Convergence of self-tuning regulators under conditional heteroscedastic noises

Traditional convergence theory of self-tuning regulators requires boundedness of the conditional variances of the systems noise processes. However, this requirement cannot be satisfied for many practical models such the well-known ARCH (Autoregressive Conditional Heteroscedasticity) model in economic systems. The aim of this paper is to provide a convergence theory of self-tuning regulators for linear uncertain systems with conditional heteroscedastic noises, where the conditional variance is unbounded. To be specific, we consider weighted least-squares-based self-tuning regulators, and establish both the global stability and the optimality of tracking under some natural conditions on the system models as well as on the conditional heteroscedastic noises. To the best of the authors' knowledge, this is the first paper that investigates this kind of problems with a convergence theory, and makes the self-tuning regulators applicable to systems with noised modeled by ARCH.