Robust FOSMC of a Quadrotor in the Presence of Parameter Uncertainty

Abstract

This study addresses the problem of attitude and altitude tracking for a quadrotor system in the presence of parameter uncertainties. The goal is to develop a robust control strategy that can handle the nonlinear, strongly coupled dynamics of the quadrotor. To achieve this, we propose a fractional-order sliding mode control (FOSMC) scheme, which is specifically designed to improve system performance under uncertain parameters. The FOSMC approach is combined with additional adaptive laws to further enhance the robustness of the control system. We derive the necessary control laws and apply them to the quadrotor’s state-space representation, ensuring that the system remains stable and performs accurately in the presence of uncertainties. Numerical simulations are conducted to evaluate the effectiveness of the proposed control strategy. The results show that the FOSMC-based controller successfully achieves precise tracking of both attitude and altitude, demonstrating significant robustness against parameter variations and disturbances. In conclusion, the proposed FOSMC scheme provides a reliable solution for controlling quadrotor systems in uncertain environments, offering the potential for real-world applications in autonomous UAV operations.

1. Introduction

The quadrotor has been used for several applications such as mapping, transportation, surveying, aerial photography, search, mobile sensor networks, imaging, transportation, rescue, etc. [1]. Its exceptional performance stems from its vertical takeoff and landing capabilities, cost-effective manufacturing, and high maneuverability. The machine can fulfill several dangerous and challenging missions in the military and civilian fields [2]. It is favored in many applications due to its mechanical simplicity compared to helicopters, which feature complex structures and require extensive maintenance [3]. Since the 1920s, quadrotor designs have evolved in shape and size to achieve mechanical simplicity. Since then, significant advancements in cost, weight, size, and other factors have taken place in the design of quadrotors, and today, the size of these machines has been reduced to the size of a palm [4].

As the quadrotor is an underactuated system, the most challenging part is the control design. The quadrotor’s operation and performance depend on the design of an optimal control strategy. Several attempts have been made to stabilize the quadrotor and track some reference trajectories. These studies have discussed control design methodologies to control quadrotors such as input–output linearization and cascade control [5], parameter-varying decoupling-based control [6], adaptive backstepping fast terminal sliding mode control [7], time-varying backstepping [8], robust generalized dynamic inversion [9], adaptive disturbance compensation, and hybrid finite-time control [10,11]. Besides the above, sliding mode control (SMC) is a robust approach that uses two essential phases: reaching and sliding. This approach has advantages like robustness against parameter uncertainties and external disturbances [12].

In recent advancements, various studies have introduced improvements to conventional sliding mode control (SMC) for quadrotors. For example, a dynamic fractional-order sliding mode control (FOSMC) method for UAVs has been proposed to reduce external disruptions and enhance reliability and performance monitoring [13]. Furthermore, integrating FOSMC with active disturbance rejection has significantly improved six-degree-of-freedom trim in UAVs, enhancing control performance and reliability in the presence of external disturbances [14]. Additionally, a flexible sliding control approach has been introduced for quadrotor balance and route monitoring, incorporating height and view balance, route monitoring, and model fitting. This method explains parameter uncertainties, sensor noise, and disruptions, showing notable enhancements through analytical simulations [15].

A quadruplet versatile backstepping control approach has also been proposed, particularly established for monitoring angular velocity and positions in underactuated UAVs. This approach assures rapid convergence and reduces control chattering, approaching both uniqueness and accuracy in UAV control [16]. A recent study explores the real-time unspecified parameter inference and flexible monitoring control for six-degree-of-freedom (6-DOF) underactuated systems. This potent flexible control strategy can be applied to practical navigational device design for feedback linearized systems with unreliability, further improving the real-time effectiveness of UAV control [17]. Additionally, in a previous study, a nonlinear quadrotor UAV dynamic model using the Newton–Euler approach has been established, with a control framework introduced for 3D route tracking. The design of the controller is divided into two parts: an inside loop for view balance and an outside loop for route tracking. The study compares several control methods such as LQR, PID, MPC, FL, and SMC, assisting in the estimation of the most adequate approaches for the control of quadrotors [18]. Furthermore, in a recent study, a controller design utilized a Lyapunov function to ensure the validity of the controller, improving control stability for quadrotors [19].

Building on these advancements, this study is motivated by the potential superiority of FOSMC over traditional methods, offering an additional degree of freedom that can enhance the performance of the SMC methodology. To put the new fractional-order sliding mode control (FOSMC) method in perspective in relation to the current advanced control methodologies, it must be contrasted with other commonly employed methods like model predictive control (MPC), nonlinear model predictive control (NMPC), and adaptive control. MPC is an optimization approach that predicts the future behavior of the system and adjusts the control inputs based on this prediction, thus being extremely effective for multivariable constrained systems. Nevertheless, it consumes heavy computational resources, which restrict its real-time utilization in rapid-response systems such as quadrotors [20]. NMPC expands MPC to nonlinear systems through successive linearization or linear parameter-varying models, enhancing precision in treating nonlinear dynamics at the cost of even higher computational complexity. In comparison, adaptive control adjusts controller parameters online to address system uncertainties so that stable performance can be achieved under varying conditions [21]. Methods like model reference adaptive control (MRAC) and adaptive gain sliding mode observers are widely applied to improve adaptability [22]. FOSMC, however, combines fractional calculus with sliding mode control, providing better robustness against disturbances and less chattering effects than classical SMC. In contrast to MPC and NMPC, FOSMC is not based on real-time optimization; hence, it is computationally light yet maintains strong disturbance rejection. In contrast to adaptive control, FOSMC naturally addresses uncertainties using its fractional-order dynamics without needing real-time adaptation of parameters [23]. With these advantages, FOSMC presents a promising alternative for quadrotor control, balancing robustness, computational efficiency, and adaptability to system uncertainties.

Theoretical Contribution: The present investigation introduces a new fractional-order sliding mode control (FOSMC) formulation that enhances stability and robustness over traditional strategies. One important innovation is the explicit incorporation of the uncertainty limits within system dynamics as well as the input gains that enhance adaptability and robust performance under changing operating conditions. Secondly, the paper includes fixed-time FOSMC to counter external disturbances to ensure improved system performance regardless of parameter uncertainty.

Application Contribution: Although FOSMC has been studied in various contexts in existing research, this research is particularly concerned with its application to quadrotors under parameter uncertainty. In contrast to existing works, this research combines altitude and attitude stabilization, trajectory tracking, and parameter estimation into a single control strategy. By comparative analysis, we show that the proposed FOSMC approach performs better compared to traditional controllers, e.g., feedback linearization (FBL), on steady-state error reduction, uncertainty robustness, and external disturbance mitigation. Empirical verification further supports the real-world applicability of FOSMC in UAV operations. In addition, the research focuses on extensive simulations and experimental validations, with empirical results demonstrating the effectiveness of the control scheme in real-world operations. This dual approach fills the gap between theoretical breakthroughs and real-world implementations, presenting tangible solutions to improve quadrotor performance and robustness.

2. Literature Review

Several studies have discussed the control design methodology, like the study conducted by Wang et al. [24] which aimed to optimize PID control of quadrotor attitude through a systematic process that included linearization of the quadrotor dynamic model and the use of a single-input single-output (SISO) method for controller tuning. Taylor’s method would be appropriate for this case as it simplifies complex dynamics around the hovering point, enabling more precise control. The direct synthesis method optimizes the efficiency of the PID controller to achieve improved stability and disturbance rejection. Another study by Hadid et al. [25] introduced the quadrotor’s feedback linearization (FBL) controller, noting that while it effectively regulates trajectory tracking, it is susceptible to noise and modeling uncertainties due to high-order derivative terms, which compromise stability and performance.

Along the same line, another study by Ali Jebelli et al. [26] introduced a control algorithm to effectively manage the state, position, and altitude of a nonlinear quadcopter model. With feedback linearization, the authors designed a two-loop control structure: the inner loop rapidly adjusts the quadcopter height and angles and the outer loop adjusts roll and pitch angles to control position. The system remains stable and convergent even in the event of rotor failure, maintaining performance under external disturbances. Moreover, Seshasayanan [27] in another paper designed an in-depth backstepping controller that addresses parameter uncertainties in a tilt-augmented quadrotor. It uses an adaptive sliding mode observer for finite-time estimation of uncertainties such that desired trajectories are achieved with exponential convergence despite external disturbances. Similarly, a study conducted by Zhu et al. [28] focused on an adaptive sliding mode controller for underactuated vessels, demonstrating improved trajectory tracking accuracy and stability under uncertainties. However, it did not specifically address the application of SMC on quadrotors or their insensitivity to parametric uncertainties.

Similarly, another study by Zheng et al. [29] suggested a second-order sliding mode controller (SOSMC) for a small quadrotor. The system was divided into two subsystems that formed the actuated and unactuated parts of the dynamics. The system was designed for underactuated details with the linear sliding manifold. A linear sliding manifold was implemented, and virtual inputs for the x and y positions were utilized to find the desired roll and pitch angles since the system was coupled and underactuated. Using Lyapunov theory, the stability was proven and tested through a simulation, which showed the error converged to zero. This work assumed a fixed payload and mass of the vehicle, as a change in the mass of the payload of the quadrotor would affect the following trajectory and system stability.

Gros et al. [30] proposed an NMPC approach utilizing multiple-shooting methods for aircraft control, effectively managing extreme maneuvers and actuator failures. Additionally, Fnadi and Plumet [31] developed an MPC-based dynamic path-tracking system for a four-wheel steering mobile robot, enhancing trajectory accuracy and stability. Integrating these references provides a comprehensive context for the proposed fractional-order sliding mode control (FOSMC) approach, highlighting its position within contemporary control strategies. In addition, another study by Vahdanipour and Khodabandeh [32] recommended using adaptive fractional-order sliding mode control in quadrotors when handling different loads within the vehicle. In this model, the adaptive law was designed, and its stability was proven through Lyapunov theory, ensuring asymptotic convergence. In addition to this, a wind disturbance was also injected into the system, and the result showed that the fractional-order sliding mode gave better performance than SMC. It performed better because of its impact on improving the control objectives.

