# Robust Finite-Time Trajectory Tracking Control of Quadrotor Aircraft via Terminal Sliding Mode-Based Active Antidisturbance Approach: A PIL Experiment

Authors: Omar Mechali; Jamshed Iqbal; Xiaomei Xie; Limei Xu; Abdelkader Senouci
Journal: International Journal of Aerospace Engineering
 (2021)

---

## Abstract

This paper presents an accurate solution of finite-time Cartesian trajectory tracking control problem of a quadrotor system by designing and implementing a novel robust flight-control algorithm. The quadrotor is subject to nonlinearities, unmodeled dynamics, parameters’ uncertainties, and external time-varying disturbances. To reject the disturbances and enhance the control system’s robustness, a terminal sliding mode-based active antidisturbance control (TSMBAADC) approach is proposed for rotational and translational subsystems. To improve the tracking performance, a nonlinear continuous terminal sliding manifold and a fast reaching law are proposed in this work to quickly drive the systems’ states to the equilibrium point even in the presence of lumped disturbances. The convergence time of the states can be pretuned based on the parameters of the sliding manifold and the reaching law. Lyapunov theorem is used to provide a rigorous stability proof for the feedback control system. Numerical simulations and processor-in-the-loop (PIL) experiments are conducted to validate and implement the designed flight control algorithm on real autopilot hardware. The novelty of the proposed research lies in hardware implementation of a sophisticated version of modern control technique that exhibits a multitude of distinguishing features including but not limited to (i) finite-time tracking stability featuring fast convergence is ensured, (ii) chattering and singularity problems in sliding mode control (SMC) are avoided, and (iii) null steady-state error is achieved along with enhanced robustness. Finally, the proposed control law is compared with two recently reported research works. Results of performance comparison in term of the integral of square error (ISE) and the absolute value of the derivative of the inputut (IADU) dictate that the proposed technique overperforms by precision and chattering alleviation.


---

## Body

## 1. Introduction

### 1.1. Context and Motivations

Nowadays, quadrotor aircraft have gained enormous interest due to their numerous merits such as low-cost manufacturability and vertical take-off and landing capability [1–3]. These interesting aircraft have been widely used in many fields to solve practical and complex missions [4–8]. However, despite their advantages, quadrotors have some critical drawbacks that are related to flight control. The quadrotor is a nonlinear multioutput multi-input system with underactuated six degrees of freedom (6-DoF) dynamics [9]. Moreover, the quadrotor’s dynamics are strongly coupled and inevitably affected by multiple disturbances [10, 11]. Meanwhile, the quadrotor is intended to operate in a challenging flight environment where it may perform aggressive maneuvers. Thus, its stabilization and control in such flight conditions is not trivial but a challenging and complex task. The robust trajectory tracking control during flight missions is one of the persistent control problems. This problem has become an important topic that should be carefully addressed by the control community. Particularly, fast convergence, strong robustness, and accuracy are considered important features of a flight control algorithm to safely and effectively drive the quadrotor during the mission. Therefore, a reliable flight control system essentially relying on a modern control technique is required to achieve good tracking performance. Hence, this paper focuses on the design of a new flight-control system to deal with the robust Cartesian trajectory tracking control problem for the quadrotor system.


### 1.2. Literature Review and Contributions

Recently, a large and growing body of literature has been focused on the Cartesian trajectory tracking control problem of the quadrotor system using finite-time control, as stated in [12, 13]. In contrast to classical asymptotically stable controllers, finite-time stable control systems ensure fast convergence of the system’s trajectories to the origin along with higher accuracy and enhanced robustness. Motivated by the mentioned works, the finite-time tracking control problem for the quadrotor system is investigated in this study.Multiple works have been reported on the trajectory tracking control problem of the quadrotor helicopter. Linear control such as proportional-integral-derivative (PID) [14–16] and linear quadratic regulator (LQR) [17, 18] have been initially used to design flight control systems for the quadrotor. However, linear control can only ensure good performance around a specific equilibrium point of the linearized model of the quadrotor. In practice, the quadrotor is intended to operate in a challenging flight environment and may exhibit aggressive maneuvers leading to strong nonlinear behavior. Therefore, the linear control seems to be unable to ensure the required flight performance during the mission. Hence, to design an adequate flight control system that can ensure a safe flight for the quadrotor, nonlinear control is considered a reliable tool that can overcome the shortcoming of classical linear control.Backstepping (BS) is Lyapunov’s theory-based recursive and flexible nonlinear control design methodology that can be used to deal with the control problem of high-order nonlinear systems such as the quadrotor aircraft [19]. For instance, the work reported in [20] employs a nonlinear adaptive BS method for a trajectory tracking control of the quadrotor. A robust adaptive BS is designed for the position loop in [21]. However, the BS design technique suffers from three main issues: the “explosion of complexity”, lack of robustness against disturbances, and only asymptotic stability in infinite-time is guaranteed, as reported in [22, 23]. Unfortunately, many reported works, e.g., [21, 24] do not address these issues. In practice, the implementation of a control law with such issues could lead to system instability and mission failure. The present research is aimed at solving the problem inherently present in classical BS control design so as to further improve its performance. In particular, (a) a sliding-mode-based filter (SMBF) is introduced in the recursive BS design to restore the derivative of the virtual control to avoid the “explosion of complexity” problem. (b) Terminal sliding mode control (TSMC) is combined with the BS technique to ensure finite-time stability featuring fast transient response. (c) A disturbance observer is designed to enhance the disturbance rejection capability of the compounded controller.SMC control has been effectively applied for control design and stabilization of a variety of nonlinear dynamical systems such as the underactuated quadrotor system. This is motivated by its robustness against uncertainties and disturbances and simplicity of control design [25–28]. The design procedure of the SMC control systems mainly consists of two steps: the choice of a sliding surface with desirable dynamic characteristics and the design of the SMC controller. The controller is designed such that the system’s states reach and remain on the sliding manifold and consequently converge to the origin. Recently, many works are concerned with the robust control of the quadrotor system subjected to disturbances using SMC theory. As an example of sliding mode combination with BS, the work in [29] presents an integral backstepping sliding mode control (IBSMC) method for a perturbated quadrotor system. In [30], a regular SMC is combined with the BS technique to design a robust nonlinear controller. Nevertheless, all these methods are based on linear sliding mode control (LSMC). The most serious disadvantage of this control approach is that the switching manifold is linear. Thus, only asymptotic convergence can be achieved. Also, LSMC inevitably suffers from the chattering problem. The chattering impact is reflected by the presence of disrupting high switching frequencies in the control input of the system [31]. Such a control signal will cause low control accuracy and degrade the control performance [26]. Furthermore, it can also damage the actuators of the quadrotor (the brushless motors). To deal with the issues of LSMC, this study proposes to use advanced continuous-SMC techniques. The proposed terminal sliding manifold in our work allows that tracking errors are stabilized to the origin in fast finite-time by contrast to the linear switching manifold that can only guarantee asymptotic convergence in an infinite time. Moreover, the control law designed based on our terminal sliding surface is continuous; hence, the chattering problem inherent in SMC and switching control methods can be effectively mitigated, which makes the controller applicable in practice.In order to ensure better tracking control performance, finite-time control is considered in many works. For instance, in the interesting work [32], Mobayen and Ma have innovatively proposed a robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time delay. The same authors have investigated a novel nonsingular fast terminal sliding-mode control method for the stabilization of the uncertain time-varying and nonlinear third-order systems in [25]. A recursive singularity-free fast terminal sliding mode control is used in [33] to design a finite-time tracker for nonholonomic systems including a wheeled mobile robot and an underactuated surface vessel. In [34], by designing an LMI-based sliding mode controller, the state trajectories of a class of underactuated systems are shown to be directed toward the sliding manifold in a finite-time with exponential policy and thereafter remained on it. Although the above studies have interesting and promising results from theoretical and practical aspects, they have not been applied specifically to solve the trajectory tracking problem quadrotor system, which motivates us to further investigate the finite-time control of quadrotor aircraft.In our recently published study [27], a finite-time observer-based robust continuous twisting control is proposed for an uncertain quadrotor system subjected to disturbances. Another work reported in [35] comes up with a modified super twisting algorithm that can ensure robustness and finite-time convergence for the quadrotor helicopter. In the work [36], the tracking errors are driven to zero in finite-time by employing a fractional-order controller based on nonsingular control law. However, these studies are limited to the finite-time control of the 3-DoF attitude dynamics, and they did not address the finite-time control of the full 6-DoF dynamics of the quadrotor. Also, the attitude dynamics are fully actuated with three inputs and three outputs which makes the control design much easier. Our study includes finite-time control for the 6-DoF dynamics, i.e., attitude and position subsystems, where the underactuated problem inherent in the quadrotor dynamics has been addressed.To address the finite-time control of attitude and position of the quadrotor, a modified super twisting fast nonlinear sliding mode controller is proposed to stabilize a quadrotor system under time-varying disturbances in [37], in finite-time. Also, a finite-time trajectory tracking control for a quadrotor aircraft with unknown external disturbances is investigated in [38]. However, together with [21], the works [37, 38] have presented the design of the control laws by considering a perfect model with precise knowledge of the model parameters. In practice, it is difficult to identify and estimate the exact parameters of the quadrotor system, notably, the aerodynamic coefficients. Besides, model uncertainties are inevitably present in the dynamic model of the quadrotor. Therefore, unlike these works, model uncertainties are also considered in our study besides external time-varying disturbances, which is more realistic in practice. Some reported works on finite-time control have considered the model imperfection and uncertainties in the control design; however, they have not provided an upper bound on the settling-time, e.g., [37, 39]. It is well-known that one of the most interesting features of finite-time stability is that an upper bound on the settling-time can be provided and tuned in the function of the control parameters. Therefore, in contrast to [37, 39], we have established a clear estimation of the settling-time during the sliding motion which allows tuning the convergence-time by adjusting the parameters of the sliding surface and the reaching law.In [40], a nonsingular terminal sliding mode control law with finite-time convergence is designed for the quadrotor. In [41], a finite-time convergent nonsingular terminal sliding mode control for a quadrotor with a total rotor failure is proposed. In [42], finite-time adaptive integral backstepping fast terminal sliding mode control is applied for the tracking of the attitude and position of the quadrotor. Although [21, 40–42] provide an estimation of the settling-time, the sign· function appears explicitly in the control law, which may cause the chattering and thus making the controller inapplicable in practice. To reduce the chattering effect, the authors have replaced the sign· function with smooth approximating functions. However, this may degrade the robustness of the controller leading to a chattering-robustness tradeoff. By contrast to that, no smooth functions are required to approximate thesign· function in our control law. Hence, the robustness of the sliding mode is well preserved along with a chattering-free control.Overall, the abovementioned finite-time control methods are limited in terms of performance and their application since full-state measurements are required. The signals provided by the gyroscope and accelerometer sensors for the velocity measurement are affected by noise. This issue may lead to degrading the performance of full-state feedback-based controllers. Also, the sensors are exposed to faults that inevitably compromise the stability of the quadrotor leading to mission failure [43]. To deal with this practical shortcoming of full-state feedback-based finite-time controllers, we propose in our study to design an output-feedback control method (velocity-free control) that contributes to enhancing the quadrotor’s robustness and in estimating the unmeasurable states (velocities) at the same time within an active disturbance rejection control (ADRC) framework. Our approach shows its effectiveness and superiority over classical passive antidisturbance control laws that fail to deal with strong disturbances leading to the degradation of the nominal control’s performance which may threaten the system’s stability, as mentioned in [44, 45].Moreover, most of the reported works on finite-time control of the quadrotor aircraft may look very promising as evidenced by simulation results. However, owing to lack of practical implementation, their real-world significance may be a valid concern. In contrast, the present paper presents design, simulation, and hardware realization of modern robust finite-time control laws. The physical implementation in autopilot hardware while attempting to address the abovementioned problems essentially attributes to the scientific contribution of our work.Motivated by the previously reported studies and inspired by the works developed in [46, 47], this paper focuses on addressing all the issues discussed above by designing a new robust flight-control system to deal with the Cartesian trajectory tracking control problem of a quadrotor system subject to lumped disturbances. To solve the underactuation problem, a hierarchical control structure with a position-attitude loop is adopted. To reject the disturbances and enhance the control system robustness, a TSMBAADC approach is employed in each loop. In this context, a disturbance observer-based control (DOBC) law is designed for the attitude loop while the position loop is controlled by an innovatively designed ADRC scheme. The compounded DOBC control structure integrates a finite-time observer (FTO) and a continuous nonsingular terminal sliding mode control (CNTSMC). In practice, it is more reliable to use the technique of differentiators in the implementation process of the control algorithm. Thus, a tracking differentiator is designed to supply the estimates of desired attitude signals. The ADRC control is designed within an output-feedback scheme to ensure a velocity-free control. It combines a fixed-time extended state observer (FXESO) and a backstepping integral terminal sliding mode control (BSITSMC). To avoid the “explosion of complexity” problem inherent in the classic BS design, a sliding-mode-based filter (SMBF) is introduced in the recursive BS design to restore the derivative of the virtual control. By designing nonlinear continuous terminal sliding surfaces, fast finite-time convergence of the tracking errors is ensured for both rotational and translational subsystems. Lyapunov theorem is used to analyze the stability of the feedback control system. Numerical simulations and PIL experiments are conducted to validate and implement the designed flight control algorithm in real autopilot hardware. Compared with relevant reported works, the suggested control strategy has the overall superiority in practice because (i) null steady-state error is achieved along with enhanced robustness, (ii) chattering and singularity problems of SMC and switching control are avoided, and (iii) proposed sliding manifolds and reaching law allows finite-time tracking stability featuring fast convergence.The remaining manuscript is arranged in four sections detailed here. Section2 presents fundamentals mathematical formulation. Section 3 details the design of the proposed flight control system. Rigorous stability analysis of the feedback loop system is also discussed. Section 4 illustrates implementation results to investigate the theoretical findings. The paper is concluded in Section 5 with possible future research directions.


## 1.1. Context and Motivations

Nowadays, quadrotor aircraft have gained enormous interest due to their numerous merits such as low-cost manufacturability and vertical take-off and landing capability [1–3]. These interesting aircraft have been widely used in many fields to solve practical and complex missions [4–8]. However, despite their advantages, quadrotors have some critical drawbacks that are related to flight control. The quadrotor is a nonlinear multioutput multi-input system with underactuated six degrees of freedom (6-DoF) dynamics [9]. Moreover, the quadrotor’s dynamics are strongly coupled and inevitably affected by multiple disturbances [10, 11]. Meanwhile, the quadrotor is intended to operate in a challenging flight environment where it may perform aggressive maneuvers. Thus, its stabilization and control in such flight conditions is not trivial but a challenging and complex task. The robust trajectory tracking control during flight missions is one of the persistent control problems. This problem has become an important topic that should be carefully addressed by the control community. Particularly, fast convergence, strong robustness, and accuracy are considered important features of a flight control algorithm to safely and effectively drive the quadrotor during the mission. Therefore, a reliable flight control system essentially relying on a modern control technique is required to achieve good tracking performance. Hence, this paper focuses on the design of a new flight-control system to deal with the robust Cartesian trajectory tracking control problem for the quadrotor system.


