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

CHAPTER 5 NORMS OF SYSTEMS

5.3

Norms of MIMO LTI Systems

Some of the common norms for multiple-input, multiple-output (MIMO) LTI sys-

tems can be expressed in terms of the singular values of the

transfer matrix

n

n

z

w

, which, roughly speaking, give information analogous to the magnitude of a SISO

H

transfer function. The singular values of a matrix

n

n

are de ned by

z

w

M

2

C

( ) = ( (

))1 2=

= 1 ... min

(5.20)

M

M

M

i

fn

n

g

i

i

z

w

where ( ) denotes the th largest eigenvalue. The largest singular value (i.e., 1)

i

i

is also denoted max. A plot of ( ( )) is called a singular value plot, and is

H

j

!

i

analogous to a Bode magnitude plot of a SISO transfer function (an example is

given in gure 5.17).

5.3.1

RMS Response to a Particular Noise Input

Suppose is stable (i.e., each of its entries is stable), and

is the power spectral

H

S

w

density matrix of . Then

w

1

1 2

Z

=

1

rms =

( ) ( ) ( )

kH

k

T

r

H

j

!

S

j

!

H

j

!

d!

:

w

2

w

;1

5.3.2

Norm: RMS Response to White Noise

H

2

If ( )

for those frequencies for which ( ) is signi cant, then this norm is

S

!

I

H

j

!

w

approximately the 2 norm of a MIMO system:

H

1

1 2

Z

=

1

rms

2 =

( ) ( )

(5.21)

kH

k

kH

k

T

r

H

j

!

H

j

!

d!

:

w

2 ;1

The 2 norm of is therefore the RMS value of the output when the inputs are

H

H

driven by independent white noises.

By Parseval's theorem, the 2 norm can be expressed as

H

1 2

Z

=

1

2 =

( ) ( )T

kH

k

T

r

0 h t h t dt

1 2

!

=

= n n

z

w

X

X

22

kH

k

=1

ik

=1

i

k

where is the impulse matrix of . Thus, the 2 norm of a transfer matrix is

h

H

H

H

the square root of the sum of the squares of the 2 norms of its entries.

H

5.3 NORMS OF MIMO LTI SYSTEMS

111

The 2 norm can also be expressed in terms of the singular values of the transfer

H

matrix :

H

1 2

!

=

Z

n

1

X

2 =

1

( ( ))2

(5.22)

kH

k

2

H

j

!

d!

=1 i

;1

i

where = min

. Thus, the square of the 2 norm is the total area under

n

fn

n

g

H

z

w

the squared singular value plots, on a linear frequency scale. This is shown in

gure 5.16.

80

70

60

50

40

21 + 22 + 23

30

;

;

20

10

0

0

5

10

15

20

25

30

35

40

!

For a MIMO transfer matrix,

22 is proportional to the area

Figure

5.16

kH

k

under a plot of the sum of the squares of the singular values of (shown

H

here for min

= 3).

fn

n

g

z

w

5.3.3

Peak Gain

The peak gain of a MIMO system is

Z

n

w

1

X

pk gn = sup kHwk1 = max

( )

(5.23)

kH

k

=0

1

0

jh

t

j

dt:

ij

i

n

kw

k

z

1

=1

kw

k

6

1

j

For MIMO systems, the peak gain is not the same as the average-absolute gain

(c.f. (5.7)). The average-absolute gain is

aa gn = sup

aa

kH

w

k

= T pk gn

kH

k

kH

k

aa=0

aa

kw

k

kw

k

6

the peak gain of the LTI system whose transfer matrix is the transpose of .

H

112