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Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

Anew finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chan can be left out of the atlas without losing the property that the atlas covers the manifold. (c) 2007 Elsevier Inc. All rights reserved.

In this paper we study a novel parametrization for state-space systems, namely separable least squares data driven local coordinates (slsDDLC). The parametrization by slsDDLC has recently been successfully applied to maximum likelihood estimation of linear dynamic systems. In a simulation study, the use of slsDDLC has led to numerical advantages in comparison to the use of more conventional parametrizations, including data driven local coordinates (DDLC). However, an analysis of properties of slsDDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the slsDDLC construction and show a number of results which are important for actual identification, in particular for maximum likelihood estimation. We also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations.

In this paper, we show that a recently defined cepstral norm for ARMA models equals, up to a constant factor, the square root of the area enclosed by the polar plot of the logarithm of the transfer function.

We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

Through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension. A possible problem of these parametrizations is that they are not overlapping. This could be a drawback for the application of balanced parametrizations in such areas as system identification, model reduction and optimization. It is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations. We first introduce input-normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix. These canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered. Then we introduce so-called block-balanced input normal forms which use the previously constructed input normal forms as building blocks. The classes of systems for which such parametrizations are given are the stable minimal systems, positive-real minimal systems, bounded-real minimal systems and the class of all minimal systems of given McMillan degree. The results include both the single-input single-output and the multivariable case. In the single-input single-output case, however, the issue of choosing nice selections of columns does not occur. Therefore in this case the derivation and presentation of the results is considerably simplified.

The balanced canonical form and parametrization of Ober for the case of SISO stable systems are extended to block-balanced canonical forms and related input-normal forms and parametrizations. They form an overlapping atlas of parametrizations of the manifold of stable SISO systems of given order. This extends the usefulness of these parametrizations, e.g., in gradient algorithms for system identification. As an implication of our construction it follows that each of the subsets of the parametrization of [R. Ober, Internat. J. Control, 46 (1987), pp. 643--670] corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable SISO systems of fixed order.

We investigate the feasibility of using computer algebra for solving L 2 system approximation problems. The first-order optimality conditions yield a set of polynomial equations, which can, in principle, be solved using Grobner basis methods. A general solution along these lines would be tremendously useful, although it does not appear to be feasible at present, except for rather low McMillan degrees. We demonstrate that it can be feasible for specific examples; in such cases global optima can be found reliably. We show that the use of a Schwarz-like canonical form simplifies the structure of the problem.

Lyapunov and Sylvester equations play an important role in linear systems theory. This paper deals with a method of solving such equations of the form AP + PB = K and P − APB = K with A ∈ Rm × m, B ∈ Rn × n, and P, K ∈ Rm × n, by exploiting the matrix-algebra structure of the problem. No use is made of Kronecker products and the largest matrices occurring in the algorithms are of sizes m × m, n × n, and m × n. The Faddeev method for matrix inversion lies at the very heart of the algorithms presented. It occurs on several levels of the problem: for the matrices A and B and for the Lyapunov and Sylvester operators. The resulting algorithms are capable of solving the equations in a finite number of recursion steps. They are very much apt for symbolic calculation. It is shown how a solution can be quickly obtained for an equation with an arbitrary right-hand side K, provided a solution is known for a right-hand side xyT of rank 1, where (A, x) and (BT, y) are reachable pairs. The concept of a Faddeev reachability matrix introduced here turns out to be very useful. It establishes a close connection between the controller canonical (companion) form of a reachable pair (A, b) and the Faddeev sequence of A. If A is already on controller form, then its Faddeev sequence takes on an especially simple form. Also in the symmetric case where A = BT, many important simplifications arise. For this case alternative algorithms that require less iterations are developed. The paper concludes with some examples concerning the symbolic solution of the Lyapunov equation AP + PAT = bbT with (A, b) on controller form, showing the potential of the algorithms.

In this paper, we show that the following classes of linear dynamical systems are diffeomorphic to each other: fixed order, asymptotically stable, strictly bounded real, strictly positive real, stable and minimum phase, and invertible. These results are then used to prove the diffeomorphisms between the state bundles associated with these classes of systems, and between their associated principal fibre bundles. We shall conclude this paper by discussing how these results can be used to deduce geometric properties of other classes of linear systems, and to construct overlapping parametrizations.

We investigate the feasibility of using computer algebra for solving L 2 system approximation problems. The first-order optimally conditions yield a set of polynomial equations which can, in principle, be solved using Groebner basis methods. A general solution along these lines would be tremendously useful, although it does not appear to be feasible at present except for rather low McMillan degrees. We demonstrate that it can be feasible for specific examples; in such cases global optima can be found reliably. We show that the use of a Schwarz-like canonical form simplifies the structure of the problem, and the required computations.

Semi-plenary conference by Ralf L.M. Peeters; International audience; Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification , system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

International audience; In this paper, the connections are investigated between two different approaches towards the parametrization of multivariable stable all-pass systems in discrete-time. The first approach involves the tangential Schur algorithm, which employs linear fractional transformations. It stems from the theory of reproducing kernel Hilbert spaces and enables the direct construction of overlapping local parametrizations using Schur parameters and interpolation points. The second approach proceeds in terms of state-space realizations. In the scalar case, a balanced canonical form exists that can also be parametrized by Schur parameters. This canonical form can be constructed recursively, using unitary matrix operations. Here, this procedure is generalized to the multivariable case by establishing the connections with the first approach. It gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential Schur algorithm.

