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Disturbance Rejection
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Transcript of Disturbance Rejection
(c) 2001 P. D. Olivier Disturbance Rejection 1
Disturbance Rejection
Disturbances only mentioned on page 3 of Ogata
(c) 2001 P. D. Olivier Disturbance Rejection 2
Disturbance Rejection
C ( s ) G 1 ( s )G R ( s ) G 2 ( s )
H( s )
R ( s )E ( s ) U( s )
L ( s )
Y( s )++
-
+
+
+
N ( s )R(s)=0
N(s)=0
2
2 1
( ) ( )
( ) 1 ( ) ( ) ( ) ( )c
Y s G s
L s G s G s G s H s
2
2 1
( )( ) ( )
1 ( ) ( ) ( ) ( )Lc
G sY s L s
G s G s G s H s
2
2 1
( ) 1( )
1 ( ) ( ) ( ) ( )Lc
G sY s
G s G s G s H s s
( ) 0
( ) 0L
L
Y s
y t
(c) 2001 P. D. Olivier Disturbance Rejection 3
Disturbance Rejection
• Red is input signal• Blue is specified hardware• Green is unspecified
hardware
• When will A be close to zero?
• When will A be exactly zero?
• What does this mean for the hardware?
2
2 1
( ) 1( )
1 ( ) ( ) ( ) ( )Lc
G sY s
G s G s G s H s s
{ }
Aterms with LHP
s
( ) 1( )Lssy t A t 0 2
02 1
( )
1 ( ) ( ) (( ) )
1
c sG
G
sA
s
G s G s H ss
s
Homework 1 and 2 are now assigned.
Why?
(c) 2001 P. D. Olivier Disturbance Rejection 4
Final Value TheoremWe have been performing a particular computation over-and-over again. Assume that Y(s) has •no poles in the RHP, and •has no complex poles on the imaginary axis, except for •perhaps a simple pole at the origin, then
This is called the Final Value Theorem.See page 233 of FC&NSee pp. 76 of DS&WSee page19 of FE Reference, 2nd to last entry of table
0 0( ) ( ) ( ) ( ) lim ( )ss s s t
y t sY s t sY s y t
(c) 2001 P. D. Olivier Disturbance Rejection 5
Proof of FVTAssumptions as on previous slide.
( )A
Y s transient termss
0( )
sA sY s
Notice the similarity in the work we have done several time.Many (most?) Theorems come from such observations.Notice that if Y(s) does not have a pole at s=0, thenA evaluates to 0.
End of Proof.