MATLAB for Controls

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Log of MATLAB commands I am learning for EE 447 Control Systems Analysis I.

Website: http://brl.ee.washington.edu/LegacyHtml/EE447/
Book: "Control Systems Engineering," N.S. Nise, 5th Edition, Wiley 2008
Companion site: http://bcs.wiley.com/he-bcs/Books?action=index&itemId=0471794759&bcsId=4132
MATLAB Central: http://www.mathworks.com/matlabcentral/fileexchange

Contents

1 Preface
2 Requirements
3 References
4 Links
5 Appendix B

Preface

(Summary of Section 1.6)
We can perform analysis, design, and simulation.
Alternative way of solving control system problems

Tool we are using:
MATLAB and MATLAB Control Systems Toolbox, which expands MATLAB to include control system-specific commands. Also contains:

Simulink: GUI
LTI Viewer: allows measurements directly from time and frequency response curves
SISO Design Tool: analysis and design tool
Symbolic Math Toolbox: saves labor when making symbolic calculations in analysis and design

Requirements

MATLAB Version 7
Control System Toolbox 8

References

Appendix B: MATLAB/Control Systems Toolbox tutorials and code
Appendix C: Simulink tutorials and diagrams
Appendix D: MATLAB GUI tools, tutorials, and examples. LTI Viewer and SISO Design Tool
Appendix E: Symbolic Math Toolbox tutorials and code

ch*sp* : Symbolic Math Toolbox

Links

Useful MATLAB Commands/Tips
[1] [2]

How to plot Ramp response
[1]

Appendix B

Code is available in Control Systems Engineering Toolbox folder at Companion site and at MATLAB Central

To access m-files from MATLAB, either:

Add file to the search path

File->Set Path...

Navigate 'Current Directory' window to the file location

To run m-files in MATLAB, either:

Type the m-file name in the Command Window
In the Current Directory window, right-click -> Run

To view m-files in Command Window, either:

Enter:

type <file name>
help <file name>

Double-click file in Current Window to bring up file in editor

To create m-files:

Type code into a .m file

Chapter 2

Represent polynomials
find roots of polynomials
multiply polynomials
partial-fraction expansion

ch2p1 - Bit strings, Comments, Numbers, Operators, Variables

Bit strings: text enclosed in apostrophes
Comments: Begin with %
Numbers: Just enter as-is
Arithmetic operators: Use proper operator such as +,*
Variables: Use left-hand argument and = sign

abs(Q)
Magnitude of complex number Q

angle(Q)
Angle of complex number Q

m-file:

'(ch2p1)'                           % Display label.
'How are you?'                      % Display string.
-3.96                               % Display scalar number -3.96.
-4+7i                               % Display complex number -4+7i.
-5-6j                               % Display complex number -5-6j.
(-4+7i)+(-5-6i)                     % Add two complex numbers and display sum.
(-4+7j)*(-5-6j)                     % Multiply two complex numbers and display product.
M=5                                 % Assign 5 to M and display.
N=6                                 % Assign 6 to N and display.
P=M+N                               % Assign M+N to P and display.
Q=3+4j                              % Define complex number, Q.
MagQ=abs(Q)                         % Find magnitude of Q.
ThetaQ=(180/pi)*angle(Q)            % Find the angle of Q in degrees.

output:

ans =

(ch2p1)

ans =

How are you?

ans =

   -3.9600

ans =

-4.0000 + 7.0000i

ans =

-5.0000 - 6.0000i

ans =

-9.0000 + 1.0000i

ans =

62.0000 -11.0000i

M =

     5

N =

     6

P =

    11

Q =

   3.0000 + 4.0000i

MagQ =

     5

ThetaQ =

   53.1301

Variables:

ch2p2 - Polynomials

Polynomials: Polynomials in s can be represented as row vectors containing coefficients

spaces in between: row vector
commas in between: column vector

m-file:

P1=[1 7 -3 23] %Store polynomial s^3 + 7s^2 -3s + 23
P2=[1,7,-3,23] % column vector

