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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

where is the classical closed-loop I/O transfer function. Hence if is the closed-

T

H

loop transfer matrix realized by some controller , i.e.,

rlzbl, then we must

K

H

2

H

have

12 = 21

(7.2)

H

H

0 22 = 21

(7.3)

P

H

H

11

0 21 = 0

(7.4)

H

;

P

H

P

13 =

12

(7.5)

H

;H

23 =

22

(7.6)

H

;H

:

It is not hard to show that the ve explicit speci cations on given in (7.2{7.6)

H

are not just implied by realizability they are equivalent to it. Roughly speaking,

we have only one transfer function that we can design, , whereas the closed-loop

K

transfer matrix contains six transfer functions. The ve constraints (7.2{7.6)

H

among these six transfer functions make up the missing degrees of freedom.

The speci cations (7.2{7.6) are ane, so at least for the classical SASS 1-DOF

controller, realizability is an a ne speci cation. In fact, we will see that in the

general case, rlzbl is a ne. Thus, realizability is a closed-loop convex speci cation.

H

To establish that rlzbl is a ne in the general case, we use a simple trick that

H

replaces the inverse appearing in (7.1) with a simpler expression. Given any n n

u

y

transfer matrix , we de ne the

transfer matrix by

K

n

n

R

u

y

= (

) 1

;

(7.7)

R

K

I

;

P

K

:

y

u

This correspondence is one-to-one: given any

transfer matrix , the

n

n

R

n

n

u

y

u

y

transfer matrix given by

K

= ( +

) 1

(7.8)

;

K

I

R P

R

y

u

makes sense and satis es (7.7).

Hence we can express the realizability speci cation as

rlzbl =

=

+

for some

(7.9)

H

fH

j

H

P

P

R P

n

n

R g

:

z

w

z

u

y

w

u

y

This form of rlzbl can be given a simple interpretation, which is shown in gure 7.1.

H

The transfer matrix can be thought of as the \controller" that would realize the

R

closed-loop transfer matrix if there were no feedback through our plant, i.e.,

=

H

P

y

u

0 (see gure 7.1(b)). Our trick above is the observation that we can reconstruct

the controller that has the same e ect on the true plant as the controller that

K

R

operates on the plant with

set to zero. Variations on this simple trick will

P

y

u

appear again later in this chapter.

From (7.9) we can establish that rlzbl is a ne. Suppose that

~

rlzbl.

H

H

H

2

H

Then there are two

transfer matrices and ~ such that

n

n

R

R

u

y

=

+

H

P

P

R P

z

w

z

u

y

w

~ =

+

~

H

P

P

R

P

:

z

w

z

u

y

w

7.1 REALIZABILITY

149

+

q

r

w

z

P

z

w

+

+

r

q

P

P

K

y

w

z

u

+

P

y

u

R

(a)

+

q

r

w

z

P

z

w

+

P

P

R

y

w

z

u

(b)

The closed-loop transfer matrix can be realized by the feed-

Figure

7.1

H

back system (a) for some if and only if it can be realized by the system

K

(b) for some transfer matrix . In (b) there is no feedback.

R

Let

. We must show that the transfer matrix

=

+ (1

) ~ is also

;

H

2

R

H

H

realizable as the closed-loop transfer matrix of our plant with some controller. We

note that = +

H

P

P

R

P

z

w

z

u

y

w

where

=

+ (1 ) ~

R

R

;

R

:

This shows that

rlzbl.

H

2

H

We can nd the controller

that realizes the closed-loop transfer matrix

K

H

using the formula (7.8) with . If and ~ are controllers that yield the closed-

R

K

K

loop transfer matrices and ~, respectively, the controller that realizes the closed-

H

H

loop transfer matrix

is not

+ (1 ) ~ it is

H

K

;

K

= ( + ) 1( + )

(7.10)

;

K

A

B

C

D

where = + ~(

~) 1;

A

I

K

I

;

P

K

P

y

u

y

u

150