Model order reduction by dominance of mode sets and optimal modal span

Many industrial chemical processes are complex, multi-phase and large scale in nature. These processes are characterized by various nonlinear physiochemical effects and fluid flows. Such processes often show coexistence of fast and slow dynamics during their time evolutions. The increasing demand for a flexible operation of a complex process, a pressing need to improve the product quality, an increasing energy cost and tightening environmental regulations make it rewarding to automate a large scale manufacturing process. Mathematical tools used for process modeling, simulation and control are useful to meet these challenges. Towards this purpose, development of process models, either from the first principles (conservation laws) i.e. the rigorous models or the input-output data based models constitute an important step. Both types of models have their own advantages and pitfalls. Rigorous process models can approximate the process behavior reasonably well. The ability to extrapolate the rigorous process models and the physical interpretation of their states make them more attractive for the automation purpose over the input-output data based identified models. Therefore, the use of rigorous process models and rigorous model based predictive control (R-MPC) for the purpose of online control and optimization of a process is very promising. However, due to several limitations e.g. slow computation speed and the high modeling efforts, it becomes difficult to employ the rigorous models in practise. This thesis work aims to develop a methodology which will result in smaller, less complex and computationally efficient process models from the rigorous process models which can be used in real time for online control and dynamic optimization of the industrial processes. Such methodology is commonly referred to as a methodology of Model (order) Reduction. Model order reduction aims at removing the model redundancy from the rigorous process models. The model order reduction methods that are investigated in this thesis, are applied to two benchmark examples, an industrial glass manufacturing process and a tubular reactor. The complex, nonlinear, multi-phase fluid flow that is observed in a glass manufacturing process offers multiple challenges to any model reduction technique. Often, the rigorous first principle models of these benchmark examples are implemented in a discretized form of partial differential equations and their solutions are computed using the Computational Fluid Dynamics (CFD) numerical tools. Although these models are reliable representations of the underlying process, computation of their dynamic solutions require a significant computation efforts in the form of CPU power and simulation time. The glass manufacturing process involves a large furnace whose walls wear out due to the high process temperature and aggressive nature of the molten glass. It is shown here that the wearing of a glass furnace walls result in change of flow patterns of the molten glass inside the furnace. Therefore it is also desired from the reduced order model to approximate the process behavior under the influence of changes in the process parameters. In this thesis the problem of change in flow patterns as result of changes in the geometric parameter is treated as a bifurcation phenomenon. Such bifurcations exhibited by the full order model are detected using a novel framework of reduced order models and hybrid detection mechanisms. The reduced order models are obtained using the methods explained in the subsequent paragraphs. The model reduction techniques investigated in this thesis are based on the concept of Proper Orthogonal Decompositions (POD) of the process measurements or the simulation data. The POD method of model reduction involves spectral decomposition of system solutions and results into arranging the spatio-temporal data in an order of increasing importance. The spectral decomposition results into spatial and temporal patterns. Spatial patterns are often known as POD basis while the temporal patterns are known as the POD modal coefficients. Dominant spatio-temporal patterns are then chosen to construct the most relevant lower dimensional subspace. The subsequent step involves a Galerkin projection of the governing equations of a full order first principle model on the resulting lower dimensional subspace. This thesis can be viewed as a contribution towards developing the databased nonlinear model reduction technique for large scale processes. The major contribution of this thesis is presented in the form of two novel identification based approaches to model order reduction. The methods proposed here are based on the state information of a full order model and result into linear and nonlinear reduced order models. Similar to the POD method explained in the previous paragraph, the first step of the proposed identification based methods involve spectral decomposition. The second step is different and does not involve the Galerkin projection of the equation residuals. Instead, the second step involves identification of reduced order models to approximate the evolution of POD modal coefficients. Towards this purpose, two different methods are presented. The first method involves identification of locally valid linear models to represent the dynamic behavior of the modal coefficients. Global behavior is then represented by ‘blending’ the local models. The second method involves direct identification of the nonlinear models to represent dynamic evolution of the model coefficients. In the first proposed model reduction method, the POD modal coefficients, are treated as outputs of an unknown reduced order model that is to be identified. Using the tools from the field of system identification, a blackbox reduced order model is then identified as a linear map between the plant inputs and the modal coefficients. Using this method, multiple local reduced LTI models corresponding to various working points of the process are identified. The working points cover the nonlinear operation range of the process which describes the global process behavior. These reduced LTI models are then blended into a single Reduced Order-Linear Parameter Varying (ROLPV) model. The weighted blending is based on nonlinear splines whose coefficients are estimated using the state information of the full order model. Along with the process nonlinearity, the nonlinearity arising due to the wear of the furnace wall is also approximated using the RO-LPV modeling framework. The second model reduction method that is proposed in this thesis allows approximation of a full order nonlinear model by various (linear or nonlinear) model structures. It is observed in this thesis, that, for certain class of full order models, the POD modal coefficients can be viewed as the states of the reduced order model. This knowledge is further used to approximate the dynamic behavior of the POD modal coefficients. In particular, reduced order nonlinear models in the form of tensorial (multi-variable polynomial) systems are identified. In the view of these nonlinear tensorial models, the stability and dissipativity of these models is investigated. During the identification of the reduced order models, the physical interpretation of the states of the full order rigorous model is preserved. Due to the smaller dimension and the reduced complexity, the reduced order models are computationally very efficient. The smaller computation time allows them to be used for online control and optimization of the process plant. The possibility of inferring reduced order models from the state information of a full order model alone i.e. the possibility to infer the reduced order models in the absence of access to the governing equations of a full order model (as observed for many commercial software packages) make the methods presented here attractive. The resulting reduced order models need further system theoretic analysis in order to estimate the model quality with respect to their usage in an online controller setting.

