 Research article
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Multivariate momentmatching for model order reduction of quadraticbilinear systems using error bounds
Advanced Modeling and Simulation in Engineering Sciences volume 9, Article number: 23 (2022)
Abstract
We propose an adaptive momentmatching framework for model order reduction of quadraticbilinear systems. In this framework, an important issue is the selection of those shift frequencies where momentmatching is to be achieved. So far, the choice often has been random or linked to the linear part of the nonlinear system. In this paper, we extend the use of an existing a posteriori error bound for general linear time invariant systems to quadraticbilinear systems and develop a greedytype framework to select a good choice of interpolation points for the construction of the projection matrices. The results are compared with standard quadraticbilinear projection methods and we observe that the approximations obtained by the proposed method yield high accuracy.
Introduction
There are different applications where the dynamics of the system can be represented by quadraticbilinear differential algebraic equations (QBDAEs). These include simulation of distribution networks [1], fluid flow problems [2] and nonlinear VLSI circuits [3, 4]. In addition, a large class of nonlinear systems can be written in quadraticbilinear form by using liftings to higherdimensional statespaces [4]. Most of these applications involve a large number of equations i.e., a highdimensional statespace. This makes simulation, control and optimization computationally inefficient. A remedy to this issue is the use of model order reduction (MOR).
We consider the problem of MOR for a singleinput singleoutput quadraticbilinear descriptor system of the form:
where \(E, ~\!A, ~\!N\in \mathbb {R}^{n\times n}\), \(Q\in \mathbb {R}^{n\times n^2}\), \(B,~\!C^T\in \mathbb {R}^{n}\) are the coefficient matrices and vectors. \(x(t)\in \mathbb {R}^n\) is the state vector and \(u(t),~\!y(t)\in \mathbb {R}\) are the input and output of the system. The matrix E may or may not be singular but the pencil is assumed to be regular, i.e., \(\lambda EA\) is singular only for finitely many values \(\lambda \in \mathbb {C}\) [5].
The goal of MOR is to construct a reduced system of dimension \(r\ll n\):
with the output response \(y_r(t)\) approximately equal to y(t). In case of linear systems (where Q and N are zero matrices), there are various techniques in the literature to compute reducedorder models (ROMs), cf., [6,7,8]. Among these methods, projectionbased momentmatching methods [9, 10] are well used and have been extended to quadraticbilinear systems [4, 11, 12]. Projection involves approximating the state vector x(t) in an rdimensional subspace spanned by the column vectors of \(V\in \mathbb {R}^{n{\times }r}\), so that the residual in the state equation is orthogonal to another rdimensional subspace spanned by the column vectors of \(W\in \mathbb {R}^{n{\times }r}\). That is, we approximate \(x(t)\approx Vx_r(t)\) such that the following PetrovGalerkin orthogonality condition holds:
If \(W=V\), the projection is orthogonal and is often called onesided projection, otherwise it is oblique and is called twosided projection. The oblique projection framework leads to a set of reduced system matrices of the form:
In case of linear systems, a suitable choice of the basis matrices V and W implicitly ensures momentmatching, where moments are the coefficients of the series expansion of the transfer function at some predefined shift frequencies. Thus for projectionbased momentmatching, the choice of V and W is related to the transfer function of the system. However, nonlinear systems have no universal inputoutput representation. For some classes of nonlinear systems, though, including QBDAE systems, it is possible to generalise the transfer function concept by utilising the Volterra theory [13], where the inputoutput relationship is represented by a set of highorder transfer functions. This makes the concept of momentmatching more complex in the nonlinear case, since the structure of the basis matrices V and W in (4) now depends on multiple highorder transfer functions. To achieve momentmatching, simplifications are often made in the literature [4, 11] for computing the ROMs. For example, [11] constructs V and W such that the reduced system matches the moments of the first and secondorder transfer functions. In [12], simplified forms of highorder transfer functions are derived, which also enable the projectionbased techniques to match moments of highorder transfer functions. In addition, all the existing momentmatching/interpolation approaches [4, 11, 12] are based on the simplification that the interpolation points for each frequency variable are the same. We discuss these results further in “Background”.
