Strong controlled-invariance of behavioural nD systems

In this paper we give a stronger version of the notion of behavioral controlled invariance introduced in (Pereira & Rocha, 2017) in the context of regular partial interconnections. In such interconnections, the variables are divided into two sets: the variables to-be-controlled and the variables on which it is allowed to enforce restrictions (control variables); moreover, regularity means that the restrictions of the controller do not overlap with the ones already implied by the laws of the original behavior. A complete characterization of strong controlled invariance for nD behaviors is derived making use of a special controller behavior known as the canonical controller.publishe

Key problems in the extension of module-behaviour duality

AbstractThe duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. We demonstrate the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients

Controllable and autonomous nD linear systems

The 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 (k q) (Z n) is precisely one which is minimal in its transfer class