Nonlinear Control Sy.. - Free

10.6. THE ZERO DYNAMICS 285
Definition 10.11 Given a dynamical system of the form (10.37)-(10.39) (i.e., represented
in normal form), the autonomous equation
is called the zero dynamics.
1 = fo(y,0)
Because of the analogy with the linear system case, systems whose zero dynamics is stable
are said to be minimum phase.
Summary: The stability properties of the zero dynamics plays a very important role
whenever input-output linearization is applied. Input-output linearization is achieved via
partial cancelation of nonlinear terms. Two cases should be distinguished:
r = n: If the relative degree of the nonlinear system is the same as the order of the
system, then the nonlinear system can be fully linearized and, input-output linearization
can be successfully applied. This analysis, of course, ignores robustness issues
that always arise as a result of imperfect modeling.
r < n: If the relative degree of the nonlinear system is lower than the order of the
system, then only the external dynamics of order r is linearized. The remaining n - r
states are unobservable from the output. The stability properties of the internal
dynamics is determined by the zero dynamics. Thus, whether input-output linearization
can be applied successfully depends on the stability of the zero dynamics. If the
zero dynamics is not asymptotically stable, then input-output linearization does not
produce a control law of any practical use.
Before discussing some examples, we notice that setting y - 0 in equations (10.37)-
(10.39) we have that
y-0 ty t;-0
u(t) _ O(x)
f7 = fo(mI, 0)
Thus the zero dynamics can be defined as the internal dynamics of the system when the
output is kept identically zero by a suitable input function. This means that the zero
dynamics can be determined without transforming the system into normal form.
Example 10.18 Consider the system
xl = -kxl - 2x2u
4 22 = -x2 + xiu
y = X2