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Category: ECSE 307

Description

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1 Objectives

In this lab, we will design a controller for angular speed. We will use the identified transfer function

of the Qnet DC Motor in lab-03 to design a controller that matches some desired perfermances. We

will then use Matlab to drive the Qnet DC Motor and evaulate the designed controller.

2 The Open-loop System

2.1 Ressource files

Similary 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_05_matlab.zip.

3. Unzip the content in your Matlab workspace.

4. Open Matlab and navigate to the path where you have Lab_05_SpeedControl.m.

5. Run Lab_05_SpeedControl.m. You shouldn’t have any errors, you will have warnings that you can

ignore.

2.2 Qnet DC Motor model

The general form of the transfer function between the rotor angular speed Ω and the input voltage V

of a DC Motor (Figure 1) can be written as

H(s) = Ω(s)

V(s) =

Kt

(Js + b)(Ls + R) + KtKe

−

+ v(t)

R

i

L

−

+ e(t)

+

−

T, ω

Figure 1 A schematic diagram of a DC motor

We have seen in lab-03 that the Qnet DC Motor transfer function can be approximated to a first

order transfer function with DC gain G and time constant τ.

Ω(s)

V(s) =

G

τs + 1

ECSE 307: Linear Systems and Control

Fall 2017

Instructor: Aditya Mahajan

TA: Mohammad Afshari and Anas El Fathi

Page 2 of 4 November 6, 2017

where G = Kt

bR + KtKe

and τ =

JR

bR + KtKe

. The values of the G and τ might change from Qnet DC

Motor to another, however, we will use the following values for the controller design.

gain = 28.5; % DC gain

tau = 0.16; % Time constant

In open-loop form, we can represent the block diagram of this system as follow (Figure 2)

V(s) G

τs + 1 Ω(s)

Figure 2 Block diagram of the open-loop system.

We can simulate the open-loop system in Matlab by using lsim. For instance, in order to simulate

the response to a defined voltage signal one can do:

dt = 0.01; % Sampling time

T = 10; % Duration of simulation

time = 0:dt:T; % Define time vector

% Generate a voltage input signal

% equals 0 from [0 sec, 1 sec],

% equals 2 from [1 sec, 9 sec],

% equals 0 from [9 sec, 10 sec]

uSim = 0*(time<1) + 2*(time>=1 & time<T-1) + 0*(time>=T-1);

H = tf(gain, [tau 1]); % Define the transfer function.

ySim = lsim(H, uSim, time); % Simulate the transfer function H.

2.3 Question 1

We want to drive the Qnet DC Motor in the following square wave form Figure 3.

5 10 15 20 25

25

75

Time

Angular speed

Ω

Figure 3 Square wave with frequency 0.1 Hz.

ECSE 307: Linear Systems and Control

Fall 2017

Instructor: Aditya Mahajan

TA: Mohammad Afshari and Anas El Fathi

Page 3 of 4 November 6, 2017

1. In Matlab generate and plot a signal that mimics the shape of the square wave presented in

Figure 3.

2. What is the value of the voltage V that you should input to the Qnet DC Motor to get a steadystate angular speed of Ω = 25 ? Ω = 75 ?

3. Using these volage values, generate a square wave input signal that will make the Qnet DC Motor

follow the square wave form in Figure 3. Simulate and plot the response of the transfer function

H(s) to this signal.

4. Drive the Qnet DC Motor with this square wave input signal. Superpose both the test results and

the simulation results.

5. Discuss with your TA the following: Does the proposed model H(s) satisfactory describes the Qnet

DC Motor ? Do we achieve satisfactory resluts with open-loop ? What if the frequency of the

square wave was 5 times bigger ? (Include the discussion in the report).

3 Profile Tracking

3.1 Specifications

The objective of this section is to design a controller K(s) that enable us to to track the square wave

in Figure 4 with the following specifications:

• Steady-state error less than 5%.

• 5%-Settling time of 0.25 s.

• Overshoot less than 10%.

1 2 3 4 5

25

75

Time

Angular speed

Ω

Figure 4 Square wave with frequency 0.5 Hz.

The controller K(s) forms a unity feedback with the Qnet DC Motor , Figure 5 shows the block

diagram of the closed-loop system.

ECSE 307: Linear Systems and Control

Fall 2017

Instructor: Aditya Mahajan

TA: Mohammad Afshari and Anas El Fathi

Page 4 of 4 November 6, 2017

Ωref(s) ∑ K(s) G

τs + 1 Ω(s) + V(s)

−

Figure 5 Block diagram of the closed-loop system.

3.2 Question 2: P-controller

First let’s design a P-controller, i.e K(s) = kP.

1. In Matlab generate and plot a signal that mimics the shape of the square wave presented in

Figure 4.

2. Write the closed-loop transfer function between the input Ωref and the output Ω.

3. Design a controller K(s) = kP that meets the specifications. Notice that for a first order system,

the 5%-settling time (the time taken to get to 95% of the steady-state value and not oscillate after

this time) is around 3 times the time constant.

4. In Matlab simulate the system transfer function and plot both the output angular speed Ω and

the voltage V.

5. Drive the Qnet DC Motor in closed-loop with the P-controller that you have designed. Plot your

results superposed with your previously simulated results. Did your controller work as expected ?

6. The Qnet DC Motor have a voltage saturation of ±5 V, by looking at the plot of the simulated

voltage command, when does the voltage hits those saturations? Use the transfer function between

the input Ωref(s) and command voltage V(s) to give an upper-bound value of kP that keeps the

voltage between ±5 V.

7. Use the maximum value of kP to drive the Qnet DC Motor . Plot your results superposed to

simulated results.

8. What are the performance of this controller (Steady-state error, 5%-Settling time, Overshoot). How

does these perfermances compare to the desired specifications.

4 Assignement

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.

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