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

CHAPTER 4 NORMS OF SIGNALS

This is simply the 1 norm of with the time domain weight ( ) = , which serves

L

u

w

t

t

to emphasize the signal at large times, and de-emphasize the signal at small times.

u

One commonly used family of time domain weights is the family of exponential

weightings, which have the form ( ) = exp( ). If is positive, such a weighting

w

t

at

a

exponentially emphasizes the signal at large times. This may be appropriate for

measuring the size of a decaying signal. Alternatively, we can think of a speci cation

such as ~

(where ~( ) = exp( ) ( ), and

0) as enforcing a rate of

ku

k

M

u

t

at

u

t

a

>

1

decay in the signal at least as fast as exp( ). An example of a signal and the

u

;at

u

exponentially scaled signal ( ) is shown in gure 4.16.

;t

u

t

e

1

1

0 6

0 6:

e;t

:

0 2

0 2

:

:

u(t) 0 2

0 2

;

:

;

:

H

Y

u(t)e;t

H

0 6

0 6

;

:

;

:

1

1

;

0

0 5

1

1 5

2

2 5 ; 0

0 5

1

1 5

2

2 5

:

t :

:

:

t :

:

(a)

(b)

A signal u is shown in (a). The exponentially weighted 2

Figure

4.16

L

norm of u is the 2 norm of the signal ~u(t) = u(t)e , which is shown in

;t

L

(b).

If is negative, then the signal is exponentially de-emphasized at large times.

a

This might be useful to measure the size of a diverging or growing signal, where the

value of an unweighted norm is in nite.

There is a simple frequency domain interpretation of the exponentially weighted

2 norms. If the -exponentially weighted 2 norm of a signal is nite then its

L

a

L

u

Laplace transform ( ) is analytic in the region

, and in fact

U

s

fs

j

<s

>

;ag

Z

Z

~ 2

1

1

2 =

(exp( ) ( ))2 = 1

( + ) 2

(4.15)

ku

k

0

at

u

t

dt

2

jU

;a

j

!

j

d!

;1

(recall ~( ) = exp( ) ( )). The only di erence between (4.15) and (4.11) is that the

u

t

at

u

t

integral in the -exponentially weighted norm is shifted by the weight exponent .

a

a

The frequency domain calculations of the 2 norm and the -exponentially weighted

L

a

2 norm for the signal in gure 4.16(a) are shown in gure 4.17.

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4.2 COMMON NORMS OF SCALAR SIGNALS

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( + )

jU

j

!

j

!

(a)

( + ) 2

jU

j

!

j

U(0 + j!) 2

j

j

U(1 + j!) 2

!

j

j

= 0

= 1

(b)

An exponentially weighted 2 norm can be calculated in the

Figure

4.17

L

frequency domain. Consider u 2 1 = ~u 2, where ~u(t) = e u(t) is shown

;t

k

k

k

k

;

in gure 4.16(b). U( +j!) , the magnitude of the Laplace transform of u,

j

j

is shown in (a) for

0. As shown in (b), u 22 is proportional to the area

k

k

under U(j!) 2, and u 22 1 is proportional to the area under U(1+j!) 2.

j

j

k

k

j

j

;

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