Krylov-based minimization for optimal H2 model reduction (original) (raw)
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H_2 model reduction for large-scale linear dynamical systems
2008
The optimal H 2 model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focussing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal H 2 approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov and interpolation based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for H 2 model reduction. The formulation is based on finding a reduced order model that satisfies interpolation based first-order necessary conditions for H 2 optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.
Rational Krylov methods for optimal ℒ2 model reduction
49th IEEE Conference on Decision and Control (CDC), 2010
We develop and describe an iteratively corrected rational Krylov algorithm for the solution of the optimal H 2 model reduction problem. The formulation is based on finding a reduced order model that satisfies interpolation based first-order necessary conditions for H 2 optimality and results in a method that is numerically effective and suited for large-scale problems. We provide a new elementary proof of the interpolation based condition that clarifies the importance of the mirror images of the reduced system poles. We also show that the interpolation based condition is equivalent to two types of first-order necessary conditions associated with Lyapunov-based approaches for H 2 optimality. Under some technical hypotheses, local convergence of the algorithm is guaranteed for sufficiently large model order with a linear rate that decreases exponentially with model order. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.
ArXiv, 2020
In this paper, we focus on structure-preserving model order reduction (SPMOR) of the second-order system using the Iterative Rational Krylov Algorithm (IRKA). In general, the standard IRKA can be used to the second-order structure system by converting the system into an equivalent first-order form. In this case, the reduced model can not preserve the structure of the second-order form which is however necessary for further manipulation. Here we develop IRKA based algorithm which enables us to generate a reduced second-order system without explicitly converting the system into a first-order form. A challenging task in IRKA is to select a set of better interpolation points and the tangential directions. To overcome these problems, this paper discusses how to select a set of good interpolation points and the tangential directions by an internal formulation. Also, this paper talks out the H2 norm optimality of the system. The theoretical results are experimented by applying them to seve...
A Rational Krylov Iteration for OptimalH2 Model Reduction
In this paper, we address the optimal H 2 approximation of a stable, single-input single-output large-scale dynamical system. The problem we consider is as follows: Given an n th order linear dynamical system G(s) = C(sI − A) −1 B where A ∈ n×n , and B, C T ∈ n , find a stable r th order reduced system G r (s) = C r (sI r −A r ) −1 B r with r < n, such that G r (s) minimizes the H 2 error, i.e.
L2-optimal model reduction for unstable systems using iterative rational Krylov algorithm
Unstable dynamical systems can be viewed from a variety of perspectives. We discuss the potential of an inputoutput map associated with an unstable system to represent a bounded map from L2(R) to itself and then develop criteria for optimal reduced order approximations to the original (unstable) system with respect to an L2-induced Hilbert-Schmidt norm. Our optimality criteria extend the Meier-Luenberger interpolation conditions for optimal H2 approximation of stable dynamical systems. Based on this interpolation framework, we describe an iteratively corrected rational Krylov algorithm for L2 model reduction. A numerical example involving a hardto-approximate full-order model illustrates the effectiveness of the proposed approach.
Advances in Science Technology and Engineering Systems Journal , 2020
This paper focuses on exploring efficient ways to find H2 optimal Structure-Preserving Model Order Reduction (SPMOR) of the second-order systems via interpolatory projection-based method Iterative Rational Krylov Algorithm (IRKA). To get the reduced models of the second-order systems, the classical IRKA deals with the equivalent first-order converted forms and estimates the first-order reduced models. The drawbacks of that of the technique are failure of structure preservation and abolishing the properties of the original models, which are the key factors for some of the physical applications. To surpass those issues, we introduce IRKA based techniques that enable us to approximate the second-order systems through the reduced models implicitly without forming the first-order forms. On the other hand, there are very challenging tasks to the Model Order Reduction (MOR) of the large-scale second-order systems with the optimal H2 error norm and attain the rapid rate of convergence. For the convenient computations, we discuss competent techniques to determine the optimal H2 error norms efficiently for the second-order systems. The applicability and efficiency of the proposed techniques are validated by applying them to some large-scale systems extracted form engineering applications. The computations are done numerically using MATLAB simulation and the achieved results are discussed in both tabular and graphical approaches.
Krylov-based model reduction of second-order systems with proportional damping
Proceedings of the 44th IEEE Conference on Decision and Control, 2005
In this note, we examine Krylov-based model reduction of second order systems where proportional damping is used to model energy dissipation. We give a detailed analysis of the distribution of system poles, and then, through a connection with potential theory, we are able to exploit the structure of these poles to obtain an optimal single shift strategy used in rational Krylov model reduction. We show that unlike the general case that requires usage of a second-order Krylov subspace structure, one can build up approximating subspaces satisfying all required conditions much more cheaply as direct sums of standard rational Krylov subspaces within the smaller component subspaces. Numerical examples are provided to illustrate and support the analysis.
An interpolation-based approach to ℋ∞ model reduction of dynamical systems
2010
We introduce an interpolatory approach to H∞ model reduction for large-scale dynamical systems. Guided by the optimality conditions of for best uniform rational approximants on the unit disk, our proposed method uses the freedom in choosing the d-term in the reduced order model to enforce 2r + 1 interpolation conditions in the right-half plane for any given reduction order, r. 2r of these points are initialized by the Iterative Rational Krylov Algorithm of ; and then the d-term is chosen to minimize the H∞ error for this initial set of interpolation points. Several numerical examples illustrate the effectiveness of the proposed method. It consistently yields better results than balanced truncation. In all cases examined its performance is very close to or better than that of Hankel norm approximation. For the special case of state-space symmetric systems, important properties are established. Finally, we examine H∞ model reduction from a potential theoretic perspective and present a second methodology for choosing interpolation points.
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 ...
Australian journal of electrical and electronics engineering, 2019
A comparative study of two important Model Order Reduction (MOR) systems based on internal projections and H 2 norm is presented. It is about Tangential interpolation and the Linear Matrix Inequalities (LMIs) optimisation based methods. In both techniques, the resulting optimal approximants of reduced order are constructed from state space projections and constrained to verify a certain H 2 relative error norm. In this paper, its shown the applicability of these techniques as well for the continuous time as the discrete time multidimensional systems. Since the industry has always been one of the main motivating fields for the development of MOR techniques, the present work offers an expansion to state of the art low order MOR tools and their use to any industrial process to achieve three core features: simplicity of analysis, low computation efforts and an easy and feasible control.