Nonlinear Control Sy.. - Free

304 CHAPTER 11. NONLINEAR OBSERVERS
In general, nonlinear systems do not enjoy the same properties. Indeed, if the true state
is replaced by the estimate given by an observer, then exponential stability of the observer
does not, in general, guarantee closed-loop stability. To see this, consider the following
example.
Example 11.5 ([47]) Consider the following system:
-x + x4 + x2 (11.28)
k>0 (11.29)
We proceed to design a control law using backstepping. Using as the input in (11.28), we
choose the control law cb1(x) = -x2. With this control law, we obtain
Now define the error state variable
2 = -x+x4+ x2(61 (x) = -x
CC z=S-01(x)=1;+x2.
With the new variable, the system (11.28)-(11.29) becomes
Letting
and taking
we have that
i = -x +x2z
i = t: - q(x)
_ -kt +u+2xi
_ -kl; + u + 2x(-x + x4 + x2£)
_ -k + u + 2x(-x + x2z)
2(x2+z2)
-x2 + z[x3 - k + u + 2x(-x + x2z)]
(11.30)
(11.31)
(11.32)
(11.33)
(11.34)
u = -cz - x3 + kl; - 2x(-x + x2z), c > 0 (11.35)
V = -x2 - cz2
which implies that x = 0, z = 0 is a globally asymptotically stable equilibrium point of the
system
Lb = -x +x2z
i = -cz - x3