Proportional control
PID control
Tuning the gains

From the main problem, the dynamic equations in state-space form are the following:

and the system schematic looks like:

For the original problem setup and the derivation of the above equations and schematic, please refer to the Modeling a DC Motor page.

With a 1 rad/sec step reference, the design criteria are:

Now let's design a PID controller and add it into the system. First create a new m-file and type in the following commands(refer to main problem for the details of getting those commands).

J=0.01;
b=0.1;
K=0.01;
R=1;
L=0.5;
A=[-b/J   K/J
   -K/L   -R/L];
B=[0
   1/L];
C=[1   0];
D=0;
[num,den]=ss2tf(A, B, C, D);
num=num(3);

Proportional control

Let's first try using a proportional controller with a gain of 100. Add the following code to the end of your m-file:

Kp=100;
numa=Kp*num;
dena=den;

[numac,denac]=cloop(numa,dena);

Now let's see how the step response looks, add the following to the end of your m-file, and run it in the command window:

t=0:0.01:5;
step(numac,denac,t)

PID control

From the plot above we see that the steady-state error and the overshoot are both too large. Recall from PID tutorial page that adding an integral term will eliminate the steady-state error and a derivative term will reduce the overshoot. Let's try a PID controller with small Ki and Kd. Change your m-file so it looks like:

J=0.01;
b=0.1;
K=0.01;
R=1;
L=0.5;
A=[-b/J   K/J
   -K/L   -R/L];
B=[0
   1/L];
C=[1   0];
D=0;
[num,den]=ss2tf(A, B, C, D);
num=num(3);

Kp=100;
Ki=1;
Kd=1;
numc=[Kd, Kp, Ki];
denc=[1 0];
numa=conv(num,numc);
dena=conv(den,denc);
[numac,denac]=cloop(numa,dena);
step(numac,denac)

Tuning the gains

The settling time is now too long. Let's increase Ki to reduce the settling time. Go back to your m-file and change Ki to 200. Rerun the file and you should get the plot like this:

Now we see that the response is much faster than before, but the large Ki has worsened the transient response (big overshoot). Let's increase Kd to reduce the overshoot. Go back to the m-file and change Kd to 10. Rerun it and you should get this plot:

So now we know that if we use a PID controller with

all of our design requirements will be satisfied.

PID Examples

Cruise Control | DC Motor | Bus Suspension | Inverted Pendulum

DC Motor Examples

Modeling | PID | Root Locus | Frequency Response | State Space

Tutorials

Basics | PID | Modeling | Root Locus | Frequency Response | State Space | Examples