On the relative time of adaptive processes

Adaptive control systems can be classified according to the response time of the adaptive loop T
a
relative to that of the main servo loop T
m
. The response time T
a
cannot be smaller than the time needed for making the required measurement on which the adjustments are based. In a slow adaptive system,
T_{a}\ggT_{m}
, and the adjustments are made according to the estimated situation (statistical parameters of the inputs changing plant parameters, etc.) or the estimated performance (error, cost, intermediate parameters etc.). If
T_{a}\llT_{m}
, usually a reference model is chosen and the system is forced to conform to the reference model. Sometimes the reference model is dimensionless so that the fastest response can be obtained. In both cases,
T_{a}\ggT_{m}
and
T_{a}\llT_{m}
, the system can be analyzed by introducing suitable approximations. The condition of stability for the adaptive loop is derived in general terms. If
T_{a}\approxT_{m}
, both analysis and synthesis become difficult. The concept of "dual control" is introduced, but not developed in the paper.