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

CHAPTER 9 DIFFERENTIAL SENSITIVITY SPECIFICATIONS

is equivalent to the closed-loop convex speci cation

0( ) ( )

;P

j

!

K

j

!

1 +

(9.18)

0( ) ( )

2 for

bw

!

!

:

P

j

!

K

j

!

In (9.17) we might interpret

as the command input to tracking error transfer

;S

function, so that the speci cation (9.17), and hence (9.18), limits the logarithmic

sensitivity of the command input to tracking error transfer function with respect to

changes in 0.

P

9.2

MAMS Log Sensitivity

It is possible to generalize Bode's results to the MAMS case. We consider the

MAMS 2-DOF control system (see section 2.3.4), with I/O transfer matrix

= ( + 0 ~) 1

;

0

T

I

P

K

P

K

:

y

r

If the plant is perturbed so that 0 becomes 0 + 0, we have

P

P

P

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

;

( 0 + 0)

T

T

I

P

P

K

P

P

K

y

r

Retaining only rst order terms in 0 we have

P

( + 0 ~) 1

( +

) 1

( +

) 1

;

0

0 ~ ;

0 ~

0 ~ ; 0

T

'

I

P

K

P

K

;

I

P

K

P

K

I

P

K

P

K

y

r

y

y

y

r

= ( + 0 ~) 1

1

;

0( + ~ 0);

I

P

K

P

I

K

P

K

y

y

r

=

0( + ~ 0) 1

;

(9.19)

S

P

I

K

P

K

y

r

where = ( + 0 ~) 1 is the (output-referred) sensitivity matrix of the MAMS

;

S

I

P

K

y

2-DOF control system.

Now suppose that we can express the change in 0 as

P

0 =

frac

0

0

P

P

P

:

We can interpret frac

0 as an output-referred fractional perturbation of 0, as shown

P

P

in gure 9.4. Then from (9.19) we have

frac

0

0( + ~ 0) 1

frac

;

=

0

T

'

S

P

P

I

K

P

K

S

P

T

y

r

so that

frac

T

'

T

T

where frac

frac

0

(9.20)

T

'

S

P

:

This is analogous to (9.2): it states that the output-referred fractional change in

the I/O transfer matrix is, to rst order, the sensitivity matrix times the

T

S

output-referred fractional change in 0.

P

9.2 MAMS LOG SENSITIVITY

203

+

q

r

p

y

u

0

P

+

frac

0

P

An output-referred fractional perturbation of the transfer ma-

Figure

9.4

trix 0. In the SASS case, 0

log 0.

P

P

P

The design speci cation,

max( frac( )) 0 01 for

bw max( frac

)) 0 20

(9.21)

<

0 (

T

j

!

:

!

!

P

j

!

:

which limits the rst order fractional change in to 1% over the bandwidth bw,

T

!

despite variations in 0 of 20%, is therefore equivalent to the closed-loop convex

P

speci cation

max( ( )) 0 05 for

bw

(9.22)

S

j

!

:

!

!

:

We remind the reader that the inequality in (9.21) holds only to rst order in

frac

0 : its precise meaning is

P

frac

lim max(

( )) 0 01

T

j

!

:

0

bw

max( frac

)) 0 20 for ! ! :

P

!

0 (

:

P

j

!

0

9.2.1

Cruz and Perkins’ Comparison Sensitivity

Cruz and Perkins gave another generalization of Bode's log sensitivity to the MAMS

2-DOF control system using the concept of comparison sensitivity, in which the

perturbed closed-loop system is compared to an equivalent open-loop system.

The open-loop equivalent system consists of 0+ 0 driven by the unperturbed

P

P

actuator signal, as shown at the top of gure 9.5. For 0 = 0, the open-loop

P

equivalent system is identical to the closed-loop system shown at the bottom of

gure 9.5. For 0 nonzero, however, the two systems di er. By comparing the rst

P

order changes in these two systems, we can directly see the e ect of the feedback

on the perturbation 0.

P

The transfer matrix of the open-loop equivalent system is

ole = ( 0 + 0)( + ~ 0) 1

;

T

P

P

I

K

P

K

y

r

so that

ole = 0( + ~ 0) 1

;

T

P

I

K

P

K

:

y

r

204