A methodology is presented to model non-compact thermoacoustic phenomena using Reduced Order Models (ROM) based on the Linearized Navier-Stokes Equations (LNSE). The method is applicable to geometries with a complex flow field as in a gas turbine combustion chamber. The LNSE, and thus the resulting ROM, include coupling effects between acoustics and mean fluid flow, and are hence capable of describing propagation and (e.g. vortical) damping of the acoustic fluctuations within the considered volume. Such a ROM then constitutes the main building block for a novel thermoacoustic stability analysis method via a low-order hybrid approach. This method presents an expansion to state-of-the-art low-order stability tools, and is conceptually based on three core features: Firstly, the multi-dimensional and volumetric nature of the ROM establishes access to account spatial variability and non-compact effects on heat release fluctuations. As a result, it is particularly useful for high frequency phenomena such as screech. Secondly, the LNSE basis grants the ROM the capability to reconstruct complex acoustic performances physically accurate. Thirdly, the formulation of the ROM in state-space allows convenient access to the frequency and time domain. In the time domain, non-linear saturation mechanisms can be included, which reproduce the non-linear stochastic limit cycle behavior of thermoacoustic oscillations. In order to demonstrate and verify the ROM’s underlying methodology, a test case using an orifice-tube geometry as the acoustic volume is performed. The generation of the ROM of the orifice-tube is conducted in a two-step procedure. As the first step, the geometrical domain is aeroacoustically characterized through the LNSE in frequency domain, and discretized via the Finite Element Method (FEM). The second step concerns the actual derivation of the ROM. The high-order dynamical system from the LNSE discretization is subjected to a modal reduction as order reduction technique. Mathematically, this modal reduction is the projection of the high-order (N ∼200,000) system into its truncated left eigenspace. An order reduction of several magnitudes (ROM order: Nr ∼100) is achieved. The resulting ROM contains all essential information about propagation and damping of the acoustic variables, and efficiently reproduces the aeroacoustic performance of the orifice-tube. Validation is achieved by comparing ROM results against numerical and experimental benchmarks from LNSE-FEM simulations and test rig measurements, respectively. Excellent agreement is found, which grants the ROM modeling approach full eligibility for further usage in the context of thermoacoustic stability modeling. This work is concluded by a methodological demonstration of performing stability analyses of non-compact thermoacoustic systems using the herein presented ROMs.

