When Persistency is not Exciting in Data-Driven Predictive Control

What the math gives to ML

The paper identifies a useful distinction between algebraic persistency of excitation and control-relevant excitation: a data matrix can satisfy the required rank condition while placing insufficient energy at frequencies important for the desired closed-loop behavior. Its frequency-domain construction expresses the state spectrum as a linear combination of delayed input and output spectra, making the missing excitation measurable rather than purely qualitative. The transferable neural-network mechanism is task-weighted spectral experiment design: collect or generate training sequences that satisfy a rank test and allocate energy to frequencies where the learned dynamics or downstream prediction loss is sensitive.