Nonlinear Control Sy.. - Free

138 CHAPTER 5. FEEDBACK SYSTEMS
5.1 Basic Feedback Stabilization
In this section we look at several examples of stabilization via state feedback. These examples
will provide valuable insight into the backstepping design of the next section.
Example 5.1 Consider the first order system given by
we look for a state feedback of the form
u = O(x)
that makes the equilibrium point at the origin "asymptotically stable." One rather obvious
way to approach the problem is to choose a control law u that "cancels" the nonlinear term
ax2. Indeed, setting
and substituting (5.5) into (5.4) we obtain
u = -ax 2 - x
±= -x
which is linear and globally asymptotically stable, as desired. 0
We mention in passing that this is a simple example of a technique known as feedback
linearization. While the idea works quite well in our example, it does come at a certain
price. We notice two things:
(i) It is based on exact cancelation of the nonlinear term ax2. This is undesirable since
in practice system parameters such as "a" in our example are never known exactly.
In a more realistic scenario what we would obtain at the end of the design process
with our control u is a system of the form
i=(a-a)x2-x
where a represents the true system parameter and a the actual value used in the
feedback law. In this case the true system is also asymptotically stable, but only
locally because of the presence of the term (a - a)x2.
(ii) Even assuming perfect modeling it may not be a good idea to follow this approach
and cancel "all" nonlinear terms that appear in the dynamical system. The reason
is that nonlinearities in the dynamical equation are not necessarily bad. To see this,
consider the following example.