Nonlinear Control Sy.. - Free

266 CHAPTER 10. FEEDBACK LINEARIZATION
1- The feedback law (10.4) was obtained with the sole purpose of linearizing the original
state equation. The resulting linear time-invariant system may or may not have
"desirable" properties. Indeed, the resulting system may or may not be stable, and
may or may not behave as required by the particular design.
2- Once a linear system is obtained, a secondary control law can be applied to stabilize
the resulting system, or to impose any desirable performance. This secondary law,
however, is designed using the resulting linear time-invariant system, thus taking
advantage of the very powerful and much simpler techniques available for control
design of linear systems.
Example 10.8 First consider the nonlinear mass-spring system of example 1.2
I xl = x2
which can be written in the form
(l x2 = mxl - ma2xi mx2 +
Clearly, this system is of the form (10.3) with w = 1 and O(x) = ka2xi. It then follows that
the linearizing control law is
u = ka2xi + v
Example 10.9 Now consider the system
1 it = x2
ll x2 = -axl + bx2 + cOS xl (u - x2)
Once again, this system is of the form (10.3) with w = cos xl and O(x) = x2. Substituting
into (10.4), we obtain the linearizing control law:
u=x2+cos-l X1 V
which is well defined for - z < xl < z . El