Nonlinear Control Sy.. - Free

11.4. LIPSCHITZ SYSTEMS 301
11.4 Lipschitz Systems
The nonlinear observer discussed in the previous section is inspired by the work on feedback
linearization, and belongs to the category of what can be called the differential geometric
approach. In this section we show that observer design can also be studied using a Lyapunov
approach. For simplicity, we restrict attention to the case of Lipschitz systems, defined
below.
Consider a system of the form:
J i = Ax + f (x, u)
y=Cx
(11.23)
where A E lRnxn C E Rlxn and f : Rn x R -a Rn is Lipschitz in x on an open set D C R',
i.e., f satisfies the following condition:
If(x1,u*) - f(x2,u )lI < 7IIx1 -x211 Vx E D. (11.24)
Now consider the following observer structure
x=Ax+f(x,u)+L(y-Cx) (11.25)
where L E Rnxl. The following theorem shows that, under these assumptions, the estimation
error converges to zero as t -+ oo.
Theorem 11.3 Given the system (11.23) and the corresponding observer (11.25), if the
Lyapunov equation
P(A - LC) + (A - LC)T P = -Q (11.26)
where p = PT > 0, and Q = QT > 0, is satisfied with
y<
.min (Q)
2A, (P)
then the observer error 2 = x - x is asymptotically stable.
Proof:
= [Ax + f (x, u)] - [Ax + f (x, u) + L(y - C2)]
= (A - LC)2 + f (x, u) - f (x, u).
(11.27)