Deterministic and stochastic containment control for fractional-order multi-agent systems with mixed time delays and Markovian switching topologies

For Multi-Agent Systems (MAS) to operate safely within a controlled area specified by dynamic leaders, they must be controlled in a manner that allows them to collaborate. Although integer-order systems are standard, Fractional-Order Multi-Agent Systems (FOMAS) have been found to be better models of complex memory-dependent dynamics found in other applications—such as anomalous diffusion and signal latency—in underwater robotics, smart grid frequency regulation, and others. Nevertheless, it is often impractical to simultaneously tackle the mixed time delays, stochastic noise, and multi-modal Markovian switching topologies using the existing containment protocols. This paper introduces a unified fractional-order containment control framework to fill this gap. From a generalized Lyapunov–Razumikhin method, we obtain sufficient algebraic criteria that are robust to containment over the whole range of fractional order ς∈(0,1). Our results show that the proposed protocol is able to significantly improve the transient response and noise rejection capabilities over those of conventional integer-order protocols. This paper establishes a rigorous control basis for the design of robust formation architectures in delayed, noisy and switching communication environments.