## 1.2. Literature Review and Contributions

Recently, a large and growing body of literature has been focused on the Cartesian trajectory tracking control problem of the quadrotor system using finite-time control, as stated in [12, 13]. In contrast to classical asymptotically stable controllers, finite-time stable control systems ensure fast convergence of the system’s trajectories to the origin along with higher accuracy and enhanced robustness. Motivated by the mentioned works, the finite-time tracking control problem for the quadrotor system is investigated in this study.Multiple works have been reported on the trajectory tracking control problem of the quadrotor helicopter. Linear control such as proportional-integral-derivative (PID) [14–16] and linear quadratic regulator (LQR) [17, 18] have been initially used to design flight control systems for the quadrotor. However, linear control can only ensure good performance around a specific equilibrium point of the linearized model of the quadrotor. In practice, the quadrotor is intended to operate in a challenging flight environment and may exhibit aggressive maneuvers leading to strong nonlinear behavior. Therefore, the linear control seems to be unable to ensure the required flight performance during the mission. Hence, to design an adequate flight control system that can ensure a safe flight for the quadrotor, nonlinear control is considered a reliable tool that can overcome the shortcoming of classical linear control.Backstepping (BS) is Lyapunov’s theory-based recursive and flexible nonlinear control design methodology that can be used to deal with the control problem of high-order nonlinear systems such as the quadrotor aircraft [19]. For instance, the work reported in [20] employs a nonlinear adaptive BS method for a trajectory tracking control of the quadrotor. A robust adaptive BS is designed for the position loop in [21]. However, the BS design technique suffers from three main issues: the “explosion of complexity”, lack of robustness against disturbances, and only asymptotic stability in infinite-time is guaranteed, as reported in [22, 23]. Unfortunately, many reported works, e.g., [21, 24] do not address these issues. In practice, the implementation of a control law with such issues could lead to system instability and mission failure. The present research is aimed at solving the problem inherently present in classical BS control design so as to further improve its performance. In particular, (a) a sliding-mode-based filter (SMBF) is introduced in the recursive BS design to restore the derivative of the virtual control to avoid the “explosion of complexity” problem. (b) Terminal sliding mode control (TSMC) is combined with the BS technique to ensure finite-time stability featuring fast transient response. (c) A disturbance observer is designed to enhance the disturbance rejection capability of the compounded controller.SMC control has been effectively applied for control design and stabilization of a variety of nonlinear dynamical systems such as the underactuated quadrotor system. This is motivated by its robustness against uncertainties and disturbances and simplicity of control design [25–28]. The design procedure of the SMC control systems mainly consists of two steps: the choice of a sliding surface with desirable dynamic characteristics and the design of the SMC controller. The controller is designed such that the system’s states reach and remain on the sliding manifold and consequently converge to the origin. Recently, many works are concerned with the robust control of the quadrotor system subjected to disturbances using SMC theory. As an example of sliding mode combination with BS, the work in [29] presents an integral backstepping sliding mode control (IBSMC) method for a perturbated quadrotor system. In [30], a regular SMC is combined with the BS technique to design a robust nonlinear controller. Nevertheless, all these methods are based on linear sliding mode control (LSMC). The most serious disadvantage of this control approach is that the switching manifold is linear. Thus, only asymptotic convergence can be achieved. Also, LSMC inevitably suffers from the chattering problem. The chattering impact is reflected by the presence of disrupting high switching frequencies in the control input of the system [31]. Such a control signal will cause low control accuracy and degrade the control performance [26]. Furthermore, it can also damage the actuators of the quadrotor (the brushless motors). To deal with the issues of LSMC, this study proposes to use advanced continuous-SMC techniques. The proposed terminal sliding manifold in our work allows that tracking errors are stabilized to the origin in fast finite-time by contrast to the linear switching manifold that can only guarantee asymptotic convergence in an infinite time. Moreover, the control law designed based on our terminal sliding surface is continuous; hence, the chattering problem inherent in SMC and switching control methods can be effectively mitigated, which makes the controller applicable in practice.In order to ensure better tracking control performance, finite-time control is considered in many works. For instance, in the interesting work [32], Mobayen and Ma have innovatively proposed a robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time delay. The same authors have investigated a novel nonsingular fast terminal sliding-mode control method for the stabilization of the uncertain time-varying and nonlinear third-order systems in [25]. A recursive singularity-free fast terminal sliding mode control is used in [33] to design a finite-time tracker for nonholonomic systems including a wheeled mobile robot and an underactuated surface vessel. In [34], by designing an LMI-based sliding mode controller, the state trajectories of a class of underactuated systems are shown to be directed toward the sliding manifold in a finite-time with exponential policy and thereafter remained on it. Although the above studies have interesting and promising results from theoretical and practical aspects, they have not been applied specifically to solve the trajectory tracking problem quadrotor system, which motivates us to further investigate the finite-time control of quadrotor aircraft.In our recently published study [27], a finite-time observer-based robust continuous twisting control is proposed for an uncertain quadrotor system subjected to disturbances. Another work reported in [35] comes up with a modified super twisting algorithm that can ensure robustness and finite-time convergence for the quadrotor helicopter. In the work [36], the tracking errors are driven to zero in finite-time by employing a fractional-order controller based on nonsingular control law. However, these studies are limited to the finite-time control of the 3-DoF attitude dynamics, and they did not address the finite-time control of the full 6-DoF dynamics of the quadrotor. Also, the attitude dynamics are fully actuated with three inputs and three outputs which makes the control design much easier. Our study includes finite-time control for the 6-DoF dynamics, i.e., attitude and position subsystems, where the underactuated problem inherent in the quadrotor dynamics has been addressed.To address the finite-time control of attitude and position of the quadrotor, a modified super twisting fast nonlinear sliding mode controller is proposed to stabilize a quadrotor system under time-varying disturbances in [37], in finite-time. Also, a finite-time trajectory tracking control for a quadrotor aircraft with unknown external disturbances is investigated in [38]. However, together with [21], the works [37, 38] have presented the design of the control laws by considering a perfect model with precise knowledge of the model parameters. In practice, it is difficult to identify and estimate the exact parameters of the quadrotor system, notably, the aerodynamic coefficients. Besides, model uncertainties are inevitably present in the dynamic model of the quadrotor. Therefore, unlike these works, model uncertainties are also considered in our study besides external time-varying disturbances, which is more realistic in practice. Some reported works on finite-time control have considered the model imperfection and uncertainties in the control design; however, they have not provided an upper bound on the settling-time, e.g., [37, 39]. It is well-known that one of the most interesting features of finite-time stability is that an upper bound on the settling-time can be provided and tuned in the function of the control parameters. Therefore, in contrast to [37, 39], we have established a clear estimation of the settling-time during the sliding motion which allows tuning the convergence-time by adjusting the parameters of the sliding surface and the reaching law.In [40], a nonsingular terminal sliding mode control law with finite-time convergence is designed for the quadrotor. In [41], a finite-time convergent nonsingular terminal sliding mode control for a quadrotor with a total rotor failure is proposed. In [42], finite-time adaptive integral backstepping fast terminal sliding mode control is applied for the tracking of the attitude and position of the quadrotor. Although [21, 40–42] provide an estimation of the settling-time, the sign· function appears explicitly in the control law, which may cause the chattering and thus making the controller inapplicable in practice. To reduce the chattering effect, the authors have replaced the sign· function with smooth approximating functions. However, this may degrade the robustness of the controller leading to a chattering-robustness tradeoff. By contrast to that, no smooth functions are required to approximate thesign· function in our control law. Hence, the robustness of the sliding mode is well preserved along with a chattering-free control.Overall, the abovementioned finite-time control methods are limited in terms of performance and their application since full-state measurements are required. The signals provided by the gyroscope and accelerometer sensors for the velocity measurement are affected by noise. This issue may lead to degrading the performance of full-state feedback-based controllers. Also, the sensors are exposed to faults that inevitably compromise the stability of the quadrotor leading to mission failure [43]. To deal with this practical shortcoming of full-state feedback-based finite-time controllers, we propose in our study to design an output-feedback control method (velocity-free control) that contributes to enhancing the quadrotor’s robustness and in estimating the unmeasurable states (velocities) at the same time within an active disturbance rejection control (ADRC) framework. Our approach shows its effectiveness and superiority over classical passive antidisturbance control laws that fail to deal with strong disturbances leading to the degradation of the nominal control’s performance which may threaten the system’s stability, as mentioned in [44, 45].Moreover, most of the reported works on finite-time control of the quadrotor aircraft may look very promising as evidenced by simulation results. However, owing to lack of practical implementation, their real-world significance may be a valid concern. In contrast, the present paper presents design, simulation, and hardware realization of modern robust finite-time control laws. The physical implementation in autopilot hardware while attempting to address the abovementioned problems essentially attributes to the scientific contribution of our work.Motivated by the previously reported studies and inspired by the works developed in [46, 47], this paper focuses on addressing all the issues discussed above by designing a new robust flight-control system to deal with the Cartesian trajectory tracking control problem of a quadrotor system subject to lumped disturbances. To solve the underactuation problem, a hierarchical control structure with a position-attitude loop is adopted. To reject the disturbances and enhance the control system robustness, a TSMBAADC approach is employed in each loop. In this context, a disturbance observer-based control (DOBC) law is designed for the attitude loop while the position loop is controlled by an innovatively designed ADRC scheme. The compounded DOBC control structure integrates a finite-time observer (FTO) and a continuous nonsingular terminal sliding mode control (CNTSMC). In practice, it is more reliable to use the technique of differentiators in the implementation process of the control algorithm. Thus, a tracking differentiator is designed to supply the estimates of desired attitude signals. The ADRC control is designed within an output-feedback scheme to ensure a velocity-free control. It combines a fixed-time extended state observer (FXESO) and a backstepping integral terminal sliding mode control (BSITSMC). To avoid the “explosion of complexity” problem inherent in the classic BS design, a sliding-mode-based filter (SMBF) is introduced in the recursive BS design to restore the derivative of the virtual control. By designing nonlinear continuous terminal sliding surfaces, fast finite-time convergence of the tracking errors is ensured for both rotational and translational subsystems. Lyapunov theorem is used to analyze the stability of the feedback control system. Numerical simulations and PIL experiments are conducted to validate and implement the designed flight control algorithm in real autopilot hardware. Compared with relevant reported works, the suggested control strategy has the overall superiority in practice because (i) null steady-state error is achieved along with enhanced robustness, (ii) chattering and singularity problems of SMC and switching control are avoided, and (iii) proposed sliding manifolds and reaching law allows finite-time tracking stability featuring fast convergence.The remaining manuscript is arranged in four sections detailed here. Section2 presents fundamentals mathematical formulation. Section 3 details the design of the proposed flight control system. Rigorous stability analysis of the feedback loop system is also discussed. Section 4 illustrates implementation results to investigate the theoretical findings. The paper is concluded in Section 5 with possible future research directions.


## 2. Preliminaries and Problem Formulation

This section presents some relevant mathematical definitions and lemmas employed in the control design and finite-time stability proof. The control problem of our study is also formulated in this section.


### 2.1. Preliminaries

Definition 1 (see [48]) (finite-time stability).
Consider the following autonomous system:(1)ẋ=fx,x0=x0,where x∈ℝn, and the nonlinear function f:D→ℝn is continuous on an open neighborhood D⊆ℝn of the origin. The origin x=0 is an equilibrium point of system (1). x=0 is a globally finite-time convergent, if it is globally asymptotically stable, and there are an open neighborhood U⊆D of the origin and a function Tx:U\0→0,∞such that every solution xt,x0 of system (1) that starts from the initial condition x0∈U\0 is well-defined for t∈0,Txx0, and limt→Txx0xt,x0=0. Here, Txx0 is called the settling-time function, i.e., convergence-time, (w.r.t x0). x=0 is said to be a finite-time stable equilibrium if it is finite-time convergent and Lyapunov stable. If U=D=ℝn, the origin is said to be a globally finite-time stable equilibrium.Lemma 2 (see [49]).
Consider the following system:(2)σ̇0=−ρ0Ld1/n+1σ01/n+1signσ0+σ1,σ̇1=−ρ1Ld1/nσ1−σ̇0n−1/nsignσ1−σ̇0+σ2,⋮σ̇n−1=−ρn−1Ld1/2σn−1−σ̇n−21/2signσn−1−σ̇n−2+σn,σ̇n∈−ρnLdsignσn−σ̇n−1+−Ld,Ld.If the constantsLd,ρi satisfy Ld>0 and ρi>0,i=0,n¯, it results that the above system is finite-time stable.Lemma 3 (see [50]).
If the positive constantskj>0,j=1,n¯make the n-order polynomials sn+knsn−1+⋯+k2s+k1 and sn+3knsn−1+⋯+3k2s+3k1 be Hurwitz in terms of the Laplace operator s, i.e., all their roots are in the left-half plane, the origin of the following system
(3)ẋi=xi+1,i=1,n−1¯,ẋn=−∑j=1nkjxjα1,j+xj+xjα2,j,is finite-time stable equilibrium with uniform settling-time, where α1,j and α2,j are determined according to the bi-limit homogeneity reasoning as follows: α1,n−k=α/k1−α¯+1 and α2,n−k=2−α¯/kα¯−1+1, where α¯∈ϵ,1, ϵ∈n−2/n−1,1, and k=0,n−1¯.Lemma 4 (see [48]).
Suppose that there is a continuous and positive-definite Lyapunov functionVxt:ℝn→ℝ, and its derivative satisfies
(4)V̇xt≤−λVα−μVγ,where λ,μ>0,α>1, and γ<1 are some positive constants, then, the origin of system (1) is fixed-time stable. The settling-time function T0 is bounded by T⋆ as
(5)T0≤T⋆≔1λα−1+1μ1−γ.Lemma 5 (see [51]).
Consider the following system:(6)σ̇1=σ2−ρ1σ0a1−ρ¯1σ0b1,σ̇2=σ3−ρ2σ0a2−ρ¯2σ0b2,⋮σ̇n−1=σn−ρn−1σ0an−1−ρ¯n−1σ0bn−1,σ̇n∈−ρnσ0an−ρ¯nσ0bn+−Ld,Ld,where ai∈0,1, bi>1,i=1,n¯, and Ld>0 are constants. Also, if the nonnegative constants ρi,ρ¯i,i=1,n¯ are assigned to ensure the following A1,A2 matrices are Hurwitz, i.e., every eigenvalue of the matrices has a strictly negative real part:
(7)A1=−ρ110−ρ201−ρ300,A2=−ρ¯110−ρ¯201−ρ¯300.
Then, the system (6) is fixed-time stable, i.e., uniform settling-time w.r.t. initial condition.Lemma 6 (see [52]).
Consider system (1). If there exist C1 Lyapunov functionVx:ⅅ→ℝ+ and some real constants 0<c<∞ and 0<α<1, such that
(8)V̇x≤−cVxα,then, system (1) is finite-time stable for any given xt0∈D0⊆D, in which the finite settling time T∗ satisfies
(9)T∗≤Vx1−αc1−α.


### 2.2. Problem Formulation

The motion of the quadrotor in space can be described by a B-frame and an E-frame (see Figure1). The vehicle moves and changes its attitude by virtue of an appropriate set of angular speeds of the rotors. The rotors generate the lift force denoted by fi∈ℝ+,i=1,4¯.Figure 1
PIL experiment setup: quadrotor model withx-configuration represented in E-frame and B-frame.The E-frame is used to define the translational motion byP≔x,y,zT∈ℝ3. The B-frame indicates the rotational motion, i.e., Euler angles η≔Φ,θ,ψT∈ℝ3. The 6-DoF acceleration dynamics corresponding to the rotation and translation motions of the quadrotor in the presence of disturbances can be written as [48]
(10)Φ¨=Jxx−1Jyy−Jzzθ̇ψ̇−cΦΦ̇2−Jrω¯θ̇+uΦ+dΦext,θ¨=Jyy−1Jzz−JxxΦ̇ψ̇−cθθ̇2+Jrω¯Φ̇+uθ+dθext,ψ¨=Jzz−1Jxx−JyyΦ̇θ̇−cψψ̇2+uψ+dψext,(11)x¨=−m−1cΦsθcψ+sΦsψuz−kxẋ+dxext,y¨=−m−1cΦsθsψ−sΦcψuz−kyẏ+dyext,z¨=−m−1cΦcθuz−kzż+dzext+g.Since the flight of the quadcopter is driven by four propellers, the angular speeds of the four propellersΩi,i=1,4¯ determine the total lift force (thrust control input) uz and the torques uη. The quadrotor’s actuators (rotors) produce a total lift force defined as [53]
(12)uz≔∑j=14fi=ct∑j=14Ωi2.The control torquesuη developed by the quadrotor’s actuators are defined as [54]
(13)uη≔uΦuθuψ=l−f1+f2+f3−f4lf1−f2+f3−f4cd−Ω12−Ω22+Ω32+Ω42.Besides, the rotating velocities of the four propellers, i.e.,Ωi,i=1,4¯, are related to uη (the torques) and uz (the total lift force) by the means of a constant invertible matrix as [55]
(14)uzuη≔uzuΦuθuψ=ctctctct−lctlctlct−lctlct−lctlct−lct−cd−cdcdcdΩ12Ω22Ω32Ω42.Remark 7.
In practice, it is difficult to identify and to estimate the exact parameters of the quadrotor system, notably, the aerodynamic coefficients. Besides, model uncertainties are inevitably present in the dynamic model of the quadrotor. Therefore, in contrast to several reported works, e.g., [21, 37, 38] that consider a perfect model with precise knowledge of the model parameters, both model uncertainties and external time-varying disturbances are considered in the present study. Hence, the following assumption is introduced for the quadrotor dynamics given in (10) and (11).Assumption 8.
We assume in our study that the model uncertainties, i.e., internal unmodeled dynamics, are the aerodynamic and gyroscopic effect moments denoted byτa and τg, i.e.,
(15)dηunc≔dΦuncdθuncdψunc=τa+τg=−cΦΦ̇2−Jrω¯θ̇−cθθ̇2+Jrω¯Φ̇−cψψ̇2.Since it is difficult to identify the aerodynamic coefficientsKa=diagkx,ky,kz in practice, the unmodeled dynamics, for the translational subsystem, are the drag force Fa. These disturbances are defined as
(16)dPunc≔dxuncdyuncdzunc=Fa=−kxvx/m−kyvy/m−kzvz/m.Thus, the unmodeled dynamics for the quadrotor system are viewed as disturbances by the control law.Remark 9.
Since the internal unmodeled dynamics and uncertainties are considered as a part of the total disturbances, Assumption8 will not affect the system stability and control performance. Therefore, the model uncertainties dηunc,dPunc can be dealt with by the FTO and the FXESO, respectively. Thus, the simplifications of the mathematical model adopted in Assumption 8 are reasonable and can be accepted within the proposed TSMBAADC strategy.Finally, by choosingχ≔x,ẋ,y,ẏ,z,ż,Φ,Φ,̇θ,θ,̇ψ,ψ̇∈ℝ12 as a state vector, the following state-space model for the 6-DoF quadrotor dynamics is obtained:
(17)χ̇1=χ2,χ̇2=−m−1cχ7csχ9+sχ7sψuz+dx,χ̇3=χ4,χ̇4=−m−1cχ7sχ11sχ9−cχ11sχ7uz+dy,χ̇5=χ6,χ̇6=−m−1cχ7cχ9uz+dz+g,χ̇7=χ8,χ̇8=Jxx−1Jyy−Jzzχ10χ12+uΦ+dΦ,χ̇9=χ10,χ̇10=Jyy−1Jzz−Jxxχ8χ12+uθ+dθ,χ̇11=χ12,χ̇12=Jzz−1Jxx−Jyyχ8χ10+uψ+dψ.The control problem of our study is formulated mathematically in the following definition.Definition 10.
(Robust finite-time trajectory tracking control problem).
The considered control problem of our study consists of designing robust finite-time TSM control lawsuηt=uΦ,uθ,uψT and uzt for both attitude and position subsystems affected by perturbations in (17), such that
(i)
The attitude and position tracking errors tend to the origin in a fast finite-time, i.e., for∀eηt≔ηt−ηdt,∀ePt≔Pt−Pdt, there exist two constants T1,T3, such that(18)limt→T1eηt=0,∀t>T1,limt→T3ePt=0,∀t>T3,where ηdt and Pdt are the desired reference signals for the attitude and position subsystems, respectively.
(i)
The controller must ensure good robustness against lumped disturbances (model uncertainties, parameter variation, and external time-varying wind disturbances)(ii)
The control signal is nonsingular, continuous, and chattering-free


## 2.1. Preliminaries

Definition 1 (see [48]) (finite-time stability).
Consider the following autonomous system:(1)ẋ=fx,x0=x0,where x∈ℝn, and the nonlinear function f:D→ℝn is continuous on an open neighborhood D⊆ℝn of the origin. The origin x=0 is an equilibrium point of system (1). x=0 is a globally finite-time convergent, if it is globally asymptotically stable, and there are an open neighborhood U⊆D of the origin and a function Tx:U\0→0,∞such that every solution xt,x0 of system (1) that starts from the initial condition x0∈U\0 is well-defined for t∈0,Txx0, and limt→Txx0xt,x0=0. Here, Txx0 is called the settling-time function, i.e., convergence-time, (w.r.t x0). x=0 is said to be a finite-time stable equilibrium if it is finite-time convergent and Lyapunov stable. If U=D=ℝn, the origin is said to be a globally finite-time stable equilibrium.Lemma 2 (see [49]).
Consider the following system:(2)σ̇0=−ρ0Ld1/n+1σ01/n+1signσ0+σ1,σ̇1=−ρ1Ld1/nσ1−σ̇0n−1/nsignσ1−σ̇0+σ2,⋮σ̇n−1=−ρn−1Ld1/2σn−1−σ̇n−21/2signσn−1−σ̇n−2+σn,σ̇n∈−ρnLdsignσn−σ̇n−1+−Ld,Ld.If the constantsLd,ρi satisfy Ld>0 and ρi>0,i=0,n¯, it results that the above system is finite-time stable.Lemma 3 (see [50]).
If the positive constantskj>0,j=1,n¯make the n-order polynomials sn+knsn−1+⋯+k2s+k1 and sn+3knsn−1+⋯+3k2s+3k1 be Hurwitz in terms of the Laplace operator s, i.e., all their roots are in the left-half plane, the origin of the following system
(3)ẋi=xi+1,i=1,n−1¯,ẋn=−∑j=1nkjxjα1,j+xj+xjα2,j,is finite-time stable equilibrium with uniform settling-time, where α1,j and α2,j are determined according to the bi-limit homogeneity reasoning as follows: α1,n−k=α/k1−α¯+1 and α2,n−k=2−α¯/kα¯−1+1, where α¯∈ϵ,1, ϵ∈n−2/n−1,1, and k=0,n−1¯.Lemma 4 (see [48]).
Suppose that there is a continuous and positive-definite Lyapunov functionVxt:ℝn→ℝ, and its derivative satisfies
(4)V̇xt≤−λVα−μVγ,where λ,μ>0,α>1, and γ<1 are some positive constants, then, the origin of system (1) is fixed-time stable. The settling-time function T0 is bounded by T⋆ as
(5)T0≤T⋆≔1λα−1+1μ1−γ.Lemma 5 (see [51]).
Consider the following system:(6)σ̇1=σ2−ρ1σ0a1−ρ¯1σ0b1,σ̇2=σ3−ρ2σ0a2−ρ¯2σ0b2,⋮σ̇n−1=σn−ρn−1σ0an−1−ρ¯n−1σ0bn−1,σ̇n∈−ρnσ0an−ρ¯nσ0bn+−Ld,Ld,where ai∈0,1, bi>1,i=1,n¯, and Ld>0 are constants. Also, if the nonnegative constants ρi,ρ¯i,i=1,n¯ are assigned to ensure the following A1,A2 matrices are Hurwitz, i.e., every eigenvalue of the matrices has a strictly negative real part:
(7)A1=−ρ110−ρ201−ρ300,A2=−ρ¯110−ρ¯201−ρ¯300.
Then, the system (6) is fixed-time stable, i.e., uniform settling-time w.r.t. initial condition.Lemma 6 (see [52]).
Consider system (1). If there exist C1 Lyapunov functionVx:ⅅ→ℝ+ and some real constants 0<c<∞ and 0<α<1, such that
(8)V̇x≤−cVxα,then, system (1) is finite-time stable for any given xt0∈D0⊆D, in which the finite settling time T∗ satisfies
(9)T∗≤Vx1−αc1−α.


