Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

We would like to express our gratitude to Dr. Steffen W. R. Werner for providing the data and code from [16, 17].

Open Access funding enabled and organized by Projekt DEAL.

Authors and Affiliations

1. Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany

   Pawan Goyal, Igor Pontes Duff & Peter Benner

2. Otto von Guericke University, Faculty of Mathematics, Universitätsplatz 2, 39106, Magdeburg, Germany

   Peter Benner

Corresponding author

Correspondence to Pawan Goyal.

Conflict of Interest

The authors declared that they have no conflict of interest.

Communicated by: Tobias Breiten

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A   Proof of Lemma 2.1

Proof

1. a).

   First note that

   

   (48)

   Since \(x_{i_1} x_{i_2} \cdots x_{i_n} = x_{j_1} x_{j_2} \cdots x_{j_n} \), for every \((j_1,\ldots ,j_n) \in \mathcal {S}_\textbf{i}\), we can write

   

   Next, we have

   

   which proves part (a).

2. b).

   We begin with

   $$\begin{aligned} \widetilde{\textbf{H}}^{(1)} \left( {\textbf{q}}^{(1)}\otimes \cdots \otimes {\textbf{q}}^{(\mathrm N)}\right)&\nonumber \\&\hspace{-3cm}= \sum _{i_1=1}^n \cdots \sum _{i_{\mathrm N}=1}^n\widetilde{\textbf{H}}_{(1)}\left( :, \left( i_1 + \sum _{l=2}^{\mathrm N}(i_l-1)n^l )\right) \right) \left( q^{(1)}_{i_1} \cdots q^{(\mathrm N)}_{i_{\mathrm N}}\right) \nonumber \\&\hspace{-3cm} = \sum _{i_1=1}^n \cdots \sum _{i_{\mathrm N}=1}^n{\textbf{H}}_{(1)} \left( \sum _{\begin{array}{c} (j_1,\ldots ,j_n) \\ \in \mathcal {S}_\textbf{i} \end{array}}\dfrac{1}{\alpha _\textbf{i}}\left( \textbf{e}_{j_1} \otimes \textbf{e}_{j_2} \otimes \cdots \otimes \textbf{e}_{j_n} \right) \right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( q^{(1)}_{i_1} \cdots q^{(\mathrm N)}_{i_{\mathrm N}}\right) . \end{aligned}$$

   (49)

   Since \((j_1,\ldots ,j_n) \in {\mathcal {S}}_\textbf{i}\), the above equation is invariant to the interchange of the indices \(i_k\). Therefore, the Kronecker product of the \(q_i\)’s can appear in any order that would yield the same \(\widetilde{\textbf{H}}_{(1)} \left( \widetilde{\textbf{q}}_1 \otimes \cdots \otimes \widetilde{\textbf{q}}_{\mathrm N}\right) \), where \(\left( \widetilde{\textbf{q}}_1, \ldots , \widetilde{\textbf{q}}_{\mathrm N}\right) \) belongs to the set of all permutations of the set \(\{\textbf{q}_1,\ldots ,\textbf{q}_{\mathrm N}\}\). This proves the result.

3. c).

   Assuming \(l_j \in \{1,\ldots ,n\}\) for \(j \in \{1,\ldots , \mathrm N+1\}\), we have

   $$\begin{aligned}&\textbf{e}_{l_2}\widetilde{\textbf{H}}_{(2)} \left( \textbf{e}_{l_{\mathrm N +1}} \otimes \textbf{e}_{l_{\mathrm N}} \otimes \cdots \otimes \textbf{e}_{l_{3}} \otimes \textbf{e}_{l_{1}} \right) \\&\hspace{1.2cm}\text {using } (9)\\&\hspace{1cm}= \textbf{e}_{l_1}\widetilde{\textbf{H}}_{(1)} \left( \textbf{e}_{l_{\mathrm N +1}} \otimes \textbf{e}_{l_{\mathrm N}} \otimes \cdots \otimes \textbf{e}_{l_{3}} \otimes \textbf{e}_{l_{2}} \right) \\&\hspace{1.2cm}\text {using } (49)\\&\hspace{1cm}= \textbf{e}_{l_1}\widetilde{\textbf{H}}_{(1)} \left( \textbf{e}_{l_{\mathrm N +1}} \otimes \textbf{e}_{l_{\mathrm N}} \otimes \cdots \otimes \textbf{e}_{l_{m+1}} \otimes \textbf{e}_{l_2} \otimes \textbf{e}_{l_m} \otimes \textbf{e}_{l_{m-1}} \otimes \textbf{e}_{l_{3}}\right) \\&\hspace{1.2cm} \text {using } (9)\\&\hspace{1cm}= \textbf{e}_{l_2}\widetilde{\textbf{H}}_{(m)} \left( \textbf{e}_{l_{\mathrm N +1}} \otimes \textbf{e}_{l_{\mathrm N}} \otimes \cdots \otimes \textbf{e}_{l_{m+1}} \otimes \textbf{e}_{l_m} \otimes \textbf{e}_{l_{m-1}} \otimes \textbf{e}_{l_{3}} \otimes \textbf{e}_{l_1}\right) . \end{aligned}$$

