MATLAB (3)

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Mass-Spring-Damper System

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h=tf([1 4],[1 4 40])
pole(h)
zero(h)
figure
step(h)

h1=tf([10 40],[1 4 40])
pole(h1)
zero(h1)
figure
step(h1)

result-1

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h=tf([1 4],[1 4 40])
pole(h)
zero(h)
figure
step(h)

h1=tf([1 1],[1 1 40])
pole(h1)
zero(h1)
figure
step(h1)

result-2

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h=tf([1 4],[1 4 40])
pole(h)
zero(h)

h1=tf([1 1],[1 1 40])
pole(h1)
zero(h1)
step(h,h1)

result-3

Value 대입

subs함수 이용

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>> help subs
subs - Symbolic substitution

    This MATLAB function returns a copy of s, replacing all occurrences of old with
    new, and then evaluates s.

    subs(s,old,new)
    subs(s,new)
    subs(s)
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syms t y
U1=dsolve('D2y+2*Dy+y=0','y(0)=1','Dy(0)=5','t');
U2=subs(U1,t,0:0.1:10)
U3=double(U2)

U11=dsolve('D2y+2*Dy+2*y=0','y(0)=1','Dy(0)=5','t');
U22=subs(U11,t,0:0.1:10)
U33=double(U22)

U111=dsolve('D2y+2*Dy+3*y=0','y(0)=1','Dy(0)=5','t');
U222=subs(U111,t,0:0.1:10)
U333=double(U222)

figure
plot(U3)
hold on
plot(U33)
plot(U333)
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~~~
***
# Laplace transform

> Unit step function

~~~Matlab
syms t
laplace(heaviside(t))
  • 실행결과
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ans =
 
1/s

t^n

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syms t n
laplace(t^n)
  • 실행결과
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ans =
 
piecewise(-1 < real(n) | 1 <= n & in(n, 'integer'), gamma(n + 1)/s^(n + 1))
 
>> pretty(ans)
{ gamma(n + 1)
{ ------------  if  -1 < real(n) or (1 <= n and n in integer)
{     n + 1
{    s

Sin function

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syms t n
laplace(sin(n*t))
  • 실행결과
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ans =
 
n/(n^2 + s^2)

Differential

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syms t f(t) s
Df = diff(f(t),t);
laplace(Df,t,s)
  • 실행결과
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ans =
  
 s*laplace(f(t), t, s) - f(0)

Inverse Laplace transform

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help ilaplace
--- sym/ilaplace에 대한 도움말 ---

 ilaplace Inverse Laplace transform.
    F = ilaplace(L) is the inverse Laplace transform of the sym L
    with default independent variable s.  The default return is a
    function of t.  If L = L(t), then ilaplace returns a function of x:
    F = F(x).
    By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf)
    where c is a real number selected so that all singularities
    of L(s) are to the left of the line s = c, i = sqrt(-1), and
    the integration is taken with respect to s.
 
    F = ilaplace(L,y) makes F a function of y instead of the default t:
        ilaplace(L,y) <=> F(y) = int(L(y)*exp(s*y),s,c-i*inf,c+i*inf).
 
    F = ilaplace(L,y,x) makes F a function of x instead of the default t:
    ilaplace(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 f(x)
     ilaplace(1/(s-1))               returns  exp(t)
     ilaplace(1/(t^2+1))             returns  sin(x)
     ilaplace(t^(-5/2),x)            returns  (4*x^(3/2))/(3*pi^(1/2))
     ilaplace(y/(y^2 + w^2),y,x)     returns  cos(w*x)
     ilaplace(laplace(f(x),x,s),s,x) returns  f(x)