In this section, we briefly review the concept of moment-matching discussed in [11, 12] for quadratic-bilinear systems. Before going into the details of nonlinear moment-matching, we begin with the structure of high-order transfer functions.

### Multivariate transfer functions

The input-output representation for single input quadratic-bilinear systems can be expressed by the Volterra series expansion of the output *y*(*t*) with quantities analogous to the standard convolution operator. That is,

$$\begin{aligned} \begin{aligned} y(t) = \sum _{k=1}^{\infty }\int _{0}^{t}\!\!\int _{0}^{t_1}\!\!\!\! \cdots \int _{0}^{t_{k-1}}h_k(t_1,\ldots ,t_k)u(t-t_1)\cdots u(t-t_k)dt_k\cdots dt_1, \end{aligned} \end{aligned}$$

(5)

where it is assumed that the input signal is one-sided, i.e., \(u(t)=0\) for \(t<0\). In addition, each of the generalized impulse responses, \(h_k(t_1,\ldots ,t_k)\), also called the *k*-dimensional kernel of the subsystem, is assumed to be one-sided. In terms of the multivariate Laplace transform, the *k*-dimensional subsystem can be represented as,

$$\begin{aligned} Y_k(s_1,\ldots ,s_k) = H_k(s_1,\ldots ,s_k)U(s_1)\cdots U(s_k), \end{aligned}$$

(6)

where \(H_k(s_1,\ldots ,s_k)\) is the multivariate transfer function of the *k*-dimensional subsystem. The generalized transfer functions in the output expression (6) are in the so-called triangular form [13]. We denote the *k*-dimensional triangular form by \(H_{tri}^{[k]}(s_1,\ldots ,s_k)\). There are some other useful forms such as the symmetric form and the regular form of the multivariate transfer functions as discussed in [13]. The triangular form is related to the symmetric form by the following expression

$$\begin{aligned} H_{sym}^{[k]}(s_1,\ldots ,s_k)=\frac{1}{n!}\sum _{\pi (\cdot )}H_{tri}^{[k]}(s_{\pi (1)}, \ldots ,s_{\pi (k)}), \end{aligned}$$

(7)

where the summation includes all *k*! permutations of \(s_1,\ldots ,s_k\). Also, the triangular form can be connected to the regular form of the transfer function by using

$$\begin{aligned} H_{tri}^{[k]}(s_1,\ldots ,s_k)=H_{reg}^{[k]}(s_1,s_1+s_2,\ldots ,s_1+s_2+\cdots +s_k). \end{aligned}$$

(8)

According to [13], the structure of the generalized symmetric transfer functions can be identified by the growing exponential approach. The structure of these symmetric transfer functions for the first two subsystems of the quadratic-bilinear system (1) can be written as

$$\begin{aligned} \begin{aligned} H_1(s_1)&=C(s_1E-A)^{-1}B,\\ H_2(s_1,s_2)&=C((s_1+s_2)E-A)^{-1}B(s_1,s_2), \end{aligned} \end{aligned}$$

(9)

where

$$\begin{aligned} \begin{aligned}&B(s_1,s_2) = :Q(x_1(s_1)\otimes x_1(s_2))+\frac{1}{2!}N(x_1(s_1)+x_1(s_2)), \end{aligned} \end{aligned}$$

(10)

in which \(x_1(s):=(sE-A)^{-1}B\). Defining \(x_2(s_1,s_2):= ((s_1+s_2)E-A)^{-1}B(s_1,s_2)\), the first two (first- and second-order) symmetric transfer functions can be written as

$$\begin{aligned} \begin{aligned} H_1(s_1)&=Cx_1(s_1),\\ H_2(s_1,s_2)&=Cx_2(s_1,s_2). \end{aligned} \end{aligned}$$

(11)

