The influence of parametric and external noise in act-and-wait control with delayed feedback

We apply several novel semi-analytic approaches for characterizing and calculating the effects of noise in a system with act-and-wait control. For concrete illustration, we apply these to a canonical balance model for an inverted pendulum to study the combined effect of delay and noise within the act-and-wait setting. While the act-and-wait control facilitates strong stabilization through deadbeat control, a comparison of different models with continuous vs. discrete updating of the control strategy in the active period illustrates how delays combined with the imprecise application of the control can seriously degrade the performance. We give several novel analyses of a generalized act-and-wait control strategy, allowing flexibility in the updating of the control strategy, in order to understand the sensitivities to delays and random fluctuations. In both the deterministic and stochastic settings, we give analytical and semi-analytical results that characterize and quantify the dynamics of the system. These results include the size and shape of stability regions, densities for the critical eigenvalues that capture the rate of reaching the desired stable equilibrium, and amplification factors for sustained fluctuations in the context of external noise. They also provide the dependence of these quantities on the length of the delay and the active period. In particular, we see that the combined influence of delay, parametric error, or external noise and on-off control can qualitatively change the dynamics, thus reducing the robustness of the control strategy. We also capture the dependence on how frequently the control is updated, allowing an interpolation between continuous and frequent updating. In addition to providing insights for these specific models, the methods we propose are generalizable to other settings with noise, delay, and on-off control, where analytical techniques are otherwise severely scarce.