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Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs

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Abstract

This paper extends interpolatory model reduction to systems with (up to) quadratic-bilinear dynamics and quadratic-bilinear outputs. These systems are referred to as QB-QB systems and arise in a number of applications, including fluid dynamics, optimal control, and uncertainty quantification. In the interpolatory approach, the reduced order models (ROMs) are based on a Petrov-Galerkin projection, and the projection matrices are constructed so that transfer function components of the ROM interpolate the corresponding transfer function components of the original system. To extend the approach to systems with QB outputs, this paper derives system transfer functions and sufficient conditions on the projection matrices that guarantee the aforementioned interpolation properties. Alternatively, if the system has linear dynamics and quadratic outputs, one can introduce auxiliary state variables to transform it into a system with QB dynamics and linear outputs to which known interpolatory model reduction can be applied. This transformation approach is compared with the proposed extension that directly treats quadratic outputs. The comparison shows that transformation hides the problem structure. Numerical examples illustrate that keeping the original QB-QB structure leads to ROMs with better approximation properties.

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Acknowledgements

We thank the reviewers for their comments, which have led to improvements in the presentation.

Funding

This research was supported in part by AFOSR Grant FA9550-22-1-0004, NSF grants CCF-1816219 and DMS-1819144, and by a 2021 National Defense Science and Engineering Graduate (NDSEG) Fellowship Award.

Author information

Authors and Affiliations

  1. Department of Computational Applied Mathematics and Operations Research, Rice University, 6100 Main St.-MS 134, Houston, 77005, TX, USA

    Alejandro N. Diaz, Matthias Heinkenschloss & Ion Victor Gosea

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

    Ion Victor Gosea & Athanasios C. Antoulas

  3. Department of Electrical an Computer Engineering, Rice University, 6100 Main St.-MS 380, Houston, 77005, TX, USA

    Athanasios C. Antoulas

Authors
  1. Alejandro N. Diaz
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  2. Matthias Heinkenschloss
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  3. Ion Victor Gosea
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  4. Athanasios C. Antoulas
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Correspondence to Alejandro N. Diaz.

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Communicated by: Tobias Breiten

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Appendices

Appendix

A   Proof of Theorem 1

We prove Theorem 1. First observe that

$$\begin{aligned} \widehat{\varvec{\Phi }}(s)^{-1} = s\widehat{\textbf{E}}-\widehat{\textbf{A}}= \textbf{W}^*(s\textbf{E}-\textbf{A})\textbf{V}= \textbf{W}^*{\varvec{\Phi }}(s)^{-1}\textbf{V}. \end{aligned}$$

By Eq. 13a, there exist vectors \(\textbf{v}_{j, 1} \in \mathbb {C}^r\) such that

$$\begin{aligned} \textbf{V}\textbf{v}_{j, 1} = {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j, \qquad j=1, \dots , \ell . \end{aligned}$$
(1)

Multiplying Eq. 44 by \(\textbf{W}^*{\varvec{\Phi }}(\sigma _j)^{-1}\) yields \(\textbf{W}^*{\varvec{\Phi }}(\sigma _j)^{-1}\textbf{V}\textbf{v}_{j, 1} =\widehat{\varvec{\Phi }}(\sigma _j)^{-1}\textbf{v}_{j, 1} = \textbf{W}^*\textbf{B}\textbf{b}_j = \widehat{\textbf{B}}\textbf{b}_j\). Hence

$$\begin{aligned} \textbf{v}_{j, 1}=\widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j. \end{aligned}$$
(2a)

By equation Eq. 45 and the definition Eq. 44 of \(\textbf{v}_{j,1}\),

$$\begin{aligned} \widehat{\textbf{H}}_1(\sigma _j)\textbf{b}_j&= \widehat{\textbf{C}}\widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j = \textbf{C}\textbf{V}\textbf{v}_{j, 1} =\textbf{C}{\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j =\textbf{H}_1(\sigma _j)\textbf{b}_j, \end{aligned}$$

which is 14a.

