Prac 1 Control

7/27/2019 Prac 1 Control

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Transformada de Laplace

1.-() () z=20;

syms t

f=3*(t^z);

c= laplace (f)

pretty (c)

2.-() () z=20;

syms t

f= exp(z*t);

c= laplace (f)

pretty (c)


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3.-() () () ()z= 20;

w= sym (w);

t= sym (t);

f=z*sin (w*t);

c= laplace (f)

pretty (c)


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Fracciones parciales

4.-

()

=

()

z= 20;

num=[0 0 1];

den= [1 3 z];

[R,P,K]=residue (num,den)


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5.- () () Z=20;

num =[2 0 8 z];

a=3*z

den =[1 6 11 a];


primera


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6.- () ()() () z=20;

num= [0 0 z];


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den=[1 3 2];


Inversa de Laplace

7.-

() ()

()() ()

() () )

()

()()

Z=20;

syms s

f= (z*(s+2))/((s^2)*(s+1)*(s+3));


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c= ilaplace (f)

pretty (c)

or

f= (z*(s+2))/((s^4)+(4*s 3)+(3*s^2));

c= ilaplace (f)

pretty (c)

8.- () ()()() ()() () ()()()Z=20;

syms s

f= (z)/((s+1)*(s+1)*(s+3));

c= ilaplace(f)


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pretty (c)

or

f= (z)/((s^3)+(5*s^2)+(7*s)+(3));

c= ilaplace (f)

pretty (c)


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9.- () () () () Z=20;

syms s

f= ((z)*(s+2))/(s^2);

c= ilaplace (f)

pretty (c)


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or

f= ((z*s)+(z*2))/(s^2);

c= ilaplace (f)

pretty (c)


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Buscar help sintaxis

>> residue(num,den)

help residue

RESIDUE Partial-fraction expansion (residues).

[R,P,K] = RESIDUE(B,A) finds the residues, poles and direct term of

a partial fraction expansion of the ratio of two polynomials B(s)/A(s).

If there are no multiple roots,B(s) R(1) R(2) R(n)

---- = -------- + -------- + ... + -------- + K(s)

A(s) s - P(1) s - P(2) s - P(n)

Vectors B and A specify the coefficients of the numerator and

denominator polynomials in descending powers of s. The residues

are returned in the column vector R, the pole locations in column

vector P, and the direct terms in row vector K. The number of

poles is n = length(A)-1 = length(R) = length(P). The direct term

coefficient vector is empty if length(B) < length(A), otherwise

length(K) = length(B)-length(A)+1.

If P(j) = ... = P(j+m-1) is a pole of multplicity m, then the

expansion includes terms of the form

R(j) R(j+1) R(j+m-1)-------- + ------------ + ... + ------------

s - P(j) (s - P(j))^2 (s - P(j))^m

[B,A] = RESIDUE(R,P,K), with 3 input arguments and 2 output arguments,

converts the partial fraction expansion back to the polynomials with

coefficients in B and A.

Warning: Numerically, the partial fraction expansion of a ratio of


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polynomials represents an ill-posed problem. If the denominator

polynomial, A(s), is near a polynomial with multiple roots, then

small changes in the data, including roundoff errors, can make

arbitrarily large changes in the resulting poles and residues.

Problem formulations making use of state-space or zero-pole

representations are preferable.

Class support for inputs B,A,R:

float: double, single

See also poly, roots, deconv.

Reference page in Help browser

doc residue

>>laplace(f)

help laplace

--- help for sym/laplace ---

LAPLACE Laplace transform.

L = LAPLACE(F) is the Laplace transform of the scalar sym F with

default independent variable t. The default return is a function

of s. If F = F(s), then LAPLACE returns a function of t: L = L(t).

By definition L(s) = int(F(t)*exp(-s*t),0,inf), where integration

occurs with respect to t.

L = LAPLACE(F,t) makes L a function of t instead of the default s:

LAPLACE(F,t) L(t) = int(F(x)*exp(-t*x),0,inf).

L = LAPLACE(F,w,z) makes L a function of z instead of the

default s (integration with respect to w).

LAPLACE(F,w,z) L(z) = int(F(w)*exp(-z*w),0,inf).

