Two-dimensional (2D) positive systems are 2D state-space models whose state, input and output variables take only nonnegative values. In the paper we explore how linear matrix inequalities (LMIs) can be used to address the stability problem for 2D positive systems. Necessary and sufficient conditions for the stability of positive systems have been provided. The results have been obtained for most popular models of 2D positive systems, that is: Roesser model, both Fornasini-Marchesini models (FF-MM and SF-MM) and for the general model.

The paper discusses the stability problem for continuous time and discrete time positive systems. An alternative formulation of stability criteria for positive systems has been proposed. The results are based on a theorem of alternatives for linear matrix inequality (LMI) feasibility problem, which is a particular case of the duality theory for semidefinite programming problems.