Coprime fraction computation of 2-D rational matrices

This note presents a numerical method of computing a coprime fraction of a two-dimensional (2-D) rational matrix, not necessarily proper. It is achieved by searching the primary linearly dependent rows, in order from top to bottom, of the two generalized resultants. The procedure can be extended to the three- or higher dimensional case and the result can also be used to compute the greatest common divisor (GCD) of 2-D polynomial matrices without employing primitive factorizations which does not exist in the three- or higher dimensional case.     

A recursive algorithm for computing the partial fraction expansion of rational functions having multiple poles

A recursive algorithm is derived which easily permits the determination of the terms in a partial fraction expansion associated with multiple poles. The algorithm is readily programmed on a digital computer.     

Behaviors and symbols of rational matrices

Recently, we have defined rational differential equations, and derived conditions when two such equations have the same solution set. In this article, we study the same question, but present a different approach based on the new notion of symbols. Also we try to describe the relationship with the work “Behaviors defined by rational functions” of Willems and Yamamoto.     

An algorithm for inverting rational matrices

We propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices. The algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a MATLAB¹ function. Experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems. Illustrative examples are given.     

Recursive algorithm for computing the matrix partial fraction expansion of transfer matrices having multiple quadratic poles

A new method for determining matrix coefficients of partial fraction expansion (PFE) of transfer matrices (TM) with distinct and/or multiple poles is presented. The PFE coefficient matrices of TM are obtained recursively, and the matrix formulation for the matrix-PFE is amenable to computing.     

Equivalence transformations of rational matrices and applications

The known theories of transformations between polynomial matrices are extended to the case of rational matrices. Specifically, Ω-equivalence between rational matrices having possibly different dimensions is defined, and this has the property of preserving the zero structure of rational matrices in the region Ω ? C ∪ {∞}Some implications of this new equivalence transformation for linear system theory are also provided.     

Unbalanced multiway trees improved by partial expansions

Unbalanced multiway trees are generally impractical external data structures because of their poor space performance. In order to avoid this handicap, we have adapted the technique of partial expansions to these trees. Compared to partially expandedB ⁺-trees in terms of average performance, the method proposed is faster, has more compact indexes, and shows the same almostoptimal asymptotic space performance at the data bucket level.     

有理式二阶前馈型神经网络

文章提出了二阶有理式多层前馈神经网络的数学模型。有理式多层神经网络的思想来源于函数逼近理论中的有理式逼近。有理式前馈神经网络模型是传统前俯神经网络模型的推广，能有效地求解函数逼近问题。文章给出了有理式多层神经网络的学习算法，即误差反传播学习算法。就计算复杂度而言，有理式神经网络的学习算法与传统的多层神经网络反传播算法是同阶的。文章还给出了函数逼近和模式识别两个应用实例，实验结果说明二阶有理式多层神经网络在解决传统的问题上是有效的。     

A definition of column reduced proper rational matrices

The aim of this work is to develop analogue concepts of column reduced polynomial matrices for proper rational matrices. A definition of column reducedness for a class of proper rational matrices is proposed and the properties of such matrices are studied, in particular reduction to column reduced form by elementary operations over the ring of proper rational functions, and the relationship between the degrees of the invariant factors of a column reduced matrix and the so-defined column indices. The physical significance of such matrices in terms of their finite structure is explained; this interpretation completely complements the physical interpretation of a column reduced polynomial matrix. An application of the properties of column reduced proper rational matrices to the decoupling problem is also presented: the infinite structure which can be obtained while decoupling a linear multivariable system by non-regular static state feedback is completely characterized.     

Matrix fraction construction of linear compensators

A construction is given using polynomial matrix fractions of the compensator resulting from a combination of a linear feedback control law and estimator design.     

Comments on "A recursive algorithm for computing the partial fraction expansion of rational functions having multiple poles"

In the above paper, the author presents a recursive algorithm for determination of the terms in a partial fraction expansion associated with multiple poles. However, this recursive algorithm has been derived previously in a different publication [1].     

Doubly coprime decompositions of a collection of rational matrices

An N-periodic discrete-time system in z-domain given by an N-periodic collection of rational matrices is considered. Doubly coprime decomposition of these matrices is studied. For such decompositions, the block-ordered concept is used. Further, the eight matrices in the generalized Bezout identity, so that all of them are block-ordered, can be chosen. A relation among these matrices at consecutive times is established.     

On the factorization of discrete-time rational spectral density matrices

For a discrete-time rational spectral density matrixPhi(z), the relationship between the factorizationsPhi(z) = Z(z) + Z^{T} (z^{-1})andPhi(z) = W(z)QW^{T}(z), in terms of minimal state space realizations ofZ(z)andW(z), are derived in a straightforward way, without resorting to the use of the bilinear transformation. The obvious application of this result to performing a spectral factorizafion ofPhi(z)is discussed.     

Minimal partial realization from generalized orthonormal basis function expansions

A solution is presented for the problem of realizing a discrete-time LTI state-space model of minimal McMillan degree such that its first N expansion coefficients in terms of generalized orthonormal basis match a given sequence. The basis considered, also known as the Hambo basis, can be viewed as a generalization of the more familiar Laguerre and two-parameter Kautz constructions, allowing general dynamic information to be incorporated in the basis. For the solution of the problem use is made of the properties of the Hambo operator transform theory that underlies the basis function expansion. As corollary results compact expressions are found by which the Hambo transform and its inverse can be computed efficiently. The resulting realization algorithms can be applied in an approximative sense, for instance, for computing a low-order model from a large basis function expansion that is obtained in an identification experiment.