Model order reduction for real-time hybrid simulation: Comparing polynomial chaos expansion and neural network methods

Hybrid simulation is used to investigate the dynamic response of a system by combining numerical and physical substructures. To ensure high fidelity results, it is necessary to conduct hybrid simulation in real-time. One challenge in real-time hybrid simulation originates from high-dimensional nonlinear numerical substructures and, in particular, from the computational cost linked to the accurate computation of their dynamic responses. When the computation takes longer than the actual simulation time, time delays are introduced distorting the simulation timescale. In such cases, the only viable solution for performing hybrid simulation in real-time is to reduce the order of such complex numerical substructures. In this study, a model order reduction framework is proposed for real-time hybrid simulation, based on polynomial chaos expansion and feedforward neural networks. A parametric case study is used to validate the framework. Selected numerical substructures are substituted with their respective reduced-order models. To determine the framework’s robustness, parameter sets are defined covering the design space of interest. Comparisons between the full-and reduced-order hybrid model response are delivered. The attained results demonstrate the performance of the proposed framework.


Introduction
Hybrid simulation (HS) is an experimental co-simulation method used to investigate the dynamic response of a system subjected to a realistic loading scenario [1].Across different engineering disciplines, HS is also known as system-in-the-loop testing.It is one of the methods used for Model-based Development [2].The system-under-consideration consists of multiple individual sub-components, out of which one or more are selected to be tested physically, and are called the physical substructures (PS), while the remaining components, called the numerical substructures (NS), are simulated numerically.The coupled PS and NS constitute the so-called hybrid model.In more detail, the hybrid model dynamic response is evaluated online, based on a step-by-step numerical solution of the governing equations of motion of the hybrid model, combined with experimental responses obtained from the PS.The coupling of NS and PS is achieved by an arrangement of transfer systems, such as servo-controlled motors and actuators.These transfer systems are responsible for synchronizing the dynamic boundary conditions of the substructures at their interfaces throughout the entire HS.The advantage of HS lies in the fact that it can be used to examine the inner workings of a system's sub-component, beyond its linear regime, without testing the entire full-scale system.As a consequence, realistic results from the tested component are obtained at a much lower experimentation cost.A comprehensive review of HS methods and algorithms can be found in [3] and references therein.
Frequently, HS is performed on a distorted timescale, e.g. by slowing down the rate of PS testing to accommodate the power or speed of the transfer system in use.These HSs correspond to the so-called pseudodynamic tests [4].However, in order to increase the fidelity and trust in HS outcomes, HS should be conducted as close to real-time as possible.This is especially important for loading-rate-sensitive substructures.Even though real-time hybrid simulation (RTHS) has many advantages, it comes with numerous challenges.The first challenge originates from the inherent dynamics of the transfer system exciting the PS, as it introduces time delays in the overall HS that alter the dynamic response of the tested hybrid model.This problem is addressed by designing control systems to adequately compensate for these delays.Several control strategies for RTHS have been proposed and are available in the literature.Some of the recently developed novel approaches can be found in [5][6][7][8][9].Nonetheless, challenges in RTHS do not arise only from the transfer system of the PS but also from the NS side, such as from the computational power needed to compute the NS responses.In order to capture the dynamic behavior of interest of a high-dimensional nonlinear numerical model, it is often necessary to use small time-steps for the numerical solver.However, the smaller the time-step of the simulation, the larger the computational power needed to compute it.In RTHS, when the required computation time becomes longer than the actual simulation time, time delays are introduced to the hybrid model and accumulate throughout the duration of the HS, risking not only to decrease the fidelity of the HS results but also to drive the overall HS into instability.Therefore, in such cases, the only viable solution is to reduce the order of the NS to be able to conduct the HS in real-time.
Several model order reduction (MOR) techniques can be found in the literature.Proper orthogonal decomposition (POD) [10][11][12], also known as principal component analysis (PCA), and its discrete version, namely the singular value decomposition (SVD), are mostly combined with the method of snapshots [13,14] and have been extensively used for extraction of mode shapes from high-dimensional systems, among other techniques.A technique similar to POD, also using the method of snapshots but following a data-based instead of a model-based approach, is the dynamic mode decomposition (DMD) [15].In particular, the main difference between POD and DMD is that the former relies upon a time-averaged spatial correlation matrix while the latter represents the temporal dynamics with high-degree polynomials [16].An advancement of POD was also proposed based on the use of separated representations, the so-called proper generalized decomposition (PGD) [17].More recently, the component-mode synthesis (CMS) approach [18,19] has been used for non classically damped linear systems [20] as well as in HS as a method to reduce the order of the comprising NS and PS [21].Furthermore, a quadratic manifold [22] and a non-intrusive [23] approach were proposed for MOR of nonlinear structural systems.MOR techniques have also been used for Bayesian finite element model updating [24] and for reliability-based design problems [25].
In this study, a MOR framework originating from a data-driven regression is proposed to reduce the order of high-dimensional nonlinear NS in HS.Two different methodologies for MOR are addressed; a polynomial chaos expansion (PCE) and a feedforward neural network (FFNN).In both cases we treat the NS, whose order we want to reduce, as a black box, meaning that no prior knowledge of its full-order model (FOM) dynamics is required.Instead, we map the outputs of the respective NS to its inputs by simpler-to-evaluate functions than those of the original FOM.The goal of the MOR framework is to adequately capture the dynamic behavior of high-dimensional NS at a much lower computational cost and in shorter computation time, and thus enable RTHS.A parametric case study encompassing a virtual hybrid model is employed to validate the proposed MOR framework.Selected NS are replaced with their respective FFNN-and PCE-based reduced-order models, while specific parameters of the hybrid model are varied to investigate the framework's robustness to different hybrid model configurations.The corresponding reduced-order hybrid model responses are compared to the ones from the FOM.Results demonstrate the effectiveness of the proposed MOR framework.
The paper is organized as follows.Section 2 introduces the MOR framework describing the PCE and FFNN.Section 3 presents the case study with the virtual hybrid model used to validate the framework.Section 4 discusses the obtained results and Section 5 presents the overall conclusions of this study.

