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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

= (

) 1

~(

~) 1

;

;

B

K

I

;

P

K

P

;

K

I

;

P

K

P

y

u

y

u

y

u

y

u

= ~(

~) 1;

C

K

I

;

P

K

y

u

= (

) 1 ~

~ 1

;

(

);

D

K

I

;

P

K

;

K

I

;

P

K

:

y

u

y

u

The special form of

given in (7.10) is called a bilinear or linear fractional de-

K

pendence on . We will encounter this form again in section 7.2.6.

7.1.1

An Example

We consider the standard plant example of section 2.4. The step responses from the

reference input to p and for each of the controllers (a) and (b) of section 2.4

r

y

u

K

K

are shown in gures 7.2(a) and (b). Since rlzbl is a ne, we conclude that every

H

transfer matrix on the line passing through (a) and (b),

H

H

(a) + (1 ) (b)

H

;

H

2

R

can be realized as the closed-loop transfer matrix of our standard plant with some

controller. Figures 7.2(c) and (d) show the step responses from to p and of ve

r

y

u

of the transfer matrices on this line.

The average of the two closed-loop transfer matrices ( = 0 5) is realized by

:

the controller 0 5, which can be found from (7.10). Even though both (a) and

K

K

:

(b) are 3rd order, 0 5 turns out to be 9th order. From this fact we draw two

K

K

:

conclusions. First, the speci cation that be the transfer matrix achieved by a

H

controller of order no greater then ,

K

n

=

+

(

) 1

;

rlzbl =

H

P

P

K

I

;

P

K

P

z

w

z

u

y

u

y

w

H

H

n

for some with order( )

K

K

n

is not in general convex (whereas the speci cation rlzbl, which puts no limit on the

H

order of , is convex). Our second observation is that the controller 0 5, which

K

K

:

yields a closed-loop transfer matrix that is the average of the closed-loop transfer

matrices achieved by the controllers (a) and (b), would not have been found by

K

K

varying the parameters (e.g., numerator and denominator coe cients) in (a) and

K

(b).

K

7.2

Internal Stability

We remind the reader that a transfer function is proper if it has at least as many

poles as nite zeros, or equivalently, if it has a state-space realization a transfer

function is stable if it is proper and all its poles have negative real part nally, a

transfer matrix is stable if all of its entries are stable. We noted in section 6.4.1

that these are ane constraints on a transfer function or transfer matrix.

7.2 INTERNAL STABILITY

151

1:8

20

( )

1:6

a

13

s

(t)

1:4

( )b

10

;

;

1:2

23

s

(t)

1

;

;

0:8

0

@

I

@

( )

@

I

@

b

0:6

13

( )

a

s

(t)

10

;

23

0:4

s

(t)

0:20

20

;

0:2

;

0

1

2

3

4

5

0

1

2

3

4

5

t

t

(a)

(b)

1:8

20

=

;0:3

1:6

=

1:3

=

0

1:4

=

1

10

1:2

() 1

() 0

t

t

13 0:8

0:6

23

s

s

H

Y

10

@

I

H

;

=

0:5

0:4

@

=

0:5

0:2

@

I

@

@

I

=

0

H

Y

H

@

=

1

0

20

=

;0:3

;

H

Y

H

=

1:3

0:2

;

0

1

2

3

4

5

0

1

2

3

4

5

t

t

(c)

(d)

(a) shows the closed-loop step responses from to p for the

Figure

7.2

r

y

standard example with the two controllers (a) and (b). (b) shows the step

K

K

responses from to . In (c) and (d) the step responses corresponding to

r

u

ve dierent values of are shown. Each of these step responses is achieved

by some controller.

7.2.1

A Motivating Example

Consider our standard example SASS 1-DOF control system described in section 2.4,

with the controller

( ) = 36 + 33s

K

s

10

:

;

s

This controller yields the closed-loop I/O transfer function

( ) =

33 + 36

+ 36

s

T

s

3 + 10 2 + 33 + 36 = 33s

( + 3)2( + 4)

s

s

s

s

s

which is a stable lowpass lter. Thus, we will have p

provided the reference

y

r

signal does not change too rapidly the controller yields good tracking of slowly

r

K

152