problem is avoided here, however, since we consider systems which have the same number of inputs as outputs.
The second assumption is perhaps the most restrictive. It is
different from the assumption commonly made in the linear servomechanism problem; namely, that the disturbance w*(t) and the reference r*(t) both satisfy a linear differential equation of the form given by (2-4). Here we are concerned with this class of disturbance and reference signals;
however, in the nonlinear case it is important to work also with the acutal state and input trajectories which arise during tracking.
If we assume that r*(t) satisfies a differential equation of the form given by (2-4), it is actually not unreasonable to assume that x*(t) will satisfy the same equation. This is because the output y*(t), which must be identically equal to r*(t) during tracking, is taken to be a linear combination of the state x*(t). Consequently, if all elements
of the state are reflected in the output, these elements must satisfy (2-4). The assumption on the input signal u*(t) is then the assumption which needs further discussion. In the nonlinear servomechanism
problem, it is often the case that u*(t) will contain terms (e.g., sinusoids) not present in either w*(t) or r*(t). To help clarify this point, consider the following proposition.
Proposition 2.1: Given the autonomous system
M(t) = f(x(t), u(t), w(t)) (2-5)
assume there exists an input u(t) such that x(t) = xp(t) is the solution to (2-5) with initial state xp(O) and with disturbance w(t) = wP(t).