Approximate Reduction of Dynamical Systems (original) (raw)

Approximate reduction of dynamic systems

Systems & Control Letters, 2008

The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an "exact" manner-as is the case with mechanical systems with symmetry-which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples. arXiv:0707.3804v1 [math.OC]

Approximate reduction of dynamic systemsI

The reduction of dynamic systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an "exact" manner -as is the case with mechanical systems with symmetry -which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e. when it is behaviourally similar to a dynamic system on a lower dimensional space. These concepts are illustrated on a series of examples.

Reduction in Dynamical Systems: A Representational View

2010

ABSTRACT: According to the received view, reduction is a deductive relation between two formal theories. In this paper, I develop an alternative approach, according to which reduction is a representational relation between models, rather than a deductive relation between theories; more specifically, I maintain that this representational relation is the one of emulation. To support this thesis, I focus attention on mathematical dynamical systems and I argue that, as far as these systems are concerned, the emulation relation is sufficient for reduction. I then extend this representational view of reduction to the case of empirically interpreted dynamical systems, as well as to a treatment of partial, approximate, and asymptotic reduction.

A Representational Approach to Reduction in Dynamical Systems

Erkenntnis, 79(4):943-968, 2014

According to the received view, reduction is a deductive relation between two formal theories. In this paper, I develop an alternative approach, according to which reduction is a representational relation between models, rather than a deductive relation between theories; more specifically, I maintain that this representational relation is the one of emulation. To support this thesis, I focus attention on mathematical dynamical systems and I argue that, as far as these systems are concerned, the emulation relation is sufficient for reduction. I then extend this representational model-based view of reduction to the case of empirically interpreted dynamical systems, as well as to a treatment of partial, approximate, and asymptotic reduction.

Robust and Dynamically Consistent Model Order Reduction for Nonlinear Dynamic Systems

Journal of Dynamic Systems, Measurement, and Control, 2014

There is a great importance for faithful reduced order models that are valid over a range of system parameters and initial conditions. In this paper, we demonstrate through two nonlinear dynamic models (pinned-pinned beam and thin plate) that are both randomly and periodically forced that smooth orthogonal decomposition (SOD)-based reduced order models (ROMs) are valid over a wide operating range of system parameters and initial conditions when compared to proper orthogonal decomposition (POD)-based ROMs. Two new concepts of subspace robustness-the ROM is valid over a range of initial conditions, forcing functions, and system parameters-and dynamical consistency-the ROM embeds the nonlinear manifold-are used to show SODs ability in developing robust faithful ROMs that embed the nonlinear manifold in a lower dimensional space than POD when the ROM is used to approximate a system other than the one that it was created from.

A GENERALIZED REDUCTION PROCEDURE FOR DYNAMICAL SYSTEMS

Modern Physics Letters A, 1991

A generalized reduction procedure for general dynamical systems is presented in an algebraic setting. This procedure extends previously defined reduction procedures for symplectic and Poisson systems. The algebraic formulation of this procedure allow to extend it to include Nambu structures as well and a reduction theorem for Nambu rings is proved.

An interpolation-based approach to ℋ∞ model reduction of dynamical systems

2010

We introduce an interpolatory approach to H∞ model reduction for large-scale dynamical systems. Guided by the optimality conditions of for best uniform rational approximants on the unit disk, our proposed method uses the freedom in choosing the d-term in the reduced order model to enforce 2r + 1 interpolation conditions in the right-half plane for any given reduction order, r. 2r of these points are initialized by the Iterative Rational Krylov Algorithm of ; and then the d-term is chosen to minimize the H∞ error for this initial set of interpolation points. Several numerical examples illustrate the effectiveness of the proposed method. It consistently yields better results than balanced truncation. In all cases examined its performance is very close to or better than that of Hankel norm approximation. For the special case of state-space symmetric systems, important properties are established. Finally, we examine H∞ model reduction from a potential theoretic perspective and present a second methodology for choosing interpolation points.

On exact/approximate reduction of dynamical systems living on piecewise linear partition

2009

Order reduction problem for dynamical systems living on piecewise linear partitions is addressed in this paper. This problem is motivated by analysis and control of hybrid systems. The technique presented is based on the transformation of affine dynamical systems inside the cells into a new structure and it can be applied for both exact reduction and also approximate model reduction. In this method both controllability and observability of the affine system inside the polytopes are considered for the reduction purpose. The framework is illustrated with a numerical example.

A comprehensive scheme for reduction of nonlinear dynamical systems

International Journal of Dynamics and Control, 2019

Model order reduction (MOR) also known as dimension reduction is a computational tool to obtain cost-effective lower order approximations of large scale dynamical systems. This paper presents a detailed yet simplified MOR approach using nonlinear moment matching (NLMM) in conjuncture with the Discrete Empirical Interpolation Method (DEIM). NLMM avoids the expensive simulation of the underlying nonlinear Sylvester partial differential equation by reducing it to a system of nonlinear algebraic equations using proper step-by-step simplifications. This reduces the offline computational cost of generating the orthonormal projection basis substantially. This is followed by the DEIM algorithm, resulting in comprehensive savings in computational resources. The proposed algorithms are tested on two benchmark problems and the results so obtained are compared with proper orthogonal decomposition for different test inputs.