Algebraic characterization of approximate controllability of behaviours of spatially invariant systems

An algebraic characterization of the property of approximate controllability is given, for behaviours of spatially invariant dynamical systems, consisting of distributional solutions w, that are periodic in the spatial variables, to a system of partial differential equations [formula presented] corresponding to a polynomial matrix M ∈ (C[ξ1,...,ξd,τ])m×n. This settles an issue left open in Sasane (2004)

Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems

We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions, that are periodic in the spatial variables, to a system pf partial differential equations corresponding to a polynomial matrix M in (C[\xi_1,...,\xi_d, \tau])^{m \times n}.Comment: 10 page

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized

Finite dimensional interconnections and stabilization of 2D behaviors

This paper deals with discrete two-dimensional behaviors which are described by linear systems of partial difference equations with constant coefficients. Within the behavioral framework a natural concept of interconnection has been introduced by J.C.Willems, called regular interconnec- tion. We investigate regular interconnections that yield finite dimensional behaviors, and prove that when a finite dimensional behavior can be achieved from a given behavior by regular interconnection then the controllable part of is rectifiable. We apply this result to characterize all stabilizable behaviors. © 2009 EUCA

Implementation of behavioral systems

In this chapter, we study control by interconnection of a given linear differential system (the plant behavior) with a suitable controller. The problem formulations and their solutions are completely representation free, and specified only in terms of the system dynamics. A controller is a system that constrains the plant behavior through a certain set of variables. In this context, there are two main situations to be considered: either all the system variables are available for control, i.e., are control variables (full control) or only some of the variables are control variables (partial control). For systems evolving over a time domain (1D) the problems of implementability by partial (regular) interconnection are well understood. In this chapter, we study why similar results are not valid in themultidimensional (nD) case. Finally, we study two important classes of controllers, namely, canonical controllers and regular controllers

Cultivating Psychological Determinants of Flow through Autonomy-Supportive Cognitive-Behavioural Training

In educational contexts, the deep and intrinsically rewarding engagement characteristic of being in flow is invaluable to the learning process. In addition to contributing to flourishing, psychological growth and development, flow is directly related to the frequency with which a student will actively vie to continue to use and extend their highest skills. A comprehensive framework delineating how to systematically cultivate flow would prove indispensable to those who aspire to optimise their performance or facilitate this strength in others. Still, little research has examined a systematic means of actively nurturing autonomous forms of motivational regulation to engage and the psychological strengths which underlie and promote flow in academic learning contexts. Therefore, the main objective of this small-scale descriptive pilot study was to ascertain the extent to which student-athletes could learn to wilfully cultivate dispositional flow states. It was presupposed that autonomy-supportive cognitive-behavioural training in a collaborative learning environment could in fact facilitate the process. The endeavour was thus approached by establishing a multimodal cognitive-behavioural training program designed to systematically cultivate the nine dimensions of flow. The study adhered to an explanatory sequential mixed methods research design. Thus, the 13 sport science students (four females and nine males) participating in the 12-week seminar completed pretest/posttest dispositional assessments of their locus of motivational regulation, their use of cognitive-behavioural performance enhancement strategies, and flow. In addition, six months subsequent to the intervention, structured interviews were conducted with a subset of the cohort and a thematic analysis of the resultant data set was conducted in an effort to both further interpret and elucidate the results yielded from the quantitative data set. Although the psychometric test findings did not yield unequivocal results, they demonstrated posttest increases in students’ intrinsic motivational regulation as well as their use of self-talk, activation, imagery, and attention control strategies. Finally, while all but two student-athletes reported an increase in their general propensity to experience unidimensional flow, unvarying results were not yielded across the multidimensional measures thereof. However, the thematic analysis provided evidence that the student-athletes believed that if employing performance strategies including a systematic goal setting process, arousal regulation, imagery, and self-talk, one can in fact cultivate flow if one wants to. Therefore, this study contributes to scholarship pertaining to understanding how to deliberately promote flow in similar higher learning contexts

Algebraic theory of time-varying linear systems: a survey

The development of the algebraic theory of time-varying linear systems is described. The class of systems considered consists of differential-algebraic equation in kernel presentation. This class encompasses time-varying state space, descriptor systems as well as Rosenbrock systems, and time-invariant systems in the behavioural approach.One difference between time-varying and time-invariant systems is that, since the coefficients of the differential equations are time-varying function, the differential operator does not commute with the coefficients. However, the main difficulty is that solutions may exhibit a finite escape time. Hence there is a conflict between the class of time-varying coefficients and the class of admissible solution spaces. All contributions to time-varying systems have to cope with this.As an efficient tool in linear, time-invariant system theory, Kalman introduced in the 1960s elementary module theory over principal ideal rings. This tool proved efficient also for time-varying systems. Although from then on, the field of time-varying linear systems has never been a ``hot topic" in systems theory, there has been an ongoing evolution which led to a rather substantial theory. Not surprisingly, the theory is mainly restricted to linear systems and most results are on such properties as controllability, and not on stability. Recent results use successfully tools from module theory and homological algebra