Polynomial chaos expansion and neural networks as model order reduction techniques
Considering an input vector  and a computational model  = (), data-driven regression algorithms formulate a map  ∶  ↦  based on an obtained sample set of input points  = {  (1) , … ,  () }  and of the respective output values, i.e., model evaluations,  = {  (1) , … ,  () }  .The set of ,  realizations corresponds to the so-called experimental design (ED).Regression techniques encompass linear regression, kernel methods, neural networks and graphical models among others.An overview of these techniques can be found in [26,27].As mentioned above, the proposed MOR framework of this study explores use of PCE and FFNN to this end, with these methods elaborated in what follows.

Polynomial chaos expansion
PCE is a well-known uncertainty quantification spectral method used to substitute the dynamics of an expensive-to-compute numerical model with an inexpensive-to-compute surrogate (a.k.a.metamodel), representing the outputs of the model by a polynomial function of its inputs [28,29].It is proven to be a powerful surrogate technique used in a wide variety of engineering contexts to replicate the dynamic response of complex high-dimensional models [30][31][32][33][34][35].Hence, it can be seen as a promising technique for MOR of NS in HS.Furthermore, to the best of the authors' knowledge, such use of PCE within the HS context is new.
In more detail, given a random input vector  ∈ R  with statistically independent components expressed by the joint probability density function (PDF)   and a finite variance computational model The PCE function is built on the   () multivariate orthonormal polynomial basis with respect to the input vector   .The degree of the   polynomials components is identified by the  = ( 1 , … ,   ),  ∈ N  multi-index for each of the input variables, while   corresponds to the polynomial coefficients.

