Design a lag network so that the applicable static error constant has a value of 1 without significantly changing the position of the dominant poles of the system.

Engineering by Geoffrey

Project (2022)
Problem 1 (Chapter 9):
Lag controller.
Assume that for step inputs to the unity–feedback system shown exhibits 20% overshoot
when:
𝐺(𝑠) = 𝐾
(𝑠 + 1)(𝑠 + 2)(𝑠 + 6)
(a) What static error constant applies to this system, and what is its value.
(b) Design a lag network so that the applicable static error constant has a value of 1
without significantly changing the position of the dominant poles of the system.
(c) Simulate the system to verify the effects of your compensator using MATLAB.
Problem 2 (Chapter 9):
PD controller:
The unity–feedback system shown with:
𝐺(𝑠) = 𝐾(𝑠 + 6)
(𝑠 + 2)(𝑠 + 3)(𝑠 + 7)
is operating with a dominant–pole damping ratio of 0.6. Design a PD controller so that the
settling time is reduced by at least a factor of 3. Compare the transient and steady–state
performance of the uncompensated and compensated systems using MATLAB.
Problem 3 (Chapter 9):
Lead controller:
The figure part (a) shows a heat–exchanger process whose purpose is to maintain the
temperature of a liquid at a prescribed temperature.

The temperature is measured using a sensor and a transmitter, TT 22, that sends the

measurement to a corresponding controller, TC 22, that compares the actual temperature
with a desired temperature set point, SP. The controller automatically opens or closes a
valve to allow or prevent the flow of steam to change the temperature in the tank. The
corresponding block diagram for this system is shown in the figure part (b).
Assume the following transfer functions:
𝐺𝑣(𝑠) = 0.02
4𝑠 + 1
𝐺𝑝(𝑠) = 70
50𝑠 + 1
𝐻(𝑠) = 1
12𝑠 + 1
(a) Assuming Gc(s) = K, find the value of K that will result in a dominant pole with ΞΆ = 0.7.
Obtain the corresponding Ts.
(b) Design a lead compensator to obtain the same damping factor as Part (a) but with a
settling time 20% smaller.
(c) Verify your results with MATLAB simulation.
Hint: the value of GH(s) to use in the rlocus command is:
𝐺𝐻(𝑠) = 1.4
2400𝑠3 + 848𝑠2 + 66𝑠 + 1
Provide the derivation to obtain the GH(s) shown in this Hint.
Problem 4 (Chapter 11):
For the unity–feedback system shown, find the value of K required to obtain a gain margin
of 10 dB when:
𝐺(𝑠) = 𝐾
(𝑠 + 5)(𝑠 + 15)(𝑠 + 20)
Problem 5 (Chapter 11):
Use frequency response methods to find the value of K necessary to achieve a step
response with a 10% overshoot for the unity–feedback system shown when:

𝐺(𝑠) = 𝐾
𝑠(𝑠 + 5)(𝑠 + 10)
Problem 6 (Chapter 11):
Lag controller:
The unity–feedback system shown is operating with 10% overshoot when:
𝐺(𝑠) = 𝐾
𝑠(𝑠 + 5)
Design a compensator using frequency response techniques to yield Kv = 50 without
significantly changing the uncompensated system’s phase–margin frequency and phase
margin.
Problem 7 (Chapter 9):
A motor transfer function is:
πœƒπ‘œ(𝑠)
πΈπ‘Ž(𝑠) = 25
𝑠(𝑠 + 2)
(a) if this motor were the forward transfer function of a unity feedback system, calculate
the percent overshoot and settling time that could be expected.
(b) you want to improve the closed–loop response. Since the motor constants cannot be
changed and you cannot use a different motor, an amplifier and tachometer are inserted
into the loop as shown.
Find the values of K1 and Kf to yield a percent overshoot of 15% and settling time of 0.5 sec.
(c) evaluate the steady–state error specifications for both the uncompensated and
compensated systems.