Recently, a new framework [14] for quadraticbilinear systems has been proposed that is based on generalized Sylvestertype matrix equations. The approach involves truncated solution of two complex matrix equations to identify a good choice for the basis matrices V and W. Another approach is the extension of the Loewner framework from linear/bilinear systems [15, 16] to quadratic bilinear systems [17]. Also, an indirect approach for MOR of QBDAE systems is proposed in [18], where the basis matrices are constructed from the bilinear part of the quadraticbilinear system. In [19], the bilinear part of the system is viewed as a linear parametric system and an a posteriori error bound is used to select the interpolation points and construct the basis matrices adaptively. All these techniques are using the first two or three highorder transfer functions and their structure is different from the one identified in [11]. Recently, a new direction has been explored in [20] where the properties of higher order moment matching are analysed so that the nonlinear matrices are not used in the construction of basis matrices. Also, in [21] the idea of signal generator driven system is used so that univariate momentmatching can be utilised for model order reduction. These approaches are comparing their results with [11] and [19] and for some benchmark examples there behaviour is comparable. However, our target is general momentmatching for QBDAEs, so we will mainly focus on the twosided momentmatching technique from [11].
In this paper, we identify a good choice of interpolation points for quadraticbilinear systems by utilizing a greedy type framework based on error bounds for quadraticbilinear systems motivated by the recently proposed error bound for linear parametric systems in [22]. Here, we relax the restriction of using the same interpolation points for different frequency variables. The approach starts from some initial interpolation points that are iteratively updated to identify a set of interpolation points corresponding to the maximal values of certain error bounds. For each choice of interpolation points, we interpolate, not only, the original transfer function and its first derivative but also higher derivatives, so that the quadraticbilinear system is well approximated. The iteration stops when the approximation error is less than the prescribed tolerance level. Each iteration contributes to constructing a better set of basis matrices V and W, until a given error tolerance is achieved. The main difference from the work in [19] is that the quadratic part of the system is also involved in the basis construction in the proposed framework based on an a posteriori error bound for quadraticbilinear systems, whereas only the bilinear part is considered for the basis matrix computation in [19]. The error estimator used in [19] only estimates the error of the linearbilinear part.
The remaining part of the paper is organized as follows. “Background” reviews the existing projection based momentmatching techniques for quadraticbilinear systems. “Error bound for QBDAEs” presents the error bound expressions for quadraticbilinear systems and “Interpolation points using error bounds” utilises these error bounds in a greedytype algorithm to select interpolation points. Finally in “Numerical results”, numerical results are shown for some benchmark examples.
Background
In this section, we briefly review the concept of momentmatching discussed in [11, 12] for quadraticbilinear systems. Before going into the details of nonlinear momentmatching, we begin with the structure of highorder transfer functions.
Multivariate transfer functions
The inputoutput representation for single input quadraticbilinear systems can be expressed by the Volterra series expansion of the output y(t) with quantities analogous to the standard convolution operator. That is,
where it is assumed that the input signal is onesided, 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 kdimensional kernel of the subsystem, is assumed to be onesided. In terms of the multivariate Laplace transform, the kdimensional subsystem can be represented as,
where \(H_k(s_1,\ldots ,s_k)\) is the multivariate transfer function of the kdimensional subsystem. The generalized transfer functions in the output expression (6) are in the socalled triangular form [13]. We denote the kdimensional 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
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
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 quadraticbilinear system (1) can be written as
where
in which \(x_1(s):=(sEA)^{1}B\). Defining \(x_2(s_1,s_2):= ((s_1+s_2)EA)^{1}B(s_1,s_2)\), the first two (first and secondorder) symmetric transfer functions can be written as
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 mode1 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 mode2 and mode3 matricizations can be defined as
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
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 \(G(s):= sEA\), then by using
and (12), we have
where \(y_1(s):= (sEA)^{T}C^T\) and \(y_2(s_1, s_2):= (s_1EA)^{T}C(s_1,s_2)^T\) in which
Similarly
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 momentmatching properties of the ROM. In the following, we show momentmatching in the multivariate settings when \(s_1\ne s_2\) (\(s_1=\sigma _{1i}\) and \(s_2=\sigma _{2i}\)).
