Discussion on: 'Positive switched 2D linear systems described by the Roesser models'

Abstract

The paper by T. Kaczorek (see [1] in the references list for complete details) addresses the problem of establishing asymptotic stability of positive switched 2D linear systems described by the Roesser models. For this problem, the paper provides necessary and sufficient conditions that ensure asymptotic stability for any switching. Moreover, the paper illustrates the use of some of the provided conditions through numerical examples. This is a concise paper that addresses the fundamental problem of asymptotic stability at the intersection of three main areas of control systems, namely switched systems, 2D systems, and positive systems. It is well-known that these areas play a key role in the study and design of real systems. Indeed, switched systems are those systems where the coefficients switch within a certain set of values, for example due to choices of the user and/or actions of the controller. 2D systems are those systems where the state evolves along two directions rather than one only as in traditional 1D systems, which is for example the case of 2D signals. Positive systems are those systems where the state variables can take only nonnegative values due to their nature, for example as it happens in biomolecular models where the state variables are concentrations. The paper considers the case of 2D linear systems described by the Roesser models. These models extend traditional discrete-time1Dlinear systems by partitioning the state vector into two parts, typically called horizontal and vertical parts, and by defining the evolution of these parts along different directions corresponding to two indexes, typically called horizontal and vertical indexes.