Analyzing stability in 2D systems via LMIs: From pioneering to recent contributions

Abstract

2D systems, also known as doubly-indexed systems, have gained an increasingly special attention in the control community, as they allow for modeling systems with more complex dynamics than the classical so called 1D systems where the signals are indexed by one variable only usually representing the time. Like for 1D systems, stability conditions have been proposed for 2D systems in the form of a linear matrix inequality (LMI) feasibility test, as such conditions may be tested by solving a convex optimization problem, and as such conditions may open the door for a number of developments such as establishing robust stability and designing stabilizing controllers. This paper aims at presenting, under a unified framework, various LMI stability conditions for 2D systems that have been proposed in the literature, from pioneering to recent contributions, in order to provide the reader with a comprehensive collection that may serve as a source of historical information as well as a platform for comparing the major characteristics of each condition. Also, this paper proposes novel investigations of the presented conditions, in particular through conservatism and complexity analyses carried out in the best cases, in the worst cases, and for various specific numerical examples with different type of dynamics, dimensions and difficulty.