Modern Control Engineering

Section 10–5 / State Observers 769

Full-Order State Observer Minimum-Order State Observer

AAbb

Bu

y

CAab

Ke (n*1matrix) Ke [(n-1)*1matrix]

x#a-Aaa xa-Ba u

Aba xa+Bb u

x xb

Table 10–1 List of Necessary Substitutions for Writing

the Observer Equation for the Minimum-Order

State Observer

In what follows we shall present a method for designing a minimum-order observer.

The design procedure can be simplified if we utilize the design technique developed for

the full-order state observer.

Let us compare the state equation for the full-order observer with that for the

minimum-order observer. The state equation for the full-order observer is

and the “state equation” for the minimum-order observer is

The output equation for the full-order observer is

and the “output equation” for the minimum-order observer is

The design of the minimum-order observer can be carried out as follows: First, note that

the observer equation for the full-order observer was given by Equation (10–57), which

we repeat here:

(10–85)

Then, making the substitutions of Table 10–1 into Equation (10–85), we obtain

(10–86)

where the state observer gain matrix Keis an (n-1)*1matrix. In Equation (10–86),

notice that in order to estimate , we need the derivative of xa.This presents a difficulty,

because differentiation amplifies noise. If xa(=y)is noisy, the use of xis unacceptable.

a

xb

xb=AAbb-Ke AabBxb+Aba xa+Bb u+KeAx

a-Aaa^ xa-Ba^ uB

x =AA-Ke CBx +Bu+Ke y

x

a-Aaa^ xa-Ba^ u=Aab^ xb

y=Cx

x

b=Abb^ xb+Aba^ xa+Bb^ u

x

=Ax+Bu