Model Order Reduction Problem Research Articles

Regularization Paths for Re-Weighted Nuclear Norm Minimization

We consider a class of weighted nuclear norm optimization problems with important applications in signal processing, system identification, and model order reduction. The nuclear norm is commonly used as a convex heuristic for matrix rank constraints. Our objective is to minimize a quadratic cost subject to a nuclear norm constraint on a linear function of the decision variables, where the trade-off between the fit and the constraint is governed by a regularization parameter. The main contribution is an algorithm to determine the so-called approximate regularization path, which is the optimal solution up to a given error tolerance as a function of the regularization parameter. The advantage is that we only have to solve the optimization problem for a fixed number of values of the regularization parameter, with guaranteed error tolerance. The algorithm is exemplified on a weighted Hankel matrix model order reduction problem.

The Unifying Feature of Projection in Model Order Reduction

Abstract This paper considers the problem of model order reduction of linear systems with the emphasis on the common features of the main approaches. One of these features is the unifying role of operator projection in model reduction. It is shown how projections are implemented for different methods of model reduction and what their properties are. The other common feature is the subspaces where projections are defined. The main approaches for model reduction which are considered in the paper are balanced truncation, proper orthogonal decomposition and the Lanczos procedure from the Krylov subspace methods. It is shown that the range spaces of system gramians for balanced truncation and the range space of the reachability and observability matrices for the Lanczos procedure coincide. The connection between balanced truncation and the proper orthogonal decomposition method is also established. Therefore, the methods for model reduction are similar in terms of general operational principles, and differ mostly in their technical implementation. Several numerical examples are considered showing the validity of the proposed conjectures.

Forced harmonic response of sandwich plates with viscoelastic core using reduced-order model

A reduction method for forced harmonic response analysis of large-scale sandwich plates with viscoelastic core is proposed. The reduced-order model problem is developed by projecting the original problem onto low-dimensional subspaces of real and complex eigenvectors. The resulting problem is solved using the asymptotic numerical method in conjunction with automatic differentiation techniques. The response curves of the original non-reduced model are readily recovered, resulting in much faster computation, and, therefore, savings in storage and computational effort. The complex basis is effective in both low- and high-damping scenarios, while the real reduction basis performs well only for low damping.

Gramians Method in Researches of the Model Order Reduction and Dynamic LTI system Stability Problems

This paper provides a new approach to computing infinite and finite time Gramians and cross-Gramians relying on the use of the Laplacian transformation to matrix exponential time functions and expansion of the product of these functions. The expansions are bilinear and quadratic forms for sequences of Faddeev matrices generated by resolvents of original matrices. Matrix identities are obtained for bilinear and quadratic forms of these Faddeev sequences. Asymptotic expansions are found for expansion of Controllability and Observability Gramians of dynamic systems under control modes at time preceding the system arriving at the stability boundary. The common fact lies in the approach foundations: Controllability, Observability and Cross –Gramians may be represented as a quadratic or bilinear matrix forms sum, to be generated by the Lyapunov or Sylvester equation matrices.

A vertex cut algorithm for model order reduction of parasitic resistive networks

PurposeTo simulate large parasitic resistive networks, one must reduce the size of the circuit models through methods that are accurate and preserve terminal connectivity and network sparsity. The purpose here is to present such a method, which exploits concepts from graph theory in a systematic fashion.Design/methodology/approachThe model order reduction problem is formulated for parasitic resistive networks through graph theory concepts and algorithms are presented based on the notion of vertex cut in order to reduce the size of electronic circuit models. Four variants of the basic method are proposed and their respective merits discussed.FindingsThe algorithms proposed enable the production of networks that are significantly smaller than those produced by earlier methods, in particular the method described in the report by Lenaers entitled “Model order reduction for large resistive networks”. The reduction in the number of resistors achieved through the algorithms is even more pronounced in the case of large networks.Originality/valueThe paper seems to be the first to make a systematic use of vertex cuts in order to reduce a parasitic resistive network.

Fitness based Differential Evolution

Differential Evolution (DE) is a well known and simple population based probabilistic approach for global optimization. It has reportedly outperformed a few Evolutionary Algorithms and other search heuristics like Particle Swarm Optimization when tested over both benchmark and real world problems. But, DE, like other probabilistic optimization algorithms, sometimes exhibits premature convergence and stagnates at suboptimal point. In order to avoid stagnation behavior while maintaining a good convergence speed, a new position update process is introduced, named fitness based position update process in DE. In the proposed strategy, position of the solutions are updated in two phases. In the first phase all the solutions update their positions using the basic DE and in the second phase, all the solutions update their positions based on their fitness. In this way, a better solution participates more times in the position update process. The position update equation is inspired from the Artificial Bee Colony algorithm. The proposed strategy is named as Fitness Based Differential Evolution ( $$FBDE$$ ). To prove efficiency and efficacy of $$FBDE$$ , it is tested over 22 benchmark optimization problems. A comparative analysis has also been carried out among proposed FBDE, basic DE, Simulated Annealing Differential Evolution and Scale Factor Local Search Differential Evolution. Further, $$FBDE$$ is also applied to solve a well known electrical engineering problem called Model Order Reduction problem for Single Input Single Output Systems.

