Description
1 Objectives
In this lab, we want to design controllers in order to control the position and angular speed of the
Qnet DC Motor. So far we have controlled the Qnet DC Motor with a P controller. In this lab, we
observe the output and performance of the system when PI and PD controllers are in use. Precisely,
we want to design controllers to control the position and speed of the Qnet DC Motor to track a
square wave with the following specifications.
1.1 Specifications
• Zero steady-state error.
• 5%-Settling time of 0.25 s.
• Overshoot less than 10%.
2 Resource files
Similar to previous labs, we provide a Matlab class which interfaces with the Qnet DC Motor.
1. Log in to myCourses and select the content folder Labs.
2. Download lab_06_matlab.zip.
3. Unzip the content in your Matlab workspace.
4. Open Matlab and navigate to the path where you have Lab_06_Control.m.
5. Run Lab_06_Control.m.
3 Review From Lab 3
3.1 Qnet DC Motor model
The transfer function between the shaft angular speed ω(s) as well as its position θ(s) and the input
voltage V(s) in a DC Motor (Figure 1) is correspondingly given by H(s) ¯ and G(s) ¯ as follows.
H(s) = ¯ ω(s)
V(s) =
Kt
(Js + b)(Ls + R) + KtKe
G¯
(s) = θ(s)
V(s) =
1
s
Kt
(Js + b)(Ls + R) + KtKe
ECSE 307: Linear Systems and Control
Fall 2017
Instructor: Aditya Mahajan
TA: Mohammad Afshari and Anas El Fathi
Page 2 of 6 November 20, 2017
−
+ v(t)
R
i
L
−
+ e(t)
+
−
T, ω
Figure 1 A schematic diagram of a DC motor
In lab-03, we saw that this transfer function can be approximated by a first order transfer function
with a DC gain G and time constant τ. Therefore the first order transfer functions are
H(s) = ω(s)
V(s) =
G
τs + 1
G(s) = θ(s)
V(s) =
1
s
G
τs + 1
where G = 28.5 and τ = 0.16.
4 Position and Speed Controller Design in Matlab
Suppose that we want to design two different controllers in order to command the Qnet DC Motor
system to follow reference signals ωref and θref given Figure 2 and in Figure 3 below.
Note: We can not control speed and position of the rotor of a DC motor at the same time. Therefore,
we design and implement controllers separately.
1 2 3 4 5
25
75
Time
Angular speed
ωref
Figure 2 Reference angular speed of the rotor.
1 2 3 4 5
-180
180
Time
Position
θref
Figure 3 Reference position of the rotor.
ECSE 307: Linear Systems and Control
Fall 2017
Instructor: Aditya Mahajan
TA: Mohammad Afshari and Anas El Fathi
Page 3 of 6 November 20, 2017
Suppose Kω(s) controls the speed and Kθ(s) controls the position. Then, the closed loop controlled
system with controllers Kω(s) for H(s) and Kθ(s) for G(s) are given in Figure 4 and Figure 5 respectively.
ωref(s) ∑ Kω(s) H(s) ω(s) + V(s)
−
Figure 4 Block diagram of the closed-loop system for speed control.
θref(s) ∑ Kθ(s) G(s) θ(s) + V(s)
−
Figure 5 Block diagram of the closed-loop system for position control.
4.1 Question 1
In order to design controllers to control the speed and position of the Qnet DC Motor follow these
steps:
a. Define the transfer functions H(s) and G(s) in Matlab.
b. Define time interval from 1 to 5 with increments 0.01 and produce the reference signals in Figure 2
and Figure 3.
c. Use Matlab to find a PI controller for H(s) and a PD controller for G(s) and find the closed loop
controlled system using these controllers for each case.
d. Simulate the output of the system and plot it. Plot the output and the input in one figure. Are you
getting the performances given in Section 1.1? If not, tune the controller parameters based on lab
4 to meet the specifications.
5 Speed Control in Qnet DC Motor
In this section, we use the controller in the previous section for the real system. In lab 5, we designed a
P controller to control the speed of the shaft of the motor. It was shown that, it is impossible to remove
the steady state error even for large values of kP. Also, increasing the value of kP leads saturation in
the control signal, i.e., the control signal need to go over ±5 volts but it is impossible because of the
hardware. To solve this problem, we add an integrator to the system. This ensures zero steady-state
error for step inputs. The PI controller transfer function is as follows.
K(s) = kp +
kI
s
Note: We are working with a digital computer which means that every output to the hardware and
every input from the hardware is discrete in time. Hence, in order to produce discrete control signals
ECSE 307: Linear Systems and Control
Fall 2017
Instructor: Aditya Mahajan
TA: Mohammad Afshari and Anas El Fathi
Page 4 of 6 November 20, 2017
from the PI controller for Qnet DC Motor in discrete time Ts, we should implement the integrator
in a different form. This implementation sums the previous error values over time.
