Observer-based fixed-time continuous nonsingular terminal sliding mode control of quadrotor aircraft under uncertainties and disturbances for robust trajectory tracking: Theory and experiment
Graphical abstract
Introduction
The flight mission of the quadrotor aircraft is one of the most useful and practical aerospace-related applications (Sampedro et al., 2019, Sudhakar et al., 2020). The research on quadrotor’s automatic flight control is a major area of interest within the field of control science, robotics, and aeronautical engineering. Notably, the robust trajectory tracking control is a persistent control problem that has become an important topic for a wide range of industrially related academics and researchers in the control community. The autonomous flight of these unmanned aerial vehicles (UAVs) requires an effective and reliable flight control algorithm. Substantially, fast time response, strong robustness, and accuracy appear to be principal determining factors for a flight control system. Therefore, an advanced control strategy is required to achieve high performance for meeting mission requirements for the quadrotor. Position and attitude control are key elements of the flight control system, whose performance has a direct effect on the safety and stability of the flight mission (Hou, Lu and Tu, 2020). On this control problem, there are usually two factors in position and attitude dynamics that bring great difficulty and challenge for the flight control design. These factors are the high nonlinearity with a strong coupling of the translational and rotational states as well as the influence of unknown disturbances encountered frequently in practical applications (including both internal system uncertainties and external environmental disturbances) (Ai & Yu, 2019).
In this note, a new flight control system based on the CNTSMC control is designed to deal with the robust fixed-time cartesian trajectory tracking control problem for the quadrotor system affected by multiple disturbances. Within the developed FTO-CNTSMC and FXESO-CNTSMC control structures, position and attitude tracking errors can be stabilized to the origin in fixed-time independently of the system’s initial conditions despite the disturbances.
The sliding mode control (SMC) technique is known to be one of the most efficient and robust control methods (Edwards and Spurgeon, 1998, Mechali, 2019, Utkin, 1992). Such controllers are insensitive to model errors and system parameter variations (Mechali et al., 2020, Shtessel et al., 2014). 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 surface and consequently converge to the origin. More recently, many works have been devoted to the robust control of the quadrotor system subjected to disturbances using SMC theory. Among these works, a dual-loop integral sliding mode control (DLISMC) based on the linear extended state observer (LESO) is proposed in Wang, Hua, Chen, and Cai (2019) to deal with the robust trajectory tracking of the quadrotor system. In Jia et al. (2017), an integral backstepping sliding mode control (IBSMC) method has been presented to robustly track the desired attitude trajectory. To reduce the chatting effect, the authors of Jia et al. (2017) have used a boundary layer method. The technical note in Chen et al. (2016) comes up with a regular SMC controller for the attitude system of a quadrotor. In reference (Almakhles, 2020), a robust backstepping sliding mode controller has been developed considering the incorporation of the disturbances in the quadrotor model. However, all these methods are based on the LSMC. The most serious disadvantage of this control approach is that the switching manifold is linear; hence, only asymptotic convergence of the system’s states to the origin is ensured (Boukattaya, Gassara, & Damak, 2020). In addition, LSMC inevitably suffers from the undesirable chattering phenomena. The chattering impact is reflected by the presence of disrupting high switching frequencies in the control input of the system (Levant, 2010). Such a control signal will cause low control accuracy, degrade the control performance, and excite unmodeled dynamics which significantly amplify measurement noise. Furthermore, it can even damage the system’s mechanical parts (the brushless motors in the case of quadrotor). To achieve finite-time convergence for perturbated nonlinear dynamical systems, the TSMC has been introduced in which the sliding surface is designed nonlinear (Zhihong et al., 1994, Zhihong and Yu, 1997). Although TSMC exhibits faster convergence than LSMC, its control law has a singular point and suffers from the chattering. Thus, to overcome the singularity issue, nonsingular terminal sliding mode control (NTSMC) (Feng, Yu, & Man, 2002) has been developed. However, the control signal is