1. Background Review Elementary functions Complex numbers
Common test input signals
Differential equations
Laplace transform
Examples
properties
Inverse transform
Partial fraction expantion
Matlab

2. Elementary functions

3. 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

4. Elementary functions

5. Elementary functions

6. Elementary functions

7. Elementary functions

8. 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)

9. 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)

10. 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

11. 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

12. 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

13. 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.

14. 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

15. Test waveforms used in control systems

16. 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

17. Laplace Transform
Definition and examples
Unit Step Function u(t)

18. Laplace Transform

19. Name:____________
The single most important thing to remember is that whenever there is feedback, one should worry about __________

20. Laplace Transform

21. Laplace Transform

22. Laplace Transform

23. Laplace Transform

24. Laplace transform table

25. Laplace transform theorems

26. Laplace Transform

27. Laplace Transform

28. Laplace Transform

29. 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

30. 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)

31. 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)

32. 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)

33. Laplace transform Y”-2 y’-3 y=0, y(0)= 1, y’(0)= 7

34. 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

35. Partial Fraction

36. Partial Fraction

37. Partial fraction; repeated factor

38. Partial fraction; repeated factor
But No FUN

39. Partial fraction; exercise

40. 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

41. 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

42. 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)

43. 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

44. >> 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 = [ ]