By Eq. 13b, there exist vectors \(\textbf{v}_{j, 2}\in \mathbb {C}^r\) such that

$$\begin{aligned} \textbf{V}\textbf{v}_{j, 2} = {\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j)+\textbf{N}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes \textbf{b}_j) \right] , \end{aligned}$$
(2b)

\(j=1, \dots , \ell \).

Next, notice that by the definitions Eqs. 44, 46 of \(\textbf{v}_{j, 1}\) and \(\textbf{v}_{j, 2}\) and Eq. 45,

$$\begin{aligned}&\textbf{W}^*{\varvec{\Phi }}(2\sigma _j)^{-1}\textbf{V}\textbf{v}_{j, 2} =\widehat{\varvec{\Phi }}(2\sigma _j)^{-1}\textbf{v}_{j, 2} \\&=\textbf{W}^*{\varvec{\Phi }}(2\sigma _j)^{-1}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j)+\textbf{N}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes \textbf{b}_j) \right] \\&=\textbf{W}^*\textbf{Q}(\textbf{V}\otimes \textbf{V})(\textbf{v}_{j, 1}\otimes \textbf{v}_{j, 1})+ \textbf{W}^*\textbf{N}(\textbf{V}\otimes \textbf{I}_m) (\textbf{v}_{j, 1}\otimes \textbf{b}_j) \\&= \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \textbf{b}_j\right) , \qquad j=1, \dots , \ell . \end{aligned}$$

Hence

$$\begin{aligned} \textbf{v}_{j, 2} = \widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j \otimes \textbf{b}_j\right) \right] , \end{aligned}$$
(2c)

\(j=1, \dots , \ell \).

Using Eqs. 47 and 45 and then the definitions Eqs. 44, 46 of \(\textbf{v}_{j, 1}\) and \(\textbf{v}_{j, 2}\) it follows that

$$\begin{aligned} \widehat{\textbf{H}}_{2}(\sigma _j, \sigma _j)(\textbf{b}_j\otimes \textbf{b}_j)&=\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j \otimes \textbf{b}_j\right) \right] \\&\qquad + \widehat{\textbf{K}}(\widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j) + \widehat{\textbf{J}}(\widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\otimes \textbf{b}_j)\\&= \textbf{C}\textbf{V}\textbf{v}_{j, 2} + \textbf{K}(\textbf{V}\otimes \textbf{V})(\textbf{v}_{j, 1}\otimes \textbf{v}_{j, 1}) + \textbf{J}(\textbf{V}\otimes \textbf{I}_m)(\textbf{v}_{j, 1}\otimes \textbf{b}_j)\\&= \textbf{C}\textbf{V}\textbf{v}_{j, 2} + \textbf{K}(\textbf{V}\textbf{v}_{j, 1} \otimes \textbf{V}\textbf{v}_{j, 1}) + \textbf{J}(\textbf{V}\textbf{v}_{j, 1}\otimes \textbf{b}_j)\\&= \textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j)+\textbf{N}\left( {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes \textbf{b}_j\right) \right] \\&\qquad + \textbf{K}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{J}({\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j\otimes \textbf{b}_j)\\&= \textbf{H}_{2}(\sigma _j, \sigma _j)(\textbf{b}_j\otimes \textbf{b}_j), \qquad j=1, \dots , \ell , \end{aligned}$$

which is Eq. 14b.

By Eq. 13c, there exist vectors \(\textbf{w}_{j, 1} \in \mathbb {C}^r\) such that

$$\begin{aligned} \textbf{w}_{j, 1}^*\textbf{W}^* = \textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j), \qquad j=1, \dots , \ell . \end{aligned}$$
(3a)

Multiplying Eq. 48 by \({\varvec{\Phi }}(2\sigma _j)^{-1}\textbf{V}\) yields \(\textbf{w}_{j, 1}^*\textbf{W}^*{\varvec{\Phi }}(2\sigma _j)^{-1}\textbf{V}= \textbf{w}_{j, 1}^*\widehat{\varvec{\Phi }}(2\sigma _j)^{-1} = \textbf{c}_j^*\textbf{C}\textbf{V}= \textbf{c}_j^*\widehat{\textbf{C}}\). Hence,

$$\begin{aligned} \textbf{w}_{j, 1}^* = \textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j), \qquad j=1, \dots , \ell . \end{aligned}$$
(3b)