It reduces chattering, and experimental results have validated its control effect and energy efficiency. In line with this, another study by Roy and Roy [33] compared sliding mode control and fractional sliding control performance. The analysis in their research found that FOSMC performed better than SMC in terms of speed of response tracking accuracy. Moreover, a similar study carried out by Tang et al. [34] proposed FOSMC for an ABS where the main aim was to regulate the wheel slip to a reference value to obtain maximum frictional force. The controller was compared with integer-order SMC. They were tested by simulation, and FOSMC showed faster tracking to the desired slip than SMC. In addition to this, the slope of the velocity curve and the braking time for FOSMC compared to SMC were smaller and shorter.

Another study by Zhu [35] proposed a novel framework called complete model-free sliding mode control (CMFSMC) for controlling continuous-time non-affine nonlinear dynamic systems with unknown models. The key innovation lies in the introduction of two equalities to assign the derivative of the sliding functions, which bridges the gap between model-based SMC and model-free SMC designs. The study encompasses the design of a double SMC (DSMC), a state observer, and a desired reference state vector, all without relying on plant nominal models. It is revealed that the CMFSMC framework requires knowledge of the plant’s dynamic order and the boundedness of the plant and its disturbances. To configure the entire control system, U-model-based control (U-control) is incorporated, utilizing model-free double SMC for nonlinearity handling, a model-free state observer for state estimation, and an invariant controller for defining desired system performance.

Zhu et al. [36] presented a novel control design procedure called U-control-based composite nonlinear feedback (U-CNF) for generalizing and simplifying the design of CNF control systems. The U-CNF control framework incorporates a double feedback loop structure to achieve the desired control characteristics. The design involves two separate controllers: The first controller stabilizes the system and cancels nonlinearities and dynamics, effectively converting the plant into an identity matrix in the inner closed loop. The second controller improves the transient response of the system by specifying a second-order linear system with a monotonic nonlinear function to smoothly adjust the damping ratio. This approach enables the attainment of conventional CNF characteristics concisely.

In recent years, there has been a notable surge in the utilization of multi-copter systems across various fields. These systems demand high levels of stability and agility to meet their operational requirements effectively. Meeting this demand necessitates the development and incorporation of robust control strategies; therefore, Timis et al. [37] introduced an adaptive control design tailored to a physical quadrotor prototype model. This adaptive framework ensures robustness, control adaptability, and stability throughout the operational process. The results showed that fractional-order controllers are favored for their acknowledged capability to enhance overall closed-loop system robustness.

Another study by Benaddy et al. [38] investigated the implementation of a fixed-time tracking control system incorporating fractional-order dynamics for a quadrotor, which is susceptible to external disturbances. The study devised a fractional-order sliding manifold to facilitate the fixed-time convergence of the system’s state variables after outlining the problem formulation concerning a quadrotor system comprising six subsystems akin to a second-order system. To address the upper limit of the disturbances, a switching fixed-time controller is integrated into the equivalent control law that ensures fixed-time stability. The efficacy of the proposed controller, termed fixed-time fractional-order sliding mode control (FTFOSMC), was substantiated through rigorous analysis and stability assessments employing the Lyapunov approach.

Labbadi et al. [39] introduced two novel FOSMCs tailored to address trajectory challenges encountered by unmanned aerial vehicles (UAVs). In this study, the first controller design encompasses a proportional and derivative FOSMC, incorporating a novel reaching control law that mitigates disturbances and uncertainties impacting quadrotor dynamics solely through tracking errors. The second proposed fractional-order controller adopts a nonlinear approach known as fractional-order fast terminal SMC (FOFTSMC) for achieving finite-time control of the quadrotor system, mitigating the chattering phenomenon associated with traditional SMC, and enhancing tracking performance in the presence of parametric uncertainties and disturbances. The gains were fine-tuned using the Optimization Toolbox within Simulink to optimize controller performance. The efficacy of the devised fractional-order controllers was validated via extensive examination across various case scenarios involving disturbances and uncertainties.

Several methodologies have been proposed to improve performance under various operational challenges as the control design for quadrotors has been a part of significant research. Prior studies, such as that of Wang et al. [24], have used enhanced PID controllers for attitude control, attaining good results in balance and disturbance rejection. A nonlinear method, such as feedback linearization, was investigated by Hadid et al. [25] and Jabelli et al. [26], handling challenges like rotor failure and noise susceptibility. These approaches are effective and emphasize the limitations of conventional control approaches in handling disruptions and parameter unreliability. To address this, studies by Zhu [28] introduced a robust control technique that includes SMC to enhance system stability under undetermined conditions and demonstrates the efficacy of SMC in quadrotor applications. Additionally, flexible and FOSMC approaches, as explored by Vahdanipour and Khodabandeh [32] and Roy and Roy [33], have significantly improved performance, particularly by reducing chattering and enhancing robustness. These developments with novel control structures, such as U-control-based composite nonlinear feedback (U-CNF) [34] and fractional-order sliding manifolds [35], tackle the real-world challenges of outer disruptions, modeling inaccuracies, and noise.

The proposed framework offers enhanced robustness, control adaptability, and stability throughout the operational process. The emphasis on fractional-order controllers highlights their superior capability to improve overall closed-loop system robustness compared to traditional control methods. Through rigorous comparative analysis and stability assessments using the Lyapunov approach, the paper empirically demonstrates the superiority of the proposed methodologies in terms of stability, robustness, and control convergence, thereby highlighting its novelty and contribution to the field of quadrotor control.

3. Control Design and Mathematical Model

Traditional controllers like PID, feedback linearization (FBL), backstepping, and classical sliding mode control (SMC) each have significant limitations in handling these challenges. PID controllers require precise tuning and are sensitive to external disturbances, while FBL relies on exact system models, making it unsuitable for uncertain environments [13,40]. Backstepping is computationally expensive and struggles with disturbances, and classical SMC suffers from chattering, reducing efficiency and accelerating actuator wear [41]. Fractional-order sliding mode control (FOSMC) is selected for its superior robustness, ability to reject disturbances without requiring precise system models, and reduced chattering compared to classical SMC [42]. FOSMC offers enhanced stability, better tracking performance, and improved adaptation to nonlinear dynamics, making it the most suitable control strategy for quadrotors. Its fractional-order properties provide improved robustness and adaptability, ensuring stable and efficient operation under uncertain conditions.

To ensure stability in the quadrotor system, a nonlinear controller is required due to its nonlinear dynamics. Traditional linearization methods like the Taylor series approach are not effective for controlling such a system, as they may lead to divergence at different operating points. There are various nonlinear controllers available for the quadrotor system. One option is to design a feedback linearization controller, which linearizes the entire nonlinear system and applies PI control to the linearized system. However, this type of controller is susceptible to noise and model uncertainty due to the inclusion of high-order derivative terms. To address this issue, a robust controller is needed. In this paper, a fractional-order sliding mode controller will be designed and implemented to overcome these challenges.

The quadrotor system is characterized by six outputs, namely $\left(x, y, z, \phi , \theta , \mathrm{a}\mathrm{n}\mathrm{d} \psi \right)$, representing its six degrees of freedom (DOFs). However, the system only has four inputs available as manipulated variables, namely $\left(u_1, u_{\phi }, u_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} u_{\psi }\right)$. As a result, the system is classified as underactuated since the number of inputs is fewer than the number of outputs.

By examining the system closely, it becomes evident that the rotational dynamics are independent of the translational variables. This allows us to design three separate controllers for the attitude (angular rotations), treating them as a subsystem within an inner loop. The outer loop, on the other hand, will focus on controlling the position of the system. By providing controlled angles along with altitude and position inputs, it is possible to effectively achieve position control for the system. See Figure 1.

Feedback linearization is one of the most popular controllers used for nonlinear systems. The basic idea of FBL is to transform the nonlinear system into an equivalent linear system and then a linear controller can be applied to the transformed system [43]. In the quadrotor system, there are six outputs $\left(x,y,z,\varphi ,\theta , \mathrm{a}\mathrm{n}\mathrm{d} \psi \right)$ and only four inputs $\left(u_1,u_{\varphi },u_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} u_{\psi }\right)$; therefore, the system is underactuated. To solve this issue, the system outputs are divided into two parts. The inner part controls the altitude and the attitude angles $\left(\varphi ,\theta ,\psi , \mathrm{a}\mathrm{n}\mathrm{d} z\right)$, while the outer part provides the desired attitude angles to the inner part to control the position of the quadrotor $\left(x \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{y}\right)$ [43].

Inner Loop Control

The inner loop state variables are selected as $\left(\varphi ,\theta ,\psi ,p,q,r,z, \mathrm{a}\mathrm{n}\mathrm{d} z\right)$ and denoted by $\left({ x}_{inner}\right)$. The dynamics of the inner loop state variables can be written as

$${\stackrel{·}{x}}_{inner}=f\left(x_{inner}\right)+h\left(x_{inner}\right)U$$

where U is defined next.

