Modern Control Engineering

822 Chapter 10 / Control Systems Design in State Space

and

where the ai’s are coefficients of the characteristic polynomial

Define also

Show that

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

(10–143)

Referring to Problem A–10–2, we have

Hence, Equation (10–143) can be rewritten as

Therefore, we need to show that

(10–144)

The left-hand side of Equation (10–144) is

C

0

1

0

0

0

1

• a 3


• a 2


• a 1

SC

a 2

a 1

1

a 1

1

0

1

0

0

S = C

• a 3
  0
  0

0

a 1

1

0

1

0

S

C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S W=WC

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

W-^1 C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S W= C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

M-^1 AM= C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S

T-^1 AT=(MW)-^1 A(MW)=W-^1 (M-^1 AM) W= C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

T-^1 AT= G

0 0    0

• an

1 0    0

• an- 1

0 1    0

• an- 2

p

p

p

p

0 0    1

• a 1

W, T-^1 B=G

0 0    0 1

W

T=MW

∑s I-A∑=sn+a 1 sn-^1 +p+an- 1 s+an

W=G

an- 1

an- 2







a 1

1

an- 2

an- 3







1

0

p

p

p

p

a 1

1    0 0

1 0    0 0

W

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