Standard Model Reduction Techniques Research Articles

Mesoscopic model reduction for the collective dynamics of sparse coupled oscillator networks.

The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques that capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages, we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from O(N2) to O(1). We perform numerical simulations for Erdős-Rényi networks and for modified Barabási-Albert networks demonstrating remarkable quantitative agreement at and close to bifurcation.

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 Method for the Hankel-Norm Approximation of Fractional-Order Systems

A model-reduction methodology for fractional-order systems based on the Hankel-norm is presented. The methodology involves the truncation of a Laurent series associated with the fractional-order system in a transformed domain. The truncated Laurent series coefficients are used to construct a finite-order transfer function to approximate the original system. Standard model-reduction techniques are then applied to obtain a final low-order approximation. The Hankel norm of the approximation error can be specified a priori. The approximation method is applied to several fractional-order and other infinite-order systems. It is shown to be more generally applicable than standard finite-order modeling techniques.

Including inputs and control within equation-free architectures for complex systems

The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.

Time-dependent Dirichlet conditions in finite element discretizations

Abstract For the modeling and numerical approximation of problems with time-dependent Dirichlet boundary conditions, one can call on several consistent and inconsistent approaches. We show that spatially discretized boundary control problems can be brought into a standard state space form accessible for standard optimization and model reduction techniques. We discuss several methods that base on standard finite element discretizations, propose a newly developed problem formulation, and investigate their performance in numerical examples. We illustrate that penalty schemes require a wise choice of the penalization parameters in particular for iterative solves of the algebraic equations. Incidentally, we confirm that standard finite element discretizations of higher order may not achieve the optimal order of convergence in the treatment of boundary forcing problems and that convergence estimates by the common method of manufactured solutions can be misleading.

Robust Gust Alleviation and Stabilization of Very Flexible Aircraft

Robust linear control, combined with model-reduction methodologies, is investigated for gust rejection on large, very flexible aircraft using trailing-edge control surfaces. Controllers are designed on linearizations of the nonlinear aeroelastic equations, and the closed-loop response of both the linearized and nonlinear system to discrete gust distributions is compared to the open-loop dynamics. Results show that an controller performs well on a relatively large linearized system, with 9% load alleviation in root bending moments for the critical gust length. When applied to the nonlinear model of the same vehicle, the robust controller gives a good performance in response to short gusts, including the critical length, with even better load reductions than the linear case. However, this performance gap decreases as the gust length is increased. It is also shown how standard model-reduction techniques can provide metrics for the selection of a minimum size of the aeroelastic system. Finally, it is shown that both the full and reduced controllers are capable of stabilizing a large-amplitude phugoid mode on the nonlinear model of the vehicle.

Parameter preserving model order reduction for MEMS applications

Model order reduction techniques are known to work reliably for finite element-type simulations of micro-electro-mechanical systems devices. These techniques can tremendously shorten computational times for transient and harmonic analyses. However, standard model reduction techniques cannot be applied if the equation system incorporates time-varying matrices or parameters that are to be preserved for the reduced model. However, design cycles often involve parameter modification, which should remain possible also in the reduced model. In this article we demonstrate a novel parameterization method to numerically construct highly accurate parametric ordinary differential equation systems based on a small number of systems with different parameter settings. This method is demonstrated to parameterize the geometry of a model of a micro-gyroscope, where the relative error introduced by the parameterization lies in the region of . We also present recent developments on semi-automatic order reduction methods that can preserve scalar parameters or functions during the reduction process. The first approach is based on a multivariate Padé-type expansion. The second approach is a coupling of the balanced truncation method for model order reduction of (deterministic) linear, time-invariant systems with interpolation. The approach is quite flexible in allowing the use of numerous interpolation techniques like polynomial, Hermite, rational, sinc and spline interpolation. As technical examples we investigate a micro anemometer as well as the gyroscope. Speed-up factors of 20–80 could be achieved, while retaining up to six parameters and keeping typical relative errors below 1%.

