# Fault-Tolerant Optimal Consensus Control for Heterogeneous Multi-Agent Systems
## 1 Introduction  
Due to rapid advancements in science, technology, and evolving societal needs, researchers have consistently prioritized the collaborative control of multi-agent systems. Especially in recent years, rapid development has been seen. It has achieved a large number of application results in the field of engineering, such as UAV formation [1; 2], intelligent city traffic [3], smart grid [4; 5], etc. Consistency is the core issue of collaborative control, and corresponding control protocols are designed according to different controlled objects and working environments to achieve the convergence and stability of the controlled system states. Reference [6] proposed a control method based on the bat algorithm, using an improved bat algorithm to find the optimal control gains. It aims to enhance the robot's trajectory tracking accuracy and compares the trajectory tracking accuracy and control energy consumption of two robots. Reference [7] addresses the control problem of a single-link mechanical arm and introduces a practical control method. It proposes a finite-time disturbance observer to estimate external disturbances and ensures that tracking errors converge to a specified bounded set within a finite time. It demonstrates that the closed-loop system is globally finite-time stable. Reference [8] investigates the finite-time consensus problem for a class of general linear leader-follower multi-agent systems using event-triggered mechanisms. A distributed event-triggering strategy based on observers is proposed. Under given conditions, it ensures that the tracking error converges to zero within a finite time. Multi-agent systems are primarily categorized into homogeneous and heterogeneous types. The consistency inquiry of homogeneous multi-agent systems has achieved fruitful results [9; 10; 11; 12]. During actual operation or use, most systems exhibit heterogeneous characteristics. Heterogeneous multi-agent systems are structurally constrained and mainlyunable to achieve complete state consistency. Based on this, Scholars both domestically and internationally are increasingly concentrating on the study of consensus in heterogeneous multi-agent systems [13; 14; 15; 16; 17; 18; 19]. In conjunction with the increasing working time of intelligent agents and the complex and changeable actual environment, individual intelligent agents will inevitably encounter failures, and multi-agent systems cannot solve this problem through their attributes. Therefore, ensuring multi-agent systems achieve consistency during failures has become a subject of great concern. Reference [20] proposed an approach utilizing inner and outer loops to accomplish the control objective of multiple uncrewed aerial vehicles following a leader aircraft in the presence of actuator failures. The outer loop is responsible for designing position state control laws to uphold the formation flight, while the inner loop addresses actuator effects through adaptive control laws for UAV attitudes. Reference [21] implemented a distributed sliding mode observer (SMO) for each trailing UAV, estimating necessary reference values through information exchange among neighboring UAVs. Adaptive neural fault-tolerant control (FTC) units were subsequently devised for the UAV to converge toward the benchmark value produced by the distributed SMO in case of actuator failures. Reference [22] examined the coordination of drones in formation under conditions involving leader and actuator failures. The study applied boundary layer theory and devised a control protocol featuring adaptive mechanisms for compensating actuator deficiencies. Reference [23] devised a formation control law with adaptive mechanisms for multi-octocopter drone models, taking into account uncertainties and actuator failures. Reference [24] proposed a new adaptive backstepping method related to command filters to address model uncertainties and input saturation problems for multiple UAVs under input saturation, actuator failures, and external disturbances, designing a robust controller.