## 2.2. Problem Formulation

The motion of the quadrotor in space can be described by a B-frame and an E-frame (see Figure1). The vehicle moves and changes its attitude by virtue of an appropriate set of angular speeds of the rotors. The rotors generate the lift force denoted by fi∈ℝ+,i=1,4¯.Figure 1
PIL experiment setup: quadrotor model withx-configuration represented in E-frame and B-frame.The E-frame is used to define the translational motion byP≔x,y,zT∈ℝ3. The B-frame indicates the rotational motion, i.e., Euler angles η≔Φ,θ,ψT∈ℝ3. The 6-DoF acceleration dynamics corresponding to the rotation and translation motions of the quadrotor in the presence of disturbances can be written as [48]
(10)Φ¨=Jxx−1Jyy−Jzzθ̇ψ̇−cΦΦ̇2−Jrω¯θ̇+uΦ+dΦext,θ¨=Jyy−1Jzz−JxxΦ̇ψ̇−cθθ̇2+Jrω¯Φ̇+uθ+dθext,ψ¨=Jzz−1Jxx−JyyΦ̇θ̇−cψψ̇2+uψ+dψext,(11)x¨=−m−1cΦsθcψ+sΦsψuz−kxẋ+dxext,y¨=−m−1cΦsθsψ−sΦcψuz−kyẏ+dyext,z¨=−m−1cΦcθuz−kzż+dzext+g.Since the flight of the quadcopter is driven by four propellers, the angular speeds of the four propellersΩi,i=1,4¯ determine the total lift force (thrust control input) uz and the torques uη. The quadrotor’s actuators (rotors) produce a total lift force defined as [53]
(12)uz≔∑j=14fi=ct∑j=14Ωi2.The control torquesuη developed by the quadrotor’s actuators are defined as [54]
(13)uη≔uΦuθuψ=l−f1+f2+f3−f4lf1−f2+f3−f4cd−Ω12−Ω22+Ω32+Ω42.Besides, the rotating velocities of the four propellers, i.e.,Ωi,i=1,4¯, are related to uη (the torques) and uz (the total lift force) by the means of a constant invertible matrix as [55]
(14)uzuη≔uzuΦuθuψ=ctctctct−lctlctlct−lctlct−lctlct−lct−cd−cdcdcdΩ12Ω22Ω32Ω42.Remark 7.
In practice, it is difficult to identify and to estimate the exact parameters of the quadrotor system, notably, the aerodynamic coefficients. Besides, model uncertainties are inevitably present in the dynamic model of the quadrotor. Therefore, in contrast to several reported works, e.g., [21, 37, 38] that consider a perfect model with precise knowledge of the model parameters, both model uncertainties and external time-varying disturbances are considered in the present study. Hence, the following assumption is introduced for the quadrotor dynamics given in (10) and (11).Assumption 8.
We assume in our study that the model uncertainties, i.e., internal unmodeled dynamics, are the aerodynamic and gyroscopic effect moments denoted byτa and τg, i.e.,
(15)dηunc≔dΦuncdθuncdψunc=τa+τg=−cΦΦ̇2−Jrω¯θ̇−cθθ̇2+Jrω¯Φ̇−cψψ̇2.Since it is difficult to identify the aerodynamic coefficientsKa=diagkx,ky,kz in practice, the unmodeled dynamics, for the translational subsystem, are the drag force Fa. These disturbances are defined as
(16)dPunc≔dxuncdyuncdzunc=Fa=−kxvx/m−kyvy/m−kzvz/m.Thus, the unmodeled dynamics for the quadrotor system are viewed as disturbances by the control law.Remark 9.
Since the internal unmodeled dynamics and uncertainties are considered as a part of the total disturbances, Assumption8 will not affect the system stability and control performance. Therefore, the model uncertainties dηunc,dPunc can be dealt with by the FTO and the FXESO, respectively. Thus, the simplifications of the mathematical model adopted in Assumption 8 are reasonable and can be accepted within the proposed TSMBAADC strategy.Finally, by choosingχ≔x,ẋ,y,ẏ,z,ż,Φ,Φ,̇θ,θ,̇ψ,ψ̇∈ℝ12 as a state vector, the following state-space model for the 6-DoF quadrotor dynamics is obtained:
(17)χ̇1=χ2,χ̇2=−m−1cχ7csχ9+sχ7sψuz+dx,χ̇3=χ4,χ̇4=−m−1cχ7sχ11sχ9−cχ11sχ7uz+dy,χ̇5=χ6,χ̇6=−m−1cχ7cχ9uz+dz+g,χ̇7=χ8,χ̇8=Jxx−1Jyy−Jzzχ10χ12+uΦ+dΦ,χ̇9=χ10,χ̇10=Jyy−1Jzz−Jxxχ8χ12+uθ+dθ,χ̇11=χ12,χ̇12=Jzz−1Jxx−Jyyχ8χ10+uψ+dψ.The control problem of our study is formulated mathematically in the following definition.Definition 10.
(Robust finite-time trajectory tracking control problem).
The considered control problem of our study consists of designing robust finite-time TSM control lawsuηt=uΦ,uθ,uψT and uzt for both attitude and position subsystems affected by perturbations in (17), such that
(i)
The attitude and position tracking errors tend to the origin in a fast finite-time, i.e., for∀eηt≔ηt−ηdt,∀ePt≔Pt−Pdt, there exist two constants T1,T3, such that(18)limt→T1eηt=0,∀t>T1,limt→T3ePt=0,∀t>T3,where ηdt and Pdt are the desired reference signals for the attitude and position subsystems, respectively.
(i)
The controller must ensure good robustness against lumped disturbances (model uncertainties, parameter variation, and external time-varying wind disturbances)(ii)
The control signal is nonsingular, continuous, and chattering-free


## 3. Flight Control System Design

The control system design is aimed at realizing a robust Cartesian trajectory tracking for the quadrotor system subjected to lumped disturbances. This can be attained through a robust tracking of the position and attitude references.The quadrotor system (17) is a nonlinear system with underactuated dynamics. This system has six output variables x,y,z,Φ,θ,ψ but only four control inputs are available uz,uΦ,uθ,uψ. Notably, the translational movements of the vehicle are directly achieved by the rotational motions. To deal with this problem, the hierarchical control structure is adopted as depicted in Figure 2. In the context of the hierarchical control scheme, the flight control of the quadrotor system is divided into two control loops: an inner loop for the rotational subsystem and an outer loop for the translational subsystem. The inner loop corresponds to the CNTSMC that ensures attitude stability of the quadrotor by controlling the angular variables. The input of this control loop is the reference angles Φd,θd,ψd, and the output is the appropriate roll, pitch, and yaw torques uΦ,uθ,uψ. A BSITSMC is synthesized for the outer loop to achieve robust position tracking. This loop takes the desired position signals xd,yd,zd as input and generates the reference angles Φd,θd for the inner loop and also the total thrust force control uz.Figure 2
Block diagram of the quadrotor aircraft depicting the proposed flight control structure.


### 3.1. Attitude Control Design

In this subsection, a tracking differentiator (TD) is introduced to estimate the desired attitude target signals and their first and second derivatives. Subsequently, the DOBC structure is designed for the rotational subsystem so that the target signals are tracked in finite time.


#### 3.1.1. Tracking Differentiator

In practice, it is more reliable to use the technique of differentiators in the implementation process of the control algorithm. Thus, a TD is designed to supply the estimates ofΦd,θd,Φ̇d,θ̇d, and Φ¨d,θ¨d for the controller. To this end, a robust exact differentiator-based TD (REDBTD) is adopted [56], which is defined as
(19)ϰ^̇d,0=k0ϰd−ϰ∧d,02/3+ϰ^d,1,ϰ^̇d,1=k1ϰd−ϰ∧d,01/3+ϰ^d,2,ϰ^̇d,2=k2signϰd−ϰ^d,0,where ϰd is the signal to be differentiated and ϰd=Φd,θd. Then, ϰ^d,0,ϰ^d,1 and ϰ^d,2 are the finite-time estimates of ϰd,ϰ̇d and ϰ¨d, respectively, i.e., ϰ^d,0→ϰd,ϰ^d,1→ϰ̇d, andϰ^d,2→ϰ¨d.


#### 3.1.2. Disturbance Observer Design for the Attitude Loop

A convenient model is established to facilitate the design of the controller and disturbance observer. Therefore, from the quadrotor model given in (17), the following attitude dynamic model can be established:
(20)Ẋ1t=X2t,Ẋ2t=fηX2,t+gηtuηt+dηt,Y1t=X1t.Here, Xη≔X1,X2T∈ℝ3×2 is the states vector, where X1≔η=χ7,χ9,χ11T=Φ,θ,ψT∈ℝ3,X2≔η̇=Φ̇,θ̇,ψ̇T=χ8,χ10,χ12T∈ℝ3, uη≔uΦ,uθ,uψT∈ℝ3 is the control inputs vector,Y1≔Φ,θ,ψT∈ℝ3is the controlled outputs vector, and the uncertain function dη≔dηunc+dηext stands for the total disturbances. The external disturbances are modeled as sinusoidal signals with different frequencies. Thus, they are considered Lipschitz continuous matched disturbances with bounded derivatives [57]. Hence, the unknown disturbances dη behave as a sufficiently smooth uncertain function with its first-time derivative satisfying ḋη≤lη, where lη is a nonnegative bounded constant, i.e., 0<lη<∞ [51, 58]. The functions fηX2,t,gηt are defined as
(21)fη≔fΦfθfψ=Jxx−1Jyy−Jzzχ10χ12Jyy−1Jzz−Jxxχ8χ12Jzz−1Jxx−Jyyχ8χ10,gη≔gΦgθgψT=Jxx−1Jyy−1Jzz−1T.The model that we have established in (20) is a general description of the attitude system with nonlinear second-order dynamics associated with roll, pitch, and yaw motions in the presence of disturbances.Theorem 11.
Given the attitude dynamic model described in the presence of disturbances (20), an FTO observer is designed as
(22)Γ̇0η=ʌ0+fη+gηuη,ʌ0=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η,Γ̇1η=ʌ1,ʌ1=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η,Γ̇2η=−ρ3ηlηsignΓ2η−ʌ1,where Γ2η is the estimate of dη. Then, the disturbance dη can be precisely identified within a finite-time T0, i.e., dη≡d^η.Proof.
Let us define the observation errors as(23)ℓ1η≔Γ0η−η̇,ℓ2η≔Γ1η−dη,ℓ3η≔Γ2η−ḋη.
By differentiating these errors w.r.t time and substitutingη¨,Γ̇0η,Γ̇1η,Γ̇2η by their expressions, the corresponding error dynamics can be written as
(24)ℓ̇1η=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η−dη,ℓ̇2η=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η−ḋηℓ̇3η=−ρ3ηlηsignΓ2η−ʌ1−d¨η.
Then, by substitutingʌ0,ʌ1 by their expressions, and after some manipulations, we get
(25)ℓ̇1η=−ρ1ηlη1/3ℓ1η2/3signℓ1η+ℓ2η,ℓ̇2η=−ρ2ηlη1/2ℓ2η−ℓ̇1η1/2signℓ2η−ℓ̇1η+ℓ3η,ℓ̇3η=−ρ3ηlηsignℓ3η−ℓ̇2η−d¨η.
Based on Lemma2, the observation errors ℓ1η,ℓ2η,ℓ3η are finite-time convergent to the origin within the time T0, i.e., Γ1η≡d^η.


#### 3.1.3. Finite-Time Continuous Nonsingular Terminal Sliding Mode Control Design

LetX1d≔ηd=χ7d,χ9d,χ11dT=Φd,θd,ψdT∈ℝ3 be the reference attitude and X1 be the actual attitude. Let us define the attitude tracking errors as
(26)eη≔X1−X1d,ϱη≔Ẋ1−Ẋ1d,where eη≔eχ7,eχ9,eχ11T,ϱη≔ėη=ϱχ7,ϱχ9,ϱχ11T. The tracking errors dynamics is given as
(27)ėη=ϱη,ϱ̇η=X¨1−X¨1d.The control objective for the attitude system is to make theX1 states track the desired reference X1d by designing a continuous-SMC law uη=uΦ,uθ,uψT. Then, the tracking errors vector can be stabilized to zero, i.e., eη≡0. Let e1η≔eη and e2η≔ėη. To ensure fast convergence of attitude variables to their reference signals, the following CNTSM surface is proposed for the rotational system (19)
(28)sη≔e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j,where the nonnegative parameters kjη,α1,j, and α2,j are chosen based on Remark 13 which is drawn hereafter. The control action is applied to establish the reaching phase of the sliding surface sη, and the sliding motion on sη=0 is determined as follows [59]:
(29)uη≔ueqη+gη−1urη.This control structure consists of two parts; the termueqη is the equivalent control part, which maintains the variables on the sliding surface, and urη is the reaching control part, which ensures faster convergence. On the one hand, the ueqη control can be obtained from the sliding motion sη=0. Thus, when sη=0, we get
(30)e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j=0,where e¨η≔X¨1−X¨1d. Substituting X¨1 by its expression from (20) into (30), and after some manipulations, the equivalent control can be obtained as
(31)ueqη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d,where η¨d=X¨1d, d^η is estimated by the FTO observer given by (22). On the other hand, theurη control is chosen to ensure the finite-time reaching of the sliding surface, and it is proposed as
(32)u̇nη≔−ληsηξ−μηsηε.Therefore, using (31) and (32), the final attitude control is given as
(33)uη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d+unη.Finally, by replacingη with Φ,θ,ψ, the corresponding roll, pitch, and yaw control laws can be deduced.Remark 12.
In contrast to many reported control laws suffering from the singularity problem, our control law is designed to be singularity-free. It is worth mentioning that since the derivatives of the termsejηα1,j,sηε with fractional power α1,j,ε<1 are not required in the expression of the control law (33), it results that the singularity problem is avoided, i.e., the magnitude of the control signal does not tend to the infinity. The singularity problem may occur when there exists a term with a negative power in the control signal. For example, the singularity will happen if the derivative of ejηα1,j exists in the control, that is, limejη→0dejηα1,j/dt≔limejη→0α1,jejηα1,j−1ėi,jη=∞, since α1,j−1<0.Remark 13.
In this study, rigorous conditions are established for the choice of the control parameters. On one hand, the exponentsα1,j,α2,j,j=1,2¯ are chosen based on homogeneity theory as follows [50]: α1,n−k=α¯/k1−α¯+1 and α2,n−k=2−α¯/kα¯−1+1, where α¯∈ϵ,1, ϵ∈n−2/n−1,1 and k=0,n−1¯. To ensure the finite-time convergence feature, α1,j,α2,j,j=1,2¯ should satisfy α1,1,α1,2<1, and α2,1,α2,2>1. On the other hand, the positive constants kjη>0,j=1,n¯should be selected to make the n-order polynomials sn+knsn−1+⋯+k2s+k1 and sn+3knsn−1+⋯+3k2s+3k1 be Hurwitz in terms of the Laplace operator s. Besides, the positive constants λη,μη>0, ξ>1 and ε<1 are chosen to preadjust a settling-time for the reaching phase of the sliding mode as it is shown in Lemma 4. In addition, the positive parameters of the observer ρiη>0,i=1,3¯ should be selected to satisfy the following condition ρ1η>ρ2η>ρ3η. The parameter lη is the upper bound of the total disturbances. It is a bounded constant 0<lη<∞.


#### 3.1.4. Stability Analysis for the Attitude Closed-Loop System

Theorem 14.
For the nonlinear perturbated attitude system given by (20), if the control law uη is designed by (33), and employing disturbance observer (22), then, the attitude system is finite-time stable within a bounded time T1, i.e., eηt≡0,∀t≥T1.Proof.
The proof of the theorem is based on two consecutive steps. First, we prove that the sliding manifold is reached in finite time. Second, we show that the tracking errors of the attitude system tend to zero along with the sliding manifold in finite-time.
Step 1: By substitutingη¨ from (20) into the sliding surface sη (28), we get
(34)sη=fη+gηuη+dη−η¨d+∑j=1nkjηejηα1,j+ejη+ejηα2,j.
Then, by substituting the designed control lawuη (33) into sη given in (34), the sliding surface dynamics becomes as
(35)sη=urη+dη−d^η.
By Theorem11, we have dη≡d^η for all t≥T0. Hence, we get
(36)sη=urη.
By differentiating (36) and substituting (32), it yields
(37)ṡη=u̇rη=−ληsηξ−μηsηε.
Subsequently, let us define the following positive-definite Lyapunov function:(38)Vηsη≔sη22,
By differentiatingVη and substituting (37), it yields
(39)V̇ηsη=sηṡη=sη−ληsηξ−μηsηε,=−ληsηsηξ−μηsηsηε.
By usingsη=sη and sηsηξ=sηsηξ, we get
(40)V̇ηsη≤−ληsηξ+1−μηsηε+1,≤−ληVηξ+1/2−μηVηε+1/2.
By considering thatξ>1,ε∈0,1, it yields ξ+1/2>1 and ε+1/2<1. According to Lemma 4, the finite-time reaching of sliding surface sη=0 is guaranteed within the following bounded reaching-time Tr≤T⋆≔1/ληξ+1/2−1+1/μη1−ε+1/2.
Step 2: Recalling the errors dynamics for the angular signals given in (27)
(41)ėη=ϱη,ϱ̇η=η¨−η¨d.
We definex¯1≔eη=e1η,x¯2≔ėη=ϱη=e2η and x¯≔x¯1,x¯2T. Consequently, the closed-loop dynamics (41) can be rewritten as
(42)x¯̇1=x¯2,x¯̇2=η¨−η¨d.
Whensη=0, from (28) and considering x¯1=eη,x¯2=ϱη, we have
(43)η¨=η¨d−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
Substituting (43) into error dynamics expressions (42), we get
(44)x¯̇1=x¯2,x¯̇2=−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
According to Lemma3, we can deduce that the tracking error dynamics (44) can be stabilized to zero during the sliding motion sη=0 within a finite bounded time. Thus, there exists a constant Ts such that eη→0 and ϱη→0 for all t≥T1≔T0+Tr+Ts. This completes the proof.


### 3.2. Position Control Design

The synthesis of the position controller is divided into two steps. In the first step, the disturbance observer is addressed; while in the second step, the robust backstepping sliding mode controller is developed.


#### 3.2.1. FXESO Observer Design for the Position Loop

The quadrotor translational system is divided into three second-order subsystems, the altitudez, and the horizontal position x,y. Thus, the following translational system subjected to disturbances can be obtained from (17) as
(45)Ẋ3t=X4t,Ẋ4t=fPX4,t+FPX1,t+dPt,Y2t=X4t.Here, XP≔X3,X4T∈ℝ3×2 is the states vector, where X3≔P=χ1,χ3,χ5T=x,y,zT∈ℝ3,X4≔Ṗ=Y=ẋ,ẏ,żT=χ2,χ6,χ8T∈ℝ3,Y2≔x,y,zT is the controlled outputs vector, and the uncertain function dPt≔dPunc+dPext summarizes the total lumped disturbances including model uncertainty effects, parameters’ uncertainties, and external disturbances, where ḋPt≤lP and 0<lP<∞. The functions fPX4,t,FPX1,t are defined as
(46)fPX4,t≔Fa=KaX4,(47)FP≔FxFyFz=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d−uzm−1cχ11dcχ9d+g.Given the disturbed position dynamic model described in (45), the FXESO observer is designed as
(48)Γ̇1P=Γ2P+ρ1Pℓ1Pa1+ρ¯1Pℓ1Pb1,Γ̇2P=Γ3P+FP+ρ2Pℓ1Pa2+ρ¯2Pℓ1Pb2,Γ̇3P=ρ3Pℓ1Pa3+ρ¯3Pℓ1Pb3+lPsignℓ1P.Then, the disturbances and the velocities can be estimated within a fixed bounded timeT2 as in (52); hereafter, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y. The exponents ai,bi are selected as follows: ai∈0,1,i=1,3¯ satisfy the recurrent relations ai=ia¯−i−1,i=2,3¯, and a1=a¯ where a¯∈1−ε1,1 for a sufficiently small ε1>0. Also, bi>1,i=1,3¯, and bi=ib¯−i−1,i=2,3¯, and b=b¯ where b¯∈1,1+ε2 for a sufficiently small ε2>0. Besides, ℓ1P≔P−P^ is the observation error and FP≔fP+FP.In (48), the terms ℓ1Pai and ℓ1Pbi are continuous and differentiable. However, the term signℓ1P is discontinuous. Thus, to avoid the chattering effect and to realize the observer in practice, the following sigmoid function is used:
(49)sgmfℓ1P≔21+exp−hℓ1P−1,ℓ1P≤δ,signℓ1P,ℓ1P>δ,where h is a constant that is inversely proportional to δ, i.e.,δ=1/h.Theorem 15.
The FXESO observer designed in (48) can precisely estimate the disturbance and the velocities within a fixed bounded time T2, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y.Proof.
Let us define the observation errors as(50)ℓ1P≔P−P^,ℓ2P≔Y−Y^,ℓ3P≔dP−d^P.
By differentiating these errors w.r.t time, it yields(51)ℓ̇1P=ℓ2P−ρ1Pℓ1Pa1−ρ¯1Pℓ1Pb1,ℓ̇2P=ℓ3P−ρ2Pℓ1Pa2−ρ¯2Pℓ1Pb2,ℓ̇3P=ḋP−ρ3Pℓ1Pa3−ρ¯3Pℓ1Pb3−lPsgmfℓ1P.
Based on Lemma5, the observation errors ℓ1P,ℓ2P,ℓ3P are guaranteed to converge to the origin in fixed-time bounded as
(52)T2≤λmaxΛP1Ξ1Λ+1Ξ2σϖ¯σ,where Ξ1=λminQ1/λmaxP1, Ξ2=λminQ2/λmaxP2, Λ=1−a¯, σ=b¯−1, and ϖ≤λminP2. Q1,Q2,P1 and P2 are nonsingular, symmetric, and positive-definite matrices and satisfy P1A1+A1T=Q1,P2A2+A2T=Q2.


