| Linear periodic systems have wide applications in many fields. In addition, the antilinear mapping has been paid more attention. In this dissertation, antilinear mappings and linear periodic systems are combined, and thus discrete antilinear periodic system is proposed. This class of systems can be viewed as a type of complex dynamic systems with structure constraint, which can be used to model the energy generation and distribution in the actual systems. Complex dynamic systems with structure constraint are ubiquitous in our real life. Therefore, the antilinear discrete periodic system is worthy of our deep study. In this dissertation, the main research contents and results include the following aspects.Firstly, the concept and two kinds of mathematical descriptions of the antilinear discrete periodic system are introduced. Besides, state-transition matrix and state response of the system are proposed. By comparing with the concept of monodromy matrix of linear periodic system, the monodromy matrix of discrete antilinear periodic system and relevant expressions are given.Secondly, the concept and related criteria of controllability and observability of discrete antilinear periodic system are given. It has been shown that for the discrete antilinear periodic system, the controllability and reachability are not equivalent.Further, some criteria of stability are proposed by the second differentiable method of Lyapunov for discrete antilinear periodic systems in terms of the so-called coupled periodic anti-Lyapunov matrix equation. Numerical examples are given to show the effectiveness of the presented criteria.Finally, some algorithms are proposed for solving the coupled periodic anti-Lyapunov matrix equations. Based on the iterative algorithm, new iterative algorithm is presented by making full use of the latest estimation information. The effectiveness of the new iterative algorithm has been verified by theoretical analysis and numerical simulations. |
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