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Unformatted text preview: CARLETON UNIVERSITY Department of Systems and Computer Engineering SYSC 3600 Lab #3: Control of an Inverted Pendulum Before coming to Lab #3: • Read and understand the below pre-lab for Lab #3. • Read the “S IMULINK Extras” in Appendix A. • Familiarize yourself with the S IMULINK models given in Appendix B. Instructions: Read through and complete the entire lab. Your report should address the points made in the Report suggestion boxes and demonstrate your understanding of the material. 1 Purpose The purpose of this lab is: (a) To look at the inverted pendulum problem, which is an unstable, non-linear system. (b) To study a proportional-plus-derivative (PD) controller design using S IMULINK . (c) To study the validity of a linear approximation for a non-linear system . 2 Pre-Lab 2.1 Introduction A useful design problem in control theory is known as the inverted pendulum problem. The basic problem is to balance a long object on end in a vertical position. One example would be to balance an inverted broom on your hand (where the brain acts as the control system to do the balancing). Another example is the attitude control used to balance a space booster in a vertical position during takeoff (where the space booster is balanced on top of the thrust generated by the booster). The difficulty with an inverted pendulum is that it is naturally an unstable and non-linear system. In this lab we will model the inverted pendulum illustrated in Fig. 1 and simulate this system when a proportional-plus-derivative (PD) controller is used to attempt to keep the inverted pendulum balanced 1 . The inverted pendulum is attached to a cart with a bracket that allows the pendulum to swing only to the left or to the right. Similarly, the cart can move only to the left or to the right. The object is to design a control system that keeps the inverted pendulum in a vertical position by applying a force f to the cart under the pendulum. Note that for the inverted pendulum we are assuming no friction for the cart or the pivot of the pendulum to simplify the problem. Also, we are assuming no extra forces on the pendulum, such as air resistance or wind, with the exception of gravity itself. Small external forces, such as air resistance or small amounts of wind, would ultimately be compensated for by the PD controller which is trying to keep the inverted pendulum vertical. 1 Design problem taken from Ogata, System Dynamics , Prentice Hall, 1978, pp. 531–536. L3.1 SYSC 3600: Lab #3 2 PRE-LAB Figure 1: Inverted pendulum system. Figure 2: Block diagram of inverted pendulum system controlled with a PD controller. For the purpose of this lab, we will take that M = 1000 kg, m = 200 kg, and ‘ = 10 m. We wish to design a suitable controller that generates a force f such that the overall system has a damping factor of ζ = 0 . 7 and an undamped natural frequency of ω n = 0 . 5 rad/sec. Choosing a PD controller for this problem, the overall system with the controller has a block diagram as shown in Fig. 2.problem, the overall system with the controller has a block diagram as shown in Fig....
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- Winter '08
- John Bryant
- Cutoff frequency, PD controller, Inverted pendulum, S IMULINK