The singular-value decomposition of an infinite Hankel matrix
TL;DR: In this paper, a new algorithm for the numerical calculation of the singular values and vectors of an infinite Hankel matrix of known finite rank r is presented, which proceeds by reduction to the singular value problem for an r × r matrix, achieved without solving for the poles of the symbol of H.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-04-01 and is currently open access. It has received 38 citations till now. The article focuses on the topics: Hankel matrix & Singular solution.
Citations
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01 Jul 1992
TL;DR: In this article, the authors propose a new algorithm which deploys most of the ideas of generalized predictive control (GPC) but yields control configurations with guaranteed stability and provides a systematic means of optimising robust stability margins.
Abstract: Generalised predictive control (GPC) has many attributes which make it an effective practical tool for the control of scalar systems. The method is easy to understand/implement and has won popularity among many practicising engineers. Despite its success it does not have a general stability and robust stability theory. The paper proposes a new algorithm which deploys most of the ideas of GPC but yields control configurations with guaranteed stability and provides a systematic means of optimising robust stability margins. It achieves this through the use of a particular stabilising feedback controller which replaces the relevant transfer function operators by simple polynomial operators. This simplification also reduces the computational burden of a computer implementation of the method. The superiority of the results of the paper are illustrated by means of numerical examples.
191 citations
TL;DR: In this article, a numerically well-behaved method for computing Hankel optima (CF) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials.
Abstract: The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...
58 citations
TL;DR: Approximative modelling of stable continuous-time, possibly infinite dimensional, systems is studied based on an optimal approximation approach and Laguerre-Fourier series methods and Hankel operator techniques can be utilized to develop fully practical identification methods.
41 citations
TL;DR: In this article, a unified approach to Hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces.
39 citations
Cites background from "The singular-value decomposition of..."
...A closely related derivation can be found in Young (1983) and Lindquist and Picci (1985)....
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TL;DR: In this paper, efficient algorithms for two variants of the Nevanlinna-Pick problem are devised based on the formulation in [14] and their implementation is validated by several examples, based on establishing the equivalence of two different operator-theoretic approaches to the solution of classical interpolation problems.
Abstract: Based on the formulation in [14], efficient algorithms are devised for two variants of the Nevanlinna-Pick problem. Their implementation is validated by several examples. The method depends on establishing the equivalence of two different operator-theoretic approaches to the solution of classical interpolation problems [1, 14].
33 citations
References
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808 citations
01 Jan 1978
646 citations
TL;DR: It is proved that entries in the Pade table can be computed by the Extended Euclidean Algorithm, and an algorithm EMGCD (Extended Middle Greatest Common Divisor) is described which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log2 n).
419 citations
193 citations