Since the inner loop outputs are selected as $\left(\varphi ,\theta ,\psi , \mathrm{a}\mathrm{n}\mathrm{d} z\right)$ and denoted by $\left(Y_{inner}\right)$, the dynamics of these outputs are expressed as

$$\underset{{\stackrel{·}{Y}}_{inner}}{\underbrace{\left[\begin{array}{c}\stackrel{·}{\varphi }\\ \stackrel{·}{\theta }\\ \stackrel{·}{\psi }\\ \stackrel{·}{z}\end{array}\right]}}=\underset{\widehat{f}\left(x_{inner}\right)}{\underbrace{\left[\begin{array}{c}p+q sin\phi tan\theta +r cos\phi tan\theta \\ q cos\phi -r sin\phi \\ {\displaystyle \frac{1}{cos\theta }}\left[q sin\phi +r cos\phi \right]\\ V_z\end{array}\right]}}+\left[\underset{h\left(Y_{inner}\right)}{\underbrace{Zero}}\ast U\right]$$

Since there is no input appearing in the above output dynamics, the second derivative is then computed as follows:

$${\ddot{Y}}_{inner}={\displaystyle \frac{d}{dt}}\widehat{f}\left(x_{inner}\right)=\underset{F_{inner}}{\underbrace{ {\displaystyle \frac{\partial \widehat{f}\left(x_{inner}\right)}{\partial x_{inner}}}}} {\stackrel{·}{x}}_{inner}={\displaystyle \frac{\partial \widehat{f}\left(x_{inner}\right)}{\partial x_{inner}}}\left[f\left(x_{inner}\right)+h\left(x_{inner}\right)\ast U\right]$$

Denoting ${\displaystyle \frac{\partial \widehat{f}\left(x_{inner}\right)}{\partial x_{inner}}}$ as $F_{inner}$, the above equation can be rewritten as

$${\ddot{Y}}_{inner}=F_{inner} f\left(x_{inner}\right)+F_{inner} h\left(x_{inner}\right) U$$

The control inputs can be calculated using the error dynamics of the outputs as follows:

$$U=\left[\begin{array}{c}{U}_{\varphi }\\ U_{\theta }\\ U_{\psi }\\ U_1\end{array}\right]=-\left[{\left(F_{inner} h\left(x_{inner}\right)\right)}^{-1}\ast F_{inner} f\left(x_{inner}\right)\right]+{\left(F_{inner} h\left(x_{inner}\right)\right)}^{-1}\left[{\ddot{Y}}_{inne{r}_d}-{\lambda }_P{e}_y-{\lambda }_d{\stackrel{·}{e}}_y\right]$$

where

$$e_y=\left[\begin{array}{c}{e}_{\varphi }\\ e_{\theta }\\ e_{\psi }\\ e_z\end{array}\right]=\left[\begin{array}{c}\varphi -{\varphi }_d\\ \theta -{\theta }_d\\ \psi -{\psi }_d\\ z-z_d\end{array}\right]$$

$$F_{inner}=\left[\begin{array}{cccccccc}q cos\varphi tan\theta -r sin\varphi tan\theta & {sec}^2\theta \left[q sin\varphi +r cos\varphi \right]& 0& 1& sin\varphi tan\theta & cos\varphi tan\theta & 0& 0\\ -q sin\varphi -r cos\varphi & 0& 0& 0& cos\varphi & -sin\varphi & 0& 0\\ {\displaystyle \frac{1}{cos\theta }}\left[q cos\varphi -r sin\varphi \right]& tan\theta sec\theta \left[q sin\phi +r cos\phi \right]& 0& 0& {\displaystyle \frac{sin\varphi }{cos\theta }}& {\displaystyle \frac{cos\varphi }{cos\theta }}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$

$$h\left(x_{inner}\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\displaystyle \frac{1}{I_x}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{I_y}}& 0& 0\\ 0& 0& {\displaystyle \frac{1}{I_z}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{cos\varphi cos\theta }{m}}\end{array}\right]$$

The desired values $\left({\varphi }_d \mathrm{a}\mathrm{n}\mathrm{d} {\theta }_d\right)$ are provided to the inner loop by the outer loop, such that the quadrotor position is controlled and stabilized.

Outer Loop Control

The outer loop is responsible for providing the inner loop with the desired angles $\left({\varphi }_d \mathrm{a}\mathrm{n}\mathrm{d} {\theta }_d\right)$ to control the position $\left(x \mathrm{a}\mathrm{n}\mathrm{d} y\right)$ of the quadrotor.

The dynamics of the quadrotor position are expressed as follows:

$$\ddot{x}={\displaystyle \frac{1}{m}}\left[\left(cos\phi sin\theta cos\psi +sin\phi sin\psi \right) u_1-K_t\stackrel{·}{x}\right]$$

$$\ddot{y}={\displaystyle \frac{1}{m}}\left[\left(cos\phi sin\theta sin\psi -sin\phi cos\psi \right) u_1-K_t\stackrel{·}{y}\right]$$

It is assumed that $(\varphi \mathrm{a}\mathrm{n}\mathrm{d} \theta )$ are small angles. Using small angle approximation,

$$sin\varphi =\varphi , sin\theta =\theta , cos\varphi =1, and cos\theta =1$$

As such, ($\ddot{x}={\displaystyle \frac{1}{m}}\left[\left(cos\phi sin\theta cos\psi +sin\phi sin\psi \right)u_1-K_t\stackrel{·}{x}\right]$) and ($\ddot{y}={\displaystyle \frac{1}{m}}\left[\left(cos\phi sin\theta sin\psi -sin\phi cos\psi \right)u_1-K_t\stackrel{·}{y}\right]$) can be rewritten using the above approximation:

$$\ddot{x}={\displaystyle \frac{1}{m}}\left[\left(\theta cos\psi +\phi sin\psi \right) u_1-K_t\stackrel{·}{x}\right]$$

$$\ddot{y}={\displaystyle \frac{1}{m}}\left[\left(\theta sin\psi -\phi cos\psi \right) u_1-K_t\stackrel{·}{y}\right]$$

($\ddot{x}={\displaystyle \frac{1}{m}}\left[\left(\theta cos\psi +\phi sin\psi \right)u_1-K_t\stackrel{·}{x}\right]$) can be multiplied by $cos\psi$, and ($\ddot{y}={\displaystyle \frac{1}{m}}\left[\left(\theta sin\psi -\phi cos\psi \right)u_1-K_t\stackrel{·}{y}\right]$) by $sin\psi$ to get the following:

$${\displaystyle \frac{m}{U_1}}\left(\ddot{x}+{\displaystyle \frac{K_t}{m}}\stackrel{·}{x}\right)Cos\psi =\theta Co{s}^2\psi +\varphi Sin\psi Cos\psi$$

$${\displaystyle \frac{m}{U_1}}\left(\ddot{y}+{\displaystyle \frac{K_t}{m}}\stackrel{·}{y}\right)Sin\psi =\theta {Sin}^2\psi -\varphi Cos\psi Sin\psi$$

By solving the two equations and replacing the variables with the desired values, we can get

$${\varphi }_d={\displaystyle \frac{m}{u_1}}\left[{\ddot{x}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{x}-K_p{e}_x-K_d{\stackrel{·}{e}}_x\right]Sin\psi -{\displaystyle \frac{m}{u_1}}\left[{\ddot{y}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{y}-K_p{e}_y-K_d{\stackrel{·}{e}}_y\right]Cos\psi$$

$${\theta }_d={\displaystyle \frac{m}{u_1}}\left[{\ddot{x}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{x}-K_p{e}_x-K_d{\stackrel{·}{e}}_x\right]Cos\psi +{\displaystyle \frac{m}{u_1}}\left[{\ddot{y}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{y}-K_p{e}_y-K_d{\stackrel{·}{e}}_y\right]Sin\psi$$

3.1. Fractional Calculus

Fractional-order control, a branch of control theory, incorporates the principles of fractional calculus into control systems. The application of fractional control methods can improve control strategies and techniques [44]. In this context, the fractional differential operator is represented by the symbol $\left({\mathfrak{D}}^{\alpha }\right)$, where $\left(\alpha \in \mathfrak{R}\right)$ denotes a non-integer order for differentiation or integration. The definition of the differ-integral operator ${\mathfrak{D}}^{\alpha }$ is provided in the following equation [44,45]:

$${\mathfrak{D}}_t^{\alpha }=\left\{\begin{array}{c}{\displaystyle \frac{d^{\alpha }}{{dt}^{\alpha }}} for \alpha \succ 0 \\ 1 for \alpha =0 \\ {\int }_0^t({d\tau )}^{\alpha } for \alpha \prec 0 \end{array}\right.$$

(1)

Multiple definitions exist for the fractional operator, each with its own distinct characteristics. Several commonly used definitions are available in the literature [45,46,47].

▪

The Reimann–Liouville (RL) definition:

$${\mathfrak{D}}_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(n-\alpha \right)}}{\left({\displaystyle \frac{d}{dt}}\right)}^n{\int }_0^t{\displaystyle \frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{\left(1-n+\alpha \right)}}}d\tau$$

(2)

▪

The Caputo definition:

$${\mathfrak{D}}_t^{\alpha } f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(n-\alpha \right)}}{\int }_0^t{\displaystyle \frac{f^n\left(\tau \right)}{{\left(t-\tau \right)}^{\left(1-n+\alpha \right)}}}d\tau$$

(3)

In the above definitions, $\left(n-1 \prec \alpha \prec n\right)$, and $\Gamma \left(.\right)$ is the Euler gamma function:

$$\Gamma \left(x\right)={\int }_0^{\infty }{e}^{-t} t^{\left(x-1\right)}dt, x>0$$

(4)

• Properties of fractional-order derivative $\left(\mathrm{f}\mathrm{o}\mathrm{r} n=1\right)$ [33,48,49].

3.1.1. Commutative Property

The fractional-order derivative $\left({\mathfrak{D}}^{\alpha }\right)$ of the fractional-order integral $\left({\mathfrak{D}}^{-\alpha }\right)$ for any function is equal to the function itself.

$${\mathfrak{D}}^{\alpha }\left({\mathfrak{D}}^{-\alpha } f\left(t\right)\right)=f\left(t\right)$$

(5)

Conversely, the fractional-order integral $\left({\mathfrak{D}}^{-\alpha }\right)$ of the fractional-order derivative $\left({\mathfrak{D}}^{\alpha }\right)$ applied to any function is given by

$${\mathfrak{D}}^{-\alpha }\left({\mathfrak{D}}^{\alpha } f\left(t\right)\right)=f\left(t\right)-{\sum }_{k=0}^{n-1}{\displaystyle \frac{f^k\left(a\right)}{k!}}{\left(t-a\right)}^k$$

(6)

3.1.2. Linearity Property

$${\mathfrak{D}}^{\alpha }\left(f_1+f_2\right)={\mathfrak{D}}^{\alpha } f_1+{\mathfrak{D}}^{\alpha } f_2$$

(7)

$${\mathfrak{D}}^{\alpha }\left(c f_1\right)=c {\mathfrak{D}}^{\alpha }\left(f_1\right)$$

(8)

3.1.3. Zero-Rule Property

$${\mathfrak{D}}^0 f\left(t\right)=f\left(t\right)$$

(9)

These properties hold significance and prove useful in the design of control laws.