still not continuous which causes the chattering effect. On the other hand, the so-called continuous-SMC techniques are known for their continuous control signal providing an effective solution to eliminating the chattering problem (Rabiee et al., 2019, Torres-González et al., 2017). They are also classified by the researchers as the fifth generation of SMC (Fridman, Moreno, Bandyopadhyay, Kamal, & Chalanga, 2015). CNTSMC control is a kind of continuous-SMC techniques that can achieve finite-time convergence and improve precision. Besides, its control law is singularity-free and chattering-free. The work in Wang, Han, Feng, and Xia investigates the robust control of two-link flexible manipulator systems in uncertain conditions by using a CNTSMC controller. Authors of Jin, Lee, and d. K. K. Ahn (2015) offer a successful implementation of a CNTSMC with time-delay estimation for shape memory alloy actuators. In the brief (Wang, Li, & Li, 2019), a robust output voltage regulation problem of the DC–DC boost converter system is addressed by a CNTSMC technique. The article (Du, Fang, & Liu, 2019) provides a continuous full-order NTSMC for systems under matched and mismatched disturbances. To our best knowledge, few works in the literature (Falcón et al., 2019, Ríos et al., 2019) are devoted to investigating the design of the CNTSMC control laws for the quadrotor system. The technical notes (Falcón et al., 2019) present a comprehensive experimental study for a set of continuous-SMC algorithms for a quadrotor robust tracking, including those developed in Torres-González et al. (2017) and (Fridman et al., 2015), but these controllers are finite-time stable and do not show the fixed-time tracking stability property. Similar work is presented in Ríos et al. (2019). Although finite-time stable systems provide better performance than asymptotically stable systems, finite-time control design suffers from an inevitable drawback. It has been shown that the convergence time of finite-time controllers grows unboundedly along with the deviation of initial conditions from the equilibrium point. To deal with this issue, fixed-time stabilization has been introduced (Polyakov, 2012). Fixed-time stability aims to predefine and adjust a uniformly bounded settling-time. Such a method allows for stabilizing the system’s states in a fixed-time independently of initial conditions. This excellent property is very useful in practical scenarios of the quadrotor since it enhances the system’s robustness. An up-to-date publication (Hou, Yu, Xu, Rsetam and Cao, 2020) develops a continuous-TSMC control for servo motor systems. Therein, the authors propose a novel full-order TSM surface based on the bi-limit homogeneous property, which inspires us in this paper.
In terms of enhancing the robustness of a control system, the DOBC and ADRC control methods are considered as active anti-disturbance control approaches that have drawn much attention. These methods have been considered as a promising solution to cope with the strong disturbances affecting the system. Moreover, they show their effectiveness and superiority over classical passive anti-disturbance control that fails in dealing with strong disturbances leading to the degradation of nominal control performance which may compromise the system’s stability (Wang and Li, 2018, Zolotas, 2014). For instance, the work in Shtessela, Shkolnikovb, and Levant (2007) provides an FTO observer with a smooth second-order SMC for missile guidance application. The same observer has been used in Yang, Li, Su, and Yu (2013) within the control of a permanent magnet synchronous motor (PMSM) subjected to mismatched disturbances. Similarly, in Wang, Lv, Zhang, Liu, and Er (2017) unknown disturbances affecting a marine vehicle are estimated and rejected. In the context of the ADRC control, the work in (Zhang, Yu, & Yan, 2019) comes up with a feedback trajectory tracking control for a marine surface vessel based on a new fixed-time extended state observer (FXESO).
A key observation from the above studies is that owing to the advantages of continuous-SMC techniques, continued efforts are needed in the research of the advanced theory and practice of this new and young fifth-generation of sliding mode techniques. There is, therefore, a definite need for improving and applying these control methods to a wide range of physical dynamical systems such as the quadrotor aircraft which is intended to operate in challenging flight conditions. Furthermore, more real-time implementations on dedicated hardware and embedded systems should be made available to validate the newly designed flight control algorithms. Such experiments are strongly recommended to bridge the gap between the theoretical findings and the practice, which is in the scope of the present note.
