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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

signal , and we must include and in the regulated variables vector , in which

w

u

y

z

case the four transfer matrices (7.11{7.14) appear as submatrices of the closed-loop

transfer matrix . Internal stability is then expressed as the speci cation that

H

these entries of be stable, which we mentioned in chapter 6 (section 6.4.1) is an

H

a ne speci cation.

It seems clear that any sensible set of design speci cations should limit the e ect

of sensor and process noise on and . Indeed, a sensible set of speci cations will

u

y

constrain these four transfer matrices more tightly than merely requiring stability

some norm of these transfer matrices will be constrained. So sensible sets of spec-

i cations will generally be strictly tighter than internal stability internal stability

will be a redundant speci cation. We will see examples of this in chapter 12: for

the LQG and

problems, niteness of the objective will imply internal stability,

H

1

provided the objectives satisfy certain sensibility requirements. See the Notes and

References at the end of this chapter.

7.2.3

Closed-loop Affineness of Internal Stability

We now consider the speci cation that is the closed-loop transfer matrix achieved

H

by a controller that stabilizes the plant:

=

(

) 1

;

stable =

zw + zu

yu

yw

H

P

P

K

I

;

P

K

P

(7.15)

H

H

for some that stabilizes

:

K

P

Of course, this is a stronger speci cation than realizability.

Like rlzbl, stable is also a ne: if and ~ each stabilize , then for each

H

H

K

K

P

, the controller

given by (7.10) also stabilizes . In the example of

2

R

K

P

section 7.1.1, the controllers (a) and (b) each stabilize the plant. Hence the ve

K

K

controllers that realize the ve step responses shown in gure 7.2 also stabilize the

plant.

We can establish that stable is a ne by direct calculation. Suppose that con-

H

trollers and ~ each stabilize the plant, yielding closed-loop transfer matrices

K

K

H

and ~, respectively. We substitute the controller

given in (7.10) into the four

H

K

transfer matrices (7.11{7.14), and after some algebra we nd that

(

yu ) 1

(

) 1

) ~(

) 1

;

;

;

yu =

yu

yu + (1

yu ~

yu

K

I

;

P

K

P

K

I

;

P

K

P

;

K

I

;

P

K

P

(

yu ) 1 =

(

) 1

1

;

+ (1 ) ~(

);

;

yu

yu ~

K

I

;

P

K

K

I

;

P

K

;

K

I

;

P

K

(

yu ) 1

) 1

1

;

)(

);

;

yu = (

yu

yu + (1

yu ~

yu

I

;

P

K

P

I

;

P

K

P

;

I

;

P

K

P

(

yu ) 1 = (

) 1 + (1 )(

) 1

;

;

;

yu

yu ~

I

;

P

K

I

;

P

K

;

I

;

P

K

:

Thus, the four transfer matrices (7.11{7.14) achieved by

are a ne combinations

K

of those achieved by and ~. Since the right-hand sides of these equations are all

K

K

stable, the left-hand sides are stable, and therefore

stabilizes .

K

P

We can use the same device that we used to simplify our description of rlzbl.

H

The four transfer matrices (7.11{7.14) can be expressed in terms of the transfer

7.2 INTERNAL STABILITY

155

matrix given in (7.7):

R

(

yu ) 1

;

yu =

yu

(7.16)

K

I

;

P

K

P

R P

(

yu ) 1

;

=

(7.17)

K

I

;

P

K

R

(

yu ) 1

+

)

;

yu = (

yu

yu

(7.18)

I

;

P

K

P

I

P

R

P

(

yu ) 1 = +

(7.19)

;

yu

I

;

P

K

I

P

R :

Hence we have

8

yu

9

R P

>

>

>

>

<

=

stable =

zw + zu yw

R

are stable

(7.20)

H

P

P

R P

+ yu

:

I

P

R

>

>

>

>

:

( + yu ) yu

I

P

R

P

We will nd this description useful.

7.2.4

Internal Stability for a Stable Plant

If the plant is stable, then in particular yu is stable. It follows that if is stable,

P

P

R

then so are yu, + yu , and ( + yu ) yu. Hence we have

R P

I

P

R

I

P

R

P

stable = zw + zu yw

stable

(7.21)

H

fP

P

R P

j

R

g

:

This is just our description of rlzbl, with the additional constraint that be stable.

H

R

Given any stable , the controller that stabilizes and yields a closed-loop

R

P

transfer matrix = zw + zu yw is

H

P

P

R P

= ( + yu) 1

(7.22)

;

K

I

R P

R :

Conversely, every controller that stabilizes can be expressed by (7.22) for some

P

stable .

R

7.2.5

Internal Stability via Interpolation Conditions

For the classical SASS 1-DOF control system (see section 2.3.2), the speci cation of

internal stability can be expressed in terms of the I/O transfer function , although

T

the speci cation is not as simple as stability of alone (recall our motivating

T

example). We have already noted that a closed-loop transfer matrix

that is

H

realizable has the form

=

0(1

)

P

;

T

;T

T

H

0

0

;T

;T

=P

T

=P

=

0 0 0

0

1

1

P

0 0 0 +

;P

;

(7.23)

T

1

1 0 1 0

;

;

=P

=P

where is the I/O transfer function. We will describe stable as the set of transfer

T

H

matrices of the form (7.23), where satis es some additional conditions.

T

156