Description
1 Purpose
The purpose of this laboratory is to design a cruise control system for a car using satisfying certain
technical specifications. In particular, using the model estimation you completed in the previous
lab, you will design a PI controller using the root locus method.
2 Introduction
+
+
Plant
Disturbance
Input Output
a
s b +
V s( ) ( ) V s m
( ) D s d
s
=
Figure 1: Block diagram of the plant
Recall the mathematical model of a car moving on a straight road with unknown slope θ derived
in the introduction to laboratory 2. In the block diagram above, d =
g
a
sin θ, where g is the
gravitational constant.
+
+
Plant
+
−
a
s b +
Disturbance
( ) D s d
s
=
V s( ) R s( )
E s( ) ( ) V s m C s( )
Figure 2: Block diagram of the closed-loop system
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Problem Statement: Consider the feedback control system in Figure 2. Design the controller
C(s) to meet the specifications below.
• (SPEC1) The output v(t) of the closed-loop system should asymptotically track reference step
signal r(t) despite the presence of the unknown disturbance d(t) = d · 1(t)
• (SPEC2) The closed-loop system (input R(s), output V (s), and no disturbance, D(s) = 0)
should be BIBO stable.
• (SPEC3) All poles of the closed-loop system should lie on the real axis, so that the output
v(t) does not have oscillatory behavior.
• (SPEC4) When D(s) = 0 (i.e., when the cart track is horizontal), the settling time Ts should
be less than 0.2s.
• (SPEC5) The magnitude voltage vm(t) imparted by the control system to the DC motor
should be less than 11.75V.
Concretely, you are required to design a control system making the cart, track constant reference
speeds even when the track is not horizontal, and meeting certain transient performance specifications.
The experimental setup used in the lab is shown in figure 3.
y
Cart System
u
Controller
Simulink Environment
Figure 3: Block diagram of the experimental setup
The design controller will be implemented in Simulink, in analogous way as in Lab 2.
3 Preparation
1. Referring to Figure 2, verify that the plant has type 0 (i.e. the plant has no pole in the
origin). Hence, in order to meet SPEC1, you need a controller with a pole at zero. Letting
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E(s) = R(s) − V (s), define the PI controller
Vm(s)
E(s)
= K(1 + 1
TIs
), K, TI > 0.
Write the transfer function from R(s) to E(s) (assuming D(s) = 0), and from D(s) to E(s)
(assuming R(s) = 0).
2. Let R(s) = ¯v/s and recall that D(s) = ¯d/s. Using superposition, write the expression for
E(s) in the closed-loop system.
3. Applying the final value theorem to E(s), show that if limx→∞ e(t) exists, then it must be
zero. You have thus shown that a PI controller is capable of meeting SPEC1.
Submit your derivations for points 1-3 with your pre-lab report.
4 Experiment
4.1 Identification of model parameters a and b
For this part you can use the same plant parameters a and b you estimated in your lab 2.
4.2 Controller design using Matlab
As in the previous lab, write a unique MATLAB script named script4 2.m for all the steps required
in this section.
1. In your preparation you’ve shown that a PI controller is capable of meeting SPEC1. We now
focus on SPEC2-SPEC4. Let θ = 0 or, equivalently, D(s) = 0. Recall that an approximate
expression for the settling time in a second-order system with complex conjugate poles is
Ts =
4
σ
, where σ is the real part of the poles of the closed-loop system . In order to satisfy
SPEC4, you thus need 4
σ
≤ 0.2, or σ ≥ 20. You’ll achieve this by an appropriate choice of K
and TI .
Pick a guess for TI , TI = 1. Using Matlab, you will now draw the root locus of the system,
i.e., the locus of the poles of the closed-loop system as the gain K is varied between 0 and ∞.
First, define the transfer function G(s) of the open-loop system without the gain K. This is
given by
G(s) = TIs + 1
TIs
a
s + b
.
In Matlab, you can define G(s) as follows
>> TI=1; a=(your value); b=(your value);
>> G=tf([TI 1],[Ti 0])*tf([a],[1 b]);
The root locus is the locus of the poles of KG(s)/(1 + KG(s)) as K is varied. Plot the root
locus by issuing the command.
