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Robustness with Respect to Generation of the Nominal Signals
It was previously indicated that, in practice, the nominal signals
x (t) and (t) used as open-loop commands may not be generated
correctly. This could be due to modeling errors in the nonlinear system
or even to imperfections in the actual generating mechanism.
First consider the case when only the input is not generated
correctly. This is the more important case since often the nominal
state trajectory is known exactly while the corresponding nominal input
is only approximate due to modeling errors. Suppose that the nominal
input actually supplied to the system is a(t) while, as before, the
*
input needed to obtain the nominal trajectory is u (t). Now make the
definition
d(t) = *(t) a(t) (5-15)
where lid(t) will be referred to as an input disturbance.
In terms of the definition given by equation (5-15) we may look at
the problem from a different point of view. That is, assume the input
u (t) is being generated correctly, however, also assume that there is a
disturbance d(t) acting in the input channel so that, effectively,
(t) ua(t) = Iia(t) is the true signal supplied to the system. This
is shown in Figure 5-3. By looking at the problem from the new perspec
tive it is evident that the method described in the previous section can
still be applied by simply modeling d(t) as part of the
disturbance. It can readily be deduced from Figure 5-3 that this
j
translates into Tr(t) satisfying a linear differential equation of the
form given in assumption (A.2). Consequently, the dynamics associated