Abstract
The Loewner framework (LF) in combination with Volterra series (VS) offers a non-intrusive approximation method that is capable of identifying bilinear models from time-domain measurements. This method uses harmonic inputs which establish a natural way for data acquisition. For the general class of nonlinear problems with VS representation, the growing exponential approach allows the derivation of the generalized kernels, namely, symmetric generalized frequency response functions (GFRFs). In addition, the homogeneity of the Volterra operator determines the accuracy in terms of how many kernels are considered. For the weakly nonlinear setup, only a few kernels are needed to obtain a good approximation. In this direction, the proposed adaptive scheme is able to improve the estimations of the computationally nonzero kernels. The Fourier transform associates these measurements with the derived 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 in the time domain. This is done by fitting first a linear model, and afterward, by fitting the corresponding bilinear operator.
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Notes
- 1.
The state-output equation often is represented as \(y(t)=\mathbf{c}\mathbf{x}(t)+du(t)\).
- 2.
\((h*u)(t)=\int _{-\infty }^{\infty }h(\tau )u(t-\tau )d\tau \).
- 3.
With \((j^2=-1)\), as the frequency \(s=j\omega \) lies on the imaginary axis, the Laplace transform simplifies in most cases to Fourier transform (e.g., for square-integrable functions).
- 4.
\(\mathbf{G}_{n}^{p,q}=\mathbf{G}(\underbrace{j\omega ,...,j\omega }_{p-times};\underbrace{-j\omega ,...,-j\omega }_{q-times})\).
- 5.
The Hadamard product is denoted with “\(\odot \)”; the matrix multiplication is performed element-wise.
- 6.
The vectorization is row-wise, \(vec(\mathbf{N})=\left[ \begin{array}{ccc} \mathbf{N}(1,1:n)&\cdots&\mathbf{N}(n,1:n) \end{array}\right] ^T\in {\mathbb R}^{n^2\times 1}\).
- 7.
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Karachalios, D.S., Gosea, I.V., Antoulas, A.C. (2021). On Bilinear Time-Domain Identification and Reduction in the Loewner Framework. In: Benner, P., Breiten, T., Faßbender, H., Hinze, M., Stykel, T., Zimmermann, R. (eds) Model Reduction of Complex Dynamical Systems. International Series of Numerical Mathematics, vol 171. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-72983-7_1
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