Polynomial basis
The multivariate polynomials are constructed as a tensor product of their univariate orthonormal polynomials  ()  (  ) such that   () = ∏  =1  ()   (  ).The latter meet the (  )   (  )   =   orthonormality criteria, where  corresponds to the input variable,  and  to the polynomial degree,    (  ) to the th-input marginal PDF and   to the Kronecker symbol.The selection of the univariate orthonormal polynomial families depends on the marginal PDF of each input variable to which they are orthogonal, e.g., if an input variable follows the uniform/ Gaussian distribution then the Legendre/ Hermite orthogonal polynomial family is used respectively for this specific input variable [29,36].

Truncation schemes
Once the univariate polynomials families are selected for each input variable, the next step is the construction of the PCE following Eq.( 1).However, because the terms of the sum in Eq. ( 1) are infinite, it is often truncated to a finite number of terms for practical reasons.Hence, the truncated basis is defined as  ⊂ N  and the PCE of () admits: The performance of PCE is closely connected with the chosen truncation scheme.Underfitting or overfitting is possible when too many terms are discarded or introduced, respectively [33].In the standard basis truncation scheme [29] the maximum degree  ∈ N + is defined, such that the degree of each polynomial is capped by this value.Therefore, the standard basis truncation scheme consists of ( +  ) elements and follows: where  corresponds to the number of input variables of () and || = ∑  =1   to the total degree of all polynomials in   .Alternate truncation schemes have been developed namely, the maximum interaction and hyperbolic truncation [37], and are used based on the scope of each application.In this study the standard basis truncation scheme of Eq. ( 3) is employed.

Feedforward neural network
Neural networks are an extremely versatile group of methods used both for regression and classification problems.The most straightforward and widely used variety of neural networks are the fully connected FFNN.Such a FFNN consists of layers of ''neurons'', where the value of the neurons in each layer is calculated as a matrix multiplication between a weight matrix and the vector of neuron values at the previous layer, followed by the application of an element-wise nonlinear ''activation'' function.FFNNs have proven highly capable at approximating a broad range of functions and can be shown to be, in the limit of infinite neurons, universal approximators [39,40].As such, they are an obvious candidate for metamodelling, and have been widely demonstrated in many such contexts [41][42][43].Much of the popularity of FFNNs is due to their scalability to very large and high dimensional datasets, and, especially in the scope of deep learning methods, power for dealing with highly nonlinear relations [44,45].This also makes the FFNNs a very attractive option for metamodelling in a hybrid testing context, in which they have been demonstrated in limited number of cases to date [46,47].

Network architecture
Much of the power of neural networks comes from the ease with which their architectures can be adapted to deal with systems of different sizes and complexity [48].The basic architecture of a 3 layer FFNN is demonstrated in Fig. 1.Such a network consists of the input layer, where the neurons take the values of the input vector .The number of neurons in this layer is naturally equal to the dimensionality of the input vector.The final layer of the network is the output layer, at which the neuron values are the elements of the output vector  .The number of neurons in the output layer is equal to the dimensionality of the output of the network.Between these two layers lies a hidden layer which can have an arbitrary number of neurons, the selection of which greatly affects the performance of the network.The hidden layer neurons are characterized by the activation values, which are represented in the vector .It is possible and often desirable to use multiple hidden layers between the input and the output layers, creating a so-called deep neural network.Deep neural networks are even more powerful for representing highly complex functions [49], however, only a single hidden layer is used in this work since such a FFNN is deemed adequate.
The single-hidden-layer FFNN diagram in Fig. 1 is mathematically described in Eqs. ( 7) and (8).Eq. ( 7) represents the calculation of the activation values in the hidden layer of the FFNN, the input vector  is matrix multiplied by the weight matrix   and the bias vector   is added.The activation vector  is then calculated by the element-wise application of the activation function  1 .In Eq. ( 8), the output vector of the FFNN is calculated in a similar fashion, via matrix multiplication of the activation vector  with a second weight matrix   , to which a second bias vector   is added before the element-wise application of a second activation function  2 .In these equations, both weight matrices   and   , and both bias vectors,   and   , are populated by parameters which are learned during the training of the FFNN.The activation functions  1 and  2 , on the other hand, are hyperparameters and must be chosen by the network designer.Common candidates for these activation functions include the tanh,  and  activation functions, each with different associated properties [50]. )

