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LAB 4: Control Design Using the Root Locus

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CONTROL SYSTEMS LABORATORY
ECE356
LAB 4: Control Design Using the Root Locus
1 Purpose
The purpose of this laboratory is to design a cruise control system for a car using the root locus.
Root locus is a design technique which allows you to sketch the locus of the closed loop poles as
a function of a parameter, usually the gain of the loop transfer function (hence the name root
locus). Whereas the Routh-Hurwitz criterion determines whether a fixed polynomial is stable, the
root locus provides more qualitative information about how the poles and zeros of the loop transfer
function affect the poles of the closed loop system, as the gain parameter varies. While we do not
teach root locus as a topic in this course since, unlike the design techniques that we do teach, it is
not based on frequency response, it is still something worth knowing. This laboratory gives you an
appreciation of the use of root locus using Matlab tools.
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 3. In the block diagram above, d =
g
a
sin θ, where g is the
gravitational constant.
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)
1
+
+
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
• (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 0.3s.
• (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.
PWM Encoder
Arduino
D/A Plant A/D
Controller
a
s b +
Figure 3: Block diagram of the experimental setup
The design controller will be implemented on the Arduino board. As shown in the figure, the
micro-controller determines the velocity of the cart from the encoder (A/D) and computes the
control input to be applied to the cart. Given the desired control input, the motor shield generates
a PWM signal that approximates the desired control value (D/A).
2
3 Preparation
1. Referring to Figure 2, verify that the plant has type 0. Hence, in order to meet SPEC1, you
need a controller with a pole at zero. Letting 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 limt→∞ 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 to your lab TA at the beginning of the lab. Thoroughly read
the next section prior to the lab.
4 Experiment
Note that the software files referred to in this lab have the filenames *lab3*. This is because this
used to be “Lab 3”. For simplicity, we have not changed the filenames.
4.1 Identification of model parameters a and b
If you are using a different cart and track unit than the one used in Lab 3, you need to re-stimate
the plant parameters a and b following the procedure outlined in Lab 3. If instead you are working
with the same experimental setup, use the values of a and b you estimated in Lab 3.
4.2 Controller design using Matlab
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
.
3
In Matlab, you can define G(s) as follows
>> 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.
>> 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
>> 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.
Have your TA sign here before proceeding to the next step.
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 lab3.mdl from Blackboard and open it. Begin by
double-checking that your controller indeed meets SPEC4. Enter the values of θ, TI and K
into the Matlab workspace
>> VMAX UPM=11.75;
Run the Simulink block by clicking on Simulation > Start. 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.
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.
Have your TA sign here before proceeding to the next step.
4
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 rlocfind
to find the value of K for which the closed-loop system has two poles close to -30.
Enter the values of K and TI you just found in the Matlab workspace and run the Simulink
diagram lab3.mdl. 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. 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?
What is the cause of the differences you observe?
Have your TA sign here before proceeding to the next step.
4.3 Controller Implementation
To perform the practical experiment for each part, open its Arduino file with “.ino” suffix and the
associated Matlab m-file. You will enter your controller in the Arduino code and will record the
data using Matlab. Follow your TA’s instructions on how to compile the Arduino code and how to
perform the experiment. Perform the following steps:
• Open the Arduino code file named as “Lab3.ino”, The velocity reference is set to a square
wave form with the amplitude of 0.2 m/s and frequency of 0.5 Hz.
• In the beginning of the code, modify and adjust the variables “K” and “Ti”.
• Enter the values of K and TI associated with the less “aggressive” controller. Upload the code
and launch the Matlab m-file located in the same directory named “real time data plot Lab3”.
• Save the obtained plot. Does your controller satisfy the design specifications?
Have your TA sign here before proceeding to the next step.
• Now you’ll test the performance of your controller against the unknown disturbance d(t).
Using one or more books, raise one end of the track. This corresponds to setting θ 6= 0. While
holding the track in a firm position, run the experiment again and compare the results.
• Now enter the values of K and TI associated with the more “aggressive” controller and repeat
all the above steps.
Have your TA sign here before proceeding to the next step.
5 Report