On exponentially discounted adaptive control

In many applications of adaptive control it may be clear if the unknown parameters are constants or are time varying functions. It may be especially difficult to determine if some of the parameters are constants or slowly varying functions of time. To determine parameter variations it is necessary to "forget" the past of the state. The approach of exponential discounting of past information has been studied for some practical applications ([1], [2]) and has often been used in other control problems. If it is unclear whether the unknown parameters are constants or time varying, then an estimator must compromise between the conflicting goals of accuracy in the case of constant parameters and response to changes in the parameters in the case of time varying parameters. In this paper a least squares estimator with exponential discounting is formed. Properties of this estimator are investigated for the case where the parameters are periodic functions and the rate of the exponential discounting approaches zero. This asymptotic behaviour of the estimator provides a quantitative way to compromise between the possibilities of constant or periodic parameters via discounting. The model that is considered here is a linear stochastic differential equation where the unknown parameters appear affinely in the drift. The asymptotic distribution of the estimator is obtained for an identification problem (where the feedback control is fixed) as the discount rate approaches zero. For an