Simultaneous Local ARMA Identification and Data-Driven Model Predictive Control Persistent Excitation, Eventual Exact Recovery, and Asymptotic Output Regulation

This preprint develops an exact-data theory for simultaneous local ARMA identification and data-driven model predictive control. At each decision time, the realized input-output window of the unknown plant is assumed to admit an exact order-nₐ monic ARMA representation, while the representing coefficient matrix may depend on the window. The assumption concerns only the realized finite trajectory and does not postulate one global LTI plant model. The admissible coefficient set is compact and every direct-input block has full row rank. The controller identifies an ARMA matrix from the measured past and optimizes the nominal control action lexicographically: exact fitting has absolute priority, robust first-step regulation has second priority, and finite-horizon performance has third priority. After the nominal first action is fixed, a bounded perturbation is selected. Its radius is positive at every finite time and vanishes asymptotically. The perturbation can be chosen to preserve a prescribed sliding-window persistence-of-excitation rank condition of the actual input whenever the corresponding one-step rank-reachability condition holds. After the closed-loop scheme has been defined, an asymptotic nondegeneracy assumption is imposed on the exact local ARMA models associated with the realized closed-loop trajectory: every accumulation point has maximal McMillan degree nₐny. Compactness then implies eventual maximal degree. For a maximal-degree local ARMA model, an input that is persistently exciting of order nₐ+1+nₐny makes the past ARMA regression matrix full row rank. The proof follows from the finite-horizon rank form of Willems' Fundamental Lemma and an exact row-space identity between the input-output Hankel matrix and the ARMA regression matrix. Consequently the exact fitting set becomes a singleton after a finite time. The common overlap of consecutive windows then forces the exact local matrices to become identical along the realized trajectory. Finally, whole-window exactness makes the implemented first prediction equal to the actual plant output, and vanishing excitation gives \|yₖ\|_∞≤ b₀ρₖ→0. The conclusion is asymptotic output regulation and eventual LTI input-output behavior along the realized closed-loop trajectory; it does not assert that the underlying unknown plant is globally LTI.