J. Wood

Modules and Behaviours in nD Systems Theory

This paper is intended both as an introduction to the behavioural theory of nD systems, in particular the duality of Oberst and its applications, and also as a bridge between the behavioural theory and the module-theoretic approach of Fliess, Pommaret and others. Our presentation centres on Pommarets notion of a system observable, and uses this concept to provide new interpretations of known behavioural results. We discuss among other subjects autonomous systems, controllable systems, observability, transfer matrices, computation of trajectories, and system complexity.

Trajectory Control and Interconnection of 1D and nD Systems

In this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems. We show that, for one-dimensional linear time-invariant systems, the ability to obtain a given subsystem by regular interconnection (a prerequisite for any feedback-type structure) is equivalent to the ability to drive any trajectory into that subsystem. However, in the case of multidimensional systems, the former is a stronger property than the latter. Trajectory controllability can, however, be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables. This leads as a by-product to the notion of controlling a system by means of latent variables.

A formal theory of matrix primeness

Primeness of nD polynomial matrices is of fundamental importance in multidimensional systems theory. In this paper we define a quantity which describes the “amount of primeness” of a matrix and identify it as the concept of grade in commutative algebra. This enables us to produce a theory which unifies many existing results, such as the Bézout identities and complementation laws, while placing them on a firm algebraic footing. We also present applications to autonomous systems, behavioural minimality of regular systems, and transfer matrix factorization.

Controllable and Autonomous nD Linear Systems

AbstractThe theory of multidimensional systems suffers in certain areas from a lack of development of fundamental concepts. Using the behavioural approach, the study of linear shift-invariant nD systems can be encompassed within the well-established framework of commutative algebra, as previously shown by Oberst. We consider here the discrete case. In this paper, we take two basic properties of discrete nD systems, controllability and autonomy, and show that they have simple algebraic characterizations. We make several non-trivial generalizations of previous results for the 2D case. In particular we analyse the controllable--autonomous decomposition and the controllable subsystem of autoregressive systems. We also show that a controllable nD subsystem of

Notes on the definition of behavioural controllability

(k^q )^{(Z^n )}

Controllable and Uncontrollable Poles and Zeros of nD Systems

is precisely one which is minimal in its transfer class.

Key problems in the extension of module-behaviour duality

We use the tools of behavioral theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional (nD) system. We make a natural division of the poles into controllable and uncontrollable poles. When the behavior in question has latent variables, we make a further division into observable and unobservable poles. In the case of a one-dimensional (1D) state-space model, the uncontrollable and unobservable poles correspond, respectively, to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices, and their left and right matrix fraction descriptions (MFDs). We find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behavior as the sum of subbehaviors associated with various poles. This is related to the integral representation theorem, which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

Singular 2D behaviors: Fornasini-Marchesini and Givone-Roesser Models

We take another look at the behavioural definition of controllability for discrete 1D systems, and its extension to multidimensional (nD) systems defined on