Modern Control Engineering

828 Chapter 10 / Control Systems Design in State Space

The right-hand side of Equation (10–148) is

(10–150)

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

Hence,

Thus, the right-hand side of Equation (10–150) becomes the same as the right-hand side of

Equation (10–149). Consequently,

which is Equation (10–148). This last equation can be modified to

The derivation presented here can be extended to the general case of any positive integer n.

A–10–8. Consider a completely observable system defined by

(10–151)

(10–152)

Define

and

where the a’s are coefficients of the characteristic polynomial

Define also

Q=(WN*)-^1

∑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

N= CC*A* C*p(A*)n-^1 C*D

y =Cx+Du

x# =Ax+Bu

NA(N)-^1 = C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

N*A= C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S N*

• a 1 CA^2 - a 2 CA-a 3 C=CA^3

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

=C

CA

CA^2

• a 3 C-a 2 CA-a 1 CA^2

S

C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S N*=C

0

0

• a 3

1

0

• a 2

0

1

• a 1

SC

C

CA

CA^2

S

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