Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

CHAPTER 11 A PICTORIAL EXAMPLE

which

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The controller stability plot in gure 11.22 was produced by nding points where the

transfer function ( ) 2 vanished for some frequency . Since = 1 (1 + std

0

)

S

j

!

=!

!

S

=

P

K

vanishes wherever std

0 or

has a

axis pole, the

axis poles of are exactly the

P

K

j

!

j

!

K

axis zeros of ( ) 2 the factor of 2 cancels the two zeros at = 0 that inherits

j

!

S

j

!

=!

!

s

S

from the two = 0 poles of std

0 . At each frequency , the linear equations in and

s

P

!

1

(a)

(b)

(c)

2

13 ( ) +

13 ( ) + (1

) 13 ( ) . = 0

<

;

H

j

!

H

j

!

;

;

H

j

!

!

1

(a)

(b)

(c)

2

13 ( ) +

13 ( ) + (1

) 13 ( ) . = 0

=

;

H

j

!

H

j

!

;

;

H

j

!

!

may be dependent, independent, or inconsistent their solution in the rst two cases gives

either a line or point in the (

) plane. When these lines and points are plotted over

all frequencies they determine subsets of slice over which the controller has a constant

H

K

number of unstable (right half-plane) poles. By checking any one controller inside each

subset of slice for open-loop stability, each subset of slice can be labeled as being achieved

H

H

by stable or unstable controllers.