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Published byAdela O’Connor’ Modified over 9 years ago
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Background Review Elementary functions Complex numbers
Common test input signals Differential equations Laplace transform Examples properties Inverse transform Partial fraction expantion Matlab
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Elementary functions
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The most beautiful equation
It contains the 5 most important numbers: 0, 1, i, p, e. It contains the 3 most important operations: +, *, and exponential. It contains equal sign for equations
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions F(t)=3sin 3t +4cos 3t
F(t)=Asin(3t-d)=Acosd sin3t –Asin d cos3t Acos d =3 Asin d =-4 A2=25, A=5 tan d =-4/3, d=-53.13o F(t)=5sin(3t+53.13o)
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Complex Numbers X2+1=0 x=i where i2=-1 X2+4=0, then x=2i, or 2j
If z1=x1+iy1, z2=x2+iy2 Then z1+ z2= (x1+ x2)+i(y1 + y2) z1 z2=(x1+iy1)(x2+iy2)=(x1x2 -y1y2) +i(x1y2 +x2y1)
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Polar form of Complex Numbers
z=x+iy, let’s put x=rcosq, y= rsinq Then z = r(cosq+i sinq) = r cisq = rq Absolute value (modulus) r2=x2+y2 Argument q= tan-1(y/x) Example z=1+i
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Euler Formula z=x+iy ez =ex+iy= ex eiy= ex (cos y+i sin y)
eix =cos x+i sin x = cis x | eix | = sqrt(cos2 x+ sin2 x) = 1 z=r(cosq+i sinq)=r eiq Find e1+i Find e-3i
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In Matlab >> z1=1+2*i z1 = i >> z2=3+i*5 z2 = i >> z3=z1+z2 z3 = i >> z4=z1*z2 z4 = i >> z5=z1/z2 z5 = i >> r1=abs(z1) r1 = >> theta1=angle(z1) theta1 = >> theta1=angle(z1)*180/pi theta1 = >> real(z1) ans = 1 >> imag(z1) ans = 2
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Poles and zeros Pole of G(s) is a value of s near which the value of G goes to infinity Zero of G(s) is a value of s near which the value of G goes to zero.
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Poles and zeros in Matlab
>> s=tf(‘s’) Transfer function: s >> G=exp(-2*s)/s/(s+1) Transfer function: 1 exp(-2*s) * s^2 + s >> pole(G) ans = 0, -1 >> zero(G) ans = Empty matrix: 0-by-1
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Test waveforms used in control systems
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1st order differential equations
y’ + a y = 0; y(0)=C, and zero input Solution: y(t) = Ce-at y’ + a y = d(t); y(0)=0, input = unit impulse Unit impulse response: h(t) = e-at y’ + a y = f(t); y(0)=C, non zero input Total response: y(t) = zero input response + zero state response = Ce-at + h(t) * f(t) Higher order LODE: use Laplace
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Laplace Transform Definition and examples Unit Step Function u(t)
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Laplace Transform
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Name:____________ The single most important thing to remember is that whenever there is feedback, one should worry about __________
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Laplace Transform
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Laplace Transform
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Laplace Transform
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Laplace Transform
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Laplace transform table
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Laplace transform theorems
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Laplace Transform
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Laplace Transform
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Laplace Transform
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Laplace Transform y”+9y=0, y(0)=0, y’(0)=2
L(y”)=s2Y(s)-sy(0)-y’(0)= s2Y(s)-2 L(y)=Y(s) (s2+9)Y(s)=2 Y(s)=2/ (s2+9) y(t)=(2/3) sin 3t
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Matlab F=2/(s^2+9) F = 2/(s^2+9) >> f=ilaplace(F) f =
2/9*9^(1/2)*sin(9^(1/2)*t) >> simplify(f) ans = 2/3*sin(3*t)
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Laplace Transform y”+2y’+5y=0, y(0)=2, y’(0)=-4
L(y”)=s2Y(s)-sy(0)-y’(0)= s2Y(s)-2s+4 L(y’)=sY(s)-y(0)=sY(s)-2 L(y)=Y(s) (s2+2s+5)Y(s)=2s Y(s)=2s/ (s2+2s+5)=2(s+1)/[(s+1)2+22]-2/[(s+1)2+22] y(t)= e-t(2cos 2t –sin 2t)
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Matlab >> F=2*s/(s^2+2*s+5) F = 2*s/(s^2+2*s+5)
>> f=ilaplace(F) f = 2*exp(-t)*cos(2*t)-exp(-t)*sin(2*t)
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Laplace transform Y”-2 y’-3 y=0, y(0)= 1, y’(0)= 7
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Y”+2 y’+ y=0, y(0)= 1, y’(0)= -2 >> A=[0 1;-1 -2]; B=[0;1]; C=[1 0]; D=0; >> x0=[1;-2]; >> t=sym('t'); >> y=C*expm(A*t)*x0 y = exp(-t)-t*exp(-t) Y”+2 y’+ y=f(t)=u(t), y(0)= 2, y’(0)= 3
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Partial Fraction
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Partial Fraction
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Partial fraction; repeated factor
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Partial fraction; repeated factor
But No FUN
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Partial fraction; exercise
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Matlab >> [r p k]=residue(n,d) r = >> d=[1 -1 0] 1 d = 2
k = [] >> d=[1 -1 0] d = >> n=[3 -2] n = 1/(s-1) + 2/s
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Matlab >> [r p k]=residue(n,d) r = 1.5000 >> n=[1 9 -9]
1.0000 p = 3 -3 k = [] >> n=[1 9 -9] n = >> d=[ ] d = 1.5/(s-3)-1.5/(s+3)+1/s
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Matlab >> [r p k]=residue(n,d) r = 2.0000 -3.0000
1.0000 p = k = [] >> n=[11 -14] n = >> d=[ ] d = 2/(s-2)-3/(s+2)+1/(s-1)
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Matlab >> [r p k]=residue(a,b) r = 1 >> b=[1 2 1] -1 b =
[] >> b=[1 2 1] b = >> a=[1 0] a = 1/(s+1)-1/(s+1)2
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>> Y=(s^4-7*s^3+13*s^2+4*s-12)/s^2/(s-3)/(s^2-3*s+2)
Transfer function: s^4 - 7 s^ s^2 + 4 s - 12 s^5 - 6 s^ s^3 - 6 s^2 >> [n,d]=tfdata(Y,'v') n = d = >> [r,p,k]=residue(n,d) r = 3.0000 2.0000 p = 1.0000 k = [ ]
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