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The Loewner Framework is a frequency-domain system identification and model reduction method, named after Charles Loewner, who introduced the Loewner Matrix which is essential to this method.
Introduction
Model order reduction is commonly used to simulate and control complex physical processes. This is done by replacing the large-scale system model described in terms of differential equations by a system of much lower dimension that has similar response characteristics. In many situations, a model (the collection of differential equations written in matrix format) for the underlying dynamical process is not explicitly available. Hence, the system matrix realization can not be directly obtained. Instead, a black box is provided that produces input-output measurements. These situations include VLSI modeling from chips or real time simulation of multi-body dynamics with constraints.
We study reduction methods based on moment matching, that is, matching of the coefficients of power series expansions of the transfer function at selected points in the complex plane. More precisely, values of the underlying transfer function, together with the values of its derivatives are interpolated. Hence, the main underlying problem is rational interpolation. Different methods based on Krylov iteration, as well as the Arnoldi or the Lanczos procedures, and multi-point (rational) versions thereof.
Interpolatory model reduction methods compute ROMs (reduced order models) by means of interpolating the response of the original system at selected points (usually the transfer function). The Loewner framework (introduced in [1]) falls into this category. It is a data-driven method that directly constructs a ROM using only measured data (samples of the transfer function). Additionally, by compressing the (usually large) data set, it extracts the dominant features and eliminates the inherent redundancies. For a recent survey on the Loewner framework for linear systems, see [2]. For the nonlinear case, this method requires appropriate definition of transfer functions, but the main procedure does not differ too much from the one applied for the linear case. Some of the extensions include the bilinear systems case in [3], quadratic-bilinear systems [4] and linear switched systems [5]. The dissertation [6] represents a comprehensive collection of these methods.
The motivation behind developing generalized versions of the Loewner framework to new systems is that such classes of systems can be viewed as a bridge between linear and nonlinear systems. For example, one can always write an approximation of a nonlinear system by means of a bilinear system (Carleman linearization [7]). Furthermore, for certain types of nonlinear systems, one can always find an equivalent quadratic-bilinear model without performing any approximation (McCormick relaxation [8]) whatsoever. Linear switched systems can be viewed as a special class of hybrid systems that are used to model coupled or distributed processes characterized by both discrete and continuous dynamics.
Loewner Framework for Linear Systems
Loewner Framework for Bilinear Systems
Loewner Framework for Quadratic-Bilinear Systems
Loewner Framework for Linear Switched Systems
Loewner Framework for Parametric Systems
References
- ↑ A.J. Mayo, A.C. Antoulas. A framework for the generalized realization problem. Linear Algebra and Its Applications 426(2--3): 634--662, 2007.
- ↑ A.C. Antoulas, S. Lefteriu, A.C. Ionita. A tutorial introduction to the Loewner Framework for model reduction. In: Model Reduction and Approximation for Complex Systems: 335--376, SIAM, 2017.
- ↑ A.C. Antoulas, I.V. Gosea, A.C. Ionita. Model reduction of bilinear systems in the Loewner framework. SIAM J. Sci. Comput. 38(5): B889--B916, 2016.
- ↑ I.V. Gosea, A.C. Antoulas. Data-driven model order reduction of quadratic-bilinear systems. Numercial Linear Algebra with Applications: Accepted, 2018.
- ↑ I.V. Gosea, M. Petreczky, A.C. Antoulas. Model reduction of bilinear systems in the Loewner framework. SIAM J. Sci. Comput. 40(2): B572--B610, 2018.
- ↑ I.V. Gosea. Model order reduction of linear and nonlinear systems in the Loewner framework. Ph.D. thesis, Jacobs University Bremen, 2017.
- ↑ T. Carleman. Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires. Acta Math. 59: 63--87, 1932.
- ↑ G.P. McCormick. Computability of global solutions to factorable nonconvex programs: Part I -- Convex underestimating problems. Mathematical Programming 10(1): 147--175 1976.
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