Momentmatching for QBDAE
The goal of a momentmatching based reduction approach is to ensure that the highorder transfer functions are well approximated. In case of symmetric transfer functions, we can represent it as
with \(\hat{H}_k(s_1,\ldots ,s_k)\) being the kth 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 secondorder transfer function becomes
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 \(\lambda EA\). 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
then the reduced QBDAE satisfies the following (Hermite) interpolation conditions:
See [11] for a proof. Next, we present momentmatching 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
Then the reduced QBDAE satisfies the following (Hermite) interpolation conditions:
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 \(\sigma _{2i}\).
Error bound for QBDAEs
In this section, we show how the error bound expression, derived initially in [22] for parametric linear time invariant systems, can be extended to quadraticbilinear DAEs. We begin with a brief overview of the error bound for the first subsystem, as in [22] and then discuss the extension to the second subsystem of the QBDAE (1).
Error bound for \(H_1(s_1)\)
Here the error bound provides an estimate for the error between \(H_1(s_1)\) and \(\hat{H}_1(s_1)\). To this end, we define the primal and the dual systems as:
respectively, where T denotes the transpose of a matrix. The error bound is constructed so that it is based on two residuals, which result from MOR of the primal and the dual system, respectively. The primal system is reduced using the matrix pair \(V_{1}\) and \(W_{1}\),
where
As a result, the reduced primal system is,
where \(\hat{E}_1= W_1^T EV_1\), \(\hat{A}_1= W_1^T AV_1\), \(\hat{B}_1= W_1^T B\) and \(\hat{C}_1= CV_1\). Here \(\hat{x}_1(s_1):=V_1z_1(s_1)\) is the approximation of \(x_1(s_1)\). Due to the dual relation between (16) and (17), the dual system can be reduced by using \(V_1^{du} = W_{1}\) and \(W_{1}^{du} = V_1\). The reduced dual system is
where \(\tilde{E}_1= V_1^T EW_1\), \(\tilde{A}_1= V_1^T AW_1\), \(\tilde{C}_1= W_1^T C^T\). Also \(\tilde{x}_1^{du}(s_1):=W_1z_1^{du}(s_1)\) is the approximation of \(x_1^{du}(s_1)\). The residuals associated with the reduction of the primal and the dual systems can be written as
With these quantities, the following result provides an a posteriori upper bound on the approximation error, \(H_1(s_1)\hat{H}_1(s_1)\):
Theorem 1
[22] The upper bound on the approximation of the transfer function \(H_1(s_1)=C(s_1EA)^{1}B\) can be written as \(H_1(s_1)\hat{H}_1(s_1) \le \Delta _1(s_1)\), where
in which \(\beta _1(s_1)=\sigma _{\min } (G(s_1))\), where \(\sigma _{\min }\) indicates the smallest singular value of \(G(s_1)\).
