The stability of a proportional rate extremum regulator

When the signal component of the input to a linear unity feedback sampled-data system is additively contaminated by a relatively wide-band noise process, it has been shown previously that the mean-square values of the true and apparent error quantities are minimized by essentially the same value of loop gain. This paper presents an investigation into the stability of a proportional-rate extremum regulator, which utilizes this fact to establish the optimum gain setting from finite estimates of the measurable mean-square apparent error. Assuming a parabolic operating characteristic, the stability and dynamic behavior of the adaptive loop is studied using path tangent curves in the incremental phase plane. The ability of the regulator to establish the optimum gain setting for a given system is verified experimentally for a number of different operating conditions. In addition, the experimental results show that the transient disturbances generated by the corrections to the gain parameter may adversely affect the regulator stability if they exist over a sufficient fraction of the measurement period.