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.

Application and Comparative Analysis of Various Classical and Soft Computing Techniques for Model Reduction of MIMO Systems

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.