Showing papers on "Model order reduction published in 2013"
TL;DR: In this paper, a simple approach to load frequency control (LFC) design for the power systems having parameter uncertainty and load disturbance is proposed, which is based on two-degree-of-freedom, internal model control (IMC) scheme, which unifies the concept of model-order reduction like Routh and Pade approximations.
Abstract: The large-scale power systems are liable to performance deterioration due to the presence of sudden small load perturbations, parameter uncertainties, structural variations, etc. Due to this, modern control aspects are extremely important in load frequency control (LFC) design of power systems. In this paper, the LFC problem is illustrated as a typical disturbance rejection as well as large-scale system control problem. For this purpose, simple approach to LFC design for the power systems having parameter uncertainty and load disturbance is proposed. The approach is based on two-degree-of-freedom, internal model control (IMC) scheme, which unifies the concept of model-order reduction like Routh and Pade approximations, and modified IMC filter design, recently developed by Liu and Gao [24]. The beauty of this paper is that in place of taking the full-order system for internal-model of IMC, a lower-order, i.e., second-order reduced system model, has been considered. This scheme achieves improved closed-loop system performance to counteract load disturbances. The proposed approach is simulated in MATLAB environment for a single-area power system consisting of single generating unit with a non-reheated turbine to highlight the efficiency and efficacy in terms of robustness and optimality.
258 citations
TL;DR: In this paper, a model-order reduction procedure based on the Pade approximation method is used to reduce the partial differential equation model to a low-order system of ordinary differential equations.
182 citations
TL;DR: The reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries are extended, focusing on the role played by the Brezzi’s and Babuška's stability constants.
Abstract: In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi's and Babuska's stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi's saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuska's inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.
142 citations
TL;DR: It is shown that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates.
124 citations
01 Jan 2013
TL;DR: The balanced truncation method as mentioned in this paper is based on transforming the state-space system into a balanced form so that its controllability and observability Gramians become diagonal and equal, and the states that are difficult to reach or to observe, are truncated.
Abstract: Optimal control problems for partial differential equation are often hard to tackle numerically because their discretization leads to very large scale optimization problems. Therefore, different techniques of model reduction were developed to approximate these problems by smaller ones that are tractable with less effort. Balanced truncation [2, 66, 81] is one well studied model reduction technique for state-space systems. This method utilizes the solutions to two Lyapunov equations, the so-called controllability and observability Gramians. The balanced truncation method is based on transforming the state-space system into a balanced form so that its controllability and observability Gramians become diagonal and equal. Moreover, the states that are difficult to reach or to observe, are truncated. The advantage of this method is that it preserves the asymptotic stability in the reduced-order system. Furthermore, a-priori error bounds are available. Recently, the theory of balanced truncation model reduction was extended to descriptor systems; see, e.g., [50] and [21]. Recently the application of reduced-order models to linear time varying and nonlinear systems, in particular to nonlinear control systems, has received
118 citations
TL;DR: The methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable is developed and recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems.
Abstract: We propose a suitable model reduction paradigm -- the certied reduced basis method (RB) -- for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations (PDEs). In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable. Firstly, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimate on the state, control and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.
116 citations
TL;DR: A novel approach for the numerical simulation of nonlinear, hyperelastic soft tissues at kilohertz feedback rates necessary for haptic rendering based upon the use of proper generalized decomposition techniques, a generalization of PODs is introduced.
Abstract: We introduce here a novel approach for the numerical simulation of nonlinear, hyperelastic soft tissues at kilohertz feedback rates necessary for haptic rendering. This approach is based upon the use of proper generalized decomposition techniques, a generalization of PODs. Proper generalized decomposition techniques can be considered as a means of a priori model order reduction and provides a physics-based meta-model without the need for prior computer experiments. The suggested strategy is thus composed of an offline phase, in which a general meta-model is computed, and an online evaluation phase in which the results are obtained at real time. Results are provided that show the potential of the proposed technique, together with some benchmark test that shows the accuracy of the method.
80 citations
TL;DR: A new method of model order reduction is introduced by combining the merits of big bang big crunch (BBBC) optimization technique and stability equation (SE) method, which reduces a linear-continuous single-input single-output system of higher order to a lower order system.
Abstract: A new method of model order reduction is introduced by combining the merits of big bang big crunch (BBBC) optimization technique and stability equation (SE) method. A linear-continuous single-input single-output system of higher order is considered and reduced to a lower order system. The denominator polynomial of the reduced system is obtained by SE method, whereas the numerator terms are generated using BBBC optimization. Furthermore, step and frequency responses of the original reduced system are plotted. The superiority of the proposed method is justified by solving numerical examples from the available literature and comparing the reduced systems in terms of error indices.