Examples:

syms a s t w x

laplace(t^5) returns 120/s^6

laplace(exp(a*s)) returns 1/(t-a)

laplace(sin(w*x),t) returns w/(t^2+w^2)

laplace(cos(x*w),w,t) returns t/(t^2+x^2)

laplace(x^sym(3/2),t) returns 3/4*pi^(1/2)/t^(5/2)

laplace(diff(sym('F(t)'))) returns laplace(F(t),t,s)*s-F(0)

See also sym/ilaplace, sym/fourier, sym/ztrans.


doc sym/laplace

>>syms

help syms

SYMS Short-cut for constructing symbolic objects.

SYMS arg1 arg2 ...

is short-hand notation for

arg1 = sym('arg1');

arg2 = sym('arg2'); ...


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SYMS arg1 arg2 ... real


arg1 = sym('arg1','real');

arg2 = sym('arg2','real'); ...

SYMS arg1 arg2 ... positive


arg1 = sym('arg1','positive');

arg2 = sym('arg2','positive'); ...

SYMS arg1 arg2 ... clear


arg1 = sym('arg1','clear');

arg2 = sym('arg2','clear'); ...

Each input argument must begin with a letter and must contain only

alphanumeric characters.

By itself, SYMS lists the symbolic objects in the workspace.

Examples:

syms x beta real

is equivalent to:

x = sym('x','real');

beta = sym('beta','real');

syms k positive

is equivalent to:

k = sym('k','positive');

To clear the symbolic objects x and beta of 'real' or 'positive' status, type

syms x beta clear

See also sym.


doc syms

>>pretty

help pretty

--- help for sym/pretty ---

PRETTY Pretty print a symbolic expression.

PRETTY(S) prints the symbolic expression S in a format that

resembles type-set mathematics.

See also sym/subexpr, sym/latex, sym/ccode.


doc sym/pretty


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>>sym(variable)

help sym

SYM Construct symbolic numbers, variables and objects.

S = SYM(A) constructs an object S, of class 'sym', from A.If the input argument is a string, the result is a symbolic number

or variable. If the input argument is a numeric scalar or matrix,

the result is a symbolic representation of the given numeric values.

If the input is a function handle the result is the symbolic form

of the body of the function handle.

x = sym('x') creates the symbolic variable with name 'x' and stores the

result in x. x = sym('x','real') also assumes that x is real, so that

conj(x) is equal to x. alpha = sym('alpha') and r = sym('Rho','real')

are other examples. Similarly, k = sym('k','positive') makes k a

positive (real) variable. x = sym('x', 'clear') restores x to a

formal variable with no additional properties (i.e., insures that x

is NEITHER real NOR positive). Defining the symbol 'i' will use

sqrt(-1) in place of the imaginary i until 'clear' is used.

See also: SYMS.

Statements like pi = sym('pi') and delta = sym('1/10') create symbolic

numbers which avoid the floating point approximations inherent in the

values of pi and 1/10. The pi created in this way temporarily replaces

the built-in numeric function with the same name.

S = sym(A,flag) converts a numeric scalar or matrix to symbolic form.

The technique for converting floating point numbers is specified by

the optional second argument, which may be 'f', 'r', 'e' or 'd'.

The default is 'r'.

'f' stands for 'floating point'. All values are transformed fromdouble precision to exact numeric values N*2^e for integers N and e.

'r' stands for 'rational'. Floating point numbers obtained by

evaluating expressions of the form p/q, p*pi/q, sqrt(p), 2^q and 10^q

for modest sized integers p and q are converted to the corresponding

symbolic form. This effectively compensates for the roundoff error

involved in the original evaluation, but may not represent the floating

point value precisely. If no simple rational approximation can be

found, the 'f' form is used.

'e' stands for 'estimate error'. The 'r' form is supplemented by a

term involving the variable 'eps' which estimates the difference

between the theoretical rational expression and its actual floating

point value. For example, sym(3*pi/4,'e') is 3*pi/4-103*eps/249.

'd' stands for 'decimal'. The number of digits is taken from the

current setting of DIGITS used by VPA. Using fewer than 16 digits

reduces accuracy, while more than 16 digits may not be warranted.

For example, with digits(10), sym(4/3,'d') is 1.333333333, while

with digits(20), sym(4/3,'d') is 1.3333333333333332593,

which does not end in a string of 3's, but is an accurate decimal


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representation of the double-precision floating point number nearest

to 4/3.

See also syms, class, digits, vpa.

