# Formation Control of Nonlinear Multi-Agent Systems with Nested Input Saturation
## 1 Introduction
### State of the Art and Contributions  
The literature on formation control of MAS demonstrates a rich amount of methodologies facilitating coordinated navigation and adaptive configurations among multiple autonomous agents. Strategies like behavior-based schemes [5; 6; 7], virtual structures [8; 9] and leader-follower approaches [10; 11; 12; 13; 14; 15] have significantly influenced the shaping of formations and the interaction control among agents. In region-based algorithms the agents move within a dynamic area, defining the desired formation shape via potential functions [16]. In ref. [17] the authors address the challenge of elliptical coverage by employing an elliptical potential function to achieve area coverage. An adaptive leader-follower formation tracking protocol for spacecraft formation was recently proposed in ref. [18], addressing limited sensing and external disturbances. This study introduces an adaptive leader-following technique for spacecraft, guaranteeing stability, connectivity maintenance, collision avoidance, and disturbance rejection within a settled time. In ref. [19] the appointed-time formation control for multiple spacecrafts under resource constraints was investigated by introducing an event-triggered protocol. Using a sliding mode manifold and output performance constraints, the authors devise a controller ensuring desired convergence and performance guarantees. Additionally, the fixed-time fault-tolerant formation control for a leader-follower heterogeneous MAS, considering actuator faults and disturbances was investigated into ref. [20]. Specifically, a finite-time performance function was introduced to ensure the convergence of formation errors within a specified time frame in combination with a distributed formation control algorithm using sliding mode control. Utilizing the digital twin framework authors in refs. [21; 22] focus on a decentralized adaptive attack-resilient control scheme designed to address unbounded actuation attacks within heterogeneous MAS, ensuring the system's resilience and stability despite adversarial disruptions. Despite these advancements, a critical challenge persists in ensuring robust performance when confronted with input constraints. While these control strategies excel in shaping formations, they often struggle to maintain stability, convergence, and performance guarantees under real-world circumstances and limitations. In particular, actuation constraints, frequently lead to degraded tracking accuracy, stability issues, and reduced convergence rates, affecting the performance and safe operation of the MAS in practical applications. Extensive research efforts have focused on addressing these challenges within the MAS control framework. Adaptive sliding-mode control methods [23; 24] and optimal control algorithms [25; 26] aimed to both maintain formations and address uncertainties present in agent dynamics. Robust adaptive controllers [27], have also been introduced to counteract external disturbances, parameter uncertainties, and input nonlinearities within the control loop. Nevertheless, complex MAS, (e.g., swarms of quadrotors), often exhibit highly nonlinear behavior influenced by both internal uncertainties and external disturbances. Consequently, achieving high-performance formation tracking in such nonlinear MAS remains a fundamental challenge for researchers. Recent advances in control methodologies have leveraged approximation techniques, driven by advancements in computational capabilities. In [28] the authors propose an adaptive fuzzy backstepping controller for the formation control of under-actuated spacecraft teams, addressing unknown nonlinearities and actuator saturation. They introduce a path-following mechanism for the leader and a follower control design based on Lyapunov theory to maintain the desired formation. Furthermore, ref. [29] proposes a neuro-adaptive leader-follower formation control scheme for autonomous surface vehicles, ensuring scalable formation sizes, while addressing uncertainties and input saturation.