Model Reduction Of Systems Research Articles

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.

Comparison of uncertainty quantification process using statistical and data mining algorithms

Uncertainty quantification has always been an important topic in model reduction and simulation of complex systems. In this aspect, global sensitivity analysis (GSA) methods such as Fourier amplitude sensitivity test (FAST) are well recognized as effective algorithms. Recently, some data-based metamodeler such as Random Forest (RF) also developed their own variable importance selection solutions for parameters with perturbations. This paper proposes a visual comparison of these two uncertainty quantification methods, using datasets retrieved from vibroacoustic models. Their results have a lot in common and are capable to explain many results. The remarkable agreement between methods under fundamentally different definitions can potentially improve their compatibility in various occasions.

Structure‐preserving Galerkin approximation for a class of nonlinear port‐Hamiltonian partial differential equations on networks

AbstractThe development of structure‐preserving approximation methods, which regard fundamental underlying physical principles, is an active field of research. Especially when the application of model reduction is desirable, systematic and rather generic approaches are of great interest. In this contribution we discuss a structure‐preserving Galerkin approach for a prototypical class of nonlinear partial differential equations on networks. Its derivation is guided by port‐Hamiltonian‐type modeling and appropriate variational principles. Also complexity‐reduction schemes can be integrated in a structure‐preserving way, which becomes crucial in the context of model reduction for nonlinear systems.

A bilinear identification‐modeling framework from time domain data

AbstractAn ever‐increasing need for improving the accuracy includes more involved and detailed features, thus inevitably leading to larger‐scale dynamical systems [1]. To overcome this problem, efficient finite methods heavily rely on model reduction. One of the main approaches to model reduction of both linear and nonlinear systems is by means of interpolation. The Loewner framework is a direct data‐driven method able to identify and reduce models derived directly from measurements. For measured data in the frequency domain, the Loewner framework is well established in linear case [2] while it has already extended to nonlinear [6]. On the other hand in the case of time domain data, the Loewner framework was already applied for approximating linear models [3]. In this study, an algorithm which uses time domain data for nonlinear (bilinear) system reduction and identification is presented.

Time domain model reduction of time-delay systems via orthogonal polynomial expansions

This paper investigates model reduction of time-delay systems in the time domain based on the expansion of systems over orthogonal polynomials. Time-delay systems are first expanded over generalized Laguerre polynomials. The nice properties of generalized Laguerre polynomials lead to a direct system expansion and a Sylvester equation with special structures which enables an efficient calculation of Laguerre coefficients of systems. Projection methods are then adopted to generate reduced models, and we show that a desired number of Laguerre coefficients are preserved by reduced models. Further, we extend the proposed method to general orthogonal polynomials, where the relationship between Taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time-delay systems. Systems with multiple delays and delays appearing in the derivative of the states are also discussed. Two numerical examples are simulated to showcase the efficiency of our approach.

Model reduction in power systems using a structure-preserving balanced truncation approach

This paper contributes to the development of a computationally efficient technique for reducing large-scale power system models. The proposed method comprises two steps: the partitioning of the power system into a study area and a set of coherent clusters, and the model reduction of the remote clusters. The focus of this work is on the second step. Every remote cluster is handled separately during the reduction process. For each of them, a simple and effective model of the surrounding network is developed in order to preserve a proper dynamic behavior while limiting the computing load. The reduced clusters are then mutually reconnected to one another, which enables to keep the structural information through the lines linking these clusters. Numerical results on the IEEE 118-bus system model show that the new method is able to build a reduced model suitable for performing rotor angle stability studies.

A Tangential Block Lanczos Method for Model Reduction of Large-Scale First and Second Order Dynamical Systems

In this paper, we present a new approach for model reduction of large scale first and second order dynamical systems with multiple inputs and multiple outputs. This approach is based on the projection of the initial problem onto tangential subspaces to produce a simpler reduced-order model that approximates well the behaviour of the original model. We present an algorithm named: adaptive block tangential Lanczos-type algorithm. We give some algebraic properties and present some numerical experiences to show the effectiveness of the proposed algorithms.

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.

Balanced Model Reduction for Linear Time-Varying Symmetric Systems

The goal of this paper is to develop balancing theory for the linear time-varying (LTV) symmetric systems. To this end, first, we extend the concept of symmetry in terms of the dual system. Then, we define the cross Gramian for the LTV systems. For LTV symmetric systems, we establish a connection among the controllability, observability, and cross Gramians. In particular, if one of these three Gramians is obtained, the other two Gramians can be constructed. Based on this fact, we show that the symmetry structure is preserved under balanced truncation if the Hankel singular values are pointwise distinct.