Model Order Reduction for Neutral Systems by Moment Matching

Circuits with delay elements are very popular and important in the simulation of very-large-scale integration (VLSI) systems. Neutral systems (NSs) with multiple constant delays (MCDs), for example, can be used to model the partial element equivalent circuits (PEECs), which are widely used in high-frequency electromagnetic (EM) analysis. In this paper, the model order reduction (MOR) problem for the NS with MCDs is addressed by moment matching method. The nonlinear exponential terms coming from the delayed states and the derivative of the delayed states in the transfer function of the original NS are first approximated by a Padé approximation or a Taylor series expansion. This has the consequence that the transfer function of the original NS is exponential-free and the standard moment matching method for reduction is readily applied. The Padé approximation of exponential terms gives an expanded delay-free system, which is further reduced to a delay-free reduced-order model (ROM). A Taylor series expansion of exponential terms lets the inverse in the original transfer function have only powers-of-s terms, whose coefficient matrices are of the same size as the original NS, which results in a ROM modeled by a lower-order NS. Numerical examples are included to show the effectiveness of the proposed algorithms and the comparison with existing MOR methods, such as the linear matrix inequality (LMI)-based method.

Model order-reduction for discrete-time switched linear systems

This article describes the problem of model order-reduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. This article investigates two novel approaches to model order-reduction. The first approach consists in evaluating the error approximation performance; the problems are solved using the robust stability results of the switched systems. The second approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. A numerical example shows the effectiveness of the proposed approaches.

Cognitive learning in differential evolution and its application to model order reduction problem for single-input single-output systems

Differential evolution (DE) is a well known and simple population based probabilistic approach for global optimization over continuous spaces. It has reportedly outperformed a few evolutionary algorithms and other search heuristics like the particle swarm optimization when tested over both benchmark and real world problems. DE, like other probabilistic optimization algorithms, has inherent drawback of premature convergence and stagnation. Therefore, in order to find a trade-off between exploration and exploitation capability of DE algorithm, a new parameter namely, cognitive learning factor (CLF) is introduced in the mutation process. Cognitive learning is a powerful mechanism that adjust the current position of individuals by the means of some specified knowledge (previous experience of individuals). The proposed strategy is named as cognitive learning in differential evolution (CLDE). To prove the efficiency of various approaches of CLF in DE, CLDE is tested over 25 benchmark problems. Further, to establish the wide applicability of CLF, CLDE is applied to two advanced DE variants. CLDE is also applied to solve a well known electrical engineering problem called model order reduction problem for single input single output systems.

On a Finiteness Result for the Number of Critical Points of the H2 Approximation Criterion

The long-standing open problem about whether the number of critical points in the H2 SISO real model order reduction problem is finite is answered in the positive in the case the transfer function of the to-be-reduced model has distinct poles (ie. only has poles of algebraic multiplicity one). This has important implications for various search methods for finding critical points or local minima of the criterion function for this model reduction problem. In fact more is shown namely that in a particular parametrization the number of complex solutions of the first order conditions of the H2 real model order reduction problem is finite and lies in a bounded set of which the bound can be determined from information about the to-be-reduced model. This implies that the H2 model order reduction problem can be solved in principle by constructive algebra methods (such as Groebner basis methods) in case the to-be-reduced model has distinct poles. It simplifies the methods for co-order three reduction as described in a companion paper.

Interpolation-Based ${\cal H}_2$-Model Reduction of Bilinear Control Systems

In this paper, we will discuss the problem of optimal model order reduction of bilinear control systems with respect to the generalization of the well-known ${\cal H}_2$-norm for linear systems. We revisit existing first order necessary conditions for ${\cal H}_2$-optimality based on the solutions of generalized Lyapunov equations arising in bilinear system theory and present an iterative algorithm which, upon convergence, will yield a reduced system fulfilling these conditions. While this approach relies on the solution of certain generalized Sylvester equations, we will establish a connection to another method based on generalized rational interpolation. This will lead to another way of computing the ${\cal H}_2$-norm of a bilinear system and will extend the pole-residue optimality conditions for linear systems, also allowing for an adaption of the successful iterative rational Krylov algorithm to bilinear systems. By means of several numerical examples, we will then demonstrate that the new techniques outperform the method of balanced truncation for bilinear systems with regard to the relative ${\cal H}_2$-error.