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

Fluid-Structure interaction in a collapsible channel with a flexible wall segment which deforms under the fluid dynamic loads arising from the flow within the channel is considered by using a reduced order model based on the method of principal component analysis developed for unsteady aerodynamic pressures and the structural dynamic state variables by using a series of discrete time results. A computational fluid dynamics approach, based on the finite volume solver is used to solve the discretized Navier-Stokes equations. The structural model of the flexible wall is based on the membrane equation and the membrane tension is varied to study the effects of the structural modeling parameters on the instabilities that arise due to fluid-structu re interaction. It is well known in aeroelasticit y, that fluid-structural coupling based on lagging the fluid dynamics code with the structural dynamics solver can produce spurious numerical phenomena. On the other hand, tightly coupled aeroelastic solvers can be difficult to implement due to numerical ill-conditioning. Using a series of discrete time results, a reduced order model is developed for the unsteady aerodynamic pressure and the structural dynamic state variables based on the method of principal component analysis. This reduced order model now serves as a de-coupler for the fluidstructure interaction. The effects of varying the tension and the inertia of the membrane on the unsteady fluid dynamics are investigated using the reduced order model. The principal component analysis captures the fluid dynamic damping at every time instant in the system dynamics and this knowledge is used to detect the onset of instabilities. A regularization technique is also used to improve the conditioning of the matrices involved in the reduced-order fluid-structure interaction model.

Efficient aerodynamic design using BézierGAN and model order reduction: A computational study

This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system’s dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases.

The need for reduced order models (ROMs) has become considerable higher with the increasing technological advances that allows one to model complex dynamical systems. When using ROMs, the following two questions always arise: 1) “What is the lowest dimensional ROM?” and 2) “How well does the ROM capture the dynamics of the full scale system model?” This paper considers the newly developed concepts the authors refer to as subspace robustness — the ROM is valid over a range of initial conditions, forcing functions, and system parameters — and dynamical consistency — the ROM embeds the nonlinear manifold — which quanitatively answers each question. An eighteen degree-of-freedom pinned-pinned beam which is supported by two nonlinear springs is forced periodically and stochastically for building ROMs. Smooth and proper orthogonal decompositions (SOD and POD, respectively) based ROMs are dynamically consistent in four or greater dimensions. In the strictest sense POD-based ROMs are not considered coherent whereas, SOD-based ROMs are coherent in roughly five dimesions and greater. Is is shown that in the periodically forced case, the full scale dynamics are captured in a five-dimensional POD and SOD-based ROM. For the randomly forced case, POD and SOD-based ROMs need three dimensions but SOD captures the dynamics better in a lower-dimensional space. When the ROM is developed from a different set of initial conditions and forcing values, SOD outperforms POD in periodic forcing case and are equal in the random forcing case.

Dynamic modeling and model order reduction of compliant mechanisms

The consideration of the thermals of piezoelectric transducers has gained relevance due to the increasing demands on the performance of actuators as well as the special applications such as high-power, high-precision or high-temperature applications. In the state of the art, many works deal with the representation of the temperature-dependence of piezoelectric performance and the development of suitable systems that are optimized in terms of self-heating. The prerequisite for this, is to characterize the actuators thermally and to have thermal models of piezoelectric actuators, that represent the temperature of the actuator. In this paper a workflow is presented for the thermal characterization and modeling of piezoelectric transducers in a fast and non-destructive manner. Aim of this work is to generate Finite-Element (FE) Models and Reduced Order Models (ROM) of the thermal behavior of piezoelectric transducers for many different sizes of actuators. For this purpose an experimental setup and characterization workflow are developed. The advantage of the workflow lies in the speed of characterization through direct comparison of measurement results with a Metamodel and automated model order reduction. Parametric model order reduction is also investigated in this context for faster model building and compared to the results of FE and ROM. Model order reduction offers the advantage that the resulting models are low in computing effort, can be used without FE software and can directly be included in for example control or system models. Finally by direct comparison with a simple circuit model as in the state of the art the advantages of FE modeling can be shown. The workflow can be used for many different test specimen that vary in size and shape. The method is exemplarily presented using a Lead Zirconate Titanate (PZT) multilayer actuator.

