Patrick Dewilde

Subspace model identification Part 1. The output-error state-space model identification class of algorithms

In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme)...

Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences

An exact and approximate realization theory for estimation and model filters of second-order stationary stochastic sequences is presented. The properties of J -lossless matrices as a unifying framework are used. Necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a J -lossless system are deduced. An extension of the so-called Schur algorithm yields an approximate J -lossless realization based on partial past information about the process. The geometric properties of such partial realizations and their convergence are studied. Finally, connections with the Nevanlinna-Pick problem are made, and how the techniques presented constitute a generalization of many aspects of the Levinson-Szego theory of partial realizations is shown. As a consequence generalized recursive formulas for reproducing kernels and Christoffel-Darboux formulas are obtained. In this paper the scalar case is considered. The matrix case will be considered in a separate publication.

The eigenstructure of an arbitrary polynomial matrix: computational aspects

We give a new numerical method to compute the eigenstructure—i.e. the zero structure, the polar structure, and the left and right null space structure—of a polynomial matrix P(λ). These structural elements are of fundamental importance in several systems and control problems involving polynomial matrices. The approach is more general than previous numerical methods because it can be applied to an arbitrary m × n polynomial matrix P(λ) with normal rank r smaller than m and/or n. The algorithm is then shown to compute the structure of the left and right null spaces of P(λ) as well. The speed and accuracy of this new approach are also discussed.

Pipelined cordic architectures for fast VLSI filtering and array processing

The paper presents a revised functional description of Volders Coordinate Rotation Digital Computer algorithm (CORDIC), as well as allied VLSI implementable processor architectures. Both pipelined and sequential structures are considered. In the general purpose or multi-function case, pipeline length (number of cycles), function evaluation time and accuracy are all independent of the various executable functions. High regularity and minimality of data-paths, simplicity of control circuits and enhancement of function evaluation speed are ensured, partly by mapping a unified set of micro-operations, and partly by invoking a natural encoding of the angle parameters. The approach benefits the execution speed in array configurations, since it will allow pipelining at the bit level, thereby providing fast VLSI implementations of certain algorithms exhibiting substantial structural pipelining or parallelism.

A Fast Solver for HSS Representations via Sparse Matrices

In this paper we present a fast direct solver for certain classes of dense structured linear systems that works by first converting the given dense system to a larger system of block sparse equations and then uses standard sparse direct solvers. The kind of matrix structures that we consider are induced by numerical low rank in the off-diagonal blocks of the matrix and are related to the structures exploited by the fast multipole method (FMM) of Greengard and Rokhlin. The special structure that we exploit in this paper is captured by what we term the hierarchically semiseparable (HSS) representation of a matrix. Numerical experiments indicate that the method is probably backward stable.

Lossless inverse scattering, digital filters, and estimation theory

Methods to construct rational solutions of the lossless inverse scattering (LIS) problem for one-port passive digital systems are described. The first method is recursive and can be viewed as a generalization of the celebrated Schur algorithm. The second method is global and leads to a parametrization of what we call fundamental solutions from which all LIS solutions may be constructed. Quite a few classical problems in estimation theory and network theory may be viewed as special cases of the LIS problem. With each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process.

Lossless Inverse Scattering and Reproducing Kernels for Upper Triangular Operators

In this paper the topics mentioned in the title are studied in the algebra of upper triangular bounded linear operators acting on the space l N 2 of “square summable” sequences f = (..., f −1, f 0, f l,...) with components f i in a complex separable Hilbert space N. Enroute analogues of point evaluation (where in this case the points are operators and the functions are operator valued functions of operators) and simple Blaschke products are developed in this more general context. These tools are then used to establish a theory of structured reproducing kernel Hilbert spaces in the class of upper triangular Hilbert Schmidt operators on l N 2. (An application of these tools and spaces to solve a Nevanlinna Pick interpolation problem wherein both the interpolation points and the values assigned at those points are block diagonal operators, will be considered in a future publication.)

Some Fast Algorithms for Sequentially Semiseparable Representations

An extended sequentially semiseparable (SSS) representation derived from time-varying system theory is used to capture, on the one hand, the low-rank of the off-diagonal blocks of a matrix for the purposes of efficient computations and, on the other, to provide for sufficient descriptive richness to allow for backward stability in the computations. We present (i) a fast algorithm (linear in the number of equations) to solve least squares problems in which the coefficient matrix is in SSS form, (ii) a fast algorithm to find the SSS form of X such that AX=B, where A and B are in SSS form, and (iii) a fast model reduction technique to improve the SSS form.

On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion

A novel method is presented to determine the SmithMacmillan form of a rational m \times n matrix R(p) from Laurent expansions in its poles and zeros. Based on that method, a numerically stable algorithm is deduced, which uses only a minimal number of terms of the Laurent expansion, hence providing a shortcut with respect to cumbersome and unstable procedures based on elementary transformations with unimodular matrices. The method can be viewed as a generalization of Kublanovkayas algorithm for the complete solution of the eigenstructre problem for \lambda I - A . From a systems point of view it provides a handy and numerically stable way to determine the degree of a zero of a transfer function and unifies a number of results from multivariable realization and invertibility theory. The paper presents a systematic treatment of the relation between the eigen-information of a transfer function and the information contained in partial fraction or Laurent expansions. Although a number of results are known, they are presented in a systematic way which considerably simplifies the total picture and introduces in a natural way a number of novel techniques.

A design methodology for fixed-size systolic arrays

The authors present a methodology to design fixed-size systolic arrays. It allows a systematic and hierarchical mapping of full-size arrays to fixed-size arrays. Two processor-clustering techniques are described. They can be used to achieve the following design objectives: (1) transforming inefficient arrays into efficient arrays, (2) reducing the size of an array, (3) reducing the dimension of an array, and (4) balancing local memory and external communication of processors. A technique is described to cluster processors in such a way that the number of I/O pins of the resulting processor is independent of the number of processors that are clustered. The approach presented unifies and generalizes array reduction techniques.<<ETX>>