CHAPTER ONE
INTRODUCTION
One of the most important problems in applications of feedback
control is to provide output tracking in the presence of external
disturbances. This is commonly referred to as the servomechanism
problem. More precisely, given a certain system, the servomechanism
problem involves the design of a controller which enables the output to
asymptotically track a reference signal r(t), in the presence of a
disturbance w(t), where r(t) and w(t) belong to a certain class of
functions. The class of functions might be, for example, combinations
of step, ramp and sinusoidal signals. The frequency of the signals is
usually assumed to be known. Typically, enough freedom is allowed,
however, so that it is not necessary to have apriori knowledge of the
amplitude or phase of either the disturbance or the reference.
The assumption of known frequency but unknown amplitude and phase
provides a realistic model for many reference and disturbance signals
encountered in practice. For example, an imbalance in a piece of
rotating machinery might cause a sinusoidal disturbance force to act on
a certain system. Although the frequency of this force might be easy to
predict, it is doubtful that the exact amplitude could be determined.
Even if the amplitude was known exactly, modeling errors in the plant
would make such schemes as open-loop compensation unreliable. This
leads to an important feature of a controller design to solve the
servomechanism problem. Namely, there should be a certain amount of
robustness with respect to plant variations and with respect to
variations in signal level.
1