(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.