chapter 4 transfer function and block diagram operations § 4.1 linear time-invariant systems § 4.2...
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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:
y(t)r(t)LTI
I.C.
Differential Equation Formulation
input) system, .,C.I(y)t(y
)0(r),......0(r r(0),
)0(y),......0( y y(0),I.C.
system causal for mn
rp......rprpyq...... yqy
)2m((1)
)1n((1)
0)2m(
2m)1m(
1m0)1n(
1n)n(
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
Eigen value
Solution modes
0q...... λ qλ 01n
1nn
iβ±α=λ roots Complex
......=λ=λ roots repeated Real
1,2,......=j ,λ roots distinct Real
jjj
21
j
ii
i
tλk
λ eigenvalue ofty Multiplici :n
1n ......, 2, 1, ,0k
et i
0λ roots alRe j
4
§ 4.1 Linear Time-Invariant Systems (4)
exponentialdecay
t
Amplitude
exponentialgrowth
t
Amplitude
sinusoidal
t
Amplitude
modulated decaysinusoidal
t
Amplitude
modulated growthsinusoidal
t
Amplitude
0λ roots Real j 0λ j
constant
t
Amplitude
0λ j
iβλ rootsmaginary I jj
iβαλ rootsomplex C jjj
0α j 0α j
5
§ 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)
yt(t): Transient solution
ys(t): Steady state solution
Transient solution is contributed by initial condition and forcing function.
0)t(ylim tt
6
§ 4.2 Transfer Function and Dynamic Systems (1)
Input is transfered through system G to output.
• Definition:
Key points: Linear, Time-Invariant, Zero initial condition
y(t)r(t)
InputSystem
Output
G
system. a of relaxation Initial
ignored. is condition initial of nInformatio :0I.C.
Transform Laplace L
input of Transform
response forced of Transform
trL
tyL
inputL
outputLsG
0CI0CI
t - Domain s - DomainL
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• Definition:
Time function
• Existence Condition
• Inverse Laplace Transform
Signals that are physically realizable (causal) always has a Laplace transform.
0t f(t),
f(t)
0t ,0
st
0L[f(t)] F(s) f(t)e dt
s j
0σ for ,dte)t(f 1
0
tσ1
j1 st
j
1L [F(s)] F(s)e ds
2 j
0+0- t
f(t)
0
9
§ 4.2 Transfer Function and Dynamic Systems (4)
• Important Properties: t – Domain s – Domain Linearity Time shift
Scaling
Final value theorem
Initial value theorem
Convolution
Differentiation
Integration
1 2 1 2 af (t) bf (t) aF (s) bF (s)
a)F(s tfe -at t
f( ) aF(as)a
s f(0 ) lim sF(s)
t t
0 0f(t- )g( )d or f( )g(t )d F(s)G(s)
df(t) sF(s)-f(0)
dt
t t
0 0 t 0
F(s) 1 f( )d f( )d
s s
t s 0 lim f(t) lim sF(s)
10
§ 4.2 Transfer Function and Dynamic Systems (5)• Signal:
Unit impulse Unit step
Ramp
Exponential decay
Sine wave
Cosine wave
1 t
s
1 )t(U s
2s
1 t
as
1 0a ,e at-
20
20
0ωs
ω tωins
20
20ωs
s tωcos
F(s) 0t f(t), ≥
t0
t0
1
t0
1
t0
t0
t0
1
0ω
π2
0ω
π2
11
§ 4.2 Transfer Function and Dynamic Systems (6)
• Fundamental Transfer Function of Mechanical System:
Elements Function Block Diagram T.F. Example Static element(Proportional element)
Integral element
Differential element
Transportation lag
12
§ 4.2 Transfer Function and Dynamic Systems (7)
Flow(r) Quantity Effort(E) Flux Edt
E Electrical i Q e
M Mechanical v x F Impulse H Fluid q Q P T Thermal qt Qt T
Effort (E)
=f(r)=Lr
Flow (r)
Quantity (Q)
Q=f(E)C=Q/E
R=E/rE=f(r)
RC L
Flux( )s
1
s
1
• States and Constitutive Law of Physical Systems:
13
§ 4.2 Transfer Function and Dynamic Systems (8)
• Analog Physical Systems:
C
K M
y=x
f(t)
frictionless
0 ( )tdv
M cv k vdt f tdt
01
( )tdi
L Ri idt e tdt c
R L C
e(t)
i
14
§ 4.2 Transfer Function and Dynamic Systems (9)
• Inverse Laplace Transform and Partial Fraction Expansion:
Rb ,a n,m ,asa......sas
bsb......sbsb
)s(D
)s(N)s(F
ii01
1n1n
n01
1m1m
mm
Roots of D(s)=0:
(1) Real and distinct roots
From Laplace transform pairs
)]s(F)rs[(limc
rs
c......
