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

CHAPTER 4 NORMS OF SIGNALS

Thus, the weight emphasizes the power spectral density of where ( ) is large,

u

jW

j

!

j

meaning it contributes more to the integral, and de-emphasizes the power spectral

density of where ( ) is small. We note from (4.12) that the weighted norm

u

jW

j

!

j

depends only on the magnitude of the weighting transfer function , and not its

W

phase. This is not true of all weighted norms.

We can view the e ect of the weight as changing the relative importance of

W

di erent frequencies in the total power integral (4.12). We can also interpret the

-weighted RMS norm in terms of the average power conceptual model shown in

W

gure 4.5, by changing the resistive load to a frequency-dependent load, i.e., a

R

more general passive admittance ( ). The load admittance is related to the

G

s

G

weight by the spectral factorization

W

( ) + ( )

G

s

G

;s

2

= ( ) ( )

(4.13)

W

s

W

;s

so that ( ) = ( ) 2. Since the average power dissipated in at a frequency

<G

j

!

jW

j

!

j

G

is proportional to the real part (resistive component) of ( ), we see that the

!

G

j

!

total average power dissipated in is given by the square of (4.12), or the square

G

of the -weighted RMS norm of .

W

u

For example, suppose that ( ) = (1 + p2 ) (1 + ), which gives up to 3dB

W

s

s

=

s

emphasis at frequencies above p2. The load admittance for this weight is ( ) =

G

s

(1+2 ) (1+ ), which we realize as the parallel connection of a 1 resistor and the

s

=

s

series connection of a 1 resistor and a 1 capacitor, as shown in gure 4.14.

F

q

+

u

1F

(t)

1

1

;

q

The average power dissipated in the termination admittance

Figure

4.14

G(s) = (1 + 2s)=(1 + s) is u 2 rms, the square of the W-weighted RMS

k

k

W

norm of the driving voltage u, where W(s) = (1 + p2s)=(1 + s).

Frequency domain weights are used less often with the other norms, mostly

because their e ect is harder to understand than the simple formula (4.12). One

common exception is the maximum slew rate, which is the peak norm used with the

weight ( ) = , a di erentiator:

W

s

s

slew rate = _

(4.14)

kuk

kuk

:

1

Maximum slew-rate speci cations occur frequently, especially on actuator signals.

For example, an actuator signal may represent the position of a large valve, which

4.2 COMMON NORMS OF SCALAR SIGNALS

83

is opened and closed by a motor that has a maximum speed. A graphical interpre-

tation of the slew-rate constraint

slew rate 1 is shown in gure 4.15.

kuk

The peak norm is sometimes used with higher order di erentiators:

2

acc = d u

kuk

2

dt

1

3

jerk = d u

kuk

3

:

dt

1

Weights that are successively higher order di erentiators yield the amusing snap,

crackle, and pop norms.

4

3

2

1

()t

0

u

;1

;2

;3

;4

0

1

2

3

4

5

6

7

8

9

10

t

An interpretation of the maximum slew-rate specication

Figure

4.15

_u

1: at every time t the graph of u must evolve within a cone, whose

k

k

sides have slopes of 1. Examples of these cones are shown for t = 0 3 6

u is slew-rate limited at t = 0 and t = 6, since _u(0) = 1 and _u(6) = 1.

;

4.2.9

Time Domain Weights

If the initial linear transformation consists of multiplying the signal by some given

function of time ( ), we refer to as a time domain weight. One example is the

w

t

w

ITAE (integral of time multiplied by absolute error) norm from classical control,

de ned as

Z

1

itae =

( )

kuk

0 tju t j dt:

84