Output:

P1 =

1 7 -3 23

P2 =

1 7 -3 23

Variables:

ch2p3 - Semicolons

Ending commands with a semicolon supresses the display of results.

source:

P3=[1 7 -3 23]; %Store polynomial s^3 + 7s^2 -3s + 23
%But don't display output

ch2p4 - poly()

poly(V)
Represent a function from factored form to polynomial form

factored form: a row vector containing the roots of polynomial

(s+2)(s+5)(s+6) -> [-2 -5 -6]

polynomial form: a row vector containing the coefficients of polynomial (same as ch2p2)

m-file:

P3=poly([-2 -5 -6])                  % Store polynomial
                                     % (s+2)(s+5)(s+6) as P3 and
                                     % display the coefficients

Output:

P3 =

1 13 52 60

ch2p5 - roots()

roots(V)
Go from polynomial form to factored form
root of polynomial returned as a column vector

m-file:

P4=[5 7 9 -3 2]                     % Form 5s^4+7s^3+9s^2-3s+2 and
                                    % display.
rootsP4=roots(P4)                   % Find roots of 5s^4+7s^3+9s^2
                                    % -3s+2,
                                    % assign to rootsP4, and display.

output:

P4 =

     5     7     9    -3     2

rootsP4 =

-0.8951 + 1.2351i
-0.8951 - 1.2351i
   0.1951 + 0.3659i
   0.1951 - 0.3659i

ch2p6 - conv()

conv(a,b)
Multiply polynomials

P5=conv([1 7 10 9],[1 -3 6 2 1])    % Form (s^3+7s^2+10s+9)(s^4-
                                    % 3s^3+6s^2+2s+1), assign to
                                    % P5, and display.

output:

P5 =

1 4 -5 23 48 81 28 9

ch2p7 - residue()

[K, p, k] = residue(b, a)

Partial-fraction expansion for F(s) = b(s)/a(s)

K - residue
p = roots of denominator
k = direct quotient

found by dividing polynomials prior to performing expansion??

m-file:

% We expand F(s) = (7s^2+9s+12)/[s(s+7)(s^2+10s+100)].

% Result: F(s) = [(0.2554 - 0.3382i)/(s+5.0000 - 8.6603i)] + [(0.2554 + 0.3382i)/(s+5.0000 + 8.6603i)] - [0.5280/(s+7)] +[ 0.0171/s].

numf=[7 9 12];                      % Define numerator of F(s).
denf=conv(poly([0 -7]),[1 10 100]);
                                    % Define denominator of F(s).
[K,p,k]=residue(numf,denf)          % Find residues and assign to K;
                                    % find roots of denominator and
                                    % assign to p; find
                                    % constant and assign to k.

output:

K =

   0.2554 - 0.3382i
   0.2554 + 0.3382i
-0.5280
   0.0171

p =

-5.0000 + 8.6603i
-5.0000 - 8.6603i
-7.0000
        0

k =

     []

ch2p8 - example (TODO)

example 2.3 (TODO)

ch2p9 - Creating Transfer functions - Using Vectors

Use row vectors in either polynomial form or factored form to create LTI object/transfer function.

Polynomial form
Transfer function F(s) = N(s) / D(s)

N(s) = row vector containing coefficients of the numerator
D(s) = row vector containing coefficients of the denominator

F = tf(numf/denf)

source:

'Vector Method, Polynomial Form'
                                    % Display label.
numf=150*[1 2 7]                    % Store 150(s^2+2s+7) in numf and
                                    % display.
denf=[1 5 4 0]                      % Store s(s+1)(s+4) in denf and
                                    % display.
'F(s)'                              % Display label.
F=tf(numf,denf)                     % Form F(s) and display.
clear                               % Clear previous variables from workspace.

output:

Vector Method, Polynomial Form

numf =

150 300 1050

denf =

1 5 4 0

ans =

F(s)

Transfer function:
150 s^2 + 300 s + 1050
----------------------
s^3 + 5 s^2 + 4 s

