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1 Objective
The main goal of this assignment is to design and implement LEAD and LAG compensators for
cart’s position system.
2 Your responsibility
Your responsibility is to answer all questions which have been asked throughout this assignment
and submit all your answers in addition to Matlab codes and Simulink results.
3 Questions
1. Using the coefficients you have found in Lab3-Q4 (estimated coefficients of the system),
define a transfer function in Matlab and find Bode diagram of the system (consider position
as the output of the system). Using the bode diagram find the phase margin and gain margin
of the system.
2. Use the value of Kp, in Lab4-Q3, plot the open-loop bode diagram of the system. Design a
LEAD Controller, which can achieve phase margin of 60◦
.
Hint: If you do not remember design method for LEAD compensator you can follow these
steps:
1
(a) Suppose LEAD compensator is of the form:
Clead(s) = Klead
α.T s + 1
T s + 1
(b) Set Klead = 1, since there is no requirement on steady-state error.
(c) Find the current open-loop phase margin of the system.
(d) Calculate required phase for reaching 60◦
, phase margin.(also include a small safety
factor)
(e) Find coefficient α, using following formula:
φmax = sin−1
α − 1
α + 1
(f) Find the desired wm by looking at the Bode plot of the system and find the frequency
at which gain equals:
−20 log(K
√
α) dB
(g) Find T using following equation:
T =
1
wm
√
α
3. Using transfer function blocks in simlulink, simulate the step response of your estimated
system in presence of proportional feedback controller, with and without the compensator.
(In your Simulink model instead of physical system insert the estimated transfer function)
4. Using the cart system (physical system), find the step response of the system to proportional
feedback controller with and without compensator(plot both step responses in one figure).
Compare the results with simulations.
5. Describe the effect of LEAD compensator on rise-time, steady state error, and bandwidth of
the closed loop system.
6. Add a LAG controller to the LEAD you just designed to get a LEAD-LAG system.
Your LAG controller should reduce the steady state error to a ramp input by a factor of at
least 10. Your combined system should have the specified phase margin of 60◦
.
Hint: If you do not remember design method for LEAD compensator you can follow these
steps:
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