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

CHAPTER 5 NORMS OF SYSTEMS

an algorithm that computes

with guaranteed accuracy, by trying di erent

kH

k

1

values of see the Notes and References at the end of this chapter.

The result (5.41) can be understood as follows.

is true if and only if

kH

k

<

1

for all

, 2

( ) ( ) is invertible, or equivalently, the transfer matrix

!

2

R

I

;

H

j

!

H

j

!

( ) =

1

;

2

;

( )T ( ) ;

G

s

I

;

H

;s

H

s

has no

axis poles. We can derive a realization of as follows. A realization of

j

!

G

( ) (which is the adjoint system) is given by

T

H

;s

( )

1

;

;

;

;

T

= ; T

T

T

H

;s

B

sI

;

;A

;C

:

Using this and the block diagram of shown in gure 5.22, a realization of is

G

G

given by ( ) = (

) 1 + , where

;

G

s

C

sI

;

A

B

D

G

G

G

G

=

A

M

G

= B

B

G

0

= 0

1

;

T

B

C

G

=

D

I

:

G

Since

= , and it can be shown that this realization of is minimal, the

A

M

G

j

!

G

axis poles of are exactly the imaginary eigenvalues of

, and (5.41) follows.

G

M

+

r

y

u

+

q

( )

( )

T

H

s

H

;s

A realization of ( ) = 2( 2

( ) ( )) 1. has no

T

;

Figure

5.22

G

s

I

;

H

;s

H

s

G

imaginary axis poles if and only if the inequality specication kHk <

1

holds.

The condition that

not have any imaginary eigenvalues can also be expressed

M

in terms of a related algebraic Riccati equation (ARE),

+

+ 1

+ 1

= 0

(5.42)

T

;

T

;

T

A

X

X

A

X

B

B

X

C

C

:

(Conversely,

is called the Hamiltonian matrix associated with the ARE (5.42).)

M

This equation will have a positive de nite solution if and only if

has no

X

M

imaginary eigenvalues (in which case there will be only one such ). If

,

X

kH

k

<

1

we can compute the positive de nite solution to (5.42) as follows. We compute any

matrix such that

T

1

11 ~12

A

;

= ~A

T

M

T

0 ~22

A

5.6 STATE-SPACE METHODS FOR COMPUTING NORMS

123

where ~11 is stable (i.e., all of its eigenvalues have negative real part). (One good

A

choice is to compute the ordered Schur form of

see the Notes and References

M

at the end of this chapter.) We then partition as

T

=

11

12

T

T

T

21

22

T

T

and the solution is given by

X

= 21 1

;

11

X

T

T

:

The signi cance of is discussed in the Notes and References for chapter 10.

X

5.6.4

Computing the -Shifted

Norm

a

H

1

The results of the previous section can be applied to the realization

_ = ( + ) +

=

x

A

aI

x

B

w

z

C

x

of the -shifted transfer matrix (

). We nd that

holds if and

a

H

s

;

a

kH

k

<

1

a

only if the eigenvalues of all have real part less than

and the matrix

A

;a

+

1

;

T

A

aI

B

B

1

;

T

T

;

C

C

;A

;

aI

has no imaginary eigenvalues.

5.6.5

Computing the Entropy

To compute the -entropy of , we rst form the matrix

in (5.40). If

H

M

M

has any imaginary eigenvalues, then by the result (5.41),

and hence

kH

k

1

( ) = . If

has no imaginary eigenvalues, then we nd the positive de nite

I

H

1

M

solution of the ARE (5.42) as described above. The -entropy is then

X

( ) =

;

T

(5.43)

I

H

T

r

B

X

B

:

From (5.42), the matrix ~ =

satis es the ARE

X

X

~

2 ~

~

T

+ ~ + ;

T

+ T = 0

A

X

X

A

X

B

B

X

C

C

:

Note that as

, this ARE becomes the Lyapunov equation for the observability

!

1

Gramian (5.36), so the solution ~ converges to the observability Gramian obs.

X

W

From (5.43) and (5.35) we see again that ( )

22 as

(see (5.24) in

I

H

!

kH

k

!

1

section 5.3.5).

124