Modern Control Engineering

Show that

wherea 1 ,a 2 ,p,anare the coefficients of the characteristic polynomial

Solution.Let us consider the case where n=3.We shall show that

(10–140)

The left-hand side of Equation (10–140) is

The right-hand side of Equation (10–140) is

(10–141)

The Cayley–Hamilton theorem states that matrix Asatisfies its own characteristic equation or, in

the case of n=3,

(10–142)

Using Equation (10–142), the third column of the right-hand side of Equation (10–141) becomes

Thus, Equation (10–141) becomes

Hence, the left-hand side and the right-hand side of Equation (10–140) are the same. We have

thus shown that Equation (10–140) is true. Consequently,

The preceding derivation can be easily extended to the general case of any positive integer n.

A–10–3. Consider a completely state controllable system

Define

M= CB  AB  p  An-^1 BD

x# =Ax+Bu

M-^1 AM= C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S

CBABA^2 BDC

0

1

0

0

0

1

• a 3


• a 2


• a 1

S =CABA^2 BA^3 BD

• a 3 B-a 2 AB-a 1 A^2 B=A-a 3 I-a 2 A-a 1 A^2 BB=A^3 B

A^3 +a 1 A^2 +a 2 A+a 3 I= 0

CBABA^2 BDC

0

1

0

0

0

1

• a 3


• a 2


• a 1

S =CABA^2 B-a 3 B-a 2 AB-a 1 A^2 BD

AM=ACB  AB  A^2 BD=CAB  A^2 B  A^3 BD

AM=MC

0

1

0

0

0

1

• a 3


• a 2


• a 1

S

∑s I-A∑=sn+a 1 sn-^1 +p+an- 1 s+an

M-^1 AM= G

0 1 0    0

0 0 1    0

p

p

p

p

0 0 0    1

• an


• an- 1


• an- 2
  
  
  


• a 1

W

Example Problems and Solutions 821