as well as demonstrate the concepts of local stability, observability, controllability and detectability of linearized systems under varying parameters. Some examples of constant disturbances and effects are considered. The ultimate goal is to examine stabilizability, through both static and dynamic feedback controllers, of mechanical systems Chapter 1: Introduction Section 1.1: What is Feedback Control? Why Study it? Any system, be it of mechanical, electrical or biological origins, is often subject to perturbations from a desired normal state. Feedback control utilizes the current state of the system to determine an appropriate feedback to return the system to this desired state, i.e. to stabilize the system. Compensating for unwanted perturbations is necessary to ensure proper functioning of a system; in many cases the margin of error for a system is very small. Perturbations can cause severe problems with a sensitive system. In a radio antenna, for example, a few degrees off axis can result in the antenna pointing at the wrong section of sky, resulting in a complete lack of functionality. Other mechanical systems such as bipedal robots and rockets also require feedback control in order to ensure optimal efficiency. These mechanical systems all utilize the same fundamental control principles and techniques. In fact, bipedal locomotion and rockets are directly modeled off of inverted pendulum models. Feedback control methodologies are used to determine the current state of a mechanical system and induce a mechanical feedback that will return it to the desired state. This paper will be dealing solely with feedback control and stability within mechanical systems. 2 The State Space representation of a mechanical system is a methodology of writing the system equations. Definition 1.1: The State Space Representation of a linear time invariant system is: (1.1) ) ( ) ( ) ( ) ( ) ( t Cx t y t Bu t Ax dt t dx = + = In the future we will write dt dx x = . In equations (1.1): • x(t) is the system state at time t. In mechanical systems this usually refers to quantities like position and velocity. This is where the name State Space originates. • A is an n x n matrix that determines the system dynamics in the absence of any inputs (u=0) • B is an n x m matrix that determines the interaction between the inputs, u(t), and the system state x(t). • C is a p x n matrix that determines the interaction between x(t), the state space variables and y(t) the observable output.