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

Chapter 7

Realizability and Closed-Loop

Stability

In this chapter we consider the design specications of realizability and internal

(closed-loop) stability. The central result is that the set of closed-loop transfer

matrices realizable with controllers that stabilize the plant is ane and readily

described. This description is referred to as the parametrization of closed-loop

transfer matrices achieved by stabilizing controllers.

7.1

Realizability

An important constraint on the transfer matrix

is that it should be the closed-

H

2

H

loop transfer matrix achieved by some controller , in other words, should have

K

H

the form

+

(

) 1

for some . We will refer to this constraint

;

P

P

K

I

;

P

K

P

K

z

w

z

u

y

u

y

w

as realizability:

rlzbl =

=

+

(

) 1

for some

(7.1)

;

H

H

H

P

P

K

I

;

P

K

P

K

:

z

w

z

u

y

u

y

w

We can think of rlzbl as expressing the dependencies among the various closed-loop

H

transfer functions that are entries of .

H

As an example, consider the classical SASS 1-DOF control system described in

section 2.3.2. The closed-loop transfer matrix is given by

2

0

0

0

3

P

P

K

P

K

;

= 6 1 + 0

1 + 0

1 + 0 7

P

K

P

K

P

K

6

7

H

6

0

7

P

K

K

K

4

5

;

1 + 0

;

1 + 0

1 + 0

P

K

P

K

P

K

=

0(1

)

P

;

T

;T

T

0

0

;T

;T

=P

T

=P

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