Model Reduction Of Dynamical Systems Research Articles

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.

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

Robustness guarantees for structured model reduction of dynamical systems with applications to biomolecular models

AbstractModel reduction methods usually focus on the error performance analysis; however, in presence of uncertainties, it is important to analyze the robustness properties of the error in model reduction as well. This problem is particularly relevant for engineered biological systems that need to function in a largely unknown and uncertain environment. We give robustness guarantees for structured model reduction of linear and nonlinear dynamical systems under parametric uncertainties. We consider a model reduction problem where the states in the reduced model are a strict subset of the states of the full model, and the dynamics for all of the other states are collapsed to zero (similar to quasi‐steady‐state approximation). We show two approaches to compute a robustness guarantee metric for any such model reduction—a direct linear analysis method for linear dynamics and a sensitivity analysis based approach that also works for nonlinear dynamics. Using the robustness guarantees with an error metric and an input‐output mapping metric, we propose an automated model reduction method to determine the best possible reduced model for a given detailed system model. We apply our method for the (1) design space exploration of a gene expression system that leads to a new mathematical model that accounts for the limited resources in the system and (2) model reduction of a population control circuit in bacterial cells.

Projection-based model reduction of dynamical systems using space–time subspace and machine learning

This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional spatiotemporal output quantities of interest, such as pressure, temperature and displacement fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using space–time bases combined with machine learning methods. In particular, the space–time solutions are sought in a low-dimensional space–time linear trial subspace that can be obtained by computing tensor decompositions of usual state-snapshots data. The mapping between the input parameters and the basis expansion coefficients (or generalized coordinates) is approximated using four different machine learning techniques: multivariate polynomial regression, k-nearest-neighbors, random forests and neural networks. The relative costs and effectiveness of the four machine learning techniques are explored through three engineering problems: steady heat conduction, unsteady heat conduction and unsteady advective–diffusive–reactive system. Numerical results demonstrate that the proposed method performs well in terms of both accuracy and computational cost, and highlights the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.

Preserving general physical properties in model reduction of dynamical systems via constrained‐optimization projection

AbstractModel‐reduction techniques aim to reduce the computational complexity of simulating dynamical systems by applying a (Petrov–)Galerkin projection process that enforces the dynamics to evolve in a low‐dimensional subspace of the original state space. Frequently, the resulting reduced‐order model (ROM) violates intrinsic physical properties of the original full‐order model (e.g., global conservation, Lagrangian structure, state‐variable bounds) because the projection process does not generally ensure preservation of these properties. However, in many applications, ensuring the ROM preserves such intrinsic properties can enable the ROM to retain physical meaning and lead to improved accuracy and stability properties. In this work, we present a general constrained‐optimization formulation for projection‐based model reduction that can be used as a template to enforce the ROM to satisfy specific properties on the kinematics and dynamics. We introduce constrained‐optimization formulations at both the time‐continuous (i.e., ODE) level, which leads to a constrained Galerkin projection, and at the time‐discrete level, which leads to a least‐squares Petrov–Galerkin projection, in the context of linear multistep schemes. We demonstrate the ability of the proposed formulations to equip ROMs with desired properties such as global energy conservation and bounds on the total variation.

Windowed least-squares model reduction for dynamical systems

This work proposes a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space–time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov–Galerkin (LSPG), and space–time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space–time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space–time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space–time residuals than Galerkin and LSPG projection.

Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evolve in a linear subspace imposes a fundamental limitation to the accuracy of the resulting reduced-order model (ROM). In particular, linear-subspace ROMs can be expected to produce low-dimensional models with high accuracy only if the problem admits a fast decaying Kolmogorov n-width (e.g., diffusion-dominated problems). Unfortunately, many problems of interest exhibit a slowly decaying Kolmogorov n-width (e.g., advection-dominated problems). To address this, we propose a novel framework for projecting dynamical systems onto nonlinear manifolds using minimum-residual formulations at the time-continuous and time-discrete levels; the former leads to manifold Galerkin projection, while the latter leads to manifold least-squares Petrov–Galerkin (LSPG) projection. We perform analyses that provide insight into the relationship between these proposed approaches and classical linear-subspace reduced-order models; we also derive a posteriori discrete-time error bounds for the proposed approaches. In addition, we propose a computationally practical approach for computing the nonlinear manifold, which is based on convolutional autoencoders from deep learning. Finally, we demonstrate the ability of the method to significantly outperform even the optimal linear-subspace ROM on benchmark advection-dominated problems, thereby demonstrating the method's ability to overcome the intrinsic n-width limitations of linear subspaces.

