Martin Scheicher

Time-autonomy and time-controllability of discrete multidimensional behaviours

Predecessors and starting point of this article on multidimensional discrete time-autonomy (ta) and time-controllability (tc) were three papers from 2002 to 2004 by Sasane, Controneo, Thomas and Willems on the continuous case, a paper from 2011 by Napp, Rapisarda and Rocha on time-relevant two-dimensional (2D)-behaviours and a submitted paper by Bisiacco and Valcher on 2D-tc and dead-beat control. We also acknowledge interesting talks on this subject by Bisiacco and Valcher in Innsbruck. We develop a general framework for discrete multidimensional behaviours in which the discrete time variable in ℕ plays a special part among all independent variables. Following the models of our predecessors we define ta and tc of a behaviour by properties of its trajectories. The goal of this article is the characterisation of these properties by constructive algebraic conditions which is fully achieved for ta and for tc in dimension two and partially for tc in dimensions higher than two.

Multidimensional Discrete Stability by Serre Categories and the Construction and Parametrization of Observers via Gabriel Localizations

Immediate predecessors of this work were a paper on two-dimensional deadbeat observers by Bisiacco and Valcher [Multidimens. Systems Signal Process., 19 (2008), pp. 287--306] and one on one-dimensional functional observers by Blumthaler [Linear Algebra Appl., 432 (2010), pp. 1560--1577] (compare also Fuhrmanns comprehensive paper [Linear Algebra Appl., 428 (2008), pp. 44--136]). The present paper extends Blumthalers results to continuous or discrete multidimensional behaviors, i.e., constructs and parametrizes all controllable observers of a given multidimensional behavior, and for this purpose also discusses the required multidimensional stability. Such an observer produces a signal that approximates or estimates a desired component of the behavior such that the signal difference is negligible in a suitable sense. This definition thus presupposes that of negligible or stable autonomous systems. In the standard one-dimensional case these are the asymptotically stable behaviors. We define and investigate...

Controllability up to negligible trajectories of discrete multidimensional behaviours

We define the controllability of the title algebraically and characterise it by a corresponding concatenability of trajectories. The main result is motivated by and extends the characterisation of standard controllability by Wood and Zerz (Notes on the definition of behavioural controllability, Syst Control Lett 37(1):31–37, 1999). Negligibility is defined with respect to a suitably chosen Serre category of modules over a given operator domain. A behaviour is called negligible if its module belongs to this category. Standard examples of negligible behaviours with respect to suitably chosen Serre categories are asymptotically stable and nilpotent behaviours according to Bisiacco and Valcher. A behaviour is called controllable up to negligible trajectories if its factor behaviour modulo its largest controllable subbehaviour is negligible. This controllability notion generalises multidimensional stabilisability and coincides with it in dimension one. The preceding definitions also apply to continuous systems. The domain of the independent discrete variables of the considered discrete behaviours is an arbitrary finitely generated submonoid of a free abelian group.

Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids

Abstract For finitely generated submonoids of the integer lattice and submodules over the associated monoid algebra, we investigate Grobner bases with respect to generalised term orders. Up to now, this theory suffered two disadvantages: The algorithm for computing the Grobner bases was slow and it was not known whether there existed generalised term orders for arbitrary finitely generated submonoids. This limited the applicability of the theory. Here, we describe an algorithm which transports the problem of computing the Grobner bases to one over a polynomial ring and use the conventional Grobner theory to solve it, thus making it possible to apply known, optimised algorithms to it. Furthermore, we construct generalised term orders for arbitrary finitely generated submonoids. As an application we solve the Cauchy problem (initial value problem) for systems of linear partial difference equations over finitely generated submonoids.

A Generalized Tracking and Disturbance Rejection Problem for Multidimensional Behaviors

In this paper we study a generalized tracking and disturbance rejection problem for multidimensional behaviors. Given a multidimensional plant, our first goal is to design a compensator to be connected to the plant through regular partial interconnection, in such a way that the overall controlled system is autonomous and stable, when no exogenous signal acts on the system. On the other hand, when exogenous signals affect the controlled system evolution, we want to impose that a suitable linear combination of the overall system trajectories is “negligibile” in a sense we will clarify within the paper. This problem setup formalizes a number of classical control problems, first of all tracking of some (reference) signal together with rejection of another (disturbance) signal. The adopted approach is extremely general and it is based on the idea of describing all behavior trajectories as the sum of a “transient signal” and a “steady state” component, a decomposition that relies on Gabriels localization theor...

The asymptotic stability of stable and time-autonomous discrete multidimensional behaviors

We generalize the important paper of Napp et al. (Automatica 47, 2373–2382, 2011), to discrete time-autonomous (ta) (=time-relevant), but not necessarily square-autonomous behaviors in arbitrary dimensions. This paper and therefore also the present one were essentially influenced by the papers of Wood et al. (SIAM J Control Optim 43, 1493–1520 2005), and Willems (Proceedings of the International Conference on Multidimensional (nD) Systems, Aveiro, 2007). In the present paper the discrete domain of the independent variables is the lattice of vectors of integers of arbitrary (but fixed) length whose first component is a natural number and interpreted as a discrete time instant. The stability of an autonomous behavior is defined by a spectral condition on its characteristic variety. The behavior is time autonomous if each trajectory is determined by a fixed number of its initial values. Under a weak additional condition, a discrete stable and time-autonomous behavior is asymptotically stable in the sense that under suitable initial conditions its trajectories converge to zero when the time tends to infinity. We derive algorithms for the constructive verification of the assumptions of most of our results and in particular establish a constructive normal form of ta behaviors in arbitrary dimensions. The Fourier transform on finitely generated free abelian groups plays an important part in the derivations as it already did in the quoted papers. Stability and stabilization of multidimensional discrete behaviors were previously discussed by various colleagues, for instance by Bisiacco, Bose, Fornasini, Lin, Marchesini, Pillai, Quadrat, Rogers, Shankar, Sule, Valcher, Wood, but only partly from the analytic point of view.