Network training
When using a FFNN to create a metamodel, the first step involves generating a dataset with which to train, validate and test the network.Such a dataset consists of input and output vector pairs (points) for the system being metamodelled.For training a neural network, this dataset is then usually further broken down into training, validation and testing datasets.The training dataset contains those input-output pairs used for optimizing the learned parameters of the system in the weight matrices and bias vectors.The validation dataset is a set of held-back input-output pairs which are used to compare FFNNs created with differing hyperparameters, such as hidden layer size or activation functions.This yields FFNNs with better performance on the training dataset.The use of a separate validation dataset informs on the ability of the trained FFNN to generalize, i.g., demonstrate whether the FFNN performance is maintained for input-output pairs outside of the training dataset, as well as indicate whether the FFNN is ''overfitting'' to the training points [48].The testing dataset finally, comprises those points on which the final performance of the trained network of the chosen architecture is tested.
In this work, the neural network was created and trained using the MATLAB deep learning toolbox [51].Since, in the context of metamodelling, the problem is one of regression, i.e., predicting a continuous output variable, the loss function of the network to be minimized was selected as the mean squared error value, as demonstrated in Eq. (9).In this equation, the loss value  is calculated as the mean value of the squared error between the true output value   for the training point  and the predicted output value from the network Ŷ for all  training data points.The network weights were trained using the Bayesian regularization algorithm.The latter is more resistant to overfitting and tends to train networks with greater generalization [52].A validation dataset was used to select the final architecture of the network as that which gave the best validation performance.The final network architecture selected for this problem was a 3 layer network with a single hidden layer.The activation functions for the hidden and output layers are tanh and  functions, respectively.A single hidden layer network was used as it was found that deeper networks gave no performance benefit and even demonstrated worse generalization performance on the validation dataset.The final single hidden layer network used had 30 neurons and a validation normalized mean squared error (NMSE) loss of 0.357.For comparison, a 2 hidden layer network, in which each hidden layer consists of 30 neurons, resulted in a NMSE loss of 0.541.As such the single hidden layer network was deemed sufficient.