This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe- spot analysis, control and decision-making in the presence of uncertainty. Of particular interest to Sandia is the use of ROMs for the quantification of the compressible captive-carry environment, simulated for the design and qualification of nuclear weapons systems. It is an unfortunate reality that many ROM techniques are computationally intractable or lack an a priori stability guarantee for compressible flows. For this reason, this LDRD project focuses on the development of techniques for building provably stable projection-based ROMs. Model reduction approaches based on continuous as well as discrete projection are considered. In the first part of this report, an approach for building energy-stable Galerkin ROMs for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. The key idea is to apply a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. It is shown that, for many PDE systems including the linearized compressible Euler and linearized compressible Navier-Stokes equations, the desired transformation is induced by a special inner product, termed the “symmetry inner product”. Attention is then turned to nonlinear conservation laws. A new transformation and corresponding energy-based inner product for the full nonlinear compressible Navier-Stokes equations is derived, and it is demonstrated that if a Galerkin ROM is constructed in this inner product, the ROM system energy will be bounded in a way that is consistent with the behavior of the exact solution to these PDEs, i.e., the ROM will be energy-stable. The viability of the linear as well as nonlinear continuous projection model reduction approaches developed as a part of this project is evaluated on several test cases, including the cavity configuration of interest in the targeted application area. In the second part of this report, some POD/Galerkin approaches for building stable ROMs using discrete projection are explored. It is shown that, for generic linear time-invariant (LTI) systems, a discrete counterpart of the continuous symmetry inner product is a weighted L2 inner product obtained by solving a Lyapunov equation. This inner product was first proposed by Rowley et al., and is termed herein the “Lyapunov inner product“. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases. Also in the second part of this report, a new ROM stabilization approach, termed “ROM stabilization via optimization-based eigenvalue reassignment“, is developed for generic LTI systems. At the heart of this method is a constrained nonlinear least-squares optimization problem that is formulated and solved numerically to ensure accuracy of the stabilized ROM. Numerical studies reveal that the optimization problem is computationally inexpensive to solve, and that the new stabilization approach delivers ROMs that are stable as well as accurate. Summaries of “lessons learned“ and perspectives for future work motivated by this LDRD project are provided at the end of each of the two main chapters.

In this proposal, order diminution of self-balanced linear-momentum based bicycle-robot (LMBR) controller is proposed with the help of time-moments (TMs) and Markov-parameters (MPs) of higher-order (HO) LMBR controller and its desired reduced-order (RO) model by utilizing jaya algorithm. Firstly, the TMs and MPs of HO LMBR controller and its RO model are obtained. The TMs of HO LMBR controller and RO model are exploited for steady state matching. The MPs are utilized to construct the objective function which is weighted summation of errors in between MPs of HO LMBR controller and RO model. The framed objective function is minimized with the help of jaya algorithm. The minimization of errors between MPs of HO LMBR controller and RO model are done for ascertainment of unknown coefficients of desired RO model. The proposed RO model for HO LMBR controller is compared with the model obtained by exploiting other order reduction methods available in the literature.

