Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Abstract

In the past decades, Model Order Reduction (MOR) has demonstrated its robustness and wide applicability for simulating large-scale mathematical models in engineering and the sciences. Recently, MOR has been intensively further developed for increasingly complex dynamical systems. Wide applications of MOR have been found not only in simulation, but also in optimization and control. In this survey paper, we review some popular MOR methods for linear and nonlinear large-scale dynamical systems, mainly used in electrical and control engineering, in computational electromagnetics, as well as in micro-and nanoelectro-mechanical systems design. This complements recent surveys on generating reduced-order models for parameterdependent problems (Benner et al. in 2013; Boyaval et al. in Arch Comput Methods Eng 17(4):435-454, 2010; Rozza et al. Arch Comput Methods Eng 15(3):229-275, 2008) which we do not consider here. Besides reviewing existing methods and the computational techniques needed to implement them, open issues are discussed, and some new results are proposed.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…The reduced basis community largely focuses on formulation of the model reduction problem in the continuous (PDE) domain; however, the resulting numerical algorithms build a projection basis as the span of a set of discrete snapshots over the parameter domain, just as in the POD. As discussed in section 3.3, duality between time and frequency domain formulations for linear systems also reveals the connections between POD and balanced truncation, and between POD and rational interpolation methods; see [30] for a more detailed discussion of these connections. Nonetheless, some important differences remain among the methods, most notably the error bounds associated with balanced truncation (for a fixed parameter sample) and the applicability of POD to general nonlinear systems.…”

“…,ŝ k as opposed to a single expansion pointŝ, leading to the concept of multipoint momentmatching, also called multipoint rational interpolation. Surveys on this class of model reduction methods can be found in [22,101]; see also [14,15,30,44,202].…”

“…Despite the analogy between the two methods, there is no asymptotic stability guarantee associated with balanced POD, unlike balanced truncation. The relation between frequency-domain POD and approximate balanced truncation is further discussed in [30], where also relations to balanced truncation via low-rank Gramians computed by the ADI method and to moment-matching methods are studied.…”

“…When the initial condition is nonzero but known, this information could be easily included in the time-domain POD method as the initial condition for the numerical simulation. For rational interpolation and balanced truncation approaches, the nonzero initial conditions can be appended to the input-state-matrix B(p) representing an initial impulse as done in [30,57,128]. However, if the initial condition is unknown and/or the reduced model needs to provide a good approximation for a wide range of initial conditions, the corresponding model reduction problem can no longer be easily handled by simple modification of the zero-initial condition case.…”

A Two-Step Global Maximum Error Controller-Based TPWL MOR with POD Basis Vectors and Its Applications to MEMS

“…By contrast, the nonlinear MOR methods are still in their development stage [4][5][6], although a great deal of effort has been devoted. Owing to good properties of linear problems and maturity of linear MOR methods, the most obvious approach in dealing with nonlinear problems is linearization.…”