Model Order Reduction for Magneto-Quasistatic Equations

The problem of model reduction of linear systems with certain interconnection structure is considered in this paper. To preserve the interconnection structure between subsystems in the reduction, special care needs to be taken. This problem is important and timely because of the recent focus on complex networked systems in control engineering. Two different model reduction methods are introduced and compared in this paper. Both methods are extensions to the well‐known balanced truncation method. Compared with earlier work in the area these methods use a more general linear fractional transformation framework, and utilize linear matrix inequalities. Furthermore, new approximation error bounds that reduce to classical bounds in special cases are derived. The so‐called structured Hankel singular values are used in the methods, and indicate how important states in the subsystems are with respect to a chosen input–output map for the entire interconnected system. It is shown how these structured Hankel singular values can be used to select an approximation order. Finally, the two methods are applied to a model of a mechanical device. Copyright © 2008 John Wiley & Sons, Ltd.

This paper is concerned with the problem of ℋ ∞ model reduction for linear distributed-delay systems. For a given stable system containing both discrete and distributed delays in the state, our attention is focused on the construction of reduced-order model of the same structure, which guarantees the corresponding error system to be asymptotically stable and has a prescribed H∞ norm error performance. This problem is solved through a delay-dependent approach. By defining a new Lyapunov functional and making use of the Newton-Leibniz formula, a delay-dependent bounded real lemma is first established in terms of linear matrix inequalities (LMIs). Based on this performance criterion and by employing the projection lemma, sufficient conditions are formulated for the existence of admissible reduced-order models. Owing to the presence of matrix inverse constraints, these conditions are intrinsically nonconvex. An algorithm based on the sequential linear programming matrix method is proposed for verifying these conditions, which can be readily implemented by using standard numerical software. Desired reduced-order models can be constructed if these conditions are satisfied. The effectiveness of the proposed model reduction method is illustrated via a numerical example.

We discuss algorithms for balanced truncation (BT) based model reduction of linear systems. BT is known to have good global approximation properties and to preserve important system properties. A computable error bound allows to choose the order of the reduced-order model adaptively. We will emphasize those aspects that makes the application of BT to models arising in circuit simulation a non-straightforward task. In recent years, these issues have been addressed by several authors. We will survey some of these developments and demonstrate that BT is now suitable for linear descriptor systems encountered in circuit simulation.

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time cases are considered. A numerical example is provided to show the effectiveness of the proposed theory.

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

This article is concerned with the problem of the Hankel-norm model reduction for stochastic discrete-time nonlinear systems in interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy framework. The IT2 T-S fuzzy model is an efficient model for describing uncertain nonlinear systems, and the model reduction is to simplify the high-order complex systems by reducing the order of the original system. The aim of this article is to reduce the order of the original stochastic discrete-time IT2 fuzzy system into lower order system without ignoring the influence of IT2 membership functions. First, the Hankel-norm performance of the stochastic discrete-time IT2 fuzzy model is analyzed. Then, based on the projection theorem and cone complementary linearization approach, a convex Hankel-norm-based model reduction approach subject to conditions in the form of linear matrix inequalities (LMIs) is obtained. A membership-functions-dependent (MFD) technique is applied to capture the information of IT2 membership functions and further reduce the conservativeness. A numerical example is presented to illustrate the effectiveness of the proposed results.

In this paper a new balanced realization method for nonlinear systems is proposed which is based on singular value analysis of Hankel operators. The proposed method balances the relationship between the input-to-state behavior and the state-to-output behavior of nonlinear dynamical systems, whereas the existing results only balance the relationship among the coordinate axes of the state-space. This result is expected to be a basis for new model reduction and system identification of nonlinear systems.

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

This work proposes a model-reduction methodology that both preserves Lagrangian structure and leads to computationally inexpensive models, even in the presence of high-order nonlinearities. We focus on parameterized simple mechanical systems under Rayleigh damping and external forces, as structural-dynamics models often t this description. The proposed model-reduction methodology directly approximates the quantities that dene the problem’s Lagrangian structure: the Riemannian metric, the potential-energy function, the dissipation function, and the external force. These approximations preserve salient properties (e.g., positive deniteness), behave similarly to the functions they approximate, and ensure computational eciency. Results applied to a simple parameterized trussstructure problem demonstrate the importance of preserving Lagrangian structure and illustrate the method’s ability to generate speedups while maintaining observed stability, in contrast with other model-reduction techniques that do not preserve structure.

An effective nonlinear model reduction approach, empirical Gramians balanced reduction approach, is studied, in order to reduce the computation complexity in nonlinear power system model application. The realization procedure is: firstly, computing the empirical controllable and observable Gramians matrices, secondly, computing the balance transformation matrix to obtain the balanced system model of the original model, then, and obtaining the diagonal Hankel singular values of balance transformation matrix. Finally, deciding the lower-order subspace to obtain the reduced model. A 15-machine nonlinear power system model is taken as an example to perform the reduction simulation analysis. Simulation result shows that, under the condition to maintain the stability and input-output dynamic behavior of the original system, the order of the reduced system model could be as low as 1/5 of that of the original system model, verifying the effectiveness of the proposed approach.

Mathematical models of networked systems usually take the form of large-scale, nonlinear differential equations. Model reduction is a commonly used technique for understanding and analyzing systems of this size, by producing simplified yet accurate descriptions for them. Most available reduction methods work well for linear system descriptions or small-scale nonlinear system descriptions but they usually involve a state transformation to 'balance' the system before truncation. However, linear or nonlinear state combinations destroy the system structure that is important for drawing conclusions about the original networked system from the reduction. In this paper we propose an algorithmic methodology for model order reduction of nonlinear systems, without inducing state transformations. A priority list of states to be collapsed according to the estimated worst-case 2-norm of the error between the outputs of the original and reduced systems is produced. The main advantage of the method is that the states of the reduced system are a subset of the states of the original system.

The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. Results are derived on subsystem truncation of such systems and it is shown that this truncation preserves the physical realizability property of linear quantum stochastic systems. It is also shown that the property of complete passivity of linear quantum stochastic systems is preserved under subsystem truncation. A necessary and sufficient condition for the existence of a balanced realization of a linear quantum stochastic system under sympletic transformations is derived. Such a condition turns out to be very restrictive and will not be satisfied by generic linear quantum stochastic systems, thus necessary and sufficient conditions for relaxed notions of simultaneous diagonalization of the controllability and observability Gramians of linear quantum stochastic systems under symplectic transformations are also obtained. The notion of a quasi-balanced realization is introduced and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. Moreover, an explicit bound for the subsystem truncation error on a quasi-balanceable linear quantum stochastic system is provided. The results are applied in an example of model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities.

Model reduction by walsh function techniques

An identification approach to three-phase power system modeling and model reduction taking the power system network as multi-input, multi-output (MIMO) process is presented. The model estimate can be obtained in discrete-time input-output form, discrete- or continuous-time state-space variable form, or frequency-domain impedance transfer function matrix form. An algorithm for determining the model structures of this MIMO process is described. The effect of measurement noise on the approach is also discussed. This approach has been applied to a sample system. Simulation results are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Dimensionally reduced Krylov subspace model reduction for large scale systems