Summary
We introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments.
We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes.We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode.In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians. † Data-Driven System Reduction and Identification Group,
Abstract. The Loewner framework for model reduction is extended to the class of linear switched systems. One advantage of this framework is that it introduces a trade-off between accuracy and complexity. Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system. Hence, another advantage of the framework is that it does not require the initial system matrices. More exactly, the data used in this framework consists in frequency domain samples of input-output mappings of the original system. The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems. A key role is played by the coupling matrices, which ensure the transition from one active mode to another. 1. Introduction. Model order reduction (MOR) seeks to transform large, complicated models of time dependent processes into smaller, simpler models that are nonetheless capable of accurately representing the behavior of the original process under a variety of operating conditions. The goal is an efficient, methodical strategy that yields a dynamical system evolving in a substantially lower dimension space (hence requiring far fewer computational resources for realization) yet retaining response characteristics close to the original system. Such reduced order models could be used as efficient surrogates for the original model, replacing it as a component in larger simulations.Hybrid systems are a class of nonlinear systems which result from the interaction of continuous time dynamical subsystems with discrete events. More precisely, a hybrid system is a collection of continuous time dynamical systems. The internal variable of each dynamical system is governed by a set of differential equations. Each of the separate continuous time systems is labeled as a discrete mode. The transitions between the discrete states may result in a jump in the continuous internal variable. Linear switched systems (LSSs) constitute a subclass of hybrid systems; the main property is that these systems switch among a finite number of linear subsystems. Also, the discrete events interacting with the subsystems are governed by a piecewise continuous function called the switching signal.Hybrid and switched systems are powerful models for distributed embedded sys-
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