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This work solves the analysis and protocol design problems of the prescribed-time formation-containment (FC) control of double-integrator multi-agent systems (MASs) on directed graphs in the presence of unknown and time-varying actuator faults. First, a novel kind of distributed prescribed-time observers (DPTOs) is designed for the leaders and the followers, respectively, which could achieve distributed zero-error estimation on directed graphs within a predefined time interval. Based on this DPTO, a decentralized protocol equipped with a time-scaling function is proposed, which possesses both prescribed-time performance and fault-tolerant merits against both additive and multiplicative faults. This protocol ensures that the MAS achieves fault-tolerant FC in a cooperatively ultimately uniformly bounded sense after a prescribed time interval, while the formation error and containment error further converge to zero at an exponentially predefined speed. Also, the upper bounds of the tracking error, formation error, and containment error have been given explicitly. The effectiveness of the proposed DPTO-based protocol is proven via an illustrative simulation example.
Gong et al. (Tue,) studied this question.