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

CHAPTER 6 GEOMETRY OF DESIGN SPECIFICATIONS

is convex. Let and ~ be any two transfer matrices, and let 0

1. Now,

H

H

using the triangle inequality and homogeneity property for norms, together with

the fact that and 1

are nonnegative, we see that

;

( + (1 ) ~) =

xx + (1

) ~xx

des

xx

H

;

H

k

H

;

H

;

H

k

= ( xx

des

xx ) + (1

)( ~xx

des

xx )

k

H

;

H

;

H

;

H

k

( xx

des

xx ) + (1

)( ~xx

des

xx )

k

H

;

H

k

k

;

H

;

H

k

=

xx

des

xx + (1

) ~xx

des

xx

kH

;

H

k

;

kH

;

H

k

= ( ) + (1 ) ( ~)

H

;

H

so the functional is convex. Since is convex, the norm-bound speci cation (6.3)

is convex. In chapters 8{10 we will see that many speci cations can be expressed

in the form (6.3), and are therefore closed-loop convex. (We note that the entropy

functional de ned in section 5.3.5 is also convex, although it is not a norm.)

An important variation on the norm-bound speci cation (6.3) is the speci cation

xx

(6.4)

kH

k

<

1

which requires that an entry or submatrix of have a nite norm, as measured by

H

. This speci cation is a ne, since if xx and ~xx each satisfy (6.4) and

,

k

k

H

H

2

R

then we have

xx + (1

) ~xx

xx + 1

~xx

k

H

;

H

k

j

jkH

k

j

;

jkH

k

<

1

so that xx + (1 ) ~xx also satis es (6.4).

H

;

H

If we use the

norm for

, (6.4) is the speci cation that xx be stable,

H

k

k

H

1

i.e., the poles of xx have negative real parts. Similarly, if we use the -shifted

H

a

H

1

norm as

, (6.4) is the speci cation that the poles of xx have real parts less

k

k

H

than . These speci cations are therefore a ne.

;a

6.4

Some Examples

In chapters 7{10 we will encounter many speci cations and functionals that are

a ne or convex in most cases we will simply state that they are a ne or convex

without a detailed justi cation. In this section we consider a few typical speci ca-

tions and functionals for our standard example plant (see section 2.4), and carefully

establish that they are a ne or convex.

6.4.1

An Affine Specification

Consider the speci cation that the closed-loop transfer function from the reference

input to p have unity gain at = 0, so that a constant reference input results in

r

y

!

6.4 SOME EXAMPLES

137

p = in steady-state. The set of 2 3 transfer matrices that corresponds to this

y

r

speci cation is

dc =

13(0) = 1

(6.5)

H

f

H

j

H

g

:

We will show that this set is a ne. Suppose that

~

dc, so that 13(0) =

H

H

2

H

H

~13(0) = 1, and

. Then the transfer matrix

=

+ (1 ) ~ satis es

H

2

R

H

H

;

H

13(0) =

13(0) + (1

) ~13(0) = + (1 ) = 1

H

H

;

H

;

and hence

dc.

H

2

H

Alternatively, we note that the speci cation dc can be written as the functional

H

equality constraint

dc =

dc( ) = 1

H

f

H

j

H

g

where dc( ) = 13(0) is an a ne functional.

H

H

6.4.2

Convex Specifications

Consider the speci cation act e introduced in section 3.1: \the RMS deviation of

D

due to the sensor and process noises is less than 0.1". The corresponding set of

u

transfer matrices is

act e =

act e ( ) 0 1

H

fH

j

H

:

g

where we de ned the functional

1 2

Z

=

1

act e ( ) =

1

;

21( ) 2 proc( ) + 22( ) 2 sensor( )

H

2

jH

j

!

j

S

!

jH

j

!

j

S

!

d!

;1

and proc and sens are the power spectral densities of the noises proc and sensor,

S

S

n

n

respectively.

To show that act e is a convex functional we rst express it as a norm of the

submatrix 21 22] of :

H

H

H

2

1 0 3

T

act e ( ) =

0

0 1

proc

0

W

4

5

H

1

H

0 0

0

sensor

W

2

where proc( ) 2 = proc( ) and sensor( ) 2 = sensor( ). From the results

jW

j

!

j

S

!

jW

j

!

j

S

!

of sections 6.2.3 and 6.3.2 we conclude that act e is a convex functional, and

therefore act e is a convex speci cation.

H

As another example, consider the speci cation os introduced in section 3.1:

D

\the step response overshoot from the command to p is less than 10%". We

y

can see that the corresponding set of transfer matrices, os, is convex, using the

H

argument in section 6.2.3 with = 1, = 3 and the convex subset of scalar signals

i

k

= : +

( ) 1 1 for

0

(6.6)

V

f

s

R

!

R

j

s

t

:

t

g

:

This is illustrated in gure 6.6.

138