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

CHAPTER 7 REALIZABILITY AND CLOSED-LOOP STABILITY

+

q

r

w

1

z

U

+

3

U

v

2

e

U

U

Q

Figure 7.5 redrawn.

Figure

7.6

+

q

r

w

1

z

U

+

3

e

v

Q

U

2

U

Figure 7.6 redrawn.

Figure

7.7

7.3.1

Modified Controller Paradigm for a Stable Plant

As an example of the modi ed controller paradigm, we consider the special case of

a stable plant (see section 7.2.4). Since the plant is stable, the nominal controller

nom = 0 stabilizes the plant.

K

How do we modify the zero controller to produce and accept ? One obvious

e

v

method is to add into , and let be the di erence between and yu , which

v

u

e

y

P

u

ensures that the closed-loop transfer matrix from to is zero, as required by the

v

e

modi ed controller paradigm. This is shown in gure 7.8.

From gure 7.8 we see that

1 = zw

U

P

2 = zu

U

P

3 = yw

U

P

:

To apply the second step of the modi ed controller paradigm, we connect a stable

7.3 MODIFIED CONTROLLER PARADIGM

161

w

z

P

y

u

+

r

nom = 0

q

K

+

q

yu

+

r

P

;

v

e

One method of extracting and injecting when the plant is

Figure

7.8

e

v

stable.

w

z

P

y

u

+

r

nom = 0

q

K

+

q

yu

+

r

P

;

Q

v

e

K

The modied controller paradigm, for a stable plant, using the

Figure

7.9

augmented controller shown in gure 7.8.

162