Adaptive output-feedback control for exponential regulation

To date, most schemes of adaptive stabilization are confined to convergence, and cannot specify the convergence rate. As such, the possibility of excessively slow convergence could not be ruled out, which undermines the application scope of adaptive schemes. Such imperfection is mainly due to the limited capability of compensation/learning mechanisms which lack the ingredients affording specific convergence rate. For stronger applicability, more comprehensive compensation/learning mechanisms need to be exploited, inevitably aggravating control synthesis and analysis. This paper aims to enhance adaptive control in terms of convergence rate, and particularly, seeks exponential regulation via adaptive output feedback for stochastic nonlinear systems allowing large uncertainties coupling to unmeasurable states and dynamic uncertainties. To this end, a refined adaptive output-feedback scheme is established in the stochastic framework, though no non-stochastic counterpart exists. Critically, a novel dynamic high gain is introduced in a universal manner, with its updating law delicately incorporating the gain-dependent time-varying information in exponential form. The dynamic gain would not only become sufficiently large to capture the large uncertainties, but also render the desired convergence with the rate online regulated to match the dynamic uncertainties. In this way, an adaptive output-feedback controller based on dynamic-gain observer is designed, which guarantees the system and observer states to almost surely converge to zero at an exponential rate. The stochastic framework complicates the validity verification of the established scheme, entailing subtle martingale-based analysis. Correspondingly, a distinctive analysis pattern is developed for the resulting closed-loop system, to attain expected stochastic convergence and boundedness.