Data‐driven (Bilinear) identification and reduction

Abstract

AbstractIdentifying dynamical systems from measured data is an important step towards accurate modeling and control. Model order reduction (MOR) constitutes a class of methods that can be used to replace large, complex models of dynamical processes with simpler, smaller models. The reduced‐order models (ROMs) can be then used for further tasks such as control, design, and simulation. One typical approach for projection‐based model reduction for both linear and nonlinear dynamical systems is by employing interpolation. Projection‐based methods require access to the internal dynamics of the system which is not always available. The aim here is to compute ROMs without having access to the internal dynamics, by means of a realization independent method. The proposed methodology will fall into the broad category of data‐driven approaches. The method under consideration, which will be referred to as the Loewner framework (LF), was originally introduced by the third author. Based on data, LF identifies state‐space models in a direct way. In the original setup, the framework relies on compressing the full data set to extract dominant features and, at the same time, to eliminate the inherent redundancies. In the broader class of nonlinear control systems, the LF has been already extended to certain classes with a special structure such as quadratic or bilinear systems. The LF in combination with Volterra series‐(VS) offers a non‐intrusive approximation method that is capable of identifying bilinear models from input‐output time‐domain measurements. This method uses harmonic inputs which establish a natural way for data acquisition. The Fourier transform associates these measurements with the derived generalized frequency response functions GFRFs and the LF makes the connection with system theory. In the linear case, the LF associates the so‐called S‐parameters with the linear transfer function by interpolating in the frequency domain. The goal of the proposed method is to extend identification to the case of bilinear systems from time‐domain measurements and to approximate other general nonlinear systems by means of the Carleman bilinearizarion scheme. By identifying the linear contribution with the LF, a considerable reduction is achieved by means of the SVD. The fitted linear system has the same McMillan degree as the original linear system. Then, the performance of the linear model is improved by augmenting a special nonlinear structure. In a nutshell, we learn reduced‐dimension bilinear models directly from a potentially large‐scale system that is simulated/sampled in the time domain. This is done by fitting first a linear model, and afterwards, by fitting the corresponding bilinear operator which will improve the fitting performance.