L₁-Gain Controller Design for 2-D Markov Jump Positive Systems With Directional Delays

Abstract

This article is concerned with the stochastic stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> -gain control of two-dimensional (2-D) positive Markov jump systems (PMJSs) with directional delays based on the Roesser model. First, necessary and sufficient conditions (NSCs) for the stochastic stability of the addressed system are established by constructing a deterministic “equivalent” system and applying a stochastic copositive Lyapunov function. This reveals that the stochastic stability of 2-D PMJSs with delays is affected by the size of directional delays, the transition matrix, and system matrices. Second, the exact <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> -gain index is calculated and NSCs in the form of linear programming (LP) are established for the addressed system. Systematic methods for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> -gain controller design are proposed so that the closed-loop system (CLS) is positive and stochastically stable and has an optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> -gain performance, which is achieved using an iterative algorithm and an analytical calculation method for a single-input case. Finally, the potency and accuracy of the theoretical results are verified using two examples.