## 1 Introduction
### Contributions  
The present study addresses a crucial gap in the research on FOS, where the combined effects of time delay and input saturation have been less explored in prior literature. While these factors are highly relevant in practical applications, previous studies have largely neglected their combined influence on system behavior. This paper introduces several key contributions to overcome these challenges by proposing a novel approach to stability analysis and control design for nonlinear FOSs under both saturation control and time delay constraints. The key contributions of this paper are as follows:  
* Addressing saturation limitations: Two innovative approaches are presented to manage the challenges posed by input saturation, a prevalent constraint in practical control and robotic systems. By incorporating the Lipchitz condition into the saturation model, this paper ensures that the FOSs remain stable even when control inputs are constrained. This method offers a robust solution to the issues related to input saturation. * Addressing time-delay limitations: Recognizing the shortcomings of traditional Lyapunov-like function methods in analyzing time-delayed FOS, this paper utilizes the M-matrix and vector Lyapunov function techniques as critical tools. These methods create a rigorous framework for overcoming the stability challenges induced by time delays, ensuring system stability under these complex conditions. * Development of a stable controller: This paper makes a substantial contribution by developing an optimized stable controller that accounts for practical system constraints. The controller is designed to guarantee stability in the presence of both saturation and time-delay conditions. Furthermore, a new stability criterion for delayed nonlinear FOSs with saturation constraints is introduced, which significantly broadens the scope of existing stability analysis techniques. * Adaptation of the Lyapunov-like function: This research modifies the Lyapunov-like function by incorporating diffusive realization and the Lipchitz condition, providing a rigorous criterion for asymptotic stability. This approach effectively resolves the stability challenges of interconnected FOSs operating under saturation control, ensuring robustness in practical applications. Application of vector Lyapunov functions with M-matrix methods: The study applies vector Lyapunov functions in combination with M-matrix methods to establish strong stability conditions. This integration enhances the analytical framework for assessing system stability, particularly in cases involving both time delays and saturation constraints, thereby expanding the potential applicability of the proposed methodologies. * Application to two significant FO systems: FO Lorenz and FO Liu as illustrative examples: The study of stability and control of the FO Lorenz and FO Liu systems is essential not only for understanding their theoretical behavior but also for investigating their practical relevance in automation and robotics, where stability and precise coordination between system components are critical. By applying the proposed analysis to these FO chaotic systems, we aim to deepen the understanding of their complex dynamics and validate the effectiveness of the advanced stabilization techniques introduced in this work. These contributions collectively offer a comprehensive stability analysis and control framework for nonlinear FOSs subjected to both time delay and saturation control. The proposed methodologies provide significant insights for future research and practical applications and demonstrate the potential to advance the state-of-the-art in FO control strategies. This paper is structured into six key sections, each contributing to a rigorous analysis of fractional-order (FO) nonlinear systems under saturation control. Section 2 establishes the theoretical groundwork with essential theorems and definitions. Section 3 presents the primary contributions, including stability conditions and the impact of time delays on FO systems.