i need help with the third problem .any help will be appreciated
MMAE 443 HOMEWORK #10 Fall 2016 Due: Wednesday, November 16, 2016
Problem #1 (30%)
You are given the following system 20
+ 12 + 20
We now are interested in achieving a settling time that is less than 1 sec. You will have to use
feedback control to do this.
a. Draw the block diagram of the system with a proportional feedback controller (Kp) and
calculate the closed-loop-transfer function (CLTF).
b. Determine the poles of the CLTF with Kp=0, 0.2, 0.4, 0.6, 0.8, 1, 2, 4 , and plot them on
a complex plane.
c. Now determine the Kp value that will give you a 1 sec settling time (Make your slow
decaying pole settle at 1 sec). Plot the unit step response of the resulting system with this
Kp. The settling time should be fast enough, but you should see some steady state error.
d. To help mitigate the steady-state-error problem, look at your closed loop transfer function
and calculate the steady state response to a unit step using the final value theorem. What
do you need to do to Kp to ?help? with the steady state error problem?
e. Crank your Kp way up: make it 1000. Now plot the step response again. What happened
to the steady state error? Did using a large Kp cause any potential problems?
( )= Problem #2 (30%)
Consider the DC motor drive system shown in the block diagram. Instead of speed control, we are
interested in the position control of the motor. The block diagram of the system with a feedback
controller Gc (s ) is shown below: K LTL ( s )
- R(s) - Gc (s)
Controller U(s) Km
ms 1 ( s ) 1
s Y ( s) (s ) Motor
Fig3.1 Dynamics Assume K m 147, m 0.1 and the controller is a proportional plus derivative (PD) controller
given by Gc ( s ) K p K d s , which can be implemented in reality by applying the following
control input voltage:
u (t ) K p e(t ) K d e(t ), e(t ) r (t ) y (t )
a. Obtain the closed-loop transfer functions from the reference input R ( s) and the disturbance
input TL ( s) to the output Y ( s ) .
b. Determine the proportional gain K p and the derivative gain K d of the controller so that the
unit step response of the closed-loop system to the reference input has an overshoot of 15%
and a 2% setting time of 0.1 sec.
_1_ Problem #3 (40%)
Re-consider the DC motor drive system ( K m 147, m 0.1 ) that you examined in problem 2.
The goal is to design a feedback controller Gc ( s ) to meet the following performance
P1. The steady-state error caused by a unit constant disturbance torque TL ( s) should be
zero (Steady-state performance specifications).
P2. The unit step response of the closed-loop system to the reference input should have
an overshoot of 15% and a 2% setting time of 0.1 sec (Transient performance
a. Using the closed-loop transfer function GYT ( s) from the disturbance torque TL ( s) to the
output Y ( s ) , obtain the steady-state error caused by a unit constant disturbance torque
TL ( s) when a PD feedback controller Gc ( s ) K P K D s is used. If you are doing correctly,
you will find that the steady-state error obtained is not zero and thus a PD controller cannot
satisfy the steady-state performance requirement P1.
b. To satisfy the steady-state performance requirement P1, we can use the following
proportional plus integral and derivative (PID) feedback controller:
Gc ( s ) K P K D s K I
which can implemented by applying the following physical control input to the plant:
1 u (t ) K P e(t ) K D e(t ) K I e(t )dt or U ( s) K P K D s K I E ( s )
Obtain the closed-loop transfer functions GYR ( s) and GYT ( s ) when the above PID
feedback controller is used.
c. Show that with the above PID controller, as long as the closed-loop system is stable, the
steady-state performance requirement P1 will be satisfied.
d. To determine the PID gains K P , K D and K I to satisfy the transient performance requirement
P2, we will use the pole placement technique. Your denominator of the closed loop
transfer function should be third order so you will need to place three poles. Determine a
desired form for the denominator of the closed loop transfer function such that the
dominant poles will be second order with an overshoot of 15% and a 2% setting time of
0.1 sec (as prescribed by requirement P2). Place a third pole at -100.
e. Determine the PID gains K P , K D and K I which will produce the desired pole locations from
part d. _2_
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