Structure-preserving model reduction for dynamical systems with a first integral

Since the expense of the numerical integration of large scale dynamical systems is often computationally prohibitive, model reduction methods, which approximate such systems by simpler and much lower order ones, are often employed to reduce the computational effort. In this paper, for dynamical systems with a first integral, new structure-preserving model reduction approaches are presented that yield reduced-order systems while preserving the first integral. We apply energy-preserving integrators to the reduced-order systems and show some numerical experiments that demonstrate the favourable behaviour of the proposed approaches.

Optimal Hankel norm model reduction for discrete-time descriptor systems

Optimal Hankel norm model reduction for dynamical systems is of great significance in model-based simulation and design. For the class of linear time-invariant systems, it is among the few optimal reduction methods for which a prior error bound between the original system and its approximation is known. However, for descriptor systems, this optimal approximation technique no longer applies. In this paper, we propose several definitions of the Hankel operator for dynamical discrete-time descriptor systems. We investigate the implications of these definitions for the problem of optimal model approximation of descriptor systems in the sense of the Hankel norm. Novel reduction algorithms are derived for this class of systems with and without preservation of the DAE-index. The performance of the proposed methods is illustrated by numerical examples.

Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives

Abstract We consider two proper orthogonal decomposition (POD) methods for dimension reduction of dynamical systems. The first method (M1) uses only time snapshots of the solution, while the second method (M2) augments the snapshot set with time-derivative snapshots. The goal of the paper is to analyze and compare the approximation errors resulting from the two methods by using error bounds. We derive several new bounds of the error from POD model reduction by each of the two methods. The new error bounds involve a multiplicative factor depending on the time steps between the snapshots. For method M1 the factor depends on the second power of the time step, while for method 2 the dependence is on the fourth power of the time step, suggesting that method M2 can be more accurate for small between-snapshot intervals. However, three other factors also affect the size of the error bounds. These include (i) the norm of the second (for M1) and fourth derivatives (M2); (ii) the first neglected singular value and (iii) the spectral properties of the projection of the system’s Jacobian in the reduced space. Because of the interplay of these factors neither method is more accurate than the other in all cases. Finally, we present numerical examples demonstrating that when the number of collected snapshots is small and the first neglected singular value has a value of zero, method M2 results in a better approximation.

Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries.

We demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters. First, we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low-rank coherent structures. Here, libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points. The selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics, stability, and bifurcations of nonlinear dynamical systems. The use of empirical interpolation points and sparse representation facilitates a family of local reduced-order models for each physical regime, rather than a higher-order global model, which has the benefit of physical interpretability of energy transfer between coherent structures. The method advocated also allows for orders-of-magnitude improvement in computational speed and memory requirements. To illustrate the method, the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex Ginzburg-Landau equation. It is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present.

New structural damage-identification method using modal updating and model reduction

Early prediction of damages using vibration signal is essential in avoiding the failure in structures. Among different damage-detection approaches, the finite-element model updating and modal analysis-based methods are of most importance due to their applicability and feasibility. Owing to some restrictions in nodal measurements in experimental cases, finite-element model reduction is an indispensable part of fault-detection methods. Even though model reduction of dynamic systems leads to the less complicated models, an improved convergence rate and acceptable accuracy are highly required for a successful structural health monitoring of the real complex systems. In this paper, the aim is to design a damage-detection algorithm based on a new model updating method, which has a faster rate of convergence and higher accuracy. Then the proposed method is applied on a simulated damaged beam considering different noise levels to see how capable the method is in dealing with noise-corrupted data. Finally, the experimentally extracted data from a cracked beam in a real noisy condition are used to evaluate the efficiency of the proposed method in identifying the damages in a beam-like structure. It is concluded that the identification of the damages by the proposed method is encouraging and robust to the noise compared with the traditional method. Also, the proposed method converges faster and is more accurate in identifying damage than the traditional method.

