A Novel Approach for Adaptive Tracking Control of Nonlinear Systems Subject to Long-Range-Dependent Stochastic Disturbances

While extensive research exists on adaptive control for stochastic systems driven by standard Brownian motion (sBm), little attention has been paid to systems subject to long-range-dependent stochastic disturbances modeled by fractional Brownian motion (fBm), despite its relevance to batteries, vehicles, and robotics. In this article, we develop an adaptive tracking control scheme for unmeasured stochastic nonlinear systems (SNSs) driven by fBm. To achieve this, we address two key technical obstacles. First, novel stability criteria are needed for nonlinear systems driven by fBm, as existing ones apply only to linear systems or to nonlinear systems driven by sBm. Second, the inherent complexity of fBm with long-range dependence and nonsemimartingale properties may compromise the applicability of existing observer-based control schemes to such systems. To overcome these obstacles, we develop novel stability criteria (including stochastic and practical stochastic stability) for system analysis and control synthesis, and employ the Bolzano-Weierstrass and Heine theorems for observer stability proofs. It is demonstrated that, with the proposed stability criteria and the observer-based adaptive tracking control schemes, all closed-loop signals remain bounded in the mean-square sense. The effectiveness of the proposed strategy is validated using the vehicle model.