Factored form
Transfer function G(s) = K * N(s) / D(s)

N(s) = row vector containing roots of the numerator
D(s) = row vector containing roots of the denominator

G = zpk(numg,deng,K)

zpk stands for zeros, poles, and gain

source:

'Vector Method, Factored Form'      % Display label.
numg=[-2 -4]                        % Store (s+2)(s+4) in numg and
                                    % display.
deng=[-7 -8 -9]                     % Store (s+7)(s+8)(s+9) in deng and
                                    % display.
K=20                                % Define K.
'G(s)'                              % Display label.
G=zpk(numg,deng,K)                  % Form G(s) and display.
clear                               % Clear previous variables from workspace.

output:

Vector Method, Factored Form

numg =

    -2    -4

deng =

    -7    -8    -9

K =

    20

ans =

G(s)

Zero/pole/gain:
20 (s+2) (s+4)
-----------------
(s+7) (s+8) (s+9)

ch2p9 - Creating Transfer functions - Using Rational Expression in s Method

Type the transfer function the same way we write it

Polynomial form

Create a symbolic ??? (TODO). Define 's' as an object
s = tf('s')

Enter F(s) similar to typing it on a computer

source:

'Rational Expression Method, Polynomial Form'
                                    % Display label.
s=tf('s')                           % Define 's' as an LTI object in
                                    % polynomial form.
F=150*(s^2+2*s+7)/[s*(s^2+5*s+4)]
                                    % Form F(s) as an LTI transfer
                                    % function in polynomial form.
G=20*(s+2)*(s+4)/[(s+7)*(s+8)*(s+9)]
                                    % Form G(s) as an LTI transfer
                                    % function in polynomial form.
clear                               % Clear previous variables from
                                    % workspace.

output:

Rational Expression Method, Polynomial Form

Transfer function:
s

Transfer function:
150 s^2 + 300 s + 1050
----------------------
s^3 + 5 s^2 + 4 s

Transfer function:
20 s^2 + 120 s + 160
--------------------------
s^3 + 24 s^2 + 191 s + 504

Factored form

To create a transfer function in factored form instead:
s = zpk('s')

source:

'Rational Expression Method, Factored Form'
                                    % Display label.
s=zpk('s')                          % Define 's' as an LTI object
                                    % in factored form.
F=150*(s^2+2*s+7)/[s*(s^2+5*s+4)]
                                    % Form F(s)as an LTI transfer
                                    % function in factored form.
G=20*(s+2)*(s+4)/[(s+7)*(s+8)*(s+9)]
                                    % Form G(s) as an LTI transfer
                                    % function in factored form.

output:

Rational Expression Method, Factored Form

Zero/pole/gain:
s

Zero/pole/gain:
150 (s^2 + 2s + 7)
------------------
s (s+1) (s+4)

Zero/pole/gain:
20 (s+2) (s+4)
-----------------
(s+7) (s+8) (s+9)

ch2p10- Obtain values from Transfer functions - tf2zp(), zp2tf()

Obtain factors of numerator and denominator from numerator & denominator of transfer function in polynomial form

[outNum, outDen] = tf2zp(numf, denf)

source:

'Coefficients for F(s)'             % Display label.
numftf=[10 40 60]                   % Form numerator of F(s) =
                                    % (10s^2+40s+60)/(s^3+4s^2+5s+7).
denftf=[1 4 5 7]                    % Form denominator of
                                    % F(s) =
                                    % (10s^2+40s+60)/(s^3+4s^2+5s+7).
'Roots for F(s)'                    % Display label.
[numfzp,denfzp]=tf2zp(numftf,denftf)

output:

Coefficients for F(s)

numftf =

    10    40    60

denftf =

     1     4     5     7

ans =

Roots for F(s)

numfzp =

-2.0000 + 1.4142i
-2.0000 - 1.4142i

denfzp =

-3.1163
-0.4418 + 1.4321i
-0.4418 - 1.4321i

Obtain coefficients of numerator and denominator from transfer function in factored form

