Model Reduction Of Systems Research Articles

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Model reduction of dynamical systems with a novel data-driven approach: The RC-HAVOK algorithm.

This paper introduces a novel data-driven approximation method for the Koopman operator, called the RC-HAVOK algorithm. The RC-HAVOK algorithm combines Reservoir Computing (RC) and the Hankel Alternative View of Koopman (HAVOK) to reduce the size of the linear Koopman operator with a lower error rate. The accuracy and feasibility of the RC-HAVOK algorithm are assessed on Lorenz-like systems and dynamical systems with various nonlinearities, including the quadratic and cubic nonlinearities, hyperbolic tangent function, and piece-wise linear function. Implementation results reveal that the proposed model outperforms a range of other data-driven model identification algorithms, particularly when applied to commonly used Lorenz time series data.

The nearest graph Laplacian in Frobenius norm

AbstractWe address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.

Parameterization of nonlinear aeroelastic reduced order models via direct interpolation of Taylor partial derivatives

The identification of optimally sparse Taylor partial derivatives presents a new opportunity in efficient nonlinear model reduction for complex aeroelastic systems. Unfortunately, for this class of reduced order model (ROM), the robustness that is observed in the linear regime to parameters including; dynamic pressure, control hinge linear stiffness, or even freeplay, can be quickly compromised in the nonlinear regime. In this paper, the nonlinear sensitivity of selected critical parameters is addressed by interpolating a library of nonlinear unsteady aerodynamic ROMs across a compact subspace in dynamic pressure and freeplay magnitude. The ROM, based on Lagrange interpolation of sparse higher-order Taylor partial derivatives, demonstrates excellent precision in modelling high amplitude transonic limit cycle oscillations for an all-movable wing with freeplay, capturing the LCO region (up to 96% of the linear flutter boundary), and for a range of freeplay values.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

GIẢM BẬC MÔ HÌNH ĐỘNG HỌC SỬ DỤNG CẮT NGẮN PHƯƠNG THỨC TRONG BÀI TOÁN CHUYỂN ĐỘNG TÒA NHÀ

Modal truncation, an advanced algorithm for model reduction in dynamic systems, efficiently simplifies complex models by selectively discarding less influential eigenmodes, maintaining a balance between computational efficiency and model accuracy. This paper explores the algorithm's application to a 48th order building model. Proceed to reduce this model to lower orders, then analyze errors in time and frequency domains. Modal truncation algorithm systematically reduces model dimensions while preserving critical dynamic attributes. Numerical simulations reveal a favorable reduction order range (from order 6th to order 25th) for optimal balance, with sensitivity observed at order 25th. From the results obtained, depending on specific requirements, users can use a lower-order model corresponding to the allowed error to replace the original system. Recommendations include iterative refinement for adaptive reduction orders and in-depth analysis around critical points. This algorithm becomes an effective method for researchers dealing with high-dimensional dynamic systems, offering simpler yet accurate model representations. As technology develops, continued refinements and applications of modal truncation are expected, solidifying its role in the realm of model reduction.

Wireless Power Transfer System Model Reduction with Split Frequency Matching

Reduced-order dynamic models of wireless power transfer (WPT) systems are desired to simplify the analysis and design of power control, phase synchronization, and maximum efficiency tracking. The reduced-order dynamic phasor model is a good choice because of its straightforward physical meaning and concise mathematical formula. However, the model relies on the assumption of loose coupling and loses accuracy when the coupling becomes stronger. In this paper, a model reduction method with split frequency matching is proposed to improve model accuracy under relatively strong coupling conditions, which is suitable for most short-distance WPT applications, such as wireless electrical vehicle charging. Split frequency matching is achieved through a pair of conjugate equivalent mutual inductances, which are derived from the asymmetry characteristics of the full-order dynamic phasor model in the positive and negative frequency domains. The proposed model retains the advantages of the existing model while significantly improving the accuracy under strong coupling conditions. Its characteristics are verified by comparing the experimental results and model predictions under both large step changes and small-signal perturbations.