Case study
Given that the focus of this work is a MOR framework for RTHS, the case study employs a virtual hybrid model to conduct virtual RTHSs (vRTHSs).All of the components of the virtual hybrid model are implemented numerically.Thus, virtual PSs (vPSs) are used in place of physical specimens.In this regard, without loss of generality and for the sake of clarity, transfer systems that would be required between NS and PS in a physical HS, as well as the necessary tracking controller to compensate for the respective time delays and tracking errors, are omitted from this case study.
The virtual hybrid model used in this case study represents a prototype motorcycle.This model consists of one vPS, the electronically-controlled continuously variable transmission (eCVT) of the motorcycle, and four NS: (i) the engine dynamics model, (ii) the motorcycle body dynamics model including the tires and the road, (iii) the rear wheel braking systems, and (iv) the front wheel braking system.The interconnections between NS and the vPS are displayed in Fig. 2, whereas Fig. 3 illustrates the real eCVT PS that would be utilized in a non-virtual RTHS.In essence, the four NS of the hybrid model correspond to the remaining parts of the motorcycle.
The components of the virtual hybrid model were developed using the Simcenter Amesim software [53].The latter is a multiphysics system simulation tool that can be used to virtually model and optimize the performance of engineering structures.Simcenter Amesim was also used to export the aforementioned models into Functional Mock-up Interfaces (FMIs) [54], which were later used to conduct the vRTHS.In this regard, to simulate the FMIs in real-time and to allow data exchange between them, a real-time Simcenter simulation platform was used.It should be noted that in case a physical RTHS was to be performed, the exact same Simcenter simulation platform would be used to run and coordinate the HS in real-time.The substructures of the hybrid model are briefly introduced below.For a more detailed description of the aforementioned models the reader is encouraged to consult [55].
In particular, the eCVT vPS consist of a multi-input-multi-output (MIMO) model with two sets of inputs/outputs.The first set is connected to the engine dynamics model NS and the second to the motorcycle body dynamics NS.The latter connection corresponds to the transmission output shaft of the motorcycle.The engine dynamics model NS aims to simulate the behavior of the motorcycle combustion engine.It is a multi-input-single-output (MISO) model, the inputs being the angular velocity of the engine crankshaft   and the throttle level ℎ, and the output being the torque of the engine crankshaft   .The parameters of this model are the engine moment of inertia and coefficient of viscous friction.The motorcycle body NS addresses the inner body dynamics of the motorcycle along with the dynamics of the suspension and the tires as well as the road profile and the environmental driving conditions.The parameters of this model include the motorcycle mass, as well as damping and stiffness of the front and rear tires and suspensions.The motorcycle body NS is a MIMO model with 3 sets of inputs/outputs.The first set is connected to the eCVT vPS with the input and the output being the torque   and the angular velocity   of the transmission output shaft, respectively.The second and third sets are connected to the rear and front wheel braking system NS, respectively.Both braking systems are represented as MISO models.The inputs of the rear wheel braking system NS are the angular velocity of the rear wheel   and the applied force in the brake pedal    , while the output is the generated braking torque in the rear wheel   .Similarly, the inputs for the front wheel braking system NS are the angular velocity of the front wheel    and the applied force on the front brake lever     , while the output is the braking torque in the front wheel    .
It should be noted that, as shown in Fig. 2, for the coupling between the NS and vPS of the hybrid model four signals were used, namely the torque   and the angular velocity   of the engine crankshaft and the torque   and the angular velocity   of the transmission output shaft.In particular, the signals   and   were sent to the vPS from the respective NS, while the signals   and   correspond to the vPS computed measurements which are fed back to the according NS.

Motorcycle testing scenario
The performance of the motorcycle prototype is examined by testing its virtual hybrid model under predefined driving, road and wind scenarios.These tests are, in fact, vRTHSs.
The case study involves a 45 s long motorcycle driving scenario on a road with a defined profile and in defined wind conditions.The driving scenario is described by the variation of the throttle ℎ and the braking forces    =     (front and rear wheel braking forces are assumed to be equal) described by Eqs.(10) and (11), respectively, and shown in Fig. 4. Note that the braking forces are measured in Newton and the maximum applied throttle is 0.5, corresponding to a half-open throttle.The road profile, i.e. height variation of the road, is defined by the sinusoidal function ℎ() in Eq. ( 12), where  denotes the current position of the motorcycle in meters.The ambient wind velocity is assumed to be zero.
Each case study motorcycle driving test involves ''driving'' the motorcycle virtual hybrid model following the predefined driving scenario along the given road in the given wind conditions.The position of the motorcycle is computed by solving its equation of motion.In this case study, the RK4 (fourth-order Runge-Kutta) method with a fixed time-step of 0.1 msec was used to numerically solve this equation of motion.In addition to the position of the motorcycle, a number of indicative performance parameters are computed and monitored.These include global response parameters, such as the motorcycle velocity , as well as the input and output parameters of the NSs, i.e. the angular velocities and torques shown in Fig. 2.

Table 1
Marginal PDFs and their characteristics per input variable of the front and rear brake system RO-NSs.Used to construct the EDs of Eqs. ( 13) and ( 14).