#### 3.2.2. Backstepping Integral Terminal Sliding Mode Control Design

The control law is designed for the positionx here. Without loss of generality, the appropriate control laws for the y and z positions can be derived similarly.Letxd be the desired trajectory of the position x. The tracking error is defined as
(53)ex≔x−xd.To ensure fast finite-time convergence to the desired reference signal and to simultaneously ensure null steady-state error, the following integral terminal sliding surface is designed:(54)sx,1≔kx,1ex+kx,2∫exαsignexdt.Let us define the following Lyapunov function:(55)Vex≔ex22.The time derivative ofVex is given as
(56)V̇ex=exėx.In (56), the term ėx can be calculated from the sliding motion ṡx,1=0 as
(57)ėx=−kx,2kx,1exαsignex.Therefore,V̇exbecomes
(58)V̇ex=−kx,2kx,1exαsignexex≤−kx,2kx,1exα+1,≤−kx,2kx,1ex2α+1/2=−2α+1/2kx,2kx,1ex22α+1/2,=−2α+1/2kx,2kx,1Vexα+1/2≤0.Let us defineς and ϑ, respectively, as follows: ς≔2α+1/2kx,2/kx,1 and ϑ≔α+1/2. Then, we can get
(59)V̇ex≤−ςVexϑ.It follows from Lemma6 that the convergence of the tracking errors for the position x can be achieved within a finite-time T3, bounded as
(60)T3≤Vex,01−ϑς1−ϑ,where Vex,0ϑ is the initial value of the Lyapunov function (55). The first time derivative of the sliding surface (54) is given as
(61)ṡx,1=kx,1ėx+kx,2exαsignex.Subsequently, the time derivative of (61) is calculated as
(62)s¨x,1=kx,1e¨x−αkx,22kx,1ex2α−1signex.Putting (61) and (62) together, the following second-order system is established [46]:
(63)ṡx,1=sx,2,ṡx,2=kx,1e¨x−αkx,22kx,1ex2α−1signex.Let us definewx,1 as
(64)wx,1≔∫sx,1dt.To investigate the stability of the sliding surfacesx,1, the following Lyapunov function is defined as
(65)V1≔12sx,12+λx,12wx,12.Then, the time derivative ofV1 can be obtained as
(66)V̇1=sx,1ṡx,1+λx,1wx,1ẇx,1,=sx,1sx,2+λx,1wx,1sx,1.The desiredsx,2, i.e.,sx,2d, should be chosen to make V̇1 negative. Thus, from (66), the sx,2d is given as
(67)sx,2d≔−ξx,1sx,1−λx,1wx,1,where φx,1≔sx,2d is the virtual variable. The derivative of the virtual control signalφx,1, i.e., φ̇x,1, is required for the controller. Therefore, the problem of “explosion of complexity” should be addressed. For this reason, the following SMBF is employed to estimate φ̇x,1:
(68)ż−1=−Θ2L1/3z−12/3+z0−φx,1,ż0=−Θ1L2/3z−11/3+z1,ż1=−Θ0Lz−10,where z0,z1 are the estimates of φx,1 and φ̇x,1, respectively. Then, the derivative φ̇x,1 can be accurately restored in a finite time. Hence, z0≡φx,1 and z1≡φ̇x,1.Let us define the error betweensx,2 and sx,2d as
(69)ϵx,1≔sx,2−sx,2d.Then, equation (69) can be written as
(70)sx,2=sx,2d+ϵx,1=−ξx,1sx,1−λx,1wx,1+ϵx,1.Substituting (70) into (66), the time derivative of Lyapunov function V1 becomes
(71)V̇1=sx,1−ξx,1sx,1−λx,1wx,1+ϵx,1+λx,1wx,1sx,1,=−ξx,1sx,12+sx,1ϵx,1.Next, we define the integral of the error variableϵx,1 as
(72)wx,2≔∫ϵx,1dt.To make both the errorϵx,1 and its integration wx,2 converge to zero, the following Lyapunov function candidate is chosen:
(73)V2≔V1+12ϵx,12+λx,22wx,22,where λx,2∈ℝ+. The time derivative of V2 is given as
(74)V̇2=V̇1+ϵx,1ϵ̇x,1+λx,2wx,2ẇx,2.Using (71) and (72) into (74), it yields
(75)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1ϵ̇x,1=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1e¨x−αkx,22kx,1ex2α−1signex−φ̇x,1.Consideringe¨x≔x¨−x¨d in (75), it becomes
(76)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1fx+Fx+d^x−x¨d−αkx,22kx,1ex2α−1signex−φ̇x,1.Then, using (76) and taking Assumption 8 into consideration, i.e., fx=0, the desired virtual controls Fxd that can stabilize the system is designed as
(77)Fxd≔−d^x−x¨d+1kx,1αkx,22kx,1ex2α−1signex+φ̇x,1−ξx,2ϵx,1−sx,1−λx,2wx,2.Remark 16.
Many reported works on finite-time control do not provide an estimation of the settling-time. In contrast, we have established rigorous proofs to show that the tracking error dynamics can be stabilized to the origin within a finite bounded time where the upper bound of the convergence time is provided.Remark 17.
In contrast to some reported works on finite-time control, no discontinuous terms appear in the control law (77). The term ex2α−1signex in the control law is continuous; hence the chattering effect is avoided in the control input. Moreover, to avoid the singularity problem, the fractional exponent 2α−1 should be tuned to be greater than 0, i.e., 2α−1>0. Hence, the parameter α should be chosen to be greater than 0.5, i.e., α>0.5.Theorem 18.
Given the disturbed translational dynamic model described in (45), the control law designed by (77) can guarantee that the position tracking error ex is finite-time stable.Proof.
By substituting the control law designed in (77) into (76), one gets
(78)V̇2<−ξx,1sx,12−ξx,2ϵx,12≤0.
Hence,V̇2 is negative definite, which implies that sx,1 and ϵx,1 will converge to zero equilibrium. Therefore, when the states are confined on the sliding manifold after the reaching phase, it turns that the position error ex to approach the origin in a finite time during the sliding phase, i.e., when sx,1→0 and ṡx,1→0. This completes the proof.Remark 19.
In the previously reported studies on finite-time control, thesign· function appears explicitly in the control law which may cause the chattering and thus making the controller inapplicable in practice. To reduce the chattering effect, some works adopted approximating functions, i.e., sat· and tanh· functions, to replace the sign· function. However, this may degrade and threaten the robustness of the control law leading to a chattering-robustness tradeoff. In contrast to that, no smooth functions are required to approximate the sign· function in our control law. Hence, the robustness of the sliding mode is well preserved along with a chattering-free control.


#### 3.2.3. Thrust Force Control and Desired Attitude Computation

The thrust forceuz and the desired attitude angles Φd,θd are defined in functions of the designed force control FPd≔Fxd,Fyd,FzdT for the translational subsystem. The physical meaning of the control vector FPd is that it corresponds to the desired forces that make the quadrotor moving along the directions x,y, and z. The thrust force uz constitutes the magnitude of these forces, whereas their corresponding orientations are determined by the desired attitude χ7d=Φd,χ9d=θd,χ11d=ψd, where Φd, θd, and ψd are the desired roll, pitch, and yaw angles, respectively.Recalling the expressions ofFP=Fx,Fy,FzT from (47) and combining them with FPd=Fxd,Fyd,FzdT, we get
(79)Fx=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d≔Fxd,Fy=−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d≔Fyd,Fz=−uzm−1cχ11dcχ9d+g≔Fzd.Therefore, after some algebraic manipulations of the above expressions, the thrust forceuz and the desired roll and pitch angles χ7d=Φd,χ9d=θd for the attitude subsystem can be, respectively, written as
(80)uz≔mFxd2+Fyd2+Fzd−g2,Φd≔arcsin−muzFxdsψd−Fydcψd,θd≔arctan1Fzd−gFxdcψd+Fydsψd.


## 3.1. Attitude Control Design

In this subsection, a tracking differentiator (TD) is introduced to estimate the desired attitude target signals and their first and second derivatives. Subsequently, the DOBC structure is designed for the rotational subsystem so that the target signals are tracked in finite time.


### 3.1.1. Tracking Differentiator

In practice, it is more reliable to use the technique of differentiators in the implementation process of the control algorithm. Thus, a TD is designed to supply the estimates ofΦd,θd,Φ̇d,θ̇d, and Φ¨d,θ¨d for the controller. To this end, a robust exact differentiator-based TD (REDBTD) is adopted [56], which is defined as
(19)ϰ^̇d,0=k0ϰd−ϰ∧d,02/3+ϰ^d,1,ϰ^̇d,1=k1ϰd−ϰ∧d,01/3+ϰ^d,2,ϰ^̇d,2=k2signϰd−ϰ^d,0,where ϰd is the signal to be differentiated and ϰd=Φd,θd. Then, ϰ^d,0,ϰ^d,1 and ϰ^d,2 are the finite-time estimates of ϰd,ϰ̇d and ϰ¨d, respectively, i.e., ϰ^d,0→ϰd,ϰ^d,1→ϰ̇d, andϰ^d,2→ϰ¨d.


### 3.1.2. Disturbance Observer Design for the Attitude Loop

A convenient model is established to facilitate the design of the controller and disturbance observer. Therefore, from the quadrotor model given in (17), the following attitude dynamic model can be established:
(20)Ẋ1t=X2t,Ẋ2t=fηX2,t+gηtuηt+dηt,Y1t=X1t.Here, Xη≔X1,X2T∈ℝ3×2 is the states vector, where X1≔η=χ7,χ9,χ11T=Φ,θ,ψT∈ℝ3,X2≔η̇=Φ̇,θ̇,ψ̇T=χ8,χ10,χ12T∈ℝ3, uη≔uΦ,uθ,uψT∈ℝ3 is the control inputs vector,Y1≔Φ,θ,ψT∈ℝ3is the controlled outputs vector, and the uncertain function dη≔dηunc+dηext stands for the total disturbances. The external disturbances are modeled as sinusoidal signals with different frequencies. Thus, they are considered Lipschitz continuous matched disturbances with bounded derivatives [57]. Hence, the unknown disturbances dη behave as a sufficiently smooth uncertain function with its first-time derivative satisfying ḋη≤lη, where lη is a nonnegative bounded constant, i.e., 0<lη<∞ [51, 58]. The functions fηX2,t,gηt are defined as
(21)fη≔fΦfθfψ=Jxx−1Jyy−Jzzχ10χ12Jyy−1Jzz−Jxxχ8χ12Jzz−1Jxx−Jyyχ8χ10,gη≔gΦgθgψT=Jxx−1Jyy−1Jzz−1T.The model that we have established in (20) is a general description of the attitude system with nonlinear second-order dynamics associated with roll, pitch, and yaw motions in the presence of disturbances.Theorem 11.
Given the attitude dynamic model described in the presence of disturbances (20), an FTO observer is designed as
(22)Γ̇0η=ʌ0+fη+gηuη,ʌ0=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η,Γ̇1η=ʌ1,ʌ1=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η,Γ̇2η=−ρ3ηlηsignΓ2η−ʌ1,where Γ2η is the estimate of dη. Then, the disturbance dη can be precisely identified within a finite-time T0, i.e., dη≡d^η.Proof.
Let us define the observation errors as(23)ℓ1η≔Γ0η−η̇,ℓ2η≔Γ1η−dη,ℓ3η≔Γ2η−ḋη.
By differentiating these errors w.r.t time and substitutingη¨,Γ̇0η,Γ̇1η,Γ̇2η by their expressions, the corresponding error dynamics can be written as
(24)ℓ̇1η=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η−dη,ℓ̇2η=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η−ḋηℓ̇3η=−ρ3ηlηsignΓ2η−ʌ1−d¨η.
Then, by substitutingʌ0,ʌ1 by their expressions, and after some manipulations, we get
(25)ℓ̇1η=−ρ1ηlη1/3ℓ1η2/3signℓ1η+ℓ2η,ℓ̇2η=−ρ2ηlη1/2ℓ2η−ℓ̇1η1/2signℓ2η−ℓ̇1η+ℓ3η,ℓ̇3η=−ρ3ηlηsignℓ3η−ℓ̇2η−d¨η.
Based on Lemma2, the observation errors ℓ1η,ℓ2η,ℓ3η are finite-time convergent to the origin within the time T0, i.e., Γ1η≡d^η.


### 3.1.3. Finite-Time Continuous Nonsingular Terminal Sliding Mode Control Design

LetX1d≔ηd=χ7d,χ9d,χ11dT=Φd,θd,ψdT∈ℝ3 be the reference attitude and X1 be the actual attitude. Let us define the attitude tracking errors as
(26)eη≔X1−X1d,ϱη≔Ẋ1−Ẋ1d,where eη≔eχ7,eχ9,eχ11T,ϱη≔ėη=ϱχ7,ϱχ9,ϱχ11T. The tracking errors dynamics is given as
(27)ėη=ϱη,ϱ̇η=X¨1−X¨1d.The control objective for the attitude system is to make theX1 states track the desired reference X1d by designing a continuous-SMC law uη=uΦ,uθ,uψT. Then, the tracking errors vector can be stabilized to zero, i.e., eη≡0. Let e1η≔eη and e2η≔ėη. To ensure fast convergence of attitude variables to their reference signals, the following CNTSM surface is proposed for the rotational system (19)
(28)sη≔e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j,where the nonnegative parameters kjη,α1,j, and α2,j are chosen based on Remark 13 which is drawn hereafter. The control action is applied to establish the reaching phase of the sliding surface sη, and the sliding motion on sη=0 is determined as follows [59]:
(29)uη≔ueqη+gη−1urη.This control structure consists of two parts; the termueqη is the equivalent control part, which maintains the variables on the sliding surface, and urη is the reaching control part, which ensures faster convergence. On the one hand, the ueqη control can be obtained from the sliding motion sη=0. Thus, when sη=0, we get
(30)e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j=0,where e¨η≔X¨1−X¨1d. Substituting X¨1 by its expression from (20) into (30), and after some manipulations, the equivalent control can be obtained as
(31)ueqη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d,where η¨d=X¨1d, d^η is estimated by the FTO observer given by (22). On the other hand, theurη control is chosen to ensure the finite-time reaching of the sliding surface, and it is proposed as
(32)u̇nη≔−ληsηξ−μηsηε.Therefore, using (31) and (32), the final attitude control is given as
(33)uη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d+unη.Finally, by replacingη with Φ,θ,ψ, the corresponding roll, pitch, and yaw control laws can be deduced.Remark 12.
In contrast to many reported control laws suffering from the singularity problem, our control law is designed to be singularity-free. It is worth mentioning that since the derivatives of the termsejηα1,j,sηε with fractional power α1,j,ε<1 are not required in the expression of the control law (33), it results that the singularity problem is avoided, i.e., the magnitude of the control signal does not tend to the infinity. The singularity problem may occur when there exists a term with a negative power in the control signal. For example, the singularity will happen if the derivative of ejηα1,j exists in the control, that is, limejη→0dejηα1,j/dt≔limejη→0α1,jejηα1,j−1ėi,jη=∞, since α1,j−1<0.Remark 13.
In this study, rigorous conditions are established for the choice of the control parameters. On one hand, the exponentsα1,j,α2,j,j=1,2¯ are chosen based on homogeneity theory as follows [50]: α1,n−k=α¯/k1−α¯+1 and α2,n−k=2−α¯/kα¯−1+1, where α¯∈ϵ,1, ϵ∈n−2/n−1,1 and k=0,n−1¯. To ensure the finite-time convergence feature, α1,j,α2,j,j=1,2¯ should satisfy α1,1,α1,2<1, and α2,1,α2,2>1. On the other hand, the positive constants kjη>0,j=1,n¯should be selected to make the n-order polynomials sn+knsn−1+⋯+k2s+k1 and sn+3knsn−1+⋯+3k2s+3k1 be Hurwitz in terms of the Laplace operator s. Besides, the positive constants λη,μη>0, ξ>1 and ε<1 are chosen to preadjust a settling-time for the reaching phase of the sliding mode as it is shown in Lemma 4. In addition, the positive parameters of the observer ρiη>0,i=1,3¯ should be selected to satisfy the following condition ρ1η>ρ2η>ρ3η. The parameter lη is the upper bound of the total disturbances. It is a bounded constant 0<lη<∞.


### 3.1.4. Stability Analysis for the Attitude Closed-Loop System

Theorem 14.
For the nonlinear perturbated attitude system given by (20), if the control law uη is designed by (33), and employing disturbance observer (22), then, the attitude system is finite-time stable within a bounded time T1, i.e., eηt≡0,∀t≥T1.Proof.
The proof of the theorem is based on two consecutive steps. First, we prove that the sliding manifold is reached in finite time. Second, we show that the tracking errors of the attitude system tend to zero along with the sliding manifold in finite-time.
Step 1: By substitutingη¨ from (20) into the sliding surface sη (28), we get
(34)sη=fη+gηuη+dη−η¨d+∑j=1nkjηejηα1,j+ejη+ejηα2,j.
Then, by substituting the designed control lawuη (33) into sη given in (34), the sliding surface dynamics becomes as
(35)sη=urη+dη−d^η.
By Theorem11, we have dη≡d^η for all t≥T0. Hence, we get
(36)sη=urη.
By differentiating (36) and substituting (32), it yields
(37)ṡη=u̇rη=−ληsηξ−μηsηε.
Subsequently, let us define the following positive-definite Lyapunov function:(38)Vηsη≔sη22,
By differentiatingVη and substituting (37), it yields
(39)V̇ηsη=sηṡη=sη−ληsηξ−μηsηε,=−ληsηsηξ−μηsηsηε.
By usingsη=sη and sηsηξ=sηsηξ, we get
(40)V̇ηsη≤−ληsηξ+1−μηsηε+1,≤−ληVηξ+1/2−μηVηε+1/2.
By considering thatξ>1,ε∈0,1, it yields ξ+1/2>1 and ε+1/2<1. According to Lemma 4, the finite-time reaching of sliding surface sη=0 is guaranteed within the following bounded reaching-time Tr≤T⋆≔1/ληξ+1/2−1+1/μη1−ε+1/2.
Step 2: Recalling the errors dynamics for the angular signals given in (27)
(41)ėη=ϱη,ϱ̇η=η¨−η¨d.
We definex¯1≔eη=e1η,x¯2≔ėη=ϱη=e2η and x¯≔x¯1,x¯2T. Consequently, the closed-loop dynamics (41) can be rewritten as
(42)x¯̇1=x¯2,x¯̇2=η¨−η¨d.
Whensη=0, from (28) and considering x¯1=eη,x¯2=ϱη, we have
(43)η¨=η¨d−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
Substituting (43) into error dynamics expressions (42), we get
(44)x¯̇1=x¯2,x¯̇2=−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
According to Lemma3, we can deduce that the tracking error dynamics (44) can be stabilized to zero during the sliding motion sη=0 within a finite bounded time. Thus, there exists a constant Ts such that eη→0 and ϱη→0 for all t≥T1≔T0+Tr+Ts. This completes the proof.


