Robust Finite-Time Trajectory Tracking Control of Quadrotor Aircraft via Terminal Sliding Mode-Based Active Antidisturbance Approach: A PIL Experiment
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 input (IADU) dictate that the proposed technique overperforms by precision and chattering alleviation.
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 . 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 . For instance, the work reported in  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 . 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  presents an integral backstepping sliding mode control (IBSMC) method for a perturbated quadrotor system. In , 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 . Such a control signal will cause low control accuracy and degrade the control performance . 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 , 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 . A recursive singularity-free fast terminal sliding mode control is used in  to design a finite-time tracker for nonholonomic systems including a wheeled mobile robot and an underactuated surface vessel. In , 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 , a finite-time observer-based robust continuous twisting control is proposed for an uncertain quadrotor system subjected to disturbances. Another work reported in  comes up with a modified super twisting algorithm that can ensure robustness and finite-time convergence for the quadrotor helicopter. In the work , 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 , in finite-time. Also, a finite-time trajectory tracking control for a quadrotor aircraft with unknown external disturbances is investigated in . However, together with , 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 , a nonsingular terminal sliding mode control law with finite-time convergence is designed for the quadrotor. In , a finite-time convergent nonsingular terminal sliding mode control for a quadrotor with a total rotor failure is proposed. In , 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 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 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 the 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 . 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. Section 2 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.
Definition 1 (see ) (finite-time stability). Consider the following autonomous system: where , and the nonlinear function is continuous on an open neighborhood of the origin. The origin is an equilibrium point of system (1). is a globally finite-time convergent, if it is globally asymptotically stable, and there are an open neighborhood of the origin and a function such that every solution of system (1) that starts from the initial condition is well-defined for , and . Here, is called the settling-time function, i.e., convergence-time, (w.r.t ). is said to be a finite-time stable equilibrium if it is finite-time convergent and Lyapunov stable. If , the origin is said to be a globally finite-time stable equilibrium.
Lemma 2 (see ). Consider the following system:
If the constants satisfy and , it results that the above system is finite-time stable.
Lemma 3 (see ). If the positive constants make the -order polynomials and be Hurwitz in terms of the Laplace operator , i.e., all their roots are in the left-half plane, the origin of the following system is finite-time stable equilibrium with uniform settling-time, where and are determined according to the bi-limit homogeneity reasoning as follows: and , where , , and .
Lemma 4 (see ). Suppose that there is a continuous and positive-definite Lyapunov function , and its derivative satisfies where , and are some positive constants, then, the origin of system (1) is fixed-time stable. The settling-time function is bounded by as
Lemma 5 (see ). Consider the following system: where , , and are constants. Also, if the nonnegative constants are assigned to ensure the following matrices are Hurwitz, i.e., every eigenvalue of the matrices has a strictly negative real part: Then, the system (6) is fixed-time stable, i.e., uniform settling-time w.r.t. initial condition.
Lemma 6 (see ). Consider system (1). If there exist Lyapunov function and some real constants and , such that then, system (1) is finite-time stable for any given , in which the finite settling time satisfies
2.2. Problem Formulation
The motion of the quadrotor in space can be described by a B-frame and an E-frame (see Figure 1). 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 .
The E-frame is used to define the translational motion by. The B-frame indicates the rotational motion, i.e., Euler angles . The 6-DoF acceleration dynamics corresponding to the rotation and translation motions of the quadrotor in the presence of disturbances can be written as 
Since the flight of the quadcopter is driven by four propellers, the angular speeds of the four propellers determine the total lift force (thrust control input) and the torques . The quadrotor’s actuators (rotors) produce a total lift force defined as 
The control torques developed by the quadrotor’s actuators are defined as 
Besides, the rotating velocities of the four propellers, i.e., , are related to (the torques) and (the total lift force) by the means of a constant invertible matrix as 
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 by and , i.e.,
Since it is difficult to identify the aerodynamic coefficients in practice, the unmodeled dynamics, for the translational subsystem, are the drag force . These disturbances are defined as
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, Assumption 8 will not affect the system stability and control performance. Therefore, the model uncertainties 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 choosing as a state vector, the following state-space model for the 6-DoF quadrotor dynamics is obtained:
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 laws and 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 , there exist two constants , such thatwhere and 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 but only four control inputs are available . 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 , and the output is the appropriate roll, pitch, and yaw torques . A BSITSMC is synthesized for the outer loop to achieve robust position tracking. This loop takes the desired position signals as input and generates the reference angles for the inner loop and also the total thrust force control .
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 of ,, and for the controller. To this end, a robust exact differentiator-based TD (REDBTD) is adopted , which is defined as where is the signal to be differentiated and . Then, and are the finite-time estimates of and , respectively, i.e., ,, and.
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: Here, is the states vector, where , is the control inputs vector,is the controlled outputs vector, and the uncertain function 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 . Hence, the unknown disturbances behave as a sufficiently smooth uncertain function with its first-time derivative satisfying , where is a nonnegative bounded constant, i.e., [51, 58]. The functions are defined as
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 where is the estimate of . Then, the disturbance can be precisely identified within a finite-time , i.e., .
Proof. Let us define the observation errors as By differentiating these errors w.r.t time and substituting by their expressions, the corresponding error dynamics can be written as Then, by substituting by their expressions, and after some manipulations, we get Based on Lemma 2, the observation errors are finite-time convergent to the origin within the time , i.e., .