Using Eqs. 49 and 48 gives

$$\begin{aligned} \textbf{c}_j^*\widehat{\textbf{H}}_1(2\sigma _j)&= \textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j) \widehat{\textbf{B}}= \textbf{w}_{j, 1}^* \textbf{W}^* \textbf{B}= \textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\textbf{B}= \textbf{c}_j^*\textbf{H}_1(2\sigma _j), \qquad j=1, \dots , \ell , \end{aligned}$$

which is Eq. 14c.

By Eq. 13d, there exist vectors \(\textbf{w}_{j, 2} \in \mathbb {C}^r\) such that

$$\begin{aligned} \textbf{w}_{j, 2}^*\textbf{W}^*&= \textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\Big [\textbf{Q}({\varvec{\Phi }}(\sigma _j)\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j ) + \textbf{N}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j\Big ] \nonumber \\&\quad + \textbf{c}_j^*\textbf{K}({\varvec{\Phi }}(\sigma _j) \otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{c}_j^*\textbf{J}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j). \end{aligned}$$
(3c)

In the next equalities we use the identity \(\textbf{M}= \textbf{M}\otimes 1\) for any matrix \(\textbf{M}\) to apply the Kronecker product property Eq. 1. Using Eq. 50, the identity \(\textbf{M}= \textbf{M}\otimes 1\), Eqs. 44, 48, 49, 45 gives

$$\begin{aligned} \textbf{w}_{j, 2}^*\widehat{\varvec{\Phi }}(\sigma _j)^{-1}&= \textbf{w}_{j, 2}^*\textbf{W}^* {\varvec{\Phi }}(\sigma _j)^{-1}\textbf{V}\\&=\Big (\textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j) \otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{N}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j\right] \\&\quad + \textbf{c}_j^*\textbf{K}({\varvec{\Phi }}(\sigma _j) \otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{c}_j^*\textbf{J}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j)\Big )\left( {\varvec{\Phi }}(\sigma _j)^{-1}\textbf{V}\otimes 1\right) \\&= \textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}(\textbf{V}\otimes \textbf{V}\textbf{v}_{j,1}) + \textbf{N}(\textbf{V}\otimes \textbf{b}_j)\right] + \textbf{c}_j^*\textbf{K}(\textbf{V}\otimes \textbf{V}\textbf{v}_{j,1}) + \textbf{c}_j^*\textbf{J}(\textbf{V}\otimes \textbf{b}_j) \\&= \textbf{w}_{j, 1}^*\textbf{W}^*\left[ \textbf{Q}(\textbf{V}\otimes \textbf{V})(\textbf{I}_r \otimes \textbf{v}_{j,1}) + \textbf{N}(\textbf{V}\otimes \textbf{I}_m)(\textbf{I}_r \otimes \textbf{b}_j)\right] \\&\quad + \textbf{c}_j^*\textbf{K}(\textbf{V}\otimes \textbf{V})(\textbf{I}_r\otimes \textbf{v}_{j, 1}) + \textbf{c}_j^*\textbf{J}(\textbf{V}\otimes \textbf{I}_m)(\textbf{I}_r \otimes \textbf{b}_j) \\&= \textbf{w}_{j, 1}^*\left[ \textbf{W}^*\textbf{Q}(\textbf{V}\otimes \textbf{V})(\textbf{I}_r \otimes \textbf{v}_{j,1}) + \textbf{W}^*\textbf{N}(\textbf{V}\otimes \textbf{I}_m)(\textbf{I}_r \otimes \textbf{b}_j)\right] \\&\quad + \textbf{c}_j^*\textbf{K}(\textbf{V}\otimes \textbf{V})(\textbf{I}_r \otimes \textbf{v}_{j,1}) + \textbf{c}_j^*\textbf{J}(\textbf{V}\otimes \textbf{I}_r)(\textbf{I}_r \otimes \textbf{b}_j) \\&= \textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \textbf{I}_r \otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}(\textbf{I}_r \otimes \textbf{b}_j)\right] \\&\quad + \textbf{c}_j^*\widehat{\textbf{K}}\left( \textbf{I}_r \otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \textbf{c}_j^*\widehat{\textbf{J}}(\textbf{I}_r \otimes \textbf{b}_j). \end{aligned}$$

