Nonlinear Control Sy.. - Free

11.5. NONLINEAR SEPARATION PRINCIPLE 305
This control law, of course, assumes that both x and 1; are measured. Now assume that only
x is measured, and suppose that an observer is used to estimate l;. This is a reduced-order
observer, that is, one that estimates only the nonavailable state 1;. Let the observer be given
by
1;=-kl;+u.
The estimation error is t; = l; - l;. We have
It follows that
_ -kl;+u+kl-u
-kl;
which implies that l; exponentially converges toward . Using the estimated in the control
law (11.35), we obtain
&)e-kt
-x+x2z+x21;
-cz-x3+2x31;
Even though l; exponentially converges to zero, the presence of the terms x21; and 2x31; leads
to finite escape time for certain initial conditions. To see this, assume that the error variable
z - 0. We have
Solving for x, we obtain
x(t) =
_ -x +x21;
&)e-kt
xo(1 + k)
(1 + k - oxo)et + Eoxoe-kt
which implies that the state x grows to infinity in finite time for any initial condition lxo >
1+k.