Modern Control Engineering

Section 10–5 / State Observers 771

u x y

x





B C

• K

Minimum-order observer

Transformation

~

~

h h~

.

++

+

++

+

+

D

C

F

A

B

x.

^

^

^

^

A

^

By subtracting Equation (10–92) from Equation (10–84), we obtain

(10–93)

Define

Then Equation (10–93) becomes

(10–94)

This is the error equation for the minimum-order observer. Note that eis an (n-1)-

vector.

The error dynamics can be chosen as desired by following the technique developed

for the full-order observer, provided that the rank of matrix

isn-1.(This is the complete observability condition applicable to the minimum-order

observer.)

F

Aab

Aab Abb







Aab Anbb-^2

V

e

=AAbb-Ke AabBe

e=xb- xb=H-H

x

b- x



b=AAbb-Ke^ AabBAxb- x



bB

Figure 10–17
System with
observed-state
feedback, where the
observer is the
minimum-order
observer.