Proposed Model Reduction Method Research Articles

A mixed model reduction method based on the symmetrizer for two classes of dynamical systems

In regard of the cross Gramian and the block Arnoldi process, this paper focuses on mixed model reduction for two classes of dynamical systems. By studying the correlation between the cross Gramian and the Hankel singular values, model reduction via the cross Gramian is investigated for symmetric regular linear systems. Considering the non-symmetric regular linear system, associated with the symmetrizer, a convex optimization problem is established to obtain a symmetric system. Then, the non-symmetric regular linear system is divided into two parts, in which one is determined by the symmetric system and another is a non-symmetric counterpart. Model reduction via the cross Gramian is employed to reduce the first part. According to the structure of the non-symmetric counterpart, a modified version of the block Arnoldi process is developed to reduce the non-symmetric counterpart. The H2 error bound between the original system and the reduced system is discussed. This model reduction approach avoids partitioning the non-symmetric system into several single-input and single-output systems. Moreover, the proposed model reduction method can also be applied to the non-symmetric linear systems. Finally, the proposed model reduction approach is illustrated by a 2D convective thermal flow benchmark and an optimal cooling problem of steel profiles.

Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method

This paper studies frequency-limited model reduction for linear positive systems. Specifically, the objective is to develop a reduced-order model for a high-order positive system that preserves the positivity, while minimizing the approximation error within a given H∞ upper bound over a limited frequency interval. To characterize the finite-frequency H∞ specification and stability, we first present the analysis conditions in the form of bilinear matrix inequalities. By leveraging these conditions, we derive convex surrogate constraints by means of an inner-approximation strategy. Based on this, we construct a novel iterative algorithm for calculating and optimizing the reduced-order model. Finally, the effectiveness of the proposed model reduction method is illustrated with a numerical example.

A new reduced model of damped moment frame for rapid optimization of viscous damper placement

AbstractA new model reduction method is presented for moment‐resisting frames with viscous dampers. The proposed method considers (1) the interactive effect of added damping between the horizontal‐vertical directions and (2) the direct effect with regard to the vertical direction different from the conventional static condensation method. When brace‐type dampers are treated, these effects cannot be neglected originally. The proposed method gives a better correspondence of the fundamental eigenmode with that of the non‐reduced moment frame than the static condensation method. In addition, the proposed reduced model can accurately evaluate the maximum interstory drifts of the non‐reduced moment frame. The proposed model reduction method is formulated in the frequency domain. The frequency dependency of the stiffness matrix and the damping matrix of the reduced model is investigated. The influences of the added damping matrix by viscous dampers on the natural circular frequencies, the damping ratios and the participation vectors are also investigated. Finally, it is demonstrated through a numerical example that the displacement‐based optimization of the damper allocation can be accurately and rapidly conducted by using the reduced models throughout the optimization process.

First Order Plus Dead Time Model Approximation and Proportional-Integral-Derivative Controllers Tuning for Multi-Volume Process

Analysis and controller design for the FOPDT model is widely used because of its simpleness and convenience. In this paper, the problem of the multi-volume process (MVP) being approximated to first order plus dead time (FOPDT) model, as well as its proportional-integral-derivative (PID) controller arguments tuning is studied. These have been heavily studied in recent years and the methods developed for its optimal design rely on the idea of including several robust performance specifications in the objective function the method presents fast convergence and consists of mentioning a desired closed-loop transfer function. Particle swarm optimization (PSO) algorithm and performance index of the integral of the time-weighted absolute error (ITAE) minimum is presented to determine the approximate FOPDT model coefficients of MVP processes, where the order is from two to fifteen. In addition, the approximate FOPDT model is used to design the PID controller, which is used to control MVP. A large number of tuning methods are provided to analyze and compare the closed-loop control performances. At the end of the paper, two simulation examples illustrate the superiority and effectiveness of the PID controller design based on the proposed model reduction method. The simulation results show that the reduced-order controller can control a high-order system well, but the process of order reduction is complicated and it needs a long computation time. The FOPID is a generalization of the conventional PID controller. This is based on an extension calculus. A new method for approximating MVP to the FOPDT model is presented in this paper with more effectiveness.