3.2. Fractional-Order Sliding Mode Control (FOSMC)

Fractional order can be added to SMC [40]. SMC is a variable structure control system that enables discontinuous feedback by switching once the system has reached the sliding manifold and then enforcing the system to remain on the sliding surface. The system’s function is the sliding variable denoted by (S) in this study design. Whenever the controlled states enter the sliding surface, it will show the controlled system’s behavior and decrease the dynamics compared to the original system. That is why SMC is considered a robust control since the reduced order has the advantage of becoming insensitive to parameter uncertainties [50]. In this part, a fractional-order sliding mode controller is designed to control the altitude (z) of the quadrotor. The switching function $S_z$ is defined as follows, and the sliding manifold is given by setting $S_z=0$.

$$S_z={\mathfrak{D}}^{\beta } {\stackrel{·}{e}}_z+{\lambda }_z{e}_z$$

(10)

In the sliding mode control literature, Equation (10) represents a switching function, while the sliding manifold is formally defined as $S_z=0$. To ensure finite-time convergence to this manifold, different formulations for $S_z$ can be used.

The tracking errors for altitude and attitude control are defined as follows:

Altitude tracking error: $ez=z_d-z.$

Roll angle tracking error: $e_{\varphi }={\varphi }_d-\varphi$.

Pitch angle tracking error: $e_{\theta }={\theta }_d-\theta$.

Yaw angle tracking error: $e_{\psi }={\psi }_d-\psi$, where $z_d, {\varphi }_{d, } {\theta }_d, \mathrm{a}\mathrm{n}\mathrm{d} {\psi }_d$ are the desired reference values, and $z, \varphi , \theta and$Ψ are the actual quadrotor states.

Once the system reaches the manifold $S_z=0$, the system dynamics are governed by the sliding mode, ensuring that the tracking errors asymptotically converge to zero. The reduced-order dynamics are defined by the equivalent control $S_z=-k{S}_z$, where $k>0$ ensures system stability [51]. By Lyapunov-based stability criteria, the tracking errors $z_d,{\varphi }_{d,}{\theta }_d,\mathrm{a}\mathrm{n}\mathrm{d} {\psi }_d$ will converge to zero as $t\to \infty t$.

Taking the derivative of the sliding surface would result in the following equation:

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } {\ddot{e}}_z+{\lambda }_z {\stackrel{·}{e}}_z=-\left({\eta }_z{S}_z+k_z sgn\left(S_z\right)\right)$$

(11)

where $\left[-\left({\eta }_z{S}_z+k_{z}sgn\left(S_z\right)\right)\right]$ represents the discontinuous part, which satisfies the reachability condition and ${\eta }_z>0,k_z>0$:

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\ddot{Z}\right)+{\lambda }_z {\stackrel{·}{e}}_z=-\left({\eta }_z{S}_z+k_z sgn\left(S_z\right)\right)$$

(12)

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left({\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta -{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}-g\right)\right)+{\lambda }_z {\mathfrak{D}}^1 e_z=-\left({\eta }_z{S}_z+k_z sgn\left(S_z\right)\right)$$

(13)

Using the linear property of the fractional-order operator,

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } {\ddot{Z}}_d-{\mathfrak{D}}^{\beta } \left[{\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta -{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}-g\right]+{\lambda }_z {\mathfrak{D}}^1 e_z=-\left({\eta }_z{S}_z+k_z sgn\left(S_z\right)\right)$$

(14)

Now, we can solve for the controller $\left(U_1\right)$, which is equal to the sum of the equivalent control part and the discontinuous part.

$${\mathfrak{D}}^{\beta } {\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta ={\mathfrak{D}}^{\beta } {\ddot{Z}}_d-{\mathfrak{D}}^{\beta }\left[-{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}-g\right]+{\lambda }_z {\mathfrak{D}}^1 e_z+{\eta }_z{S}_z+k_z sgn\left(S_z\right)$$

(15)

Using the commutative property of the fractional-order operator, we multiply by ${\mathfrak{D}}^{-\beta }$ to cancel the existing ${\mathfrak{D}}^{\beta }$.

$${\mathfrak{D}}^{-\mathit{\beta }}{\mathfrak{D}}^{\beta } {\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta ={\mathfrak{D}}^{-\mathit{\beta }}{\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left(-{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}-g\right)\right)+{\lambda }_z{\mathfrak{D}}^{-\mathit{\beta }} {\mathfrak{D}}^1 e_z+{\mathfrak{D}}^{-\mathit{\beta }} \left({\eta }_z{S}_z+k_z sgn\left(S_z\right)\right)$$

(16)

→

$${\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta ={\ddot{Z}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}+g+{\lambda }_z {\mathfrak{D}}^{1-\beta } e_z+{\eta }_z {\mathfrak{D}}^{-\beta } S_z+k_z {\mathfrak{D}}^{-\beta } sgn\left(S_z\right)$$

(17)

→

$${\mathit{u}}_1={\displaystyle \frac{m}{Cos\varphi Cos\theta }}\left[{\ddot{Z}}_d+{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}+g+{\lambda }_z {\mathfrak{D}}^{1-\beta } e_z+{\eta }_z {\mathfrak{D}}^{-\beta } S_z+k_z {\mathfrak{D}}^{-\beta } sgn\left(S_z\right)\right]$$

(18)

It is important to analyze the conditions under which the control input ${\mathit{u}}_{\mathbf{1}}$ becomes singular. The denominator of Equation (33) contains the term $Cos\varphi Cos\theta$, which becomes zero when either $\varphi =\pm \mathbf{90}\mathbf{°} \mathrm{o}\mathrm{r} \theta =\pm \mathbf{90}\mathbf{°}$. At these points, ${\mathit{u}}_{\mathbf{1}}$ becomes undefined due to division by zero. Physically, this corresponds to the quadrotor tilting too much in the roll or pitch direction, making altitude control ineffective. To prevent singular behavior, constraints on the roll and pitch angles $(e.g., \left|\varphi \right|<85° \mathrm{o}\mathrm{r} \theta <85°$) can be applied. Alternatively, regularization methods such as adding a small constant ϵ to the denominator can be used to maintain numerical stability. Adaptive control strategies can also help mitigate the issue by adjusting thrust distribution dynamically when the quadrotor nears singular configurations.

The convergence of the system trajectories to the sliding manifolds $S_z=0, S_{\varphi }=0, S_{\theta }=0, { S}_{\psi }=0$ implies that the tracking errors $e_z,e_{\varphi }, e_{\theta }, {\mathrm{a}\mathrm{n}\mathrm{d} e}_{\psi }$ satisfy fractional-order differential equations. To formally establish their convergence to zero, we analyze the stability of these equations.

Once the system reaches the sliding manifolds, the tracking error dynamics are governed by fractional-order differential equations of the following form:

$$D^{\alpha }{e}_i+k_i{e}_i=0, \mathrm{i}\in \left\{z,\varphi ,\theta ,\psi \right\}$$

where $D^{\alpha }$ denotes the fractional derivative of order $k_i=0$, which represents the control gains. According to stability conditions for fractional-order systems, solutions of such equations asymptotically converge to zero if the system matrices satisfy Matignon’s stability condition, which requires that the real parts of all eigenvalues lie in the left half-plane. Given that the control gains $k_i$ are positive, this condition is met, ensuring that the tracking errors decay over time [52].

3.3. Stability Analysis

The primary objective when designing a control system is to ensure that the system reaches a stable equilibrium despite uncertainties in dynamics and input gains. This is achieved by proving the stability of the control law through Lyapunov stability analysis, which ensures that system trajectories converge to the equilibrium point over time, even in the presence of nonlinearities and uncertainties. The Lyapunov function is a key tool for assessing the stability of nonlinear systems. In this study, the Lyapunov function is constructed to be a positive definite function, and its time derivative must be negative definite or semi-negative definite to prove system stability. This method is well established for systems under parameter uncertainty and disturbances.

The Lyapunov function for the altitude control is given as follows:

$$V_z={\displaystyle \frac{1}{2}}{S}_z^2$$

(19)

The time derivative of the Lyapunov function can be taken as

$$\stackrel{·}{V_z}=S_z{\stackrel{·}{S}}_z$$

(20)

$$\stackrel{·}{V_z}=S_z\left[{\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left({\displaystyle \frac{U_1}{m}} Cos\varphi Cos\theta -{\displaystyle \frac{K_t}{m}}\stackrel{·}{Z}-g\right)\right)+{\lambda }_z {\mathfrak{D}}^1 e_z \right]$$

(21)

Then, $U_1$ can be substituted and the result simplified.

$$\stackrel{·}{V_z}=S_z\left[{\mathfrak{D}}^{\beta } \left(-\left[\underset{{\mathrm{U}}_1\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{{\lambda }_z {\mathfrak{D}}^{1-\beta } e_z+{\eta }_z {\mathfrak{D}}^{-\beta } S_z+k_z {\mathfrak{D}}^{-\beta } sgn\left(S_z\right)}}\right]\right)+{\lambda }_z {\mathfrak{D}}^1 e_z \right]$$

(22)

$$\stackrel{·}{V_z}=S_z\left[-{\lambda }_z {\mathfrak{D}}^1 e_z+{\lambda }_z {\mathfrak{D}}^1 e_z-{\eta }_z{S}_z-k_z sgn\left(S_z\right) \right]$$

(23)

→

$${\stackrel{·}{V}}_z=-{\eta }_z{S}_z^2-k_z \left|S_z\right| \le 0$$

(24)

which is negative $\forall t$, since ${\eta }_z,k_z>$ 0.

Since ${\eta }_z,k_z$ > 0, it is a negative definite function at all times, meaning that the system will converge to the desired equilibrium point. The system is stable under bounded uncertainties in system dynamics and input gains.

The Lyapunov function was selected because it expresses the square error between the desired and actual altitude, thus making it appropriate for the control of altitude. A positive definite function guarantees that the error of the system is always non-negative, and its time derivative is negative definite when the bounded uncertainties are subjected to the conditions specified.

