Agenda for transforms (1 of 2)

1
Agenda for transforms (1 of 2)

• 1. System response
• 2. Transforms
• 3. Partial fractions
• 4. Laplace transforms
• 5. Transfer functions
• 6. Laplace applications
• 7. Frequency response

2
1. System response

• Introduction
• Example
• Discrete convolution
• Continuous convolution

1. System response
3
Introduction

• The response of the system to inputs and
  disturbances is important in design
• Differential equations provide insight into this
  response
• Transform methods provide a simpler way of
  solving differential equations
• We will assume linear differential equations
  with constant coefficients

1. System response
4
Example (1 of 4)
system, h
input, r
output, y
h(t)
.99
.95
1
.75
.4
.3
.25
time
0
0 1 2 3 4 5 6
Response of system to a unit impulse ? (t) 1 at
t0
1. System response
5
Example (2 of 4)

• What is the response of the system at t 6 to
  the following inputs
• At t1, input(1) 2 ? (1)
• At t3, input(3) 3 ? (3)
• At t5, input(5) 1 ? (5)

input
y ?
3 2 1 0

time
0 1 2 3 4 5 6
1. System response
6
Example (3 of 4)
? (1) 2
.99
.95
1
.75
.4
.3
.25
y(6) 2 0.3 0.6
0
time
0 1 2 3 4 5 6
.99
.95
? (3) 3
1
.75
.4
.3
.25
0
y(6) 3 0.75 2.25
time
0 1 2 3 4 5 6
.99
.95
1
? (5) 1
.75
.4
.3
.25
y(6) 1 0.99 0.99
time
0
0 1 2 3 4 5 6
y(6) 0.6 2.25 0.99 3.84
7
Example (4 of 4)

• Response of system
• y(6) h(6 -1) 2 ? (1) h(6 -3) 3 ? (3)
  h(6 -5) 1 ? (5)
• This relationship is an example of convolution

1. System response
8
Discrete convolution
?
y(n) ? h(n - k) r(k)
k -?
1. System response
9
Continuous convolution
t
y(t) h(t-?) r(?) d ?
0
1. System response
10
2. Transforms

• Definition
• Examples

2. Transforms
11
Definition

• Transforms -- a mathematical conversion from one
  way of thinking to another to make a problem
  easier to solve

solution in original way of thinking
problem in original way of thinking
transform
solution in transform way of thinking
inverse transform
2. Transforms
12
Example 1
solution in English
problem in English
English to algebra
solution in algebra
algebra to English
2. Transforms
13
Example 2
solution in English
problem in English
English to matrices
solution in matrices
matrices to English
2. Transforms
14
Example 3
solution in time domain
problem in time domain
Laplace transform
solution in s domain
inverse Laplace transform

• Other transforms
• Fourier
• z-transform
• wavelets

2. Transforms
15
3. Partial fractions

• Definition
• Different terms of 1st degree
• Repeated terms of 1st degree
• Different quadratic terms
• Repeated quadratic terms

3. Partial fractions
16
Definition

• Definition -- Partial fractions are several
  fractions whose sum equals a given fraction
• Example --
• (11x - 1)/(x2 - 1) 6/(x1) 5/(x-1)
• 6(x-1) 5(x1)/(x1)(x-1))
• (11x - 1)/(x2 - 1)
• Purpose -- Working with transforms requires
  breaking complex fractions into simpler fractions
  to allow use of tables of transforms

3. Partial fractions
17
Different terms of 1st degree

• To separate a fraction into partial fractions
  when its denominator can be divided into
  different terms of first degree, assume an
  unknown numerator for each fraction
• Example --
• (11x-1)/(x2 - 1) A/(x1) B/(x-1)
• A(x-1) B(x1)/(x1)(x-1))
• AB11
• -AB-1
• A6, B5

3. Partial fractions
18
Repeated terms of 1st degree (1 of 2)

• When the factors of the denominator are of the
  first degree but some are repeated, assume
  unknown numerators for each factor
• If a term is present twice, make the fractions
  the corresponding term and its second power
• If a term is present three times, make the
  fractions the term and its second and third powers