[numgtf,dengtf]=zp2tf(numgzp',dengzp',K)

source:

'Roots for G(s)'                    % Display label.
numgzp=[-2 -4]                      % Form numerator of
K=10                                % G(s) = 10(s+2)(s+4)/[s(s+3)(s+5)].
dengzp=[0 -3 -5]                    % Form denominator of
                                    % G(s) = 10(s+2)(s+4)/[s(s+3)(s+5)].
'Coefficients for G(s)'             % Display label.
[numgtf,dengtf]=zp2tf(numgzp',dengzp',K)
                                    % Convert G(s) to polynomial form.

output:

Roots for G(s)

numgzp =

    -2    -4

K =

    10

dengzp =

     0    -3    -5

ans =

Coefficients for G(s)

numgtf =

     0    10    60    80

dengtf =

     1     8    15     0

Transpose: ' (TODO)

ch2p11- Convert Transfer functions between polynomial and factored form

use command tf() to convert a transfer function expressed in
factored form -> polynomial form

source:

'Fzpk1(s)'                          % Display label.
Fzpk1=zpk([-2 -4],[0 -3 -5],10)     % Form Fzpk1(s)=
                                    % 10(s+2)(s+4)/[s(s+3)(s+5)].
'Ftf1'                              % Display label.
Ftf1=tf(Fzpk1)                      % Convert Fzpk1(s) to
                                    % coefficients form.

output:

ans =

Fzpk1(s)

Zero/pole/gain:
10 (s+2) (s+4)
--------------
s (s+3) (s+5)

ans =

Ftf1

Transfer function:
10 s^2 + 60 s + 80
------------------
s^3 + 8 s^2 + 15 s

use command zpk() to convert a transfer function expressed in
polynomial form -> factored form

source:

'Ftf2'                              % Display label.
Ftf2=tf([10 40 60],[1 4 5 7])       % Form Ftf2(s)=
                                    % (10s^2+40s+60)/(s^3+4s^2+5s+7).
'Fzpk2'                             % Display label.
Fzpk2=zpk(Ftf2)                     % Convert Ftf2(s) to
                                    % factored form.

output:

Ftf2

Transfer function:
10 s^2 + 40 s + 60
---------------------
s^3 + 4 s^2 + 5 s + 7

ans =

Fzpk2

Zero/pole/gain:
10 (s^2 + 4s + 6)
---------------------------------
(s+3.116) (s^2 + 0.8837s + 2.246)

ch2p12- Plotting functions - plot()

Plot functions by specifying:

variable and it's range
functions
plot characteristics

Specifying range and increment:
'variable' = 'start point':'increment':'end point'

Specifying functions:
'function' = "function i.t.o variable defined earlier"

Plotting:
plot('variable', 'function', 'plot characteristics')

we can also specify multiple functions on the same plot:
Adding 3 more parameters to the plot() for each function

source:

'(ch2p12)'                % Display label.
t=0:0.01:10;            % Specify time range and increment.
f1=cos(5*t);            % Specify f1 to be cos(5t).
f2=sin(5*t);            % Specify f2 to be sin(5t)
plot(t,f1,'r',t,f2,'g')    % Plot f1 in red and f2 in green.
pause

output:

ch2sp1 - Symbolic objects, Inverse Laplace transform

Any symbolic calculations require defining the symbolic objects. For example:

Laplace transform variable s
Time variable t

Defining symbolic objects allows us to write functions of the 'variable' as we would normally write it; no need to use vectors

Only variables/objects given as input to the program needs to be defined. Variable/object outputted by the program need not be defined

To define a 'variable' as symbolic object:

syms 'variable'

To print functions on the command window as we would normally write it:
pretty(F)

