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This paper formulates and studies the distributed formation problems of multi-agent systems (MAS) with randomly switching topologies and time-varying delays. The nonlinear dynamic of each agent at different time-interval corresponds to different switching mode which reflects the changing of traveling path in practical systems. The communication topology of the system is switching among finite modes which are governed by a finite-state Markov process. On the basis of artificial potential functions (APFs), a formation controller is designed in a general form. Sufficient conditions for stochastic formation stability of the multi-agent system are obtained in terms of Lyapunov functional approach and linear matrix inequalities (LMIs). Some heuristic rules to design a formation controller for the MAS are then presented. Finally, specific potential functions are discussed and corresponding simulation results are provided to demonstrate the effectiveness of the proposed approach.

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This paper is concerned with the stability analysis of discrete-time systems with time-varying state delay. By defining a new Lyapunov functional and by making use of novel techniques to achieve delay dependence, a new criterion is obtained for the asymptotic stability of these systems. The proposed criterion is less conservative as well as needs fewer decision variables than some pervious ones, which is established theoretically. Two examples are given to illustrate the advantages of our method.