Stochastic adaptive system theory for identification, filtering, prediction and control

This thesis examines the basic asymptotic properties of various stochastic adaptive systems for identification, filtering, prediction and control. These include the convergence of long-term averages of signals of interest (self-optimality), the convergence of adaptive filters or controllers (self-tuning property), the convergence of parameter estimates, and the rates of convergence.This thesis divides itself naturally into two parts. The first part considers identification, adaptive prediction and control based on the ARMAX model, while the second part considers general stochastic parallel model adaptation problems, which include output error identification, adaptive IIR filtering, adaptive noise cancelling, and adaptive feedforward control with or without input contamination.In the first part, the use of a generalized certainty equivalence approach in which the estimates of disturbance as well as parameters are utilized is proposed. Based on this, the self-optimality of adaptive minimum variance prediction and model reference adaptive control is established for systems with general delay and colored noise. Both direct and indirect approaches based on the extended least squares as well as the stochastic gradient algorithms are considered. For the direct approach, it is shown that interlacing is not necessary for convergence, thus resolving this long-standing open problem. Concerning the self-tuning property, it is established that self-optimality in the mean square sense, in general, implies self-tuning, by exhibiting the convergence of the parameter estimates to the null space of a certain covariance matrix, and by characterizing this null space. It is found that adaptive minimum variance regulators self-tune because of the "internal excitation" due to the plant disturbance alone. Finally, the exact order of external excitation required for the parameter estimates to converge to the true parameter is determined.In the second part of the thesis, the convergence of several parallel model adaptation schemes in the presence of nonstationary colored noise is established. A special case of our results resolves the long-standing problem of the convergence and unbiasedness of the output error identification scheme in the presence of colored noise. We also develop a simple general technique for analyzing the strong consistency of parameter estimation with projection.Of pedagogical interest is the deterministic reduction viewpoint we adopt in which all relevant properties of stochastically modeled disturbances are characterized deterministically by some long-term average properties. Readers more familiar with deterministic theory may well find this viewpoint to be more enlightening with respect to understanding the goals and results of stochastic adaptive system theory.U of I OnlyETDs are only available to UIUC Users without author permissio