Chapter 4 Transfer Function and Block Diagram
Operations
§ 4.1 Linear Time-Invariant Systems
§ 4.2 Transfer Function and Dynamic Systems
§ 4.3 Transfer Function and System Response
§ 4.4 Block Diagram Operations for Complex Systems
1
§ 4.1 Linear Time-Invariant Systems (1)
• LTI Systems:
I.C.
r(t)
y(t)
LTI
Differential Equation Formulation
y (n )  qn1y (n1)  ......  q0 y  pm1r (m1)  pm2r (m2 )  ......  p0r
n  m for causal system
I.C. y(0),y (1) (0),......y (n1) (0)
r(0),r (1) (0),......r (m2 ) (0)
y( t )  y(I.C., system,input)
2
§ 4.1 Linear Time-Invariant Systems (2)
• Solution Decomposition:
y(t)=y(I.C., system)+y(system, input)
y(I.C., system)=yh(t)
I.C.-dependent solution
Homogeneous solution
Natural response
Zero-input response
y(system, input)=yp(t)
Forcing term dependent solution
Particular solution
Forced response
Zero-state response
3
§ 4.1 Linear Time-Invariant Systems (3)
• Solution Modes:
Characteristic equation
λn  qn1 λn1  ......  q0  0
Re al roots λ j  0
Eigen value
Real distinct roots λ j , j = 1,2,......
Real repeated roots λ 1 = λ 2 = ......
Complex roots λ j = α j ± β ji
Solution modes
t k eλ it
k  0, 1, 2, ......, ni  1
ni : Multiplici ty of eigenvalue λ i
4
§ 4.1 Linear Time-Invariant Systems (4)
Real roots λ j  0
λj 0
λj 0
Amplitude
Amplitude
Amplitude
constant
exponential
growth
exponential
decay
t
t
t
Imaginary roots λ j  β ji
Amplitude
sinusoidal
t
Complex roots λ j  α j  β ji
Amplitude
αj  0
modulated decay
sinusoidal
t
5
Amplitude
modulated growth
sinusoidal
αj  0
t
§ 4.1 Linear Time-Invariant Systems (5)
• Output Response:
(1) y(t)=yh(t)+yp(t)
yh(t): Linear combination of solution modes
yp(t): Same pattern and characteristics as the forcing function
The RH side of LTI model affects only the coefficients of solution modes.
The LH side of LTI model dominates the solution modes of the transient
response.
(2) y(t)=ys(t)+yt(t)
lim y t ( t )  0
t 
yt(t): Transient solution
ys(t): Steady state solution
Transient solution is contributed by initial condition and forcing function.
6
§ 4.2 Transfer Function and Dynamic Systems (1)
Input
System
r(t)
Output
y(t)
G
Input is transfered through system G to output.
• Definition:
G s 
Loutput 
L y t 
Transformof forcedresponse


Linput  IC0 Lr  t  IC0
Transformof input
L   Laplace Transform
I.C.  0 : Informatio n of initial condition is ignored.
Initial relaxation of a system.
Key points: Linear, Time-Invariant, Zero initial condition
L
t - Domain
s - Domain
L-1
7
§ 4.2 Transfer Function and Dynamic Systems (2)
Pierre-Simon Laplace (1749 ~ 1827)
Monumental work “ Traite de mécanique céleste ”
8
§ 4.2 Transfer Function and Dynamic Systems (3)
Laplace Transform
•
f(t)
Definition:
0, t  0
Time function f (t) 
f (t),t  0
L[f(t)]  F(s) 


0
 st
0- 0 0+
t
f(t)e dt
s    j
•
Existence Condition

0
•

f ( t ) e σ1t dt  , for σ1  0
Inverse Laplace Transform
1   j
L [F(s)] 
F(s)est ds