3. Partial fractions
19
Repeated terms of 1st degree (2 of 2)

• Example --
• (x23x4)/(x1)3 A/(x1) B/(x1)2 C/(x1)3
• x23x4 A(x1)2 B(x1) C
• Ax2 (2AB)x (ABC)
• A1
• 2AB 3
• ABC 4
• A1, B1, C2

3. Partial fractions
20
Different quadratic terms

• When there is a quadratic term, assume a
  numerator of the form Ax B
• Example --
• 1/(x1) (x2 x 2) A/(x1) (Bx C)/ (x2
  x 2)
• 1 A (x2 x 2) Bx(x1) C(x1)
• 1 (AB) x2 (ABC)x (2AC)
• AB0
• ABC0
• 2AC1
• A0.5, B-0.5, C0

3. Partial fractions
21
Repeated quadratic terms

• Example --
• 1/(x1) (x2 x 2)2 A/(x1) (Bx C)/ (x2
  x 2) (Dx E)/ (x2 x 2)2
• 1 A(x2 x 2)2 Bx(x1) (x2 x 2)
  C(x1) (x2 x 2) Dx(x1) E(x1)
• AB0
• 2A2BC0
• 5A3B2CD0
• 4A2B3CDE0
• 4A2CE1
• A0.25, B-0.25, C0, D-0.5, E0

3. Partial fractions
22
4. Laplace transform

• Laplace transformation
• Definition
• Transforms

4. Laplace transforms
23
Laplace transformation
time domain
linear differential equation
time domain solution
integration
Laplace transform
inverse Laplace transform
Laplace transformed equation
Laplace solution
algebra
Laplace domain or complex frequency domain
4. Laplace transforms
24
Definition

• The Laplace transform of the function f(t) is

?
F(s)
e-st f(t) dt
0
4. Laplace transforms
25
Transforms (1 of 11)

• Impulse -- ? (to)

?
e-st ? (to) dt
F(s)
0
e-sto
f(t)
? (to)
t
4. Laplace transforms
26
Transforms (2 of 11)

• Step -- u (to)

?
F(s)
e-st u (to) dt
0
e-sto/s
f(t)
u (to)
1
t
4. Laplace transforms
27
Transforms (3 of 11)

• tn

?
F(s)
e-st tn dt
0
n!/sn1
4. Laplace transforms
28
Transforms (4 of 11)

• e-at

?
F(s)
e-st e-at dt
0
1/(sa)
4. Laplace transforms
29
Transforms (5 of 11)

• e-atcos ?t

?
e-st e-atcos ?t dt
F(s)
0
(sa)/(sa)2 ?2
4. Laplace transforms
30
Transforms (6 of 11)

• e-atsin ?t

?
e-st e-atsin ?t dt
F(s)
0
? /(sa)2 ?2
4. Laplace transforms
31
Transforms (7 of 11)
f1(t) ? f2(t) a f(t) eat f(t) f(t T) f(t/a)
F1(s) F2(s) a F(s) F(s-a) eTs F(s) a
F(as)
Linearity Constant multiplication Complex
shift Real shift Scaling
4. Laplace transforms
32
Transforms (8 of 11)

• n-th derivative

Dn f(t)
sn F(s) - Dn-1 f(0) - s Dn-2 f(0) - - sn-1
f(0)
4. Laplace transforms
33
Transforms (9 of 11)

• first derivative

t
1/s F(s)
f(t) dt
0
4. Laplace transforms
34
Transforms (10 of 11)

• convolution integral

t
f1(?) f2(t-?) t) d ?
F1(s)
F2(s)
0
4. Laplace transforms
35
Transforms (11 of 11)

• Most mathematical handbooks have tables of
  Laplace transforms

4. Laplace transforms
36
Solution process (1 of 8)

• Any nonhomogeneous linear differential equation
  with constant coefficients can be solved with the
  following procedure, which reduces the solution
  to algebra

4. Laplace transforms
37
Solution process (2 of 8)

• Step 1 Put differential equation into standard
  form
• D2 y 2D y 2y cos t
• y(0) 1
• D y(0) 0