Overloaded methods:

inline/sym


doc sym

>>ilaplace(F)

help ilaplace

--- help for sym/il aplace ---

I LAPLACE I nverse Laplace transform.

F = I LAPLACE(L ) is the inverse Laplace transform of the scalar sym L

with defaul t independent vari able s. The defaul t return is a

function of t. I f L = L(t), then IL APLACE r eturns a function of x:F = F(x).

By defin ition, F(t) = int(L(s)*exp(s* t),s,c-i*i nf,c+i* inf )

where c is a real number selected so that al l singul ari ties

of L (s) are to the left of the li ne s = c, i = sqrt( -1), and

the integration is taken with respect to s.

F = I LAPLACE(L ,y) makes F a function of y i nstead of the default t:

I LAPLACE(L ,y) F(y) = int(L (y)*exp(s* y),s,c-i* inf ,c+i*i nf).

Here y is a scalar sym.

F = I LAPLACE(L ,y,x) makes F a function of x instead of the default t:

IL APLACE(L,y,x) F(y) = int(L(y)*exp(x*y),y,c-i* inf,c+i*inf),

integration is taken with respect to y.

Examples:

syms s t w x y

il aplace(1/(s-1)) return s exp(t)

il aplace(1/(t^ 2+1)) returns sin (x)

il aplace(t^ (-sym(5/2)),x) returns 4/3/pi^ (1/2)*x ^ (3/2)

il aplace(y/(y^2 + w^ 2),y,x) returns cos(w*x )

ilaplace(sym('laplace(F(x),x,s)'),s,x) returns F(x)

See also sym/laplace, sym/i fou ri er, sym/i ztrans.


doc sym/ilaplace


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Practica 2

Polinomio y races de una matriz con >>sym2poly y >>roots .

1.- [ ]z= 20;

s= sym (s);

A= [1 2 0; z 2 2; 0 -1 1];

I= eye (3);

p= det(s*I-A)

pretty (p)

p1=sym2poly(p)

roots (p1)

2.- [ ]


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z= 20;

s= sym (s);

A= [1 -3 4; 2 -2 z; 2 -1 0];

I= eye (3);

p= det(s*I-A)

pretty (p)

p1=sym2poly(p)

roots (p1)

Polinomio y raices con eig ()

1.- [ ]z= 20;

A= [1 2 0; z 2 2; 0 -1 1];


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eig(A)

2.- [ ]z= 20;

A= [1 -3 4; 2 -2 z; 2 -1 0];

eig(A)


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Matrices de Vectores propios sus valores propios

1.- [ ]z= 20;

A= [1 2 0; z 2 2; 0 -1 1];

[X, D]= eig (A)


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2.- [ ]z= 20;

A= [1 -3 4; 2 -2 z; 2 -1 0];

[X, D]= eig(A)


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Resolver si 1.- s= sym(s);

a= sym(a);

c= sym(c);

d= sym(d);

b= sym(b);

A=[-a c; d -b]

I=eye(2);

C=inv(s*I-A)

pretty (C)


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2.-det s= sym(s);

a= sym(a);

c= sym(c);

d= sym(d);

b= sym(b);

A=[-a c; d -b]

I=eye(2);

C=det(s*I-A)

pretty (C)


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Explique cada comando utilizado en la prctica. Utilice help si es necesario revisar la sintaxis

Eye()

>> help eye

EYE Identity matrix.

EYE(N) is the N-by-N identity matrix.

EYE(M,N) or EYE([M,N]) is an M-by-N matrix with 1's on

the diagonal and zeros elsewhere.

EYE(SIZE(A)) is the same size as A.

EYE with no arguments is the scalar 1.

EYE(M,N,CLASSNAME) or EYE([M,N],CLASSNAME) is an M-by-N matrix with 1's


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of class CLASSNAME on the diagonal and zeros elsewhere.

Note: The size inputs M and N should be nonnegative integers.

Negative integers are treated as 0.

Example:

x = eye(2,3,'int8');

See also speye, ones, zeros, rand, randn.

Overloaded methods:

distributed/eye

codistributor2dbc/eye

codistributor1d/eye

codistributed/eye


doc eye

det()

>> help det

DET Determinant.

DET(X) is the determinant of the square matrix X.

Use COND instead of DET to test for matrix singularity.


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See also cond.

