Download Extracting a Linear Model into MATLAB

UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
LAB MANAUAL #07
DC Motor Speed Modeling in MATLAB
and Simulink
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Physical setup and system equations:
A common actuator in control systems is the DC motor. It directly provides rotary motion
and, coupled with wheels or drums and cables, can provide transitional motion. The electric
circuit of the armature and the free body diagram of the rotor are shown in the following
figure:
For this example, we will assume the following values for the physical parameters.
* moment of inertia of the rotor (J) = 0.01 kg.m^2/s^2
* damping ratio of the mechanical system (b) = 0.1 Nms
* electromotive force constant (K=Ke=Kt) = 0.01 Nm/Amp
* electric resistance (R) = 1 ohm
* electric inductance (L) = 0.5 H
* input (V): Source Voltage
* output (theta): position of shaft
The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back
emf, e, is related to the rotational velocity by the following equations:
In SI units (which we will use), Kt (armature constant) is equal to Ke (motor constant).
From the figure above we can write the following equations based on Newton's law
combined with Kirchhoff's law:
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Transfer Function:
Using Laplace Transforms, the above modeling equations can be expressed in terms of s.
By eliminating I(s) we can get the following open-loop transfer function, where the rotational
speed is the output and the voltage is the input.
MATLAB representation of Transfer Function:
We can represent the above transfer function into MATLAB by defining the numerator and
denominator matrices as follows:
J=0.01;
b=0.1;
K=0.01;
R=1;
L=0.5;
num=K;
den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)];
motor=tf(num,den);
step(motor,0:0.1:3);
title ('Step Response for the Open Loop System');
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Step Response
0.1
0.09
0.08
0.07
Amplitude
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.5
1
1.5
2
2.5
3
Time (sec)
DC Motor Speed Modeling in Simulink:
The motor torque, T, is related to the armature current, i, by a constant factor Kt. The back
emf, e, is related to the rotational velocity by the following equations:
In SI units (which we will use), Kt (armature constant) is equal to Ke (motor constant).
Building the Model :
This system will be modeled by summing the torques acting on the rotor inertia and
integrating the acceleration to give the velocity. Also, Kirchhoff’s laws will be applied to the
armature circuit.

Open Simulink and open a new model window.
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
First, we will model the integrals of the rotational acceleration and of the rate of change of
armature current.




Insert an Integrator block (from the linear block library) and draw lines to and from
its input and output terminals.
Label the input line "d2/dt2(theta)" and the output line "d/dt (theta)" as shown below.
To add such a label, double click in the empty space just above the line.
Insert another Integrator block above the previous one and draw lines to and from its
input and output terminals.
Label the input line "d/dt(i)" and the output line "i".
Next, we will start to model both Newton's law and Kirchoff's law. These laws applied to the
motor system give the following equations:
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
The angular acceleration is equal to 1/J multiplied by the sum of two terms (one pos., one
neg.). Similarly, the derivative of current is equal to 1/L multiplied by the sum of three terms
(one pos., two neg.).







Insert two Gain blocks, (from the Linear block library) one attached to each of the
integrators.
Edit the gain block corresponding to angular acceleration by double-clicking it and
changing its value to "1/J".
Change the label of this Gain block to "inertia" by clicking on the word "Gain"
underneath the block.
Similarly, edit the other Gain's value to "1/L" and it's label to Inductance.
Insert two Sum blocks (from the Linear block library), one attached by a line to each
of the Gain blocks.
Edit the signs of the Sum block corresponding to rotation to "+-" since one term is
positive and one is negative.
Edit the signs of the other Sum block to "-+-" to represent the signs of the terms in
Kirchoff's equation.
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Now, we will add in the torques which is represented in Newton's equation. First, we will add
in the damping torque.




Insert a gain block below the inertia block, select it by single-clicking on it, and select
Flip from the Format menu (or type Ctrl-F) to flip it left-to-right.
Set the gain value to "b" and rename this block to "damping".
Tap a line (hold Ctrl while drawing) off the rotational integrator's output and connect
it to the input of the damping gain block.
Draw a line from the damping gain output to the negative input of the rotational Sum
block.
Next, we will add in the torque from the armature.



Insert a gain block attached to the positive input of the rotational Sum block with a
line.
Edit its value to "K" to represent the motor constant and Label it "Kt".
Continue drawing the line leading from the current integrator and connect it to the Kt
gain block.
Now, we will add in the voltage terms which are represented in Kirchoff's equation. First, we
will add in the voltage drop across the coil resistance.


Insert a gain block above the inductance block, and flip it left-to-right.
Set the gain value to "R" and rename this block to "Resistance".
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT


Tap a line (hold Ctrl while drawing) off the current integrator's output and connect it
to the input of the resistance gain block.
Draw a line from the resistance gain output to the upper negative input of the current
equation Sum block.
Next, we will add in the back emf from the motor.



Insert a gain block attached to the other negative input of the current Sum block with
a line.
Edit it's value to "K" to represent the motor constant and Label it "Ke".
Tap a line off the rotational integrator output and connect it to the Ke gain block.
The third voltage term in the Kirchhoff’s equation is the control input, V. We will apply a
step input.



Insert a Step block (from the Sources block library) and connect it with a line to the
positive input of the current Sum block.
To view the output speed, insert a Scope (from the Sinks block library) connected to
the output of the rotational integrator.
To provide a appropriate unit step input at t=0, double-click the Step block and set the
Step Time to "0".
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Open-loop response:
To simulate this system, first, an appropriate simulation time must be set. Select Parameters
from the Simulation menu and enter "3" in the Stop Time field. 3 seconds is long enough to
view the open-loop response. The physical parameters must now be set. Run the following
commands at the MATLAB prompt:
J=0.01;
b=0.1;
K=0.01;
R=1;
L=0.5;
Run the simulation (Ctrl-t or Start on the Simulation menu). When the simulation is finished,
double-click on the scope and hit its autoscale button. You should see the following output.
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Extracting a Linear Model into MATLAB
A linear model of the system (in state space or transfer function form) can be extracted from
a Simulink model into MATLAB. This is done through the use of In and Out Connection
blocks and the MATLAB function linmod. First, replace the Step Block and Scope Block with
an In Connection Block and an Out Connection Block, respectively (these blocks can be
found in the Connections block library). This defines the input and output of the system for
the extraction process.
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
MATLAB will extract the linear model from the saved model file, not from the open model
window. At the MATLAB prompt, enter the following commands:
[A,B,C,D]=linmod ('motormodel')
[num,den]=ss2tf(A,B,C,D)
You should see the following output, providing both state-space and transfer function models
of the system.
A =
-10.0000
-0.0200
1.0000
-2.0000
B =
0
2
C =
1
0
D =
0
num =
0
0.0000
2.0000
1.0000
12.0000
20.0200
den =
To verify the model extraction, we will generate an open-loop step response of the extracted
transfer function in MATLAB. Enter the following command in MATLAB.
step(num,den);
You should see the following plot which is equivalent to the Scope's output.
Control Engineering
Lab Instructor: Engr. Romana
UNIVERSITY OF ENGINEERING AND TECHNOLOGY, TAXILA
FACULTY OF TELECOMMUNICATION AND INFORMATION ENGINEERING
COMPUTER ENGINEERING DEPARTMENT
Step Response
0.1
0.09
0.08
0.07
Amplitude
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.5
1
1.5
2
2.5
3
Time (sec)
LAB TASK:
Read the entire manual carefully implements each step one by one in Simulink and complete
the Model. Save the model as”motorspeed.mdl” and execute it to verify the Results.
Control Engineering
Lab Instructor: Engr. Romana