## 3.1.1. Tracking Differentiator

In practice, it is more reliable to use the technique of differentiators in the implementation process of the control algorithm. Thus, a TD is designed to supply the estimates ofΦd,θd,Φ̇d,θ̇d, and Φ¨d,θ¨d for the controller. To this end, a robust exact differentiator-based TD (REDBTD) is adopted [56], which is defined as
(19)ϰ^̇d,0=k0ϰd−ϰ∧d,02/3+ϰ^d,1,ϰ^̇d,1=k1ϰd−ϰ∧d,01/3+ϰ^d,2,ϰ^̇d,2=k2signϰd−ϰ^d,0,where ϰd is the signal to be differentiated and ϰd=Φd,θd. Then, ϰ^d,0,ϰ^d,1 and ϰ^d,2 are the finite-time estimates of ϰd,ϰ̇d and ϰ¨d, respectively, i.e., ϰ^d,0→ϰd,ϰ^d,1→ϰ̇d, andϰ^d,2→ϰ¨d.


## 3.1.2. Disturbance Observer Design for the Attitude Loop

A convenient model is established to facilitate the design of the controller and disturbance observer. Therefore, from the quadrotor model given in (17), the following attitude dynamic model can be established:
(20)Ẋ1t=X2t,Ẋ2t=fηX2,t+gηtuηt+dηt,Y1t=X1t.Here, Xη≔X1,X2T∈ℝ3×2 is the states vector, where X1≔η=χ7,χ9,χ11T=Φ,θ,ψT∈ℝ3,X2≔η̇=Φ̇,θ̇,ψ̇T=χ8,χ10,χ12T∈ℝ3, uη≔uΦ,uθ,uψT∈ℝ3 is the control inputs vector,Y1≔Φ,θ,ψT∈ℝ3is the controlled outputs vector, and the uncertain function dη≔dηunc+dηext stands for the total disturbances. The external disturbances are modeled as sinusoidal signals with different frequencies. Thus, they are considered Lipschitz continuous matched disturbances with bounded derivatives [57]. Hence, the unknown disturbances dη behave as a sufficiently smooth uncertain function with its first-time derivative satisfying ḋη≤lη, where lη is a nonnegative bounded constant, i.e., 0<lη<∞ [51, 58]. The functions fηX2,t,gηt are defined as
(21)fη≔fΦfθfψ=Jxx−1Jyy−Jzzχ10χ12Jyy−1Jzz−Jxxχ8χ12Jzz−1Jxx−Jyyχ8χ10,gη≔gΦgθgψT=Jxx−1Jyy−1Jzz−1T.The model that we have established in (20) is a general description of the attitude system with nonlinear second-order dynamics associated with roll, pitch, and yaw motions in the presence of disturbances.Theorem 11.
Given the attitude dynamic model described in the presence of disturbances (20), an FTO observer is designed as
(22)Γ̇0η=ʌ0+fη+gηuη,ʌ0=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η,Γ̇1η=ʌ1,ʌ1=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η,Γ̇2η=−ρ3ηlηsignΓ2η−ʌ1,where Γ2η is the estimate of dη. Then, the disturbance dη can be precisely identified within a finite-time T0, i.e., dη≡d^η.Proof.
Let us define the observation errors as(23)ℓ1η≔Γ0η−η̇,ℓ2η≔Γ1η−dη,ℓ3η≔Γ2η−ḋη.
By differentiating these errors w.r.t time and substitutingη¨,Γ̇0η,Γ̇1η,Γ̇2η by their expressions, the corresponding error dynamics can be written as
(24)ℓ̇1η=−ρ1ηlη1/3Γ0η−η̇2/3signΓ0η−η̇+Γ1η−dη,ℓ̇2η=−ρ2ηlη1/2Γ1η−ʌ01/2signΓ1η−ʌ0+Γ2η−ḋηℓ̇3η=−ρ3ηlηsignΓ2η−ʌ1−d¨η.
Then, by substitutingʌ0,ʌ1 by their expressions, and after some manipulations, we get
(25)ℓ̇1η=−ρ1ηlη1/3ℓ1η2/3signℓ1η+ℓ2η,ℓ̇2η=−ρ2ηlη1/2ℓ2η−ℓ̇1η1/2signℓ2η−ℓ̇1η+ℓ3η,ℓ̇3η=−ρ3ηlηsignℓ3η−ℓ̇2η−d¨η.
Based on Lemma2, the observation errors ℓ1η,ℓ2η,ℓ3η are finite-time convergent to the origin within the time T0, i.e., Γ1η≡d^η.


## 3.1.3. Finite-Time Continuous Nonsingular Terminal Sliding Mode Control Design

LetX1d≔ηd=χ7d,χ9d,χ11dT=Φd,θd,ψdT∈ℝ3 be the reference attitude and X1 be the actual attitude. Let us define the attitude tracking errors as
(26)eη≔X1−X1d,ϱη≔Ẋ1−Ẋ1d,where eη≔eχ7,eχ9,eχ11T,ϱη≔ėη=ϱχ7,ϱχ9,ϱχ11T. The tracking errors dynamics is given as
(27)ėη=ϱη,ϱ̇η=X¨1−X¨1d.The control objective for the attitude system is to make theX1 states track the desired reference X1d by designing a continuous-SMC law uη=uΦ,uθ,uψT. Then, the tracking errors vector can be stabilized to zero, i.e., eη≡0. Let e1η≔eη and e2η≔ėη. To ensure fast convergence of attitude variables to their reference signals, the following CNTSM surface is proposed for the rotational system (19)
(28)sη≔e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j,where the nonnegative parameters kjη,α1,j, and α2,j are chosen based on Remark 13 which is drawn hereafter. The control action is applied to establish the reaching phase of the sliding surface sη, and the sliding motion on sη=0 is determined as follows [59]:
(29)uη≔ueqη+gη−1urη.This control structure consists of two parts; the termueqη is the equivalent control part, which maintains the variables on the sliding surface, and urη is the reaching control part, which ensures faster convergence. On the one hand, the ueqη control can be obtained from the sliding motion sη=0. Thus, when sη=0, we get
(30)e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j=0,where e¨η≔X¨1−X¨1d. Substituting X¨1 by its expression from (20) into (30), and after some manipulations, the equivalent control can be obtained as
(31)ueqη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d,where η¨d=X¨1d, d^η is estimated by the FTO observer given by (22). On the other hand, theurη control is chosen to ensure the finite-time reaching of the sliding surface, and it is proposed as
(32)u̇nη≔−ληsηξ−μηsηε.Therefore, using (31) and (32), the final attitude control is given as
(33)uη=gη−1−fη−e¨2η+∑j=1nkjηejηα1,j+ejη+ejηα2,j−d^η+η¨d+unη.Finally, by replacingη with Φ,θ,ψ, the corresponding roll, pitch, and yaw control laws can be deduced.Remark 12.
In contrast to many reported control laws suffering from the singularity problem, our control law is designed to be singularity-free. It is worth mentioning that since the derivatives of the termsejηα1,j,sηε with fractional power α1,j,ε<1 are not required in the expression of the control law (33), it results that the singularity problem is avoided, i.e., the magnitude of the control signal does not tend to the infinity. The singularity problem may occur when there exists a term with a negative power in the control signal. For example, the singularity will happen if the derivative of ejηα1,j exists in the control, that is, limejη→0dejηα1,j/dt≔limejη→0α1,jejηα1,j−1ėi,jη=∞, since α1,j−1<0.Remark 13.
In this study, rigorous conditions are established for the choice of the control parameters. On one hand, the exponentsα1,j,α2,j,j=1,2¯ are chosen based on homogeneity theory as follows [50]: α1,n−k=α¯/k1−α¯+1 and α2,n−k=2−α¯/kα¯−1+1, where α¯∈ϵ,1, ϵ∈n−2/n−1,1 and k=0,n−1¯. To ensure the finite-time convergence feature, α1,j,α2,j,j=1,2¯ should satisfy α1,1,α1,2<1, and α2,1,α2,2>1. On the other hand, the positive constants kjη>0,j=1,n¯should be selected to make the n-order polynomials sn+knsn−1+⋯+k2s+k1 and sn+3knsn−1+⋯+3k2s+3k1 be Hurwitz in terms of the Laplace operator s. Besides, the positive constants λη,μη>0, ξ>1 and ε<1 are chosen to preadjust a settling-time for the reaching phase of the sliding mode as it is shown in Lemma 4. In addition, the positive parameters of the observer ρiη>0,i=1,3¯ should be selected to satisfy the following condition ρ1η>ρ2η>ρ3η. The parameter lη is the upper bound of the total disturbances. It is a bounded constant 0<lη<∞.


## 3.1.4. Stability Analysis for the Attitude Closed-Loop System

Theorem 14.
For the nonlinear perturbated attitude system given by (20), if the control law uη is designed by (33), and employing disturbance observer (22), then, the attitude system is finite-time stable within a bounded time T1, i.e., eηt≡0,∀t≥T1.Proof.
The proof of the theorem is based on two consecutive steps. First, we prove that the sliding manifold is reached in finite time. Second, we show that the tracking errors of the attitude system tend to zero along with the sliding manifold in finite-time.
Step 1: By substitutingη¨ from (20) into the sliding surface sη (28), we get
(34)sη=fη+gηuη+dη−η¨d+∑j=1nkjηejηα1,j+ejη+ejηα2,j.
Then, by substituting the designed control lawuη (33) into sη given in (34), the sliding surface dynamics becomes as
(35)sη=urη+dη−d^η.
By Theorem11, we have dη≡d^η for all t≥T0. Hence, we get
(36)sη=urη.
By differentiating (36) and substituting (32), it yields
(37)ṡη=u̇rη=−ληsηξ−μηsηε.
Subsequently, let us define the following positive-definite Lyapunov function:(38)Vηsη≔sη22,
By differentiatingVη and substituting (37), it yields
(39)V̇ηsη=sηṡη=sη−ληsηξ−μηsηε,=−ληsηsηξ−μηsηsηε.
By usingsη=sη and sηsηξ=sηsηξ, we get
(40)V̇ηsη≤−ληsηξ+1−μηsηε+1,≤−ληVηξ+1/2−μηVηε+1/2.
By considering thatξ>1,ε∈0,1, it yields ξ+1/2>1 and ε+1/2<1. According to Lemma 4, the finite-time reaching of sliding surface sη=0 is guaranteed within the following bounded reaching-time Tr≤T⋆≔1/ληξ+1/2−1+1/μη1−ε+1/2.
Step 2: Recalling the errors dynamics for the angular signals given in (27)
(41)ėη=ϱη,ϱ̇η=η¨−η¨d.
We definex¯1≔eη=e1η,x¯2≔ėη=ϱη=e2η and x¯≔x¯1,x¯2T. Consequently, the closed-loop dynamics (41) can be rewritten as
(42)x¯̇1=x¯2,x¯̇2=η¨−η¨d.
Whensη=0, from (28) and considering x¯1=eη,x¯2=ϱη, we have
(43)η¨=η¨d−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
Substituting (43) into error dynamics expressions (42), we get
(44)x¯̇1=x¯2,x¯̇2=−k1ηx¯1α1,1+x¯1+x¯1α2,1−k2ηx¯2α1,2+x¯2+x¯2α2,2.
According to Lemma3, we can deduce that the tracking error dynamics (44) can be stabilized to zero during the sliding motion sη=0 within a finite bounded time. Thus, there exists a constant Ts such that eη→0 and ϱη→0 for all t≥T1≔T0+Tr+Ts. This completes the proof.


## 3.2. Position Control Design

The synthesis of the position controller is divided into two steps. In the first step, the disturbance observer is addressed; while in the second step, the robust backstepping sliding mode controller is developed.


### 3.2.1. FXESO Observer Design for the Position Loop

The quadrotor translational system is divided into three second-order subsystems, the altitudez, and the horizontal position x,y. Thus, the following translational system subjected to disturbances can be obtained from (17) as
(45)Ẋ3t=X4t,Ẋ4t=fPX4,t+FPX1,t+dPt,Y2t=X4t.Here, XP≔X3,X4T∈ℝ3×2 is the states vector, where X3≔P=χ1,χ3,χ5T=x,y,zT∈ℝ3,X4≔Ṗ=Y=ẋ,ẏ,żT=χ2,χ6,χ8T∈ℝ3,Y2≔x,y,zT is the controlled outputs vector, and the uncertain function dPt≔dPunc+dPext summarizes the total lumped disturbances including model uncertainty effects, parameters’ uncertainties, and external disturbances, where ḋPt≤lP and 0<lP<∞. The functions fPX4,t,FPX1,t are defined as
(46)fPX4,t≔Fa=KaX4,(47)FP≔FxFyFz=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d−uzm−1cχ11dcχ9d+g.Given the disturbed position dynamic model described in (45), the FXESO observer is designed as
(48)Γ̇1P=Γ2P+ρ1Pℓ1Pa1+ρ¯1Pℓ1Pb1,Γ̇2P=Γ3P+FP+ρ2Pℓ1Pa2+ρ¯2Pℓ1Pb2,Γ̇3P=ρ3Pℓ1Pa3+ρ¯3Pℓ1Pb3+lPsignℓ1P.Then, the disturbances and the velocities can be estimated within a fixed bounded timeT2 as in (52); hereafter, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y. The exponents ai,bi are selected as follows: ai∈0,1,i=1,3¯ satisfy the recurrent relations ai=ia¯−i−1,i=2,3¯, and a1=a¯ where a¯∈1−ε1,1 for a sufficiently small ε1>0. Also, bi>1,i=1,3¯, and bi=ib¯−i−1,i=2,3¯, and b=b¯ where b¯∈1,1+ε2 for a sufficiently small ε2>0. Besides, ℓ1P≔P−P^ is the observation error and FP≔fP+FP.In (48), the terms ℓ1Pai and ℓ1Pbi are continuous and differentiable. However, the term signℓ1P is discontinuous. Thus, to avoid the chattering effect and to realize the observer in practice, the following sigmoid function is used:
(49)sgmfℓ1P≔21+exp−hℓ1P−1,ℓ1P≤δ,signℓ1P,ℓ1P>δ,where h is a constant that is inversely proportional to δ, i.e.,δ=1/h.Theorem 15.
The FXESO observer designed in (48) can precisely estimate the disturbance and the velocities within a fixed bounded time T2, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y.Proof.
Let us define the observation errors as(50)ℓ1P≔P−P^,ℓ2P≔Y−Y^,ℓ3P≔dP−d^P.
By differentiating these errors w.r.t time, it yields(51)ℓ̇1P=ℓ2P−ρ1Pℓ1Pa1−ρ¯1Pℓ1Pb1,ℓ̇2P=ℓ3P−ρ2Pℓ1Pa2−ρ¯2Pℓ1Pb2,ℓ̇3P=ḋP−ρ3Pℓ1Pa3−ρ¯3Pℓ1Pb3−lPsgmfℓ1P.
Based on Lemma5, the observation errors ℓ1P,ℓ2P,ℓ3P are guaranteed to converge to the origin in fixed-time bounded as
(52)T2≤λmaxΛP1Ξ1Λ+1Ξ2σϖ¯σ,where Ξ1=λminQ1/λmaxP1, Ξ2=λminQ2/λmaxP2, Λ=1−a¯, σ=b¯−1, and ϖ≤λminP2. Q1,Q2,P1 and P2 are nonsingular, symmetric, and positive-definite matrices and satisfy P1A1+A1T=Q1,P2A2+A2T=Q2.


### 3.2.2. Backstepping Integral Terminal Sliding Mode Control Design

The control law is designed for the positionx here. Without loss of generality, the appropriate control laws for the y and z positions can be derived similarly.Letxd be the desired trajectory of the position x. The tracking error is defined as
(53)ex≔x−xd.To ensure fast finite-time convergence to the desired reference signal and to simultaneously ensure null steady-state error, the following integral terminal sliding surface is designed:(54)sx,1≔kx,1ex+kx,2∫exαsignexdt.Let us define the following Lyapunov function:(55)Vex≔ex22.The time derivative ofVex is given as
(56)V̇ex=exėx.In (56), the term ėx can be calculated from the sliding motion ṡx,1=0 as
(57)ėx=−kx,2kx,1exαsignex.Therefore,V̇exbecomes
(58)V̇ex=−kx,2kx,1exαsignexex≤−kx,2kx,1exα+1,≤−kx,2kx,1ex2α+1/2=−2α+1/2kx,2kx,1ex22α+1/2,=−2α+1/2kx,2kx,1Vexα+1/2≤0.Let us defineς and ϑ, respectively, as follows: ς≔2α+1/2kx,2/kx,1 and ϑ≔α+1/2. Then, we can get
(59)V̇ex≤−ςVexϑ.It follows from Lemma6 that the convergence of the tracking errors for the position x can be achieved within a finite-time T3, bounded as
(60)T3≤Vex,01−ϑς1−ϑ,where Vex,0ϑ is the initial value of the Lyapunov function (55). The first time derivative of the sliding surface (54) is given as
(61)ṡx,1=kx,1ėx+kx,2exαsignex.Subsequently, the time derivative of (61) is calculated as
(62)s¨x,1=kx,1e¨x−αkx,22kx,1ex2α−1signex.Putting (61) and (62) together, the following second-order system is established [46]:
(63)ṡx,1=sx,2,ṡx,2=kx,1e¨x−αkx,22kx,1ex2α−1signex.Let us definewx,1 as
(64)wx,1≔∫sx,1dt.To investigate the stability of the sliding surfacesx,1, the following Lyapunov function is defined as
(65)V1≔12sx,12+λx,12wx,12.Then, the time derivative ofV1 can be obtained as
(66)V̇1=sx,1ṡx,1+λx,1wx,1ẇx,1,=sx,1sx,2+λx,1wx,1sx,1.The desiredsx,2, i.e.,sx,2d, should be chosen to make V̇1 negative. Thus, from (66), the sx,2d is given as
(67)sx,2d≔−ξx,1sx,1−λx,1wx,1,where φx,1≔sx,2d is the virtual variable. The derivative of the virtual control signalφx,1, i.e., φ̇x,1, is required for the controller. Therefore, the problem of “explosion of complexity” should be addressed. For this reason, the following SMBF is employed to estimate φ̇x,1:
(68)ż−1=−Θ2L1/3z−12/3+z0−φx,1,ż0=−Θ1L2/3z−11/3+z1,ż1=−Θ0Lz−10,where z0,z1 are the estimates of φx,1 and φ̇x,1, respectively. Then, the derivative φ̇x,1 can be accurately restored in a finite time. Hence, z0≡φx,1 and z1≡φ̇x,1.Let us define the error betweensx,2 and sx,2d as
(69)ϵx,1≔sx,2−sx,2d.Then, equation (69) can be written as
(70)sx,2=sx,2d+ϵx,1=−ξx,1sx,1−λx,1wx,1+ϵx,1.Substituting (70) into (66), the time derivative of Lyapunov function V1 becomes
(71)V̇1=sx,1−ξx,1sx,1−λx,1wx,1+ϵx,1+λx,1wx,1sx,1,=−ξx,1sx,12+sx,1ϵx,1.Next, we define the integral of the error variableϵx,1 as
(72)wx,2≔∫ϵx,1dt.To make both the errorϵx,1 and its integration wx,2 converge to zero, the following Lyapunov function candidate is chosen:
(73)V2≔V1+12ϵx,12+λx,22wx,22,where λx,2∈ℝ+. The time derivative of V2 is given as
(74)V̇2=V̇1+ϵx,1ϵ̇x,1+λx,2wx,2ẇx,2.Using (71) and (72) into (74), it yields
(75)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1ϵ̇x,1=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1e¨x−αkx,22kx,1ex2α−1signex−φ̇x,1.Consideringe¨x≔x¨−x¨d in (75), it becomes
(76)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1fx+Fx+d^x−x¨d−αkx,22kx,1ex2α−1signex−φ̇x,1.Then, using (76) and taking Assumption 8 into consideration, i.e., fx=0, the desired virtual controls Fxd that can stabilize the system is designed as
(77)Fxd≔−d^x−x¨d+1kx,1αkx,22kx,1ex2α−1signex+φ̇x,1−ξx,2ϵx,1−sx,1−λx,2wx,2.Remark 16.
Many reported works on finite-time control do not provide an estimation of the settling-time. In contrast, we have established rigorous proofs to show that the tracking error dynamics can be stabilized to the origin within a finite bounded time where the upper bound of the convergence time is provided.Remark 17.
In contrast to some reported works on finite-time control, no discontinuous terms appear in the control law (77). The term ex2α−1signex in the control law is continuous; hence the chattering effect is avoided in the control input. Moreover, to avoid the singularity problem, the fractional exponent 2α−1 should be tuned to be greater than 0, i.e., 2α−1>0. Hence, the parameter α should be chosen to be greater than 0.5, i.e., α>0.5.Theorem 18.
Given the disturbed translational dynamic model described in (45), the control law designed by (77) can guarantee that the position tracking error ex is finite-time stable.Proof.
By substituting the control law designed in (77) into (76), one gets
(78)V̇2<−ξx,1sx,12−ξx,2ϵx,12≤0.
Hence,V̇2 is negative definite, which implies that sx,1 and ϵx,1 will converge to zero equilibrium. Therefore, when the states are confined on the sliding manifold after the reaching phase, it turns that the position error ex to approach the origin in a finite time during the sliding phase, i.e., when sx,1→0 and ṡx,1→0. This completes the proof.Remark 19.
In the previously reported studies on finite-time control, thesign· function appears explicitly in the control law which may cause the chattering and thus making the controller inapplicable in practice. To reduce the chattering effect, some works adopted approximating functions, i.e., sat· and tanh· functions, to replace the sign· function. However, this may degrade and threaten the robustness of the control law leading to a chattering-robustness tradeoff. In contrast to that, no smooth functions are required to approximate the sign· function in our control law. Hence, the robustness of the sliding mode is well preserved along with a chattering-free control.


