Necessary and sufficient conditions for solving consensus problems of double‐integrator dynamics via sampled control

In this paper, consensus problems of double‐integrator dynamics via sampled control are investigated. The sampled control protocol is induced from continuous‐time linear consensus protocol by using periodic sampling technology and zero‐order hold circuit. With the obtained sampled control protocol, the continuous‐time multi‐agent system is equivalently transformed into a linear discrete‐time system. Necessary and sufficient conditions are given to guarantee that all the agents asymptotically travel with zero relative positions and common velocities. Furthermore, consensus problem with continuous‐time consensus protocol is re‐analyzed. A necessary and sufficient condition is also obtained which is consistent with the special case when the sampling period tends to zero. The effectiveness of these algorithms is demonstrated through simulations.