71 citations
TL;DR: In this paper, the balanced truncation model order reduction for symmetric second-order systems is considered, where the large-scale generalized and structured Lyapunov equations are solved with a specially adapted low-rank alternating directions implicit (ADI) type method.
Abstract: We consider balanced truncation model order reduction for symmetric second-order systems. The occurring large-scale generalized and structured Lyapunov equations are solved with a specially adapted low-rank alternating directions implicit (ADI) type method. Stopping criteria for this iteration are investigated, and a new result concerning the Lyapunov residual within the low-rank ADI method is established. We also propose a goal-oriented stopping criterion which tries to incorporate the balanced truncation approach already during the ADI iteration. The model reduction approach using the ADI method with different stopping criteria is evaluated on several test systems.
69 citations
TL;DR: This paper replaces the standard first-order condition by the relaxed first- order condition, which is more suitable when algebraic reduced models are used as surrogate models, and proposes two optimization algorithms that uses the error bound to define a trust region and penalizes the objective with theerror bound.
Abstract: Design optimization problems are often formulated as an optimization problem whose objective is a function of the output of a large-scale parametric linear system, obtained from the discretization of a PDE. To reduce the high computational cost of the objective and its gradient, model order reduction techniques can be used. This paper uses interpolatory reduced models as surrogate models in an optimization procedure. We replace the standard first-order condition by the relaxed first-order condition, which is more suitable when algebraic reduced models are used as surrogate models. The relaxed first-order condition imposes that the approximation quality of the surrogate model at the interpolation point can be measured and refined and that the surrogate model is equipped with an error bound on the entire parameter space. We propose two optimization algorithms: one uses the error bound to define a trust region and the other penalizes the objective with the error bound. We prove convergence of both methods un...
64 citations
TL;DR: In this article, the normal modes of the linearized structure are used as boundary conditions on a detailed model reduced to the joints only, since contact non-linearities alter mode shapes, they are corrected as vibrational energy increases.
TL;DR: By applying the new approach, the POD reduced order model is implemented on an unstructured mesh finite element fluid flow model, and is applied to 3D flows, and the feasibility and accuracy of the reduced order models applied to3D fluid flows are demonstrated.
Abstract: A new scheme for implementing a reduced order model for complex mesh-based numerical models (e.g. finite element unstructured mesh models), is presented. The matrix and source term vector of the full model are projected onto the reduced bases. The proper orthogonal decomposition (POD) is used to form the reduced bases. The reduced order modeling code is simple to implement even with complex governing equations, discretization methods and nonlinear parameterizations. Importantly, the model order reduction code is independent of the implementation details of the full model code. For nonlinear problems, a perturbation approach is used to help accelerate the matrix equation assembly process based on the assumption that the discretized system of equations has a polynomial representation and can thus be created by a summation of pre-formed matrices. In this paper, by applying the new approach, the POD reduced order model is implemented on an unstructured mesh finite element fluid flow model, and is applied to 3D flows. The error between the full order finite element solution and the reduced order model POD solution is estimated. The feasibility and accuracy of the reduced order model applied to 3D fluid flows are demonstrated.
TL;DR: In this paper, the interpolation technique on Grassmann manifolds is used to reduce the computational complexity of model order reduction for models whose system matrices depend affinely on parameters by analyzing the structure of sums of singular value decompositions.
Abstract: SUMMARY
Model order reduction helps to reduce the computational time in dealing with large dynamical systems, for example, during simulation, control, optimization. In many cases, the considered model depends on parameters; Model order reduction techniques are, therefore, preferred to symbolically preserve this dependence or to be adaptive to the change of the model caused by the variation in the values of the parameters. In this paper, we first present the application of the interpolation technique on Grassmann manifolds to this problem. We then improve the method for the models whose system matrices depend affinely on parameters by considerably reducing the computational complexity on the basis of analyzing the structure of sums of singular value decompositions and decomposing the whole procedure into offline and online stages. A numerical example is shown to illustrate the method as well as to prove its effectiveness. Copyright © 2012 John Wiley & Sons, Ltd.
TL;DR: It is shown that the computational time can be reduced when using the model reduction based on the present method, and the adequate number of snapshots automatically is determined automatically.
Abstract: The model reduction based on the method of snapshots is applied to the finite element analysis of three-dimensional transient eddy current problems. It is known that accuracy of the reduced model highly depends on the number of snapshots. In this paper, we introduce a novel method which determines the adequate number of snapshots automatically. It is shown that the computational time can be reduced when using the model reduction based on the present method.