### 3.2.3. Thrust Force Control and Desired Attitude Computation

The thrust forceuz and the desired attitude angles Φd,θd are defined in functions of the designed force control FPd≔Fxd,Fyd,FzdT for the translational subsystem. The physical meaning of the control vector FPd is that it corresponds to the desired forces that make the quadrotor moving along the directions x,y, and z. The thrust force uz constitutes the magnitude of these forces, whereas their corresponding orientations are determined by the desired attitude χ7d=Φd,χ9d=θd,χ11d=ψd, where Φd, θd, and ψd are the desired roll, pitch, and yaw angles, respectively.Recalling the expressions ofFP=Fx,Fy,FzT from (47) and combining them with FPd=Fxd,Fyd,FzdT, we get
(79)Fx=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d≔Fxd,Fy=−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d≔Fyd,Fz=−uzm−1cχ11dcχ9d+g≔Fzd.Therefore, after some algebraic manipulations of the above expressions, the thrust forceuz and the desired roll and pitch angles χ7d=Φd,χ9d=θd for the attitude subsystem can be, respectively, written as
(80)uz≔mFxd2+Fyd2+Fzd−g2,Φd≔arcsin−muzFxdsψd−Fydcψd,θd≔arctan1Fzd−gFxdcψd+Fydsψd.


## 3.2.1. FXESO Observer Design for the Position Loop

The quadrotor translational system is divided into three second-order subsystems, the altitudez, and the horizontal position x,y. Thus, the following translational system subjected to disturbances can be obtained from (17) as
(45)Ẋ3t=X4t,Ẋ4t=fPX4,t+FPX1,t+dPt,Y2t=X4t.Here, XP≔X3,X4T∈ℝ3×2 is the states vector, where X3≔P=χ1,χ3,χ5T=x,y,zT∈ℝ3,X4≔Ṗ=Y=ẋ,ẏ,żT=χ2,χ6,χ8T∈ℝ3,Y2≔x,y,zT is the controlled outputs vector, and the uncertain function dPt≔dPunc+dPext summarizes the total lumped disturbances including model uncertainty effects, parameters’ uncertainties, and external disturbances, where ḋPt≤lP and 0<lP<∞. The functions fPX4,t,FPX1,t are defined as
(46)fPX4,t≔Fa=KaX4,(47)FP≔FxFyFz=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d−uzm−1cχ11dcχ9d+g.Given the disturbed position dynamic model described in (45), the FXESO observer is designed as
(48)Γ̇1P=Γ2P+ρ1Pℓ1Pa1+ρ¯1Pℓ1Pb1,Γ̇2P=Γ3P+FP+ρ2Pℓ1Pa2+ρ¯2Pℓ1Pb2,Γ̇3P=ρ3Pℓ1Pa3+ρ¯3Pℓ1Pb3+lPsignℓ1P.Then, the disturbances and the velocities can be estimated within a fixed bounded timeT2 as in (52); hereafter, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y. The exponents ai,bi are selected as follows: ai∈0,1,i=1,3¯ satisfy the recurrent relations ai=ia¯−i−1,i=2,3¯, and a1=a¯ where a¯∈1−ε1,1 for a sufficiently small ε1>0. Also, bi>1,i=1,3¯, and bi=ib¯−i−1,i=2,3¯, and b=b¯ where b¯∈1,1+ε2 for a sufficiently small ε2>0. Besides, ℓ1P≔P−P^ is the observation error and FP≔fP+FP.In (48), the terms ℓ1Pai and ℓ1Pbi are continuous and differentiable. However, the term signℓ1P is discontinuous. Thus, to avoid the chattering effect and to realize the observer in practice, the following sigmoid function is used:
(49)sgmfℓ1P≔21+exp−hℓ1P−1,ℓ1P≤δ,signℓ1P,ℓ1P>δ,where h is a constant that is inversely proportional to δ, i.e.,δ=1/h.Theorem 15.
The FXESO observer designed in (48) can precisely estimate the disturbance and the velocities within a fixed bounded time T2, i.e., Γ3P≔d^P≡dP and Γ2P≔Y^≡Y.Proof.
Let us define the observation errors as(50)ℓ1P≔P−P^,ℓ2P≔Y−Y^,ℓ3P≔dP−d^P.
By differentiating these errors w.r.t time, it yields(51)ℓ̇1P=ℓ2P−ρ1Pℓ1Pa1−ρ¯1Pℓ1Pb1,ℓ̇2P=ℓ3P−ρ2Pℓ1Pa2−ρ¯2Pℓ1Pb2,ℓ̇3P=ḋP−ρ3Pℓ1Pa3−ρ¯3Pℓ1Pb3−lPsgmfℓ1P.
Based on Lemma5, the observation errors ℓ1P,ℓ2P,ℓ3P are guaranteed to converge to the origin in fixed-time bounded as
(52)T2≤λmaxΛP1Ξ1Λ+1Ξ2σϖ¯σ,where Ξ1=λminQ1/λmaxP1, Ξ2=λminQ2/λmaxP2, Λ=1−a¯, σ=b¯−1, and ϖ≤λminP2. Q1,Q2,P1 and P2 are nonsingular, symmetric, and positive-definite matrices and satisfy P1A1+A1T=Q1,P2A2+A2T=Q2.


## 3.2.2. Backstepping Integral Terminal Sliding Mode Control Design

The control law is designed for the positionx here. Without loss of generality, the appropriate control laws for the y and z positions can be derived similarly.Letxd be the desired trajectory of the position x. The tracking error is defined as
(53)ex≔x−xd.To ensure fast finite-time convergence to the desired reference signal and to simultaneously ensure null steady-state error, the following integral terminal sliding surface is designed:(54)sx,1≔kx,1ex+kx,2∫exαsignexdt.Let us define the following Lyapunov function:(55)Vex≔ex22.The time derivative ofVex is given as
(56)V̇ex=exėx.In (56), the term ėx can be calculated from the sliding motion ṡx,1=0 as
(57)ėx=−kx,2kx,1exαsignex.Therefore,V̇exbecomes
(58)V̇ex=−kx,2kx,1exαsignexex≤−kx,2kx,1exα+1,≤−kx,2kx,1ex2α+1/2=−2α+1/2kx,2kx,1ex22α+1/2,=−2α+1/2kx,2kx,1Vexα+1/2≤0.Let us defineς and ϑ, respectively, as follows: ς≔2α+1/2kx,2/kx,1 and ϑ≔α+1/2. Then, we can get
(59)V̇ex≤−ςVexϑ.It follows from Lemma6 that the convergence of the tracking errors for the position x can be achieved within a finite-time T3, bounded as
(60)T3≤Vex,01−ϑς1−ϑ,where Vex,0ϑ is the initial value of the Lyapunov function (55). The first time derivative of the sliding surface (54) is given as
(61)ṡx,1=kx,1ėx+kx,2exαsignex.Subsequently, the time derivative of (61) is calculated as
(62)s¨x,1=kx,1e¨x−αkx,22kx,1ex2α−1signex.Putting (61) and (62) together, the following second-order system is established [46]:
(63)ṡx,1=sx,2,ṡx,2=kx,1e¨x−αkx,22kx,1ex2α−1signex.Let us definewx,1 as
(64)wx,1≔∫sx,1dt.To investigate the stability of the sliding surfacesx,1, the following Lyapunov function is defined as
(65)V1≔12sx,12+λx,12wx,12.Then, the time derivative ofV1 can be obtained as
(66)V̇1=sx,1ṡx,1+λx,1wx,1ẇx,1,=sx,1sx,2+λx,1wx,1sx,1.The desiredsx,2, i.e.,sx,2d, should be chosen to make V̇1 negative. Thus, from (66), the sx,2d is given as
(67)sx,2d≔−ξx,1sx,1−λx,1wx,1,where φx,1≔sx,2d is the virtual variable. The derivative of the virtual control signalφx,1, i.e., φ̇x,1, is required for the controller. Therefore, the problem of “explosion of complexity” should be addressed. For this reason, the following SMBF is employed to estimate φ̇x,1:
(68)ż−1=−Θ2L1/3z−12/3+z0−φx,1,ż0=−Θ1L2/3z−11/3+z1,ż1=−Θ0Lz−10,where z0,z1 are the estimates of φx,1 and φ̇x,1, respectively. Then, the derivative φ̇x,1 can be accurately restored in a finite time. Hence, z0≡φx,1 and z1≡φ̇x,1.Let us define the error betweensx,2 and sx,2d as
(69)ϵx,1≔sx,2−sx,2d.Then, equation (69) can be written as
(70)sx,2=sx,2d+ϵx,1=−ξx,1sx,1−λx,1wx,1+ϵx,1.Substituting (70) into (66), the time derivative of Lyapunov function V1 becomes
(71)V̇1=sx,1−ξx,1sx,1−λx,1wx,1+ϵx,1+λx,1wx,1sx,1,=−ξx,1sx,12+sx,1ϵx,1.Next, we define the integral of the error variableϵx,1 as
(72)wx,2≔∫ϵx,1dt.To make both the errorϵx,1 and its integration wx,2 converge to zero, the following Lyapunov function candidate is chosen:
(73)V2≔V1+12ϵx,12+λx,22wx,22,where λx,2∈ℝ+. The time derivative of V2 is given as
(74)V̇2=V̇1+ϵx,1ϵ̇x,1+λx,2wx,2ẇx,2.Using (71) and (72) into (74), it yields
(75)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1ϵ̇x,1=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1e¨x−αkx,22kx,1ex2α−1signex−φ̇x,1.Consideringe¨x≔x¨−x¨d in (75), it becomes
(76)V̇2=−ξx,1sx,12+sx,1ϵx,1+λx,2wx,2ϵx,1+ϵx,1kx,1fx+Fx+d^x−x¨d−αkx,22kx,1ex2α−1signex−φ̇x,1.Then, using (76) and taking Assumption 8 into consideration, i.e., fx=0, the desired virtual controls Fxd that can stabilize the system is designed as
(77)Fxd≔−d^x−x¨d+1kx,1αkx,22kx,1ex2α−1signex+φ̇x,1−ξx,2ϵx,1−sx,1−λx,2wx,2.Remark 16.
Many reported works on finite-time control do not provide an estimation of the settling-time. In contrast, we have established rigorous proofs to show that the tracking error dynamics can be stabilized to the origin within a finite bounded time where the upper bound of the convergence time is provided.Remark 17.
In contrast to some reported works on finite-time control, no discontinuous terms appear in the control law (77). The term ex2α−1signex in the control law is continuous; hence the chattering effect is avoided in the control input. Moreover, to avoid the singularity problem, the fractional exponent 2α−1 should be tuned to be greater than 0, i.e., 2α−1>0. Hence, the parameter α should be chosen to be greater than 0.5, i.e., α>0.5.Theorem 18.
Given the disturbed translational dynamic model described in (45), the control law designed by (77) can guarantee that the position tracking error ex is finite-time stable.Proof.
By substituting the control law designed in (77) into (76), one gets
(78)V̇2<−ξx,1sx,12−ξx,2ϵx,12≤0.
Hence,V̇2 is negative definite, which implies that sx,1 and ϵx,1 will converge to zero equilibrium. Therefore, when the states are confined on the sliding manifold after the reaching phase, it turns that the position error ex to approach the origin in a finite time during the sliding phase, i.e., when sx,1→0 and ṡx,1→0. This completes the proof.Remark 19.
In the previously reported studies on finite-time control, thesign· function appears explicitly in the control law which may cause the chattering and thus making the controller inapplicable in practice. To reduce the chattering effect, some works adopted approximating functions, i.e., sat· and tanh· functions, to replace the sign· function. However, this may degrade and threaten the robustness of the control law leading to a chattering-robustness tradeoff. In contrast to that, no smooth functions are required to approximate the sign· function in our control law. Hence, the robustness of the sliding mode is well preserved along with a chattering-free control.


## 3.2.3. Thrust Force Control and Desired Attitude Computation

The thrust forceuz and the desired attitude angles Φd,θd are defined in functions of the designed force control FPd≔Fxd,Fyd,FzdT for the translational subsystem. The physical meaning of the control vector FPd is that it corresponds to the desired forces that make the quadrotor moving along the directions x,y, and z. The thrust force uz constitutes the magnitude of these forces, whereas their corresponding orientations are determined by the desired attitude χ7d=Φd,χ9d=θd,χ11d=ψd, where Φd, θd, and ψd are the desired roll, pitch, and yaw angles, respectively.Recalling the expressions ofFP=Fx,Fy,FzT from (47) and combining them with FPd=Fxd,Fyd,FzdT, we get
(79)Fx=−uzm−1cχ7dcχ11dsχ9d+sχ7dsχ11d≔Fxd,Fy=−uzm−1cχ7dsχ11dsχ9d−cχ11dsχ7d≔Fyd,Fz=−uzm−1cχ11dcχ9d+g≔Fzd.Therefore, after some algebraic manipulations of the above expressions, the thrust forceuz and the desired roll and pitch angles χ7d=Φd,χ9d=θd for the attitude subsystem can be, respectively, written as
(80)uz≔mFxd2+Fyd2+Fzd−g2,Φd≔arcsin−muzFxdsψd−Fydcψd,θd≔arctan1Fzd−gFxdcψd+Fydsψd.


## 4. Results and Discussions

To verify the effectiveness of our proposed flight control system, numerical simulations, as well as hardware implementation on a real autopilot, have been conducted. In addition, a comparative study has been carried out. In the following, the software and hardware setup for simulation and implementation is explained.


### 4.1. Software and Hardware Configuration

The procedure that we have followed to verify and implement the proposed control algorithm is known as the model-based design (MBD) framework (see Figure3) [48]. This procedure is accomplished in two steps. First, the control algorithm is simulated in MATLAB®/Simulink®. Second, the control algorithm is simulated as a C++ code in the real autopilot hardware (PIL simulation). To facilitate the implementation process and conduct the PIL experiment, a specific MATLAB® toolbox called “UAV Toolbox Support Package for PX4® Autopilots” is used. This toolbox allows to automatically generate C++ code for the controller from the MATLAB® code and also to deploy the generated code to the autopilot. This toolbox also allows the communication between the autopilot and Simulink® for data visualization and monitoring by using a USB wire. The control inputs computed by the Pixhawk flight control board are sent back to Simulink® to control the quadrotor model. The hardware set up for the PIL experiment is shown in Figure 1. In both simulation steps (Simulink®, PIL), the position and orientation of the quadrotor are sent to the FlightGear® flight simulator to visualize the flight of the quadrotor in a realistic environment. MATLAB® code and C++ code of the control algorithm are represented by two different blocks in Simulink®. We can switch the choice between these two blocks as can be seen in Figure 3.Figure 3
Block diagram of the implementation process from the simulation phase to the real autopilot hardware following an MBD procedure.The Pixhawk® autopilot includes a main 32-bit STM32F427 system-on-chip, 180 MHz, ARM® Cortex® M4 CPU, 2 MB flash, and 256 KB SRAM with L1 cache. Moreover, STM32F100 value line microcontroller with ARM® Cortex® M3 CPU running on 24 MHz and an SRAM of 8 KB is also included. The processor is running NuttX real-time operating system (RTOS). Besides, the autopilot board includes the following sensors: inertial measurement unit (IMU), Invensense MPU 6000 3-axis accelerometer/gyroscope, ST Micro L3GD20H 16-bit gyroscope, MEAS MS5611 barometer, and ST Micro LSM303D 14-bit accelerometer/magnetometer.Remark 20.
Most of the reported works on finite-time control of the quadrotor aircraft may look very promising as evidenced by simulation results. However, owing to lack of practical implementation, their real-world significance may be a valid concern. In contrast, our work presents design, simulation, and hardware realization of modern robust finite-time control laws. The physical implementation in autopilot hardware while attempting to address the abovementioned problems essentially attributes to the scientific contribution of our work.


