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

Abstract

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.

Set-Theoretic Model Reference Adaptive Control for Performance Guarantees in Human-in-the-Loop Systems; A Pilot Study

“…In this paper, we consider human-in-the-loop physical systems with inner and outer feedback control loops as also considered in [15]. The control objective is to have the uncertain machine at the inner loop to behave close to an ideal system that corresponds to the case where there is no uncertainties present in the system.…”

“…In this section, we overview the results presented in [15] for establishing the stability and guaranteeing a user-defined performance constraint on the norm of system error trajectories. Specifically, the proposed update law in ( 15) is predicated on the set-theoretic model reference adaptive control architecture presented in [18] in order to impose a user-defined performance bound on the system error vector.…”

“…Specifically, the proposed update law in ( 15) is predicated on the set-theoretic model reference adaptive control architecture presented in [18] in order to impose a user-defined performance bound on the system error vector. Unlike a common update law in standard model reference adaptive control, the term φ d ( e(t) P ) in (15), defined in Definition 2, serves as an error dependent learning rate. As a result, the user-defined performance guarantee at the inner loop can be shown by considering the energy function V(e,W) = φ( e P ) + γ −1 tr (WΛ 1/2 ) T (WΛ 1/2 ) as shown in [18].…”

“…Furthermore, it follows from the asymptotically stability of the nominal system in (35) that there exist an α > 0 such that Ψ(t) 2 ≤ Ke −αt for some K > 1. We can now apply Comment 4 of [15] to bound the error dynamics in (34) given by…”

“…Furthermore, recent results on set-theoretic MRAC further our understanding in terms of designing controllers that can bring the behavior of nonlinear machine dynamics within a tolerance of the behavior of a reference model; that is, such controllers can make nonlinear and uncertain dynamics behave like a "nominal model" [14]. In [15], it is shown that guaranteeing a user-defined performance constraint on the norm of system trajectory error ultimately results in an acceptable human performance in accomplishing a given task. Within the context of the cited study, the results are theoretically proven assuming LTI models for humans.…”