Section 10–5 / State Observers 761
Equation (10–70) describes the dynamics of the observed-state feedback control system.
The characteristic equation for the system is
or
Notice that the closed-loop poles of the observed-state feedback control system consist
of the poles due to the pole-placement design alone and the poles due to the observer
design alone. This means that the pole-placement design and the observer design are
independent of each other. They can be designed separately and combined to form the
observed-state feedback control system. Note that, if the order of the plant is n,then the
observer is also of nth order (if the full-order state observer is used), and the resulting
characteristic equation for the entire closed-loop system becomes of order 2n.
Transfer Function of the Observer-Based Controller. Consider the plant defined by
Assume that the plant is completely observable. Assume that we use observed-state
feedback control Then, the equations for the observer are given by
(10–71)
(10–72)
where Equation (10–71) is obtained by substituting into Equation (10–57).
By taking the Laplace transform of Equation (10–71), assuming a zero initial
condition, and solving for we obtain
By substituting this into the Laplace transform of Equation (10–72), we obtain
(10–73)
Then the transfer function U(s)/Y(s) can be obtained as
Figure 10–13 shows the block diagram representation for the system. Notice that the
transfer function
acts as a controller for the system. Hence, we call the transfer function
(10–74)
U(s)
- Y(s)
=
num
den
=KAs I-A+Ke C+BKB-^1 Ke
KAs I-A+Ke C+BKB-^1 Ke
U(s)
Y(s)
=-KAs I-A+Ke C+BKB-^1 Ke
U(s)=-KAs I-A+Ke C+BKB-^1 Ke Y(s)
X
(s)
X
(s)=As I-A+Ke C+BKB
- 1
Ke Y(s)
X
(s),
u=-Kx
u =-Kx
x =AA-Ke C-BKBx +Ke y
u=-Kx.
y =Cx
x# =Ax+Bu
@s I-A+BK@@s I-A+Ke C@ = 0
2
s I-A+BK
0
- BK
s I-A+Ke C
(^2) = 0