This is an introductory problem. Click
here for the complete problem. In the problem, Dr. Abner Mallity
needs to control a system, and the data he has for the system is shown
below. That data was the result of applying a one volt step to the
motor, and the output is in radians/sec. Here's the data.
Actually, in the problem, there were two sets
of data. Here is the result of the other measurement.
Using the data, Mallity's reasoning goes like
the following.
The DC gain is somewhere
around 7. In the first data set it looks to be right around 7, while
the second data set makes it look closer to 6, maybe around 6.2.
Take an average of 6.6, but be prepared for variations in this system.
The control system will apparently have to work when the gain runs anywhere
from 6 to 7.
The time constant seems
to be around 2 seconds in the first data set. Mallity gets that by
taking .63 x 7 to get 4.4, and noting that the response seems to go through
4.4 at 2 seconds. Doing the same kind of thing for the second data
set, he looks for where the response goes through .63 x 6.2 (or 3.9) and
he thinks that's more like 2.2 seconds or so. Again, he takes the
time constant as 2.1 sec. and knows he has to be prepared for variation.
His transfer function
is:
G(s) = 6.6/(2.1s + 1)
Now,
Mallity decides to try proportional control, "wrapping" a loop around the
system, and using a control effort that is proportional to the error in
the system. Here is the system he is thinking about.
One of Mallity's graduate students, Willy
Nilly, has provided him with a simulation of the system above. In
the simulation (of the entire closed loop system) you can set the following.
DC Gain
Time Constant
Proportional Controller
Gain
Desired Output (Speed)
Be cautious using the simulation. If you
try to change numbers on the fly, you may find oddball happenings in the
plots, especially if you manage to make one of the entries NaN (Not-a-Number!).
For example, if you backspace leaving only a decimal point in a field,
strange things will happen. Also, you can click the button to reset
after a simulation, then click again to start again.
Problem
Using the simulation above, do the following. (Click
here to get a copy of the simulator in a separate window.)
P1.
Determine the proportional gain, Kp, that will give a
SSE (Steady State
Error) for a
step input (The simulation is set up only for a step input.) that is under
5%.
After you have run the simulation, explain the results.
P2.
Determine the gain that will give a SSE that is less than 2% for a step
input.
P3.
Determine the gain that will give a SSE that is less than 2% for a step
input for any DC gain in the system (Assume anything between 6 and 7 is
possible.).
Now, at this point, you should be have a better feel for how this system
operates. Here is what you have done.
You started with measured
step response data for a system. From that data you determined a
transfer function for the system.
Using the system transfer
function you calculated responses for a closed loop control system that
"wrapped" a control loop around the system.
In calculating those responses
you found that the performance of the system changes as the gain in the
control loop changes.
The changes in the response
were changes in response time
and accuracy.
As you learn more about control systems you
will address these issues in more detail. As you encounter more complex
systems there will also be issues related to stability
in addition to the issues of response time and accuracy noted above.
Send
us your comments on these lessons.
