Section 10–5 / State Observers 773
where
Observed-State Feedback Control System with Minimum-Order Observer.
For the case of the observed-state feedback control system with full-order state observer,
we have shown that the closed-loop poles of the observed-state feedback control system
consist of the poles due to the pole-placement design alone, plus the poles due to the
observer design alone. Hence, the pole-placement design and the full-order observer
design are independent of each other.
For the observed-state feedback control system with minimum-order observer, the
same conclusion applies. The system characteristic equation can be derived as
(10–98)
(See Problem A–10–11for the details.) The closed-loop poles of the observed-state feed-
back control system with a minimum-order observer comprise the closed-loop poles
due to pole placement Cthe eigenvalues of matrix (A-BK)Dand the closed-loop poles
due to the minimum-order observer Cthe eigenvalues of matrix (Abb-KeAab)D. There-
fore, the pole-placement design and the design of the minimum-order observer are
independent of each other.
Determining Observer Gain Matrix Kewith MATLAB. Because of the duality
of pole-placement and observer design, the same algorithm can be applied to both the
pole-placement problem and the observer-design problem. Thus, the commands acker
andplacecan be used to determine the observer gain matrix Ke.
The closed-loop poles of the observer are the eigenvalues of matrix A-KeC.The
closed-loop poles of the pole-placement are the eigenvalues of matrix A-BK.
Referring to the duality problem between the pole-placement problem and observer-
design problem, we can determine Keby considering the pole-placement problem for the
dual system. That is, we determine Keby placing the eigenvalues of A-CKeat the
desired place. Since Ke=K*, for the full-order observer we use the command
Ke= acker(A',C',L)'
whereLis the vector of the desired eigenvalues for the observer. Similarly, for the full-
order observer, we may use
Ke= place(A',C',L)'
providedLdoes not include multiple poles. [In the above commands, prime (') indicates
the transpose.] For the minimum-order (or reduced-order) observers, use the following
commands:
Ke= acker(Abb',Aab',L)'
or
Ke= place(Abb',Aab',L)'
@s I-A+BK@@s I-Abb+Ke Aab@ = 0
fAAbbB=Abbn-^1 +aˆ 1 Anbb-^2 +p+aˆn- 2 Abb+aˆn- 1 I