Extracting second-order structures from single-input state-space models: Application to model order reduction

Extracting second-order structures from single-input state-space models: Application to model order reductionThis paper focuses on the model order reduction problem of second-order form models. The aim is to provide a reduction procedure which guarantees the preservation of the physical structural conditions of second-order form models. To solve this problem, a new approach has been developed to transform a second-order form model from a state-space realization which ensures the preservation of the structural conditions. This new approach is designed for controllable single-input state-space realizations with real matrices and has been applied to reduce a single-input second-order form model by balanced truncation and modal truncation.

Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.

Parametric Model Reduction in the Loewner framework

AbstractWe address the problem of parametric model order reduction given measurements of the response of a device performed with respect to frequency, and also with respect to a design parameter. The proposed approach is based on a generalization of the Loewner matrix framework [1] to the case where data depend on two variables. The purpose of this note is to address issues arising in MIMO systems.

MODEL REDUCTION FOR DISCRETE-TIME SWITCHED LINEAR TIME-DELAY SYSTEMS VIA THE H∞ ROBUST STABILITY

This paper describes the problem of model order-reduction for a class of hybrid switched linear time-delay systems composed of linear discrete-time delayed invariant subsystems with a switching rule. The paper investigates a novel delay term-dependent approach to model reduction. It consists of evaluating the error approximation performance between the original system and the order-reduced system; the stability problem is solved under arbitrary switching rule. The model order-reduction conditions are given in the form of linear matrix inequalities with a term-independent delay. A numerical example shows the effectiveness of the proposed approach.

Krylov subspace methods for model order reduction of bilinear control systems

We discuss the use of Krylov subspace methods with regard to the problem of model order reduction. The focus lies on bilinear control systems, a special class of nonlinear systems, which are closely related to linear systems. While most existent approaches are based on series expansions around zero, we will extend the underlying ideas to a more general context and show that there exist several ways to reduce bilinear systems. Besides, we will briefly address the problem of stability preserving model reduction and further explain the benefit of using two-sided projection methods. By means of some numerical examples, we will illustrate the performance of the presented reduction methods.

Model and Controller Order Reduction for Infinite Dimensional Systems

This paper presents a reduced order model problem using reciprocal transformation and balanced truncation followed by low order controller design of infinite dimensional systems. The class of systems considered is that of an exponentially stable state linear systems

On the application of Karhunen–Loève transform to transient dynamic systems

The Karhunen–Loève transform (KLT) has become a popular method in various fields of engineering science. Due to its ability to identify the most prominent features in the underlying system dynamics the KLT is a favorable method for such tasks as process monitoring, model order reduction or optimum control. However, it is a well-known fact that the KLT is ‘case sensitive’. That is that changes in the dynamic system behavior can decisively affect the KLT results. As much as this property is desired for monitoring problems, it limits the performance of KLT in model order reduction or optimum control problems, if systems are subject to structural changes. Recent research interest focuses on extending applications of KLT to systems with transient dynamic behavior or changing boundary conditions. Approaches have been published that circumvent the limitations of KLT by either assuming reasonable comparability of system dynamics or by measuring the representative performance of KLT-bases a posteriori. However, such methods require additional simulations of the full size system and thus jeopardize the idea of model order reduction. In this paper, we introduce a novel a priori measure to evaluate the performance of the current KLT-basis. This procedure can be of great help in either monitoring or adaptive control of systems that show intermittent transient and (quasi-)stationary dynamic behavior. This a priori measure prepares the path for adaptive model order reduction schemes. Moreover, it can be used to measure the stationarity of multidimensional dynamic processes.

Noniterative ${\cal H}_{\infty }$-Based Model Order Reduction of LTI Systems Using LMIs

This brief addresses the model order reduction problem for linear time invariant (LTI) systems. The problem is to minimize the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> norm of the error between the original system and the reduced system, which is known to be a nonconvex optimization problem. Based on this, the brief proposes a convex suboptimal solution which is expressed in terms of linear matrix inequalities (LMIs). The performance of the proposed method is assessed through application to several large order systems and compared with the well-known Hankel Norm model order reduction method.

Efficient solution of large scale Lyapunov and Riccati equations arising in model order reduction problems

AbstractModel order reduction of large–scale linear time–invariant systems is an omnipresent task in control and simulation of complex dynamical processes. The solution of large scale Lyapunov and Riccati equations is a major task, e.g., in balanced truncation and related model order reduction methods, in particular when applied to semi–discretized partial differential equations constraint control problems. The software package LyaPack has shown to be a valuable tool in the task of solving these equations since its introduction in 2000.Here we want to discuss recent improvements and extensions of the underlying algorithms and their implementation. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)