First Order Plus Dead Time (FOPDT) Model Approximation and Proportional-Integral-Derivative Controllers Tuning for Multi-Volume Process

Analysis and controller design for the FOPDT model is widely used because of its simpleness and convenience. In this paper, the problem of the multi-volume process (MVP) being approximated to first order plus dead time (FOPDT) model, as well as its proportional-integral-derivative (PID) controller arguments tuning is studied. These have been heavily studied in recent years and the methods developed for its optimal design rely on the idea of including several robust performance specifications in the objective function the method presents fast convergence and consists of mentioning a desired closed-loop transfer function. Particle swarm optimization (PSO) algorithm and performance index of the integral of the time-weighted absolute error (ITAE) minimum is presented to determine the approximate FOPDT model coefficients of MVP processes, where the order is from two to fifteen. In addition, the approximate FOPDT model is used to design the PID controller, which is used to control MVP. A large number of tuning methods are provided to analyze and compare the closed-loop control performances. At the end of the paper, two simulation examples illustrate the superiority and effectiveness of the PID controller design based on the proposed model reduction method. The simulation results show that the reduced-order controller can control a high-order system well, but the process of order reduction is complicated and it needs a long computation time. The FOPID is a generalization of the conventional PID controller. This is based on an extension calculus. A new method for approximating MVP to the FOPDT model is presented in this paper with more effectiveness.

A model reduction method for nonlinear analysis of materials and structures with tension–compression asymmetric properties

A non-smooth tension–compression asymmetric constitutive law has important applications in modelling of bio-composite structures with a unilateral contact behavior and no-tension/compression structures. Quadratic programming algorithms based on Parametric Variational Principle (PVP) have good numerical robustness on this kind of issue. However, the challenge of computational cost prevents its application into some practical engineering structures with a large-scale degree of freedoms (DOFs). The paper proposes an efficient model reduction method based on the Extended Multiscale FEM (EMsFEM) for mechanical analysis of composite structures with tension–compression asymmetric properties. At the macro-scale, a reduced-order model is constructed and solved to get the displacement field, while a complementarity finite element formulation is proposed to get the stress field at the micro-scale. A two-scale iterative scheme bridging upscale and downscale computations is developed to deal with material nonlinearity. Numerical simulations show satisfactory computational accuracy of the proposed model reduction method, which can find applications in nacre-like composites and membrane antennae. Compared with the previous FEM-PVP, the proposed EMsFEM-PVP method saves computer’s memory and CPU time greatly. The study provides a generalized numerical way to handle large-scale non-smooth mechanics.

Characterization of single-cell biophysical properties and cell type classification using dielectrophoresis model reduction method

Characterization of single-cell biophysical properties and cell type classification are significant to study cytopathology, cell sorting, drug screening, etc. Existing methods usually suffer from unsatisfied measurement accuracy and low efficiency. We propose a new dielectrophoresis (DEP) model reduction method to characterize biophysical properties and classify cell types of single cells in situ by directly solving a reduced DEP model from DEP-based cell moving velocities. Cell radius, specific membrane capacitance and cytoplasm conductivity of single cells are directly calculated by using a solution method rather than traditional nonlinear DEP spectrum fitting. The frequencies of DEP signals used for determining the cellular parameters are optimized to achieve minimum errors. By this way, the measurement accuracy and efficiency are greatly improved. Using a homemade electrode-array chip allows simultaneous measurements on multiple single-cells and facilitates high-throughput analyses. We perform experimental measurements on four typical cell lines, including adherent cells (HeLa and MCF-7) and suspension cells (Jurkat and GM12878), cancer cells (HeLa, MCF-7 and Jurkat) and normal cells (GM12878), and demonstrate statistically significant differences in their biophysical properties, which can be used to classify cell types. The proposed model reduction method provides an accurate and efficient approach for characterizing single-cell biophysical properties and classifying cell types, while with real-time, in-situ, label-free, and less invasive advantages.

An adaptive reduced basis ANOVA method for high-dimensional Bayesian inverse problems

In Bayesian inverse problems sampling the posterior distribution is often a challenging task when the underlying models are computationally intensive. To this end, surrogates or reduced models are often used to accelerate the computation. However, in many practical problems, the parameter of interest can be of high dimensionality, which renders standard model reduction techniques infeasible. In this paper, we present an approach that employs the ANOVA decomposition method to reduce the model with respect to the unknown parameters, and the reduced basis method to reduce the model with respect to the physical parameters. Moreover, we provide an adaptive scheme within the MCMC iterations, to perform the ANOVA decomposition with respect to the posterior distribution. With numerical examples, we demonstrate that the proposed model reduction method can significantly reduce the computational cost of Bayesian inverse problems, without sacrificing much accuracy.

