Page 792 - Mechanical Engineers' Handbook (Volume 2)

6 Observer-Based Controllers 783

If the problem as stated is nonsingular
det[V (k)] 0 k k 0 (110)
2
the optimal-observer gain matrix is given by the recurrence relations
T
T
L(k) [F(k)Q(k)C (k) V (k)][V (k) C(k)Q(k)C (k)] (111)
2
12
T
Q(k 1) [F(k) L(k)C(k)]Q(k)F (k) V (k) L(k)V (k) (112)
T
1
12
with the initial condition
Q(k ) Q 0 (113)
0
and k k . The initial condition on the observer state should be
0
ˆ x(k ) x 0 (114)
0
Again, the similarity of the optimal-observer equations (111)–(113) to the optimal-controller
equations (33)–(35) for the LQR problem results from the duality of state estimation and
state feedback control problems.
The similarity extends to the steady-state behavior of the optimal observer and the
3
optimal controller. If the time-varying system
x(k 1) F(k)x(k) S(k)w (k) (115)
1
y(k) C(k)x(k) w (k) (116)
2
is uniformly completely controllable by w (k) and uniformly completely reconstructible, the
1
solution Q(k) of Eqs. (111) and (112) converges to a steady-state solution Q (k)as k →
s
0
for any positive-semidefinite Q . The corresponding steady-state observer
0
ˆ x(k 1) F(k) ˆx(k) L (k)[y(k) C(k) ˆx(k)] (117)
s
where L (k) is obtained using Q (k) for Q(k) in Eq. (111) is exponentially stable. Also, if
s s
the system and the noise statistics are time invariant, the matrix difference equations (111)
and (112) become algebraic equations as k → . If the corresponding time-invariant system
0
is completely controllable by the input w (k) and completely observable, the resulting steady-
1
state optimal observer is asymptotically stable. Again, note that the measurable input u(k)
has been omitted from Eqs. (115) and (116) for simplicity but does not change the substance
of the results.
32
Interested readers are referred to Kwakernaak and Sivan for a more complete consid-
eration of the optimal observer. There is also extensive literature available on Kalman fil-
ters. 31,33,34
6 OBSERVER-BASED CONTROLLERS
The observers described in the preceding section can be used to provide estimates of system
state that, in turn, can be used to provide state feedback as described in Sections 2–4. The
resulting controllers are dynamic compensators and are referred to as observer-based con-
trollers.
The design of such observer-based controllers for LTI systems is simplified somewhat
by the fact that their modes or eigenvalues satisfy the separation property, that is, the eigen-
values of the observer-controller are the same as the eigenvalues of the observer and the
eigenvalues of the controller, the latter evaluated assuming perfect state measurement. For