• Effects of Bounded Uncertainties on Stability:

The bounded errors in dynamics and input gains ensure that the control inputs fall within a confined range and, hence, prevent instability. The Lyapunov-based proof of stability ensures that even with changes in the system dynamics (such as payload, aerodynamic forces, or sensor inaccuracies), the error will not become unbounded. Instead, it will asymptotically converge to zero. This means that the system reaches fixed-time convergence at the equilibrium point.

• Effect of Input Gain Uncertainty:

There are uncertainties in the input gains, such as roll, pitch, and yaw gains. They are known to have bounded physical limits. The proof of stability also takes into account these uncertainties, which adapt the control law for variations in the input gain. Hence, under real-world conditions, the controller will continue to be effective and maintain accurate control despite variations in the system parameters.

3.4. Attitude Control Design

The angles of rotation control the orientation of the quadrotor. Three separate controllers can be designed to control the roll, pitch, and yaw angles. Similarly to the altitude control design, the sliding manifold is designed, and the control law can be expressed as follows:

Roll Angle

The dynamics of the roll angles $\varphi$ can be expressed as follows [52]:

$$\stackrel{·}{\mathit{p}}={\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega +U_{\varphi }\right]$$

(25)

The sliding manifold can be designed as

$$S_{\varphi }={\mathfrak{D}}^{\beta } {\stackrel{·}{e}}_{\varphi }+{\lambda }_{\varphi }{e}_{\varphi }$$

(26)

$${\stackrel{·}{S}}_{\varphi }={\mathfrak{D}}^{\beta } {\ddot{e}}_{\varphi }+{\lambda }_{\varphi }{\stackrel{·}{e}}_{\varphi }=-\left({\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)\right)$$

(27)

→

$${\stackrel{·}{S}}_{\varphi }={\mathfrak{D}}^{\beta } \left({\ddot{\varphi }}_d-\ddot{\varphi }\right)+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }=-\left({\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)\right)$$

(28)

$${\stackrel{·}{S}}_{\varphi }={\mathfrak{D}}^{\beta } \left[{\ddot{\varphi }}_d-{\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega +u_{\varphi }\right]\right]+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }=-\left({\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)\right)$$

(29)

$${\stackrel{·}{S}}_{\varphi }={\mathfrak{D}}^{\beta } {\ddot{\varphi }}_d-{\mathfrak{D}}^{\beta } \left[{\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega +u_{\varphi }\right]\right]+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }=-\left({\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)\right)$$

(30)

The following equation can be used to solve for $U_{\varphi }:$

$${\mathfrak{D}}^{\beta }{\displaystyle \frac{1}{I_x}}{U}_{\varphi }={\mathfrak{D}}^{\beta } {\ddot{\varphi }}_d-{\mathfrak{D}}^{\beta }\left[{\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega \right]\right]+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }+{\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)$$

(31)

Using the commutative property of the fractional-order operator, we can multiply by ${\mathfrak{D}}^{-\beta }$ to cancel the existing ${\mathfrak{D}}^{\beta }$.

$${\mathfrak{D}}^{-\mathit{\beta }}{\mathfrak{D}}^{\beta }{\displaystyle \frac{1}{I_x}}{u}_{\varphi }={\mathfrak{D}}^{-\mathit{\beta }}{\mathfrak{D}}^{\beta }\left({\ddot{\varphi }}_d-{\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega \right]\right)+{\lambda }_{\varphi }{\mathfrak{D}}^{-\mathit{\beta }} {\mathfrak{D}}^1 e_{\varphi }+{\mathfrak{D}}^{-\mathit{\beta }}\left({\eta }_{\varphi }{S}_{\varphi }+k_{\varphi } sgn\left(S_{\varphi }\right)\right)$$

(32)

→

$$u_{\varphi }=I_x {\ddot{\varphi }}_d-\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega \right]+I_x\left[{\lambda }_{\varphi } {\mathfrak{D}}^{1-\beta } e_{\varphi }+{\eta }_{\varphi } {\mathfrak{D}}^{-\beta } S_{\varphi }+k_{\varphi } {\mathfrak{D}}^{-\beta } sgn\left(S_{\varphi }\right)\right]$$

(33)

3.5. Stability Analysis

A Lyapunov function is designed as

$$V_{\varphi }={\displaystyle \frac{1}{2}}{S}_{\varphi }^2$$

(34)

The time derivative of the Lyapunov function can be taken as

$$\stackrel{·}{V_{\varphi }}=S_{\varphi }{\stackrel{·}{S}}_{\varphi }$$

(35)

$$\stackrel{·}{V_{\varphi }}=S_{\varphi }\left[{\mathfrak{D}}^{\beta } \left[{\ddot{\varphi }}_d-{\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega +u_{\varphi }\right]\right]+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }\right]$$

(36)

Then, the result can be simplified using the value obtained in (35) in (36).

$$\stackrel{·}{V_{\varphi }}=S_{\varphi }\left[{\mathfrak{D}}^{\beta } \left[-\underset{U_{\varphi }\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\underbrace{\left[{\lambda }_{\varphi } {\mathfrak{D}}^{1-\beta } e_{\varphi }+{\eta }_{\varphi } {\mathfrak{D}}^{-\beta } S_{\varphi }+k_{\varphi } {\mathfrak{D}}^{-\beta } sgn\left(S_{\varphi }\right)\right]}}\right]+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi } \right]$$

(37)

$$\stackrel{·}{V_{\phi }}=S_{\phi } \left[-{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi }-{\eta }_{\varphi }{S}_{\varphi }-k_{\varphi } sgn\left(S_{\varphi }\right)+{\lambda }_{\varphi } {\mathfrak{D}}^1 e_{\varphi } \right]$$

(38)

→

$${\stackrel{·}{V}}_{\phi }=-{\eta }_{\varphi }{S}_{\varphi }^2-k_{\varphi } \left|S_{\varphi }\right| \le 0$$

(39)

Similarly, the dynamics of the pitch angle $\theta$ can be expressed as follows [53]:

$$\stackrel{·}{\mathit{q}}={\displaystyle \frac{1}{I_y}}\left[-p r \left(I_x-I_z\right)-K_r q+J_r p \Omega +u_{\theta }\right]$$

(40)

The designed controller $U_{\theta }$ is

$${\mathit{u}}_{\mathit{\theta }}=I_y {\ddot{\theta }}_d-\left[-p r \left(I_x-I_z\right)-K_r q+J_r p \Omega \right]+I_y\left[{\lambda }_{\theta } {\mathfrak{D}}^{1-\beta } e_{\theta }+{\eta }_{\theta } {\mathfrak{D}}^{-\beta } S_{\theta }+k_{\theta } {\mathfrak{D}}^{-\beta } sgn\left(S_{\theta }\right)\right]$$

(41)

The Lyapunov function is designed as

$$V_{\theta }={\displaystyle \frac{1}{2}}{S}_{\theta }^2$$

(42)

The time derivative after simplification is

$${\stackrel{·}{V}}_{\theta }=-{\eta }_{\theta }{S}_{\theta }^2-k_{\theta } \left|S_{\theta }\right| \le 0$$

(43)

The dynamics of the yaw angle $\psi$ can be expressed as follows [52]:

$$\stackrel{·}{\mathit{r}}={\displaystyle \frac{1}{I_z}}\left[-p q \left(I_y-I_x\right)-K_r r+u_{\psi }\right]$$

(44)

The designed controller $U_{\psi }$ is

$$u_{\psi }=I_z {\ddot{\psi }}_d-\left[-p q \left(I_y-I_x\right)-K_r r\right]+I_z\left[{\lambda }_{\psi } {\mathfrak{D}}^{1-\beta } e_{\psi }+{\eta }_{\psi } {\mathfrak{D}}^{-\beta } S_{\psi }+k_{\psi } {\mathfrak{D}}^{-\beta } sgn\left(S_{\psi }\right)\right]$$

(45)

The Lyapunov function is designed similarly to the above functions as

$$V_{\psi }={\displaystyle \frac{1}{2}}{S}_{\psi }^2$$

(46)

The time derivative after simplification is

$${\stackrel{·}{V}}_{\psi }=-{\eta }_{\psi }{S}_{\psi }^2-k_{\psi } \left|S_{\psi }\right| \le 0$$

(47)

Stability Analysis of Tracking Errors

The previous analysis has established that the system trajectories converge to the sliding surfaces $S_z=0, S_{\theta }=0, {\mathrm{a}\mathrm{n}\mathrm{d} S}_{\psi }=0$ in finite time. However, it is also necessary to analyze the behavior of the tracking errors $ez, e_{\varphi }, e_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} e_{\psi }$ in the closed-loop system. Using Lyapunov-based stability analysis, we show that in the absence of uncertainties $\Delta =0,$ the tracking errors asymptotically converge to zero. When uncertainties $\Delta \ne 0$ are present, the system remains stable, but the tracking errors are confined within a bounded neighborhood around zero.

By defining the Lyapunov function as a quadratic function of the tracking errors and analyzing its time derivative, it is shown that the error convergence region is upper-bounded by

$${e^2}_z+{e^2}_{\varphi }+{e^2}_{\theta }+{e^2}_{\psi }=\le {\displaystyle \frac{\mathsf{\Delta }}{\mathrm{m}\mathrm{i}\mathrm{n}(kz,k_{\varphi }, k_{\theta }, k_{\psi })}}$$

This result provides a precise characterization of how uncertainties affect the tracking performance, ensuring that errors remain small and predictable. The analysis confirms that the proposed FOSMC guarantees stability even in the presence of system uncertainties, thereby reinforcing the robustness of the control scheme.

