Pre-filter design for non-iterative data-driven controller parameters tuning using closed-loop step response data

Non-iterative data-driven controller parameters tuning, such as VRFT(Virtual reference feedback tuning), FRIT(Fictitious Reference Iterative Tuning), and NCbT (Non-iterative Correlation based Tuning), improves control performance by tuning controller parameters from one-shot experimental data. In such approaches, the pre-filter design plays an important role for improving control performance. Originally, Campi et al. proposed the optimal pre-filter for VRFT that makes data-driven cost function close to the cost function representing original control objective. However, the study handled open-loop random one-shot experimental data, and gave the condition of the optimal pre-filter consisting of power spectral density of initial input data. Since estimating power spectral density of deterministic signals is a difficult task, it is not applicable to closed-loop step response data. Thus, the paper considers a pre-filter design method for closed-loop step response data. The present work focuses on our previous work on VRFT, and extends the method to FRIT. In addition, the present work clarifies the derivation of the pre-filter from theoretical point of view. To this end, the paper assumes that the controller class includes integrator explicitly so that tracking error asymptotically goes to zero for a step reference signal, and uses an inverse filter that transfers initial output data to a discrete-time impulse signal for deriving pre-filter. Using the formulations, the pre-filter is designed by comparing the second Taylor expansions between the data-driven cost function and the original cost function. The pre-filter design is applied to NCbT, and the difference between these pre-filtered data-driven cost functions, such as FRIT, VRFT, and NCbT is discussed. Finally, the effectiveness of the pre-filter designs for these methods is shown through a numerical example.