1
s
(ωref(s) − ω(s)) ⇔ ∫
NTs
0
(ωref(t) − Ω(t))dt =
N−1
∑
k=0 ∫
(k+1)Ts
kTs
(ωref(t) − ω(t))dt
=
N−1
∑
k=0
(ωref(k) − Ω(k))Ts
5.1 Question 2
1. Using the controller gains you found in the previous part for speed control, drive the closed loop
controlled output as well as the command signal (output of the controller) for Qnet DC Motor .
Plot your results superposed with your previously simulated results.
Hint 1: The solution to this part is similar to the solution of P controller in lab 5.
Hint 2: In order to implement the integrator in the controller we use sum function in Matlabsum returns the sum of the elements of its input. To find more about sum type the following in
the command window:
doc sum
Hence, the implementation of the PI controller is as follows:
u_ = Kp*w_error(n) + Ki*sum(w_error)*dt;
% w_error(n) is the error between the reference and the output at time n.
% w_error is the entire error vector.
2. What is the performance of this controller compared to the specifications in Section 1.1 (Steadystate error, 5%-Settling time, Overshoot)? Tune the values of kI and kP to get the desired performance. Plot and superpose your test results with the one in the previous part.
3. Do we have saturation again?
6 Position Control for Qnet DC Motor
Suppose that an arm is connected to the shaft of this motor and our goal is to control this arm. This can
be done with the position control of the Qnet DC Motor . In this part, we implement the controller
we designed in Question 1 to control the position of the shaft of the real Qnet DC Motor . The PD
controller transfer function is as follows.
K(s) = kp + kDs
Note: Again, because we are working with a digital computer we have to implement the derivative
part in a discrete way. In order to use a PD controller for a real system like Qnet DC Motor in
discrete time with increments dt we approximate the derivative as follows.
1
dt (Δ(t) − Δ(t − dt))
where Δ(t) = θref(t) − θ(t). The implementation of the derivative in Matlab is as follows:
ECSE 307: Linear Systems and Control
Fall 2017
Instructor: Aditya Mahajan
TA: Mohammad Afshari and Anas El Fathi
Page 5 of 6 November 20, 2017
u_ = Kp*p_error(n) + Kd/dt*(p_error(n)-p_error(n-1));
% p_error(n) is the error between the reference and the output at time n.
6.1 Question 3
1. Using the controller gains you found in Question 1 to control a continuous time transfer function in
Matlab , drive the closed loop controlled output position as well as the command signal (output
of the controller) for Qnet DC Motor .
Hint: The Qnet DC Motor unit for input position signal of the Qnet DC Motor is radian.
Hence, we have to change the input reference signal from degree to radian. This can be done as
follows:
radian = degree × π
180.
2. What is the performance of this controller (Steady-state error, 5%-Settling time, Overshoot). Tune
the values of kD and kP to get the desired performance. Plot and superpose your test results and
simulated results.
Note: If you want to plot the results in degree you have to change the unit of the output and
input to degree again.
6.2 Question 4
1. Suppose that the disk which is installed on the shaft of the Qnet DC Motor is the wheel of a
car that moves only in one direction: forward and backward. We want to design a controller that
controls the car to go forward for 0.4 meter and then move backward 0.2 meter. Design a PD
controller and plot the output position of the system as well as the control signal of the car.
Hint 1: The diameter of the disk is 0.05 meter and the circumference of a circle is πd. The
circumference of a circle shows the distance it can pass in one cycle (considering that the road is
not slippery). Therefore, for a specified distance in meter we can find the number of cycles the
rotor has to turn from the following calculation.
Number of cycles =
distance(m)
π × 0.05 .
Hint 2: Each cycle is 360∘ or 2π rad. Therefore, for a specific number of turns we can find the
angle in radian for the motor input.
2. In this question, what is the physical interpretation of rise time, settling time, and overshoot?
3. In 1 second and with the gains you found from Question 3, what is the maximum distance that
your car can traverse?
ECSE 307: Linear Systems and Control
Fall 2017
Instructor: Aditya Mahajan
TA: Mohammad Afshari and Anas El Fathi
Page 6 of 6 November 20, 2017
7 Assignment
In a report format, answer the laboratory questions. The report should contain:
• An introduction and a conclusion, outlining the purpose of the laboratory and what you have
learned.
• Explanation of the steps to answer the laboratory questions.
• All figures should have a legend and a caption.
• Include your code in the report appendix.
The assignment is due 7 days after your lab.