2j  j
1
Signals that are physically realizable (causal) always has a Laplace transform.
9
§ 4.2 Transfer Function and Dynamic Systems (4)
•
Important Properties:
t – Domain
s – Domain
Linearity
af1(t)  bf2 (t)
aF1(s)  bF2 (s)
Time shift
e-at f  t 
F(s  a)
Scaling
t
f( )
a
aF(as)
Final value theorem
lim f(t)
lim sF(s)
Initial value theorem
f(0 )
t 
t
Convolution
Differentiation
Integration
 0 f(t-)g()d or
df(t)
dt

t
0
f( )d
s 0
lim sF(s)
s

t
0
f()g(t  )d F(s)G(s)
sF(s)-f(0)
F(s) 1 t
  f( )d
s
s 0
t  0
10
§ 4.2 Transfer Function and Dynamic Systems (5)
• Signal:
Unit impulse
f(t),t ≥ 0
F(s)
 t 
1
t
0
1
Unit step
Us ( t )
t
0
Ramp
t
1
t
0
e -at , a  0
1
Exponential decay
t
0
Sine wave
0
t
sinω0 t
t
cosω0 t
2π
ω0
Cosine wave
0
2π
ω0
11
1
s
1
s2
1
sa
ω0
s2  ω0
s
2
s2  ω0
2
§ 4.2 Transfer Function and Dynamic Systems (6)
• Fundamental Transfer Function of Mechanical System:
Elements
Static element
(Proportional element)
Integral element
Differential element
Transportation lag
12
Function
Block Diagram
T.F.
Example
§ 4.2 Transfer Function and Dynamic Systems (7)
• States and Constitutive Law of Physical Systems:
Flux    Edt
Flow(r) Quantity Effort(E)
E
M
H
T
Electrical
Mechanical
Fluid
Thermal
i
v
q
qt
Q
x
Q
Qt
Effort
(E)
13
Impulse
Flux
()
1
s
E=f(r)
Q=f(E)
C=Q/E

e
F
P
T
R=E/r
C
R
L
Quantity
(Q)
1
s
Flow
(r)
 =f(r)
 =Lr
§ 4.2 Transfer Function and Dynamic Systems (8)
• Analog Physical Systems:
R
L
C
i
e(t)
L
di
1 t
 Ri   idt  e(t )
dt
c 0
y=x
C
K
f(t)
M
frictionless
t
dv
M
 cv  k  vdt  f (t )
0
dt
14
§ 4.2 Transfer Function and Dynamic Systems (9)
• Inverse Laplace Transform and Partial Fraction Expansion:
F(s) 
N(s)
D(s)
bmsm  bm1sm1  ......  b1s  b0

, m  n, ai , bi  R
n
n1
s  an1s  ......  a1s  a0
Roots of D(s)=0:
(1) Real and distinct roots
s  r1,  r2 , ......  rn
F(s) 

N(s)
(s  r1 )(s  r2 )......(s  rn )
c1
c
c
 2  ......  n
s  r1 s  r2
s  rn
c i  lim [(s  ri )F(s)]
s  ri
From Laplace transform pairs
n
f ( t )  L [F(s)]   c ierit
1
i1
15
§ 4.2 Transfer Function and Dynamic Systems (10)
(2) Real repeated roots
s  r1, with multiplici ty k
F(s) 

N(s)
(s  r1 )k (s  r2 )......(s  rn )
c1
c k 1
ck

......


s  r1
(s  r1 )k (s  r1 )k 1

dn-1
d1
 ...... 
(s  rn )
(s  r2 )
c k  lim [(s  r1 )k F(s)]
s  r1
dk i
1
[(s  r1 )k F(s)]},i  1, 2, ......k
c i  lim {
)
i

k
(
s  r1 (k  i)! ds
d1, ......dn-1 are obtained by follow ing(1)
From Laplace transform pairs
k
c i i1 r1t n
f (t )  
t e  di1erit
i1 (i  1)!
i2
16
§ 4.2 Transfer Function and Dynamic Systems (11)
(3) Complex conjugate pairs with real distinct roots
s2  as  b  0, a 2  4b  0
N(s)
F(s)  2
(s  as  b)(s  r1 )......(s  rn )

c 1s  c 2
d1
dn


......

s2  as  b s  r1
s  rn
c1, c 2 , di are obtained by balancing equation
w ith
A(s  c )  Bω0
c 1s  c 2

