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Summary
In this chapter, a nonlinear system was considered with the number
of inputs equal to the number of outputs and with the output taken as a
linear combination of the system's state. In the last part of the
chapter, conditions were given so that a nonlinear system with more
inputs than outputs could be treated.
Prior to developing the theory for the nonlinear servomechanism
problem, two major assumptions were made. The first of these, assump
tion (A.l), was absolutely necessary since without it, the
servomechanism problem could not be solved under any circumstance. This
being the case, the primary attention was focused on the second assump
tion, assumption (A.2). Here, the requirement was made that the input
and state trajectories which occured during tracking were to satisfy a
linear differential equation. It was noted that in practice, such an
assumption may only be approximate, however, a design based on the
approximation could be perfectly adequate. Typically, truncated Fourier
series expansions approximating the true signals would be used for
design purposes.
In the first part of the controller design we dealt with the
development of an internal model system. It was indicated that this
internal model system would have to contain dynamics which matched the
dynamics of the state and input which are necesary for tracking. The
importance of such an internal model system becomes evident when it is
compared to a standard alternative. A typical approach to solving the
nonlinear servomechanism problem is to first linearize the nonlinear
system and then design a controller using linear servomechanism
theory. This leads to an internal model system containing dynamics