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In this paper, we focus on human-in-the-loop physical systems with inner and outer feedback control loops. Specifically, our problem formulation considers that inner loop control laws use a model reference adaptive control approach to suppress the effect of system uncertainties such that the overall physical system operates close to its ideal behavior as desired in the presence of adverse conditions due to failures and/or modeling inaccuracies. Moreover, we consider that the outer loop control laws exist owing to employing either sequential loop closure and/or high-level guidance methods. As it is true in practice, in addition, humans are considered to inject commands directly to the outer loop dynamics in response to the changes in the physical system, where the outer loop commands affect inner loop dynamics in response to the commands received from the humans as well as in response to the changes in the physical system. The presence of humans can result in system instability, even when the resulting physical system augmented with inner and outer feedback control loops yield to stable trajectories in the absence of humans. This paper addresses this problem by proving a sufficient stability condition for the overall physical system with human dynamics modeled as a linear timeinvariant system with human reaction time-delay, where this condition does not depend on system uncertainties similar to our recent theoretical results. Furthermore, inner loop system errors during the transient phase of adaptively suppressing system uncertainties can severely affect the human-outer loop interactions. We also address this issue by utilizing a recently proposed set-theoretic model reference adaptive control approach at the inner loop for enforcing a user-defined performance constraint on the norm of the system error trajectories, where we show how the selection of this constraint affects the overall physical system. Finally, the efficacy of our results is demonstrated through an illustrative numerical example for an adaptive flight control application with a Neal-Smith pilot model.

Human-in-the-Loop Systems with Inner and Outer Feedback Control Loops: Adaptation, Stability Conditions, and Performance Constraints