Before going into the partial differentiation of these multivariate transfer functions, we introduce the concept of matricization. The process of reshaping a tensor into a matrix is called matricization. In [11], the matrix \(Q\in \mathbb {R}^{n\times n^2}\) is considered as the mode-1 matricization of a 3 dimensional tensor \(\mathcal {Q}\in \mathbb {R}^{n\times n\times n}\). The \(n\times n\) components of *Q* are the frontal slices \(\mathcal {Q}_i \in \mathbb {R}^{n\times n}\), \(i=1,\ldots ,n\) of the tensor \(\mathcal {Q}\), i.e. \(Q = \begin{bmatrix} \mathcal {Q}_1&\cdots&\mathcal {Q}_n\end{bmatrix}\). The mode-2 and mode-3 matricizations can be defined as

$$\begin{aligned} \begin{aligned} Q^{(2)}&= \begin{bmatrix} \mathcal {Q}_1^T&\cdots&\mathcal {Q}_n^T\end{bmatrix},\\ Q^{(3)}&= \begin{bmatrix} vec(\mathcal {Q}_1)&\cdots&vec(\mathcal {Q}_n)\end{bmatrix}^T. \end{aligned} \end{aligned}$$

Note that the concept of matricization allows us to symmetrize *Q* to \(\tilde{Q}\) so that \(Q(x\otimes x) = \tilde{Q}(x\otimes x)\) holds and the commutativity property \(\tilde{Q}(u\otimes v)=\tilde{Q}(v\otimes u)\) for arbitrary choices of \(u,v\in \mathbb {R}^n\) is enforced. In addition, the property

$$\begin{aligned} w^TQ(u\otimes v)=u^TQ^{(2)}(v\otimes w), \end{aligned}$$

(12)

also holds, where \(w,u,v\in \mathbb {R}^{n}\) are arbitrary and *Q* is assumed to be in the symmetrized form, see [23]. Let \(G(s):= sE-A\), then by using

$$\begin{aligned} \frac{\partial G(s)^{-1} }{\partial s}= -G(s)^{-1}\frac{\partial G(s)}{\partial s} G(s)^{-1}, \end{aligned}$$

and (12), we have

$$\begin{aligned} \begin{aligned} \frac{\partial H_2(s_1,s_2) }{\partial s_1}= -y_1(s_1+s_2)^{T}Ex_2(s_1,s_2)-x_1(s_1)^TE^Ty_2(s_1,s_2), \end{aligned} \end{aligned}$$

(13)

where \(y_1(s):= (sE-A)^{-T}C^T\) and \(y_2(s_1, s_2):= (s_1E-A)^{-T}C(s_1,s_2)^T\) in which

$$\begin{aligned} C(s_1,s_2) = Q^{(2)}\big (x_1(s_2)\otimes y_1(s_1+s_2)\big )+\frac{1}{2!}N^Ty_1(s_1+s_2). \end{aligned}$$

Similarly

$$\begin{aligned} \begin{aligned} \frac{\partial H_2(s_1,s_2) }{\partial s_2}= -y_1(s_1+s_2)^{T}Ex_2(s_1,s_2)-x_1(s_2)^TE^Ty_2(s_2,s_1). \end{aligned} \end{aligned}$$

(14)

Notice that when \(s_1=s_2=\sigma \), the two partial differentiations are the same. This condition on interpolation points is assumed in [11] to show the moment-matching properties of the ROM. In the following, we show moment-matching in the multivariate settings when \(s_1\ne s_2\) (\(s_1=\sigma _{1i}\) and \(s_2=\sigma _{2i}\)).

### Moment-matching for QBDAE

The goal of a moment-matching based reduction approach is to ensure that the high-order transfer functions are well approximated. In case of symmetric transfer functions, we can represent it as

$$\begin{aligned} H_k(s_1,\ldots ,s_k)\approx \hat{H}_k(s_1,\ldots ,s_k), \quad \text{ for } k=1,\ldots ,K, \end{aligned}$$

(15)

with \(\hat{H}_k(s_1,\ldots ,s_k)\) being the k-th order multivariate transfer function of the reduced system (2). With the task in (15) achieved for some *K*, we can expect that the output *y*(*t*) is well approximated by \(\hat{y}(t)\). To get recursive relations between vectors for approximation subspaces, it is assumed in [11] that \(s_1=s_2=\sigma \). With these settings, the second-order transfer function becomes

$$\begin{aligned} H_2(\sigma ,\sigma )= y(2\sigma )^T\Big ( Q \left( x_1(\sigma )\otimes x_1(\sigma )\right) + Nx_1(\sigma )\Big ). \end{aligned}$$

The following Lemma summarizes the result introduced in [11].

