Constrained optimal reduced-order models from input/output data

Abstract

Abstract-Model reduction by moment matching does not preserve, in a systematic way, the transient response of the system to be reduced, thus limiting the use of this model reduction technique in control problems. With the final goal of designing reduced-order models which can effectively be used (not just for analysis but also) for control purposes, we determine, using a data-driven approach, an estimate of the moments and of the transient response of an unknown system. We compute the unique, up to a change of coordinates, reducedorder model which possesses the estimated transient and, simultaneously, achieves moment matching at the prescribed interpolation points. The error between the output of the system and the output of the reduced-order model is minimized and we show that the resulting system is a constrained optimal (in a sense to be specified) reduced-order model. The results of the paper are illustrated by means of a simple numerical example.

Simulation-driven fixed-order controller tuning via moment matching

“…Based on our assumptions, the interconnection between S and C −1 o has an invariant manifold described by {(x c , ω) ∈ R nx c × R ν : x c = Πω}. The output of this interconnection is thus given by (18) with e tr (t) = e Act (x c (0)−Πω(0)) an exponentially decaying signal. Under the additional hypothesis that the number N of available samples is sufficiently large so that the transient of C o −1 is negligible, equation 18can be approximated by…”

“…To estimate this model from data, a new data-setD N = {e tr (t i )} N i=1 has to be built that comprises samples of the transient response of (27). By using the definition of the error provided in (18) and the estimate Ξ obtained through Algorithm 1, the elements ofD N can be built from D N as:…”

“…Moreover, continuous effort is put into the investigation of model-reduction policies for nonlinear systems, see e.g., [2], [8], [10], [19]. In the last decade some data-driven approaches have also been proposed to directly derive reduced-order models from data, see e.g., [13], [16], which rely upon frequency domain measurements, and [16], [18], that exploit time-domain data.…”

Data-driven constrained optimal model reduction

“…A preliminary version of this paper has been published in [55]. The additional contributions of the present paper are as follows: first, while [55] considers only the 2 norm, in this paper we discuss also other p norms and, in addition, we establish a result for the H 2 norm.…”

“…A preliminary version of this paper has been published in [55]. The additional contributions of the present paper are as follows: first, while [55] considers only the 2 norm, in this paper we discuss also other p norms and, in addition, we establish a result for the H 2 norm. Second, we compute error bounds and we provide results to determine the optimal initial state of the reduced-order model in various cases.…”