Data-Driven Sparse Identification and Robust Tube-Based Predictive Control for Multivariable Chaotic Systems

This paper addresses the data-driven synchronization and robust control of multivariable chaotic systems subject to strict actuation constraints, modeling uncertainties and bounded input disturbances. Using empirical data collected from a four-dimensional chaotic R¨ossler response system, Sequentially Thresholded Least Squares (STLSQ) sparse regression is employed to recover a nominal control-affine model, achieving an identification accuracy of R2 = 1.0000 across all channels with root-mean-square error within [4.15 × 10−6, 8.33 × 10−4]. To establish a mathematically rigorous bridge between data-driven identification and robust synthesis, all structural discrepancies in the drift vector field and input matrix are unified into a single lumped uncertainty field δ(x, u). A computable scalar bound ¯δ is established based on library truncation error, regression residuals, input bounds, and Lipschitz constants (Lemma 1). We subsequently prove that input–output feedback linearization maps δ(x, u) into a state- and control-dependent additive disturbance d(z, ν) in Brunovsky canonical coordinates, bounded by ∥d(z, ν)∥2 ≤ α1¯δ + α2¯δ∥ν∥2 with closed-form expressions for α1 and α2 (Theorem 1). This certified bound, together with a bounded input disturbance and a zero-order-hold margin, yields a compact disturbance set that seeds a robust tube-based model predictive controller. We prove that the resulting tightened quadratic program is recursively feasible, that the physical actuator limit (|ui| ≤ 1100) remains inactive under an explicit sufficient condition, and that the closedloop tracking error is Input-to-State Stable (ISS) with an asymptotic gain that is linear in ¯δ plus a term due to the exogenous input disturbance (Theorems 2 and 3).