Distributed Adaptive Optimal Fault-Tolerant Formation Containment Control for High-Order Nonlinear Multi-Agent Systems via a Fully Actuated System Approach

This paper proposes a distributed adaptive optimal fault-tolerant formation containment control scheme for high-order nonlinear multi-agent systems (MASs) subject to actuator faults and unknown dynamics. Radial basis neural networks are employed to approximate unknown nonlinear functions. Subsequently, an extended observer is designed to estimate unmeasurable system states. Based on the fully actuated system (FAS) approach, an adaptive reinforcement learning algorithm is developed within an identifier–critic–actor framework to handle high-order dynamics and derive the control input, thereby achieving nearly optimal control. Moreover, an (n-1)-order sliding mode surface is constructed to regulate the tracking error, and an adaptive optimal fault-tolerant control strategy is developed. Theoretical analysis shows that even in the presence of actuator faults, MASs can maintain their outputs within the convex hull formed by the leaders, while achieving high-precision formation trajectory tracking and optimized overall performance. Finally, the feasibility and effectiveness of the proposed control scheme are verified through two simulation examples. Note to Practitioners—Formation containment control of high-order nonlinear MASs represents a critical research area in control engineering, with substantial practical implications. In MASs, actuator faults can lead to severe performance degradation or even a complete loss of control, potentially resulting in system instability or mission failure. These issues pose significant challenges to the safety and reliability of real-world engineering systems. Under resource-constrained conditions, achieving optimal control performance with minimal control cost has become an essential design objective. Moreover, physical systems governed by Lagrangian equations or Newton’s second law often exhibit complex high-order dynamic behaviors in practice. Traditional approaches commonly employ reduced-order modeling methods to transform such systems into first-order forms for analysis and controller synthesis. However, this method tends to obscure the original physical meaning of the system and increases the complexity of control design. Motivated by these challenges, this paper proposes a distributed adaptive optimal fault-tolerant formation containment control scheme for nonlinear MASs with high-order dynamics. Within the FAS framework, actuator faults are effectively compensated for, and control costs are minimized, while desired formation containment performance is ensured. Furthermore, an (n-1)-order sliding mode surface is constructed to enable precise adjustment of formation tracking accuracy. By employing the FAS approach, high-order nonlinear terms are directly addressed without conventional model reduction, thereby significantly reducing design and computational complexity. This approach facilitates the practical application of formation containment control strategies for high-order MASs in real engineering systems.