Consensus of discrete-time linear multi-agent systems with Markov switching topologies and time-delay

This paper investigates the average consensus problems of the discrete-time linear multi-agent systems (LMAS) with Markov switching topologies. The average consensus protocol is a time-delay feedback switching controller. Compared with existing controllers, it is switching with time-delay. The constant time-delay exists in the signal feedback, and the time-varying time-delay exists in the state feedback. Firstly, we introduce a concept of the average consensus in this stochastic system. Then, we develop a new signal mode to simplify this challenging problem. By Lyapunov technique, two LMIs determinate theorems of average consensus are provided. Then we can find a controller to solve such problems effectively by these theorems. And the last theorem also reveals that we can determinate consensus by the maximum and minimum nonzero eigenvalues of the Laplacian matrices. Finally, a numerical example is given to illustrate the efficiency of our results.