The main contributions of this paper can be summarized from both theoretical and practical aspects as follows:
• Two robust nonlinear controllers are designed for the attitude and position subsystems of the quadrotor UAV to deal with the trajectory tracking control problem under various disturbances. These disturbances include internal unmodeled dynamics, plant parameters variation, and external time-varying wind gusts. To reject disturbances and enhance the control system robustness, the DOBC and ADRC approaches are adopted by contrast to passive anti-disturbance control methods which are not robust enough when encountering strong disturbances. Thus, passive methods do not provide satisfactory results in disturbance rejection. An FTO is used within the DOBC while an FXESO is adopted within the ADRC control framework to establish an output-feedback control (velocity-free control). Our control strategy is based on a fixed-time CNTSMC control scheme combined with observers. A homogeneous full-order TSM surface is designed to develop a tailored CNTSMC control algorithm for quadrotor system control. The choice of such a sliding surface together with an appropriate reaching law in control design allows that the sliding motion is fixed-time stable independently of the system’s initial conditions. Which contributes to enhancing the system’s robustness against disturbances. Unlike the works (Du et al., 2019, Falcón et al., 2019, Jin et al., 2015, Ríos et al., 2019, Wang et al., Wang, Li et al., 2019), we provide a clear estimation of the settling-time during the reaching phase of the sliding motion. It is shown that the sliding surface admits a fixed settling-time uniform with respect to the initial conditions. Moreover, within the synthesized control framework, the control law is continuous, hence it overcomes the chattering problem of LSMC, TSMC, and NTSMC. In addition, the singularity problem encountered in traditional TSMC is avoided. Thus, the designed control strategy is effective and more applicable to the quadrotor system in practice. Besides, fixed-time convergence of both sliding surface and tracking error can be ensured to achieve fast convergence by contrast to asymptotic methods (Almakhles, 2020, Chen et al., 2016, Jia et al., 2017, Wang, Hua et al., 2019). Also, the closed-loop system stability is strictly proved by using the Lyapunov theorem and bi-limit homogeneous theory. Meanwhile, based on the ISS notion, it is shown that the closed-loop system is robust to nonlinear disturbance inputs.
• An experimental comparative analysis is carried out for our designed control algorithms and other control strategies. Extensive PIL experiments and real-time tests are conducted on a real quadrotor platform to illustrate the difference between some quantitative indexes for all the controllers regarding the error signals and the control efforts. The following well-known quantitative performance indexes are used, root-mean-square error (RMSE), integral of square error (ISE), integral of the absolute value of the input (IAU), and integral of the absolute value of the derivative of the input (IADU). This comprehensive study is made by paying more attention to the differences between some sliding mode-based controllers. This aims to highlight the attained improvements regarding chattering-free control, accuracy, and transient response. The most obvious improvement to emerge from our synthesized control strategy is that it has overall superiority than the other controllers. The control performances are enhanced concerning fast transient response, high precision, strong robustness, and disturbance rejection capability. Moreover, in addition to the finite-time SMC strategies presented in Falcón et al. (2019) and (Ríos et al., 2019), the present work provides one of the few investigations on new fixed-time CNTSMC control algorithms design and real-time implementation for the quadrotor system. Finally, for the control algorithms design, validation, and real-time implementation, an effective and reliable MBD framework is used. This framework allows for time-saving and avoiding coding errors.
The remainder of this paper is organized as follows. Section 2 is devoted to the preliminaries and problem statement. The proposed control strategy together with the stability analysis of the closed-loop system are presented in Section 3. Experimental results and discussions are illustrated in Section 4 to examine the theoretical findings. The paper is concluded in Section 5 with possible future research directions.
Section snippets
Preliminaries and problem statement
This section will introduce the notations used in the paper and some relevant mathematical definitions and lemmas employed in the control design and fixed-time stability proof. Our main results are exploited on these fundamental facts. Besides, the control problem is formulated in this section.
Main results
The control objective is to realize a robust cartesian trajectory tracking for the quadrotor system in the presence of multiple disturbances. This can be attained through a robust tracking of the position and attitude references.
The quadrotor’s dynamics (24) are nonlinear, underactuated, and strongly coupled. This aircraft system has six output variables but only four control inputs are available . Notably, the translational movements are directly achieved by the
Experimental results and discussions
To validate the obtained theoretical results and demonstrate the feasibility of our study, the proposed control approach has been applied to a real quadrotor system. The obtained results provide important insights into two major aspects. First, the chattering has been remarkably alleviated. Second, convergence speed, accuracy, and robustness have been improved.
To highlight the improvement attained with the proposed control strategy, comparative experiments are performed for the attitude control
Conclusion
In this paper, a solution has been worked out for the robust fixed-time attitude and position control problems of a quadrotor system subject to multiple disturbances. Thus, a new flight control system has been proposed to ensure an accurate cartesian trajectory tracking control in 3D state-space. The 6-DoF equations of motion of the quadrotor are derived based on the Newton–Euler formula. Subsequently, DOBC and ADRC control approaches have been creatively proposed. Basically, a homogeneous
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to express the sincerest gratitude to the Editor-in-Chief, the Associate Editor, and the anonymous reviewers whose insightful comments have helped to improve the quality of this paper considerably.
The authors would like to thank Pr. Jamshed Iqbal, Department of Computer Science and Technology, Faculty of Science and Engineering, University of Hull, HU6 7RX, UK, for his assistance and suggestions.
This work was supported by the National Natural Science Foundation of China
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