### 4.2. Trajectory Tracking Results and Robustness Test

The numerical simulations and PIL experiments are conducted by using the Euler integration method with a fixed integration step equal to4×10−2 s. The parameters of the quadrotor model are based on a real experimental platform where we have identified them experimentally. These parameters are summarized in Table 1.Table 1
Physical parameters of the quadrotor aircraft.
ParameterValueUnitm1.72kgl0.225mg9.81m/s2Jxx0.0232kg.m2Jyy0.0249kg.m2Jzz0.0342kg.m2We have estimated the moments of inertiaJxx,Jyy,Jzz by the bifilar pendulum experiment as shown in Figure 4. The moment of inertia is calculated by (81) as
(81)JPP≔mgdw2Toscil2/16π2lwkg.m2,where JPP=Jxx,Jyy,Jzz, lw is the length of the two suspending wires (filars) in meters, dw is the distance between the two wires in meters, and Toscil is the period of one oscillation in seconds. The period of 10 oscillations is measured using a stopwatch for 5 trials with an initial swing angle equal to 10 degrees. The average value was then used to calculate Toscil.Figure 4
Bifilar pendulum setup for the estimation of the moments of inertia of the quadrotor: (a)Jxx estimation and (b) Jzz estimation.
(a)(b)The parameters of the CNTSMC controller are chosen ask1Φ=k2Φ=10,k1θ=k2θ=10,k1ψ=k2ψ=15, λη=3.5,μη=12. The exponents of the attitude controller are tuned by considering α¯=0.85 and ξ=1.5,ε=0.001. The parameters of the FTO observer are set to be ρ1η=20,ρ2η=15,ρ3η=2,lη=3. The parameters of the BSITSMC controller are chosen to be ξi,1=0.5,kx,1=0.25,ξi,2=1.2,ky,1=0.6,,λi,1=0.012,kz,1=0.71,kx,2=0.15,,α=0.8,ky,2=0.12,λi,2=0.01,kz,2=0.2. The parameters of SMBF are Θ0=1.2,Θ1=1.6,Θ2=2.5,L=1.5,r=3. Whereas the FXESO observer gains are selected to be ε1=0.26,a¯=0.74,ε2=0.2,b¯=1.2,ρix=ρ¯ix=ρiΦ=ρ¯iΦ=5,ρiy=ρ¯iy=ρiθ=ρ¯iθ=ρiz=10,ρ¯iz=ρiψ=ρ¯iψ=12,lP=5.To highlight the improvement attained with the proposed control strategy, a comparative study is conducted among four different flight control systems (FCS) (see Table2). FCS 1 is the default control system of the Pixhawk® autopilot which is based on the PID controller, FCS 2 is based on an adaptive nonsingular fast terminal sliding mode control (ANFTSMC), FCS 3 is based on IBSMC, and FCS 4 is our proposed flight control system.Table 2
Flight control systems compared for the Cartesian trajectory tracking.
Flight control systemPosition controllerAttitude controllerFCS 1 (Pixhawk®)PIDPIDFCS 2 [53]ANFTSMCANFTSMCFCS 3 [29]IBSMCIBSMCFCS 4 (proposed)BSITSMC-FXESOCNTSMC-FTOTo better evaluate the FCSs, two different flight scenarios for the Cartesian trajectory tracking are performed. To test the robustness and disturbance rejection capabilities of the FCSs, Cartesian trajectory tracking control has been conducted under the influence of parameter variation and internal model uncertainties as well as external time-varying wind disturbances. This approach is more realistic in practice since all these disturbances and uncertainties are simultaneously present in the dynamic model of the quadrotor. The overall objective is to test the control system in the most complex situation that could be encountered in a real scenario. The total disturbances are given as follows:(i)
External disturbancesWind disturbances are applied att=15 s. The disturbances in the position dynamics are generated by the Dryden wind. Their component dPext along each axis P∈x,y,z is expressed as [60]
(82)dPextt≔−kd,PvP−vw,P2signvP−vw,P,and the disturbances on the attitude dynamics are defined by the following expressions:
(83)dΦextt=0.5cos0.4t,dθextt=0.5sin0.5t,dψextt=0.5sin0.7t.(ii)
Model uncertaintiesConsidering the simplifications in the Assumption8 model uncertainties (unmodeled dynamics) are also present as disturbances.
(iii)
Parameters’ uncertaintiesUncertainties of +50% and+30% are introduced in the nominal values of the moments of inertiaJxx,Jyy,Jzz, and the total mass 𝑚, respectively.Furthermore, to make the scenario closer to reality so as to adequately evaluate the control algorithm performance, the measurement noise effects are included while considering the gyroscope sensor data. Also, the measurement errors are added to the data corresponding to GPS and barometer sensors as follows:(i)
Measurement noiseThe output of the gyroscope sensor is modeled as(84)ζt=ζ~t+βζ+Ntrad/s,where ζ=p,q,rTrad/s is the final signal used in control, ζ~ is the true measurement inrad/s, βζ is the measurement bias in rad/s, and Nt is a random noise with the standard deviation σp in rad/s and the mean 0, i.e., N=σprand·, where rand· is a MATLAB® function that generates a random number in the interval (0,1).
(ii)
Position errorThe GPS and barometer signals are obtained as(85)Pt=P~t+rand·Pem,where P~ is the true measurement and Pe is the position error.


#### 4.2.1. Flight Scenario 1

For this trajectory tracking scenario, we consider a practical and more realistic example. The quadrotor is required to accomplish a flight mission by following a set of waypoints listed in the form ofxwp,ywp,zwp triples as follows: (0,0,0), (0,0,1), (0,0,1), (10,0,5), (20,20,5), (25,35,5), (25,60,5), (20,80,10), (5,90,15), (-10,90,15), (-17,70,15), (-20,40,15), (-15,10,1), (0,0,1), (0,0,1). In practice, these waypoints can be set manually or can be generated by a specific path planning algorithm such as the well-known rapidly exploring random tree (RRT) algorithm [8] or A∗ Algorithm [61]. The B-spline polynomial algorithm is used to generate a dynamically feasible and smooth Cartesian trajectory that passes through the given waypoints. The initial conditions of the quadrotor’s states are P0=x0,y0,z0T=0,0,0T m, η0=Φ0,θ0,ψ0T=0,0,0T deg.The results of the PIL simulation for the trajectory tracking are presented in Figures5–16. The way in which the aircraft follows the reference flight trajectory for the different control methods is presented in the 3D space in Figure 5. The evolution of the quadrotor’s trajectory in the x−y plane is presented in Figure 6. The first observation from Figures 5 and 6 is that all the controllers can track the reference trajectory. However, it is clear that the PID controller is affected by the disturbances. The position tracking errors are displayed in Figure 7, whereas the actual and desired positions are shown in Figure 8. It can be seen from Figure 7 that all the controllers successfully converged the translational variables to the required set-points in the presence of the abruptly changing reference. However, a closer inspection of the tracking errors in Figure 7 shows that the proposed control method is more accurate regarding the reference trajectory tracking on the x,y, and z axes as compared to the other controllers. It can be seen that the tracking errors converge to the origin and continue staying in the vicinity of the origin. All the controllers are affected by the wind gust at t=20sfor the position x and at t=22s for the position y except our proposed controller, which exhibits an effective disturbance rejection capability. This enhanced robustness in our suggested FCS is due to the adopted TSMBAADC control strategy including the robust ADRC and the DOBC control structures based on the nonlinear terminal SMC. It is evident that the PID controller is more sensitive to disturbances and suffers from a lack of robustness with large tracking errors. This is because the control law of the PID does not consider the quadrotor’s dynamics. Also, we can see that the FCS 2 which is based on the ANFTSMC provides noticeable disturbance rejection.Figure 5
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 1.Figure 6
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 1.Figure 7
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 1.Figure 8
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 1.Figure 9
The output signals obtained by REDBTD for the roll angleΦd,Φ̇d,Φ¨d and pitch angle θd: flight scenario 1.Figure 10
Profiles of the attitude of the quadrotor: Rotational variablesΦ,θ,ψ: flight scenario 1.Figure 11
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 1.Figure 12
The derivative of virtual lawφ̇P,1 obtained by the SMBF: flight scenario 1.Figure 13
Estimated lumped disturbances for the translational subsystem by the FXESO: flight scenario 1.Figure 14
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 1.Figure 15
Velocities of the simulated Dryden wind.Figure 16
Estimated lumped disturbances for the rotational subsystem by the FTO observer.To provide a more precise quantitative comparison of the achieved results, the integral of square error (ISE) index is used for error signal analysis. The ISE is defined as(86)ISE=∫t0tfePτ2dτ,where P=x,y,z. This performance index is computed and summarized in Table 3 where the best performances are shown in boldface.Table 3
ISE performance index analysis: flight scenario 1.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)172.528.470.83FCS 2 [53]6.021.390.05FCS 3 [29]28.321.470.32FCS 4 (proposed)0.100.220.003We can see from Table3 that the proposed control strategy has the smallest ISE index for the x,y, and z states. The PID controller is observed to have the largest ISE index. This can be explained by the fact that the proposed FCS has a strong disturbance rejection feature which leads to enhancing the robustness.Meanwhile, the desired attitude signals, their first and second derivative provided by the REDBTD, are depicted in Figure9. It is clear that the REDBTD can provide a continuous, smooth, and bounded estimate of the desired attitude signal and its derivative. Such that the high frequency due to the traditional differentiation can be filtered out.The attitude tracking of the proposed robust CNTSMC controller is shown in Figure10. It can be observed that the proposed attitude controller guarantees a robust and accurate tracking of the attitude reference. The control signals for the SMC-based controllers, i.e., IBSMC, ANFTSMC, and the proposed controller are depicted in Figure 11 where we can see that the magnitudes of the inputs are within the admissible ranges in the case of the proposed controller. Moreover, unlike the IBSMC that is affected by chattering, the control inputs of the proposed control system have no chattering effect. We can also observe that the FCS2 which is based on the ANFTSMC controller allows mitigating the chattering effect; however, our proposed controller has the overall superiority since the control signals provided by the control law are continuous.The well-known integral of the absolute value of the derivative of the inputut (IADU) criterion is used for the control signal (shown in Figure 11) analysis. It is given as
(87)IADU=∫titfduτdτdτ.The IADU index is very appropriate to check the control signal smoothness and thereby indicates chattering alleviation capability for the control input [62]. Taken together, the control signal results are compared in Table 4.Table 4
IADU performance index for chattering analysis.
Control strategyPerformance indexIADUuzuΦuθuψFCS 3 [29]0.1080.08050.1220.996FCS 2 [53]0.0370.0100.0310.03FCS 4 (proposed)0.0290.0060.0100.007It is apparent from this table that the smoothness is improved by the proposed control strategy for all control signals compared to the IBSMC and ANFTSMC controllers. Thus, the chattering effect has been effectively alleviated.The outputφ̇i,1 of the SMBF which is employed to tackle the “explosion of complexity” issue is shown in Figure 12. It is evident that the filter provides a bounded and smooth estimate of the virtual control signal. We can see that the output signal provided by the differentiator is not affected by the noise; hence, the controller is more applicable in practice.Figures13 and 14 display the outputs provided by the designed FXESO observer. Figure 13 shows the estimated lumped disturbances along the three axes. It can be observed that the disturbances can be exactly estimated along the three axes after they have been applied to the quadrotor system at t=15s, which leads to enhancing the robustness of the control system against strong lumped disturbances. The estimates of the velocities are presented in Figure 14 from which it is clear that the FXESO observer provides a precise velocity estimate within a very short convergence time. By estimating the velocities, the velocity-free control can be achieved where only the position of the quadrotor is assumed to be measurable. This prominent result of the adopted output-feedback control contributes to avoiding the shortcoming of the full-state feedback-based controllers. Thus, the proposed control method relying on velocity-free control is robust against the accelerometer’s faults.Figure15 depicts the velocities of the simulated Dryden wind, from which it can be seen that the wind gusts can reach a speed of around 10 m/s and 15 m/s, which is a challenging flight condition for the FCS. The estimated lumped disturbances affecting the rotational subsystem are presented in Figure 16. The designed FTO observer allows estimating the disturbances accurately within a fast convergence time, which improves the robustness of the attitude controller and thus leading to precise tracking of the attitude reference.


#### 4.2.2. Flight Scenario 2

The quadrotor is required to track the following time-varying Cartesian trajectory:(88)xd,yd,zd,ψdT=2sin0.2t,2sin0.2tcos0.2t,3,0T.The results of this flight scenario are presented in Figures17–23. The 3D trajectory tracking for different FCSs is presented in Figure 17, and the evolution of the quadrotor’s trajectory in thex−y plane is given in Figure 18. The profiles of the tracking errors and the translational variables are depicted in Figures 19 and 20, respectively. It can be seen from these figures that the designed FCS succeeds in tracking the desired flight trajectory with high accuracy while external disturbances, parameter variation, and model uncertainty effects are well compensated. Figure 21 presents the profiles of the rotational variables of the quadrotor Φ,θ,ψ. As in the first flight scenario, the proposed attitude controller ensures precise tracking of the three angles. The signals of the unmeasurable states of the quadrotor are successfully and accurately estimated by the designed FXESO observer as is shown in Figure 22. From Figure 23, we can see again that the control inputs provided by our suggested FCS are chattering-free and are within the admissible range.Figure 17
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 2.Figure 18
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 2.Figure 19
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 2.Figure 20
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 2.Figure 21
Profiles of the attitude of the quadrotor: rotational variablesΦ,θ,ψ: flight scenario 2.Figure 22
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 2.Figure 23
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 2.The results obtained from the analysis of tracking errors using the ISE index are compared in Table5. The best performances are shown in bold text. Similar to the previous flight scenario, the obtained results confirm the effectiveness of the proposed FCS in dealing with trajectory tracking even in the presence of fast-time varying external disturbances and model uncertainties. Moreover, the proposed FCS demonstrates the superior performance w.r.t the employed performance index related to the position tracking errors.Table 5
ISE performance index analysis: flight scenario 2.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)50.4115.2010.83FCS 2 [53]8.122.398.31FCS 3 [29]16.274.479.14FCS 4 (proposed)0.250.277.96


## 4.1. Software and Hardware Configuration

The procedure that we have followed to verify and implement the proposed control algorithm is known as the model-based design (MBD) framework (see Figure3) [48]. This procedure is accomplished in two steps. First, the control algorithm is simulated in MATLAB®/Simulink®. Second, the control algorithm is simulated as a C++ code in the real autopilot hardware (PIL simulation). To facilitate the implementation process and conduct the PIL experiment, a specific MATLAB® toolbox called “UAV Toolbox Support Package for PX4® Autopilots” is used. This toolbox allows to automatically generate C++ code for the controller from the MATLAB® code and also to deploy the generated code to the autopilot. This toolbox also allows the communication between the autopilot and Simulink® for data visualization and monitoring by using a USB wire. The control inputs computed by the Pixhawk flight control board are sent back to Simulink® to control the quadrotor model. The hardware set up for the PIL experiment is shown in Figure 1. In both simulation steps (Simulink®, PIL), the position and orientation of the quadrotor are sent to the FlightGear® flight simulator to visualize the flight of the quadrotor in a realistic environment. MATLAB® code and C++ code of the control algorithm are represented by two different blocks in Simulink®. We can switch the choice between these two blocks as can be seen in Figure 3.Figure 3
Block diagram of the implementation process from the simulation phase to the real autopilot hardware following an MBD procedure.The Pixhawk® autopilot includes a main 32-bit STM32F427 system-on-chip, 180 MHz, ARM® Cortex® M4 CPU, 2 MB flash, and 256 KB SRAM with L1 cache. Moreover, STM32F100 value line microcontroller with ARM® Cortex® M3 CPU running on 24 MHz and an SRAM of 8 KB is also included. The processor is running NuttX real-time operating system (RTOS). Besides, the autopilot board includes the following sensors: inertial measurement unit (IMU), Invensense MPU 6000 3-axis accelerometer/gyroscope, ST Micro L3GD20H 16-bit gyroscope, MEAS MS5611 barometer, and ST Micro LSM303D 14-bit accelerometer/magnetometer.Remark 20.
Most of the reported works on finite-time control of the quadrotor aircraft may look very promising as evidenced by simulation results. However, owing to lack of practical implementation, their real-world significance may be a valid concern. In contrast, our work presents design, simulation, and hardware realization of modern robust finite-time control laws. The physical implementation in autopilot hardware while attempting to address the abovementioned problems essentially attributes to the scientific contribution of our work.


## 4.2. Trajectory Tracking Results and Robustness Test

The numerical simulations and PIL experiments are conducted by using the Euler integration method with a fixed integration step equal to4×10−2 s. The parameters of the quadrotor model are based on a real experimental platform where we have identified them experimentally. These parameters are summarized in Table 1.Table 1
Physical parameters of the quadrotor aircraft.
ParameterValueUnitm1.72kgl0.225mg9.81m/s2Jxx0.0232kg.m2Jyy0.0249kg.m2Jzz0.0342kg.m2We have estimated the moments of inertiaJxx,Jyy,Jzz by the bifilar pendulum experiment as shown in Figure 4. The moment of inertia is calculated by (81) as
(81)JPP≔mgdw2Toscil2/16π2lwkg.m2,where JPP=Jxx,Jyy,Jzz, lw is the length of the two suspending wires (filars) in meters, dw is the distance between the two wires in meters, and Toscil is the period of one oscillation in seconds. The period of 10 oscillations is measured using a stopwatch for 5 trials with an initial swing angle equal to 10 degrees. The average value was then used to calculate Toscil.Figure 4
Bifilar pendulum setup for the estimation of the moments of inertia of the quadrotor: (a)Jxx estimation and (b) Jzz estimation.
(a)(b)The parameters of the CNTSMC controller are chosen ask1Φ=k2Φ=10,k1θ=k2θ=10,k1ψ=k2ψ=15, λη=3.5,μη=12. The exponents of the attitude controller are tuned by considering α¯=0.85 and ξ=1.5,ε=0.001. The parameters of the FTO observer are set to be ρ1η=20,ρ2η=15,ρ3η=2,lη=3. The parameters of the BSITSMC controller are chosen to be ξi,1=0.5,kx,1=0.25,ξi,2=1.2,ky,1=0.6,,λi,1=0.012,kz,1=0.71,kx,2=0.15,,α=0.8,ky,2=0.12,λi,2=0.01,kz,2=0.2. The parameters of SMBF are Θ0=1.2,Θ1=1.6,Θ2=2.5,L=1.5,r=3. Whereas the FXESO observer gains are selected to be ε1=0.26,a¯=0.74,ε2=0.2,b¯=1.2,ρix=ρ¯ix=ρiΦ=ρ¯iΦ=5,ρiy=ρ¯iy=ρiθ=ρ¯iθ=ρiz=10,ρ¯iz=ρiψ=ρ¯iψ=12,lP=5.To highlight the improvement attained with the proposed control strategy, a comparative study is conducted among four different flight control systems (FCS) (see Table2). FCS 1 is the default control system of the Pixhawk® autopilot which is based on the PID controller, FCS 2 is based on an adaptive nonsingular fast terminal sliding mode control (ANFTSMC), FCS 3 is based on IBSMC, and FCS 4 is our proposed flight control system.Table 2
Flight control systems compared for the Cartesian trajectory tracking.
Flight control systemPosition controllerAttitude controllerFCS 1 (Pixhawk®)PIDPIDFCS 2 [53]ANFTSMCANFTSMCFCS 3 [29]IBSMCIBSMCFCS 4 (proposed)BSITSMC-FXESOCNTSMC-FTOTo better evaluate the FCSs, two different flight scenarios for the Cartesian trajectory tracking are performed. To test the robustness and disturbance rejection capabilities of the FCSs, Cartesian trajectory tracking control has been conducted under the influence of parameter variation and internal model uncertainties as well as external time-varying wind disturbances. This approach is more realistic in practice since all these disturbances and uncertainties are simultaneously present in the dynamic model of the quadrotor. The overall objective is to test the control system in the most complex situation that could be encountered in a real scenario. The total disturbances are given as follows:(i)
External disturbancesWind disturbances are applied att=15 s. The disturbances in the position dynamics are generated by the Dryden wind. Their component dPext along each axis P∈x,y,z is expressed as [60]
(82)dPextt≔−kd,PvP−vw,P2signvP−vw,P,and the disturbances on the attitude dynamics are defined by the following expressions:
(83)dΦextt=0.5cos0.4t,dθextt=0.5sin0.5t,dψextt=0.5sin0.7t.(ii)
Model uncertaintiesConsidering the simplifications in the Assumption8 model uncertainties (unmodeled dynamics) are also present as disturbances.
(iii)
Parameters’ uncertaintiesUncertainties of +50% and+30% are introduced in the nominal values of the moments of inertiaJxx,Jyy,Jzz, and the total mass 𝑚, respectively.Furthermore, to make the scenario closer to reality so as to adequately evaluate the control algorithm performance, the measurement noise effects are included while considering the gyroscope sensor data. Also, the measurement errors are added to the data corresponding to GPS and barometer sensors as follows:(i)
Measurement noiseThe output of the gyroscope sensor is modeled as(84)ζt=ζ~t+βζ+Ntrad/s,where ζ=p,q,rTrad/s is the final signal used in control, ζ~ is the true measurement inrad/s, βζ is the measurement bias in rad/s, and Nt is a random noise with the standard deviation σp in rad/s and the mean 0, i.e., N=σprand·, where rand· is a MATLAB® function that generates a random number in the interval (0,1).
(ii)
Position errorThe GPS and barometer signals are obtained as(85)Pt=P~t+rand·Pem,where P~ is the true measurement and Pe is the position error.


