  Both fixed-gain control and adaptive learning architectures aim to mitigate
the effects of uncertainties. In particular, fixed-gain control offers more
predictable closed-loop system behavior but requires the knowledge of
uncertainty bounds. In contrast, while adaptive learning does not necessarily
require such knowledge, it often results in less predictable closed-loop system
behavior compared to fixed-gain control. To this end, this paper presents a
novel symbiotic control framework that offers the strengths of fixed-gain
control and adaptive learning architectures. Specifically, this framework
synergistically integrates these architectures to mitigate the effects of
uncertainties in a more predictable manner as compared to adaptive learning
alone and it does not require any knowledge on such uncertainties. Both
parametric and nonparametric uncertainties are considered, where we utilize
neural networks to approximate the unknown uncertainty basis for the latter
case. Counterintuitively, the proposed framework has the ability to achieve a
desired level of closed-loop system behavior even with an insufficient number
of neurons (e.g., when the neural network approximation error is large) or in
the face of injudiciously selected adaptive learning parameters (e.g., high
leakage term parameters).