Reduced-order numerical substructures
To demonstrate the proposed MOR framework, two of the four NS in the virtual hybrid model of the motorcycle, specifically the rear and front wheel braking systems, were replaced with their respective reduced-order models, applying the PCE and FFNN techniques addressed in Section 2. The braking systems were selected because they include the brake hydraulic systems and friction pads.Adequately emulating the dynamic behavior of such systems using numerical models requires small integration time-steps, making them ideal candidates for MOR.The reduced-order models of the two braking system NSs will be referred to as reduced-order NS (RO-NS) hereafter.
To train the front and rear brake system PCE and FFNN RO-NSs, two sets of 200 points were generated for the input variables listed in Table 1, assuming the inputs are statistically independent, using the Latin hypercube sampling (LHS) method [56].For each of the 200 generated points, full-order models (FOMs) of the front and rear braking systems were used to (accurately but expensively) compute the resulting front and rear brake torques   and    , respectively.The obtained EDs for the front and the rear braking systems are: Two types of RO-NSs were developed for each braking system, one featuring a single PCE and the other being a FFNN.The same ED was used to train, validate and test the PCE and the FFNN RO-NSs for each braking system, as discussed in Section 2.

Virtual hybrid models
Four different virtual hybrid models of the motorcycle were constructed in the case study, differing in the selection of braking system NSs.The base-line virtual hybrid model features the FOM NSs of both the front and the rear braking systems.The remaining three virtual hybrid models have RO-NSs.In the PCE virtual hybrid model, both braking systems are represented using PCE RO-NSs.Analogously, the FFNN virtual hybrid model uses FFNN RO-NSs.Finally, the mixed virtual hybrid model of the motorcycle uses the FFNN RO-NS for the front brakes and the PCE RO-NSs for the rear brakes.
The performance of the thus generated motorcycle virtual hybrid models was tested statistically.To this end, the 11 parameters of the motorcycle model, listed in Table 2, were assigned uniform distributions with parameters identified from [57][58][59] to represent realistic variations of these mechanical motorcycle properties.A total of 30 different samples of the 11 motorcycle parameters were generated using the LHS method, essentially resulting in 30 different motorcycle prototypes.These prototypes were modeled using four virtual hybrid models described above.Using the driving scenario described above, a total of 120 motorcycle model response time histories were computed in this case study.
The variability of motorcycle prototype response in the selected driving scenario, computed using the (accurate but expensive) FOM virtual hybrid model, is demonstrated in Figs. 5.The 30 time histories of two indicative virtual hybrid model response variables, the front wheel brake torque    and the motorcycle velocity  = ẋ, are compared to their mean value.All results presented hereafter correspond to the mean values of the 30 virtual hybrid model response simulations.
Additionally, the time history responses of Fig. 5 can be intuitively interpreted.Fig. 5a depicts the time history of the front wheel brake torque    , generated by the employed driving scenario.As can be appreciated from Fig. 4 and Eq. ( 11), the motorcycle brakes are activated between the 20th and 37th second of the simulation and therefore the braking torque is non-zero only inside this time interval.Fig. 5b illustrates the evolution of the motorcycle's velocity response  using the aforementioned driving scenario.According to Fig. 4 and Eq. ( 10), for the first five seconds of the simulation, no throttle is applied to the motorcycle and hence its velocity is zero.Between the 5th and 17th second of the simulation, the driving scenario involves stepping on the throttle and therefore the motorcycle starts to accelerate.As a result, its velocity is increasing.Following Eq. ( 11), as mentioned before, after the 20th second, the driving scenario involves pressing on the brakes and thus the motorcycle's velocity begins to decrease and it is brought to almost a full stop in the 45th second of the simulation.Second, the response of the four virtual hybrid models is compared by examining the NRMSEs and the time histories of the indicative model response quantities, namely the angular velocities and torques that couple the substructures of the hybrid models (  ,    ,   ,   ,   and   ) and the motorcycle velocity .The variables that couple the substructures of the hybrid models were investigated since deviations from their nominal values, would risk the HS outcomes and could drive the HS loop into instability.In addition, for non-virtual RTHS that include real test benches and PS, large deviations could damage the corresponding physical specimen and laboratory equipment.On the contrary, the motorcycle velocity  is chosen to be examined as it corresponds to a global response quantity that informs about the overall dynamic behavior of the motorcycle during the driving scenario.Table 4 provides an overview of the NRMSE and Figs. 7, 8 show the response quantity time histories for the four virtual hybrid models.From Figs. 7, 8 it can be shown that the discrepancies between the full-order and reduced-order models are mainly concentrated approximately beyond the 20th second of the simulation.This can be intuitively explained since in the driving scenario (Fig. 4), the braking of the motorcycle is initiated in the 20th second of the simulation.Evidently, the difference between the virtual hybrid models is small: the response time history graphs virtually overlap, while the NRMSEs are all smaller than 2%.Therefore, both the PCE and the FFNN MOR techniques described in Section 2 produce valid and efficient reduced-order models for RTHS.It is also notable that a combination of RO-NSs developed using different MOR approaches also works, giving a much-needed ability to combine reduced-order models in practice.
Finally, given that the performance of PCE-based and FFNN-based reduced-order models is practically the same, model designers need to make their modeling choice based on additional consideration.High-dimensional models, with long input and output vectors, are more suitable for FFNN-based MOR, since PCEs may require utilization of sparse schemes and thus incur additional errors.On the other hand, PCE-based MOR offers a distinct advantage in that the PCE coefficients can be used to compute the statistics of the relevant model response quantities, as well as assess their sensitivity to model input quantities via the Sobol' indices approach, at no additional cost [60,61].The latter is not a feature of neural networks.Notably, the software for developing both FFNN and PCE reduced-order models exists in the form of MATLAB toolboxes [34,51], making practical implementation of these to MOR approaches relatively easy.