01 Jan 2013
TL;DR: An introductory survey of both methods is given, their application to gas transport problems is discussed, and both methods are compared by means of a simple test case from industrial practice.
Abstract: CPU-intensive engineering problems such as networks of gas pipelines can be modelled as dynamical or quasi-static systems. These dynamical systems represent a map, depending on a set of control parameters, from an input signal to an output signal. In order to reduce the computational cost, surrogates based on linear combinations of translates of radial functions are a popular choice for a wide range of applications. Model order reduction, on the other hand, is an approach that takes the principal structure of the equations into account to construct low-dimensional approximations to the problem. We give an introductory survey of both methods, discuss their application to gas transport problems and compare both methods by means of a simple test case from industrial practice.
TL;DR: In this article, a model order reduction (MOR) technique for a linear multivariable system is proposed using invasive weed optimization (IWO), which is applied with the combined advantages of retaining the dominant poles and the error minimization.
TL;DR: A new approach is implemented that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in Newton strategies and results in an approximation of the problem solution in the form of a series expansion.
Abstract: This paper deals with the extension of proper generalized decomposition methods to non-linear problems, in particular, to hyperelasticity. Among the different approaches that can be considered for the linearization of the doubly weak form of the problem, we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in Newton strategies. This approach results in an approximation of the problem solution in the form of a series expansion. Each term of the series is expressed as a finite sum of separated functions. The advantage of this approach is the presence of only one tangent operator, identical for every term in the series. The resulting approach has proved to render very accurate results that can be stored in the form of a meta-model in a very compact format. This opens the possibility to use these results in real-time, reaching kHz feedback rates, or to be used in deployed, handheld devices such as smartphones and tablets.
TL;DR: In this article, the Euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest is used to increase the number of interpolation points and number of matched frequency derivatives.
Abstract: Frequency sweep problems arise in many structural dynamic, acoustic, and structural acoustic applications. In each case, they incur the evaluation of a frequency response function for a typically large number of frequencies. Because each function evaluation requires the solution of an often large-scale system of equations, frequency sweep problems are computationally intensive. Interpolatory model order reduction is a powerful tool for reducing their cost. However, the performance of this tool depends on the location and number of the interpolation frequency points. It also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point. So far, these two choices have been made in the literature in a heuristic manner. In contrast, this paper proposes an automatic adaptive strategy based on monitoring the Euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest. More specifically, the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global Euclidean norm of the relative residual is reduced below a user-specified tolerance. The robustness, accuracy, and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large-scale structural dynamic, acoustic, and structural acoustic finite element models. Copyright © 2012 John Wiley & Sons, Ltd.
TL;DR: This work develops hybrid and goal-oriented adaptive reduced basis methods to tackle challenges by accurately and efficiently computing the failure probability of a stochastic PDE system with solution of low regularity in probability space.
30 Mar 2013
TL;DR: In this paper, a mixed method of interval systems is proposed for model order reduction of systems with uncertain parameters, where two separate methods are used for finding parameters of the numerator and denominator.
Abstract: Mixed method of interval systems is a combination of classical reduction methods and stability preserving methods of interval systems. This paper proposed a new method for model order reduction of systems with uncertain parameters. The bounds on the uncertain parameters are known a priori. Two separate methods are used for finding parameters of the numerator and denominator. The numerator parameters are obtained by either of these methods such as differentiation method, factor division method, cauer second form, moment matching method or Pade approximation method. The denominator is obtained by the differentiation method in all the cases. A numerical example has been discussed to illustrate the procedures. From the above mixed methods, differentiation method and cauer second form as resulted in better approximation when compared with other methods. The errors between the original higher order and reduced order models have also been highlighted to support the effectiveness of the proposed methods.
TL;DR: This paper presents a framework for multi-resolution simulation of switching converter circuits by providing an appropriate amount of detail based on the time scale and phenomenon being considered, and demonstrates orders of magnitudes improvement in simulation speed.
Abstract: Highly detailed models of power-electronic converter circuits can be slow to simulate due to the wide disparity in transient time scales. This paper presents a framework for multi-resolution simulation of switching converter circuits by providing an appropriate amount of detail based on the time scale and phenomenon being considered. In this approach, a detailed full-order model that accounts for the higher-order effects of components, parasitics, switching nonlinearity (e.g., saturated inductors), switching event detection, etc., is constructed first. Efficient order-reduction techniques are then used to extract several lower order models for the desired resolution of the simulation. The simulation resolution can be adjusted as needed even during a simulation run time. The state continuity across different resolutions and switching events is ensured using appropriate similarity transforms. The proposed high-fidelity model of converter is verified with hardware measurement and is used to verify different simulation resolutions. The proposed methodology is demonstrated to achieve orders of magnitudes improvement in simulation speed.