Error bound for \(H_2(s_1,s_2)\)
Analogous to \(H_1(s_1)\), we define the primal and dual systems as:
respectively. The interpolation points for \(H_1(s_1)\) can be identified through the error bound \(\Delta _1(s_1)\) by using a greedy framework as presented in [22]. This means that we can select \(\sigma _{1i}\) for \(i=1,\ldots ,r\) as the interpolation points corresponding to the maximal values of the error bound at subsequent iterations of the greedy algorithm in [22]. With these interpolation points fixed for \(s_1\), we can also express the error bound for the second subsystem. The error bound is constructed based on two residuals, which result from MOR of the primal and the dual systems in (21) (22), respectively. The primal system is reduced using the matrix pair \(V_{2}\) and \(W_{2}\), where
As a result, the reduced primal system is
where \(\hat{E}_2= W_2^T EV_2\), \(\hat{A}_2= W_2^T AV_2\), \(\hat{B}(s_1,s_2)= W_2^T B(s_1,s_2)\) and \(\hat{C}_2= CV_2\). Similarly, the dual system is reduced using the matrix pair \(V_{2}^{du}\) and \(W_{2}^{du}\),
The reduced dual system is
where \(\tilde{E}_2= (W_2^{du})^T EV_2^{du}\), \(\tilde{A}_2= (W_2^{du})^T AV_2^{du}\), \(\tilde{C}^T_2= (V_2^{du})^T C^T\). The residuals associated with the reduction of the primal and dual systems can be written as
With these quantities, the following result provides an a posteriori upper bound on the approximation error, \(H_2(s_1,s_2)\hat{H}_2(s_1,s_2)\):
Theorem 2
The upper bound on the approximation of
can be written as \(H_2(s_1,s_2)\hat{H}_2(s_1,s_2) \le \Delta _2(s_1,s_2)\),
where
in which \(\beta _2(s_1,s_2)=\sigma _{\min } (G(s_1+s_2))\), where \(\sigma _{\min }\) indicates the smallest singular value of \(G(s_1+s_2)=(s_1+s_2)EA\).
The proof is similar to Theorem 1 and therefore is omitted.
Interpolation points using error bounds
As discussed in “Background”, the projection matrices V and W defined in Lemma 2 require a good choice of interpolation points \(\sigma _{1i}\) and \(\sigma _{2i}\) which also serve as interpolation points for MOR of the primal and dual systems in (16)(17) and (21)(22). In this section, we show the use of the error bound expressions derived previously to select the interpolation points.
The idea is to identify interpolation points corresponding to the maximal bound \(\Delta _1 (s_1)\). Assuming that \(\sigma _{1i}\) are the selected interpolation points for \(s_1\), the remaining interpolation points for \(s_2\) correspond to the maximal bound \(\Delta _2(\sigma _{1i},s_2)\) for each value of \(\sigma _{1i}\). In this way, the error bound can be used iteratively to select a good choice of interpolation points in a predefined sample space, starting from an initial choice of sigma’s. The sample spaces \(S_1\) and \(S_2\) can be arbitrarily selected with some fixed size. One possible choice is to use the \(\mathcal {H}_2\)(sub)optimal interpolation points obtained from IRKA applied to the linear part of (1) and some other random interpolation points in the complex plane around IRKA points. The selected interpolation points are then used to construct and update the required basis matrices V and W, by using the multimomentmatching technique described before. It is interesting to see that although we need to construct the ROMs for the primal and the dual systems in (16), (17) and (21), (22), the projection matrices for those ROMs are obtained without extra computations, since \(V_1, W_1\) and \(V_2, W_2\) are part of V, W by definition. Therefore, V, W can be obtained by orthogonalizing \(V_1\) with \(V_2\) and \(W_1\) with \(W_2\) as indicated in Step of Algorithm 1, where a greedy framework for selecting interpolation points is presented. For an initial pair of interpolation points, the ROMs of the primal and the dual systems in (16), (17) and (21), (22) are constructed and the error bounds \(\Delta _1, \Delta _2\) are computed. A new pair is selected such that the corresponding error bounds \(\Delta _1\) and \(\Delta _2\) are maximized at these points. With the selected interpolation points, we enrich the projection matrices V, W for MOR of the original quadraticbilinear system iteratively during the greedy algorithm. Finally, the reduced quadratic bilinear system is constructed using V, W that are derived upon convergence of Algorithm 1. Algorithm 1 stops when \(\Delta :=\Delta _1+\Delta _2\) is below the tolerance \(\epsilon _{tol}\), where \(\Delta \) includes the errors introduced by approximating the first and second transfer functions. Since the interpolation points are selected according to the error bounds \(\Delta _1\) and \(\Delta _2\), it is important that the error bounds dynamically reflect the decay of the true error with the iteration of the greedy algorithm. Ideally, the error bounds should be very close to the true error. Numerical tests in the next section show that the error bounds really control the true error robustly.
