Ball & Beam Control System - Rough Draft

AENG-556 MODERN CONTROL TERM PROJECT

CONTROL OF BALL ON A ROTATING BEAM

AKINO JOHNKENNEDY

TABLE OF CONTENTS1. ABSTRACT2. INTRODUCTION 3. PROBLEM DESCRIPTION4. SYSTEM PARAMETERS5. MODELING6. EIGEN VALUES AND EIGEN VECTORS7. MODAL ANALYSIS8. LYAPUNOV STABILITY ANALYSIS9. CONDITION NUMBER10. CONTROLLABILITY AND OBSERVABILITY11. BALL AND BEAM CONTROL THEORY 12. DYNAMIC SHAPING13. SERVOMECHANISM FULL STATE FEEDBACK POLE PLACEMENT DESIGN14. OBSERVER BASED COMPENSATOR DESIGN15. CONCLUSION

APPENDIXMATLAB CODE

1. ABSTRACT

The ball and beam system is a very simple and powerful control system problem. The easy construction of this system combined with its challenging control design requirement makes it one of the most interesting models. The model contains a horizontal beam which can pivot about one end; a servo motor whose shaft is connected to the other end of the beam; and a ball that can freely roll on top of the beam. 2. INTRODUCTION

The ultimate goal of this project is to develop a control system for the ball and beam control system. In this project, a ball is free to roll along the track of a beam. The control problem involves moving the ball from one position on the track to another by controlling the beam angle. The control job is to automatically regulate the position of the ball on the beam byChanging the angle of the beam. This is a difficult control task because the ball does not stay in one place on the beam but moves with acceleration that is approximately proportional to the tilt of the beam. In control terminology the system is open loop unstable because the system output (the ball position) increases without limit for a fixed input (beam angle). Feedback control must be used to stabilize the system and to keep the ball in a desired position on the beam. The control system is simulated in Matlab where ball and beam system is modeled. Data from the Matlab is analyzed.

3. PROBLEM DESCRIPTION

The ball is modeled by a mass, m, a radius, r, and a mass moment of inertia, Jb. The beam is modeled by a mass moment of inertia, J. A torque, is applied at the beam center to change the angle of the beam to adjust the position of the ball. The major control objective is to position the ball anywhere along the beam. This is a difficult problem because the only actuator available is the beam itself and its rotation. Moreover, this is a nonlinear and unstable system.

The inherent dynamics of the ball and beam system are described by a set of nonlinear differential equations. Instead of designing a single nonlinear control law, a collection of linear controls can be designed about various trim conditions. This project focuses on the control of the ball position about a trim condition.

4. SYSTEM PARAMETERS

Physical parameters of the ball:Mass of the ball, mball:0.065 kgRadius of the ball, rball:0.025 mBall mass moment of Inertia, Jball:(2/5)*mball*(rball)2 kg-m2Physical parameters of the beam:Mass of the beam, mbeam:0.65 kgLength of the beam, lbeam:0.425 mBeam mass moment of Inertia, Jbeam: (1/12)*mbeam*(lbeam)2 kg-m2

5. MODELING:

The ball and beam system modeling involves two steps 1. Developing a nonlinear mathematical model and2. Linearizing the non linear model

Analysis of motion of ball:

xyax

Analysis of motion of beam (in xyz):

Therefore the nonlinear model of the system is:

And its state space elements are:

Nonlinear State Space Model:

Initial conditions for Steady, level beam; constant ball position

Obtaining the A, B, C, D matrices:

6. EIGEN VALUES AND EIGEN VECTORS:

The Eigen values of the system areEig =

Plot Of Eigen Values

Right Eigen Vector of the system is

R_Eig =

Left Eigen Vector of the system is

L_Eig =

7. MODAL ANALYSIS: In this, the function computes the mode sensitivities for each state associated with the matrix A. This gives an indication of which states influence each mode. This function also computes a set of metrics for each mode. Note that frequency components of a mode exist if eigenvectors have complex conjugate pairs.so, the mode sensitivity matrix and mode metrics are as follows:

Msens =

0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500

Mmetric =

1.0e+15 *

0.0000 -1.0406 -1.0406 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.7532 -0.7532 -0.0000

8. LYAPUNOV STABILITY ANALYSIS:Lyapunov stability is a very mild requirement on equilibrium points. In particular, it does not require that trajectories starting close to the origin tend to the origin asymptotically. Also, stability is defined at a time instant t0. Uniform stability is a concept which guarantees that the equilibrium point is not losing stability. The Sylvester's Method was performed to determine if P is positive definite.

For this ball and beam model, P is not positive definite. System is unstable.

Phase portraits to enforce stability analysis

9. CONDITION NUMBER:

The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables.A problem with a low condition number is said to bewell-conditioned, while a problem with a high condition number is said to beill-conditioned.For this ball and beam system, the Conditional number is Cn = 61.1140.

10. Controllability and Observability Analysis:

A system is said to be completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in a finite time, to