Modern Control Engineering

Section 10–5 / State Observers 767

uyx

y



Plant

C

A

B

• K

x~ Minimum-order

observer

Transformation

+

+

Figure 10–16
Observed-state
feedback control
system with a
minimum-order
observer.

Minimum-Order Observer. The observers discussed thus far are designed to

reconstruct all the state variables. In practice, some of the state variables may be accu-

rately measured. Such accurately measurable state variables need not be estimated.

Suppose that the state vector xis an n-vector and the output vector yis an m-vector

that can be measured. Since moutput variables are linear combinations of the state

variables,mstate variables need not be estimated. We need to estimate only n-m

state variables. Then the reduced-order observer becomes an (n-m)th-order observ-

er. Such an (n-m)th-order observer is the minimum-order observer. Figure 10–16

shows the block diagram of a system with a minimum-order observer.

Response to Initial Condition Response to Initial Condition

state variable

x^1

state variable

x^2

error state variable

e^1

error state variable

e^2

1.5

1

0.5

0

01234

01234

01234

012

t (sec) t (sec)

34

−0.5

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

−1.5

− 1

−0.5

0

0.5

0

−0.5

−1.5

− 1

− 2

−2.5

Figure 10–15

Response curves to

initial condition.