Coprime Factor Model Reduction of Multidimensional Systems

In this work, the right coprime factor for balancing and truncating unstable systems is extended to parameter-varying multidimensional systems employing recent works on coprime factor model reduction of one-dimensional uncertain systems. Since the balanced truncation method cannot be applied directly to unstable systems, state feedback gains should be computed and incorporated in order to stabilize the given system and to be able to apply the balanced truncation technique via defining the so-called stable coprime factor, as coprime factorization overcomes the stability condition required for model reduction. Parameter-dependent state feedback and parameter-dependent Gramians are considered in this work yielding less conservative techniques. In addition, the Gramians are defined as block diagonal matrices, which are partitioned according to the structure of the multidimensional systems. The application to a simulation example demonstrates the applicability and validity of the proposed reduction approach leading to small error bounds between the full and the reduced models.

A new method for model reduction and controller design of large-scale dynamical systems

A new method for model reduction and controller design of large-scale dynamical systems

Dynamic–quadratic balancing: A computational approach to balancing and model reduction for affine nonlinear systems

AbstractIn this paper, we discuss a new approach to balancing (known as dynamic–quadratic balancing) and model reduction for affine nonlinear system. We give a fresh look to balancing in terms of the dynamics of the system, rather than simply a structural property. With this perspective, we also develop a new approach to obtaining the balancing transformation in one step, instead of a three‐step process as proposed in earlier methods. Further, we explore the relationship between quadratic balancing and input–output stability. In addition, we also develop a computational approach for obtaining the balancing transformation by solving a coupled system of PDEs or inequalities (Lyapunov/Hamilton–Jacobi type). After that, model reduction can be carried out in the conventional way using Hankel singular‐value functions or using a new criterion. Finally, we present some examples to clarify the results.

Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

Accurate error estimation is crucial in model order reduction, to obtain small reduced-order models as well as to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.

Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs

Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs

Modular model reduction of interconnected systems: A robust performance analysis perspective

Many complex engineering systems consist of multiple subsystems that are developed by different teams of engineers. To analyse, simulate and control such complex systems, accurate yet computationally efficient models are required. Modular model reduction, in which the subsystem models are reduced individually, is a practical and an efficient method to obtain accurate reduced-order models of such complex systems. However, when subsystems are reduced individually, without taking their interconnections into account, the effect on stability and accuracy of the resulting reduced-order interconnected system is difficult to predict. In this work, a mathematical relation between the accuracy of reduced-order linear-time invariant subsystem models and (stability and accuracy of) resulting reduced-order interconnected linear time-invariant model is introduced. This result can subsequently be used in two ways. Firstly, it can be used to translate accuracy characteristics of the reduced-order subsystem models directly to accuracy properties of the interconnected reduced-order model. Secondly, it can also be used to translate specifications on the interconnected system model accuracy to accuracy requirements on subsystem models that can be used for fit-for-purpose reduction of the subsystem models. These applications of the proposed analysis framework for modular model reduction are demonstrated on an illustrative structural dynamics example.

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications

Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications—symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems—with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization algorithm.

Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov N-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.

Multi‐fidelity error estimation accelerates greedy model reduction of complex dynamical systems

AbstractModel order reduction usually consists of two stages: the offline stage and the online stage. The offline stage is the expensive part that sometimes takes hours till the final reduced‐order model is derived, especially when the original model is very large or complex. Once the reduced‐order model is obtained, the online stage of querying the reduced‐order model for simulation is very fast and often real‐time capable. This work concerns a strategy to speed up the offline stage of model order reduction for large and complex systems. In particular, it is successful in accelerating the greedy algorithm that is often used in the offline stage for reduced‐order model construction. We propose to replace the high‐fidelity error estimator in the greedy algorithm with multi‐fidelity error estimation. Consequently, the computational complexity of the greedy algorithm is reduced and the algorithm converges more than two times faster without incurring noticeable accuracy loss.

Canonical and noncanonical Hamiltonian operator inference

A method for the nonintrusive and structure-preserving model reduction of canonical and noncanonical Hamiltonian systems is presented. Based on the idea of operator inference, this technique is provably convergent and reduces to a straightforward linear solve given snapshot data and gray-box knowledge of the system Hamiltonian. Examples involving several hyperbolic partial differential equations show that the proposed method yields reduced models which, in addition to being accurate and stable with respect to the addition of basis modes, preserve conserved quantities well outside the range of their training data.