Conclusions
In this study, several model order reduction techniques are bench-marked for virtual real-time hybrid simulation.The use of model order reduction techniques in hybrid simulation is especially important for hybrid models encompassing high-dimensional nonlinear numerical substructures, risking distorting the timescale of hybrid simulation by introducing time delays due to the extended computational power needed.The selected model order reduction techniques are based on data-driven regression techniques and particularly on polynomial chaos expansion and feedforward neural networks.A parametric case study consisting of a  virtual hybrid model is used to validate the framework.Feedforward neural networks and polynomial chaos expansion surrogates are trained to replicate the dynamic response of selected numerical substructures.Substitution of their respective full-order components forms the reduced-order hybrid model.Specific parameters of the hybrid model are varied and multiple virtual hybrid simulation evaluations are conducted to determine the robustness of the model order reduction framework.Comparisons with the full-order hybrid model responses are made to assess whether the reduced-order hybrid models can accurately replicate the initial dynamic responses while the corresponding errors are proven to be negligibly small.Results confirm the validity, benefits and performance of the proposed model order reduction framework.In particular, for the case study presented in this work, model order reduction techniques allowed for a ≈ 52% reduction of the computational cost, for a maximum of 6.54% loss on absolute accuracy versus full-order model.Model order reduction techniques are therefore capable of producing hybrid models for more reliable real-time hybrid simulations without a significant loss of accuracy.The use of model order reduction techniques pushes the boundaries of hybrid simulation further by enabling the use of more complex and computationally involved numerical substructures.Future work will focus on quantifying how much model order reduction techniques contribute to improving real-time hybrid simulation with physical substructures.

Fig. 3 .
Fig. 3.The eCVT PS that would be utilized in a non-virtual RTHS.

Fig. 6 .
Fig. 6.Front and rear wheel braking torque for the cases of FOM, FFNN-based and PCE-based MOR.Mean values of the 30 vRTHS evaluations.

Fig. 7 .
Fig. 7. Selected hybrid model responses evaluated in each testing scenario.Mean values of the 30 vRTHS evaluations.Zoomed axes are added to highlight the difference between the shown response time histories.

Table 3
NRMSE of rear and front wheel braking torque between FOM NS and RO-NSs.The values are in percent and correspond to the mean values of the 30 vRTHS evaluations.

Table 4
NRMSE for the mean values of the selected virtual hybrid model response quantities.The values are in percent.