echo on
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%%% Laplace transforms of time signals x(t)
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syms t n a b c w J s %%% "syms"
defines symbolic variables
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%%% "laplace" computes the Laplace transform of the given function x(t)
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disp(laplace(t^n*exp(-a*t))) %%% Time function t^n*exp(a*t):
pretty(laplace(sym(1))) %%%
Unitary step function:
pretty(laplace(t)) %%%
Unitary ramp function:
pretty(laplace(t^2/2)) %%%
Unitary parabola function:
pretty(laplace(exp(-a*t))) %%%
Exponential function:
pretty(laplace(sin(w*t))) %%% Sin function:
pretty(laplace(cos(w*t))) %%% Cos function:
pause %%% Press a keybord key to continue
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%%% Laplace transforms of other funcitons
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pretty(simplify(laplace(exp(-a*t)*cos(w*t))))
%%% Exponential-cosinus function:
pretty(simplify(laplace((1-exp(-b*t/J))/b)))
%%% Time function:
pretty(laplace(2*(1+t^2)*exp(5*t)))
%%% Time function:
pretty(laplace(4+3*exp(-3*t)*sin(7*t)))
%%% Time function:
pretty(laplace(2*exp(5*t)*sin(8*t)))
%%% Time function:
pretty(laplace(2*t^2*exp(-4*t)))
%%% Time function:
pause %%% Press a keybord key to continue
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%%% "step" computes the step response of the given s function
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s=tf('s'); %%%
Defines "s" as the Laplace variable
Gs=(3*s^2+5*s+2)/(s^3+4*s^2+3*s+1) %%%
Defines the Gs function:
figure(1) %%% Open a new figure
step(Gs) %%%
Compute the step response of function Gs
grid on %%% Add
the grid to the figure
%%% See the figure.
pause %%% Press a keybord key to continue
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Gs=(6*s+8)/(2*s+1) %%%
Defines the Gs function:
figure(1) %%% Open a new figure
step(Gs) %%%
Compute the step response of function Gs
grid on %%% Add the grid to the figure
ylim([0 10]) %%%
Defines the limits of the y axis
%%% See the figure.
pause %%% Press a keybord key to continue
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%%% "ilaplace(Ys)" computes the inverse Laplace transform of function Ys
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syms s t %%%
Defines the symbolic variable s
Ys=(5*s+3)/((s+1)*(s+2)*(s+3)) %%%
Defines the Ys function:
yt=ilaplace(Ys); %%%
Inverse Laplace transform y(t):
pretty(yt) %%% Nice
graphical representation
%%%%%%
Ys=(7*s^2-8*s+5)/(s^3+2*s^2+5*s) %%%
Defines the Ys function:
yt=ilaplace(Ys); %%%
Inverse Laplace transform y(t):
pretty(yt) %%% Nice graphical representation
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Ys=1/(s+2)/(s+1)^2 %%%
Defines the Ys function:
yt=ilaplace(Ys); %%%
Inverse Laplace transform y(t):
pretty(yt) %%% Nice graphical representation
pause %%% Press a keybord key to continue
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%%% Two functions plotted on the same figure
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figure(2); clf %%% Open a new figure
t=(0:0.01:1)*5; %%%
Defines the time axis
yt=6*exp(-t).*(cos(2*t) - (4*sin(2*t))/3) + 1; %%% Defines the y(t) function
y1t=1+10*exp(-t).*cos(2*t+atan(4/3));
%%% Defines the y1(t) function
plot(t,yt,'b') %%% Plot in blue the y(t) function
hold on %%%
Mantains the previous plots
plot(t,y1t,'r.') %%% Plot in red the points of function y1(t)
grid on %%% Add the grid to the figure
pause %%% Press a keybord key to continue
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%%% The impulsive response is equal to the inverse Laplace transform
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syms s t %%%
Defines the symbolic variable s
Ys=(5*s+3)/((s+1)*(s+2)*(s+3)) %%%
Defines the Ys function:
yt=ilaplace(Ys); %%%
Inverse Laplace transform y(t):
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s=tf('s'); %%%
Defines "s" as the Laplace variable
Y1s=(5*s+3)/((s+1)*(s+2)*(s+3)) %%%
Defines the Ys function:
figure(3); clf %%% Open a new figure
impulse(Y1s) %%%
Compute the impulse response of function Y1s hold on %%%
Mantains the previous plots
V=axis; %%% Read the ranges of the axis
t=(0:0.005:1)*V(2); %%%
Defines the time axis
yt=eval(yt); %%%
Defines the time axis
plot(t,yt,'r.') %%% Plot in red the points of function y1(t)
grid on %%% Add the grid to the figure
pause %%% Press a keybord key to continue
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%%% Dominant poles
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syms s t %%%
Defines the symbolic variable s
Ys=(s+37)*(s+225)/((s+350)*(10*s+15)*(s^2+2*s +20)*(40*s+2)*s);
yt=ilaplace(Ys) y0=sym(111/2800);
y1=- (313*exp(-350*t))/16639696400880;
y2=(10579*exp(-(3*t)/2))/7782005;
y3=(111656197*exp(-t)*(cos(19^(1/2)*t) +
(5150758*19^(1/2)*sin(19^(1/2)*t))/111656197) )/746752945400;
y4=- (66495220*exp(-t/20))/1615852131;
figure(3); clf %%% Open a new figure
t=(0:0.002:1)*60;
plot(t,eval(yt),'r') hold on
plot(t,eval(y0)+0*t,'m') plot(t,eval(y1),'b')
plot(t,eval(y2),'b')
plot(t,eval(y3),'b') plot(t,eval(y4),'m') grid on
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