Nonlinear Control Sy.. - Free

11.1. OBSERVERS FOR LINEAR TIME-INVARIANT SYSTEMS 295
or
x = (A - LC).i. (11.10)
Thus, x -+ 0 as t --> oo, provided the so-called observer error dynamics (11.10) is asymptotically
(exponentially) stable. This is, of course, the case, provided that the eigenvalues
of the matrix A - LC are in the left half of the complex plane. It is a well-known result
that, if the observability condition (11.7) is satisfied, then the eigenvalues of (A - LC) can
be placed anywhere in the complex plane by suitable selection of the observer gain L.
11.1.4 Separation Principle
Assume that the system (11.1) is controlled via the following control law:
u = Kx (11.11)
i = Ax+Bu+L(y-Cx) (11.12)
i.e. a state feedback law is used in (11.11). However, short of having the true state x
available for feedback, an estimate x of the true state x was used in (11.11). Equation
(11.12) is the observer equation. We have
i = Ax+BKx
Ax+BKx - BK±
(A + BK)x - BK±.
Also
Thus
where we have defined
(A - LC)i
A 0BK ABLC] [x]
x=Ax
The eigenvalues of the matrix A are the union of those of (A + BK) and (A - LC).
Thus, we conclude that
(i) The eigenvalues of the observer are not affected by the state feedback and vice versa.
(ii) The design of the state feedback and the observer can be carried out independently.
This is called the separation principle.