Abstract
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.
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