MESB374 System Modeling and Analysis Transfer Function Analysis
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Transcript of MESB374 System Modeling and Analysis Transfer Function Analysis
MESB374 System Modeling and Analysis
Transfer Function Analysis
Transfer Function Analysis• Dynamic Response of Linear Time-Invariant (LTI)
Systems– Free (or Natural) Responses
– Forced Responses
• Transfer Function for Forced Response Analysis– Poles
– Zeros
• General Form of Free Response – Effect of Pole Locations
– Effect of Initial Conditions (ICs)
• Obtain I/O Model based on Transfer Function Concept
( ) ( ) (0)Y s U s y
( )
y y Ku
y y u
K
£ £
£ £ K £
Dynamic Responses of LTI SystemsEx: Let’s look at a stable first order system:
Y s
y y Ku
( ) ( ) (0)1 1
KY s U s y
s s
– Solve for the output:
– Take LT of the I/O model and remember to keep tracks of the ICs:
– Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other:
0sY s y U s
1s K
Free ResponseForced Response
Time constant
Free & Forced Responses• Free Response (u(t) = 0 & nonzero ICs)
– The response of a system to zero input and nonzero initial conditions.
– Can be obtained by
• Let u(t) = 0 and use LT and ILT to solve for the free response.
• Forced Response (zero ICs & nonzero u(t))– The response of a system to nonzero input and zero initial conditions.
– Can be obtained by
• Assume zero ICs and use LT and ILT to solve for the forced response (replace differentiation with s in the I/O ODE model).
In Class ExerciseFind the free and forced responses of the car suspension system without tire model:
2 4 2 4 , , ry y y u u y z u x
2 2 4 ( ) 2 4 ( ) 2 (0) (0)s s Y s s U s s y y
2
2 4 2 4
2 4 2 4
0 0 2 0 4 2 0 4
y y u u
y y y u u
s Y s sy y sY s y Y s sU s u U s
£ £
£ £ £ £ £
2 2
2 (0) (0)2 4( ) ( )
2 4 2 4
s y ysY s U s
s s s s
– Solve for the output:
– Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other:
Free ResponseForced Response
– Take LT of the I/O model and remember to keep tracks of the ICs:
Given a general n-th order system model:
The forced response (zero ICs) of the system due to input u(t) is:– Taking the LT of the ODE:
Forced Response & Transfer Function
( ) ( 1) ( ) ( 1)1 1 0 1 1 0
n n m mn m my a y a y a y b u b u b u b u
( ) ( ) WHY?n ny s Y s £1
1 1 01
1 1 0
1 11 1 0 1 1 0
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
n nn
m mm m
n n m mn m m
D s N s
s Y s a s Y s a sY s a Y s
b s U s b s U s b sU s b U s
s a s a s a Y s b s b s b s b
( )U s
ForcedResponse Transfer Function Inputs Transfer
FunctionInputs
11 1 0
11 1 0
( )
( )( ) ( ) ( ) ( ) ( )
( )
m mm m
n nn
G s
b s b s b s b N sY s U s U s G s U s
s a s a s a D s
= =
Transfer FunctionGiven a general nth order system:
The transfer function of the system is:
– The transfer function can be interpreted as:
( ) ( 1) ( ) ( 1)1 1 0 1 1 0
n n m mn m my a y a y a y b u b u b u b u
11 1 0
11 1 0
( )m m
m mn n
n
b s b s b s bG s
s a s a s a
DifferentialEquation
u(t)
Input
y(t)
Output
Time Domain
G(s)U(s)
Input
Y(s)
Output
s - Domain
zero I.C.s
( )Y s
G sU s
Static gain
Transfer Function Matrix
For Multiple-Input-Multiple-Output (MIMO) System with m inputs and p outputs:
( ) ij p mG s G s
zero I.C.s
iij
j
Y sG s
U s
Inputs
1U s
2U s
mU s
Outputs
1Y s
2Y s
pY s
1, ,k p
1
1 1 2 2
m
k kj jj
k k km m
Y s G s U s
G s U s G s U s G s U s
1 11 1 1
1
m
p p pm m
U sY s G s
Y s G s G s U s
Y s G s G s U s
Poles and Zeros
• PolesThe roots of the denominator of the TF, i.e. the roots of the characteristic equation.
Given a transfer function (TF) of a system:1
1 1 01
1 1 0
( )( )
( )
m mm m
n nn
b s b s b s b N sG s
s a s a s a D s
11 1 0
1 2
( )
( )( ) ( ) 0
n nn
n
D s s a s a s a
s p s p s p
1 2 , , , np p p
1 0 1 2
1 0 1 2
( )( ) ( )( )( )
( ) ( )( ) ( )
mm m m
nn
b s b s b b s z s z s zN sG s
s a s a D s s p s p s p
n poles of TF
• ZerosThe roots of the numerator of the TF.N s b s b s b s b
b s z s z s zm
mm
m
m m
( )
( )( ) ( )
11
1 0
1 2 0
1 2 , , , mz z z
m zeros of TF
(2) For car suspension system:
Find TF and poles/zeros of the system.
