Modern Control Engineering

Thus,

Hence,

Next, referring to Equation (10–138), we have

(10–139)

Referring to Equation (10–136), notice that vector Bcan be written in terms of qlinearly

independent column vectors f 1 ,f 2 ,p,fq.Thus, we have

Consequently, Equation (10–139) may be written as follows:

Thus,

where

A–10–2. Consider a completely state controllable system

Define the controllability matrix as M:

M= CB  AB  p  An-^1 BD

x

=Ax+Bu

B 11 = F

b 11

b 21







bq1

V

Bˆ = c

B 11

0

d

b 11 f 1 +b 21 f 2 +p+bq1 fq=Cf 1 f 2 pfqvq+ 1 pvnD

b 11

b 21







bq1

0







0

B=b 11 f 1 +b 21 f 2 +p+bq1 fq

B= Cf 1 f 2 pfqvq+ 1 pvnD Bˆ

P-^1 AP=Aˆ = c

A 11

0

A 12

A 22

d

AP=Pc

A 11

0

A 12

A 22

d

820 Chapter 10 / Control Systems Design in State Space

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