Prescott, Arizona Campus
Department of Electrical and Computer Engineering
EE 402 Control Systems Laboratory
Spring Semester 2016
Lab Section 01
Thursday 1:25 – 4:05 pm
King Eng. Bldg. Rm 122
Lab Instructor:
Dr. Stephen Bruder
Lab 01 – Pre-lab
Control Systems Modeling & Analysis with MATLAB
Date Experiment Performed:
Thursday, Jan 21, 2016

Instructor’s Comments:
Comment #1

Comment #2
Date Report Submitted:
Monday, Jan 25
Group Members:
Student # 1 Name & Email
Student # 2 Name & Email
Grade:
EE 402 Control Systems Lab
Spring 2016
1. BASIC MATLAB FUNCTIONALITY
In the prelab, we will preview some of the basic functionality of MATLAB. In particular, you
can enter “>> help topic” or “>> doc topic” or click on the
icon to get more information.
You can enter MATLAB commands in the command window for an interactive session or create
scripts to be saved as *.m files. To create a *.m file, type “>> edit filename.m” in the
command window, or select File  New  Script from the pull-down menu, or select the
icon.
1.a.
Built in Capabilities:
Some particularly useful built-in variables are:

Both i and j (  1 ) can be used to indicate complex numbers
>> 2 + sqrt(-1) + 2i +3j
ans =
2.0000 + 6.0000i

pi has the value 3.1415926…

Building matrices/vectors
>> row = [1 2 5.4 -6.6]
row =
1.0000 2.0000 5.4000 -6.6000
>> column = [4; 2; 7; 4]
column =
4
2
7
4
>> size(row)
ans =
1 4
>> size(column)
ans =
4 1
>> x = 1:4
x=
1 2 3 4
Some of the particularly relevant built-in MATLAB functions are listed in Table 1 below.
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Table 1 Some Common MATLAB Functions
Description
Function
abs(x)
sqrt(x)
round(x)
fix(x)
floor(x)
ceil(x)
sign(x)
rem(x, y)
exp(x)
log(x)
Computes the absolute value of x
Computes the positive square root of x
Rounds x to the nearest integer
Rounds x to the nearest integer towards 0
Rounds x to the nearest integer toward - ∞
Rounds x to the nearest integer toward + ∞
Returns: -1 if x < 0, 0 if x = 0, and 1 otherwise
Returns the remainder of x/y (i.e. modulus function)
log10(x)
Computes
x
Computes e , where e is the base of the natural log
Computes the natural log of x to base e
sin(x), cos(x),
tan(x)
asin(x), acos(x),
atan(x)
>> log10(100) = 2
Computes the sine, cosine, & tangent of the angle x,
where x is given in radians
Computes the arcsine, arccosine, & arctangent of x
returning an angle in radians
Computes the arctangent (or inverse tangent) of
atan2(y, x)
y
x
retuning an angle in radians
Returns the complex conjugate of the complex
number x
Returns the real part of the complex number x
Returns the imaginary part of the complex number x
Returns the angle =atan2(imag(x), real(x))
conj(x)
real(x)
imag(x)
angle(x)
1.b.
log10 ( x)
( i.e.
Example
>> abs(-2.1) = 2.1000
>> sqrt(2.1) = 1.4491
>> round(2.1) = 2
>> fix(-2.9) =-2
>> floor(-2.9) = -3
>> ceil(-2.9) = -2
>> sign(2.1) = 1
>> rem(17, 4) = 1
>> exp(1) = 2.7183
>> log(exp(1)) = 1
x  r e j  r  cos( )  j sin( )  )
>> cos(60*pi/180) = 0.5000
>> acos(0.5)*180/pi = 60.0000
>> atan2(1,-1)*180/pi = 135
>> conj(2+3*j) = 2.000 - 3.000i
>> real(2 + 3*j) = 2
>> imag(2 + 3*j) = 3
angle(1 + j)*180/pi = 45
Practice:
1. Compute the following using the MATLAB command window (2 decimal digits please):
17

15
  5 log  e   ln  e  
5 1
2
 13
2

7
3
10
4
 121
11 =
2. Write a MATLAB script to compute:
x
for
tan( )  sin( )
 log10  a 5  b 2 
cos( )
a.    / 4 , a  5 , and b  3 => x =
b.    / 3 , a  2 , and b  3 => x =
1.c.
Working with Polynomials
To build the polynomial p(s)  s3  2s  5 in MATLAB, enter
>> p = [1 0 -2 -5];
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Note that the coefficient of the highest power is first in the list and all coefficients (i.e., the
0s 2 term) must be entered!! To solve for the roots of the polynomial use
>> r = roots(p)
r=
2.0946
-1.0473 + 1.1359i
-1.0473 - 1.1359i
To recover the polynomial from its roots
>> poly([2.0946, -1.0473 + 1.1359i, -1.0473 - 1.1359i])
ans =
1.0000
0 -2.0002 -5.0000
i.e., p(s)  s3  2s  5   s  2.0946 s  1.0473  1.1359i  s  1.0473 - 1.1359i 
2. USING THE SYMBOLIC TOOLBOX
The symbolic toolbox can be used to perform algebraic computations and render a symbolic
(i.e., not only numeric) result.
A particularly useful tool is the Laplace transform (and Inverse Laplace transform) functions.
For example, if we had modeled a simple system and performed the forward LT to determine the
system’s transfer function as G ( s ) 
b1s  b0
a2 s  a1s  a0
2
(see Figure 1) and now wanted to determine the
F (s)
Input
Y (s)
Output
systems output response due to a unit step input, we
could use the Symbolic toolbox to compute the
Figure 1 A simple system model
result.
>> syms s
>> a2 = 1; a1 = 5; a0 = 6;
% Values for the denominator coeffs
>> b1 = 2; b0 = 5;
% Values for the numerator coeffs
>> Gs = (b1*s + b0)/(a2*s^2+a1*s+a0); % G(s) = The Transfer Function
>> Ys = (1/s)*Gs;
% Y(s) = F(s) G(s)
>> yt = ilaplace(Ys)
% y(t) = InverseLT(Y(s))
yt =
5/6 - exp(-3*t)/3 - exp(-2*t)/2
>> ezplot(yt, [0 5 0 1])
% Plot a symbolic function
The last line of code returns the plot. Past a copy of your plot in Figure 2 (below).
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EE 402 Control Systems Lab
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Figure 2 A plot of y (t ) vs time for t=0,…, 5 seconds
3. USING SIMULINK
The previous example could also be analyzed using
the Simulink graphical simulation tool.
1. Start MATLAB and enter
>> simulink
2. Select File  New  Model
3. Select “Continuous” and drag a “Transfer Fcn”
to your blank model window.
4. Select “Sources” and drag a “Step” to your
model window.
5. Select “Sinks” and drag a “Scope” to your
model window.
6. Connect the “Step” to the “Transfer Fcn”
and “Transfer Fcn” to the “Scope.”
7. Double click on the “Step” and enter
a “Step Time” of 0.
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8. Double click on the “Transfer Fcn” and
set the “Numerator coefficients” to [b1 b0]
and “Denominator coefficients” [a2 a1 a0].
9. Set the simulation time to 5.0 sec.
10. Save the model as “first.mdl”.
11. In the command window enter
>> a2 = 1; a1 = 5; a0 = 6;
>> b1 = 2; b0 = 5;
12. Click on the play button ().
13. Double click on the “Scope”
to view the results (you might want
to autoscale
the plot).
Past a copy of your plot in Figure 3 (below).
Figure 3 A plot of y (t ) vs time from Simulink modeling
QUESTION:
How does Figure 2 compare with Figure 3?
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