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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

Instead of stability, we consider the requirement that each pole of a transfer

function should satisfy

0 1 and

. We will call such transfer

<p

;

:

j=pj

;<p

functions -stable ( stands for generalized). In classical terminology, -stability

G

G

G

guarantees a stability degree and a minimum damping ratio for a transfer function,

as illustrated in gure 7.11.

( )

=

p

( )

<

p

;

;

= 0 1

p

;

:

A transfer function is -stable if its poles lie in the region

Figure

7.11

G

to the left. In classical control terminology, such transfer functions ha

p

ve a

stability degree of at least 0 1 and a damping ratio of at least 1 2. All

:

=

of the results in this chapter can be adapted to this generalized notion of

stability.

We say that a controller

-stabilizes the plant if every entry of the four

K

G

P

transfer matrices (7.11{7.14) is -stable (c.f. de nition 7.1). It is not hard to show

G

that -stable, the speci cation that the closed-loop transfer matrix is achievable

H

G

by a -stabilizing controller, is a ne in fact, we have

G

8

yu

9

R P

>

>

>

>

<

=

-stable =

zw + zu yw

R

are -stable

H

P

P

R P

G

G

+ yu

I

P

R

>

>

>

>

:

( + yu ) yu

I

P

R

P

which is just (7.20), with \ -stable" substituted for \stable".

G

For the SASS 1-DOF control system, the speci cation -stable can be expressed

H

G

in terms of the interpolation conditions described in section 7.2.5, with the following

modi cations: condition (1) becomes \ is -stable", and the list of poles and zeros

T

G

of 0 must be expanded to include any poles and zeros that are -unstable, i.e., lie

P

G

in the right-hand region in gure 7.11.

7.5 SOME GENERALIZATIONS OF CLOSED-LOOP STABILITY

167

There is a free parameter representation of -stable:

H

G

-stable = 1 + 2 3

is a -stable u y transfer matrix

H

fT

T

QT

j

Q

G

n

n

g

G

which is (7.25) with \ -stable" substituted for \stable". This free parameter rep-

G

resentation can be developed from state-space equations exactly as in section 7.4,

provided the state-feedback and estimator gains are chosen such that P

u sfb

A

;

B

K

and P

est y are -stable, i.e., their eigenvalues lie in the left-hand region of

A

;

L

C

G

gure 7.11.

168