To find inverse laplace transform of F(s):
ilaplace(F)

source:

syms s                       % Construct symbolic object for
                             % Laplace variable 's'.
'Inverse Laplace transform' % Display label.
F=2/[(s+1)*(s+2)^2];         % Define F(s) from Case 2 example.
'F(s) from Case 2'           % Display label.
pretty(F)                    % Pretty print F(s).
f=ilaplace(F);               % Find inverse Laplace transform.
'f(t) for Case 2'            % Display label.
pretty(f)                    % Pretty print f(t) for Case 2.
F=3/[s*(s^2+2*s+5)];         % Define F(s) from Case 3 example.
'F(s) for Case 3'            % Display label.
pretty(F)                    % Pretty print F(s) for Case 3.
f=ilaplace(F);               % Find inverse Laplace transform.
'f(t) for Case 3'            % Display label.
pretty(f)                    % Pretty print f(t) for Case 3.

output:

ans =

Inverse Laplace transform

F =

2/((s + 1)*(s + 2)^2)

ans =

F(s) from Case 2

         2
----------------
                 2
(s + 1) (s + 2)

f =

2/exp(t) - 2/exp(2*t) - (2*t)/exp(2*t)

ans =

f(t) for Case 2

    2         2         2 t
------ - -------- - --------
exp(t)   exp(2 t)   exp(2 t)

F =

3/(s*(s^2 + 2*s + 5))

ans =

F(s) for Case 3

         3
----------------
      2
s (s + 2 s + 5)

f =

3/5 - (3*(cos(2*t) + sin(2*t)/2))/(5*exp(t))

ans =

f(t) for Case 3

        /            sin(2 t) \
      3 | cos(2 t) + -------- |
3     \               2     /
- - -------------------------
5           5 exp(t)

ch2sp2 - Laplace transform

Find laplace transform, in partial fractions form, of time function. The time function should be defined in terms of symbolic object of time.
F=laplace(f)

source:

syms t                       % Construct symbolic object for 
                             % time variable 't'.
'Laplace transform'             % Display label.                     
'f(t) from Case 2'           % Display label.
f=2*exp(-t)-2*t*exp(-2*t)-2*exp(-2*t)
                             % Define f(t) from Case 2 example.
pretty(f)                    % Pretty print f(t) from Case 2 example.
'F(s) for Case 2'            % Display label.
F=laplace(f)                % Find Laplace transform.
pretty(F)                    % Pretty print partial fractions of
                             % F(s) for Case 2.

output:

f(t) from Case 2

f =

2/exp(t) - 2/exp(2*t) - (2*t)/exp(2*t)

    2         2         2 t
------ - -------- - --------
exp(t)   exp(2 t)   exp(2 t)

ans =

F(s) for Case 2

F =

2/(s + 1) - 2/(s + 2) - 2/(s + 2)^2

    2       2        2
----- - ----- - --------
s + 1   s + 2          2
                  (s + 2)

Modify the form of resulting laplace transform:

Collect common coefficient terms of F
collect(F)

expands product of factors of F
expand(F)

factors F
factor(F)

finds simplest form of F with the least number of terms
simple(F)

simplifies F - combine the partial fractions
simplify(F)

source:

F=simplify(F) % Combine partial fractions.
pretty(F) % Pretty print combined partial fractions.

output:

F =

2/((s + 1)*(s + 2)^2)

2
----------------
2
(s + 1) (s + 2)

Convert fractional symbolic terms into decimal terms with specified no.of decimal places
vpa(expression, places)

source:

syms t

f=3/5-3/5*exp(-t)*[cos(2*t)+(1/2)*sin(2*t)];

                             % Define f(t) from Case 3 example.
pretty(f)                    % Pretty print f(t) for Case 3.
'F(s) for Case 3 - Symbolic fractions'
                             % Display label.
F=laplace(f);                % Find Laplace transform.
pretty(F)                    % % Pretty print partial fractions of
                             % F(s) for Case 3.
'F(s) for Case 3 - Decimal representation'
                             % Display label.
F=vpa(F,3);                  % Convert symbolic numerical fractions to
                             % 3-place decimal representation for F(s).
pretty(F)                    % Pretty print decimal representation.

output:

        /            sin(2 t) \
      3 | cos(2 t) + -------- |
3     \               2     /
- - -------------------------
5           5 exp(t)

ans =

F(s) for Case 3 - Symbolic fractions

   3       3 (s + 1)              3
--- - ---------------- - ----------------
5 s             2                  2
        5 ((s + 1) + 4)   5 ((s + 1) + 4)

ans =

F(s) for Case 3 - Decimal representation

0.6         0.6           0.6 (s + 1.0)
--- - ---------------- - ----------------
   s             2                  2
        (s + 1.0) + 4.0   (s + 1.0) + 4.0

ch2sp3 - Input Transfer functions using Symbolic Math

Entering functions symbolically makes it easy to enter complicated transfer functions.

