Model Reduction Of Nonlinear Systems Research Articles

Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

Nonlinear Model Reduction in the Loewner Framework

We introduce a novel method of model reduction for nonlinear systems by extending the Loewner framework developed for linear time-invariant systems. This objective is achieved by defining Loewner functions obtained by utilizing a state-space interpretation of the Loewner matrices. A Loewner equivalent model using these functions is derived. This allows constructing reduced order models achieving interpolation in the Loewner sense.

Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

This paper presents a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.

Model order reduction for gas and energy networks

To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model reduction, yet, large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic) equation systems, modeling natural gas transport, are a challenging application for model order reduction algorithms.For this industrial application, we bring together the scientific computing topics of: mathematical modeling of gas transport networks, numerical simulation of hyperbolic partial differential equation, and parametric model reduction for nonlinear systems. This research resulted in the morgen (Model Order Reduction for Gas and Energy Networks) software platform, which enables modular testing of various combinations of models, solvers, and model reduction methods. In this work we present the theoretical background on systemic modeling and structured, data-driven, system-theoretic model reduction for gas networks, as well as the implementation of morgen and associated numerical experiments testing model reduction adapted to gas network models.

Spatial Dynamics Representation and Recoverable One-degree Model Reduction for Nonlinear Systems

Abstract A new concept of a dynamical system representation is proposed. The idea of the representation is simple: combination of nonlinear coordinate transformation and time scale transformation. Introducing a projected referential trajectory and a moving frame on it, the coordinate transformation separates a system state into sub-states tangent to the trajectory and orthogonal one. Then time scale transformation using the arc length of the trajectory yields a state equation in which the state evolves spatially around the trajectory. As one of the advantages of the proposed method, time scale transformation enables us to ignore dynamics of the tangent sub-state at the same time. Therefore, the proposed representation realizes one-degree model reduction for smooth control systems. Since all transformation used in this article is invertible, this reduction is recoverable.

Truncated model reduction methods for linear time-invariant systems via eigenvalue computation

This paper provides three model reduction methods for linear time-invariant systems in the view of the Riemannian Newton method and the Jacobi-Davidson method. First, the computation of Hankel singular values is converted into the linear eigenproblem by the similarity transformation. The Riemannian Newton method is used to establish the model reduction method. Besides, we introduce the Jacobi-Davidson method with the block version for the linear eigenproblem and present the corresponding model reduction method, which can be seen as an acceleration of the former method. Both the resulting reduced systems can be equivalent to the reduced system originating from a balancing transformation. Then, the computation of Hankel singular values is transformed into the generalized eigenproblem. The Jacobi-Davidson method is employed to establish the model reduction method, which can also lead to the reduced system equivalent to that resulting from a balancing transformation. This method can also be regarded as an acceleration of a Riemannian Newton method. Moreover, the application for model reduction of nonlinear systems with inhomogeneous conditions is also investigated.

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.

Model Reduction of Nonlinear Systems by Trajectory Piecewise Linear Based on Output-Weighting Models: A Balanced-Truncation Methodology

This paper extends some methods in model order reduction (MOR) of nonlinear models by trajectory piecewise-linear models based on output weighting (TPWLOW). The devised methods generalize the balanced-truncation method as the core for the proposed algorithms. These algorithms preserve the stability in some mentioned conditions. An error bound of reduction is calculated for these algorithms. In addition, a Krylov–TBR combinational MOR is also casted which benefits from the advantages of both balanced-truncation and Krylov subspace methods. The results show that in balanced-truncation-based algorithms, the efficiency of the first proposed algorithm fairly exceeds the second one. In addition, it is shown that the Krylov–TBR algorithm can reach the same efficiency despite remarkable lower order, provided that the designer can find a stable subspace. The robustness of TPWLOW-reduced models to linearization point movement is also evaluated, which shows the less sensitivity of the proposed methods in comparison with formerly introduced trajectory piecewise-linear models.

A Reduced Order Modeling Methodology for Steam Turbine Clearance Control Design

Finite element models (FEMs) are extensively used in the design optimization of utility scale steam turbines. As an example, by simulating multiple startup scenarios of steam power plants, engineers can obtain turbine designs that minimize material utilization, and at the same time, avoid the damaging effects of large thermal stresses or rubs between rotating and stationary parts. Unfortunately, FEMs are computationally expensive and only a limited amount of simulations can be afforded to get the final design. For this reason, numerous model reduction techniques have been developed to reduce the size of the original model without a significant loss of accuracy. When the models are nonlinear, as is the case for steam turbine FEMs, model reduction techniques are relatively scarce and their effectiveness becomes application dependent. Although there is an abundant literature on model reduction for nonlinear systems, many of these techniques become impractical when applied to a realistic industrial problem. This paper focuses on a class of nonlinear FEM characteristic of thermo-elastic problems with large temperature excursions. A brief overview of popular model reduction techniques is presented along with a detailed description of the computational challenges faced when applying them to a realistic problem. The main contribution of this work is a set of modifications to existing methods to increase their computational efficiency. The methodology is demonstrated on a steam turbine model, achieving a model size reduction by four orders of magnitude with only 4% loss of accuracy with respect to the full order FEMs.

Generalized Differential Balancing for Variationally Symmetric Systems

Abstract: In this paper, we introduce the notion of variational symmetry for nonlinear prolonged systems (a variational extension of a nonlinear system) by using corresponding variational systems and by developing the recently introduced generalized differential balancing method for this type of system. Generalized differential balancing is proposed for model reduction of nonlinear systems in the contraction framework and is based on solutions to two nonlinear differential matrix inequalities. Our aim of this paper is to show that for variationally symmetric systems, these two inequalities reduce to one inequality. Finally, we show that variational symmetry is preserved under generalized balanced truncation if the variational Hankel singular values are distinct.

Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates

United States. Air Force Office of Scientific Research. Dynamic Data-Driven Application Systems Program (Grant FA9550-11-1-0339)

Model Reduction for Nonlinear Systems by Incremental Balanced Truncation

In this paper, the method of incremental balanced truncation is introduced as a tool for model reduction of nonlinear systems. Incremental balanced truncation provides an extension of balanced truncation for linear systems towards the nonlinear case and differs from existing nonlinear balancing techniques in the definition of two novel energy functions. These incremental observability and incremental controllability functions form the basis for a model reduction procedure in which the preservation of stability properties is guaranteed. In particular, the property of incremental stability, which provides a notion of stability for systems with nonzero inputs, is preserved. Moreover, a computable error bound is given. Next, an extension towards so-called generalized incremental balanced truncation is proposed, which provides a reduction technique with increased computational feasibility at the cost of a (potentially) larger error bound. The proposed reduction technique is illustrated by means of application to an example of an electronic circuit with nonlinear elements.

Model reduction for nonlinear systems with incremental gain or passivity properties

In this paper, model reduction techniques for a class of nonlinear systems are proposed. Specifically, nonlinear systems are considered that can be decomposed as the feedback interconnection of a high-order linear subsystem and a nonlinear subsystem of relatively low order, allowing for the application of well-developed reduction techniques for linear systems. In this setting, conditions are given under which internal stability, as well as passivity or a bound on the L2 gain are preserved for the reduced-order nonlinear model. Additionally, a priori error bounds are given. In the derivation of the error bound, an incremental gain (or incremental passivity) property of the nonlinear subsystem is shown to be instrumental. Additionally, the techniques developed in this paper are applied in the scope of controller reduction, as is illustrated by means of an industrial temperature control benchmark example.

Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds

A methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented. The approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations. Due to the existence of external and parametric periodic excitations, however, the geometry of the manifold varies with time. As a result, the manifold is constructed in terms of temporal and dominant state variables. The governing partial differential equation (PDE) for the manifold is nonlinear and contains time-varying coefficients. An approximate technique to find solution of this PDE using a multivariable Taylor–Fourier series is suggested. It is shown that, in certain cases, it is possible to obtain various reducibility conditions in a closed form. The case of time-periodic systems is handled through the use of Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant; however, the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric excitation frequencies. This warrants some structural changes in the proposed manifold, but the solution procedure remains the same. Two examples; namely, a 2-dof mass–spring–damper system and an inverted pendulum with periodic loads, are used to illustrate applications of the technique for systems with constant and periodic coefficients, respectively. Results show that the dynamics of these 2-dof systems can be accurately approximated by equivalent 1-dof systems using the proposed methodology.

Impedance Identification by POD Model Reduction Techniques (Impedanz-Identifikation mittels POD Modellreduktion)

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of nonlinear systems. It is based on a Galerkin type discretization with basis elements created from the nonlinear system itself. In this paper an identification problem arising in car vehicle acoustics is considered. To estimate the impedance from point-wise sound pressure values the identification problem is formulated in terms of an optimal control problem, which is solved by a globalized quasi-Newton method. For the efficient and fast numerical realization, a POD based model reduction is utilized.

Proper orthogonal decomposition for optimality systems

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled dynamics in the proper orthogonal decomposition approach to optimal control. It is referred to as optimality system proper orthogonal decomposition (OS-POD).

Model reduction of nonlinear systems

AbstractThe paper is concerned with the model reduction of nonlinear systems. Such methods are required in all branches of engineering. Often the level of detail in system models can cloud the essential behaviour and hence, methods are required to identify the behaviour of interest to the designer. In this contribution, the particular focus is on the empirical balanced truncation method of model reduction. A standard test case will illustrate the efficacies of the suggested approach. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Model reduction of nonlinear systems

This paper presents the application in circuit simulation of a method for model reduction of nonlinear systems that has recently been developed for chemical systems. The technique is an extension of the well‐known balanced truncation method that has been applied extensively in the reduction of linear systems. The technique involves the formation of controllability and observability gramians either by simulated results or by measurement data. The empirical gramians are subsequently employed to determine a subspace of the full state‐space that contains the most significant dynamics of the system. A Galerkin projection is used to project the system onto the subspace to form a lower‐dimensional nonlinear model. The method is applied to a nonlinear resistor network which is a standard example for exemplifying the effectiveness of a nonlinear reduction strategy.

On the exploitation of chaos to build reduced-order models

The present study focuses on the model reduction of non-linear systems. The proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement. These eigenmodes, called the proper orthogonal modes, are optimal with respect to energy content and are used to build a low-dimensional model of the non-linear system. For this purpose, the proper orthogonal modes obtained from a chaotic orbit are considered. Indeed, such an orbit is assumed to cover a portion of the phase space of higher dimension, and hence of greater measure. This higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower-dimensional periodic orbit. In an example, it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non-chaotic response. This is applied to a buckled beam with two permanent magnets and the reduced-order model is validated using both qualitative and quantitative comparisons.

Trajectory-based model reduction of nonlinear systems

A nonlinear system is studied around some specific trajectories. Linearization around the trajectories is used to estimate the state error caused by omitting some components in the model. A method for adjustment of the remaining model is also suggested. The main results are applied to an exemple in terms of a rotating pendulum.