A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

AbstractIn this paper, we develop a reduced generalized multiscale basis method for efficiently solving the parametrized groundwater flow problems in heterogeneous porous media. Recently proposed generalized multiscale finite element (GMsFE) method is one of the accurate and efficient methods to solve the multiscale problems on a coarse grid. However, the GMsFE basis functions usually depend on the random parameters in the parametrized model, which substantially impacts on the computation efficiency when the parametrized equation needs to be solved many times for many instances of parameters. To enhance computation efficiency, we construct reduced generalized multiscale basis functions independent of parameters from the high‐dimensional GMsFE space and generate a reduced‐order multiscale model by projecting the parameterized flow governing equation onto the low‐dimensional reduced GMsFE space. Then, in order to improve the online computation of reduced‐order model, we apply a matrix discrete empirical interpolation method to affinely decompose the nonaffine parametrized systems arising from the discretization of governing equation. Finally, a few numerical experiments are carried out for the parametrized transient flow problems in heterogeneous porous media to illustrate the efficiency and accuracy of the proposed model reduction method. The results show that the proposed reduced generalized multiscale basis method can significantly improve computation efficiency while maintaining comparative accuracy for the groundwater flow problems by selecting a suitable coarse mesh size and an optimal number of reduced GMsFE basis functions at each of coarse nodes.

Moment matching model reduction for negative imaginary systems

In this paper, moment matching model reduction problem for negative imaginary systems is considered. For a given negative imaginary system with poles at the origin, our goal is to find a reduced-order negative imaginary system such that a prescribed number of the moments and the poles at the origin are preserved. Firstly, the original negative imaginary system is split into an asymptotically stable subsystem, a lossless negative imaginary subsystem and an average subsystem. Then, moment matching model reduction is implemented on the asymptotically stable subsystem and the lossless negative imaginary subsystem. The resulting reduced-order system preserves the negative imaginary structure and the poles at the origin. Also, the proposed model reduction method is extended to the positive real systems. Numerical examples demonstrate the effectiveness of the proposed model reduction method.

H2 model reduction for negative imaginary systems

This paper studies the model reduction problem for negative imaginary (NI) systems. For a given high-order system that is stable and NI, our goal is to approximate it by a low-order NI system so that the norm of the approximation error system is minimised. By using the Galerkin projection, the model reduction problem is formulated as a minimisation problem over the Stiefel manifold. The first-order necessary condition is derived for the construction of a local optimal reduced-order system. A gradient descent algorithm is provided to solve the first-order necessary condition. The resulting reduced-order system preserves the stability and the NI structure of the original system. Finally, two examples are presented to demonstrate the effectiveness of the proposed model reduction method.

Local–global model reduction method for stochastic optimal control problems constrained by partial differential equations

In this paper, a local–global model reduction method is presented to solve stochastic optimal control problems constrained by stochastic partial differential equations (stochastic PDEs). If the optimal control problems involve uncertainty, we need to use a few random variables to parameterize the uncertainty. The stochastic optimal control problems require solving coupled optimality system for a large number of samples in the stochastic space to quantify the statistics of the system response and explore the uncertainty quantification. Thus the computation is prohibitively expensive. To overcome the difficulty, model reduction is necessary to significantly reduce the computation complexity. We exploit the advantages from both reduced basis method and Generalized Multiscale Finite Element Method (GMsFEM) and develop the local–global model reduction method for stochastic optimal control problems with stochastic PDE constraints. This local–global model reduction can achieve much more computation efficiency than using only local model reduction approach and only global model reduction approach. We recast the stochastic optimal problems in the framework of saddle-point problems and analyze the existence and uniqueness of the optimal solutions of the reduced model. In the local–global approach, most of computation steps are independent of each other. This is very desirable for scientific computation. Moreover, the online computation for each random sample is very fast via the proposed model reduction method. This allows us to compute the optimality system for a large number of samples. To demonstrate the performance of the local–global model reduction method, a few numerical examples are provided for different stochastic optimal control problems.

An efficient iterative model reduction method for aeroviscoelastic panel flutter analysis in the supersonic regime

Abstract The flutter boundary prediction of complex aeroelastic systems is not an easy task. In some cases, these analyses may become prohibitive due to the high computational cost and time associated with the large number of degrees of freedom of the aeroelastic models, particularly when the aeroelastic model incorporates a control strategy with the aim of suppressing the flutter phenomenon, such as the use of viscoelastic treatments. In this situation, the use of a model reduction method is essential. However, the construction of a modal reduction basis for aeroviscoelastic systems is still a challenge, owing to the inherent frequency- and temperature-dependent behavior of the viscoelastic materials. Thus, the main contribution intended for the present study is to propose an efficient and accurate iterative enriched Ritz basis to deal with aeroviscoelastic systems. The main features and capabilities of the proposed model reduction method are illustrated in the prediction of flutter boundary for a thin three-layer sandwich flat panel and a typical aeronautical stiffened panel, both under supersonic flow.