<p>Reduced-order modeling is an emerging technique for cutting down the computational expenses incurred in terms of CPU time and usage associated with repetitive simulation of flow dynamics of natural aquifer systems. Identifying the patterns related to the evolution of aquifer response with time is the key to model order reduction methodology. However, the accuracy of reduced-order groundwater models is dependent on several factors. It has been observed that the accuracy decreases while accounting for random heterogeneity of natural aquifers. Besides, the imposition of different boundary conditions also tends to influence the accuracy of reduced-order models. In this work, we study the effects of Dirichlet and Neumann boundary conditions on reduced-order modeling of groundwater flow through randomly distributed heterogeneous porous media. For low dimensional modeling, we have performed Singular Value Decomposition of the ‘Snapshot Matrix’ to obtain a set of orthonormal basis functions. The ‘Snapshot Matrix’ is formed from the solution of a Finite Volume Method based full system groundwater flow model at some exponentially distributed time instants. The governing groundwater flow equation is then projected onto the reduced sub-space of orthonormal basis functions via Galerkin Projection to obtain the solution at each time-step. We have carried out the study on a two-dimensional square-shaped synthetic heterogeneous aquifer with multiple pumping wells operating simultaneously within the domain. Four illustrative case studies have been performed with the aquifer being subjected to: (1) Dirichlet condition on all boundaries; (2) Dirichlet condition on three boundaries while the remaining boundary is impermeable; (3) Dirichlet condition on two parallel boundaries while the other two boundaries are impermeable; (4) Three impermeable boundaries and Dirichlet condition on the remaining boundary. The study shows that the accuracy of the reduced-order model is maximum when all the four boundaries of the aquifer are subjected to a constant specified head (Dirichlet) boundary condition. The accuracy starts to go down as we start introducing impermeable boundaries withdrawing the Dirichlet boundaries. The error analysis is performed by comparing the error statistic parameters: Maximum Absolute Error, Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Normalized Root Mean Square Error (NRMSE) for the four case studies with respect to the results obtained from corresponding full system model runs. However, if we look into the computational expenses, the model takes lesser computation time per iteration as the complexity of boundary conditions increases. Although the reduction in the accuracy of the reduced-order model is observed with the introduction of impermeable boundaries, the error statistic parameters are within desirable limits. Hence, the proposed reduced-order modeling methodology can potentially be accepted as an accurate and efficient alternative for replication of high-dimensional full system groundwater flow models, and can also be applied for natural aquifers on a watershed scale.</p>

In recent years, autonomously guided parachute systems have been developed with the goal of increasing the landing accuracy of airdrop operations. These precision airdrop systems utilize an onboard global positioning system (GPS) coupled with a guidance, navigation and control (GNC) system to continuously monitor and correct the flight path of the parachute system during descent. A critical component of the GNC system is an algorithm that predicts the response of the system to various aerodynamic forces in real-time that are typically based on a low-order multi-degree of freedom (DOF) dynamics model. Reduced order modeling is the process of representing a complex dynamic system with many DOF with a model having relatively few DOF. The goal of this study is to examine the use of large scale finite element models to construct and calibrate Reduced Order Models (ROM) for parachute systems. The resulting ROM would provide parachute designers with a simple tool to examine parachute behavior under various operating conditions. The use of numerical simulations to calibrate the ROM would help to eliminate the number of physical tests needed to characterize a parachute system’s properties. In this study, a 6 degree of freedom ROM is adopted. The system dynamic properties for the ROM and the resultant forces and moments acting on the ROM are obtained from finite element simulations. The 6 DOF model originally proposed by Cockrell and Doherr and later used by Dobrokhodov is adopted for the ROM. The finite element (FE) code, TENSION, is used to perform the large scale simulations and extract the ROM dynamic properties and resultant forces. Several examples are performed that verify the ROM procedure by comparing ROM predictions to FE simulations for various parachute system trajectories. The performance of the ROM is shown to be excellent for in-plane motion under an oscillating wind field. It is shown, however, that the ROM procedure deteriorates as the FE model is refined by adding more DOF. It is demonstrated that, for highly deformable FE models with many DOF, more sophisticated ROM procedures are needed.

