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

CHAPTER 9 DIFFERENTIAL SENSITIVITY SPECIFICATIONS

ole

T

0 + 0

ole

p

y

P

P

q

p

u

y

+

K

0

r

0

P

;

q

r

pert

T

pert

q

pert

u

+

0

r

p

y

K

0 + 0

P

P

;

In the open-loop equivalent system, shown at top, the actuator

Figure

9.5

signal drives 0+ 0, so there is no feedback around the perturbation 0.

u

P

P

P

The benet of feedback can be seen by comparing the rst order changes in

the transfer matrices from to ole

p and pert

p , respectively (see (9.23)).

r

y

y

Comparing this to the rst order change in the I/O transfer matrix, given in (9.19),

we have

ole

(9.23)

T

'

S

T

:

This simple equation shows that the rst order variation in the I/O transfer

matrix is equal to the sensitivity transfer matrix times the rst order variation in

the open-loop equivalent system. It follows that the speci cation (9.22) can be

interpreted as limiting the sensitivity of the I/O transfer matrix to be no more than

5% of the sensitivity of I/O transfer matrix of the open-loop equivalent system.

9.3

General Differential Sensitivity

The general expression for the rst order change in the closed-loop transfer matrix

due to a change in the plant transfer matrix is

H

+

(

) 1

+

(

) 1

(9.24)

;

;

H

'

P

P

K

I

;

P

K

P

P

K

I

;

P

K

P

z

w

z

u

y

u

y

w

z

u

y

u

y

w

+

(

) 1

(

) 1

;

;

P

K

I

;

P

K

P

K

I

;

P

K

P

:

z

u

y

u

y

u

y

u

y

w

9.3 GENERAL DIFFERENTIAL SENSITIVITY

205

The last term shows that

(which is a complicated object with four indices)

@

H

=@

P

y

u

has components that are given by the product of two closed-loop transfer functions.

It is usually the case that design speci cations that limit the size of

are

@

H

=@

P

y

u

not closed-loop convex, since, roughly speaking, a product can be made small by

making either of its terms small.

9.3.1

An Example

Using the standard example plant and controller (a), described in section 2.4, we

K

consider the sensitivity of the I/O step response with respect to gain variations in

std

0 . Since

std

0 =

std

0 , the sensitivity

P

P

P

( ) = ( )

@

s

t

s

t

=0

@

is simply the unit step response of the transfer function

std

0

(a)

P

K

(1 + std

(9.25)

0

(a))2 = ST:

P

K

This transfer function is the product of two closed-loop transfer functions, which is

consistent with our general comments above.

In gure 9.6 the actual e ect of a 20% gain reduction in std

0 on the step response

P

is compared to the step response predicted by the rst order perturbational analysis,

approx( ) = ( ) 0 2 ( )

s

t

s

t

;

:

s

t

with the controller (a). The step response sensitivity with this controller is shown

K

in gure 9.7. For plant gain changes between 20%, the rst order approximation

to the step response falls in the shaded envelope ( ) 0 2 ( ).

s

t

:

s

t

We now consider the speci cation

(1) 0 75

(9.26)

js

j

:

which limits the sensitivity of the step response at time = 1 to gain variations in

t

std

0 . We will show that this speci cation is not convex.

P

The controller (a) yields a closed-loop transfer matrix (a) with (a)(1) =

K

H

s

0 697, so (a) satis es the speci cation (9.26). The controller (b) yields a closed-

:

H

K

loop transfer matrix (b) with (b)(1) = 0 702, so (b) also satis es the speci ca-

H

s

:

H

tion (9.26). However, the average of these two transfer matrices, ( (a) + (b)) 2,

H

H

=

has a step response sensitivity at = 1 of 0 786, so ( (a) + (b)) 2 does not satisfy

t

:

H

H

=

the speci cation (9.26). Therefore the speci cation (9.26) is not convex.

206