4. Laplace transforms
38
Solution process (3 of 8)

• Step 2 Take the Laplace transform of both sides
• LD2 y L2D y L2y Lcos t

4. Laplace transforms
39
Solution process (4 of 8)

• Step 3 Use table of transforms to express
  equation in s-domain
• LD2 y L2D y L2y Lcos ? t
• LD2 y s2 Y(s) - sy(0) - D y(0)
• L2D y 2 s Y(s) - y(0)
• L2y 2 Y(s)
• Lcos t s/(s2 1)
• s2 Y(s) - s 2s Y(s) - 2 2 Y(s) s /(s2 1)

4. Laplace transforms
40
Solution process (5 of 8)

• Step 4 Solve for Y(s)
• s2 Y(s) - s 2s Y(s) - 2 2 Y(s) s/(s2 1)
• (s2 2s 2) Y(s) s/(s2 1) s 2
• Y(s) s/(s2 1) s 2/ (s2 2s 2)
• (s3 2 s2 2s 2)/(s2 1) (s2 2s 2)

4. Laplace transforms
41
Solution process (6 of 8)

• Step 5 Expand equation into format covered by
  table
• Y(s) (s3 2 s2 2s 2)/(s2 1) (s2 2s
  2)
• (As B)/ (s2 1) (Cs E)/ (s2 2s 2)
• (AC)s3 (2A B E) s2 (2A 2B C)s (2B
  E)
• 1 A C
• 2 2A B E
• 2 2A 2B C
• 2 2B E
• A 0.2, B 0.4, C 0.8, E 1.2

4. Laplace transforms
42
Solution process (7 of 8)

• (0.2s 0.4)/ (s2 1)
• 0.2 s/ (s2 1) 0.4 / (s2 1)
• (0.8s 1.2)/ (s2 2s 2)
• 0.8 (s1)/(s1)2 1 0.4/ (s1)2 1

4. Laplace transforms
43
Solution process (8 of 8)

• Step 6 Use table to convert s-domain to time
  domain
• 0.2 s/ (s2 1) becomes 0.2 cos t
• 0.4 / (s2 1) becomes 0.4 sin t
• 0.8 (s1)/(s1)2 1 becomes 0.8 e-t cos t
• 0.4/ (s1)2 1 becomes 0.4 e-t sin t
• y(t) 0.2 cos t 0.4 sin t 0.8 e-t cos t
  0.4 e-t sin t

4. Laplace transforms
44
5. Transfer functions

• Introduction
• Example
• Block diagram and transfer function
• Typical block diagram
• Block diagram reduction rules

5. Transfer functions
45
Introduction

• Definition -- a transfer function is an
  expression that relates the output to the input
  in the s-domain

y(t)
differential equation
r(t)
y(s)
transfer function
r(s)
5. Transfer functions
46
Example
R
L
v(t)
C
v(t) R I(t) 1/C I(t) dt L
di(t)/dt V(s) R I(s) 1/(C s) I(s) s L
I(s) Note Ignore initial conditions
5. Transfer functions
47
Block diagram and transfer function

• V(s)
• (R 1/(C s) s L ) I(s)
• (C L s2 C R s 1 )/(C s) I(s)
• I(s)/V(s) C s / (C L s2 C R s 1 )

C s / (C L s2 C R s 1 )
V(s)
I(s)
5. Transfer functions
48
Typical block diagram
reference input, R(s)
plant inputs, U(s)
error, E(s)
output, Y(s)
control Gc(s)
plant Gp(s)
pre-filter G1(s)
post-filter G2(s)
feedback H(s)
feedback, H(s)Y(s)
5. Transfer functions
49
Block diagram reduction rules
Series
U
Y
U
Y
G1
G2
G1 G2
Parallel
Y

U
G1
U
Y

G1 G2
G2
Feedback
Y

U
G1
G1 /(1G1 G2)
U
Y
-
G2
5. Transfer functions
50
6. Laplace applications

• Initial value
• Final value
• Poles
• Zeros
• Stability

6. Laplace applications
51
Initial value

• In the initial value of f(t) as t approaches 0 is
  given by

f(0 ) Lim s F(s)
?
s
Example
f(t) e -t
F(s) 1/(s1)
f(0 ) Lim s /(s1) 1
s
?
6. Laplace applications
52
Final value

