Examples of Control Systems Application

Modeling the Ball and Beam Experiment

Problem Setup M mass of the ball 0.11 kg R radius of the ball m d lever arm offset 0.03 m g gravitational acceleration 9.8 m/s^2 L length of the beam 1.0 m J ball's moment of inertia 9.99e-6 kgm^2 r ball position coordinate

The design criteria for this problem Settling time less than 3 seconds Overshoot less than 5%

System Equations Linearization of this equation about the beam angle, alpha = 0, gives us the following linear approximation of the system

System Equations The equation which relates the beam angle to the angle of the gear can be approximated as linear by the equation below Substituting this into the previous equation, we get

Transfer Function

State-Space

for our state-space example we will be using a slightly different model. The same equation for the ball still applies but instead of controlling the position through the gear angle, theta, we will control alpha-doubledot. This is essentially controlling the torque of the beam. Below is the representation of this system

State-Space

Matlab Representation and Open- Loop Response Transfer Function

Matlab Representation and Open- Loop Response State-Space

Closed-loop Representation

PID controller Kp = Proportional gain KI = Integral gain Kd = Derivative gain

The characteristics of P, I, and D controllers CL RESPONSE RISE TIME OVERSHOOTSETTLING TIME S-S ERROR KpDecreaseIncreaseSmall Change Decrease KiDecreaseIncrease Eliminate KdSmall Change Decrease Small Change

Proportional Control kp = 1

Proportional-Derivative Control kp = 10; kd = 10

Proportional-Derivative Control kp = 10, kd = 20

Proportional-Derivative Control kp = 15 and kd = 40

Ball & Beam Problem Using Root Locus Method Open-loop Root Locus

Ball & Beam Problem Using Root Locus Method From these equation damping ratio and natural frequency were found to be 0.7 and 1.9 respectively.

Ball & Beam Problem Using Root Locus Method The design criteria can also be plotted onto the root locus using the sgrid command. This command generates a grid of constant damping ratio and natural frequency

Animation

Questions