Adaptive sensitivity control for a feedback loop and a consensus model with unknown delay

We propose a method for estimation of a globally constant but unknown delay $$\tau >0$$ in a negative feedback loop, where only an upper bound on the possible values of $$\tau $$ is given. Also the initial datum is not revealed. The method is based on detecting the decay rate of the solution throughout its evolution and guarantees that the estimate converges to the true value of $$\tau $$ asymptotically for large times. In a second step, the estimated delay is used to adaptively control the sensitivity (feedback gain) of the loop with the goal of reaching optimal rate of convergence towards equilibrium. We stress that our approach is distinguished from traditional feedback control methods by leveraging the system’s sensitivity as a control parameter to achieve equilibrium. In the second part of the paper we adapt the method to estimate unknown delay and control the sensitivity in a linear opinion formation model. Here the estimation of the delay is based on the decay properties of the quadratic fluctuation of the agents’ opinions, employing appropriate approximations and some heuristic arguments. In both cases we present numerical examples illustrating the performance of the method.