Realization independent single time-delay dynamical model interpolation and ℋ<inf>2</inf>-optimal approximation

Abstract

In this paper, the realization-free model approximation problem, as stated in [1,2], is revisited in the case where the interpolating model might be time-delay dependent. To this aim, the Loewner framework, initially settled for delay-free realization, is firstly generalized to the single delay case. Secondly, the (infinite) model approximation H 2 optimality conditions are established through the use of the Lambert functions. Finally, a numerically effective iterative scheme, named dTF-IRKA, similar to the TF-IRKA [2], is proposed to reach a part of the aforementioned optimality conditions. The proposed method validity and interest are assessed on different numerical examples. * I. Pontes Duff is with ISAE, Onera -The French Aerospace Lab,

Data-driven interpolation of dynamical systems with delay

“…Consequently, the result obtained in [18] is a special case of the realization presented in Theorem 4.7.…”

“…Remark 5.5 Following the idea of [18], we can (formally) derive the same results by only employing the Loewner realization theorem (Theorem 2.3). Using the proportional ansatz A 2,ρ = αE ρ + βA 1,ρ with some constants α and β, we can rewrite the transfer function as…”

“…Recently, a generalization of the Loewner framework for a special class of delay systems, where A 1,ρ = 0 in (2.1), was introduced in [18]. There, the interpolant is constructed by means of Theorem 2.3 leading to the subsequent result, written here in slightly different notation.…”

Data-driven structured realization

“…Projection-based techniques are often able to retain special structural features in the reduced models that may reflect underlying physical properties of the systems under study [7,11,12,18,25,28,33]. Datadriven techniques for system identification and model reduction do not generally have this capacity, however, the recent contributions of [17,32] provide a notable exception for time-delay systems. In the present work, we build on the results of [32] and extend its domain of applicability to a wide range of structured dynamical systems.…”

“…When we seek reduced models that are structurally equivalent to standard first order realizations (that is, when we have in (2.1) K = 2, h 1 (s) = s, and h 2 (s) ≡ −1) then first order necessary conditions for optimality of the reduced order approximant with respect to the H 2 norm are known and they require that the reduced transfer function H(s) must be a Hermite interpolant of the original H(s) [20]. Even though these necessary conditions do not extend immediately to more general structured systems as appear in (2.1), it is known for some special cases such as second order systems with modal damping and port-Hamiltonian systems [9], and for systems with simple delay structures [16,17], that Hermite interpolation (in a different form then for the rational case) still plays a fundamental role in the necessary optimality conditions. Therefore, if derivative information for the transfer function H(s) is accessible then this motivates finding a structurally equivalent realization H(s) that matches both the evaluation data and the derivative data.…”