An introduction to the shift realization for finite dimensional continuous time systems

A new approach is presented for the realization of continuous-time finite dimensional linear systems. Using standard results on Laplace transforms our results are also used to present a new derivation of Fuhrmann's shift realization for rational matrix functions.     

Minimal state space realizations in Jacobson normal form

We derive a procedure for a minimal state space realization of a rational transfer matrix over an arbitrary field. The procedure is based on the Smith-McMillan form and leads to a state transition matrix in Jacobson normal form.     

Irreducible Jordan form realization of a rational matrix

This paper presents a new method for realizing a rational transfer function matrix into an irreducible Jordan canonical form state equation. The method consists of two steps, first, to form a controllable state equation and then secondly, to obtain a controllable as well as observable realization by nonsingular transformations. If every denominator of the rational matrix is given in the factored form, then the proposed method can be carried out quite easily.     

State space realizations of rational interpolants with prescribed poles

We give a generic algorithm for computing rational interpolants with prescribed poles. The resulting rational function is expressed in the so-called Newton form. State space realizations for this expression of rational functions are given. Our main tool for finding state space realizations is Fuhrmann's shift realization theory from which we obtain concrete realizations by introducing suitable bases of the state space and expressing the abstract operators with respect to these bases in matrix form.     

Approximated modeling and minimal realization of transfer function matrices with multiple time delays

The modeling and minimal realization techniques for a specific multiple time-delay continuous-time transfer function matrix with a delay-free denominator and a multiple (integer/fractional) time-delay numerator matrix have been developed in the literature. However, this is not the case for a general multiple time-delay continuous-time transfer function matrix with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. This paper presents a new approximated modeling and minimal realization technique for the general multiple time-delay transfer function matrices. According to the proposed technique, an approximated discrete-time state-space model and its corresponding discrete-time transfer function matrix are first determined, by utilizing the balanced realization and model reduction methods with the sampled unit-step response data of the afore-mentioned multiple time-delay (known/unknown) continuous-time systems. Then, the modified Z-transform method is applied to the obtained discrete-time transfer function matrix to find an equivalent specific multiple time-delay continuous-time transfer function matrix with multiple time delays in only the inputs and outputs, for which the existing control and design methodologies and minimal realization techniques can be effectively applied. Illustrative examples are given to demonstrate the effectiveness of the proposed method.     

On inner-outer factorization

The calculation of an inner-outer factorization of a proper, rational matrix by state-space methods is considered. Particular attention is given to the extraction of the inner factor, so that a minimal realization is obtained for this factor.     

A State‐Space Based New Approach To The Directional Interpolation Problem

Directional interpolation plays an important role in robust control, system realization and model reduction. Several solutions to various directional interpolation problems have been proposed. In this paper, we consider the directional interpolation problem in a general setting and present a statespace based new approach to solving the problem. The solution is simple, and its exposition is as self‐contained as possible. We describe all the (strictly) bounded real rational matrix functions that satisfy the directional interpolation requirements by means of linear fractional transformation. Moreover, we give a necessary and sufficient condition for the interpolating function to be unique and show that the unique interpolating function is an inner (a co‐inner). The main procedures used to generate the interpolating function consist of standard matrix operations consisting of easy numerical computations, so the present solution is significant from the numerical viewpoint as well as the analytical viewpoint.     

A system-theoretic appropriate realization of the empty matrixconcept

An algebraic realization of the empty matrix concept that is appropriate for system-theoretic applications is proposed. The utility of the realization of the empty matrix concept and the deficiencies of the current MATLAB realization of this concept are demonstrated using examples. These examples fully delineate how the empty matrix concept can be utilized to transparently handle static and for single-vector-input, single-vector-output systems within the more general context of dynamic, two-vector-input, two-vector-output systems     

The set of realizations of a max-plus linear sequence is semi-polyhedral

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coefficients of a commutative rational expression in one letter that yield a given max-plus linear sequence is a finite union of polyhedral sets.     

Strictly positive real matrices and the Lefschetz-Kalman-Yakubovichlemma

The authors give necessary and sufficient conditions in the frequency domain for rational matrices to be strictly positive real. Based on this result, the matrix form of the Lefschetz-Kalman-Yakubovich lemma is proved, which gives necessary and sufficient conditions for strictly positive real transfer matrices in the state-space realization form     

Computation of reachable and observable realizations of spatial filters

The input-output behaviour of a two-dimensional linear filter is defined by a formal power series in two variables- If the power series is rational, the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given.

In general reachability and observability do not imply the minimality of the realization. Nevertheless the dimension of a minimal realization is the least rank in a family of Hankel matrices.     

State space construction for continuous time transfer function matrices via Nerode equivalence

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous-time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self-contained manner. Both scalar and multivariable cases in the continuous-time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2-norm bounded analytical vector function in the open left-half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time-invariant continuous-time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.     

Neural networks,rational functions,and realization theory

The problem of parametrizing single hidden layer scalar neural networks with continuous activation functions is investigated. A connection is drawn between realization theory for linear dynamical systems, rational functions, and neural networks that appears to be new. A result of this connection is a general parametrization of such neural networks in terms of strictly proper rational functions. Some existence and uniqueness results are derived. Jordan decompositions are developed, which show how the general form can be expressed in terms of a sum of canonical second order sections. The parametrization may be useful for studying learning algorithms.This work was supported by the Australian Research Council, the Australian Telecommunications and Electronics Research Board, and the Boeing Commencai Aircraft Company (thanks to John Moore).     

Reduced-order modelling of linear discrete-time systems via continued-fraction expansion of a z transfer function about z = 1 and z = a alternately

A new time-domain procedure is suggested for obtaining reduced-order models of linear time-invariant discrete-time systems. The procedure is based on presenting a new form of continued-fraction expansion (CFE) about z = 1 and z = a alternately, and deriving a realization form for the CFE. An algorithm is presented for obtaining the new CFE of the z transfer function of a linear discrete-time system from its state-space model directly, without having to determine the corresponding rational z transfer function. Also presented is a systematic approach to deriving two similarity transformation matrices: one is used to transform a state-space model from a general form to the CFE canonical form, and the other is used to transform a state-space model from the phase-variable canonical form to the CFE canonical form. Finally, an approximate aggregation matrix is constructed to relate the state vector of the original system to that of a reduced model obtained by the present method. The proposed procedure is illustrated with examples.     

Curvature of singular Bézier curves and surfaces

This paper presents a general approach for finding the limit curvature at a singular endpoint of a rational Bézier curve and the singular corner of a rational Bézier surface patch. Conditions for finite Gaussian and mean limit curvatures are expressed in terms of the rank of a matrix.     

On the partial stochastic realization problem

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length     

Comments on ‘New algorithm for minimal balanced realization of transfer function matrix’

Gonzalez and Munro (1990) presented a new algorithm for the minimal balanced realization of a transfer function matrix. We point out that their algorithm fails generically. We also give an alternative algorithm which is both more general and computationally efficient.     

Proper rational solution to 2-D two-sided polynomial matrix equations

Necessary and sufficient conditions are established for the existence of proper rational solutions to a 2-D two-sided polynomial matrix equation of the formAXB = C. Two procedures for finding a proper rational solution to the equation are presented. The first procedure can be used ifXis a finite impulse response. The second procedure is a general one.