毕业设计（论文）倒立摆的中英文翻译--The Inverted Pendulum System.doc

The Inverted Pendulum System The inverted pendulum system is a popular demonstration of using feedback control to stabilize an open-loop unstable system. The first solution to this problem was described by Roberge [1] in his aptly named thesis, \The Mechanical Seal. Subsequently, it has been used in many books and papers as an example of an unstable system. Siebert [2, pages 177{182] does a complete analysis of this system using the Routh Criterion, by multiplying out the characteristic equation as a polynomial of s and studying the coefficients. Although correct, this approach is unnecessarily abstruse. This system is the ideal root-locus analysis example. Figure 1: Geometry of the inverted pendulum system Consider the inverted pendulum system in Figure 1. At a pendulum angle of from vertical, gravity produces an angular acceleration equal to, and a cart acceleration of produces an angular acceleration of Writing these accelerations as an equation of motion, linearizing it, and taking its Laplace Transform, we produce the plant transfer function G(s), as follows: where the time constant is defined as This transfer function has a pole in the right half-plane, which is consistent with our expectation of an unstable system. We start the feedback design by driving the cart with a motor with transfer function M(s) and driving the motor with a voltage proportional to the angle . Including the familiar motor transfer function Figure 2: Root-locus plot of pendulum and motor, L(s) = M(s)G(s) with the plant G(s), we get a root locus with one pole that stays in the right half-plane. Using normalized numbers, we get the root-locus plot as is seen in Figure 2. In order to stabilize the system, we need to get rid of the remaining zero at the origin so that the locus from the plant pole on the positive real axis moves into the left half-plane. Thus our compensator must include a pole at the origin. However, we should balance the added compensator pole with an added zero, so th