Positivity of discrete singular systems and their stability: An LP-based approach

In this paper we present an efficient approach to the analysis of discrete positive singular systems. One of our main objectives is to investigate the problem of characterizing positivity of such systems. Previously, this issue was not completely addressed. We provide easily checkable necessary and sufficient conditions for such problem to be solved. On the other hand, we study the stability of discrete positive singular systems. Note that this is not a trivial problem since the set of admissible initial conditions is not the whole space but it is represented by a special cone. All the conditions we provide are necessary and sufficient, and are based on a reliable computational approach via linear programming

Control of 2D behaviors by partial interconnection

Abstract-In this paper we study the stability of two dimensional (2D) behaviors with two types of variables: the variables that we are interested to control (the to-be-controlled variables) and the variables on which we are allowed to enforce restrictions (the control variables). We derive conditions for the stabilization of the to-be-controlled variables by regular partial interconnection, i.e., by imposing non-redundant additional restrictions to the control variables

Time-relevant stability of 2D systems

For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems 'time-relevant'. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices

An algebraic approach to multidimensional behaviors

The main aim of this thesis is the study of systems described by linear constant coefficient partial differential equations (PDEs) using commutative algebra. In order to do so we use an elegant mathematical approach called "the behavioral" framework. In this approach a system is determined by its behavior which is the set of trajectories that satisfy the laws ( i.e. equations) of the system. Fundamental questions of systems and control theory of PDEs are posed and solved in this context, e.g. feedback control, regular interconnection, partial interconnection, decomposition into controllable and autonomous part, spectral factorization, etc. In order to tackle these problems we will develop and use a powerful natural duality between these systems and finitely generated modules over the polynomial ring of differential operators. Hence, we can to translate many analytic properties of the system into algebraic properties of the corresponding module. This is very useful since it allows the use of the potent machine of computational algebra to provide constructive solutions.

Algorithms for interconnection and decomposition problems with multidimensional systems

The notion of interconnection is the basis of control in the behavioral approach. In this setting, feedback interconnection of systems is based on the still more fundamental concept of regular interconnection, which has been introduced previously. In this paper, the following problem is addressed: given a plant, under what conditions does there exist a controller such that their interconnection is regular and has finite codimension with respect to a certain desired system. If so, provide a constructive solution to the problem. The second part of the paper treats the related problem of decomposition of systems. First, the autonomous/controllable decomposition is studied, and finally we look at the decomposition of the controllable part.

Lyapunov stability of 2D finite-dimensional behaviors

In this paper we investigate a Lyapunov approach to the stability of finite-dimensional 2D systems. We use the behavioral framework and consider a notion of stability following the ideas in Pillai and Shankar (1998), Rocha (2008), Valcher (2000). We characterize stability in terms of the existence of a (quadratic) Lyapunov function and provide a constructive algorithm for the computation of all such Lyapunov functions