An Online Manifold Learning Approach for Model Reduction of Dynamical Systems

This article discusses a newly developed online manifold learning method, subspace iteration using reduced models (SIRM), for the dimensionality reduction of dynamical systems. This method may be viewed as subspace iteration combined with a model reduction procedure. Specifically, starting with a test solution, the method solves a reduced model to obtain a more precise solution, and it repeats this process until sufficient accuracy is achieved. The reduced model is obtained by projecting the full model onto a subspace that is spanned by the dominant modes of an extended data ensemble. The extended data ensemble in this article contains not only the state vectors of some snapshots of the approximate solution from the previous iteration but also the associated tangent vectors. Therefore, the proposed manifold learning method takes advantage of the information of the original dynamical system to reduce the dynamics. Moreover, the learning procedure is computed in the online stage, as opposed to being computed offline, which is used in many projection-based model reduction techniques that require prior calculations or experiments. After providing an error bound of the classical POD-Galerkin method in terms of the projection error and the initial condition error, we prove that the sequence of approximate solutions converge to the actual solution of the original system as long as the vector field of the full model is locally Lipschitz on an open set that contains the solution trajectory. Good accuracy of the proposed method has been demonstrated in two numerical examples, from a linear advection-diffusion equation to a nonlinear Burgers equation. In order to save computational cost, the SIRM method is extended to a local model reduction approach by partitioning the entire time domain into several subintervals and obtaining a series of local reduced models of much lower dimensionality. The accuracy and efficiency of the local SIRM are shown through the numerical simulation of the Navier--Stokes equation in a lid-driven cavity flow problem.

Model reduction for dynamical systems with quadratic output

SUMMARYFinite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices. For large‐scale finite element models, the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost. Padé via Krylov methods are widely used and are appreciated projection‐based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of the Krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems. Three different two‐sided model reduction approaches are formulated based on the Krylov methods. For all methods, the control (or right) Krylov space is the same. The difference between the approaches lies, thus, in the choice of the observation (or left) Krylov space. The algorithms and theory are developed for the particularly important case of structural damping. We also give numerical examples for large‐scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge. In this case, a block form of the Padé via Krylov method is used. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Model Reduction for Natural Gas Pipeline Systems

Many methods exist for model reduction of dynamical systems. However, only in a few cases a method can be picked from the literature and applied just like pushing a button. The structure and properties of the target system need to be investigated prior to model reduction. Natural gas pipeline systems are complex, large-scale systems with a special structure which must be recognized. Model reduction methods suitable for interconnected systems are tested on a true natural gas pipeline system with 600 state variables and almost 200 input variables.

Model Reduction for Dynamical Systems with Local Nonlinearities

Many kinds of nonlinear engineering systems have a large number of degrees of freedom while nonlinear components are spatially localized. Even though the local nonlinearities constitute only a small part of the system, the dynamic behavior of the system is wholly nonlinear, and the analysis of the whole model is very expensive. To reduce the computational effort signie cantly, two model reduction schemes are proposed to reduce the size of the nonlinear model before the analysis is performed. One is dee ned on the system level, and the other is on the substructure level. They are both based on the dynamic condensation technique. In the former scheme the dynamic condensation technique is implemented into the linear part of the whole system directly, whereas it is applied to the linear e exible substructures in the later scheme. The accuracy and stability of the former is usually better than the latter. However, it is more computationally expensive than the latter. Two numerical examples are also included. The results show that the present two schemes are feasible and efe cient to reduce the degrees of freedom of the model with local nonlinearities.

Model reduction for dynamical systems with local nonlinearities

Model reduction for dynamical systems with local nonlinearities

Class of stabilising decentralised controllers for interconnected dynamical systems

A method is presented for the system- atic design of stabilising decentralised controllers for large-scale interconnected dynamical systems. The design method is based on (i) decentralised implementation of global controllers obtained by using existing global controller design methods, (ii) model reduction of dynamical systems, and (iii) modelling of the interactions among the sub- systems comprising the global closed-loop control system. This is used for generating local corrective control signals, which would account for these interactions. The resulting decentralised controller uses local information only to generate local control inputs. It comprises local state feedback controllers and feedforward compensators. Three distinct schemes for the implementation of the controller are proposed. They all guarantee near optimal performance. A four-station intercon- nected dynamical system is considered. Results of time simulation studies demonstrate that the overall performance of the system under the pro- posed decentralised controller is near that of the global optimal controller.

Interaction modelling approach to distributed control with application to power systems

A class of systematic distributed control design methods suitable for large-scale interconnected power systems is presented in this paper. The design methods are based on: (i)distributed implementations of centralized control systems; (ii) model reduction of dynamical systems; and (iii) modelling of the interactions between the subsystems comprising the global control system. In these schemes the control task is decomposed into local subtasks. The function of each subtask is to generate the required local control input so that, collectively, the centralized performance of the global system is preserved. The local control input is composed of a control signal generated from local state feedback and a corrective control signal generated from low-order dynamic and, in some schemes, static compensators. The structure of these compensators is determined from the reduced-order model of the global closed-loop control system and from derived models of the interactions among the subsystems. A two-area interconnect...