L₁-gain control for 2D delayed positive continuous Markov jumping systems

This study is focused on analyzing the L₁-stochastic internal stability and the design of an L₁-gain controller for two-dimensional (2D) continuous positive Markov jumping systems (PMJSs) with constraints on the inputs and states. An algorithm is developed to explicitly determine the state feedback control law with the optional L₁-gain performance. First, by constructing a co-positive stochastic Lyapunov function for the positive system and establishing the equation for the Markov states' mathematical expectation, several sufficient and necessary conditions for L₁-stochastic internal stability are derived. This analysis indicates that 2D PMJSs with directional delays are influenced by sizes of the system matrices and magnitude of delays, and transition matrix. Second, a method for calculating the L₁-gain is formulated utilizing linear programming (LP), thus, an L₁-gain controller is designed to guarantee that the closed-loop system is positive and L₁-stochastically internally stable, while achieving optimal L₁-gain performance. Two numerical and practical examples are considered, and the influence of delays and transition probability matrices on the L₁-gain is specified to validate the preceding theoretical findings.