Model Reduction Of Large-scale Systems Research Articles

Model Reduction for Large-Scale Systems Using Riemannian Optimization Methods

Model Reduction for Large-Scale Systems Using Riemannian Optimization Methods

A novel fractional-order neural network for model reduction of large-scale systems with fractional-order nonlinear structure

In this paper, a new method for reducing the order of nonlinear large-scale fractional-order systems is presented. The considered system has a nonlinear large-scale dynamic. The proposed method is developed by introducing a new fractional-based approach for neural network learning. According to the fractional-order modeling of the system, the structure of the neural network is selected as a recurrent neural network and new design and analysis are done on this network. In order to show the proposed method for model reduction has an acceptable error, a novel fractional-order stability analysis is used to derive the neural network weighting function. Moreover, it can be concluded that the proposed reducing method can preserve the main properties of the original system like a system’s stability. Simulation examples are provided to show the effectiveness of the proposed method. Finally, the proposed method is compared with the existing methods and advantages of the proposed method are shown.

Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space

A model-constrained adaptive sampling methodology is proposed for the reduction of large-scale systems with high-dimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of high-fidelity solutions at a number of sample points throughout the parametric input space. A key challenge that must be addressed in the optimization, control, and probabilistic settings is the need for the reduced models to capture variation over this parametric input space, which, for many applications, will be of high dimension. We pose the task of determining appropriate sample points as a PDE-constrained optimization problem, which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters. The methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statistically based sampling methods. For these examples, the model-constrained adaptive sampling leads to reduced models that, for a given basis size, have error several orders of magnitude smaller than that obtained using the other methods.

Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

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. This allows a simple error analysis to be carried out for both algorithms and permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively.

An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems

In this paper, we propose a model reduction algorithm for approximation of large-scale linear time-invariant dynamical systems. The method is a two-sided projection combining features of the singular value decomposition (SVD)-based and the Krylov-based model reduction techniques. While the SVD-side of the projection depends on the observability gramian, the Krylov-side is obtained via iterative rational Krylov steps. The reduced model is asymptotically stable, matches certain moments and solves a restricted H2 minimization problem. We present modifications to the proposed approach for employing low-rank gramians in the reduction step and also for reducing discrete-time systems. Several numerical examples from various disciplines verify the effectiveness of the proposed approach. It performs significantly better than the q-cover [A. Yousouff, R.E. Skelton, Covariance equivalent realizations with applications to model reduction of large-scale systems, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 22, Academic Press, 1985, pp. 273–348; A. Yousouff, D.A. Wagie, R.E. Skelton, Linear system approximation via covariance equivalent realizations, Journal of Math. Anal. and Appl. 196 (1985) 91–115] and the least-squares [S. Gugercin, A.C. Antoulas, Model reduction of large scale systems by least squares, Linear Algebra Appl. 415(2–3) (2006) 290–321] methods that have a similar projection structure to the proposed method. Also, in terms of both the H2 and H∞ error measures, its performance is comparable to or sometimes better than that of balanced truncation. Moreover, the method proves to be robust with respect to the perturbations due to usage of approximate gramians.

Model reduction of large-scale systems by least squares

In this paper we introduce an approximation method for model reduction of large-scale dynamical systems. This is a projection which combines aspects of the SVD and Krylov based reduction methods. This projection can be efficiently computed using tools from numerical analysis, namely the rational Krylov method for the Krylov side of the projection and a low-rank Smith type iteration to solve a Lyapunov equation for the SVD side of the projection. For discrete time systems, the proposed approach is based on the least squares fit of the ( r + 1)th column of a Hankel matrix to the preceding r columns, where r is the order of the reduced system. The reduced system is asymptotically stable, matches the first r Markov parameters of the full order model and minimizes a weighted H 2 error. The method is also generalized for moment matching at arbitrary interpolation points. Application to continuous time systems is achieved via the bilinear transformation. Numerical examples prove the effectiveness of the approach. The proposed method is significant because it combines guaranteed stability and moment matching, together with an optimization criterion.

State-space truncation methods for parallel model reduction of large-scale systems

We discuss a parallel library of efficient algorithms for model reduction of large-scale systems with state-space dimension up to O (10 4). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on state-space truncation of the system matrices and includes absolute and relative error methods for both stable and unstable systems. In contrast to serial implementations of these methods, we employ Newton-type iterative algorithms for the solution of the major computational tasks. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Pentium II processors.

Active control performance enhancement for reduced-order models of large-scale systems

A scheme for model reduction of large-scale systems based on the singular value decomposition incorporating model upgrading technique for a rank 1 addition to the aggregation matrix is provided. The performance of the reduced systems is enhanced by including a procedure that accounts for modal spillovers resulting from the application of reduced-order model controllers to the large-scale systems.

On model reduction of large-scale systems

On model reduction of large-scale systems