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System Response Characteristics
ISAT 412 -Dynamic Control of Energy Systems(Fall 2005)
Review
We have overed several O.D.E. solution techniques Direct integration Exponential solutions (classical) Laplace transforms
Such techniques allow us to find the time response of systems described by differential equations
Generic 1st order model
Solution in Laplace domain
Solution comprised of Free Response (homogeneous solution) Forced Response (non-homogeneous solution)
tfcxdt
dxm
cms
sF
cms
mxsX
0
Free response of 1st order model
Free response means: Converting back to the time domain:
0 sFtf
m
ct
h
h
ex
mc
sxtx
mc
s
x
cms
mxsX
01
0
00
1L
Time constant
Define the system time constant as
Rewriting the free responsec
m
m
c
1
or
t
m
ct
h exextx
00
Free response behavior
0
0
0
Unstable
Stable
Unstable
tx
00 x
00 x
Meaning of the time constant
When t =
When t = 2, t = 3, and t = 2,
0368.000 1 xexexxh
0018.04
0045.03
0135.02
xx
xx
xx
h
h
h
Transfer Functions and Common Forcing Functions
ISAT 412 -Dynamic Control of Energy Systems(Fall 2005)
Forced response of 1st order system
The forced response corresponds to the case where x(0) = 0
In the Laplace domain, the forced response of a 1st order system is
cms
sFsX
Transfer functions
Solve for the ratio X(s)/F(s)
T(s) is the transfer function Can be used as a multiplier in the
Laplace domain to obtain the forced response to any input
cmssF
sXsT
1
sXsFsT
Using the transfer function
Now that we know the transfer function for a 1st order system, we can obtain the forced response to any input if we can express that input in the Laplace domain
Step input
Used to model an abrupt change in input from one constant level to another constant level Example: turning on a light switch
tf
b
Dt
Heaviside (unit) step function
Used to model step inputs
s
tu
t
ttu
s
s
1
0 1
0 0
L
Time shifted unit step function
For a unit step shifted in time,
Using the shifting property of the Laplace transform (property 6)
Dt
DtDtus 1
0
s
eDtu
sD
s
L
Step input model
For a step of magnitude b at time D
s
betf
DtbutfsD
s
L
Pulse input
tf
M
At Bt
Pulse input model
Use two step functions
tf
M
At Bt
0
M
Pulse input model
s
Me
s
Metf
BtMuAtMutfsBsA
ss
L
For a pulse input of magnitude M, starting at time A and ending at time B
Impulse input
tf
M
At
Examples: explosion, camera flash, hammer blow
Impulse input model
Unit impulse function
For an impulse input of magnitude M at time A
At
AtAt
0
1
sAMetf
AtMtf
L
Ramp input
tfm
At
Ramp input model
For a ramp input beginning at time A with a slope of m
2s
me
AtuAtmtf
AtmAuAtmtu
AtuAtmtf
sA
s
ss
s
LL
Other input functions
Sinusoidal inputs Combinations of step, pulse,
impulse, and ramp functions
Modeling periodic inputs
tf
A
B B2 B3 B4 B5 B6
t
Square wave input model
Addition of an infinite number of step functions with amplitudes A and -A
1
11
...3
2
ns
n
ss
ss
nBtuA
BtAuBtAu
BtAuBtAutf
Laplace transform of square wave
1
1
1
1
1
1
n
snBn
ns
n
s
e-A
nBtu-Atf LL