The Lyapunov candidate function is defined as $V={\displaystyle \frac{1}{2}}({e^2}_z+{e^2}_{\varphi }+{e^2}_{\theta }+{e^2}_{\psi })$. This function is positive definite and serves as a measure of the error magnitude. Taking the time derivative $\stackrel{·}{V}=e_z\stackrel{·}{e_z}, e_{\varphi }\stackrel{·}{e_{\varphi }}, e_{\theta }\stackrel{·}{e_{\theta }}, e_{\psi }\stackrel{·}{e_{\psi }}$, from the fractional-order sliding mode control (FOSMC) law, the error dynamics can be expressed as follows:

$$\begin{gathered} \stackrel{·}{e_z}=-k_z{e}_z+{\Delta }_z,\stackrel{·}{e_{\varphi }}=-k_{\varphi }{e}_{\varphi }+{\Delta }_{\varphi } \\ \stackrel{·}{e_{\theta }}=-k_{\theta }{e}_{\theta }+{\Delta }_{\theta },\stackrel{·}{e_{\psi }}=-k_{\psi }{e}_{\psi }+{\Delta }_{\psi } \end{gathered}$$

where $k_z,k_{\varphi }, k_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} k_{\psi }$ are positive control gains, ensuring stability, and ${\Delta }_i$ represents system uncertainties.

$\stackrel{·}{V}\le -\mathrm{m}\mathrm{i}\mathrm{n}\left(k_z,k_{\varphi }, k_{\theta }, k_{\psi }\right)\mathrm{V}+\sum {\mathrm{e}}_{\mathrm{i}}{\Delta }_i,$ using the inequality $\mathrm{a}\mathrm{b}\le {\displaystyle \frac{a^2}{2∊}}+{\displaystyle \frac{∊b^2}{2}}.$ for an appropriate choice of ϵ and assuming the uncertainties are bounded as $\mid \Delta \mathrm{i}\mid \le \Delta \mathrm{m}\mathrm{a}\mathrm{x}$, we get $\stackrel{·}{V}\le -\mathrm{m}\mathrm{i}\mathrm{n}\left(k_z,k_{\varphi }, k_{\theta }, k_{\psi }\right)\mathrm{V}+\Delta \mathrm{m}\mathrm{a}\mathrm{x}\surd \mathrm{V}$.

At steady state $(\stackrel{·}{V}=0),$ solving for V gives

$${e^2}_z+{e^2}_{\varphi }+{e^2}_{\theta }+{e^2}_{\psi }=\le {\displaystyle \frac{{ \mathsf{\Delta }}_{max}}{\mathrm{m}\mathrm{i}\mathrm{n}(kz,k_{\varphi }, k_{\theta }, k_{\psi })}}$$

Thus, the tracking errors remain bounded within a predictable region determined by the system uncertainties and control gains.

4. Robustness of FOSMC Design in the Presence of Parameter Uncertainty

In a system like a quadrotor, it is most likely that the system parameters are not known. The parametric uncertainty may affect the performance of the designed controller. Therefore, it is necessary to design a robust controller that can cover uncertainty’s impact. In FOSMC, the parametric uncertainty can be overcome if the dynamics of the system and the input gain are bounded [43]. This study proposes a novel FOSMC methodology that offers robustness against uncertainties and disturbances inherent in quadrotor dynamics, unlike the traditional linearization methods, which may fail to stabilize nonlinear systems effectively. The control design involves separate controllers for attitude and position control, recognizing the decoupled nature of rotational and translational dynamics in quadrotor systems. The use of fractional calculus principles allows for the incorporation of non-integer differentiation orders, enhancing the controller’s adaptability and performance. The stability of the proposed controllers is rigorously analyzed using Lyapunov functions, ensuring convergence to desired equilibrium points despite nonlinearities and uncertainties.

The system can be represented as

$$\ddot{x}=f\left(x,t\right)+b u\left(t\right)$$

(48)

where $f\left(x,t\right)$ is not exactly a known function; however, it can be estimated as $\widehat{f}$, and the error is bounded by $\overline{F}$, such that $\left(\left|\widehat{f}-f\right|\le \overline{F}\right)$. In addition, the input gain $b$ is unknown but with known bounds and can be chosen as $\left({\zeta }^{-1}\le {\displaystyle \frac{b}{\widehat{b}}}\le \zeta \right)$, where $\left(\widehat{b}\right)$ is the estimated gain. The robust controllers can be designed as illustrated below [33].

4.1. Altitude Control Design

The dynamics of the altitude can be expressed as follows:

$$\ddot{z}={\displaystyle \frac{1}{m}}\left[\left(cos\phi cos\theta \right) u_1-K_t\stackrel{·}{z}-mg\right]$$

(49)

Let

$$f_z=-{\displaystyle \frac{K_t}{m}}\stackrel{·}{z}-g$$

(50)

$$b_z={\displaystyle \frac{cos\varphi cos\theta }{m}}$$

(51)

Then, (49) can be rewritten using (50) and (51) as

$$\ddot{z}=f_z+b_z{u}_1$$

(52)

assuming the following bounds:

$$\left|\widehat{f_z}-f_z\right|\le {\overline{F}}_z, \mathrm{and} {\zeta }_z^{-1}\le {\displaystyle \frac{b_z}{\widehat{b_z}}}\le {\zeta }_z$$

The fractional-order sliding mode surface is

$$S_z={\mathfrak{D}}^{\beta } {\stackrel{·}{e}}_z+{\lambda }_z{e}_z$$

(53)

The derivative of the surface can be expressed as

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } {\ddot{e}}_z+{\lambda }_z {\stackrel{·}{e}}_z$$

(54)

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\ddot{Z}\right)+{\lambda }_z {\stackrel{·}{e}}_z$$

(55)

$${\stackrel{·}{S}}_z={\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left(f_z+b_z{u}_1\right)\right)+{\lambda }_z {\stackrel{·}{e}}_z$$

(56)

The equivalent control can be calculated as

$${\widehat{u}}_1={\displaystyle \frac{1}{\widehat{b_z}}} \underset{{\widehat{u}}_{1eq}}{\underbrace{\left({\ddot{z}}_d-{\widehat{f}}_z\right)+{\lambda }_z{\mathfrak{D}}^{-\beta } {\stackrel{·}{e}}_z}}$$

(57)

The overall control law is designed by adding the switching control, which satisfies the sliding mode’s reaching conditions, to the equivalent control.

$$u_1={\displaystyle \frac{1}{\widehat{b_z}}}\left[{\widehat{u}}_{1eq}+k_z {\mathfrak{D}}^{-\beta } sgn\left(S_z\right)+{\eta }_z {\mathfrak{D}}^{-\beta } S_z\right]$$

(58)

4.2. Stability Analysis

To prove the robustness of the above controller $U_1$ against parameter uncertainty within known bounds, a Lyapunov function is designed as follows:

$$V_z={\displaystyle \frac{1}{2}}{S}_z^2$$

(59)

The time derivative of the function is

$$\stackrel{·}{V_z}=S_z{\stackrel{·}{S}}_z$$

(60)

$$\stackrel{·}{V_z}=S_z\left[{\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left(f_z+b_z{U}_1\right)\right)+{\lambda }_z {\stackrel{·}{e}}_z\right]$$

(61)

By substituting (58) into (61), we get

$$\begin{gathered} \stackrel{·}{V_z}=S_z\left[{\mathfrak{D}}^{\beta } \left({\ddot{Z}}_d-\left(f_z+b_z\left[{\displaystyle \frac{1}{{\widehat{b}}_z}}\left[\left({\ddot{Z}}_d-{\widehat{f}}_z\right)+{\lambda }_z{\mathfrak{D}}^{-\beta } {\stackrel{·}{e}}_z+k_z {\mathfrak{D}}^{-\beta } sgn\left(S_z\right)+{\eta }_z {\mathfrak{D}}^{-\beta } S_z\right]\right]\right)\right)+{\lambda }_z {\stackrel{·}{e}}_z\right] \\ \stackrel{·}{V_z}=S_z\left[\left(1-{\displaystyle \frac{b_z}{{\widehat{b}}_z}}\right)\left({\mathfrak{D}}^{\beta } {\ddot{Z}}_d+{\lambda }_z {\stackrel{·}{e}}_z\right)+{\mathfrak{D}}^{\beta }\left({\displaystyle \frac{b_z}{{\widehat{b}}_z}}{\widehat{f}}_z-f\right)-{\displaystyle \frac{b_z}{{\widehat{b}}_z}}\left(k_z sgn\left(S_z\right)+{\eta }_z S_z\right)\right] \end{gathered}$$

(62)

To ensure the stability of the controller, the time derivative of the Lyapunov function must satisfy the following condition:

$$\stackrel{·}{V_z}\le 0$$

(63)

Therefore, the uncertain dynamics of the system and the control gain need to be replaced by their known bounds. Then, the parameter $k_z$ can be established such that it satisfies (63).

Now, ${\stackrel{·}{S}}_z$ can be rewritten as

$${\stackrel{·}{S}}_z=\left(1-{\displaystyle \frac{b_z}{{\widehat{b}}_z}}\right)\left({\mathfrak{D}}^{\beta } {\ddot{Z}}_d+{\lambda }_z {\stackrel{·}{e}}_z\right)+{\mathfrak{D}}^{\beta }\left({\displaystyle \frac{b_z}{{\widehat{b}}_z}}{\widehat{f}}_z-f_z\right)-{\displaystyle \frac{b_z}{{\widehat{b}}_z}}\left(k_z sgn\left(S_z\right)+{\eta }_z S_z\right)$$

(64)

Then, (56) can be multiplied by $\left({\displaystyle \frac{{\widehat{b}}_z}{b_z}}\right)$:

$${\stackrel{·}{S}}_z=\left({\displaystyle \frac{{\widehat{b}}_z}{b_z}}-1\right)\left({\mathfrak{D}}^{\beta } {\ddot{Z}}_d+{\lambda }_z {\stackrel{·}{e}}_z\right)+{\mathfrak{D}}^{\beta } \left({\widehat{f}}_z-{\displaystyle \frac{{\widehat{b}}_z}{b_z}}{f}_z\right)-k_z sgn\left(S_z\right)-{\eta }_z S_z$$

(65)

Next, we need to eliminate the unknown dynamics $\left(f_z\right)$ using the following expression:

$$f_z=\underset{{\overline{F}}_z}{\underbrace{f_z-{\widehat{f}}_z}}+{\widehat{f}}_z$$

(66)

By substituting (66) into (65), we can get

$${\stackrel{·}{S}}_z=\left({\displaystyle \frac{{\widehat{b}}_z}{b_z}}-1\right)\left({\mathfrak{D}}^{\beta } {\ddot{Z}}_d+{\lambda }_z {\stackrel{·}{e}}_z-{\mathfrak{D}}^{\beta } {\widehat{f}}_z\right)-{\displaystyle \frac{{\widehat{b}}_z}{b_z}}{\mathfrak{D}}^{\beta } {\overline{F}}_z-k_z sgn\left(S_z\right)-{\eta }_z S_z$$

(67)

Therefore, $k_z$ needs to be chosen as follows:

$$k_z\ge \left|{\zeta }_z-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_z {\widehat{U}}_{1eq}\right|-{\zeta }_z{\mathfrak{D}}^{\beta } {\overline{F}}_z$$

(68)

where $\stackrel{·}{V_z}$ satisfies the following:

$$\stackrel{·}{V_z}\le {\eta }_z S_z^2$$

(69)

In the design of the altitude control system and in its stability analysis, several assumptions are made regarding the uncertainty bounds of system dynamics and input gains. These assumptions play a crucial role in determining the robustness of the controller under parameter uncertainty.