Numerical results
We consider three benchmark examples for our results on MOR of QBDAE systems. The results are compared with the onesided and twosided projection methods, where the interpolation points are computed by IRKA, implemented on the linear part of the system. We represent the proposed method by 1s/2sgreedy (onesided/twosided projection with greedy based interpolation points) and the method from literature by 1s/2sIRKA (onesided/two sided projection with IRKA interpolation points). The use of IRKA on the linear part of the QBDAE system on convergence results in IRKA interpolation points which in the greedy framework is used to define the initial guess of the optimal points. The Max. True Error in the tables is defined as \(\max \limits _{s_1, s_2 \in S_2} H_1(s_1)\hat{H}_1(s_1)+H_2(s_1,s_2)\hat{H}_2(s_1,s_2)\) and the Max. Est. Error is \(\max \limits _{s_1, s_2 \in S_2} \Delta (s_1, s_2)\).
Nonlinear RC circuit
The nonlinear RC circuit was first considered in [24] and since then it has been used in many papers for nonlinear MOR [5]. Consider the voltage v and the current function g(v). Then the IV characteristics can be represented as: \(g(v)= e^{40v} + v 1\). The nonlinearity in the current function results in a nonlinear model. All the capacitances are fixed to \(C=1\). Figure 1 shows the complete circuit.
It is shown in [4] that the nonlinearity in the RC circuit can be written in quadraticbilinear form as in (1) by introducing some auxiliary variables. The transformation is exact, but the dimension of the system increases to \(n =2\cdot l\), where l represents the number of nodes in Figure 1, and it is also the dimension of the original nonlinear system.
For our results, we set \(l=50\), so \(n=100\) and use twosided projection to reduce the system. Table 1 shows the results with tolerance \(\epsilon _{tol}=1e^{5}\) and an initial choice of interpolation points as \(\sigma _{1}=\sigma _{20}=119.5642\).
The second column of Table 1 shows interpolation points that are identified by the greedy framework and are based on the error bound. It is clear that the error bound tightly catches the true error and can be used as a surrogate of the true error to select the interpolation points. The size of the ROM obtained from both approaches has been kept the same i.e. \(r_1 = r_2 = 12\). For the input \(u(t) = e^{t}\), the output of the original model and ROMs along with corresponding relative errors are shown in Fig. 2.
Figure 2a shows the comparison of transient response of the two approaches, while Fig. 2b plots relative errors of the two approaches. It is clearly seen that 1sgreedy and 2sgreedy outperform 1sIRKA and 2sIRKA, respectively, in terms of accuracy.
Burgers’ equation
In nonlinear MOR, the 1D Burgers’ equation is commonly used as a benchmark [2, 11]. The mathematical model of the 1D Burgers’ equation with \(\Gamma = (0,1) \times (1,T)\) is:
We use it as an example to test our proposed method. We keep the size of the original model as \(n = 1000\). Table 2 shows our results with tolerance \(\epsilon _{tol}=1e^{4}\) and an initial choice of interpolation points as \(\sigma _{10} = \sigma _{20} = 5.4124\).
The second column of the table shows the interpolation points that are based on the error bound and identified by the greedy framework. Similarly, the error bound again tightly bounds the true error and therefore is reliable for choosing the interpolation points in the greedy algorithm. The sizes of the ROMs obtained from both approaches are the same, i.e. \(r_1 = r_2 = 16\). The ROMs constructed from IRKA interpolation points and the proposed framework are shown in Fig. 3 for input \(u(t) = cos(\pi t)\).