rs
c
rs
c
)rs)......(rs)(rs(
)s(N)s(F
r...... ,r ,rs
irs
i
n
n
2
2
1
1
n21
n21
i
n
1i
tri
1 iec)]s(F[L)t(f
15
§ 4.2 Transfer Function and Dynamic Systems (10)(2) Real repeated roots
From Laplace transform pairs
(1) followingby obtained are ......d ,d
......k 2, 1,i )]},s(F)rs[(ds
d
)!ik(
1{limc
)]s(F)rs[(limc
)rs(
d ......
)rs(
d
rs
c......
)rs(
c
)rs(
c
)rs)......(rs()rs(
)s(N)s(F
kty multiplici ithw ,rs
1-n1
k1)ik(
ik
rsi
k1
rsk
n
1-n
2
1
1
11k
1
1kk
1
k
n2k
1
1
1
1
n
2i
tr1i
k
1i
tr1ii i1 edet)!1i(
c)t(f
16
§ 4.2 Transfer Function and Dynamic Systems (11)(3) Complex conjugate pairs with real distinct roots
From Laplace transform properties and pairs
20
20
221
i21
n
n
1
12
21
n12
22
ω)cs(
ωB)cs(A
bass
csc with
equation balancingby obtained are d ,c ,c
rs
d......
rs
d
bass
csc
)rs)......(rs)(bass(
)s(N)s(F
0b4a ,0bass
n
1i
tri0
ct0
ct iedtωsinBetωcosAe)t(f
17
§ 4.2 Transfer Function and Dynamic Systems (12)
• Dynamic System Equation and Transfer Function: Differential Equation and Transfer Function
Differential Equation: Transfer Function:
Problems associated with differentiation of noncontinuous functions, ex. step function, impulse function.
y(t)r(t)DifferentialEquation
I.C.
Y(s)R(s)G(s)
0r0r r(0),
0y0 y y(0),
: I.C.
rprprp
yq yqy
2m(1)
1n(1)
02m
2m1m
1m
01n
1nn
01n
1nn
02m
2m1m
1m
0CI
qsqs
pspsp
trL
tyL(s)G
18
§ 4.2 Transfer Function and Dynamic Systems (13)
Integral Equation and Transfer Function
The transfer function of a system is the Laplace transform of its impulse response
Y(s)R(s)G(s)
r(t)g(t)
y(t)
‧
* d)(r)t(g)t(y
t
0
1R(s) response, Impulse
expansion fraction partial usingby obtained is g(t) ),t(g)]s(G[L 1
(t)r(t) response, Impulse
g(t) y(t)
)s(GL[g(t)]0.C.I
G(s)Y(s)
19
§ 4.3 Transfer Function and System Response (1)
• Transfer Function G(s):
Rational T.F.
Irrational T.F.
Proper T.F.
Y(s)R(s)G(s)
s of spolynomial are N(s) D(s), , )s(D
)s(N)s(G
01n
1nn
02m
2m1m
1m
q......sqs
p......spsp)s(G
constant a is T ,eG(s) :Ex , )s(D
)s(N)s(G Ts-
n)(m
D(s) of Order N(s) of Order i.e. ,G(s) lims
20
§ 4.3 Transfer Function and System Response (2)
• Response by T.F.:
Partial fraction expansion is employed to find y(t).
