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Advanced Control Systems Analysis

This document discusses third-order systems and their step responses. It notes that the step response of higher-order systems is often approximated by the response of the dominant second-order roots, provided any other roots have a real part at least 5-10 times further from the real axis. It also notes that the step response depends greatly on how the value of a compares to b or ζωn. The document provides an example third-order transfer function and uses the final value theorem to find the steady-state value of c(t) in response to a unit step input r(t).

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SudheerKumar
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192 views2 pages

Advanced Control Systems Analysis

This document discusses third-order systems and their step responses. It notes that the step response of higher-order systems is often approximated by the response of the dominant second-order roots, provided any other roots have a real part at least 5-10 times further from the real axis. It also notes that the step response depends greatly on how the value of a compares to b or ζωn. The document provides an example third-order transfer function and uses the final value theorem to find the steady-state value of c(t) in response to a unit step input r(t).

Uploaded by

SudheerKumar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Third Order System

C (s ) a ω n2
G( s) = =
Section 4.7 & 4.8 (
R( s) (s + a ) s + 2ςω n s + ω n2
2
)

Step response will depend greatly on how value of a


Systems with additional poles compares to ζω n (the real part of complex roots)
and zeros

2nd Order Approximation Third Order System with Zero


The step response of higher order C (s ) ω n2 (s + b )
G( s) = =
systems (3rd order or more) with no
zeros is frequently approximated by the
(
R( s) (s + a ) s 2 + 2ςω n s + ω n2 )
response of the “dominant” 2nd order
roots if Step response will depend greatly on how value of a
– all other roots have a real part that is at least 5 to compares to b
10 times further away from the real axis
– see bottom of p. 187 in text

2nd Order Approximation Homework #6


The step response of higher order • You have 3 sets of transfer functions
systems (3rd order or more) is – “dominant” 1st order system with additional pole
frequently approximated by the (#1-#5)
response of the “dominant” 2nd order – “dominant” 2nd order system with additional pole
roots if (#6-#10)
– any poles closer to the origin are substantially – “dominant” 2nd order system with additional zero
cancelled by zeros (roots of the numerator) in the (#11-#15)
transfer function. • Show (by example) the validity of the
previous claims

1
Steady-state Response Laplace Transform Properties
What is the steady-state value for c(t) if r(t) is the unit
step? Final value theorem
C (s ) ω n2 (s + b ) lim f(t) = lim sF(s)
G ( s) = =
( )
t →∞ s→ 0
R( s) (s + a ) s 2 + 2ςω n s + ω n2
if the real parts of all poles of F(s) are <0
 1 ω (s + b )
2 except for possibly one pole at s=0
C (s ) = R( s)G (s) =  
( )
n

 s  (s + a ) s + 2ςωn s + ω n
2 2

Steady-state Response
What is the steady-state value for c(t) if r(t) is the unit
step?

1 ω n2 (s + b )
lim c(t) = lim sC( s) = s 
t→∞ s→0 2
(
 s  (s + a ) s + 2ςω n s + ω n
2
)
bω n2 b
c(∞) = css = =
aω n2
a

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