Time‐weighted MR for switched LPV systems via balanced realisation

A new model reduction (MR) method is developed for continuous-time switched linear parameter-varying (LPV) systems based on time-weighted gramians. Using time-weighted energy functionals and parameter-independent multiple Lyapunov functions, the time-weighted controllability and observability gramians are obtained. With a balanced transformation matrix optimally derived, the authors show that a balanced realisation can be achieved for the original switched LPV systems. A series of low-order models for different weighting factors are yielded by state truncation or singular perturbation techniques. The reduced-order models can preserve the stability. The feasibility and effectiveness of the proposed MR method are illustrated by two examples including a three spring-mass system.

A general model reduction with primal assembly in structural dynamics

A new formulation of general model reduction in structural dynamics is presented. The proposed method consists of four attributes: a hybrid method combining dynamic condensation and component mode synthesis that are arguably the most popular model reduction techniques, a precise model reduction taking the inertia effect into consideration, primal assembly at the boundary interface of substructures and general formulations of model reduction with primal assembly. Comparisons of the proposed model reduction method with the conventional methods are provided as well as an efficient model reduction procedure of the proposed method. In particular, its advantages and performance of the proposed method are thoroughly investigated with various numerical examples.

Tangential Interpolatory Projection for Model Reduction of Linear Quantum Stochastic Systems

This paper presents a model reduction method for the class of linear quantum stochastic systems often encountered in quantum optics and related fields. The approach is proposed on the basis of an interpolatory projection ensuring that specific input-output responses of the original and the reduced-order systems are matched at multiple selected points (or frequencies). Importantly, the physical realizability property of the original quantum system imposed by the laws of quantum mechanics is preserved under our tangential interpolatory projection. Error bounds are established for the proposed model reduction method and an avenue to select interpolation points is proposed. A passivity preserving model reduction method is also presented. Examples of both active and passive systems are provided to illustrate the merits of our proposed approach.

H∞ model reduction for negative imaginary systems

ABSTRACTThis paper is concerned with the H∞ model reduction for negative imaginary (NI) systems. For a given linear time-invariant system that is stable and NI, our goal is to find a stable reduced-order NI system satisfying a pre-specified H∞ approximation error bound. Sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an H∞ reduced-order NI system. Iterative algorithms are provided to solve the matrix inequalities and to minimise the H∞ approximation error. Finally, a numerical example is used to demonstrate the effectiveness of the proposed model reduction method.

A Kernel-Based PCA Approach to Model Reduction of Linear Parameter-Varying Systems

This brief presents a model reduction method for linear parameter-varying (LPV) systems using kernel-based principal component analysis (PCA). For state-space LPV models that are affine or rational in the scheduling variables and in which the variation of these variables is confined in a polytope, controller synthesis can be elegantly realized by solving the synthesis problem only at the vertices of the polytope. To exploit the computational simplicity of this approach, it is highly desirable to obtain LPV models of systems of interest in an affine or a rational form. In this respect, kernel PCA allows one to extract principal components of a given data set of scheduling variables in a high-dimensional feature space, reducing complicated coefficient dependencies that otherwise might not be easily reducible in a linear subspace; this gives kernel PCA an advantage over its linear PCA counterpart. We show that high-dimensional scheduling variables can be mapped into a set of low-dimensional variables through a nonlinear kernel PCA-based mapping. Since the kernel PCA mapping is nonlinear, finding the inverse mapping in order to represent the original scheduling variables requires solving a nonlinear optimization problem; consequently, the reduced LPV model is no longer affine in the reduced scheduling variables. To address this, we formulate an optimization problem to obtain a reduced model that is either affine or rational in the reduced scheduling variables. We apply the proposed model reduction method on a robotic manipulator system and use the reduced LPV model to design a gain-scheduled controller that satisfies an induced $\mathcal {L}_{2}$ gain performance. Numerical simulations are used to demonstrate the performance of the resulting LPV controller on the nonlinear manipulator model. The achieved performance of the LPV controller with the kernel PCA-based reduced model is also compared with that of its linear PCA-based counterpart.

Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

Summary In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather post-processes a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their proper-orthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobility-related term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization.

A two-step model reduction approach for mechanical systems with moving loads

We consider model order reduction of mechanical systems with moving loads. Such systems have a time-varying input matrix that makes the direct application of standard model reduction methods difficult. In this paper, we present a two-step model reduction approach for systems with moving loads which is based on a low-rank approximation of the input matrix and applying Krylov subspace methods to the resulting linear time-invariant system with a modified input. Numerical results demonstrate the properties of the proposed model reduction method.