Scaled containment control for first/second-order multi-agent systems in a noisy environment

This paper addresses the scaled containment control problem (SCCP) for first/second-order stochastic multi-agent systems (SMASs) in a noisy environment. The agents receive neighbor's state information subjected to nonzero scaling factors, which causes the followers to finally converge to a scaled deterministic constant formed by the leaders. A stochastic approximation protocol with time-varying gains is designed to attenuate the noise effects. Using a state decomposition method, some sufficient and necessary conditions for the SCCP of first/second-order SMASs are given under the topological requirement of containing a directed spanning forest. For first-order systems, the followers converge to the scaled deterministic constant formed by the leaders. For second-order systems, two interaction modes are analyzed: under constant velocities mode, followers' positions converge to the unbounded scaled deterministic constant formed by the leaders' positions and followers' velocities converge to the scaled deterministic constant formed by the leaders' velocities; under zero velocity mode, the convergence behavior of the second-order systems is analogous to that of first-order systems, where followers' positions converge to the scaled deterministic constant formed by the leaders' positions and followers' velocities converge to zero. The numerical examples demonstrate the validity of the theoretical findings.