A Computational Global Tangential Krylov Subspace Method for Model Reduction of Large-Scale MIMO Dynamical Systems (original) (raw)
An Adaptive Tangential Lanczos-Type Algorithm for Model Reduction in Large-Scale Dynamical Systems
2017
Large-scale simulations play a crucial role in the study of a great variety of complex physical phenomena, leading often to overwhelming demands on computational resources. Managing these demands constitutes the main motivation for model reduction: produce simpler reduced-order models, which allow for faster and cheaper simulation while accurately approximating the behaviour of the original model. The presence of multiple inputs and outputs (MIMO) systems, makes the reduction process even more challenging. In this paper, we present a new approach to treat MIMO systems, named: Adaptive Tangential Lanczos Algorithm (ATLA). We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed algorithm.
Dimensionally reduced Krylov subspace model reduction for large scale systems
Applied Mathematics and Computation, 2007
This paper introduces a new mathematical approach that combines the concepts of dimensional reduction and Krylov subspace techniques for use in the model reduction problem for large-scale systems. Krylov subspace methods for model reduction uses the Arnoldi algorithm in order to construct the bases for controllability, observability, and oblique subspaces of state space realization. The newly developed algorithm uses principal component analysis along with Krylov oblique projection model reduction technique to provide computationally efficient and inexpensive model reduction method. To demonstrate the effectiveness of the proposed hybrid scheme the residual error, forward error and stability response analyses have been performed for various randomly generated large-scale systems.
Two-sided Arnoldi in order reduction of large scale MIMO systems
2002
In order reduction of high order linear time invariant systems based on two-sided Krylov subspace methods, the Lanczos algorithm is commonly used to find the bases for input and output Krylov subspaces and to calculate the reduced order model by projection. However, this method can be numerically unstable even for systems with moderate number of states and can only find a limited number of biorthogonal vectors. In this paper, we present another method which we call Two-Sided Arnoldi and which is based on the Arnoldi algorithm. It finds two orthogonal bases for any pair of Krylov subspaces, with one or more starting vectors. This new method is numerically more robust and simpler to implement specially for nonsquare MIMO systems, and it finds a reduced order model with same transfer function as Lanczos. The Two-Sided Arnoldi algorithm can be used for order reduction of the most general case of a linear time invariant Multi-Input-Multi-Output (MIMO) systems. Furthermore, we present some suggestions to improve the method using a column selection procedure and to reduce the computational time using LU-factorization.
An adaptive block tangential method for multi-input multi-output dynamical systems
Journal of Computational and Applied Mathematics, 2019
In this paper, we present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named: Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.
Two efficient SVD/Krylov algorithms for model order reduction of large scale systems
Electronic Transactions on Numerical Analysis, 2011
We present two efficient algorithms to produce a reduced order model of a time-invariant linear dynamical system by approximate balanced truncation. Attention is focused on the use of the structure and the iterative construction via Krylov subspaces of both controllability and observability matrices to compute low-rank approximations of the Gramians or the Hankel operator. This allows us to take advantage of any sparsity in the system matrices and indeed the cost of our two algorithms is only linear in the system dimension. Both algorithms efficiently produce good low-rank approximations (in the least square sense) of the Cholesky factor of each Gramian and the Hankel operator. The second algorithm works directly on the Hankel operator, and it has the advantage that it is independent of the chosen realization. Moreover it is also an approximate Hankel norm method. The two reduced order models produced by our methods are guaranteed to be stable and balanced. We study the convergence ...
Krylov and Modal Subspace based Model Order Reduction with A-Priori Error Estimation
2018
Versatile model order reduction techniques for the reduction of dynamic systems have been presented in the last decades. Krylov subspace based methods are considered as efficient in terms of computational effort and reduction order and can be used in order to match the transfer function locally. However, they lack of a simple and efficient automatisation and error estimation. On the other hand, modal reduction is popular because it leads to exactly matching eigenfrequencies. The static behaviour and the overall accuracy of the frequency response, though, are poor. In this paper, a combination of both methods is presented and discussed. In order to characterise a method for the reduction of mechanical models for the simulation of machine tools and similar mechatronic systems, first, the requirements on the model reduction method are derived. Criteria for the relative error of the frequency response function, the transmission zeros, and the poles are defined and the idea of defining a...