Multiplying on the right by \(\widehat{\varvec{\Phi }}(\sigma _j)=\widehat{\varvec{\Phi }}(\sigma _j)\otimes 1\), we conclude that

$$\begin{aligned} \textbf{w}_{j, 2}^*&=\textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j\right) \right] \\&\quad + \textbf{c}_j^*\widehat{\textbf{K}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \textbf{c}_j^*\widehat{\textbf{J}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j\right) . \nonumber \end{aligned}$$
(3d)

Using Eqs. 51 and 50 gives

$$\begin{aligned} \textbf{c}_j^*\widehat{\textbf{H}}_2(\sigma _j, \sigma _j)(\textbf{I}_m \otimes \textbf{b}_j)&=\textbf{c}_j^*\Big [\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \textbf{I}_m\right) \right] \\&\qquad \quad + \widehat{\textbf{K}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\right) + \widehat{\textbf{J}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \textbf{I}_m\right) \Big ](\textbf{I}_m \otimes \textbf{b}_j)\\&=\textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \textbf{b}_j\right) \right] \\&\qquad \quad + \textbf{c}_j^*\widehat{\textbf{K}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \textbf{c}_j^*\widehat{\textbf{J}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\otimes \textbf{b}_j\right) \\&=\Big [\textbf{c}_j^*\widehat{\textbf{C}}\widehat{\varvec{\Phi }}(2\sigma _j)\left[ \widehat{\textbf{Q}}\left( \widehat{\varvec{\Phi }}(\sigma _j) \otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \widehat{\textbf{N}}\left( \widehat{\varvec{\Phi }}(\sigma _j) \otimes \textbf{b}_j\right) \right] \\&\qquad \quad + \textbf{c}_j^*\widehat{\textbf{K}}\left( \widehat{\varvec{\Phi }}(\sigma _j)\otimes \widehat{\varvec{\Phi }}(\sigma _j)\widehat{\textbf{B}}\textbf{b}_j\right) + \textbf{c}_j^*\widehat{\textbf{J}}\left( \widehat{\varvec{\Phi }}(\sigma _j) \otimes \textbf{b}_j\right) \Big ]\left( \widehat{\textbf{B}}\otimes 1\right) \\&= \textbf{w}_{j, 2}^* \textbf{W}^*\textbf{B}\\&= \Big [\textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j ) + \textbf{N}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j\right] \\&\qquad + \textbf{c}_j^*\textbf{K}({\varvec{\Phi }}(\sigma _j) \otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{c}_j^*\textbf{J}({\varvec{\Phi }}(\sigma _j)\otimes \textbf{b}_j)\Big ]\left( \textbf{B}\otimes 1\right) \\&=\textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j ) + \textbf{N}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes \textbf{b}_j\right] \\&\qquad + \textbf{c}_j^*\textbf{K}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}\textbf{b}_j) + \textbf{c}_j^*\textbf{J}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes \textbf{b}_j)\\&= \Big [\textbf{c}_j^*\textbf{C}{\varvec{\Phi }}(2\sigma _j)\left[ \textbf{Q}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}) + \textbf{N}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes \textbf{I}_m )\right] \\&\qquad + \textbf{c}_j^*\textbf{K}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes {\varvec{\Phi }}(\sigma _j)\textbf{B}) + \textbf{c}_j^*\textbf{J}({\varvec{\Phi }}(\sigma _j)\textbf{B}\otimes \textbf{I}_m)\Big ](\textbf{I}_m \otimes \textbf{b}_j) \\&= \textbf{c}_j^*\textbf{H}_2(\sigma _j, \sigma _j)(\textbf{I}_m \otimes \textbf{b}_j), \qquad j=1, \dots , \ell , \end{aligned}$$

which is 14d.

