Moment Matching by Recursive-Based Learning

Modeling dynamical systems is a fundamental task in the control field. The advancements of the industrial technologies demand for more complex models, which have to account for both high dimensionality and heavy nonlinearities of the systems. In practical applications, large models may represent a problem where the computational capacity is limited. Model reduction techniques have been proposed to approximate high-order systems with lower-dimensional models. Moment matching methods rely on the moments of the system, which for LTI systems are linked to the coefficients of the Laurent series of the transfer function around a point. For nonlinear systems, moments are related to a steady-state output response that results from an invariant manifold, which has been proved to be a center manifold when the system is interconnected with an input generator with specific properties. This manifold is characterized by a nonlinear mapping, which can be obtained by solving a PDE, which is known as the invariance equation. However, finding a solution to such a PDE may be impractical, since the complexity increases with model dimension and the high-order system may also be unknown. To address these challenges, data-driven methodologies have been developed, motivated by the increasing availability of input-output data from industrial processes. Recently, in, the moment is estimated by employing Tikhonov regularization on a reproducing kernel Hilbert space. While this method successfully achieves moment matching for high-order nonlinear systems, its complexity scales with the number of data points, which is necessary to increase the accuracy of the estimation. Furthermore, the accuracy decreases when data presents a transient. In this work, we present a recursive procedure that (i) updates the moment estimation using data collected online from the system and (ii) discards the oldest data point used for estimation. In this way, the number of data used for estimation is constant, thus the complexity does not increase. However, we prove that the accuracy of the estimation increases, since the effect of the transient vanishes asymptotically in time.