1. Uncertainty in System Dynamics

• Assumption: The dynamics of the quadrotor system, represented by the function $f\left(x,t\right)$ in Equation (48), are not precisely known. However, an estimate f is available, with the error bounded by $\overline{F}$.
• Rationale: Quadrotors are coupled with multiple disturbances such as aerodynamic forces, payload changes, and sensor noise.
• These factors induce drifts in the system’s dynamics, thereby making it impossible to model the system accurately. The assumption of bounded error $\overline{F}$ is rational since these uncertainties are bounded by known physical limits, for example, the maximum expected drift due to the gustiness of wind or noise in measurement. This allows the system to predictably vary only within such margins. By implication, the fact that $\overline{F}$ has a bounded error assures the controller adapts to uncertain conditions and that the real-time applications of the model stay stable.
• This assumption that errors remain within a bounded range is specifically important for the Lyapunov stability proof, as it ensures that the error function remains within a defined range, thus allowing the system to converge toward the desired equilibrium point. This adaptability improves the robustness of the controller, which will be able to reject external disturbances and ensure precise quadrotor behavior under uncertain conditions.

2. Uncertainty in Input Gains

• Assumption: The input gain b of the system is unknown but lies within known bounds ${\zeta }^{-1}$ and $\zeta$ with an estimated gain $\widehat{b}$.
• Rationale: In quadrotors, input gains are critical for determining the effectiveness of control inputs, which influence the system’s dynamics. These gains can fluctuate due to factors such as aerodynamic variations, payload changes, or sensor inaccuracies. For example, the quadrotor’s control response to pitch, roll, and yaw angles depends on these input gains, which may vary with different operational environments (e.g., wind, payload load, battery charge).
• The assumption of bounded uncertainty in input gains is reasonable because, although the exact value of B is difficult to determine precisely, the physical system constraints ensure that these variations stay within certain limits. The control system can still operate effectively within these bounds, as quadrotor systems do not experience unbounded fluctuations in input gains in practice.
• Implications: The known bounds on input gain ensure that the controller can adapt to variations in control input magnitude, ensuring that the system remains stable and responsive to environmental changes. The assumption is particularly critical for stability analysis since these uncertainties influence the control response but do not lead to instability as long as the bounds are respected. By bounding the uncertainty, the system can compensate for these variations, improving performance and ensuring precise control even in the presence of dynamic uncertainties.

3. Overall Impact on Stability and Effectiveness

• The chosen uncertainty bounds for system dynamics and input gains are essential for guaranteeing the stability and effectiveness of the altitude control system.
• By incorporating these bounds into the control design and stability analysis, the controller can effectively handle parameter uncertainties and maintain stable operation under varying conditions.
• Furthermore, the adaptability provided by these bounds enhances the controller’s robustness, ensuring reliable performance in real-world scenarios where uncertainties are prevalent.

4.3. Attitude Control Design

By applying the same procedure from Section 4.1 and using the same assumptions for the unknown dynamics and control gains, three angle controllers are designed with the following assumptions and conditions.

The dynamics of the roll angle $\left(\varphi \right)$ are given by

$$\stackrel{·}{p}={\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega +u_{\varphi }\right]$$

(70)

Let

$$f_{\varphi }={\displaystyle \frac{1}{I_x}}\left[-q r \left(I_z-I_y\right)-K_r p-J_r q \Omega \right]$$

(71)

$$b_{\varphi }={\displaystyle \frac{1}{I_x}}$$

(72)

Then, (67) can be rewritten using (68) and (69), as follows:

$$\stackrel{·}{p}=f_{ \varphi }+b_{\varphi } u_{\varphi }$$

(73)

Assuming the following bounds

$$\left|{\widehat{f}}_{\varphi }-f_{\varphi }\right|\le {\overline{F}}_{\varphi }, \mathrm{and} {\zeta }_{\varphi }^{-1}\le {\displaystyle \frac{b_{\varphi }}{\widehat{b_{\varphi }}}}\le {\zeta }_{\varphi },$$

the equivalent FOSMC can be calculated as follows:

$${\widehat{u}}_{\varphi }={\displaystyle \frac{1}{\widehat{b_{\varphi }}}} \underset{{\widehat{u}}_{\varphi eq}}{\underbrace{\left({\ddot{\varphi }}_d-{\widehat{f}}_{\varphi }\right)+{\lambda }_{\varphi }{\mathfrak{D}}^{-\beta } {\stackrel{·}{e}}_{\varphi }}}$$

(74)

The overall control law is designed by adding the switching control to the equivalent control:

$$u_{\varphi }={\displaystyle \frac{1}{\widehat{b_{\varphi }}}}\left[{\widehat{u}}_{\varphi eq}+k_{\varphi } {\mathfrak{D}}^{-\beta } sgn\left(S_{\varphi }\right)+{\eta }_{\varphi } {\mathfrak{D}}^{-\beta } S_{\varphi }\right]$$

(75)

$$where k_{\varphi }\ge \left|{\zeta }_{\varphi }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\varphi } {\widehat{u}}_{\varphi eq}\right|-{\zeta }_{\varphi }{\mathfrak{D}}^{\beta } {\overline{F}}_{\varphi }$$

(76)

Similarly, the pitch and yaw angle controller can be expressed as

$$u_{\theta }={\displaystyle \frac{1}{\widehat{b_{\theta }}}}\left[{\widehat{u}}_{\theta eq}+k_{\theta } {\mathfrak{D}}^{-\beta } sgn\left(S_{\theta }\right)+{\eta }_{\theta } {\mathfrak{D}}^{-\beta } S_{\theta }\right]$$

(77)

$$u_{\psi }={\displaystyle \frac{1}{\widehat{b_{\psi }}}}\left[{\widehat{u}}_{\psi eq}+k_{\psi } {\mathfrak{D}}^{-\beta } sgn\left(S_{\psi }\right)+{\eta }_{\psi } {\mathfrak{D}}^{-\beta } S_{\psi }\right]$$

(78)

where

$$k_{\theta }\ge \left|{\zeta }_{\theta }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\theta } {\widehat{u}}_{\theta eq}\right|-{\zeta }_{\theta }{\mathfrak{D}}^{\beta } {\overline{F}}_{\theta }$$

(79)

$$k_{\psi }\ge \left|{\zeta }_{\psi }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\psi } {\widehat{u}}_{\psi eq}\right|-{\zeta }_{\psi }{\mathfrak{D}}^{\beta } {\overline{F}}_{\psi }$$

(80)

4.4. Stability Analysis

To prove the robustness of the above controller $\left(u_{\varphi }, u_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} u_{\psi }\right)$ against the parameter uncertainty within known bounds, a Lyapunov function is designed as follows:

$$V_{\varphi ,\theta ,\psi }={\displaystyle \frac{1}{2}}{S}_{\varphi }^2+{\displaystyle \frac{1}{2}}{S}_{\theta }^2+{\displaystyle \frac{1}{2}}{S}_{\psi }^2$$

(81)

The stability of the controllers can be proven when the time derivative of the Lyapunov function satisfies the following, ensuring that the system remains stable and resilient to both parameter uncertainties and external disturbances.

$${\stackrel{·}{V}}_{\varphi ,\theta ,\psi }\le 0$$

(82)

This can be satisfied when the parameters $\left(k_{\varphi }, k_{\theta }, \mathrm{a}\mathrm{n}\mathrm{d} k_{\psi }\right)$ are designed as follows:

$$k_{\varphi }\ge \left|{\zeta }_{\varphi }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\varphi } {\widehat{u}}_{\varphi eq}\right|-{\zeta }_{\varphi }{\mathfrak{D}}^{\beta } F_{\varphi }$$

(83)

$$k_{\theta }\ge \left|{\zeta }_{\theta }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\theta } {\widehat{u}}_{\theta eq}\right|-{\zeta }_{\theta }{\mathfrak{D}}^{\beta } F_{\theta }$$

(84)

$$k_{\psi }\ge \left|{\zeta }_{\psi }-1\right|\left|{\mathfrak{D}}^{\beta }{ \widehat{b}}_{\psi } {\widehat{u}}_{\psi eq}\right|-{\zeta }_{\psi }{\mathfrak{D}}^{\beta } F_{\psi }$$

(85)

such that ${\stackrel{·}{V}}_{\varphi ,\theta ,\psi }$ satisfies the following equation:

$${\stackrel{·}{V}}_{\varphi ,\theta ,\psi }\le -\left({\eta }_{\varphi } S_{\varphi }^2+{\eta }_{\theta } S_{\theta }^2+{\eta }_{\psi } S_{\psi }^2\right)$$

(86)

In the design of the attitude control system for the quadrotor, similar to the altitude control, certain assumptions are made regarding the uncertainty bounds of system dynamics and input gains. These assumptions are critical for ensuring the stability and robustness of the controller under parameter uncertainty.