We describe a novel approach, called data driven local coordinates (DDLC), for parametrizing linear systems in state space form, and we analyze some of its properties which are relevant for e.g. maximum likelihood estimation. In addition we describe how this idea can be used for a concentrated likelihood function, obtained by a least squares type concentration step, which gives the so called sls (separable least squares) DDLC approach. Both approaches give favourable results in numerically optimizing the likelihood function in simulation studies.

The asymptotic Fisher information matrix (FIM) has several applications in linear systems theory and statistical parameter estimation. It occurs in relation to the Cramer-Rao lower bound for the covariance of unbiased estimators. It is explicitly used in the method of scoring and it determines the asymptotic convergence properties of various system identification methods. It defines the Fisher metric on manifolds of systems and it can be used to analyze questions on identifiability of parametrized model classes. For many of these applications, exact symbolic computation of the FIM can be of great use. In this paper two different methods are described for the symbolic computation of the asymptotic FIM. The first method applies to parametrized MIMO state-space systems driven by stationary Gaussian white noise and proceeds via the solution of discrete-time Lyapunov and Sylvester equations, for which a method based on Faddeev sequences is used. This approach also leads to new short proofs of certain well-known results on the structure of the FIM for SISO ARMA systems. The second method applies to parametrized SISO state-space systems and uses an extended Faddeev algorithm. For both algorithms the concept of a Faddeev reachability matrix and the solution of discrete-time Sylvester equations in controller companion form are central issues. The methods are illustrated by two worked examples. The first concerns three different parametrizations of the class of stable AR(n) systems. The second concerns a model for an industrial mixing process, in which the value of exact computation to answer identifiability questions becomes particularly clear.

A new balanced canonical form is presented for stable multivariable linear systems. Overlapping continuous block-balanced canonical forms were introduced by Hanson-Ober for the stable single-input/single-output (SISO) case as a generalisation of the balanced canonical form introduced by Ober (1987) for the SISO case. In the search for a generalization of these results to the multivariable case a new multivariable balanced canonical form was discovered, which is of interest in its own right and is presented in this paper. The new canonical form has a number of nice properties. The integer invariants that appear in the canonical form are the multiplicities of the Hankel singular values and a number of new invariants, which are in one-to-one objective correspondence with the Kronecker indexes of subsystems. Truncation of the state vector leads to stable minimal models in canonical form. In the SISO case the canonical form coincides with Ober's balanced canonical form. The reachability matrix of a system in canonical form with identical singular values is positive upper triangular. >

In this paper we are concerned with linear (stochastic) systems like dxt = (Axt+B1ut)dt + B2dwt, dyt = Cxt dt + dvt or (more or less equivalent) ARMAX models and the problem of identifying A, B1, B2, C on the basis of observations of the inputs ut and outputs yt. In particular we are interested in the problem of whether there exists a machine (a system) driven by the instant neous observations (ut, yt) which as output produces a "best" estimate of the unknown system (recursive estimation). And even more particularly we are interested on how big (in state space dimension) such a machine must be. Introducing additional state space parameters a ij , b k l 1 , b k l 2 , c r, s and equations da ij = db k l 1 = db k l 2 = dc rs = 0 converts the original problem into a nonlinear filtering problem. For such problems the socalled estimation Lie algebra contains a good deal of information (on how hard the problem is), and this is what we try to exploit in this paper.

It is shown that the (real analytic) differentiable manifold of time-invariant linear dynamical systems of fixed McMillan degree n with m′-dimensional inputs and m-dimensional outputs is (real analytically) diffeomorphic to the submanifold of i/o-stable (i.e. ‘asymptotically stable’) systems. This generalizes results of Helmke, Ober and Hanzon about the topology of such manifolds. Similar results hold for the corresponding state bundles and their associated principal bundles. This implies in fact that there is a one-to-one correspondence between smooth (local) canonical forms for finite dimensional linear systems of fixed order and for finite dimensional i/o-stable linear systems of fixed order.

The Fisher information matrix has several applications in linear systems theory. It appears in relation to the Cramer-Rao bound for unbiased estimators, methods for system identification, questions on identifiability and it defines the Fisher metric on manifolds of systems. In this paper state-space methods for the symbolic computation of explicit analytical expressions for the FIM are presented. These proceed by the solution of Lyapunov and Sylvester equations. To this end, methods based on Faddeev sequences are exploited.

A balanced canonical form for discrete-time stable SISO all-pass systems is obtained by requiring the realization to be balanced and such that the reachability matrix is upper triangular with positive diagonal entries, in analogy to the continuous-time balanced canonical form of Ober [O1]. It is shown that the resulting balanced canonical form can be parametrized by Schur parameters. The relation with the Schur parameters for stable AR systems is established. Using the structure of the canonical form it is shown that, for the space of stable all-pass systems of order less than or equal to a fixed number n, the topology of pointwise convergence and the topology induced by H(2) coincide. The topological space thus obtained has the structure of a hypersphere. Model reduction procedures based on truncation! which respect the canonical form, are discussed.