Overloaded methods:

sym/det

gf/det

laurmat/det


doc det

>>sym2poly y >>roots

>> help sym2poly

--- help for sym/sym2poly ---

SYM2POLY Symbolic polynomial to polynomial coefficient vector.

SYM2POLY(P) returns a row vector containing the coefficients

of the symbolic polynomial P.

Example:

sym2poly(x^3 - 2*x - 5) returns [1 0 -2 -5].

See also poly2sym, sym/coeffs.



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doc sym/sym2poly

>> help roots

ROOTS Find polynomial roots.

ROOTS(C) computes the roots of the polynomial whose coefficients

are the elements of the vector C. If C has N+1 components,

the polynomial is C(1)*X^N + ... + C(N)*X + C(N+1).

Note: Leading zeros in C are discarded first. Then, leading relative

zeros are removed as well. That is, if division by the leading

coefficient results in overflow, all coefficients up to the first

coefficient where overflow occurred are also discarded. This process is

repeated until the leading coefficient is not a relative zero.

Class support for input c:

float: double, single

See also poly, residue, fzero.

Overloaded methods:

gf/roots

localpoly/roots


doc roots


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eig()

>> help eig

EIG Eigenvalues and eigenvectors.

E = EIG(X) is a vector containing the eigenvalues of a square

matrix X.

[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a

full matrix V whose columns are the corresponding eigenvectors so

that X*V = V*D.

[V,D] = EIG(X,'nobalance') performs the computation with balancing

disabled, which sometimes gives more accurate results for certain

problems with unusual scaling. If X is symmetric, EIG(X,'nobalance')

is ignored since X is already balanced.

E = EIG(A,B) is a vector containing the generalized eigenvalues

of square matrices A and B.

[V,D] = EIG(A,B) produces a diagonal matrix D of generalized

eigenvalues and a full matrix V whose columns are the

corresponding eigenvectors so that A*V = B*V*D.

EIG(A,B,'chol') is the same as EIG(A,B) for symmetric A and symmetric

positive definite B. It computes the generalized eigenvalues of A and B


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using the Cholesky factorization of B.

EIG(A,B,'qz') ignores the symmetry of A and B and uses the QZ algorithm.

In general, the two algorithms return the same result, however using the

QZ algorithm may be more stable for certain problems.

The flag is ignored when A and B are not symmetric.

See also condeig, eigs, ordeig.

Overloaded methods:

sym/eig

lti/eig

codistributed/eig


doc eig

>>[X,D]=eig(A),

PRACTICA 3

Funcion transferencia () 1.- comando impulse

z=20;

num = [0 1 z];

den=[1 5 6];

t = [0: 0.1: 5];


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y=impulse(num, den, t);

plot (t, y), grid, title (' La respuesta al impulso del sistema con comando impulse ()')


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2- Funcion escalon

z=20;

num = [0 1 z];

den=[1 5 6];

t = [0: 0.1: 5];

y=step(num, den, t);

plot (t, y), grid, title (' La respuesta a escaln del sistema con comando step()')


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3- Respuesta a una entrada sen(2t)

z=20;

num = [0 1 z];

den=[1 5 6];

t = [0: 0.2: 20];

u=sin(2*t);

y= lsim(num, den, u, t);

plot (t, y), grid, title (' La respuesta del sistema a la entrada sen(2t) con comando Isim()')


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4.- Seal de salida a exp -t

z=20;

num = [0 1 z];


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den=[1 5 6];

t = [0: 0.1: 5];

u=exp(-t);

y= lsim(num, den, u, t);

plot (t, y), grid, title (' La seal de salida del sistema a una entrada et con comando Isim()')


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2. La dinmica de un sistema en tiempo discreto est representado por la siguiente funcin de

transferencia:

()

num = [0 1 -1];

den=[1 -2.5 1];

t = [0: .1; 1.5];

y= dimpulse(num, den, t);

axis([0 10 0 1.5]);

plot (t, y), grid, title ('La respuesta al impulso del sistemacon comando dimpulse()')


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2.- La respuesta a escaln del sistema. Hacer uso de dstep(num, den, t) con axis([0 10 0 1.5]).

num = [0 1 -1];

den=[1 -2.5 1];

t = [0: .1; 1.5];

y= dstep(num, den, t);

axis([0 10 0 1.5]);

plot (t, y), grid, title (' La respuesta a escaln del sistema con comando dstep()')


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Prac 1 Control