Data-driven constrained optimal model reduction

Abstract

Model reduction by moment matching can be interpreted as the problem of finding a reduced-order model which possesses the same steady-state output response of a given full-order system for a prescribed class of input signals. Little information regarding the transient behavior of the system is systematically preserved, limiting the use of reduced-order models in control applications. In this paper we formulate and solve the problem of constrained optimal model reduction. Using a data-driven approach we determine an estimate of the moments and of the transient response of a possibly unknown system. Consequently we determine a reduced-order model which matches the estimated moments at the prescribed interpolation signals and, simultaneously, possesses the estimated transient. We show that the resulting system is a solution of the constrained optimal model reduction problem. Detailed results are obtained when the optimality criterion is formulated with the time-domain 1 , 2 , ∞ norms and with the frequency-domain H 2 norm. The paper is illustrated by two examples: the reduction of the model of the vibrations of a building and the reduction of the Eady model (an atmospheric storm track model).

On Moment Matching for Stochastic Systems

“…The matrices F and G are selected as F = 0.01A and G = B. The matrix S of the signal generator is selected as in [62], [66], i.e. a matrix of order ν = 19 with eigenvalues 0, ±5.22ι, ±10.3ι, ±13.5ι, ±22.2ι, ±24.5ι, ±36ι, ±42.4ι, ±55.9ι and ±70ι (corresponding to the main frequency peaks of the deterministic model).…”

“…reduced-order model in the mean ( 34) are computed 7 . The two models have the same matrices A, B, F , H. The matrices A, B and R = I are selected using the method presented in [66], [67]. The matrices F and G are selected as G = 0.05 B and F = − GLR −1 .…”

Model Reduction by Moment Matching: Beyond Linearity A Review of the Last 10 Years

“…The notion of moment expressed in Definition 2 has led to extensive investigations in the last decade. While some of these are summarized in the remainder of this tutorial, it is worth noting the development of a dual theory in which the signal generator is driven by the system the moment of which are to be computed [18]; the definition of moments for nonlinear time-delay systems [19] and for systems driven by infinite dimensional generators [20]; the solution of structure/mode preserving model reduction problems [21], [22]; the definition of a nonlinear enhancement of the notion of phasors applicable to the study of circuits with switched mode power electronic components [23]; and a fully datadriven approach [24], [25].…”

“…E. On-line moment estimation from data [24], [25] Solving equation ( 9) with respect to the mapping π is a difficult task even when there is perfect knowledge of the dynamics of the system, i.e. the mapping f .…”

From data to reduced-order models of complex dynamical systems

“…As part of the class of moment matching methods, an algorithm for computing (a least-square approximation of) moments of linear or nonlinear systems is proposed in [156,158], building on the framework of [15]. These (estimated) moments are then used to construct a family of reduced-order models.…”