A global Arnoldi method for the model reduction of second-order structural dynamical systems
2010
In this paper we consider the reduction of second-order dynamical systems with multiple inputs and multiple outputs (MIMO) arising in the numerical simulation of mechanical structures. Undamped systems as well as systems with proportional damping are considered. In commercial software for the kind of application considered here, modal reduction is commonly used to obtain a reduced system with good approximation abilities of the original transfer function in the lower frequency range. In recent years new methods to reduce dynamical systems based on (block) versions of Krylov subspace methods emerged. This work concentrates on the reduction of second-order MIMO systems by the global Arnoldi method, an efficient extension of the standard Arnoldi algorithm for MIMO systems. In particular, a new model reduction algorithm for second order MIMO systems is proposed which automatically generates a reduced system of given order approximating the transfer function in the lower range of frequencies. It is based on the global Arnoldi method, determines the expansion points iteratively and the number of moments matched per expansion point adaptively. Numerical examples comparing our results to modal reduction and reduction via the block version of the rational Arnoldi method are presented.
Adaptive Tangential Interpolation in Rational Krylov Subspaces for MIMO Dynamical Systems
SIAM Journal on Matrix Analysis and Applications, 2014
Model reduction approaches have shown to be powerful techniques in the numerical simulation of very large dynamical systems. The presence of multiple inputs and outputs (MIMO systems), makes the reduction process even more challenging. We consider projection-based approaches where the reduction of complexity is achieved by direct projection of the problem onto a rational Krylov subspace of significantly smaller dimension. We present an effective way to treat multiple inputs, by dynamically choosing the next direction vectors to expand the space. We apply the new strategy to the approximation of the transfer matrix function and to the solution of the Lyapunov matrix equation. Numerical results confirm that the new approach is competitive with respect to state-of-the-art methods both in terms of CPU time and memory requirements.
Adaptive rational Krylov algorithms for model reduction
2007 European Control Conference (ECC), 2007
The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large scale systems. The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This paper tackles the issue of developing simple Arnoldi and Lanczos equations for the rational case that allow simple residual error expressions to be derived. This in turn permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively resulting in an optimized reduced model.
Intelligent Industrial Systems, 2015
Model reduction techniques are simplification methods based on mathematical approaches employed to realize reduced models for the original high order systems. Some existing classical model reduction techniques for multivariable system are considered and compared for their performances. Interlacing property and coefficients matching (IPCM) method gives overall minimum integral square error (ISE), integral absolute error (IAE) and integral time absolute error (ITAE) values compared to other methods. Though the IPCM method is efficient, it may not guarantee for minimization of all objective functions simultaneously. In this paper, model reduction approach based on objectives like ISE, IAE and ITAE using multi-objective differential evolution (MODE) method is proposed for reducing the numerator and the denominator is reduced by interlacing property. MODE method minimizes the small, normal and large errors persisting for long time between original and reduced models. This multi-objective approach is applied for model reduction of 10th order multivariable linear time invariant power system model. Simulation results are demonstrated for single and multi-objective model reduction and compared with multiobjective particle swarm optimization (MOPSO) method to prove the validity of proposed MODE technique.
Model order reduction of linear time invariant systems
Advances in Radio Science, 2008
This paper addresses issues related to the order reduction of systems with multiple input/output ports. The order reduction is divided up into two steps. The first step is the standard order reduction method based on the multipoint approximation of system matrices by applying Krylov subspace. The second step is based on the rejection of the weak part of a system. To recognise the weak system part, Lyapunov equations are used. Thus, this paper introduces efficient solutions of the Lyapunov equations for port to port subsystems.
A Geometric Approach to Dynamical Model Order Reduction
SIAM Journal on Matrix Analysis and Applications
Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum.
Order reduction of large scale second-order systems using Krylov subspace methods
Linear Algebra and its Applications, 2006
In order reduction of large-scale linear time invariant systems, Krylov subspace methods based on moment matching are among the best choices today. However, in many technical fields, models typically consist of sets of second-order differential equations, and Krylov subspace methods cannot directly be applied. Two methods for solving this problem are presented in this paper: (1) an approach by Su and Craig is generalized and the number of matching moments is increased; (2) a new approach via first-order models is presented, resulting in an even higher number of matching moments. Both solutions preserve the specific structure of the secondorder type model.
