Row reduced representations of behaviors over finite rings

Row reduced representations of behaviors over fields posses a number of useful properties. Perhaps the most important feature is the predictable degree property. This property allows a finite parametrization of the module generated by the rows of the row reduced matrix with prior computable bounds. In this paper we study row-reducedness of representations of behaviors over rings of the form $\mathbb{Z}_{p^r}$, where $p$ is a prime number. Using a restricted calculus within $\mathbb{Z}_{p^r}$ we derive a meaningful and computable notion of row-reducedness

State maps for linear systems

Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential-algebraic equations. As such, state-space models are not a natural starting point for modeling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state variables for systems of linear differential-algebraic equations of various forms. The point of view from which the problem is considered is the behavioral one, as put forward in [J. C. Willems, Automatica J. IFAC, 22 (1986), pp. 561–580; DynamicsReported,2(1989),pp.171–269;IEEETrans.Automat.Control,36(1991),pp. 259–294]

Observer theory

AbstractThe paper is devoted to a comprehensive exposition of the theory of partial state observers in the state space context and the elucidation of the connection between this theory and the theory of observers in the behavioral context, as developed in Valcher and Willems [1999]. For this we use several techniques, including geometric control theory, polynomial and rational models, shift realizations, coprime factorizations, partial realizations and the basic results on behaviors and behavior homomorphisms. A connection between observers and the construction of state maps is made

Realizations of behavior for generalized chain-scattering representations

This paper introduces a notion of realization of behavior which is shown to be a generalization of the classical concept of a realization of transfer function. By using this approach, the input-output structures of the generalized chain-scattering representations (GCSRs) and the dual generalized chain-scattering representations (DGCSRs) are investigated in a behavioral theory context. Subsequently the corresponding autoregressivemoving- average (ARMA) representations are proposed and are proved to be realizations of behavior for any GCSR. Realization of behavior is particularly suitable for situations in which the coefficients are symbolic rather than numerical due to the fact that no numerical computation is involved in this approach

Behavioral realizations using companion matrices and the smith form

This is the author accepted manuscript. The final version is available from Society for Industrial and Applied Mathematics via the DOI in this record.Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant be- havior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks

Distance between Behaviors and Rational Representations

In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space L2(R). In particular we generalize the “classical” gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on L2(R). We introduce a fourth behavioral metric as a generalization of the well-known ν-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compare the values they take and the topologies they induce. Finally, for all metrics we make a detailed study of necessary and sufficient conditions under which the distance between two behaviors is less than one. For this, both behavioral as well as state space conditions are derived in terms of driving variable representations of the behaviors