Modern Control Engineering

If we define

(10–130)

then Inequality (10–129) can be written as

Referring to Equation (10–128), rewritten as

notice that if we choose the generalized plant Pmatrix as

(10–131)

Then we obtain

which is exactly the same as in Equation (10–130).
We derived in Example 10–14 that if we wished to have the output yfollow the input ras
close as possible, we needed to make the norm of (s), where

(10–132)

less than 1. [See Inequality (10–126).]
Note that the controlled variable zis related to the exogenous disturbance wby

and referring to Equation (10–128)

Notice that if we choose the Pmatrix as

(10–133)

then we obtain

which is the same as £ 2 in Equation (10–132).

=Ws c

1

1 +KG

d

=Ws c 1 -

KG

1 +KG

d

=Ws-WsKG(I+KG)-^1

£=P 11 +P 12 K(I-P 22 K)-^1 P 21

P= c

Ws-WsG

I -G

d

£(s)=P 11 +P 12 K(I-P 22 K)-^1 P 21

z=£(s)w

£ 2 =

Ws

I+KG

Hq £ 2

£ 1

=WmKG(I+KG)-^1

£=P 11 +P 12 K(I-P 22 K)-^1 P 21

P= c

0 WmG

I -G

d

£=P 11 +P 12 K(I-P 22 K)-^1 P 21

7 £ 17 q 61

£ 1 =

WmKG

1 +KG

Section 10–9 / Robust Control Systems 815