On the Finite-Region Stability of 2-D Systems

Some recent papers have extended the concept of finite-time stability (FTS) to the context of 2-D linear systems, where it has been referred to as finite-region stability (FRS). FRS methodologies make even more sense than the classical FTS approach developed for 1-D systems, since the state variables of 2-D systems often depend on a pair of space coordinates, rather than on time, as it is the case, for instance, of the image processing framework. Since space coordinates clearly belong to finite intervals, FRS techniques are much more effective than the classical Lyapunov approach, which looks to the asymptotic behavior of the system over an infinite interval. To this regard, the novel contribution consists of a new sufficient condition for FRS of linear time-varying (LTV) discrete-time 2-D systems, which turns out to be less conservative than those ones provided in the existing literature. Then, a sufficient condition to solve the finite-region stabilization problem is proposed. All the results provided in the paper lead to optimization problems constrained by linear matrix inequalities (LMIs), that can be solved via widely available software. Numerical examples illustrate and validate the effectiveness of the proposed technique.