Figure 3a shows the transient responses of the Burgers’ equation computed from simulating the original model and the two different MOR approaches, while Fig. 3b compares the absolute response errors of the ROMs derived using the two approaches. The absolute error of the ROM constructed using the proposed methodology of choosing interpolation points is less than that of the ROM constructed using IRKA interpolation points, especially for the twosided projection.
FitzHugh–Nagumo system
We use the FitzHugh–Nagumo system as our third example to check our results. The FitzHugh–Nagumo system can be represented as [14]:
with \(f(\upsilon ) = \upsilon (\upsilon 0.1)(1\upsilon )\) and boundary conditions
Here, we choose \(\epsilon = 0.015\), \(h=0.5\), \(\gamma = 0.05\) and \(i_0(t) = 5 \times 10^4 t^3 e^{15t}\). When standard finite difference method is applied to numerically discretize the PDEs in (28), a system of ODEs with cubic nonlinearities is obtained. We can get a quadraticbilinear system by introducing new variables. For an original discretized system with size \(\bar{n}\), a quadraticbilinear system has the size of \(n = 3\bar{n}\). We set \(\bar{n} = 100\), which gives rise to quadraticbilinear system of order \(n = 300\). Then we choose interpolation points using the proposed greedy framework to construct a ROM of size \(r = 26\) and then compare it with the ROM of the same size, which is constructed from the interpolation points using IRKA. Table 3 shows our results with tolerance \(\epsilon _{tol}=1e^{6}\) and the interpolation points \(\sigma _{10}=\sigma _{20}=534.69\).
Table 3 shows the interpolation points that are selected by the error bound and the decay of the true error and the error bound at each iteration of the greedy algorithm. The error bound once more estimates the true error accurately, implicating that the selected interpolation points indeed nearly correspond to the largest error. The sizes of the ROMs obtained from both approaches are the same, i.e. \(r_1 = r_2 = 26\). Figure 4 shows the transient responses of the FitzHugh–Nagumo system computed from simulating the original model and two approaches.
The input signal is \(u(t) =50000t^3 e^{15t}\). It is seen that the 1sgreedy performs better than the 1sIRKA when the outputs in both cases are compared with that of the original model; however, 2sgreedy and 2sIRKA produce unstable responses.
Conclusions
In this paper, the proposed methodology of choosing interpolation points for construction of ROM of the first and secondorder transfer functions of quadraticbilinear systems has been tested for three different models. The results have also been compared with ROMs of the same size constructed using the interpolation points chosen by linear IRKA. In each case, the ROMs constructed using interpolation points from the greedy framework yield better approximation of the output than the ROMs constructed from IRKA.
Data availability statement
The model data of FitzHugh–Nagumo System and RC ladder is available at MOR https://morwiki.mpimagdeburg.mpg.de/morwiki/index.php/Category:Benchmark.
Abbreviations
 MOR:

Model order reduction
 ODE:

Ordinary differential equation
 IRKA:

Iterative rational krylov algorithm
 ROM:

Reduced ordered model
 PDE:

Partial differential equation
 QBDAE:

Quadratice blinear differential algebraic equation
 VLSI:

Very large scale integration
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Muhammad Altaf Khattak and Mian Ilyas Ahmad are supported by HEC Pakistan under NRPU Project ID 10176.
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This research is funded by HEC, Pakistan under NRPU Project ID 10176.
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Khattak, M.A., Ahmad, M.I., Feng, L. et al. Multivariate momentmatching for model order reduction of quadraticbilinear systems using error bounds. Adv. Model. and Simul. in Eng. Sci. 9, 23 (2022). https://doi.org/10.1186/s40323022002366
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DOI: https://doi.org/10.1186/s40323022002366