### 4.2.1. Flight Scenario 1

For this trajectory tracking scenario, we consider a practical and more realistic example. The quadrotor is required to accomplish a flight mission by following a set of waypoints listed in the form ofxwp,ywp,zwp triples as follows: (0,0,0), (0,0,1), (0,0,1), (10,0,5), (20,20,5), (25,35,5), (25,60,5), (20,80,10), (5,90,15), (-10,90,15), (-17,70,15), (-20,40,15), (-15,10,1), (0,0,1), (0,0,1). In practice, these waypoints can be set manually or can be generated by a specific path planning algorithm such as the well-known rapidly exploring random tree (RRT) algorithm [8] or A∗ Algorithm [61]. The B-spline polynomial algorithm is used to generate a dynamically feasible and smooth Cartesian trajectory that passes through the given waypoints. The initial conditions of the quadrotor’s states are P0=x0,y0,z0T=0,0,0T m, η0=Φ0,θ0,ψ0T=0,0,0T deg.The results of the PIL simulation for the trajectory tracking are presented in Figures5–16. The way in which the aircraft follows the reference flight trajectory for the different control methods is presented in the 3D space in Figure 5. The evolution of the quadrotor’s trajectory in the x−y plane is presented in Figure 6. The first observation from Figures 5 and 6 is that all the controllers can track the reference trajectory. However, it is clear that the PID controller is affected by the disturbances. The position tracking errors are displayed in Figure 7, whereas the actual and desired positions are shown in Figure 8. It can be seen from Figure 7 that all the controllers successfully converged the translational variables to the required set-points in the presence of the abruptly changing reference. However, a closer inspection of the tracking errors in Figure 7 shows that the proposed control method is more accurate regarding the reference trajectory tracking on the x,y, and z axes as compared to the other controllers. It can be seen that the tracking errors converge to the origin and continue staying in the vicinity of the origin. All the controllers are affected by the wind gust at t=20sfor the position x and at t=22s for the position y except our proposed controller, which exhibits an effective disturbance rejection capability. This enhanced robustness in our suggested FCS is due to the adopted TSMBAADC control strategy including the robust ADRC and the DOBC control structures based on the nonlinear terminal SMC. It is evident that the PID controller is more sensitive to disturbances and suffers from a lack of robustness with large tracking errors. This is because the control law of the PID does not consider the quadrotor’s dynamics. Also, we can see that the FCS 2 which is based on the ANFTSMC provides noticeable disturbance rejection.Figure 5
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 1.Figure 6
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 1.Figure 7
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 1.Figure 8
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 1.Figure 9
The output signals obtained by REDBTD for the roll angleΦd,Φ̇d,Φ¨d and pitch angle θd: flight scenario 1.Figure 10
Profiles of the attitude of the quadrotor: Rotational variablesΦ,θ,ψ: flight scenario 1.Figure 11
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 1.Figure 12
The derivative of virtual lawφ̇P,1 obtained by the SMBF: flight scenario 1.Figure 13
Estimated lumped disturbances for the translational subsystem by the FXESO: flight scenario 1.Figure 14
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 1.Figure 15
Velocities of the simulated Dryden wind.Figure 16
Estimated lumped disturbances for the rotational subsystem by the FTO observer.To provide a more precise quantitative comparison of the achieved results, the integral of square error (ISE) index is used for error signal analysis. The ISE is defined as(86)ISE=∫t0tfePτ2dτ,where P=x,y,z. This performance index is computed and summarized in Table 3 where the best performances are shown in boldface.Table 3
ISE performance index analysis: flight scenario 1.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)172.528.470.83FCS 2 [53]6.021.390.05FCS 3 [29]28.321.470.32FCS 4 (proposed)0.100.220.003We can see from Table3 that the proposed control strategy has the smallest ISE index for the x,y, and z states. The PID controller is observed to have the largest ISE index. This can be explained by the fact that the proposed FCS has a strong disturbance rejection feature which leads to enhancing the robustness.Meanwhile, the desired attitude signals, their first and second derivative provided by the REDBTD, are depicted in Figure9. It is clear that the REDBTD can provide a continuous, smooth, and bounded estimate of the desired attitude signal and its derivative. Such that the high frequency due to the traditional differentiation can be filtered out.The attitude tracking of the proposed robust CNTSMC controller is shown in Figure10. It can be observed that the proposed attitude controller guarantees a robust and accurate tracking of the attitude reference. The control signals for the SMC-based controllers, i.e., IBSMC, ANFTSMC, and the proposed controller are depicted in Figure 11 where we can see that the magnitudes of the inputs are within the admissible ranges in the case of the proposed controller. Moreover, unlike the IBSMC that is affected by chattering, the control inputs of the proposed control system have no chattering effect. We can also observe that the FCS2 which is based on the ANFTSMC controller allows mitigating the chattering effect; however, our proposed controller has the overall superiority since the control signals provided by the control law are continuous.The well-known integral of the absolute value of the derivative of the inputut (IADU) criterion is used for the control signal (shown in Figure 11) analysis. It is given as
(87)IADU=∫titfduτdτdτ.The IADU index is very appropriate to check the control signal smoothness and thereby indicates chattering alleviation capability for the control input [62]. Taken together, the control signal results are compared in Table 4.Table 4
IADU performance index for chattering analysis.
Control strategyPerformance indexIADUuzuΦuθuψFCS 3 [29]0.1080.08050.1220.996FCS 2 [53]0.0370.0100.0310.03FCS 4 (proposed)0.0290.0060.0100.007It is apparent from this table that the smoothness is improved by the proposed control strategy for all control signals compared to the IBSMC and ANFTSMC controllers. Thus, the chattering effect has been effectively alleviated.The outputφ̇i,1 of the SMBF which is employed to tackle the “explosion of complexity” issue is shown in Figure 12. It is evident that the filter provides a bounded and smooth estimate of the virtual control signal. We can see that the output signal provided by the differentiator is not affected by the noise; hence, the controller is more applicable in practice.Figures13 and 14 display the outputs provided by the designed FXESO observer. Figure 13 shows the estimated lumped disturbances along the three axes. It can be observed that the disturbances can be exactly estimated along the three axes after they have been applied to the quadrotor system at t=15s, which leads to enhancing the robustness of the control system against strong lumped disturbances. The estimates of the velocities are presented in Figure 14 from which it is clear that the FXESO observer provides a precise velocity estimate within a very short convergence time. By estimating the velocities, the velocity-free control can be achieved where only the position of the quadrotor is assumed to be measurable. This prominent result of the adopted output-feedback control contributes to avoiding the shortcoming of the full-state feedback-based controllers. Thus, the proposed control method relying on velocity-free control is robust against the accelerometer’s faults.Figure15 depicts the velocities of the simulated Dryden wind, from which it can be seen that the wind gusts can reach a speed of around 10 m/s and 15 m/s, which is a challenging flight condition for the FCS. The estimated lumped disturbances affecting the rotational subsystem are presented in Figure 16. The designed FTO observer allows estimating the disturbances accurately within a fast convergence time, which improves the robustness of the attitude controller and thus leading to precise tracking of the attitude reference.


### 4.2.2. Flight Scenario 2

The quadrotor is required to track the following time-varying Cartesian trajectory:(88)xd,yd,zd,ψdT=2sin0.2t,2sin0.2tcos0.2t,3,0T.The results of this flight scenario are presented in Figures17–23. The 3D trajectory tracking for different FCSs is presented in Figure 17, and the evolution of the quadrotor’s trajectory in thex−y plane is given in Figure 18. The profiles of the tracking errors and the translational variables are depicted in Figures 19 and 20, respectively. It can be seen from these figures that the designed FCS succeeds in tracking the desired flight trajectory with high accuracy while external disturbances, parameter variation, and model uncertainty effects are well compensated. Figure 21 presents the profiles of the rotational variables of the quadrotor Φ,θ,ψ. As in the first flight scenario, the proposed attitude controller ensures precise tracking of the three angles. The signals of the unmeasurable states of the quadrotor are successfully and accurately estimated by the designed FXESO observer as is shown in Figure 22. From Figure 23, we can see again that the control inputs provided by our suggested FCS are chattering-free and are within the admissible range.Figure 17
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 2.Figure 18
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 2.Figure 19
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 2.Figure 20
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 2.Figure 21
Profiles of the attitude of the quadrotor: rotational variablesΦ,θ,ψ: flight scenario 2.Figure 22
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 2.Figure 23
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 2.The results obtained from the analysis of tracking errors using the ISE index are compared in Table5. The best performances are shown in bold text. Similar to the previous flight scenario, the obtained results confirm the effectiveness of the proposed FCS in dealing with trajectory tracking even in the presence of fast-time varying external disturbances and model uncertainties. Moreover, the proposed FCS demonstrates the superior performance w.r.t the employed performance index related to the position tracking errors.Table 5
ISE performance index analysis: flight scenario 2.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)50.4115.2010.83FCS 2 [53]8.122.398.31FCS 3 [29]16.274.479.14FCS 4 (proposed)0.250.277.96


## 4.2.1. Flight Scenario 1

For this trajectory tracking scenario, we consider a practical and more realistic example. The quadrotor is required to accomplish a flight mission by following a set of waypoints listed in the form ofxwp,ywp,zwp triples as follows: (0,0,0), (0,0,1), (0,0,1), (10,0,5), (20,20,5), (25,35,5), (25,60,5), (20,80,10), (5,90,15), (-10,90,15), (-17,70,15), (-20,40,15), (-15,10,1), (0,0,1), (0,0,1). In practice, these waypoints can be set manually or can be generated by a specific path planning algorithm such as the well-known rapidly exploring random tree (RRT) algorithm [8] or A∗ Algorithm [61]. The B-spline polynomial algorithm is used to generate a dynamically feasible and smooth Cartesian trajectory that passes through the given waypoints. The initial conditions of the quadrotor’s states are P0=x0,y0,z0T=0,0,0T m, η0=Φ0,θ0,ψ0T=0,0,0T deg.The results of the PIL simulation for the trajectory tracking are presented in Figures5–16. The way in which the aircraft follows the reference flight trajectory for the different control methods is presented in the 3D space in Figure 5. The evolution of the quadrotor’s trajectory in the x−y plane is presented in Figure 6. The first observation from Figures 5 and 6 is that all the controllers can track the reference trajectory. However, it is clear that the PID controller is affected by the disturbances. The position tracking errors are displayed in Figure 7, whereas the actual and desired positions are shown in Figure 8. It can be seen from Figure 7 that all the controllers successfully converged the translational variables to the required set-points in the presence of the abruptly changing reference. However, a closer inspection of the tracking errors in Figure 7 shows that the proposed control method is more accurate regarding the reference trajectory tracking on the x,y, and z axes as compared to the other controllers. It can be seen that the tracking errors converge to the origin and continue staying in the vicinity of the origin. All the controllers are affected by the wind gust at t=20sfor the position x and at t=22s for the position y except our proposed controller, which exhibits an effective disturbance rejection capability. This enhanced robustness in our suggested FCS is due to the adopted TSMBAADC control strategy including the robust ADRC and the DOBC control structures based on the nonlinear terminal SMC. It is evident that the PID controller is more sensitive to disturbances and suffers from a lack of robustness with large tracking errors. This is because the control law of the PID does not consider the quadrotor’s dynamics. Also, we can see that the FCS 2 which is based on the ANFTSMC provides noticeable disturbance rejection.Figure 5
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 1.Figure 6
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 1.Figure 7
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 1.Figure 8
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 1.Figure 9
The output signals obtained by REDBTD for the roll angleΦd,Φ̇d,Φ¨d and pitch angle θd: flight scenario 1.Figure 10
Profiles of the attitude of the quadrotor: Rotational variablesΦ,θ,ψ: flight scenario 1.Figure 11
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 1.Figure 12
The derivative of virtual lawφ̇P,1 obtained by the SMBF: flight scenario 1.Figure 13
Estimated lumped disturbances for the translational subsystem by the FXESO: flight scenario 1.Figure 14
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 1.Figure 15
Velocities of the simulated Dryden wind.Figure 16
Estimated lumped disturbances for the rotational subsystem by the FTO observer.To provide a more precise quantitative comparison of the achieved results, the integral of square error (ISE) index is used for error signal analysis. The ISE is defined as(86)ISE=∫t0tfePτ2dτ,where P=x,y,z. This performance index is computed and summarized in Table 3 where the best performances are shown in boldface.Table 3
ISE performance index analysis: flight scenario 1.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)172.528.470.83FCS 2 [53]6.021.390.05FCS 3 [29]28.321.470.32FCS 4 (proposed)0.100.220.003We can see from Table3 that the proposed control strategy has the smallest ISE index for the x,y, and z states. The PID controller is observed to have the largest ISE index. This can be explained by the fact that the proposed FCS has a strong disturbance rejection feature which leads to enhancing the robustness.Meanwhile, the desired attitude signals, their first and second derivative provided by the REDBTD, are depicted in Figure9. It is clear that the REDBTD can provide a continuous, smooth, and bounded estimate of the desired attitude signal and its derivative. Such that the high frequency due to the traditional differentiation can be filtered out.The attitude tracking of the proposed robust CNTSMC controller is shown in Figure10. It can be observed that the proposed attitude controller guarantees a robust and accurate tracking of the attitude reference. The control signals for the SMC-based controllers, i.e., IBSMC, ANFTSMC, and the proposed controller are depicted in Figure 11 where we can see that the magnitudes of the inputs are within the admissible ranges in the case of the proposed controller. Moreover, unlike the IBSMC that is affected by chattering, the control inputs of the proposed control system have no chattering effect. We can also observe that the FCS2 which is based on the ANFTSMC controller allows mitigating the chattering effect; however, our proposed controller has the overall superiority since the control signals provided by the control law are continuous.The well-known integral of the absolute value of the derivative of the inputut (IADU) criterion is used for the control signal (shown in Figure 11) analysis. It is given as
(87)IADU=∫titfduτdτdτ.The IADU index is very appropriate to check the control signal smoothness and thereby indicates chattering alleviation capability for the control input [62]. Taken together, the control signal results are compared in Table 4.Table 4
IADU performance index for chattering analysis.
Control strategyPerformance indexIADUuzuΦuθuψFCS 3 [29]0.1080.08050.1220.996FCS 2 [53]0.0370.0100.0310.03FCS 4 (proposed)0.0290.0060.0100.007It is apparent from this table that the smoothness is improved by the proposed control strategy for all control signals compared to the IBSMC and ANFTSMC controllers. Thus, the chattering effect has been effectively alleviated.The outputφ̇i,1 of the SMBF which is employed to tackle the “explosion of complexity” issue is shown in Figure 12. It is evident that the filter provides a bounded and smooth estimate of the virtual control signal. We can see that the output signal provided by the differentiator is not affected by the noise; hence, the controller is more applicable in practice.Figures13 and 14 display the outputs provided by the designed FXESO observer. Figure 13 shows the estimated lumped disturbances along the three axes. It can be observed that the disturbances can be exactly estimated along the three axes after they have been applied to the quadrotor system at t=15s, which leads to enhancing the robustness of the control system against strong lumped disturbances. The estimates of the velocities are presented in Figure 14 from which it is clear that the FXESO observer provides a precise velocity estimate within a very short convergence time. By estimating the velocities, the velocity-free control can be achieved where only the position of the quadrotor is assumed to be measurable. This prominent result of the adopted output-feedback control contributes to avoiding the shortcoming of the full-state feedback-based controllers. Thus, the proposed control method relying on velocity-free control is robust against the accelerometer’s faults.Figure15 depicts the velocities of the simulated Dryden wind, from which it can be seen that the wind gusts can reach a speed of around 10 m/s and 15 m/s, which is a challenging flight condition for the FCS. The estimated lumped disturbances affecting the rotational subsystem are presented in Figure 16. The designed FTO observer allows estimating the disturbances accurately within a fast convergence time, which improves the robustness of the attitude controller and thus leading to precise tracking of the attitude reference.


## 4.2.2. Flight Scenario 2

The quadrotor is required to track the following time-varying Cartesian trajectory:(88)xd,yd,zd,ψdT=2sin0.2t,2sin0.2tcos0.2t,3,0T.The results of this flight scenario are presented in Figures17–23. The 3D trajectory tracking for different FCSs is presented in Figure 17, and the evolution of the quadrotor’s trajectory in thex−y plane is given in Figure 18. The profiles of the tracking errors and the translational variables are depicted in Figures 19 and 20, respectively. It can be seen from these figures that the designed FCS succeeds in tracking the desired flight trajectory with high accuracy while external disturbances, parameter variation, and model uncertainty effects are well compensated. Figure 21 presents the profiles of the rotational variables of the quadrotor Φ,θ,ψ. As in the first flight scenario, the proposed attitude controller ensures precise tracking of the three angles. The signals of the unmeasurable states of the quadrotor are successfully and accurately estimated by the designed FXESO observer as is shown in Figure 22. From Figure 23, we can see again that the control inputs provided by our suggested FCS are chattering-free and are within the admissible range.Figure 17
3D Cartesian flight trajectory tracking for different flight control systems: flight scenario 2.Figure 18
Evolution of the quadrotor’s trajectory in 2D (x−y plane) for different flight control systems: flight scenario 2.Figure 19
Evolution of the tracking errors for the translational subsystemex,ey,ez: flight scenario 2.Figure 20
Profiles of the position of the quadrotor: translational variablesx,y,z: flight scenario 2.Figure 21
Profiles of the attitude of the quadrotor: rotational variablesΦ,θ,ψ: flight scenario 2.Figure 22
Profiles of the estimated velocities of the quadrotorv^x,v^y,v^z by the FXESO observer: flight scenario 2.Figure 23
Control signals, i.e.,uz,uΦ,uθ,uψ, of the proposed FCS 4, FCS 2, and FCS3. The control signals of the proposed FCS 4 are chattering-free: flight scenario 2.The results obtained from the analysis of tracking errors using the ISE index are compared in Table5. The best performances are shown in bold text. Similar to the previous flight scenario, the obtained results confirm the effectiveness of the proposed FCS in dealing with trajectory tracking even in the presence of fast-time varying external disturbances and model uncertainties. Moreover, the proposed FCS demonstrates the superior performance w.r.t the employed performance index related to the position tracking errors.Table 5
ISE performance index analysis: flight scenario 2.
Control strategyPerformance indexISExyzFCS 1 (Pixhawk®)50.4115.2010.83FCS 2 [53]8.122.398.31FCS 3 [29]16.274.479.14FCS 4 (proposed)0.250.277.96


## 5. Conclusion

This research work presents a solution to the robust finite-time trajectory tracking control problem of a quadrotor system subjected to various disturbances. A new FCS has been proposed. The design of the control laws is based on two steps: attitude control design and position control design. First, a robust DOBC has been constructed to stabilize the inner attitude loop. In addition, a REDBTD has been introduced to supply smooth and bounded attitude reference signals. Second, an ADRC approach has been proposed by creatively combining a BSITSMC strategy and an FXESO observer. Besides, the “explosion of complexity” problem in BS design has been addressed by designing an SMBF. The most distinguishing feature of the proposed FCS lies in its ability to track the trajectory in finite-time under lumped disturbances. The control law is smooth without undesirable singularity or chattering effect. Stability analysis for the closed-loop system has been provided based on the Lyapunov theorem. PIL implementations on real Pixhawk® autopilot have corroborated the theoretical findings. Furthermore, a comparative study has been made with relevant works to evaluate the performance and show the improvements attained by the developed flight controller. The obtained results confirm the superiority of the proposed control strategy regarding the accuracy, robustness, and chattering alleviation.As an extension and improvement to the present study, fixed-time stability could be addressed in our future works. Fixed-time stability is aimed at predefining and adjusting a uniformly bounded settling time. Such a control method allows for stabilizing the system’s states in a fixed time independently of initial conditions. This important property is potentially useful in practical scenarios of the quadrotor since it enhances the system’s robustness and provides faster convergence. Besides, this interesting feature is pivotal in the cooperative control of the multiquadrotor systems, which lies in our research. Also, actuator faults and control input saturation are interesting aspects that could also be addressed. In addition, an optimization method could be usefully introduced for the tuning of the control parameters. Finally, a real outdoor flight experiment could be conducted to further evaluate the proposed flight-control system.



---

*Source: 5522379-2021-05-06.xml*