Background Review

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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
• A225, A5
• tan d -4/3, d-53.13o
• F(t)5sin(3t53.13o)

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Complex Numbers

• X210 ? xi where i2-1
• X240, then x2i, or 2j
• If z1x1iy1, z2x2iy2
• Then z1 z2 (x1 x2)i(y1 y2)
• z1 z2(x1iy1)(x2iy2)(x1x2 -y1y2) i(x1y2
  x2y1)

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Polar form of Complex Numbers

• zxiy, lets put xrcosq, y rsinq
• Then z r(cosqi sinq) r cisq r?q
• Absolute value (modulus) r2x2y2
• Argument q tan-1(y/x)
• Example z1i

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Euler Formula

• zxiy
• ez exiy ex eiy ex (cos yi sin y)
• eix cos xi sin x cis x
• eix sqrt(cos2 x sin2 x) 1
• zr(cosqi sinq)r eiq
• Find e1i
• Find e-3i

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In Matlab

• gtgt z112i
• z1 1.0000 2.0000i
• gtgt z23i5
• z2 3.0000 5.0000i
• gtgt z3z1z2
• z3 4.0000 7.0000i
• gtgt z4z1z2
• z4 -7.0000 11.0000i
• gtgt z5z1/z2
• z5 0.3824 0.0294i

• gtgt r1abs(z1)
• r1 2.2361
• gtgt theta1angle(z1)
• theta1 1.1071
• gtgt theta1angle(z1)180/pi
• theta1 63.4349
• gtgt real(z1)
• ans 1
• gtgt 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

• gtgt stf(s)
• Transfer function s
• gtgt Gexp(-2s)/s/(s1)
• Transfer function
• 1
• exp(-2s) -----------
• s2 s
• gtgt pole(G)
• ans 0, -1
• gtgt 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

• y9y0, y(0)0, y(0)2
• L(y)s2Y(s)-sy(0)-y(0) s2Y(s)-2
• L(y)Y(s)
• (s29)Y(s)2
• Y(s)2/ (s29)
• y(t)(2/3) sin 3t

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Matlab
F2/(s29) F 2/(s29) gtgt filaplace(F) f
2/99(1/2)sin(9(1/2)t) gtgt simplify(f)
ans 2/3sin(3t)
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Laplace Transform

• y2y5y0, y(0)2, y(0)-4
• L(y)s2Y(s)-sy(0)-y(0) s2Y(s)-2s4
• L(y)sY(s)-y(0)sY(s)-2
• L(y)Y(s)
• (s22s5)Y(s)2s
• Y(s)2s/ (s22s5)2(s1)/(s1)222-2/(s1)222
  
• y(t) e-t(2cos 2t sin 2t)

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Matlab
gtgt F2s/(s22s5) F 2s/(s22s5) gtgt
filaplace(F) f 2exp(-t)cos(2t)-exp(-t)sin
(2t)
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Laplace transform

• Y-2 y-3 y0, y(0) 1, y(0) 7
• Y2 y-8 y0, y(0) 1, y(0) 8
• Y2 y-3 y0, y(0) 0, y(0) 4
• 4Y4 y-3 y0, y(0) 8, y(0) 0
• Y2 y y0, y(0) 1, y(0) -2
• Y4 y0, y(0) 1, y(0) 1

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Y2 y y0, y(0) 1, y(0) -2 gtgt A0 1-1
-2 B01 C1 0 D0 gtgt x01-2 gtgt
tsym('t') gtgt yCexpm(At)x0 y
exp(-t)-texp(-t) Y2 y yf(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
gtgt r p kresidue(n,d) r 1 2 p
1 0 k
gtgt d1 -1 0 d 1 -1 0 gtgt n3
-2 n 3 -2
1/(s-1) 2/s
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Matlab
gtgt r p kresidue(n,d) r 1.5000
-1.5000 1.0000 p 3 -3 0 k

gtgt n1 9 -9 n 1 9 -9 gtgt d1 0 -9
0 d 1 0 -9 0
1.5/(s-3)-1.5/(s3)1/s
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Matlab
gtgt r p kresidue(n,d) r 2.0000
-3.0000 1.0000 p 2.0000 -2.0000
1.0000 k
gtgt n11 -14 n 11 -14 gtgt d1 -1 -4 4 d
1 -1 -4 4
2/(s-2)-3/(s2)1/(s-1)
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Matlab
gtgt r p kresidue(a,b) r 1 -1 p
-1 -1 k
gtgt b1 2 1 b 1 2 1 gtgt a1 0 a
1 0
1/(s1)-1/(s1)2
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gtgt Y(s4-7s313s24s-12)/s2/(s-3)/(s2-3s
2) Transfer function s4 - 7 s3 13 s2 4 s
- 12 ------------------------------------ s5
- 6 s4 11 s3 - 6 s2 gtgt n,dtfdata(Y,'v') n
0 1 -7 13 4 -12 d 1
-6 11 -6 0 0 gtgt r,p,kresidue(n,d
) r 0.5000 -2.0000 -0.5000
3.0000 2.0000 p 3.0000 2.0000
1.0000 0 0 k