Quiz
2: Second Order Closed Loop Systems - Solution
Name:_____________________________
This is the step response of a system to a step of magnitude 5. It
is the response of the Actuator + Plant in the closed loop system below.
The question for you
to answer is the following.
-
When you close the loop
will the closed loop system have a higher percent overshoot or a lower
percent overshoot? Assume that the proportional gain only takes on
positive values.
Dr. Abner Mallity (who
has been on extended leave from the EE Dept. - 5 yrs worth of leave) says
"Pshaw! That question can be answered for any gain, and you don't
even have to do any numerical calculations."
Given the response above, the system has to be second order, and it has
to have a transfer function of this form.
Gdcwn2/[s2
+ 2zwns
+ wn2]
The closed loop transfer
function will then be:
KpGdcwn2/[s2
+ 2zwns
+ wn2
+ KpGdcwn2]
This is also a second
order system, but with new parameters.
-
The closed loop natural
frequency is the square root of [wn2
+ KpGdcwn2]
= [1 + KpGdc]wn2
-
That means that the natural
frequency of the closed loop system is higher than the open loop system.
The closed loop natural frequency squared is higher by a factor of [1 +
KpGdc] compared to the square of the
open loop natural frequency, and the closed loop natural frequency is higher
than the open loop natural frequency by a factor of the square root of
[1 + KpGdc].
-
The closed loop DC gain
is given by KpGdc/[1 + KpGdc]
-
The closed loop DC gain
approaches 1 as the proportional gain is increased.
-
The closed loop damping
ratio is given by z/SQRT[1
+ KpGdc].
-
Since the closed loop
damping ratio is smaller, there is more overshoot.