2
s2  as  b
(s  c )2  ω0
From Laplace transform properties and pairs
ct
f ( t )  Ae cosω0 t  Be
17
ct
n
sinω0 t   die rit
i1
§ 4.2 Transfer Function and Dynamic Systems (12)
• Dynamic System Equation and Transfer Function:
Differential Equation and Transfer Function
I.C.
r(t)
Differential
Equation
y(t)
Differential Equation:
y  n   qn1y  n1   q0 y
 pm1r  m1  pm2r  m2    p0r
I.C. :
y(0),y (1) 0  y  n1 0 
Y(s)
R(s)
G(s)
Transfer Function:
G(s) 
L y t 
Lr  t  IC0
pm1sm1  pm2 sm2   p0

sn  qn1sn1   q0
r(0),r (1) 0 r  m2  0 
Problems associated with differentiation of noncontinuous functions, ex.
step function, impulse function.
18
§ 4.2 Transfer Function and Dynamic Systems (13)
Integral Equation and Transfer Function
t
y( t )   0 g( t  )r( )d
‧ R(s)
* r(t)
Y(s)
G(s)
g(t)
y(t)
Impulse response,r(t)  (t)
Impulse response,R(s)  1
y(t) g(t)
Y(s) G(s)
 L[g(t)]I.C.0  G(s)
The transfer function of a system is the Laplace transform of its
impulse response
 L1[G(s)]  g(t ), g(t) is obtained by using partial fractionexpansion
19
§ 4.3 Transfer Function and System Response (1)
• Transfer Function G(s):
R(s)
Y(s)
G(s)
N(s)
, D(s),N(s) are polynomial s of s
D(s)
pm1sm1  pm2sm2  ......  p0
G(s) 
sn  qn1sn1  ......  q0
Rational T.F. G(s) 
Irrational T.F. G(s) 
N(s)
, Ex : G(s)  e-Ts , T is a constant
D(s)
Proper T.F.
lim G(s)  , i.e. Order of N(s)  Order of D(s)
s 
(m  n)
20
§ 4.3 Transfer Function and System Response (2)
• Response by T.F.:
Y(s)  G(s)  R(s), G(s) 
 Y(s)
p(s)
q(s)
m(s) p(s)

R(s)
q(s) q(s)
Natural response Forced response
n(s)
d(s)
m(s) p(s) n(s) m(s) l(s) k(s)
 Y(s)





q(s) q(s) d(s) q(s) q(s) d(s)
m(s)
l(s)
k(s)
Y1(s) 
, Y2 (s) 
, Y3 (s) 
q(s)
q(s)
d(s)
If R(s) 
y(t)  L1[Y(s)] y1(t)  y 2 (t)  y 3 (t)
Transient response
Stedy state response
Partial fraction expansion is employed to find y(t).
21
§ 4.3 Transfer Function and System Response (3)
Ex: y  3y  2y  r (t ), r(t) is a unit step function
y(0)  1, y (0)  0
By Laplace transform
22
System : [s2 Y(s)  sy(0)  y(0)]  3[sY(s)  y(0)]  2Y(s)  R(s)
1
Input : R(s) 
s
s3
1
 Y(s)  2

s  3s  2 s(s2  3s  2)
5
1
2
1
2
2
By partial fraction expansion :Y(s)  (

)(

) 2
s 1 s  2
s 1 s  2
s
2
1
By Inverse Laplace transform: y1( t )  L1[ Y1(s)]  L1[

]
s 1 s  2
5
1
1
1  2

1

1
y 2 ( t )  L [ Y2 (s)]  L [
 2 ], y 3 ( t )  L [ Y3 (s)]  L [ 2 ]
s 1 s  2
s
5
1
 y(t)  L1[Y(s)]  2e  t  e 2t  2e  t  e 2t 
2
2
1
Steady state response:
2
§ 4.3 Transfer Function and System Response (4)
•
Poles, Zeros, and Pole-zero Diagram:
For an irreducible proper rational transfer function G(s),
a number λ (real or complex) is said to be
a pole if G(λ )  , notation " x"
a zero if G(λ )  0, notation " o"
Pole-zero diagram
Representation of poles and zeros distribution by using “x” and “o”,
respectively in complex plane along with static gain.
jω
Pole - zero diagram
s 1
1

Ex: G(s)  2
Static gain = 1/2
s  3s  2 s  2

G( 2)  , if s  -2  pole : s  -2
-2
1 s5
1
s5
Ex: G(s) 