R(s)q(s)
p(s)
q(s)
m(s) Y(s)
q(s)
p(s)G(s) ,)s(R)s(G)s(Y
(t)y(t)y(t)y[Y(s)]L y(t)
d(s)
k(s) (s) Y,
q(s)
l(s) (s) Y,
q(s)
m(s)(s) Y
d(s)
k(s)
q(s)
l(s)
q(s)
m(s)
d(s)
n(s)
q(s)
p(s)
q(s)
m(s) Y(s)
d(s)
n(s))s(R If
3211
321
response Natural response Forced
response Transient response stateStedy
21
§ 4.3 Transfer Function and System Response (3)
Ex:
3 2 ( ), r(t) is a unit step function
y(0) 1, (0) 0
By Laplace transform
y y y r t
y
2
2 2
System : [s Y(s) sy(0) y(0)] 3[sY(s) y(0)] 2Y(s) R(s)
1 Input : R(s)
ss 3 1
Y(s)s 3s 2 s(s 3s 2)
5 12 1 2 2 2 By partial fraction expansion :Y(s) ( ) ( ) s 1 s 2 s 1 s 2 s
]s2
1[L)]s(Y[L)t( y],
2s2
5
1s
2[L)]s(Y[L)t(y
]2s
1
1s
2[L)]s(Y[L)t(y:transform Laplace Inverse By
13
13
12
12
11
11
1 t 2t t 2t5 1 y(t) L [Y(s)] 2e e 2e e
2 21
Steady state response: 2
22
§ 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 (real or complex) is said to be
Pole-zero diagram Representation of poles and zeros distribution by using “x” and “o”, respectively in complex plane along with static gain.
Ex:
Ex:
Characteristic Equation i.e. characteristic roots: The roots of characteristic equation i.e. The poles of G(s).
o"" notation ,0)λG( if zero a
x"" notation ,)λ(G if pole a
-2s :pole-2s if ,)2(G2s
1
2s3s
1sG(s)
2
2
1 s 5 1 s 5G(s)
5 s s 1 5 1 3i 1 3i(s )(s )
2 2
1 3ipoles: s , zeros : s 52
0qsq......sqs ,0)s(D 011n
1nn
ωj
-2
Static gain = 1/2
ωj
-5
1
diagram zero-Pole
diagram zero-Pole
λ
23
§ 4.3 Transfer Function and System Response (5)
• Impulse Response of Poles Distribution
ωj
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
y(t)r(t) 2s
1
1
1 -2t
Sol: Y(s) G(s) R(s)
1 1
s 2 s1 12 2 2 s
1 1y(t) [Y(s)] e
2 2
s
L
Ex: Find y(t)
pattern response Find :Ex
y(t))5s)(4s)(2s(
1s
1
Y(s)s
1
y (t)
t
1/21/21/2
tt
- =2t-e2
1
-2t -4t -5t1 2 3
-2t -4t -5t1 2 3
By inspection, solution modes are
k e , k e , k e
1 y(t) k e k e k e
40
ωj
-2
1/2
ωj
-5 -4 -2 -1
1/40
diagram zero-Pole
diagram zero-Pole
Response
§ 4.3 Transfer Function and System Response (7)
response state-steady
26
§ 4.4 Block Diagram Operations for Complex Systems (1)
• Fundamental Operations: Signal operation
Summer Y(s)=X1(s)+X2(s)
Comparator Y(s)=X1(s)-X2(s)
Take-off point Y(s)=X1(s)
Component combinations
Serial
Parallel
Feedback
X Y
X Y
X Y
21 GG
21 GG
GH1
G
X1 Y
Y
X1 Y+
-
X2
X1 Y+
+
X2
X Y1G 2G
YX
1G
2G
X Y
GG
H27
§ 4.4 Block Diagram Operations for Complex Systems (2)
Moving junction / sequence
Ahead of a block
Past a block
Exchange sequence
GZ1
Z2
X
GX
+
+
XG1 G2
XG
X
GX
G
+
1/G
+
G1G2
Z2
Z1
Z1
Z2
Z2
Z1
Z3 Z2Z2Z1
28
§ 4.4 Block Diagram Operations for Complex Systems (3)
• Negative Feedback System:
GY(s)R(s) +
H
aE (s)
Y(s) G
R(s) 1 GH
H1 Y(s)R(s) +
GHH
1 E(s)
Y(s) 1 GH( )
R(s) H 1 GH
G :Gain Forward GH :Gain Forward
GH+1
G function transfer system loop-Closed
GH function transfer loop -Open :Note
True Error Signal: E(s)
H :Gain Feedback
GH :Gain Loop
aActuating Error Signal: E (s)
H
1 :Gain Cascade
GH :Gain Loop
29
§ 4.4 Block Diagram Operations for Complex Systems (4)
• Loading Effect:
Cascade
C1
R1
V1(s) C2
R2
V4(s)
KK=1
Isolated Amp
C1
R1
V1(s) V2(s) C2
R2
V3(s) V4(s)
G1 G2
21
1 1 1
V (s) 1G (s)
V (s) R C s 1
4
23 2 2
V (s) 1G (s)
V (s) R C s 1
1s)CRCR(s)CRCR(
1)s(G)s(G
22112
221121
+-
K=1
Realization Isolated Amp by 741OP
30
§ 4.4 Block Diagram Operations for Complex Systems (5)
• History of Operational Amplifier:
1965
Fairchild develops the first OpAmp (operational amplifier) generally used throughout the industry--a milestone in the linear integrated circuit field.
