The open-loop transfer function of the plant for the ball and beam experiment is given below:
The design criteria for this problem are:
To see the derivation of the equations for this problem refer to the ball and beam modeling page.
First, we will study the response of the system shown above when a proportional controller is used. Then, derivative and/or integral control will be added if necessary.
Recall, that the transfer function for a PID controller is:
m = 0.111; R = 0.015; g = -9.8; L = 1.0; d = 0.03; J = 9.99e-6; K = (m*g*d)/(L*(J/R^2+m)); %simplifies input num = [-K]; den = [1 0 0]; ball=tf(num,den); kp = 1; sys_cl=feedback(kp*ball,1);
Now, we can model the system's response to a step input of 0.25 m. Add the following line of code to your m-file and run it:
You should get the following output:
As, you can see the addition of proportional gain does not make the system stable. Try changing the value of kp and note that the system remains unstable.
m = 0.111; R = 0.015; g = -9.8; L = 1.0; d = 0.03; J = 9.99e-6; K = (m*g*d)/(L*(J/R^2+m)); %simplifies input num = [-K]; den = [1 0 0]; ball=tf(num,den); kp = 10; kd = 10; contr=tf([kd kp],1); sys_cl=feedback(contr*ball,1); t=0:0.01:5; step(0.25*sys_cl)
Your plot should be similar to the following:
Now the system is stable but the overshoot is much too high and the settling time needs to go down a bit. From the PID tutorial page in the section on characteristics of P, I, and D controllers, we see that by increasing kd we can lower the overshoot and decrease the settling time slightly. Therefore, make kd = 20 in your m-file and run it again. Your output should be:
The overshoot criterion is met but the settling time needs to come down a bit. To decrease the settling time we may try increasing the kp slightly to increase the rise time. The derivative gain (kd) can also be increased to take off some of the overshoot that increasing kp will cause. After playing with the gains a bit, the following step response plot can be achieved with kp = 15 and kd = 40:
As you can see from the above plot all the control objectives have been met without the use of an integral controller (settling time for this example is considered achieved when the response is less than 2% of its final value). Remember, that for a control problem there is usually more than one solution for the problem.
For other methods of controlling the ball and beam example, see the links below.