Part I Invited Papers. 1 The need for novel model order reduction techniques in the electronics industry .W.H.A. Schilders. 1.1 Introduction. 1.2 Mathematical problems in the electronics industry. 1.3 Passivity and realizability. 1.4 Structure preservation. 1.5 Reduction of MIMO networks. 1.6 MOR for delay equations. 1.7 Parameterized and nonlinear MOR. 1.8 Summary: present and future needs of the electronics industry. References. 2 The SPRIM Algorithm for Structure-Preserving Order Reduction of General RCL Circuits Roland W. Freund. 2.1 Introduction. 2.2 RCL Circuit Equations. 2.3 Projection-Based Order Reduction. 2.4 The SPRIM Algorithm. 2.5 Treatment of Voltage Sources. 2.6 Numerical Examples. 2.7 Concluding Remarks. References. 3 Balancing-Related Model Reduction of Circuit Equations Using Topological Structure Tatjana Stykel. 3.1 Introduction. 3.2 Circuit equations. 3.3 Balancing-related model reduction. 3.4 Numerical methods for matrix equations. 3.5 Numerical examples. 3.6 Conclusions and open problems. References. 4 Topics in Model Order Reduction with Applications to Circuit Simulation Sanda Lefteriu and Athanasios C. Antoulas. 4.1 Introduction and Motivation. 4.2 Background. 4.3 Theoretical Aspects. 4.4 Tangential interpolation for modeling Y-parameters. 4.5 Numerical Results. 4.6 Conclusion. References. Part II Contributed Papers. 5 Forward and Reverse Modeling of Low Noise Amplifiers based on Circuit Simulations L. De Tommasi, J. Rommes, T. Beelen, M. Sevat, J. A. Croon and T. Dhaene. 5.1 Introduction. 5.2 Forward and reverse modeling: problem descriptions. 5.3 Forward Modeling. 5.3.1 Performance Figures via Surrogate Models. 5.4 Reverse Modeling with the NBI method. 5.5 Reverse modeling using transistor level simulations. 5.6 Discussion and conclusions. References. 6 Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-Hand Sides Arising in Model Reduction Peter Benner and Lihong Feng. 6.1 Introduction. 6.2 Methods Based on Recycling Krylov Subspaces. 6.3 Application to Model Order Reduction. 6.4 Simulation Results. 6.5 Conclusions. References. 7 Data-driven Parameterized Model Order Reduction Using z-domain Multivariate Orthonormal Vector Fitting Technique Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom Dhaene. 7.1 Introduction. 7.2 Background. 7.3 Parametric Macromodeling. 7.4 Choice of basis functions. 7.5 Example: Double folded stub microstrip bandstop filter. 7.6 Conclusions. References. 8 Network Reduction by Inductance Elimination M.M. Gourary, S.G.Rusakov, S.L.Ulyanov, and M.M.Zharov. 8.1 Introduction. 8.2 Elimination of RC-node by TICER. 8.3 Inductance Elimination. 8.4 Elimination of Coupled Inductances. 8.5 Eliminations under LC Couplings. 8.6 Algorithmic Aspects. 8.7 Numerical Examples. 8.8 Conclusion. References. 9 Simulation of coupled oscillators using nonlinear phase macromodels and model order reduction Davit Harutyunyan and Joost Rommes. 9.1 Introduction. 9.2 Phase noise analysis of oscillators. 9.3 Oscillator coupled to a balun. 9.4 Oscillator coupling to a transmission line. 9.5 Model order reduction. 9.6 Numerical experiments. 9.7 Conclusion. References. 10 POD Model Order Reduction of Drift-Diffusion Equations in Electrical Networks Michael Hinze, Martin Kunkel and Morten Vierling. 10.1 Introduction. 10.2 Complete coupled system. 10.3 Simulation of the full system. 10.4 Model reduction. 10.5 Numerical investigation. Appendix: Proper Orthogonal Decomposition. References. 11 Model Reduction of Periodic Descriptor Systems Using Balanced Truncation Peter Benner, Mohammad-Sahadet Hossain and Tatjana Stykel. 11.1 Introduction. 11.2 Periodic Descriptor Systems. 11.3 Periodic Gramians and Matrix Equations. 11.4 Balanced Truncation Model Reduction. 11.5 Example. 