• In the final value of f(t) as t approaches ? is
  given by

f(?) Lim s F(s)
s
0
Example
f(t) e -t
F(s) 1/(s1)
f(? ) Lim s /(s1) 0
s
0
6. Laplace applications
53
Poles

• The poles of a Laplace function are the values of
  s that make the Laplace function evaluate to
  infinity. They are therefore the roots of the
  denominator polynomial
• 10 (s 2)/(s 1)(s 3) has a pole at s -1
  and a pole at s -3
• Complex poles always appear in complex-conjugate
  pairs
• The transient response of system is determined by
  the location of poles

6. Laplace applications
54
Zeros

• The zeros of a Laplace function are the values of
  s that make the Laplace function evaluate to
  zero. They are therefore the zeros of the
  numerator polynomial
• 10 (s 2)/(s 1)(s 3) has a zero at s -2
• Complex zeros always appear in complex-conjugate
  pairs

6. Laplace applications
55
Stability

• A system is stable if bounded inputs produce
  bounded outputs
• The complex s-plane is divided into two regions
  the stable region, which is the left half of the
  plane, and the unstable region, which is the
  right half of the s-plane

x
j?
s-plane
x
x
x
x
?
x
stable
unstable
x
56
7. Frequency response

• Introduction
• Definition
• Process
• Graphical methods
• Constant K
• Simple pole at origin, 1/ (j?)n
• Simple pole, 1/(1j ?)
• Simple pole, 1/(1j ?)
• Error in asymptotic approximation
• Quadratic pole

7. Frequency response
57
Introduction

• Many problems can be thought of in the time
  domain, and solutions can be developed
  accordingly.
• Other problems are more easily thought of in the
  frequency domain.
• A technique for thinking in the frequency domain
  is to express the system in terms of a frequency
  response

7. Frequency response
58
Definition

• The response of the system to a sinusoidal
  signal. The output of the system at each
  frequency is the result of driving the system
  with a sinusoid of unit amplitude at that
  frequency.
• The frequency response has both amplitude and
  phase

7. Frequency response
59
Process

• The frequency response is computed by replacing s
  with j ? in the transfer function

Example
f(t) e -t
magnitude in dB
?
F(s) 1/(s1)
F(j ?) 1/(j ? 1) Magnitude 1/SQRT(1
?2) Magnitude in dB 20 log10
(magnitude) Phase argument ATAN2(- ?, 1)

7. Frequency response
60
Graphical methods

• Frequency response is a graphical method
• Polar plot -- difficult to construct
• Corner plot -- easy to construct

7. Frequency response
61
Constant K
magnitude
60 dB
20 log10 K
40 dB
20 dB
0 dB
-20 dB
-40 dB
-60 dB
phase
180o
90o
arg K
0o
-90o
-180o
-270o
0.1 1
10 100
?, radians/sec
7. Frequency response
62
Simple pole, 1/ (j?)n
magnitude
60 dB
40 dB
20 dB
0 dB
1/ ?
-20 dB
-40 dB
1/ ?2
1/ ?3
-60 dB
phase
180o
90o
0o
1/ ?
-90o
1/ ?2
-180o
1/ ?3
-270o
0.1 1
10 100
?, radians/sec
G(s) ?n2/(s2 2? ?ns ? n2)
63
Simple pole, 1/(1j?)
magnitude
60 dB
40 dB
20 dB
0 dB
-20 dB
-40 dB
-60 dB
phase
180o
90o
0o
-90o
-180o
-270o
0.1 1
10 100
?T
7. Frequency response
64
Error in asymptotic approximation
?T 0.01 0.1 0.5 0.76 1.0 1.31 1.73 2.0 5.0 10.0
dB 0 0.043 1 2 3 4.3 6.0 7.0 14.2 20.3
arg (deg) 0.5 5.7 26.6 37.4 45.0 52.7 60.0 63.4 78
.7 84.3
7. Frequency response
65
Quadratic pole
magnitude
60 dB
40 dB
20 dB
0 dB
-20 dB
-40 dB
-60 dB
phase
180o
90o
0o
-90o
-180o
-270o
0.1 1
10 100
?T
7. Frequency response