Modern Control Engineering

This means that there are qlinearly independent column vectors in the controllability matrix. Let

us define such qlinearly independent column vectors as f 1 ,f 2 ,p,fq.Also, let us choose n-q

additionaln-vectorsvq+1,vq+2,p,vnsuch that

is of rank n.By using matrix Pas the transformation matrix, define

Show that can be given by

whereA 11 is a q*qmatrix,A 12 is a q*(n-q)matrix,A 22 is an (n-q)*(n-q)matrix, and

0 is an (n-q)*qmatrix. Show also that matrix can be given by

whereB 11 is a q*1matrix and 0 is an (n-q)*1matrix.

Solution.Notice that

or

(10–137)

Also,

(10–138)

Since we have qlinearly independent column vectors f 1 ,f 2 ,p,fq,we can use the Cayley–Hamilton

theorem to express vectors Af 1 ,Af 2 ,p,Afqin terms of these qvectors. That is,

Afq=a 1 q f 1 +a 2 q f 2 +p+aqq fq







Af 2 =a 12 f 1 +a 22 f 2 +p+aq 2 fq

Af 1 =a 11 f 1 +a 21 f 2 +p+aq 1 fq

B=PBˆ

=Cf 1 f 2 pfqvq+ 1 pvnD Aˆ

CAf 1 Af 2 pAfqAvq+ 1 pAvnD

AP=PAˆ

Bˆ = c

B 11

0

d

Bˆ

Aˆ = c

A 11

0

A 12

A 22

d

Aˆ

P-^1 AP=Aˆ, P-^1 B=Bˆ

P= Cf 1 f 2 pfqvq+ 1 vq+ 2 pvnD

818 Chapter 10 / Control Systems Design in State Space

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