Bernard Hanzon

Global Minimization of a Multivariate Polynomial using Matrix Methods

The problem of minimizing a polynomial function in several variables over Rn is considered and an algorithm is given. When the polynomial has a minimum the algorithm returns the global minimal value and finds at least one point in every connected component of the set of minimizers. A characterization of such points is given. When the polynomial does not have a minimum the algorithm computes its infimum. No assumption is made on the polynomial.

A FADDEEV SEQUENCE METHOD FOR SOLVING LYAPUNOV AND SYLVESTER EQUATIONS

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.

Approximate nonlinear filtering by projection on exponential manifolds of densities

This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the It6 or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADE An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

Constructive algebra methods for the L 2 -problem for stable linear systems

Abstract 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.

The area enclosed by the (oriented) Nyquist diagram and the Hilbert-Schmidt-Hankel norm of a linear system

It is shown that the Hilbert-Schmidt-Hankel norm (HSH-norm) of a transfer function of a stable system is equal, up to a constant factor, to the square root of the area enclosed by the oriented Nyquist diagram of the transfer function (multiplicities included). A generalization is presented for the case of systems which have no poles on the stability boundary, but otherwise have no restrictions on the pole locations. >

Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm

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.

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.

Symbolic computation of Fisher information matrices for parametrized state-space systems

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.

On some filtering problems arising in mathematical finance

Three situations in which filtering theory is used in mathematical finance are illustrated at different levels of detail. The three problems originate from the following different works: 1) On estimating the stochastic volatility model from observed bilateral exchange rate news, by R. Mahieu, and P. Schotman; 2) A state space approach to estimate multi-factors CIR models of the term structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3) Risk-minimizing hedging strategies under partial observation in pricing financial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In the first problem we propose to use a recent nonlinear filtering technique based on geometry to estimate the volatility time series from observed bilateral exchange rates. The model used here is the stochastic volatility model. The filters that we propose are known as projection filters, and a brief derivation of such filters is given. The second problem is introduced in detail, and a possible use of different filtering techniques is hinted at. In fact the filters used for this problem in 2) and part of the literature can be interpreted as projection filters and we will make some remarks on how more general and possibly more suitable projection filters can be constructed. The third problem is only presented shortly.

Balanced Parametrizations of Stable SISO All-Pass Systems in Discrete Time

Abstract. 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 H2 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.