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>> rlocus(G)
Verify, using the plot, that for all K > 0 the poles of the closed-loop system have negative
real part, and hence SPEC2 is met. However, for the present value of TI , SPEC4 cannot be
met by any choice of K > 0. Specifically, there doesn’t exist K > 0 such that the closed-loop
system has two poles on the real axis with real part ≤ −20. Prove the truth of the claim by
using the root locus plot.
Evidently, we need to choose a different value for TI . Try different (positive) values of TI . For
each choice of TI , plot the corresponding root locus. By trial and error, find the value of TI
compatible with this requirement: there exist K > 0 such that the closed-loop system has two
poles close to s = −20 on the negative real axis. Save the root locus plot.
Once you found TI > 0 satisfying the requirement above, you need to find K > 0 for which the
closed-loop system has two poles close to s = −20. First, plot the root locus corresponding
to the value of TI you just selected. Next, issue the command (be sure that the rlocus plot is
still open)
>> rlocfind(G)
Move the mouse cursor over the root locus and click on the desired location of the closed-loop
poles on the real axis as s = −20. This action will return the value of K you were looking for.
Notice that the PI controller with the values of K and TI you have just found should meet
SPEC2-SPEC4. You will include the value of K you just found in your lab report.
2. Next, you’ll double-check that using the values of K and TI you just found, SPEC2-SPEC4
are met in simulation. Download the file cart.slx from Quercus and open it. Implement your
closed loop system with a PI controller using the parameter values you just found. Save
your complete model as model 4 2.slx. You will include this file in your submission. Begin
by double-checking that your controller indeed meets SPEC4. Run the Simulink block by
clicking on Simulation > Start. Arrange your Simulink model such that the scopes depict
(i) the output v(t) versus the reference signal r(t), (ii) the tracking error e(t), and (iii) the
voltage vm(t). The reference signal is a square wave of frequency 0.5Hz and amplitude 0.2m/s.
Be sure that the input to the offset port is set to 0 since you will test on the system without
disturbances.
Recall that the settling time Ts of v(t) is the time v(t) takes to reach and stay within the
range
[v(∞) − 0.02(v(∞) − v(0)), v(∞) + 0.02(v(∞) − v(0))]
where v(∞) is the value v(t) asymptotically settles to. By zooming in on one period of the
simulation output, graphically estimate the settling time Ts. Save the plot you used to derive
your estimate. Is it true that Ts ≈ 0.2 sec.? Verify that SPEC5 is approximately met. Include
the voltage input response in your lab report.
3. Now you’ll try to design a more ”aggressive” PI controller. Similarly to what you did in
step 1, use the root locus and trial error to find the value of TI > 0 such that there exists
K > 0 such that the closed-loop system has two poles close to s = −30. Use the command
4
rlocfind to find the value of K for which the closed-loop system has two poles close to -30.
Open another copy of the file cart.slx uploaded on Quercus. Implement your closed loop
system with a PI controller using the parameter values you just found. Save your complete
model as model 4 3.slx. You will include this file in your submission. Similarly to what you
did in step 2, evaluate the settling time Ts by zooming in on one period of the simulation
output. Save the plot. Verify that SPEC5 is approximately met. Include the voltage input
response in your lab report. Compare the performance of this ”aggressive” controller to that
of the controller you evaluated earlier.
How do the settling times and overshoots compare?
Which controller is best suited to meet the specs (take into account all the specs)?
What is the cause of the differences you observe?
4.3 Rejection of disturbances
In this part of the lab you will test the two controllers you designed in the previous steps on the
system subject to disturbances. This time, use the constant value 1 to the offset port as in 4.
Figure 4: Cart model with offset enabled.
1. Open the file model 4 2.slx you previously saved. Set the input to the offset port to the
constant value as in 4. Save this model as model 4 2 offset.slx. Re run the experiment.
Does the controller properly reject the disturbance? Include the velocity response plot in your
report.
2. Open the file model 4 3.slx you previously saved. Set the input to the offset port to the
constant value as in 4. Save this model as model 4 3 offset.slx. Re run the experiment.
Does the controller properly reject the disturbance? Include the velocity response plot in your
report.
5 Submission
For this lab you are required to submit the following material:
1. Pre-lab report (first deadline);
2. Lab report;
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3. Lab files:
• model 4 2.slx
• model 4 3.slx
• model 4 2 offset.slx
• model 4 3 offset.slx
Please submit your pre-lab and final report and files before the assigned deadlines on Quercus.
Please upload your lab report and files in a unique zip folder.
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