Modern Control Engineering

Section 10–5 / State Observers 763

For the system defined by Equation (10–75), the characteristic polynomial is

Thus,

The desired characteristic polynomial for the observer is

Hence,

For the determination of the observer gain matrix, we use Equation (10–61), or

where

Hence,

(10–77)

Equation (10–77) gives the observer gain matrix Ke.The observer equation is given by Equation
(10–60):

(10–78)

Since

Equation (10–78) becomes

or

The block diagram of the system with observed-state feedback is shown in Figure 10–14(a).

= B

- 16

- 93.6

1

- 3.6

RB

x 1

x 2

R +B

16

84.6

Ry

B

x 1

x 2

R = bB

0

20.6

1

0

R - B

16

84.6

R[1 0]- B

0

1

R[29.6 3.6]rB

x 1

x 2

R + B

16

84.6

Ry

x =AA-Ke C-BKBx +Ke y

u=-Kx

x =AA-Ke CBx +Bu+Ke y

=B

0

1

1

0

RB

84.6

16

R = B

16

84.6

R

Ke=bB

0

1

1

0

RB

1

0

0

1

Rr

• 1
  B

64 +20.6

16 - 0

R

W=B

a 1

1

1

0

R =B

0

1

1

0

R

N=[CAC*]= B

1

0

0

1

R

Ke=(WN*)-^1 B

a 2 - a 2

a 1 - a 1

R

a 1 =16, a 2 = 64

=s^2 +a 1 s+a 2

As-m 1 BAs-m 2 B=(s+8)(s+8)=s^2 +16s+ 64

a 1 =0, a 2 =-20.6

∑s I-A∑=^2

s

• 20.6

- 1

s

(^2) =s^2 - 20.6=s^2 +a 1 s+a 2