Model Reduction of Markovian Jump Systems With Uncertain Probabilities

This paper studies the problem of model reduction for nonhomogeneous Markovian jump systems. The transition probability matrix of the nonhomogeneous Markovian chain has the characteristic of a polytopic structure. An asynchronous reduced-order model is considered, and the asynchronization is modeled by a hidden Markov model with a partially unknown conditional probability matrix. Under this framework, a new sufficient condition is proposed to ensure that the augmented system is stochastically mean-square stable with a specified level of $H_\infty$ performance. Finally, a numerical example is provided to show the effectiveness and advantages of the theoretic results obtained.

Frequency Limited Model Reduction Techniques for Discrete-Time Systems

The main drawback of pioneer frequency limited Gramians-based technique by Wang and Zilouchain for discrete time system is that it yields unstable reduced order models (ROMs) of stable original system. There exist some techniques in literature that produce stable ROMs, however, these techniques yield relatively large approximation error. In this brief, three modified techniques of model reduction for discrete time systems are being presented. These techniques preserve the stability in ROMs, mostly yield low approximation error (when compared to existing stability preserving techniques) and additionally contain frequency response error bounds. Several related examples are also illustrated. Comparison of results show the effectiveness of proposed techniques.

A tangential method for the balanced truncation in model reduction

In this paper, we present a new approach for large-scale Lyapunov matrix equations, where we present two algorithms named: Adaptive Block Tangential Lanczos-type and Arnoldi-type algorithms (ABTL and ABTA). This approach is based on the projection of the initial problem onto tangential Krylov subspaces to produce a low-rank approximate solution of large Lyapunov equations. These approximations are used in model reduction of large-scale dynamical systems with multiple inputs and multiple outputs (MIMO). We give some algebraic properties and present some numerical experiences to show the effectiveness of the proposed algorithms.

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.

Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation

In this article we study balanced model reduction of linear control systems using the singular perturbation approximation. Balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions, with one of their most important features being a priori L2 and H∞ bounds for the approximation error. The main focus of this work is to derive an L2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions, extending related work on balanced truncation. This L2 error bound measures the difference between the input-output maps of the original and of the reduced initial value systems. The advantages and flexibility of this approach are demonstrated with a variety of numerical examples.

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.

A trust-region method for H2 model reduction of bilinear systems on the Stiefel manifold

This paper investigates the Riemannian trust-region method for H2 model reduction of bilinear systems. The H2 error norm is treated as a cost function on the Stiefel manifold such that the orthogonality constraint for the projection matrix is plainly satisfied. The property related to the Euclidean gradient is studied. Then, the inner product associated with the Riemannian Hessian is derived, which can simplify the expression of the trust-region subproblem. The trust-region method for H2 model reduction is accordingly established and the convergence is further discussed. Finally, two numerical examples are employed to demonstrate the performance of the proposed method.

Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations

To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

Riemannian Optimal Model Reduction of Stable Linear Systems

In this paper, we develop a method for solving the problem of minimizing the $H^2$ error norm between the transfer functions of original and reduced systems on the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The stability of the reduced systems constructed using the optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. Numerical experiments demonstrate that our method considerably improves the results given by BT in the sense of the $H^2$ norm, and also provides reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty}$ error norm. Moreover, we show that our method provides a better reduced model than BT from the viewpoint of the frequency response.

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

This work proposes a space--time least-squares Petrov--Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction meth...

Interpolatory model reduction for quadratic-bilinear systems using error estimators

Purpose The purpose of this paper is to propose an interpolation-based projection framework for model reduction of quadratic-bilinear systems. The approach constructs projection matrices from the bilinear part of the original quadratic-bilinear descriptor system and uses these matrices to project the original system. Design/methodology/approach The projection matrices are constructed by viewing the bilinear system as a linear parametric system, where the input associated with the bilinear part is treated as a parameter. The advantage of this approach is that the projection matrices can be constructed reliably by using an a posteriori error bound for linear parametric systems. The use of the error bound allows us to select a good choice of interpolation points and parameter samples for the construction of the projection matrices by using a greedy-type framework. Findings The results are compared with the standard quadratic-bilinear projection methods and it is observed that the approximations through the proposed method are comparable to the standard method but at a lower computational cost (offline time). Originality/value In addition to the proposed model order reduction framework, the authors extend the one-sided moment matching parametric model order reduction (PMOR) method to a two-sided method that doubles the number of moments matched in the PMOR method.