Examples(1) Recall the first order system:
Find TF and poles/zeros of the system.
y y Ku y By K y Bu Ku
1 ( ) ( )s Y s K U s
1
Y s KG s
U s s
Pole:
Zero:
1 0s 1
s
No Zero
2 ( ) ( )s Bs K Y s Bs K U s
2
Y s Bs KG s
U s s Bs K
Pole:
Zero:
2 0s Bs K 2
1,2
4
2
B B Kp
0Bs K 1
Kz
B
System ConnectionsCascaded System
2 1G s G s G sInput Output
1G s 2G s 1U s U s 1 2Y s U s 2Y s Y s
Parallel System
1 2G s G s G s
InputOutput
1G s
2G s
1
2
U sU s
U s
1Y s
Y s++
2Y s
Feedback System
1
1 21
G sG s
G s G s
Input Output 1G s
2G s
U s 1Y s Y s+
-
2U s 2Y s
1U s
Given a general nth order system model:
The free response (zero input) of the system due to ICs is:
– Taking the LT of the model with zero input
(i.e., )
General Form of Free Response( ) ( 1) ( ) ( 1)
1 1 0 1 1 0n n m m
n m my a y a y a y b u b u b u b u
( ) ( 1)1 1 0 0n n
ny a y a y a y
1 1 2 ( 1)1 1 0 1 1
( )
( ) (0) (0)n n n n nn Free n
F s
s a s a s a Y s s a s a y y
( ) ( 1)
1 ( 1) 1 2 ( 2)1
1
( ) (0) (0) ( ) (0) (0)
( ) (0)
n n
n n n n n nn
y y
s Y s s y y a s Y s s y y
a sY s y
£ £
£0 ( ) 0
yy
a Y s
£
11 1 0
( ) ( )Free n n
n
F sY s
s a s a s a
A Polynominal of s that depends on ICsFree Response(Natural Response)
=Same Denominator as TF G(s)
Free Response (Examples)Ex: Find the free response of the car suspension system without tire model (slinker toy):
Ex: Perform partial fraction expansion (PFE) of the above free response when:
(what does this set of ICs means physically)?
2 2
2 (0) (0)2 4( ) ( )
2 4 2 4
s y ysY s U s
s s s s
2 4 2 4 , , ry y y u u y z u x
(0) 0 and (0) 1y y
Q: Is the solution consistent with your physical intuition?
2 2
2 2( ) (0)
2 4 2 4
s sY s y
s s s s
2 2 22 2 2
12 1 3( )
31 3 1 3 1 3
ssY s
s s s
11 2sin 3 cos 3 sin 3 tan 3
3 3t ty t e t t e t
Decaying rate: damping, mass
Frequency: damping, spring, mass
phase: initial conditions
The free response of a system can be represented by:
Free Response and Pole Locations
ip tiAe
11 1 0 1 2
1 2
1 2
( ) ( )( )
( )( ) ( )Free n nn n
n
n
F s F sY s
s a s a s a s p s p s p
A A A
s p s p s p
1 2
For simplicity, assume that
np p p
1 211 2( ) ( ) np tp t p t
Free Free ny t Y s A e A e A e
£
Real
Img.
,
0
is real 0
0
0
0
0
i
i i
i
i j
p
p p
p
p j
0ip
exponential decrease
iA constant
exponential increaseip tiAe
0ip 0ip
2 Re2 cos
i i i
i
p t p t p ti i i
ti i
Ae Ae AeA e t
ip j
jp j
decaying oscillation
Oscillation with constant magnitude
increasing oscillation
0
t
• Complete Response
Q: Which part of the system affects both the free and forced response ?
Q: When will free response converges to zero for all non-zero I.C.s ?
Complete Response
G sN s
D s( )
( )
( )U(s)
Input
Y(s)
Output( 1)ICs: (0) , (0) , , (0)ny y y
A Polynomial of depending on I.C.s
( ) ( ) ( )
( )
Forced Free
sD s
Y s Y s Y s
N sU s
D s
Denominator D(s)
All the poles have negative real parts.
Obtaining I/O Model Using TF Concept (Laplace Transformation Method)
• Noting the one-one correspondence between the transfer function and the I/O model of a system, one idea to obtain I/O model is to:
– Use LT to transform all time-domain differential equations into s-domain algebraic equations assuming zero ICs (why?)
– Solve for output in terms of inputs in s-domain to obtain TFs (algebraic manipulations)
– Write down the I/O model based on the TFs obtained
• Step 1: LT of differential equations assuming zero ICs
1 1 1 1 1 2 1 1 1 2
2 2 1 1 1 2 2 1 1 1 2 2 2 2
0
p p
M x B x B x K x K x
M x B x B B x K x K K x B x K x
Example – Car Suspension System
• Step 2: Solve for output using algebraic elimination method
1. # of unknown variables = # equations ?
2. Eliminate intermediate variables one by one. To eliminate one intermediate variable, solve for the variable from one of the equations and substitute it into ALL the rest of equations; make sure that the variable is completely eliminated from the remaining equations
L
21 1 1 1 1 2 1 1 1 2
22 2 1 1 1 2 2 1 1 1 2 2 2 2
0
p p
M s X s B sX s B sX s K X s K X s
M s X s B sX s B B sX s K X s K K X s B sX s K X s
21 1 1 1 1 1 2
22 1 2 1 2 2 1 1 1 2 2
0
p
M s B s K X s B s K X s
M s B B s K K X s B s K X s B s K X s
xp
1K
2K
g
2x
2B
1B
1x1M
2M
Example (Cont.)
• Step 3: write down I/O model from TFs
2
1 1 1
2 11 1
M s B s KX s X s
B s K
from first equation
Substitute it into the second equation
2
1 1 122 1 2 1 2 1 1 1 1 2 2
1 1p
M s B s KM s B B s K K X s B s K X s B s K X s
B s K
1 1 1 2 2
22 22 1 2 1 2 1 1 1 1 1p
X s B s K B s K
X s M s B B s K K M s B s K B s K
1
1 1 1 2 2
22 22 1 2 1 2 1 1 1 1 1
x pp
X s B s K B s KG
X s M s B B s K K M s B s K B s K
1 1 2 1 3 1 4 1 5 1 1 2 3p p pa x a x a x a x a x b x b x b x