### Lemma 1

Let \(\sigma _i\in \mathbb {C}\) be the interpolation points and \(\sigma _i\notin \{\Lambda (A,E), \Lambda (A_r,E_r)\}\), where \(\Lambda (A,E)\) represents the generalized eigenvalues of the matrix pencil \(\lambda E-A\). Assume that \(\hat{E}=W^TEV\) is nonsingular and \(\hat{A}\), \(\hat{Q}\), \(\hat{N}\), \(\hat{B}\), \(\hat{C}\) are as in (4) with full rank matrices \(V,W\in \mathbb {R}^{n{\times }r}\) such that

$$\begin{aligned} \begin{aligned}&\textrm{span}(V)=\textrm{span}_{i=1,\ldots ,k}\{x_1(\sigma _i), ~x_2(\sigma _i,\sigma _i)\}, \\&\textrm{span}(W)=\textrm{span}_{i=1,\ldots ,k} \{y_1(2\sigma _i),~y_2(\sigma _i,\sigma _i)\}, \end{aligned} \end{aligned}$$

then the reduced QBDAE satisfies the following (Hermite) interpolation conditions:

$$\begin{aligned} \begin{aligned} H_1(\sigma _i)&=\hat{H}_1(\sigma _i), \qquad \quad H_1(2\sigma _i)=\hat{H}_1(2\sigma _i),\\ H_2(\sigma _i,\sigma _i)&=\hat{H}_2(\sigma _i,\sigma _i), \quad \frac{\partial }{\partial s_j}H_2(\sigma _i,\sigma _i)=\frac{\partial }{\partial s_j}\hat{H}_2(\sigma _i,\sigma _i),~~ j=1,2. \end{aligned} \end{aligned}$$

See [11] for a proof. Next, we present moment-matching properties in the multivariable settings, where \(s_1\ne s_2\).

### Lemma 2

Let \(\sigma _{1i},\sigma _{2i}\in \mathbb {C}\) with \(\sigma _{1i},\sigma _{2i}\notin \{\Lambda (A,E), \Lambda (A_r,E_r)\}\). Assume that \(\hat{E}=W^TEV\) is nonsingular and \(\hat{A}\), \(\hat{Q}\), \(\hat{N}\), \(\hat{B}\), \(\hat{C}\) are as in (4) with full rank matrices \(V,W\in \mathbb {R}^{n{\times }r}\) such that

$$\begin{aligned} \begin{aligned}&\textrm{span}(V)=\textrm{span}_{i=1,\ldots ,k}\{x_1(\sigma _{1i}), ~x_1(\sigma _{2i}),~x_2(\sigma _{1i},\sigma _{2i})\}\\&\textrm{span}(W)=\textrm{span}_{i=1,\ldots ,k} \{y_1(\sigma _{1i}+\sigma _{2i}),~y_2(\sigma _{1i},\sigma _{2i}),~y_2(\sigma _{2i},\sigma _{1i})\}. \end{aligned} \end{aligned}$$

Then the reduced QBDAE satisfies the following (Hermite) interpolation conditions:

$$\begin{aligned} \begin{aligned} H_1(&\sigma _{1i})=\hat{H}_1(\sigma _{1i}), \quad H_1(\sigma _{2i})=\hat{H}_1(\sigma _{2i}), \quad H_1(\sigma _{1i}+\sigma _{2i})=\hat{H}_1(\sigma _{1i}+\sigma _{2i}),\\&H_2(\sigma _{1i},\sigma _{2i})=\hat{H}_2(\sigma _{1i},\sigma _{2i}), \quad \frac{\partial }{\partial s_1}H_2(\sigma _{1i},\sigma _{2i})=\frac{\partial }{\partial s_1}\hat{H}_2(\sigma _{1i},\sigma _{2i}),\\&\qquad \frac{\partial }{\partial s_2}H_2(\sigma _{2i},\sigma _{1i})=\frac{\partial }{\partial s_2}\hat{H}_2(\sigma _{2i},\sigma _{1i}). \end{aligned} \end{aligned}$$

The proof of the statement is similar to Lemma 1 and therefore omitted. Note that the statement in Lemma 2 reduces to Lemma 1, if \(\sigma _{1i}=\sigma _{2i}\). In the remaining part of the paper, our goal is to identify a good choice of the interpolation points \(\sigma _{1i}\) and \(\sigma _{2i}\).