Nonlinear Control Sy.. - Free

11.5. NONLINEAR SEPARATION PRINCIPLE 303
with Q = I, we obtain
1.5 -0.5
-0.5 0.5
which is positive definite. The eigenvalues of P are Amjn(P) = 0.2929, and Amax(P) _
1.7071. We now consider the function f. Denoting
we have that
x1 =
2
,
x2 =
Il f(x1) - f(x2)112 = (Q2 92)2
< 2k11x1 - x2112
[µ2
= -µ2)I
= 1(6 +µ2)(6 -{12)1
< 2121 16 - 921 = 2k112- 921
for all x satisfying 161 < k. Thus, ry = 2k and f is Lipschitz Vx = [l;1 C2]T : S I£21 < k, and
we have
or
ry=2k<
1
k<
6.8284
The parameter k determines the region of the state space where the observer is guaranteed
to work. Of course, this region is a function of the matrix P, and so a function of the
observer gain L. How to maximize this region is not trivial (see "Notes and References" at
the end of this chapter).
1
2Amax(P)
11.5 Nonlinear Separation Principle
In Section 11.1.4 we discussed the well-known separation principle for linear time-invariant
(LTI) systems. This principle guarantees that output feedback can be approached in two
steps:
(i) Design a state feedback law assuming that the state x is available.
(ii) Design an observer, and replace x with the estimate i in the control law.