To convert the symbolic function to an LTI transfer function object

extract the symbolic numerator and denominator
[num,den]=numden(G)
convert the numerator and denominator to vectors
numg=sym2poly(num)
form the LTI transfer function object using the above vectors
tf(numg,deng)

source:

syms s                        % Construct symbolic object for
                              % frequency variable 's'.
G=54*(s+27)*(s^3+52*s^2+37*s+73)...
/(s*(s^4+872*s^3+437*s^2+89*s+65)*(s^2+79*s+36));
                              % Form symbolic G(s).
'Symbolic G(s)'               % Display label.
pretty(G)                     % Pretty print symbolic G(s).
[numg,deng]=numden(G);        % Extract symbolic numerator and denominator.
numg=sym2poly(numg);          % Form vector for numerator of G(s).
deng=sym2poly(deng);          % Form vector for denominator of G(s).
'LTI G(s) in Polynomial Form' % Display label.
Gtf=tf(numg,deng)             % Form and display LTI object for G(s) in
                              % polynomial form.
'LTI G(s) in Factored Form'   % Display label.
Gzpk=zpk(Gtf)                 % Convert G(s) to factored form.

output:

Symbolic G(s)

                 3       2
((54 s + 1458) (s + 52 s + 37 s + 73)) /

        2                4        3
   (s (s + 79 s + 36) (s + 872 s +

        2
   437 s + 89 s + 65))

ans =

LTI G(s) in Polynomial Form

Transfer function:

54 s^4 + 4266 s^3 + 77814 s^2 + 57888 s

                                       + 106434

-----------------------------------------------
s^7 + 951 s^6 + 69361 s^5 + 66004 s^4

                + 22828 s^3 + 8339 s^2 + 2340 s


ans =

LTI G(s) in Factored Form

Zero/pole/gain:

54 (s+51.31) (s+27) (s^2 + 0.6934s + 1.423)
-----------------------------------------------
s (s+871.5) (s+78.54) (s+0.558) (s+0.4584)

                      (s^2 - 0.05668s + 0.1337)

Summary

List of commands learned

Chapter 3

state-space representation stuff - SKIP - (TODO)

Chapter 4

ch4p1 - calculate characteristics of second-order system

damping ratio
natural frequency
percent overshoot
settling time
peak time

Just a bunch of arithmetic used. See example below.

source:

'(ch4p1) Example 4.6'               % Display label.
p1=[1 3+7*i];                       % Define polynomial containing first
                                    % pole.
p2=[1 3-7*i];                       % Define polynomial containing second
                                    % pole.
deng=conv(p1,p2);                   % Multiply the two polynomials to
                                    % find the 2nd order polynomial,
                                    % as^2+bs+c.
omegan=sqrt(deng(3)/deng(1))        % Calculate the natural frequency,
                                    % sqrt(c/a).
zeta=(deng(2)/deng(1))/(2*omegan)   % Calculate damping ratio,
                                    % ((b/a)/2*wn).
Ts=4/(zeta*omegan)                  % Calculate settling time, (4/z*wn).
Tp=pi/(omegan*sqrt(1-zeta^2))       % Calculate peak time, pi/wn*sqrt(1-
                                    % z^2).
pos=100*exp(-zeta*pi/sqrt(1-zeta^2))
                                    % Calculate percent overshoot
                                    % (100*e^(z*pi/sqrt(1-z^2)).

output:

ans =

(ch4p1) Example 4.6

omegan =

    7.6158

zeta =

    0.3939

Ts =

    1.3333

Tp =

    0.4488

pos =

   26.0176

ch2p2 - step responses - step()

To plot step responses of a LTI transfer function T(s)=num/den
step(T)

Multiple plots can be made using:
step(T1, T2, ...)

