1、PDF外文：http:/ 第 1 页 Control of an Inverted Pendulum Johnny Lam Abstract: The balancing of an inverted pendulum by moving a cart along a horizontal track is a classic problem in the area of control. This paper will describe two methods to swing a pendulum attached to a cart from an initial
2、 downwards position to an upright position and maintain that state. A nonlinear heuristic controller and an energy controller have been implemented in order to swing the pendulum to an upright position. After the pendulum is swung up, a linear quadratic regulator state feedback optimal controller ha
3、s been implemented to maintain the balanced state. The heuristic controller outputs a repetitive signal at the appropriate moment and is finely tuned for the specific experimental setup. The energy controller adds an appropriate amount of energy into the pendulum system in order to achieve a desired
4、 energy state. The optimal state feedback controller is a stabilizing controller based on a model linearized around the upright position and is effective when the cart-pendulum system is near the balanced state. The pendulum has been swung from the downwards position to the upright position using bo
5、th methods and the experimental results are reported. 1. INTRODUCTION The inverted pendulum system is a standard problem in the area of control systems. They are often useful to demonstrate concepts in linear control such as the stabilization of unstable systems. Since the system is inherently
6、 nonlinear, it has also been useful in illustrating some of the ideas in nonlinear control. In this system, an inverted pendulum is attached to a cart equipped with a motor that drives it along a horizontal track. The user is able to dictate the position and velocity of the cart through the motor an
7、d the track restricts the cart to movement in the horizontal direction. Sensors are attached to the cart and the pivot in order to measure the cart position and pendulum joint angle, respectively. Measurements are taken with a quadrature encoder connected to a MultiQ-3 general purpose data acquisiti
8、on and control board. Matlab/Simulink is used to implement the controller and analyze data. The inverted pendulum system inherently has two equilibria, one of which is stable while the other is unstable. The stable equilibrium corresponds to a state in which the pendulum is pointing downwards.
9、 In the absence of any control force, the system will naturally return to this state. The stable equilibrium requires no control input to be achieved and, thus, is uninteresting from a control perspective. The unstable equilibrium corresponds to a state in which the pendulum points strictly upwards
10、and, thus, requires a control force to maintain this position. The basic control objective of the inverted pendulum problem is to maintain the unstable equilibrium position when the pendulum initially starts in an upright position. The control objective for this project will focus on starting from t
11、he stable equilibrium position (pendulum pointing down), swinging it up to the unstable equilibrium position (pendulum upright), and maintaining this state. 第 2 页 2. MODELLING A schematic of the inverted pendulum is shown in Figure 1. Figure 1. Inverted Pendulum Setup A cart
12、 equipped with a motor provides horizontal motion of the cart while cart position, p, and joint angle, , measurements are taken via a quadrature encoder. By applying the law of dynamics on the inverted pendulum system, the equations of motion are where mc is the car
13、t mass, mp is the pendulum mass, I is the rotational inertia, l is the half-length of the pendulum, R is the motor armature resistance, r is the motor pinion radius, Km is the motor torque constant, and Kg is the gearbox ratio. Also, for simplicity, and note that the relationship between force
14、, F, and voltage, V, for the motor is: 第 3 页 Let the state vector be defined as: Finally, we linearize the system about the unstable equilibrium(0 0 0 0)T.Note that = 0 corresponds to the pendulum being in the upright position. The linearization of the cart-pendulum system around the upright position is: Where Finally, by substituting the parameter values that correspond to the experimental setup:
