Stefano Pinzoni

Acausal models and balanced realizations of stationary processes

Abstract We study acausal realizations of stationary (or stationary-increment) processes. In particular, we characterize the family of models whose corresponding spectral factors have a fixed zero structure. Acausal models with a fixed zero structure are related to each other by a certain group of state-feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic Riccati equation. Each feedback transformation reflects some of the eigenvalues of the generator matrix A of the representation to a mirror image with respect to the imaginary axis. Dually, acausal models with a fixed “pole structure” are parametrized by a dual Riccati equation and by a corresponding family of output injection transformations. From a general standpoint the results of this study clarify the role played by dual pairs of Riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling. One natural application of the concepts discussed in the paper is to stochastic balancing. Balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework. A balancing algorithm involving the solution of a dual pair of Riccati equations is discussed.

Technical Communique: Zeros of Continuous-time Linear Periodic Systems

Zeros of continuous-time linear periodic systems are defined and their properties investigated. Under the assumption that the system has uniform relative degree, the zero-dynamics of the system is characterized and a closed-form expression of the blocking inputs is derived. This leads to the definition of zeros as unobservable characteristic exponents of a suitably defined periodic pair. The zeros of periodic linear systems satisfy blocking properties that generalize the well-known time-invariant case. Finally, an efficient computational scheme is provided that essentially amounts to solving an eigenvalue problem.

Silverman algorithm and the structure of discrete-time stochastic systems

Abstract It is a well-known, yet poorly understood fact that, contrary to the continuous-time case, the same discrete-time process y can be represented by minimal linear models (see (1.1a), (1.1b) below) which may either have a non-singular or a singular D matrix. In fact, models with D=0 have been commonly used in the statistical literature. On the other hand, for models with a singular D matrix the Riccati difference equation of Kalman filtering involves in general the pseudo-inversion of a singular matrix. This “cheap filtering” problem, dual to the better known “cheap control” problem, has been studied for several decades in connection with the so-called “invariant directions” of the Riccati equation. For a singular D, a reduction of the order of the Riccati equation is in general possible. The reasons for such a reduction do not seem to be completely clear either. In this paper we provide an explanation of this phenomenon from the classical point of view of “zero flipping” among minimal spectral factors. Changing Ds occurs whenever zeros are “flipped” from z=∞ to their reciprocals at z=0. It is well known that for finite zeros, the zero-flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all-pass matrix function. For infinite zeros, zero flipping is implemented by a dual version of the Silverman structure algorithm. Using this interpretation, we derive a new algorithm for filtering of non-regular processes, based on a reduced-order Riccati equation. We also obtain a precise characterization of the reduction of the order of the Riccati equation which is afforded by zeros either at z=∞ or at the origin. This order reduction has traditionally been associated with the study of invariant directions, a point of view which, as we show, does not capture the essence of the phenomenon.

AsymmetricalgebraicRiccatiequation:Ahomeomorphicparametrizationofthesetof solutions

Abstract In this paper, asymmetric algebraic Riccati equations are analyzed. In particular, we derive a new parametrization of the set of solutions. Generalizing on the symmetric case, the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related “feedback” matrices. Moreover, the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo-Hamiltonian matrix associated with the equation. We finally show that also the newly introduced parametrization is given by a homeomorphic map.

On the relation between additive and multiplicative decompositions of rational matrix functions

In this paper, a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented. Various extensions of the positive real lemma, as well as of other classical factorization results, both in the continuous and discrete-time cases, are established. In the case of square factorizations, the map between solutions to an asymmetric algebraic Riccati equation and pairs of factors is shown to be a homeomorphism. In this framework, we also derive a geometric characterization of non-square factorizations.

Spectral factorization and stochastic realization with zeros on the unit circle

We study the spectral factorization problem of a rational m/spl times/m spectral density matrix /spl Phi/(z) = /spl Phi/(z/sup -1/)/sup T/ which has zeros on the unit circle. In the control/estimation literature this situation is either excluded by assuming appropriate stabilizability and detectability conditions or is discussed only partially. Here we provide what seems to be a complete solution, at least in the full-rank case.