TL;DR: In this article, the authors proposed a reduced adjoint approach to variational data assimilation based on proper orthogonal decomposition (POD) which avoids the implementation of the adjoint of the tangent linear approximation of the original nonlinear model.
TL;DR: In this study, modelling of the human hearing is considered and a nonlinear elastic multibody system is derived from which the tympanic membrane and the air in the ear canal as well as in the tyMPanic cavity are considered as elastic bodies.
TL;DR: This paper presents a model approximation technique based on N-step-ahead affine representations obtained via Monte-Carlo integrations that enables simultaneous linearization and model order reduction of nonlinear systems in the original state space thus allowing the application of linear MPC algorithms to non linear systems.
TL;DR: In this article, structural dynamic simulations of a doubly clamped beam and a plane resonator subjected to an electrostatic actuation were carried out through an innovative numerical procedure, based on the combined use of a domain decomposition technique and of a proper orthogonal decomposition methodology.
17 Jul 2013
TL;DR: In this paper, a general framework for model order reduction is proposed for high-order parameter-dependent, linear time-invariant systems, which is based on matrix interpolation and consists of six steps.
Abstract: A general framework for model order reduction is proposed for high-order parameter-dependent, linear time-invariant systems. The procedure is based on matrix interpolation and consists of six steps. At first a set of high-order nonparametric systems is computed for different parameter vectors. The resulting local high-order systems are then reduced by a projection-based reduction method. Thereby, proper right and left subspaces for the reduced systems are calculated. Next the bases of the right subspaces of the reduced systems are adapted and the bases of the left subspaces are adjusted. For that the concept of duality is introduced. Finally, the precomputed matrices of the local systems are interpolated in a matrix manifold with an interpolation method. In this paper the six steps of the algorithm and the degrees of freedom which arise therein are presented. Furthermore, advantages and difficulties in the selection of the degrees of freedom are pointed out. It is additionally shown that two existing methods for parametric model order reduction by matrix interpolation are special cases of the proposed general procedure as they - often implicitly - determine a limiting selection of the degrees of freedom.
TL;DR: Two weight-free model approximation algorithms are proposed, based on recent algorithms that achieve local H2 optimal model reduction, that are validated both on a standard benchmark and on an industrial use case.
TL;DR: In this paper, the entire power system is separated into an external area and a study area, and dynamic reduction of the external area is conducted, where key input-output relationships between these areas are retained during the reduction process.
Abstract: This article demonstrates the application of balanced truncation based model order reduction to the task of dynamic reduction of power systems. The entire power system is separated into an external area and a study area; dynamic reduction of the external area is conducted. The benefit of applying the balanced truncation technique is that key input–output relationships between these areas are retained during the reduction process. For perturbations originating in the study area, patterns in the dynamic behavior of the external area, such as generator coherency, are well captured by the reduced equivalent. The efficiency of the balanced truncation algorithm is also explored in the context of changing system conditions. Illustrative applications using a test system representative of the northern grid of India have led to some key findings on the coherency of the generators in this grid.
TL;DR: In this paper, a new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem.
Abstract: The Krylov subspace projection approach is a well-established tool for the reducedorder modeling of dynamical systems in the time domain. In this paper, we address the main issues obstructing the application of this powerful approach to the time-domain solution of exterior wave problems. We use frequency-independent perfectly matched layers to simulate the extension to infinity. Pure imaginary stretching functions based on Zolotarev’s optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest. A new Krylov-based solution method via stabilitycorrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem. The ROMs are unconditionally stable and are based on a renormalized bi-Lanczos algorithm. We give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two-dimensional electromagnetic wave propagation in unbounded domains, including a photonic waveguide example. The new algorithm outperforms the conventional finitedifference time-domain method for problems on large time intervals.
TL;DR: In this article, two approaches, the proper orthogonal decomposition and the proper generalized decomposition, are applied to the vector poten-tial formulation used to solve the quasi-static problem.
Abstract: In order to reduce the computation time of a quasi-static problem solved by the finite element method, methods of model order reduction can be applied In this context, two approaches, the proper orthogonal decomposition and the proper generalized decomposition, are applied to the vector poten-tial formulation used to solve the quasi-static problem The developed methods are compared on an academic example