   This shows that the entries in \(\widetilde{\textbf{H}}_{(2)}\) and \(\widetilde{\textbf{H}}_{(m)}\) (\(m\ge 2\)) are equal, implying the result.

B   Fundamental solution using frequency domain methods

The fundamental solution of a linear operator can be defined in different ways. In this work, we follow the approaches that use the frequency domain representation of the linear operator. We refer the reader to [8, 30] for more details.

Let us start by defining the unilateral Laplace transform. Given a function \(\textbf{g}(\cdot )\), its unilateral Laplace transform is given by

where \(\textbf{G}(\cdot )\) corresponds to the frequency domain representation of \({\textbf{g}}(\cdot )\).

Now, let us consider a linear operator \({\mathcal {L}}\) such as shown in Table 1. By means of the Laplace transform, we obtain the frequency domain representation of \(({\mathcal {L}}{\textbf{x}})(t)\) as follows:

where \(\textbf{X}(s)\) corresponds to the Laplace transform of \(\textbf{x}(t)\) and \({\mathcal {K}}(s)\) corresponds to the frequency domain representation of the operator \({\mathcal {L}}\). Let us assume that the inverse of the Laplace transform of \({\mathcal {K}}^{-1}(s)\) exists and is given as

Then, \(\mathrm {\Phi }(t)\) is the fundamental solution associated to the linear operator \({\mathcal {L}}\). Indeed, \(\mathrm {\Phi }(t)\) is the solution of the functional differential equation

where \(\delta (t)\) is the Dirac delta distribution and the initial conditions are all zero. Moreover, the inhomogeneous equation

with \(\textbf{g}(\cdot )\) being a suitable function, has a solution in convolution form

C   Tangential interpolation-based MOR for MIMO systems

Here, we discuss a construction of an interpolating ROM for MIMO polynomial systems. Similar to the SISO case, the leading generalized transfer functions for a MIMO polynomial system are given as follows:

Lemma C.1

Consider the original system as given in (3). Let \(\sigma _i \in \mathbb {C}\), \(i \in \{1,\ldots ,\widetilde{r}\}\), be interpolation points such that \(\mathcal {K}(s)\) is invertible for all \(s \in \{\sigma _1,\ldots , \sigma _{\widetilde{r}}\}\), and \(\textbf{b}_i \in \mathbb {C}^m\) and \(\textbf{c}_i\in \mathbb {C}^q\) for \(i \in \{1,\ldots ,{\widetilde{r}}\}\) be right and left tangential directions corresponding to \(\sigma _i\), respectively. Let \(\textbf{V}\) and \(\textbf{W}\) be defined as follows:

If a ROM is computed as shown in (7) using the projection matrices \(\textbf{V}\) and \(\textbf{W}\), where we assume \(\textbf{V}\) and \(\textbf{W}\) to be of full rank, then the following interpolation conditions are fulfilled:

(53d)

(53e)

(53f)

(53g)

(53h)

(53i)

where \(i \in \{1,\ldots , \widetilde{r}\}\), \(\xi \in \{2,\ldots ,d\} \), \(\eta \in \{1,\ldots ,d\}\) and \(\dfrac{\partial }{\partial s_j}\) denotes the partial derivative with respect to \(s_j\) of a given function.

Proof

The proof of these interpolation conditions follows exactly the one of Theorem 3.1.

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