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
1968
Fairchild introduces an OpAmp (operational amplifier) that is one of the earliest linear integrated circuits to include temperature compensation and MOS capacitors.
31
§ 4.4 Block Diagram Operations for Complex Systems (6)• Operational Amplifier:
+
-
(+) Supply voltage,Vcc (usually DC 15V)
(-) Supply voltage, Vee(usually DC -15V)
Output
Invertinginput
inputNoninverting
741 A
Offsetnull adjust
2
3
4
7
6
1 5
iZ
oZv
A
vA
iZ
oZ
ideal
0
1
2
3
4 5
6
7
8
ln-
ln+ Out
Vcc-
Vcc+
Offset null adjust
Offset null adjust NC
32
§ 4.4 Block Diagram Operations for Complex Systems (7)
• Network 1:
• Network 2:
C1
R1
V1(s) V4(s)
R2
C2I(s)
V(s)
V2(s)C1
R1
V3(s)
R2
C2I(s)
V(s)
V1(s) V4(s)V(s)
I(s)
V3(s) V2(s)V(s)
I(s)
1s)CRCRCR(s)CRCR(
1)s(G
2122112
2211
1
⇒
Loading effect
)s(G)s(G)s(G)s(G 221
1
1+s)CR+CR+CR(+s)CRCR(
1=)s(G
1222112
2211
2⇒
Loading effect
33
§ 4.4 Block Diagram Operations for Complex Systems (8)
)s(G
)s(R1
)s(R2
1Y (s)
2Y (s)
Note: For MIMO System
Y(s) G(s)R(s)
Output
Vector
Transfer
Matrix
Input
Vector
1 11 12 1
2 21 22 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Y s G s G s R s
Y s G s G s R s
G11
G22
G 12G
21
+
+
+
+
R1(s)
R2(s) Y2(s)
Y1(s)
34
Ex: Armature control DC servomotor
Static characteristics (Ideal)
La
ia(t)
if+
eb(t)
Ra
va(t) M
-
=const.
,Ttm(t) , Jm, Bm,m(t) m(t)
or Permanent magnet
Ttd(t)
tT
max
maxv
tmaxT (Stall)
tT BT K v K
t
tmax
T maxmax 0
max Bb
max T maxT 0
tmax
i a T maxmax
T K (Stall)
v
v K K (No load)
K
T K R K
i
§ 4.4 Block Diagram Operations for Complex Systems (9)
35
§ 4.4 Block Diagram Operations for Complex Systems (10)
I/O Block Diagram Reduction
t v T
m i
a a a m m b i
m m m
(s) K
V (s) s[(R L s)(J s B ) K K ]
(s) (s) (s)
Dynamic characteristics
Total
Response
Command
Response
Disturbance
Response
BmRa
Kb
Ki
+
-
+
-a
1
L s m
1
J s1
s
m(s)ai m(s)
Electronics Mechatronics Mechanics
tdT (s) Load disturbance
aV (s) -
36
S1
1
m S
1 m(s)m(s)aV (s)Km
§ 4.4 Block Diagram Operations for Complex Systems (11)
iK
bK
S
1
SJ
1
maR
1 m(s) m(s)+
-
aV (s)
0B ,0L )2( ma
T
J=
KK
J= ,
KK
JR=
Κ
1=K
max
mmax
maxbT
m
maxmbi
mam
bm
im
a m i b
a mm
a m i b
m m
a m
K Define K , Motor gain constant
R B K K
R J , Motor time constant
R B K K
(s) K
V (s) s(1 s)
0L (1) a Model Reduction
37
§ 4.4 Block Diagram Operations for Complex Systems (12)
0 ,0B ,0L If )3( mma
S1 m(s)m(s)
KmaV (s)
Servomotor Ideal
S1 m(s)m(s)
Integrator Pure
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 inductance38
Gaintatic S