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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

as shown in gure 7.9. so that the closed-loop transfer matrix is

Q

= 1 + 2 3

H

U

U

QU

:

Thus the set of closed-loop transfer matrices achievable by the modi ed controller

shown in gure 7.9 is

mcp = zw + zu

yw

stable

H

fP

P

QP

j

Q

g

:

The expression here for mcp is the same as the expression for stable in equa-

H

H

tion (7.21) in section 7.2.4. So in this case the modi ed controller paradigm gener-

ates all stabilizing controllers: any stabilizing controller for a stable plant can

K

P

be implemented with a suitable stable as shown in gure 7.9.

Q

The reader can also verify that the connection of with the augmented nominal

Q

controller yields = ( + yu) 1 |exactly the same formula as (7.22) with

;

K

I

QP

Q

Q

substituted for .

R

7.4

A State-Space Parametrization

A general method of applying the modi ed controller paradigm starts with a nom-

inal controller that is an estimated-state feedback. The estimated-state-feedback

controller is given by

=

sfb^

(7.28)

u

;K

x

where sfb is some appropriate matrix (the state-feedback gain) and ^ is an estimate

K

x

of the component of due to , governed by the observer equation

x

u

_^ = P ^ + u + est(

y^)

(7.29)

x

A

x

B

u

L

y

;

C

x

where est is some appropriate matrix (the estimator gain). The transfer matrix of

L

this controller is thus

nom( ) =

sfb(

P + u sfb + est y) 1

;

est

K

s

;K

sI

;

A

B

K

L

C

L

:

nom will stabilize

provided sfb and est are chosen such that P

u sfb

K

P

K

L

A

;

B

K

and P

est y are stable, which we assume in the sequel.

A

;

L

C

To augment this estimated-state-feedback nominal controller, we inject into

v

, before the observer tap, meaning that (7.28) is replaced by

u

=

sfb^ +

(7.30)

u

;K

x

v

and therefore the signal does not induce any observer error. For the signal we

v

e

take the output prediction error:

=

y^

(7.31)

e

y

;

C

x :

7.4 A STATE-SPACE PARAMETRIZATION

163

D

y

w

+

+

q

q

r

w

z

D

z

w

+

r

+

D

z

u

x

B

C

w

z

(

) 1

;

sI

;

A

P

+

q

r

y

u

C

B

y

u

+

P

nom

K

+

q

^

r

C

B

y

x

u

(

) 1

;

;

sI

;

A

P

est

L

sfb

K

;

r

+

q

v

e

Q

+

x

Q

C

Q

(

) 1

;

r

+

q

B

sI

;

A

Q

Q

D

Q

The modied controller paradigm as applied to a nominal

Figure

7.10

estimated-state-feedback controller nom. is added to the actuator signal

K

v

before the observer tap, and is the output prediction error. With the

u

e

stable realization added, the modied controller is called an observer-

Q

based controller.

164