11.6 Conclusion. References. 12 On synthesis of reduced order models Roxana Ionutiu and Joost Rommes. 12.1 Introduction. 12.2 Foster synthesis of rational transfer functions. 12.3 Structure preservation and synthesis by unstamping. 12.4 Numerical examples. 12.5 Conclusions and outlook. References. 13 Model Reduction Methods for Linear Network Models of Distributed Systems with Sources Stefan Ludwig and Wolfgang Mathis. 13.1 Introduction. 13.2 Background for Model Reduction of Linear Networks. 13.3 Description of distributed sources. 13.4 Examples. 13.5 Conclusion. References. 14 Structure preserving port-Hamiltonian model reduction of electrical circuits R.V. Polyuga and A.J. van der Schaft. 14.1 Introduction. 14.2 Linear port-Hamiltonian systems. 14.3 The Kalman decomposition of port-Hamiltonian systems. 14.4 The co-energy variable representation. 14.5 Balancing for port-Hamiltonian systems. 14.6 Reduction of port-Hamiltonian systems in the general case. 14.7 Example. 14.8 Conclusions. Appendix. References. 15 Coupling of numerical and symbolic techniques for model order reduction in circuit design Oliver Schmidt Thomas Halfmann Patrick Lang. 15.1 Motivation. 15.2 Symbolic Techniques. 15.3 Hierarchical systems. 15.4 Workflow for the exploitation of the hierarchy. 15.5 Comparison to other approaches. 15.6 Summary and future work. References. 16 On Stability, Passivity and Reciprocity Preservation of ESVDMOR Peter Benner and Andre Schneider. 16.1 Introduction. 16.2 The Extended SVDMOR Approach. 16.3 Stability, Passivity, and Reciprocity. 16.4 Remarks and Outlook. References. 17 Model order reduction of nonlinear systems in circuit simulation: status and applications Michael Striebel and Joost Rommes. 17.1 Introduction. 17.2 Linear versus nonlinear model order reduction. 17.3 Some nonlinear MOR techniques. 17.4 TPWL and POD. 17.5 Numerical examples. 17.6 Discussion and outlook. References. 18 An Approach to Nonlinear Balancing and MOR Erik I. Verriest. 18.1 Static versus Dynamic Approximation. 18.2 Gramians for Linear Systems and Applications. 18.3 Metric Properties of Balanced Truncation. 18.4 Nonlinear Model Reduction. References.

This paper investigates the state consistence of parametric data-driven reduced order models (ROMs) in a state-space form obtained by various system identification methods, including autoregressive exogenous (ARX) and subspace identification (N4SID), for aeroelastic analysis in varying flight conditions. The target flight envelop is first partitioned into discrete grid points, on each of which an aerodynamic ROM is constructed using system identification to capture the dependence of the generalized aerodynamic force on the generalized displacement of structural modes. High-fidelity aeroelastic modal perturbation simulations are used to generate the ROM training and verification data. Aerodynamic ROMs not on the grid point are obtained by interpolating those at neighboring grid points. Through a thorough analysis of the model coefficients and pole migration, it is found that only the ARX-based aerodynamic ROM preserves the state consistence, and hence, allowing direct interpolation of system matrices at the non-grid point and rapid aerodynamic ROM database development in the entire flight parameter space. In contrast, N4SID-based ROM destroys the state consistence and yields physically meaningless results when ROMs are interpolated. The origin of the difference in the state consistence caused by both methods is also discussed. The interpolated ARX aerodynamic ROMs coupled with the structural ROM for parametric aeroelastic analysis exhibit excellent agreement with the high fidelity full order model (mostly <5% relative error) and salient computational efficiency.