B   System transformation

The QB-QB system Eq. 2 can be transformed into an equivalent system with linear output, but dynamics with higher nonlinearity. Specifically, we introduce the auxiliary variable

$$\begin{aligned} \textbf{z}(t) = \textbf{K}(\textbf{x}(t) \otimes \textbf{x}(t)) + \textbf{J}(\textbf{x}(t) \otimes \textbf{u}(t)). \end{aligned}$$
(3e)

To simplify the following presentation, we again assume that \(\textbf{K}\) is symmetric Eq. 17. Assuming that the input \(\textbf{u}\) is differentiable (if \( \textbf{J}\not = 0\)) and using the symmetry of \(\textbf{K}\) we obtain

$$\begin{aligned} \frac{d}{dt}\textbf{z}(t) =&2 \textbf{K}\big ( \textbf{x}(t) \otimes \frac{d}{dt}\textbf{x}(t) \big ) + \textbf{J}\big ( \frac{d}{dt}\textbf{x}(t) \otimes \textbf{u}(t) \big ) + \textbf{J}\big ( \textbf{x}(t) \otimes \frac{d}{dt}\textbf{u}(t) \big ). \end{aligned}$$

The derivative of \(\textbf{x}(t)\) is replaced using Eq. 2a. The QB-QB system Eq. 2 can be equivalently written as

$$\begin{aligned} \textbf{E}\frac{d}{dt}\textbf{x}(t) =&\textbf{A}\textbf{x}(t) + \textbf{Q}(\textbf{x}(t) \otimes \textbf{x}(t)) + \textbf{N}(\textbf{x}(t) \otimes \textbf{u}(t)) + \textbf{B}\textbf{u}(t), \quad t \in (0, T),\end{aligned}$$
(4a)
$$\begin{aligned} \frac{d}{dt}\textbf{z}(t) =&2 \textbf{K}\Big (\textbf{x}(t) \otimes \textbf{E}^{-1} \big ( \textbf{A}\textbf{x}(t) + \textbf{Q}(\textbf{x}(t) \otimes \textbf{x}(t)) + \textbf{N}(\textbf{x}(t) \otimes \textbf{u}(t)) + \textbf{B}\textbf{u}(t) \big ) \Big ) \nonumber \\&+ \textbf{J}\Big ( \textbf{E}^{-1} \big ( \textbf{A}\textbf{x}(t) + \textbf{Q}(\textbf{x}(t) \otimes \textbf{x}(t)) \ + \textbf{N}(\textbf{x}(t) \otimes \textbf{u}(t)) + \textbf{B}\textbf{u}(t) \big ) \otimes \textbf{u}(t) \Big ) \nonumber \\&+ \textbf{J}\big ( \textbf{x}(t) \otimes \frac{d}{dt}\textbf{u}(t) \big ), \hspace{32ex} t \in (0, T), \end{aligned}$$
(4b)
$$\begin{aligned} \textbf{x}(0) =&\textbf{0}, \quad \textbf{z}(0) = \textbf{0},\end{aligned}$$
(5)
$$\begin{aligned} \textbf{y}(t) =&\textbf{C}\textbf{x}(t) + \textbf{z}(t) + \textbf{D}\textbf{u}(t), \hspace{30ex} t \in (0, T), \end{aligned}$$
(6a)

One could rearrange the right hand side in Eq. 53b using Kronecker product properties like Eq. 1. While the transformed system Eq. 53 has a linear output, the price one pays are nonlinear terms of higher order (up to cubic) in the dynamics and the presence (if \( \textbf{J}\not = 0\)) of derivatives of the inputs. In general these additional terms cannot be dealt with easily by current system or interpolation-based model reduction techniques. However, if the original dynamics are linear and the output is quadratic, then the resulting transformed system is a QB system, to which in principle current model reduction techniques can be applied.

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Diaz, A.N., Heinkenschloss, M., Gosea, I.V. et al. Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs. Adv Comput Math 49, 95 (2023). https://doi.org/10.1007/s10444-023-10096-2

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