1 201134 information technologiesand control

Key Words: Cart – pendulum system; inverted pendulum; swing upcontrol; local stabilization.

Abstract. This paper considers the problem of inverted pendulumcontrol. Position control of the inverted pendulum in upright equilib-rium state can be divided into two tasks: swing up control of thependulum in upright position and local stabilization around the equi-librium point. Two main approaches for solving the first problem arepresented: the energy approach and the speed gradient approach. Atthe same time, many modern methods for control of the invertedpendulum are also introduced. The presented methods solve theinverted pendulum swing up problem and ensure its global stabiliza-tion.

1. Introduction

Laboratory exercises play an important role in controltheory and control engineering courses. The relation between thetheoretical knowledge obtained in theoretical control coursesand its practical application and implementation in laboratoryexperiments is a major part of contemporary education in thefield. Recently, quite popular have become laboratories basedon physical models of real devices and processes which arecontrolled by microprocessor regulators and programmable logiccontrollers (PLC) [24,26]. These are the so – called open labo-ratories, where the same equipment is used for carrying differentexperiments. A single well equipped laboratory supports most ofthe courses in dynamical systems analysis and control systemsdesign. Laboratory models in such laboratories use PCs withstandard input/output interface cards for data acquisition andcontrol.

A major part of these laboratory models are built to rep-resent certain characteristics and properties of the existing in-dustrial processes and systems. However, there is a large groupof models not having direct link to real systems, but neverthe-less serve as a test bed for a variety of control algorithms. Thetypical example of such a device is the physical model of theinverted pendulum. The cart-pendulum system is one of themost popular laboratory models for practical implementationand demonstration of control systems. The inverted pendulum isa classical electromechanical device for testing some compli-cated system analysis and design methods. The purpose forexploring the cart – pendulum system is to represent the diffi-culties to control an inherently unstable plant, containing numer-ous nonlinearities, characterized by many equilibrium points andserving as an example for the fundamental structural limitationsof using the feedback connection.

The cart-pendulum system in figure 1 consists of thefollowing parts [25]: i) mechanical part consisting of a cartdriven by a DC motor and a two-pole pendulum attached to thecart, ii) I/O board with built in ADC and DAC, iii) power board withbuilt – in amplifier, iv) personal computer with the appropriatesoftware tools for connecting to the hardware. The cart can move

along a horizontal rail and the pendulum is able to rotate freelyin a vertical plane parallel to the rail. In order to swing up orbalance the pendulum around its equilibrium points, the cart hasto move back and forth on the rail by a plane DC motor. Theposition of the cart on the rail and the angle of the pendulum aremeasured by two optical encoders.

The goal of the control algorithm is by using severaloscillations with increasing amplitude to bring the pendulumpoles around the upper equilibrium position without letting theangle and velocity become too large. After reaching this statethe pendulum is stabilized there while allowing to move the cartalong the rail.

2. Inverted Pendulum System ModelingThere are several cases for describing the inverted pen-

dulum control problem in the control literature: i) inverted pen-dulum with fixed end point (pendulum of Furuta)[11,12,13,20,29,38,40], ii) inverted pendulum with moving cart[8,14,23,25,27,28,33, 34,42,43,44,46], iii) double invertedpendulum with fixed end point (acrobot) [1,17,39], iv) doubleinverted pendulum with moving cart [15,18,22,47], v) tripleinverted pendulum with fixed end point [4,19,35]. The mostpopular case is the inverted pendulum with or without movingcart.

The model of the cart – pendulum system can be derivedbasically in two different ways: i) by using the formulation ofNewton – Euler that leads to effective computing of the controllaw in real time [33,21,30,17,28] and ii) by using the formu-lation of Lagrange that is based on algebraic calculations overenergy quantities using generalized coordinates and generalizedforces [33,16]. Let us consider the free body diagram of thecart-pendulum system shown in figure 2. The cart moves alonga horizontal rail and the pendulum rotates in a vertical plane

Inverted Pendulum Control: an OverviewK. Perev

Figure 1. The inverted pendulum laboratory model

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