Will self-tuning occur for general cost criteria?

Abstract

A popular approach to adaptive control consists of I.) estimating the parameters of the system at each time instant and II.) applying a control input at each time instant which is optimal with respect to a specified cost criterion if the estimated parameters are indeed the true values. The natural question for such a scheme is whether the control law based on the estimated parameters will converge asymptotically to the optimal control law with regard to the specified cost criterion for the true parameter values. In other words, will the adaptive control law self-tune to the optimal control law? Much attention has recently been paid to the problem of controlling an unknown ARMAX system where the specified cost criterion is the variance of the output process and recently it has been shown that an adaptive control law, as above, does self-tune to the minimum variance control law, see (1). Our main contention here is that the self-tuning result for a minimum variance cost criterion rests on self-tuning to an optimal control law will not generally occur for general cost criteria. The particular case of a quadratic cost criterion penalizing not only the variance of the output but also the variance of the input is analyzed by Ljung's O.D.E.'s to demonstrate this. One special situation in which self-tuning can be expected is when the ARMAX system has a large enough delay.