From data to reduced-order models via moment matching

Abstract

A new method for data-driven interpolatory model reduction is presented in this paper. Using the so-called data informativity perspective, we define a framework that enables the computation of moments at given (possibly complex) interpolation points based on time-domain input-output data only, without explicitly identifying the high-order system. Instead, by characterizing the set of all systems explaining the data, necessary and sufficient conditions are provided under which all systems in this set share the same moment at a given interpolation point. Moreover, these conditions allow for explicitly computing these moments. Reduced-order models are then derived by employing a variation of the classical rational interpolation method. The condition to enforce moment matching model reduction with prescribed poles is also discussed as a means to obtain stable reduced-order models. An example of an electrical circuit illustrates this framework.

From Data to Reduced-Order Models via Generalized Balanced Truncation

“…where Ψ is given by (27). Therefore, there exists Z such that Z = W ZV and, due to Lemma A1, Z ∈ M. Proof of Theorem 1: Since (10) holds for some S and matrix X − U − has full row rank, Σ is bounded and has nonempty interior.…”

“…In addition, the use of Loewner methods based on time-domain data and noisy frequency-domain data was pursued in [25] and [26], respectively. Besides the Loewner framework, data-driven moment matching techniques were presented in [9] and [27], where the latter exploits the so-called data informativity framework.…”

From data to reduced-order models via generalized balanced truncation

“…where Ψ ♯ is given by (27). Therefore, there exists Z such that Ẑ = W ⊤ ZV and, due to Lemma A.1, Z ∈ M.…”

“…In addition, the use of Loewner methods based on time-domain data and noisy frequency-domain data is pursued in [25] and [26], respectively. Besides the Loewner framework, data-driven moment matching techniques have been presented in [9] and [27], where the latter exploits the so-called data informativity framework.…”

Results on data-driven controllers for unknown nonlinear systems

“…Optimal and robust controllers are further explored in [41][42][43][44][45][46], all of which are based directly on measured data without any model knowledge. Other applications of such data-based methods include stabilizing linear time-varying systems [47,48] and delay systems [49], controlling linear network systems [50,51], estimating the state in presence of unknown inputs [52], and model reduction [53,54]. In [55], the informativity of data and its role in data-driven analysis and control is investigated.…”