Simultaneous Local ARMA Identification and Data-Driven Model Predictive Control Persistent Excitation, One-Step Informativity, and Asymptotic Output Regulation

This preprint develops an exact-data theory for simultaneous local ARMA fitting and data-driven model predictive control without requiring coefficient recovery. At each decision time k, the unknown plant is assumed to admit an exact order-nₐ monic ARMA representation on the complete realized controller window ₖ=-nₚ,…,k+nf\. The representing coefficient matrix may depend on the window. The admissible coefficient class is compact, and every direct-input block has full row rank. This assumption is trajectory-wise: it concerns the realized finite input-output segment and neither imposes one global LTI model on the unknown plant nor requires an eventual fixed local model. The controller has two stages. First, a lexicographic problem jointly selects an exact past-fitting ARMA matrix and a nominal MPC input sequence. Exact fitting has absolute priority, robust regulation of the first predicted output has second priority, and the remaining finite-horizon objective has third priority. Second, a bounded perturbation is selected from a full-dimensional box whose radius is positive at every finite time and vanishes asymptotically. The perturbation is used to preserve a prescribed sliding-window persistence-of-excitation rank condition of the actually implemented input whenever the corresponding one-step rank-reachability condition holds. The central result is one-step informativity rather than coefficient uniqueness. Any fixed order-nₐ ARMA relation admits a finite-dimensional LTI delay-state embedding of known dimension. Corollary~1 of Yu et al. then implies that, under sufficiently high-order persistent excitation of a measured input prefix, the realized current length-(nₐ+1) input-output segment is parameterizable by earlier segments of the same local trajectory even if the embedding is uncontrollable. Removing the current-output block from this trajectory identity shows that the implemented ARMA regressor belongs to the column space of the measured past regression matrix. Hence every exact past-fitting ARMA matrix gives the same implemented one-step prediction. The whole-window exactness assumption then identifies this common prediction with the actual output at the implemented current sample. The first-stage problem is shown to admit a global lexicographic solution in the exact-data setting. Its secondary optimum admits the explicit expressionη1,k⁼ρₖminM∈ Fₖ\|B₀(M)\|_∞, and therefore η1,k^≤ b₀ρₖ. Combining this bound with one-step informativity yields \|yₖ\|_∞≤ b₀ρₖ→0. Thus asymptotic output regulation follows without a maximal McMillan-degree condition, full-rank ARMA regression, coefficient uniqueness, overlap gluing, a separate asymptotic coefficient-identifiability assumption, or eventual LTI behavior. The convergence proof uses only the restriction of the whole-window assumption through the implemented current sample, while the stronger whole-window formulation is retained as the structural model assumption of the DDMPC framework.