Nonlinear Control Sy.. - Free

282 CHAPTER 10. FEEDBACK LINEARIZATION
which renders the exponentially stable tracking error closed-loop system
e + k2e + kle = 0. (10.32)
A glance at the result shows that the order of the closed-loop tacking error (10.32) is
the same as the relative order of the system. In our case, r = 2, whereas the original
state space realization has order n = 3. Therefore, part of the dynamics of the original
system is now unobservable after the input-output linearization. To see why, we reason as
follows. During the design stage, based on the input-output linearization principle, the state
equation (10.31) was manipulated using the input u. A look at this control law shows that u
consists of a state feedback law, and thus the design stage can be seen as a reallocation of the
eigenvalues of the A-matrix via state feedback. At the end of this process observability of the
three-dimensional state space realization (10.31) was lost, resulting in the (external) twodimensional
closed loop differential equation (10.32). Using elementary concepts of linear
systems, we know that this is possible if during the design process one of the eigenvalues of
the closed-loop state space realization coincides with one of the transmission zeros of the
system, thus producing the (external) second-order differential equation (10.32). This is
indeed the case in our example. To complete the three-dimensional state, we can consider
the output equation
Thus,
y = Pox1 +P1x2
= Poxl +pi ;i.
±1 = - xl + 1
1 y = Aid xl + Bid y.
P1 P1
(10.33)
Equation (10.33) can be used to "complete" the three-dimensional state. Indeed, using
xl along with e and a as the "new" state variables, we obtain full information about the
original state variables x1 - x3 through a one-to-one transformation.
The unobservable part of the dynamics is called the internal dynamics, and plays a
very important role in the context of the input-output linearization technique. Notice that
the output y in equation (10.33) is given by y = e + yd, and so y is bounded since e was
stabilized by the input-output linearization law, and yd is the desired trajectory. Thus, xl
will be bounded, provided that the first-order system (10.33) has an exponentially stable
equilibrium point at the origin. As we well know, the internal dynamics is thus exponentially
stable if the eigenvalues of the matrix Aid in the state space realization (10.33) are in the
left half plane. Stability of the internal dynamics is, of course, as important as the stability
of the external tracking error (10.32), and therefore the effectiveness of the input-output
linearization technique depends upon the stability of the internal dynamics. Notice also