Fast broadband modeling of frequency-domain responses by piecewise interpolation

Broadband macromodeling of frequency domain responses by Vector Fitting can be computationally demanding in CPU time and memory. This paper presents a practical solution to reduce the computational cost that is involved with the modeling of high-order frequency responses. It applies a piecewise fitting strategy that makes use of a fast rational interpolation scheme to identify a representative set of data samples and an appropriate model order. This information is exploited by the Vector Fitting algorithm to extract the poles of the broadband macromodel in a reduced amount of time. The calculation of the residues is then solved as a linear approximation problem, and standard model reduction techniques can be applied as an optional step to remove pole redundancies in the model. It is shown by an example that substantial savings are obtained in terms of computation time and memory requirements, when compared to the standard fitting procedure.

Transformation of passive electrical networks with distributed PWL-sources using a Model Order Reduction

Abstract. In modeling of distributed systems with distributed sources large networks with RLC-elements and independent sources arise. This high complexity leads to a high effort in simulations. Therefore model reduction can be used to reduce these networks, preserving the behavior at the observed nodes in the networks. For the reduction of networks with a large number of independent sources only a weak reduction is enabled with standard model reduction techniques. In this paper an efficient reduction of networks with a large number of sources with piece-wise-linear waveforms is presented, using the decomposition of piece-wise-linear functions. With the proposed method a higher reduction of the network and/or a higher accuracy can be achieved with model reduction. The validity and efficiency of the proposed method is shown by reducing a RCI-Grid model.

FE‐Model Reduction for FE‐BE Coupling with Large Fluid‐Structure Interfaces

For the finite element method model, reduction techniques exist to represent the dynamic behavior of component substructures. Depending on the type of reduction method, the reduction basis contains constraint or attachment modes, which are computed for all structural degrees of freedom on an interface. The interface can either be defined by adjacent substructures or by coupling interfaces to other physical domains, as it is the case for FE‐BE coupled systems. A large interface thus leads to an increased size of the reduced order model and limits standard model reduction techniques to applications with small interfaces. In this work, interface reduction methods are investigated. Here, the size of the reduced order model is decreased by reducing the number of retained interface modes, while marginally increasing the reduction error. A direct reduction method based on strain‐‐energy considerations is presented. Additionally, an iterative reduction scheme is proposed which only adds a basis vector to the redu...

Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations

We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with time-independent advection and the linearized Navier-Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI/Smith-type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.

Optimal control of diffusion-convection-reaction processes using reduced-order models

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process.

OPTIMAL TRANSITION CONTROL OF DIFFUSION-CONVECTION-REACTION PROCESSES

An approach for the optimal transition control of diffusion-convection- reaction processes based on finite-dimensional models is presented. The finite-dimensional state-space models are constructed directly from the process PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. The optimal controller is designed based on the finite-dimensional state-space model using continuous-time linear quadratic regulator (LQR) techniques.

Identification of a Nonproportional Damping Matrix from Incomplete Modal Information

In modeling structures the damping matrix is the most difficult to represent. This is even more difficult in complicated structures that are not lightly damped. The work presented here provides a method of modeling the damping matrix of a structure from incomplete experimental data combined with a reasonable representation of the mass and stiffness matrices developed by finite element methods and reduced by standard model reduction techniques. The proposed technique uses the reduced mass and stiffness matrices and the experimentally obtained eigenvalues and eigenvectors in a weighted least squares or a pseudo-inverse scheme (depending on the number of the equations that are available) to solve for the damping matrix. The results are illustrated through several examples. As an indication of the accuracy of the method, fictitious examples where the damping matrix is originally known are considered. The proposed method identifies the exact viscous or hysteretic damping matrix by only using a partial set, half of the system’s eigenvalues and eigenvectors. The damping matrix is assumed to be real, symmetric, and positive semidefinite.

Model Reduction and Regular Perturbation for Optimal Control of Large Linear Systems

Synthesis of optimal or suboptimal state feedbacks with the help of perturbation techniques has been handled rather extensively in the literature (Kokotovic (1972), Sandell and others (1978)). The applicability of these formalisms to large systems has remained questionable because: a) it is difficult to find a suitable perturbation structure and b) feedback of the complete state is not feasibleIn this paper a ‘reversed’ approach has been taken. Through some standard model reduction techniques a good low order model is always achieved. Then a suitable regular perturbation structure is defined through the original and reduced models. In the second stage the optimal output feedback matrix is expanded in powers of a perturbation parameter. After the initial low order optimal design any further term in the expansion is calculated successively from the same linear low order equations, although each time with different right hand sides