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译文题目： Inverted Pendulum
徐飞马 学 号： 0704111030 07自动化 机电学院 陈丽换 讲师 2011 年 2月 24日
The inverted pendulum
Key words: inverted pendulum, modeling, PID controllers,
LQRcontrollers
What is an Inverted Pendulum? Remember when you were a child and you tried to balance a broom-stick or baseball bat on your index finger or the palm of your hand? You had to constantly adjust the position of your hand to keep the object upright. An Inverted Pendulum does basically the same thing. However, it is limited in that it only moves in one dimension, while your hand could move up, down, sideways, etc. Check out the video provided to see exactly how the Inverted Pendulum works. An inverted pendulum is a physical device consisting in a cylindrical bar (usually of aluminum) free to oscillate around a fixed pivot. The pivot is mounted on a carriage, which in its turn can move on a horizontal direction. The carriage is driven by a motor, which can exert on it a variable force. The bar would naturally tend to fall down from the top vertical position, which is a position of unsteady equilibrium.
The goal of the experiment is to stabilize the pendulum (bar) on the top vertical position. This is possible by exerting on the carriage through the motor a force which tends to contrast the 'free' pendulum dynamics. The correct force has to be calculated measuring the instant values of the horizontal position and the pendulum angle (obtained e.g. through two potentiometers).
The system pendulum+cart+motor can be modeled as a linear system if all the parameters are known (masses, lengths, etc.), in order to find a controller to stabilize it. If not all the parameters are known, one can however try to 'reconstruct' the system parameters using measured data on the dynamics of the pendulum.
The inverted pendulum is a traditional example (neither difficult nor trivial) of a controlled system. Thus it is used in simulations and experiments to show the performance of different controllers (e.g. PID controllers, state space controllers, fuzzy controllers....).
The Real-Time Inverted Pendulum is used as a benchmark, to test the validity and t
he performance of the software underlying the state-space controller algorithm, i.e. the used operating system. Actually the algorithm is implement form the numerical point of view as a set of mutually co-operating tasks, which are periodically activated by the kernel, and which perform different calculations. The way how these tasks are activated (e.g. the activation order) is called scheduling of the tasks. It is obvious that a correct scheduling of each task is crucial for a good performance of the controller, and hence for an effective pendulum stabilization. Thus the inverted pendulum is very useful in determining whether a particular scheduling choice is better than another one, in which cases, to which extent, and so on.
Modeling an inverted pendulum.Generally the inverted pendulum system is modeled as a linear system, and hence the modeling is valid only for small oscillations of the pendulum.
With the use of trapezoidal input membership functions and appropriate composition and inference methods, it will be shown that it is possible to obtain rule membership functions which are region-wise affine functions of the controller input variable. We propose a linear defuzzification algorithm that keeps this region-wise affine structure and yields a piece-wise affine controller. A particular and systematic parameter tuning method will be given which allows turning this controller into a variable structure-like controller. We will compare this region-wise affine controller with a Fuzzy and Variable Structure Controller through the application to an inverted pendulum control. We will begin with system design; analyzing control behavior of a two-stage inverted pendulum. We will then show how to design a fuzzy controller for the system. We will describe a control curve and how it differs from that of conventional controllers when using a fuzzy controller. Finally, we will discuss how to use this curve to define labels and membership functions for variables, as well as how to create rules for the controller.
In the formulation of any control problem there will typically be discrepancies between the actual plant and the mathematical model developed for controller design.This mismatch may be due to unmodelled dynamics, variation in system parameters or the approximation of complex plant behavior by a straightforward model.The engineer
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