5 s2  s  1 5
1  3i
1  3i
Pole - zero diagram
(s 
)(s 
)
jω
2
2
1
poles: s  1  3i , zeros : s  5
2
Characteristic Equation
-5
i.e. D(s)  0, sn  qn1sn1  ......q1s  q0  0
characteristic roots: The roots of characteristic equation
i.e. The poles of G(s).
23

§ 4.3 Transfer Function and System Response (5)
• Impulse Response of Poles Distribution
jω
σ
24
§ 4.3 Transfer Function and System Response (6)
• Effects of Poles and Zeros
A pole of the input function generates the form of the forced response.
A pole of the transfer function generates the form of the natural response.
The zeros and poles of transfer function generate the amplitude for both
the forced and natural responses.
The growth, decay, oscillation, and their modulations determined by the
impulse response of the poles distribution.
25
§ 4.3 Transfer Function and System Response (7)
Pole - zero diagram
Ex: Find y(t)
jω
1
1
s2
r(t)
1/2
y(t)

Sol: Y(s)  G(s)  R(s)
-2
1 1

s2 s
1
1

 2 2
s2 s
1
1
 y(t)  L1[Y(s)]   e-2t 
2
2
Ex : Find responsepattern

1
s
1
s 1
(s  2)(s  4)(s  5)
Response
y (t)
1/2
1/2
1 -2t
e
2
t
t
t
jω
y(t)
1/40
k1e-2t , k 2e-4t , k 3 e-5t
 y(t)  k1e-2t  k 2e-4t  k 3 e-5t 
1/2
=
Pole - zero diagram
Y(s)
By inspection, solution modes are
26
-

-5
-4
steady - state response
1
40
-2
-1
§ 4.4 Block Diagram Operations for Complex Systems (1)
• Fundamental Operations:
Signal operation
X1
Summer
Y(s)=X1(s)+X2(s)
Comparator
Y(s)=X1(s)-X2(s)
Take-off point
Y(s)=X1(s)
+
X2
X1
X
Parallel
X
G1
G2
G1
X
GG
m
H
27
X2
X1
Y
Y
G2
Feedback
Y
+
-
Component combinations
Serial
Y
+
Y
X
G1  G 2
Y
Y
X
G1  G2
Y
Y
X

G
1  GH
Y
§ 4.4 Block Diagram Operations for Complex Systems (2)
Moving junction / sequence
X
Ahead of a block
Z1
G
X
+
Z1
G
+
Z2
Z2
1/G
Past a block
X
+
Z1
G
X
Z1
G
+
Z2
G
Z2
Exchange sequence
28
X
G1
Z1
G2
Z2
X
G2
Z3
G1
Z2
§ 4.4 Block Diagram Operations for Complex Systems (3)
• Negative Feedback System:
R(s) + Ea(s)
G
Y(s)

R(s)
1
H
+ E(s)
GH
Y(s)
H
Y(s)
G

R(s) 1  GH
Y(s) 1 GH
 (
)
R(s) H 1  GH
Forward Gain : G
Forward Gain : GH
1
Cascade Gain :
H
Feedback Gain : H
Loop Gain : GH
Loop Gain : GH
Actuating Error Signal: Ea (s)
True Error Signal: E(s)
Note : Open - loop transfer functionGH
G
Closed - loop systemtransfer function
1+ GH
29
§ 4.4 Block Diagram Operations for Complex Systems (4)
• Loading Effect:
Cascade
R1
R2
C1
V1(s)
V3(s)
V2(s)
G1
G1(s) 
G2 (s) 
V1(s)
30
Isolated Amp by 741OP
Isolated Amp
K=1
K
C1
V4 (s)
1

V3 (s) R2C2s  1
1
(R1C1R2C2 )s2  (R1C1  R2C2 )s  1
Realization
R1
V4(s)
G2
V2 (s)
1

