Exponentially weighted least squares identification of time-varying systems with white disturbances

The paper is devoted to the stochastic analysis of recursive least squares (RLS) identification algorithms with an exponential forgetting factor. A persistent excitation assumption of a conditional type is made that does not prevent the regressors from being a dependent sequence. Moreover, the system parameter is modeled as the output of a random-walk type equation without extra constraints on its variance. It is shown that the estimation error can be split into two terms, depending on the parameter drift and the disturbance noise, respectively. The first term turns out to be proportional to the memory length of the algorithm, whereas the second is proportional to the inverse of the same quantity. Even though these dependence laws are well known in very special mathematical frameworks (deterministic excitation and/or independent observations), this is believed to be the first contribution where they are proven in a general dependent context. Some idealized examples are introduced in the paper to clarify the link between generality of assumptions and applicability of results in the developed analysis