Modern Control Engineering

Note that can be determined as follows: Since

z(s)=P 11 w(s)+P 12 u(s)

y(s)=P 21 w(s)+P 22 u(s)

u(s)=K(s)y(s)

we obtain

y(s)=P 21 w(s)+P 22 K(s)y(s)

Hence

or

Therefore,

Hence,

(10–128)

EXAMPLE 10–15 Let us determine the Pmatrix in the generalized plant diagram of the control system considered

in Example 10–14. We derived Inequality (10–125) for the control system to be robust stable.

Rewriting Inequality (10–125), we have

g (10–129)

WmKG

1 +KG

g

q

61

£(s)=P 11 +P 12 K(s)[I-P 22 K(s)]-^1 P 21

={P 11 +P 12 K(s)[I-P 22 K(s)]-^1 P 21 }w(s)

z(s)=P 11 w(s)+P 12 K(s)[I-P 22 K(s)]-^1 P 21 w(s)

y(s)=[I-P 22 K(s)]-^1 P 21 w(s)

[I-P 22 K(s)]y(s)=P 21 w(s)

£(s)

814 Chapter 10 / Control Systems Design in State Space

K(s)

P(s)

w

u

z

y

P 11

P 21

P 12

P 22

Figure 10–46

A generalized plant

diagram.

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