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, assumption   (A.1),   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 assumption, 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