Modern Control Engineering

Example Problems and Solutions 827

A–10–7. Consider a completely observable system

Define the observability matrix as N:

Show that

(10–146)

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

Solution.Let us consider the case where n=3.Then Equation (10–146) can be written as

(10–147)

Equation (10–147) may be rewritten as

(10–148)

We shall show that Equation (10–148) holds true. The left-hand side of Equation (10–148) is

N*A= C (10–149)

C

CA

CA^2

S A=C

CA

CA^2

CA^3

S

N* A=C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S N*

NA(N)-^1 = C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

∑s I-A∑=sn+a 1 sn-^1 +p+an- 1 s+an

N A(N)-^1 = 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

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

y =Cx

x# =Ax

MATLAB Program 10–25

A = [0 1 0;0 0 1; -6 -11 -6];

B = [0;0;10];

J = [-2+j2sqrt(3) -2-J2Sqrt(3) -10];

K = place(A,B,J)

place: ndigits= 15

K =

15.4000 4.5000 0.8000