Abstract
In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples.
Highlights
Transfer Functions, Equivalence, Many problems from the applied sciences that involve a dynamic process under study, either to be solved, simulated, or even controlled, are described by models that could contain a considerable amount of variables and/or many inputs and outputs
We addressed the problem of approximating generic nonlinear system by means of enforcing a specific structure of the reduced-order models
We dealt with complexity reduction of large-scale lifted models of nonlinear systems
Summary
Transfer Functions, Equivalence, Many problems from the applied sciences that involve a dynamic process under study, either to be solved, simulated, or even controlled, are described by models that could contain a considerable amount of variables and/or many inputs and outputs This is true in the context of fluid dynamics. MOR methods tailored for reducing QBTI systems have been proposed in [5,6,7,8] Such approaches are fairly new, and most of them represent extensions of the techniques proposed for model reduction of linear time-invariant (LTI in short) systems and bilinear time-invariant (BTI) systems [9]. We will show applications of the approaches under consideration for two benchmarks examples (a semi-discretized classical problem of computational fluid dynamics and a nonlinear electrical circuit)
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