Bias of anticipated non-parametric closed-loop frequency response for controller gain re-tuning

In data-driven controller re-tuning, when requirements exist for the closed-loop frequency response, it is essential that the estimated closed-loop frequency response for a new controller parameter set, correctly reflects the respective true closed-loop. To quantify the difference between the true and estimated frequency response, this paper presents the bias of the non-parametric closed-loop frequency response estimate for the closed-loop system with updated controller parameters. It is calculated for the case where the estimate of the closed-loop frequency response of the system with an initial controller is used, to estimate the new closed-loop frequency response with an updated controller. Such estimates are useful in the process of gain re-tuning based on closed-loop experiments, where new controller parameters are calculated to satisfy requirements for the closed-loop frequency response. It is shown that the estimate appears similar in structure to the indirect non-parametric plant estimate, and hence the bias is reduced with an increasing excitation-to-noise-ratio for the closed-loop experiment. It is also seen that the bias tends to zero when the change in the controller parameters tends to zero or the feed-forward is increased sufficiently. Simulation results demonstrate the concept for a flight-test-based gain re-tuning of a pitch rate controller for an aircraft.