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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. Elementary functions. The most beautiful equation.

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Background Review

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Background review

Background Review

  • Elementary functions

  • Complex numbers

  • Common test input signals

  • Differential equations

  • Laplace transform

    • Examples

    • properties

    • Inverse transform

    • Partial fraction expantion

  • Matlab


Elementary functions

Elementary functions


The most beautiful equation

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


Elementary functions1

Elementary functions


Elementary functions2

Elementary functions


Elementary functions3

Elementary functions


Elementary functions4

Elementary functions


Elementary functions5

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)


Complex numbers

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)


Polar form of complex numbers

Polar form of Complex Numbers

  • z=x+iy, let’s put x=rcosq, y= rsinq

  • Then z = r(cosq+i sinq) = r cisq = rq

  • Absolute value (modulus) r2=x2+y2

  • Argument q= tan-1(y/x)

  • Example z=1+i


Euler formula

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


In matlab

In Matlab

>> z1=1+2*i

z1 = 1.0000 + 2.0000i

>> z2=3+i*5

z2 = 3.0000 + 5.0000i

>> z3=z1+z2

z3 = 4.0000 + 7.0000i

>> z4=z1*z2

z4 = -7.0000 +11.0000i

>> z5=z1/z2

z5 = 0.3824 + 0.0294i

>> r1=abs(z1)

r1 = 2.2361

>> theta1=angle(z1)

theta1 = 1.1071

>> theta1=angle(z1)*180/pi

theta1 = 63.4349

>> real(z1)

ans = 1

>> imag(z1)

ans = 2


Poles and zeros

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.


Poles and zeros in matlab

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


Test waveforms used in control systems

Test waveforms used in control systems


1 st order differential equations

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 zeroinput

  • Total response: y(t) = zero input response + zero state response = Ce-at + h(t) * f(t)

  • Higher order LODE: use Laplace


Laplace transform

Laplace Transform

  • Definition and examples

Unit Step Function u(t)


Laplace transform1

Laplace Transform


Background review

Name:____________

The single most important thing to remember is that whenever there is feedback, one should worry about __________


Laplace transform2

Laplace Transform


Laplace transform3

Laplace Transform


Laplace transform4

Laplace Transform


Laplace transform5

Laplace Transform


Laplace transform table

Laplace transform table


Laplace transform theorems

Laplace transform theorems


Laplace transform6

Laplace Transform


Laplace transform7

Laplace Transform


Laplace transform8

Laplace Transform


Laplace transform9

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


Matlab

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)


Laplace transform10

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)


Matlab1

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)


Laplace transform11

Laplace transform

  • Y”-2 y’-3 y=0, y(0)= 1, y’(0)= 7

  • Y”+2 y’-8 y=0, y(0)= 1, y’(0)= 8

  • Y”+2 y’-3 y=0, y(0)= 0, y’(0)= 4

  • 4Y”+4 y’-3 y=0, y(0)= 8, y’(0)= 0

  • Y”+2 y’+ y=0, y(0)= 1, y’(0)= -2

  • Y”+4 y=0, y(0)= 1, y’(0)= 1


Background review

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


Partial fraction

Partial Fraction


Partial fraction1

Partial Fraction


Partial fraction repeated factor

Partial fraction; repeated factor


Partial fraction repeated factor1

Partial fraction; repeated factor

But No FUN


Partial fraction exercise

Partial fraction; exercise


Background review

Matlab

>> [r p k]=residue(n,d)

r =

1

2

p =

1

0

k =

[]

>> d=[1 -1 0]

d =

1 -1 0

>> n=[3 -2]

n =

3 -2

1/(s-1) + 2/s


Background review

Matlab

>> [r p k]=residue(n,d)

r =

1.5000

-1.5000

1.0000

p =

3

-3

0

k =

[]

>> n=[1 9 -9]

n =

1 9 -9

>> d=[1 0 -9 0]

d =

1 0 -9 0

1.5/(s-3)-1.5/(s+3)+1/s


Background review

Matlab

>> [r p k]=residue(n,d)

r =

2.0000

-3.0000

1.0000

p =

2.0000

-2.0000

1.0000

k =

[]

>> n=[11 -14]

n =

11 -14

>> d=[1 -1 -4 4]

d =

1 -1 -4 4

2/(s-2)-3/(s+2)+1/(s-1)


Background review

Matlab

>> [r p k]=residue(a,b)

r =

1

-1

p =

-1

-1

k =

[]

>> b=[1 2 1]

b =

1 2 1

>> a=[1 0]

a =

1 0

1/(s+1)-1/(s+1)2


Background review

>> 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^3 + 13 s^2 + 4 s - 12

------------------------------------

s^5 - 6 s^4 + 11 s^3 - 6 s^2

>> [n,d]=tfdata(Y,'v')

n = 0 1 -7 13 4 -12

d = 1 -6 11 -6 0 0

>> [r,p,k]=residue(n,d)

r = 0.5000

-2.0000

-0.5000

3.0000

2.0000

p =3.0000

2.0000

1.0000

0

0

k = [ ]


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