1. Uncertainty in System Dynamics

• Assumption: The dynamics governing the roll, pitch, and yaw angles of the quadrotor, represented by functions f_ϕ, f_θ, and f_ψ, respectively, are subject to uncertainty arising from various external disturbances such as wind gusts and imperfections in sensor measurements. These uncertainties are bounded by error limits F_ϕ⁻, F_θ⁻, and F_ψ⁻. By incorporating these bounds into the controller’s design, the system can adapt to and mitigate the effects of these disturbances, maintaining stable operation even in the presence of unpredictable environmental factors.
• Rationale: The presence of aerodynamic disturbances, payload fluctuations, and sensor errors makes exact modeling difficult. However, the assumption that the uncertainties in roll, pitch, and yaw are bounded by $F_{\varphi }^{-}$, $F_{\theta }^{-}$, $F_{\psi }^{-}$ is reasonable because it accounts for the physical limitations of the system and external conditions. This assumption ensures that the system remains within a predictable range of behavior, even when subjected to dynamic environmental factors.
• Implications: The bounded errors $F_{\varphi }^{-}$, $F_{\theta }^{-}$, $F_{\psi }^{-}$ are crucial in ensuring that the attitude control system can effectively adapt to variations in system dynamics, maintaining precise control even in the presence of external disturbances. This assumption plays a vital role in the Lyapunov stability analysis, as it guarantees that the attitude controller can maintain stability within a defined error range, preventing instability despite the uncertainties in system dynamics.

2. Uncertainty in Input Gains

• Assumption: Similarly to system dynamics, the input gains for roll, pitch, and yaw control are subject to uncertainty due to external factors like varying aerodynamic conditions or sensor inaccuracies. These uncertainties, bounded by ζ_ϕ⁽⁻¹⁾, ζ_θ⁽⁻¹⁾, and ζ_ψ⁽⁻¹⁾, are accounted for in the controller design. By explicitly bounding the uncertainty in input gains, the system can maintain robust performance despite disturbances, ensuring precise control of the quadrotor’s attitude under varying operating conditions.
• Rationale: Variations in input gains, such as motor power fluctuations or aerodynamic changes, can affect the response of the quadrotor to control commands, leading to deviations from the desired attitude (roll, pitch, and yaw). For instance, changes in payload or battery levels may affect the thrust generation capabilities, which, in turn, affect the system’s control performance. Bounding the uncertainty in these gains ensures that the system remains effective across different conditions, maintaining precise control even in the presence of environmental disturbances.
• Implications: The known bounds on input gains allow the attitude controller to adjust its response to changes in the control input magnitude, compensating for variations in quadrotor dynamics. This adaptability is crucial for maintaining stability and control of the quadrotor under uncertain conditions. For example, if the system experiences increased payload or sensor drift, the controller can still stabilize the quadrotor and ensure accurate tracking of the desired roll, pitch, and yaw angles. By bounding the uncertainties, the system can achieve high precision in maneuvering and stabilization, enhancing overall performance in real-world applications.

Overall Impact on Stability and Effectiveness

• The explicit consideration of uncertainty bounds in system dynamics and input gains is crucial for designing a robust attitude control system for quadrotors. This approach ensures that the system can effectively handle both parameter uncertainties and external disturbances. By incorporating these bounds into the control design and stability analysis, the attitude controller not only maintains stable operation under varying conditions but also mitigates the impact of external disturbances, such as wind gusts and sensor noise, thus ensuring reliable and precise quadrotor performance in real-world environments.
• By incorporating these bounds into the control design and stability analysis, the attitude controller can effectively handle parameter uncertainties and maintain stable operation under varying conditions.
• Furthermore, the adaptability provided by these bounds enhances the controller’s robustness, ensuring reliable performance in real-world scenarios characterized by uncertainty in system dynamics and input gains.

5. Results and Discussion

This section presents the performance evaluation of the quadrotor while tracking a reference trajectory using both fractional-order sliding mode control (FOSMC) and feedback linearization (FBL) controllers. In this section, the performance of the quadrotor while tracking a reference trajectory is tested with a feedback linearization (FBL) controller.

5.1. FOSMC and Feedback Linearization (FBL) Controllers

The reference trajectory is selected to be a square-shaped path. The result is compared with the fractional-order sliding mode control result to demonstrate the differences and advantages of FOSMC. The altitude $\left(z\right)$ of both controllers is shown in Figure 2.

5.1.1. Altitude Tracking Performance

Figure 2 shows the altitude of the quadrotor when FBL is used to track the reference trajectory. However, a steady-state error of 0.077 m 1.1% is present. On the other hand, Figure 3 shows the altitude of the quadrotor under FOSMC with an error of 0.14%.

5.1.2. Robustness Against Parametric Uncertainties

Also, the robustness of the proposed controller was tested in the presence of parametric uncertainty. The following Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 show the positions and altitudes for both controllers under the same uncertain mass of the quadrotor. The quadrotor’s mass was considered uncertain and changed to 4 kg, while it was considered as 2 kg in the control design.

It can be observed from the figures above that FOSMC demonstrates better performance, particularly in terms of overshoot in the position response and minimal change in the altitude difference between the exact and uncertain mass of the quadrotor.

The selection of the parameters α and β depends on the application. They are recommended in the literature to be between zero and one.

To strengthen the discussion of the impact of the fractional-order parameter β on system performance, recent studies have demonstrated that fractional-order controllers, by appropriately tuning parameters like β, can enhance system stability, robustness, and flexibility compared to traditional integer-order controllers [53]. Conducting a sensitivity analysis by varying β and assessing its effects on performance metrics such as settling time, steady-state error, and disturbance rejection can aid in selecting an optimal β\betaβ that balances fast convergence with robustness.

The long simulation times used in the figures were intentionally selected to observe the long-term behavior of the quadrotor system under FOSMC with parameter uncertainties. The flat lines in the figures indicate that the system has reached a steady-state equilibrium after a prolonged simulation time. As expected from the control strategy, the system stabilized quickly, and the transition phase occurred earlier in the simulation. Although the flat lines might appear to show no transitions, this is indicative of the system’s robustness and ability to achieve long-term stability under the influence of parameter uncertainties.

To confirm the effectiveness of the suggested FOSMC method outside of simulations, experiments can be performed in real life on a quadrotor with onboard sensors and a flight controller. The experiments should test robustness against parameter uncertainties (e.g., changing payloads) and external disturbances (e.g., wind influences). Trajectory tracking tests along pre-programmed paths can test accuracy and stability, while comparisons with standard controllers such as FBL controllers can emphasize error reduction and disturbance rejection improvements. Major performance indices like tracking precision, response latency, and control effort will measure the effectiveness of the controller in quantifying terms, justifying its practical applicability for UAV flight in uncertain environments.

5.2. Theoretical Implications

The theoretical contributions of this paper are significant due to the novel application of fractional-order sliding mode control (FOSMC) in addressing the complex challenges of attitude and altitude tracking in quadrotor systems with parameter uncertainties. The integration of fractional calculus with sliding mode control introduces an innovative approach that combines the robustness of sliding mode techniques with the adaptability offered by fractional-order dynamics. This study’s emphasis on handling parameter uncertainties through additional adaptive laws represents a significant advancement, contributing to the development of more resilient control strategies. Furthermore, the application of FOSMC to tackle the nonlinear and strongly coupled characteristics of quadrotor systems demonstrates a distinctive theoretical perspective, offering fresh insights into the control of complex aerial vehicles.

5.3. Practical Implications

The application of FOSMC is considered promising for enhancing the control of UAVs and UGVs, offering more efficient and robust control compared to traditional integer-order control methods. Notably, it addresses the prevalent issue of parameter uncertainties in real-world systems like quadrotors, proposing a control scheme that effectively mitigates these uncertainties, ensuring reliable performance even in practical applications where accurate knowledge of system parameters may be challenging. Moreover, the study highlights the significance of reduced chattering, which can degrade control performance and cause wear on mechanical components, by implementing FOSMC to diminish chattering, thereby prolonging the lifespan of UAV components and ensuring smoother operation. Additionally, FOSMC is reported to exhibit fast convergence and eliminate steady-state error, promising quicker response times and more accurate tracking of desired attitudes and altitudes for quadrotor systems, translating to better overall performance and stability during flight operations. Crucially, the study validates its proposed control scheme through both numerical simulations and experimental tests, providing confidence in its practical effectiveness. Furthermore, it recommends the design of adaptive FOSMC to address uncertainties without known bounds, underlining the importance of adaptability in dynamic environments, where uncertainties may vary over time, suggesting avenues for further enhancing control robustness in practical implementations.

To further substantiate the feasibility of the suggested fractional-order sliding mode control (FOSMC), experimental verification in the real-world can be embraced. Experimental implementation scenarios include conducting flight experiments on a quadrotor with onboard sensors such as an IMUs, GPSs, and barometers, and a suitable flight controller such as Pixhawk or NVIDIA Jetson. The test equipment could test the controller’s behavior under real operational conditions, i.e., payload changes, external disturbances like wind, and dynamic trajectory tracking. The performance of FOSMC can be referenced against that of conventional controllers like feedback linearization (FBL) controllers using tracking accuracy, steady-state error, and uncertainty robustness. Testing can be performed in simulated indoor environments and carried over to outdoor scenarios for further confirmation. By testing the critical performance measures such as response time, control effort, and stability under disturbances, the experimental results would provide empirical substantiation of the applicability of FOSMC in UAV operations.

6. Conclusions

In this study, the efficacy of FOSMC is explored in governing quadrotor dynamics. The adoption of FOSMC for altitude, position, and attitude control stems from its demonstrated efficiency and robustness compared to traditional integer-order methods. FOSMC’s utilization of fractional-order calculus principles addresses inherent issues like chattering reduction, faster convergence, and elimination of steady-state errors, thereby enhancing overall performance. Moreover, this study reveals FOSMC’s resilience in the face of parameter uncertainties, a critical feature for real-world applications where system dynamics may vary unpredictably. Therefore, the study suggests the potential for adaptive FOSMC algorithms to further improve adaptability in scenarios where uncertainties lack known bounds. Furthermore, the study highlights FOSMC as a promising avenue for advancing control solutions in aerial robotics, with implications for a wide range of applications. Future research should focus on the practical implementation of FOSMC in outdoor flight tests or payload-carrying missions, providing valuable insights into its real-world applicability and performance.