3.1.3. Finite-Time Continuous Nonsingular Terminal Sliding Mode Control Design
Let be the reference attitude and be the actual attitude. Let us define the attitude tracking errors as where . The tracking errors dynamics is given as
The control objective for the attitude system is to make the states track the desired reference by designing a continuous-SMC law . Then, the tracking errors vector can be stabilized to zero, i.e., . Let and . To ensure fast convergence of attitude variables to their reference signals, the following CNTSM surface is proposed for the rotational system (19) where the nonnegative parameters , and are chosen based on Remark 13 which is drawn hereafter. The control action is applied to establish the reaching phase of the sliding surface , and the sliding motion on is determined as follows :
This control structure consists of two parts; the term is the equivalent control part, which maintains the variables on the sliding surface, and is the reaching control part, which ensures faster convergence. On the one hand, the control can be obtained from the sliding motion . Thus, when , we get where . Substituting by its expression from (20) into (30), and after some manipulations, the equivalent control can be obtained as where , is estimated by the FTO observer given by (22). On the other hand, the control is chosen to ensure the finite-time reaching of the sliding surface, and it is proposed as
Finally, by replacing with , 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 terms with fractional power 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 exists in the control, that is, , since .
Remark 13. In this study, rigorous conditions are established for the choice of the control parameters. On one hand, the exponents are chosen based on homogeneity theory as follows : and , where , and . To ensure the finite-time convergence feature, should satisfy , and . On the other hand, the positive constants should be selected to make the -order polynomials and be Hurwitz in terms of the Laplace operator . Besides, the positive constants , and 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 should be selected to satisfy the following condition . The parameter is the upper bound of the total disturbances. It is a bounded constant .
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 is designed by (33), and employing disturbance observer (22), then, the attitude system is finite-time stable within a bounded time , i.e., .
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 substituting from (20) into the sliding surface (28), we get Then, by substituting the designed control law (33) into given in (34), the sliding surface dynamics becomes as By Theorem 11, we have for all . Hence, we get By differentiating (36) and substituting (32), it yields Subsequently, let us define the following positive-definite Lyapunov function: By differentiating and substituting (37), it yields By using and , we get By considering that , it yields and . According to Lemma 4, the finite-time reaching of sliding surface is guaranteed within the following bounded reaching-time .
Step 2: Recalling the errors dynamics for the angular signals given in (27) We define and . Consequently, the closed-loop dynamics (41) can be rewritten as When , from (28) and considering , we have Substituting (43) into error dynamics expressions (42), we get According to Lemma 3, we can deduce that the tracking error dynamics (44) can be stabilized to zero during the sliding motion within a finite bounded time. Thus, there exists a constant such that and for all . 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 altitude , and the horizontal position . Thus, the following translational system subjected to disturbances can be obtained from (17) as Here, is the states vector, where , is the controlled outputs vector, and the uncertain function summarizes the total lumped disturbances including model uncertainty effects, parameters’ uncertainties, and external disturbances, where and . The functions are defined as
Given the disturbed position dynamic model described in (45), the FXESO observer is designed as
Then, the disturbances and the velocities can be estimated within a fixed bounded time as in (52); hereafter, i.e., and . The exponents are selected as follows: satisfy the recurrent relations , and where for a sufficiently small . Also, , and , and where for a sufficiently small . Besides, is the observation error and .
In (48), the terms and are continuous and differentiable. However, the term is discontinuous. Thus, to avoid the chattering effect and to realize the observer in practice, the following sigmoid function is used: where is a constant that is inversely proportional to , i.e.,.
Theorem 15. The FXESO observer designed in (48) can precisely estimate the disturbance and the velocities within a fixed bounded time , i.e., and .
Proof. Let us define the observation errors as By differentiating these errors w.r.t time, it yields Based on Lemma 5, the observation errors are guaranteed to converge to the origin in fixed-time bounded as where , , , , and . and are nonsingular, symmetric, and positive-definite matrices and satisfy ,.
3.2.2. Backstepping Integral Terminal Sliding Mode Control Design
The control law is designed for the position here. Without loss of generality, the appropriate control laws for the and positions can be derived similarly.
Let be the desired trajectory of the position . The tracking error is defined as
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:
Let us define the following Lyapunov function:
The time derivative of is given as
In (56), the term can be calculated from the sliding motion as
Let us define and , respectively, as follows: and . Then, we can get
It follows from Lemma 6 that the convergence of the tracking errors for the position can be achieved within a finite-time , bounded as where is the initial value of the Lyapunov function (55). The first time derivative of the sliding surface (54) is given as
Subsequently, the time derivative of (61) is calculated as
Let us define as
To investigate the stability of the sliding surface , the following Lyapunov function is defined as
Then, the time derivative of can be obtained as
The desired , i.e.,, should be chosen to make negative. Thus, from (66), the is given as where is the virtual variable. The derivative of the virtual control signal, i.e., , 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 : where are the estimates of and , respectively. Then, the derivative can be accurately restored in a finite time. Hence, and .
Let us define the error between and as
Then, equation (69) can be written as
Next, we define the integral of the error variable as
To make both the error and its integration converge to zero, the following Lyapunov function candidate is chosen: where . The time derivative of is given as
Considering in (75), it becomes