Frequency domain adaptive learning feedforward control

A learning feedforward control (LFFC) acts as an "add-on" element to an existing feedback controller (FBC). The LFFC signal is updated iteratively based on the output of a B-splines network (BSN), with the FBC signal of previous iteration as its input. There are only two parameters to tune: the B-spline support width d and the learning gain γ. Frequency stability analysis requires that d should be greater than a minimum value d_min, which determines a cutoff frequency of the low pass learning filter. Using a fixed support d for all iterations and for all time at each iteration may be conservative and the actual feedforward signal may in fact contain frequencies higher than the cutoff frequency determined by d. This is the case when there are hard nonlinearities in the system. A frequency domain adaptive scheme is proposed in this paper where d is a time function and changes from iteration to iteration. A time-frequency representation (TFR) of feedback signal is constructed using WVD (Wigner-Ville distribution) and then its first order time moment is used to obtain the time-varying support d(t) via an iterative updating scheme based on the difference of the windowed tracking error root mean squares between two previous iterations. Detailed design procedures are given.