Section 10–5 / State Observers 759
Since the desired characteristic equation is
by comparing Equation (10–66) with this last equation, we obtain
or
Method 3: We shall use Ackermann’s formula given by Equation (10–65):
where
Thus,
and
As a matter of course, we get the same Keregardless of the method employed.
The equation for the full-order state observer is given by Equation (10–57),
or
Finally, it is noted that, similar to the case of pole placement, if the system order nis 4 or
higher, methods 1 and 3 are preferred, because all matrix computations can be carried out by a
computer, while method 2 always requires hand computation of the characteristic equation
involving unknown parameters ke1,ke2,p,ken.
Effects of the Addition of the Observer on a Closed-Loop System. In the
pole-placement design process, we assumed that the actual state x(t)was available for
feedback. In practice, however, the actual state x(t)may not be measurable, so we will
need to design an observer and use the observed state for feedback as shown in Fig-
ure 10–12. The design process, therefore, becomes a two-stage process, the first stage
being the determination of the feedback gain matrix Kto yield the desired characteristic
equation and the second stage being the determination of the observer gain matrix Ke
to yield the desired observer characteristic equation.
Let us now investigate the effects of the use of the observed state rather than
the actual state x(t),on the characteristic equation of a closed-loop control system.
x(t),
x(t)
B
x 1
x 2R = B
0
1
- 100
- 20
RB
x 1
x 2R+ B
0
1
Ru+ B
120.6
20
Ry
x =AA-Ke CBx +Bu+Ke y= B
120.6
20
412
120.6
RB
0
1
1
0
RB
0
1
R = B
120.6
20
R
Ke=AA^2 + 20 A+ 100 IBB
0
1
1
0
R
- 1
B
0
1
R
f(A)=A^2 + 20 A+ 100 If(s)=As-m 1 BAs-m 2 B=s^2 +20s+ 100Ke=f(A)B
C
CA
R
- 1
B
0
1
R
Ke= B
120.6
20
R
ke1=120.6, ke2= 20
s^2 +20s+ 100 = 0