Combining Krylov subspace methods and identification-based methods for model order reduction
IEEE Transactions on Plasma Science, 2006
Many different techniques to reduce the dimensions of a model have been proposed in the near past. Krylov subspace methods are relatively cheap, but generate non-optimal models. In this paper a combination of Krylov subspace methods and Orthonormal Vector Fitting is proposed. In that way an optimal model for a large model can be generated. In the first step, a Krylov subspace method reduces the large model to a model of medium size, then an optimal model is derived with Orthonormal Vector Fitting as a second step.
2023
In this work, we are going to discuss the Structure-Preserving Model Order Reduction (SPMOR) techniques of secondorder linear time-invariant (LTI) continuous-time systems using the Iterative Rational Krylov Algorithm (IRKA). IRKA is well established for the first-order standard and/or generalized systems. Recently, the idea of this model reduction technique is generalized for second-order systems by converting the system into a first-order form. In this case, one can't return back to the original second-order system since the structure of the system is already demolished. Sometimes preservation of second-order structure is essential to perform the further simulations of the system. Also, Structure-preserving MOR allows meaningful physical interpretation and provides a more accurate approximation to the full model. We mainly focus on the SPMOR of the second-order systems using IRKA without converting the system into first-order forms. We have applied and numerically investigated the applicability and efficiency of the proposed techniques to some practical data derived from real-world models.
Model Order Reduction of Large-Scale Dynamical Systems
2015
The discretization of physical equations in complex three-dimensional domains typically leads to the solution of large-scale systems of equations. The goal of model reduction is to replace these sets of large-scale equations by an alternative set of equations of much smaller dimension that nevertheless retains the main characteristics of the original equations. Reducing the dimensionality and complexity of a given large-scale system then typically leads to a much faster solution time. Therefore, it enables routine analysis, model predictive control, data-driven systems, embedded and real-time computing, and faster design optimization, statistical studies, and uncertainty quantification.
Krylov-Based Model Order Reduction of Time-delay Systems
SIAM Journal on Matrix Analysis and Applications, 2011
We present a model order reduction method which allows the construction of a reduced, delay free model of a given dimension for linear time-delay systems, whose characteristic matrix is nonlinear due to the presence of exponential functions. The method builds on the equivalent representation of the time-delay system as an infinite-dimensional linear problem. It combines ideas from a finite-dimensional approximation via a spectral discretization on the one hand, and a Krylov-Padé model reduction approach on the other hand. The method exhibits a good spectral approximation of the original model, in the sense that the smallest characteristic roots are well approximated and the non-converged eigenvalues of the reduced model have a favorable location, and it preserves moments at zero and at infinity. The spectral approximation is due to an underlying Arnoldi process that relies on building an appropriate Krylov space for the linear infinite-dimensional problem. The preservation of moments is guaranteed, because the chosen finite-dimensional approximation preserves moments and in addition, the space on which one projects is constructed in such a way that the preservation of moments carries over to the reduced model. The implementation of the method is dynamic, since the number of grid points in the spectral discretization does not need to be chosen beforehand and the accuracy of the reduced model can always be improved by doing more iterations. It relies on a reformulation of the problem involving a companion like system matrix and a highly structured input matrix, whose structure are fully exploited.
Model order reduction of nonlinear systems: status, open issues, and applications
In this document we review the status of existing techniques for nonlinear model order reduction by investigating how well these techniques perform for typical industrial needs. In particular the TPWL-method (Trajectory Piecewise Linear-method) and the POD-approach (Proper Orthogonal Decomposion) is taken under consideration. We address several questions that are (closely) related to both the theory and application of nonlinear model order reduction techniques. The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.
Dynamical Model Reduction Method for Solving Parameter-Dependent Dynamical Systems
SIAM Journal on Scientific Computing, 2017
We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution of the full-order model at some selected parameters values. The approximation obtained by Galerkin projection is the solution of a reduced dynamical system with a modified flux which takes into account the time dependency of the reduced spaces. An a posteriori error estimate is derived and a greedy algorithm using this error estimate is proposed for the adaptive selection of parameters values. The resulting method can be interpreted as a dynamical low-rank approximation method with a subspace point of view and a uniform control of the error over the parameter set.