Information about the plots can be obtained by left-clicking on the curve

curve's label
coordinates of the point clicked

source:

clf                                 % Clear graph.
numt1=[24.542];                     % Define numerator of T1.
dent1=[1 4 24.542];                 % Define denominator of T1.
'T1(s)'                             % Display label.
T1=tf(numt1,dent1)                  % Create and display T1(s).
step(T1)                            % Run a demonstration step response
                                    % plot.
title('Test Run of T1(s)')          % Add title to graph.

output:

T1(s)

Transfer function:
24.54
-----------------
s^2 + 4 s + 24.54

To bring up the options menu, right click away from the curve:

system responses to be displayed
response characteristics to be displayed

peak response, etc
hover over these points to obtain more information

grid, normalize, properties, etc

ch2p2 - step responses - Vectors from step()

The following is an alternative way of plotting step responses.

To create vectors containing the plot's points:
[y,t] = step(T)

y = output vector
t = time vector
No plot is made

To make a plot with these vectors:
plot(t,y)

label the plot using title('blah'), xlabel('blah'), and ylabel('blah')
To place text anywhere on the graph: text(coordinateX, coordinateY, 'blah')

To clear graph:
clf

The following code shows how to plot step responses of 3 transfer functions using step() and alternatively using vectors.

source: (Continuing from the above code)

numt1=[24.542];                     % Define numerator of T1.
dent1=[1 4 24.542];                 % Define denominator of T1.
'T1(s)'                             % Display label.
T1=tf(numt1,dent1)                  % Create and display T1(s).
step(T1)                            % Run a demonstration step response
                                    % plot.
title('Test Run of T1(s)')          % Add title to graph.

'Complete Run'                      % Display label.
[y1,t1]=step(T1);                   % Run step response of T1 and collect
                                    % points.
numt2=[245.42];                     % Define numerator of T2.
p1=[1 10];                          % Define (s+10) in denominator of T2.
p2=[1 4 24.542];                    % Define (s^2+4s+24.542) in
                                    % denominator of T2.
dent2=conv(p1,p2);                  % Multiply (s+10)(s^2+4s+24.542) for
                                    % denominator of T2.
'T2(s)'                             % Display label.
T2=tf(numt2,dent2)                  % Create and display T2.
[y2,t2]=step(T2);                   % Run step response of T2 and collect
                                    % points.
numt3=[73.626];                     % Define numerator of T3.
p3=[1 3];                           % Define (s+3) in denominator of T3.
dent3=conv(p3,p2);                  % Multiply (s+3)(s^2+4s+24.542) for
                                    % denominator of T3.
'T3(s)'                             % Display label.
T3=tf(numt3,dent3)                  % Create and display T3.
[y3,t3]=step(T3);                   % Run step response of T3 and collect
                                    % points.
clf                                 % Clear graph.
plot(t1,y1,t2,y2,t3,y3)             % Plot acquired points with all three
                                    % plots on one graph.
title('Step Responses of T1(s),T2(s),and T3(s)')
                                    % Add title to graph.
xlabel('Time(seconds)')             % Add time axis label.
ylabel('Normalized Response')       % Add response axis label.
text(0.7,0.7,'c3(t)')               % Label step response of T1.
text(0.7,1.1,'c2(t)')               % Label step response of T2.
text(0.5,1.3,'c1(t)')               % Label step response of T3.

output:

Using the step() to plot T1,T2,T3:

source:

step(T1,T2,T3)                      % Use alternate method of plotting step
                                    % responses.
title('Step Responses of T1(s),T2(s),and T3(s)')
                                    % Add title to graph.

output:

ch2p3 - step response from state space

TODO

ch2p4 - Example

TODO
Table lookup - interpl()

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