P1 Matlab - Matlab in Control Systems

8/4/2019 P1 Matlab - Matlab in Control Systems

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MATLAB in Control

Systems

Practical Session 1

Commonly used commands


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Principles

MATLAB: Matrix environment i.e. matrices, vectors,constants and labels as variables.

Different toolboxes access different environments

Poly-nomial

ControlSystems

SymbolicMath


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Organising the workspace

All MATLAB commands are typed in lowercase .

Recommendation: Capitalise your variables names

Previous commands can be accessed by using theUP arrow, so that

Commands issued incorrectly, can be accessed andedited

Hitting will execute the edited command.

Clear variables with clear Var or clear Clear command window with clc no effect on stored

variables or command history


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Polynomial Domain

Rows

Matrices and vectors

321

khg

fed

cba

Matrix row elements are separated by spaces or

commas (or both), e.g. [2 3 4] or [2, 3, 4]


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Polynomial Domain

Columns Columns are created by separating rows with semi-

colons, e.g. [1, 2, 3; 4, 5, 6].

Columns may also be ceated by entering a carriagereturn after each row:

[2 4 6

8 5 3];


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Polynomial DomainMatrix Manipulations

Matrices and vectors may be transposed byadding an apostrophe after the specification,

e.g. A or B=[1 2 4].

Normal rules regarding dimensions applywhen arithmetic operations are performed.


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Polynomial DomainInverse and determinants

The inverse of a matrix may be obtained by thecommands inv(A) or A-1, i.e. A^-1

The determinant of a matrix may be found by thecommand det(A)


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Polynomial DomainVectors

Vectors are matrices that have either a singlerow of elements, or a single column ofelements.

Row vectors are used in Control Systems torepresent polynomials, i.e.

[1 0 4 6 9]represents

s4 + 0 + 4s2 + 6s + 9


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Polynomial DomainVariables

Variables can be created by setting thevariable name equal to the value e.g.

Mat = [2,6,8;2,4,6;9,7,8];

K = 5;V = [4 6 8];

L = Label;

Depending on the type , various operationsmay be performed on variables, e.g.

C=9;

D=sqrt(C) OR D=C^0.5


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Polynomial DomainCreating Labels

Labels are created by including them in singlequotes, e.g. Label


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Programming

Commands are executed as they are issuedin the command window.

A set of commands may be programmed inan m-file, and batch-executed from the m-file,by selecting (high-lighting) the commands,followed by pressing the F9 key.

M-files stored in the specified path may alsobe executed from the command window, bytyping the name of the m-file.


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Programming

Once the answer is obtained, it may becopied from the command window back into

the m-file, and commentedExample: N=[1 4]; D=[2 4 6];

tf(N,D)

% s+4% --------------

% 2s2+4s+6


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Commonly Used Commands

Nise Chapter 2 (1)

conv(A,B) Used to multiply polynomials,

represented as row vectors, with each other.(Also conv([1 3 5],[2 4 6])

poly(A) Where A=[-a, -b, -c]) Used tocreate polynomials if roots are known, i.e. findA as a polynomial if A=(s+a)(s+b)(s+c).


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Nise Chapter 2 (2) roots(A) Used to extract the value of the roots of a

polynomial.

residue(A,B) Used to find the residues

(coefficients) of the partial fractions of a function ofwhich the numerator is A (or [a b c ]) and the

denominator is B (or [f g h ]).

[c,r,d]=residue(A,B) Used to find the coefficients,

roots and direct quotient of the function A/B as abovein a single operation. The answer is given as columnvectors c (coefficients), r (roots) and row vector d(direct quotient).


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Nise Chapter 2 (3) tf(A,B) Creates the transfer function in the

classical form

zpk(N,D,k) Creates the transfer function inthe zero-pole-gain form. N are the roots of thenumerator, i.e. if the numerator is (s+4)(s -2),N = [-4, 2], and D the roots of the

denominator.k is a coefficient, known as the direct gain.


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Nise Chapter 2 (4) Once the transfer function is found in one

form, in can be converted to the other form bythe same commands, e.g.

A=tf([1,4,4],[1 3 5 7]);

B=zpk(A) and

C=tf(B) (=A)

Transfer functions found in this way arereferred to as LTI objects (Linear TimeInvariant Objects)


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Converting between domains

Control Systems to Polynomial Transfer functions (LTI objects) may be

converted to the polynomial domain with the

instruction: [N,D]=tfdata(G,v)

Example:

8145

12223

2

sss

ssG

[N,D]=tfdata(G,v) yields:

N=[0 1 2 12]

D=[1 5 14 8]


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Polynomial to Symbolic Math Row and column vectors can be converted to

symbolic math with the poly2sym(A)

command Column vectors will yield the same result as

the transposedrow vector.

The default variable is x

Example: If A=[1 3 5] and B=[1;3;5]

poly2sym(A) yields: poly2sym(B) yields:

x 2+3x+5 x 2+3x+5


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Symbolic Math to Polynomial (1) Variables must be declared for symbolic

math e.g syms s

The numerator and denominator of asymbolic math expression is firstseparated with `[N,D] = numden(H)

The numerator and denominator arethen separately converted to thepolynomial domain with sym2poly(A)


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Symbolic Math to Polynomial (2)Example: syms s

A=(s+4)/(s^2+2*s+24);

[Ns,Ds]= numden(A)% Ns=s+4

% Ds= s^2+2*s+24

N=sym2poly(Ns)

% N=[1 4]

sym2poly(Ds)

% D=[1 2 24]


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Laplace and Inverse Laplace

Laplace transforms may be converted tosymbolic math functions with ilaplace(B)

Example

T=(s+3)/(s^2+8*s+25)

Var=ilaplace(T)

% exp(-4*t)*cos(3*t)-1/3*exp(- 4*t)*sin(3*t)


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Example 1

Find the output (in the time domain) if a28-unit step is applied to a system with

the given transfer function, using theresidue() command in MATLAB

(For solution, see P1 Example1.doc file)

)4110)(7(

)9(205)(

2

sss

ssT


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Assignment 1(a)

Find the output (in the time domain) if a113-unit step is applied to a system with

the given transfer function. Verify youranswer, using the residue() command inMATLAB

)11316(15

)4(26)(

2

sss

ssT


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Assignment 1(b)

Find the output (in the time domain) if a109-unit impulse is applied to a system

with the given transfer function. Verifyyour answer, using MATLAB

)12510(15

)4(4

)2(1)( 2 ss

sss

sT

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P1 Matlab - Matlab in Control Systems