V1(s) R1C1s  1
 G1(s)G2 (s) 
C2
R2
C2
V4(s)
+
K=1
§ 4.4 Block Diagram Operations for Complex Systems (5)
• History of Operational Amplifier:
OP was first built with vacuum tubes. Originally designed by C. A. Lovell of Bell
Lab. and was used to control the movement of artillery during World War Ⅱ.
31
1965
1968
Fairchild develops the first OpAmp
(operational amplifier) generally used
throughout the industry--a milestone in
the linear integrated circuit field.
Fairchild introduces an OpAmp
(operational amplifier) that is one of the
earliest linear integrated circuits to
include temperature compensation and
MOS capacitors.
§ 4.4 Block Diagram Operations for Complex Systems (6)
• Operational Amplifier:
7
(+) Supply voltage,
Vcc (usually DC 15V)
2
Inverting
input
3
Noninverting
input
mA 741
+
Output
6
Av
Zi
Zo
ideal
15
Offset
null adjust
4
Offset null adjust
32
(-) Supply voltage, Vee
(usually DC -15V)
1
8
NC
ln-
2
7
Vcc+
ln+
3
6
Out
Vcc-
4
5
Offset null adjust
Av
Zi
Zo


0
§ 4.4 Block Diagram Operations for Complex Systems (7)
• Network 1:
R2
R1
V(s)
V4(s)
V1(s)
V1(s)
⇒ G1(s) 
C1
V(s)
I(s)
C2
V4(s)
I(s)
1
(R1C1R2C2 )s2  (R1C1  R2C2  R1C2 )s  1
Loading effect
• Network 2:
R2
R1
V(s)
V2(s)
V3(s)
V3(s)
C2
V(s)
C1
V2(s)
I(s)
⇒ G2 (s) =
1
(R1C1R2C2 )s2 + (R1C1 + R2C2 + R2C1 )s + 1
G1(s)  G1(s)G2 (s)  G2 (s)
33
I(s)
Loading effect
§ 4.4 Block Diagram Operations for Complex Systems (8)
Note: For MIMO System
Y(s)  G(s)R(s)
Output
Vector
Transfer
Matrix
Input
Vector
Y1(s)
R1(s)
G(s)
Y2 (s)
R 2 ( s)
Y1(s )  G11(s ) G12 (s )   R1(s ) 
Y (s )  G (s ) G (s ) R (s )
 2   21
22
 2 
+
G11
R1(s)
Y1(s)
+
34
1
G
12
G2
R2(s)
G22
+
+
Y2(s)
§ 4.4 Block Diagram Operations for Complex Systems (9)
Ex: Armature control DC servomotor
ia(t) Ra
if =const.
+
va(t)
or Permanent magnet
eb(t) M
La
-
Ttd(t)
m (t) ,Ttm(t) , Jm , Bm, m (t)
Static characteristics (Ideal)
Tt
t
max
T
K T max
(Stall)
v max
T t  K T v  KB
max

Kb 
t
Tmax

v max
v max
max
(Stall)
 0

T 0
t
K i  RaK T max
KB
KT
(No load)
max
t
Tmax

imax
35
§ 4.4 Block Diagram Operations for Complex Systems (10)
Dynamic characteristics
Tdt (s) Load disturbance
Va (s)
+
-
1
Las
ia
Ki
+
1
-
Ra
m (s)
Jm s
1
s
m (s)
Bm
Kb
Electronics
Mechatronics
Mechanics
I/O Block Diagram Reduction
m (s)
Ki

Va (s) s[(Ra  Las)(Jms  Bm )  K bK i ]
mt (s)  mv (s) 
Total
Response
Command
Response
mT (s)
Disturbance
Response
36
§ 4.4 Block Diagram Operations for Complex Systems (11)
Model Reduction
(1) La  0
Define K m 
m 
Ki
, Motor gain constant
RaBm  K iK b
Ra Jm
, Motor time constant
RaBm  K iK b
m (s)
Km

Va (s) s(1  ms)
(2) La  0, Bm  0
Va (s) +
1
Ra
-
m (s)
1
Jm S
Ki
Kb
Va (s)
Km
37
1
1  m S
m (s)
Km =
1
Κb
m =
R a Jm
J
, m max = m
K iK b
K TK b
1
S
=
max
m (s)
max Jm
Tmax
1
S
m (s)
§ 4.4 Block Diagram Operations for Complex Systems (12)
(3) If La  0, Bm  0, m  0
m (s)
Va (s)
Km
1
S
m (s)
Ideal Servomotor
m (s)
Static Gain
1
S
m (s)
Pure Integrator
Static gain is dominated by feedback gain Kb=1/Km in system dynamics.
Key points:
Linear time-invariant motor
No load
No delay
No damping
No inertia
No resistance
No inductance
38