Data-driven model reduction by moment matching for linear and nonlinear systems

Abstract

Theory and methods to obtain reduced order models by moment matching from input/output data are presented. Algorithms for the estimation of the moments of linear and nonlinear systems are proposed. The estimates are exploited to construct families of reduced order models. These models asymptotically match the moments of the unknown system to be reduced. Conditions to enforce additional properties, e.g. matching with prescribed eigenvalues, upon the reduced order model are provided and discussed. The computational complexity of the algorithms is analyzed and their use is illustrated by two examples: we compute converging reduced order models for a linear system describing the model of a building and we provide, exploiting an approximation of the moment, a nonlinear planar reduced order model for a nonlinear DC-to-DC converter

Steady-State Matching and Model Reduction for Systems of Differential–Algebraic Equations

“…This offers another approach that can be extended to singular systems. Finally, if the model is completely unknown, we can extend to differential-algebraic equations the data-driven techniques based on output measurements which have been presented in [37]- [39].…”

“…Consider the signal generator (13) with matrices S = [0, −1; 1, 0] and L = [0.001, −0.0098] which generates an input similar to the one used in [4]. We compute an approximation of the moment h • π (see [37], [38]), namely h(π(ω)) ≈ 10 −4 (0.0451 + 9.591ω 1 + 1.955ω 2 +0.0042ω 2 1 −0.0005ω 1 ω 2 ). We determine two reduced order models, a normal model described by the equationṡ…”

A geometric characterisation of persistently exciting signals generated by continuous-time autonomous systems

“…This is, for instance, immediately evident in the problem of data-driven model reduction by moment matching Scarciotti and Astolfi [2015]. More precisely, consider a linear, single-input, single-output, continuous-time, system described by the equationṡ…”

“…HΠ, from input-output data without any knowledge of F , G, H and x(0). This problem has been solved in Scarciotti and Astolfi [2015]. The proposed solution relies on the following assumption.…”

“…To demonstrate the significance of our result we address questions in subspace identification for linear autonomous systems Verhaegen and Verdult [2007] and nonlinear autonomous systems Padoan and Astolfi [2015]. We also envisage the use of the result to deal with issues arising in the study of the steady-state response of nonlinear systems Astolfi [2010] and in data-driven model reduction by moment matching for linear and nonlinear, possibly time-delay, systems Scarciotti and Astolfi [2015].…”