Hyperbolic tangent function-based fixed-time event-triggered control for quadrotor aircraft with prescribed performance

Abstract

In this paper, the prescribed performance trajectory tracking problem of quadrotor aircraft with six degrees of freedom is addressed. Firstly, for the sake of facilitating the construction of controller, the aircraft is decomposed into position loop and attitude loop through time scale decomposition method. A fixed-time sliding mode controller is proposed to guarantee the convergence time of the aircraft system regardless of initial states. After that, to enhance security of control system, the hyperbolic tangent performance function is designed as performance index function to maintain the error within a prescribed range. Then, the event-triggered strategy is adopted to attitude subsystem which can significantly save communication resources, and the stability of control system is analyzed by Lyapunov method. In addition, the Zeno phenomenon is avoided which can be proved by ensuring the two consecutive trigger events have a positive lower limit. Finally, the validity of the constructed controller is confirmed by simulation results.

Introduction

Quadrotor unmanned aerial vehicles (QUAVS) have simple mechanical structure, good carrying capacity, and it is widely used in military field and civil field, such as parcel delivery, geological survey and military patrols in recent years. Therefore, it has become a popular research hotspot [1], [2], [3]. However, the quadrotor aircraft is a nonlinear system with characteristics of strong coupling and under-actuation [4]. In practical application, the quadrotor has enormous demand for the control precision due to its complex work environments. Then how to design a controller with preferable tracking performance for the quadrotor aircraft is still a difficult topic.

During past decades, with the continuous improvement of control strategy and extensive application of quadrotor aircrafts, a great deal of nonlinear control algorithms such as predictive control [5], adaptive backstepping control [6] and sliding mode control (SMC) [7] have been adopted to various nonlinear systems. In [8], an integral backstepping method integrated with SMC is employed to complete the mission of trajectory tracking and to assure the stability of quadrotor aircraft. In [9], based on the characteristics of uncertainties and external disturbances, a finite-time sliding mode controller is designed for the attitude loop. A nonlinear augmented observer is discussed for the under-actuated quadrotor aircraft in [10], then the speed of the system is evaluated from the position measurement and the robustness of the system is guaranteed. In [11], a robust controller combined backstepping with adaptive interval type-2 fuzzy logic is designed for the quadrotor attitude subsystem and position subsystem. In contrast with the traditional control methods, the higher robustness against actuator failure of the controller is satisfied. In [12], to track a given reference trajectory of the quadrotor with parametric uncertainties and external disturbances, a novel adaptive neural network sliding mode controller is constructed.

Although the previous studies have achieved excellent results in reference to the trajectory tracking of quadrotors, it is remarkable that they can only ensure the asymptotic stability of the system and they cannot adjust the transient performance such as overshoot and regulation time. Thus, the prescribed performance approach is presented to address this problem in many researches, which can guarantee the error of trajectory tracking reach an arbitrarily small residual set [13], [14]. In [15], the prescribed performance strategy is first proposed and applied to multi-in multi-out (MIMO) nonlinear systems, then it is further extended to various nonlinear systems. Hyperbolic function is applied as the performance function by some researchers to avoid the primary error is beyond the performance boundary [16]. In [17], an adaptive prescribed performance fault-tolerant controller is presented to address the attitude control of a rigid spacecraft with unknown disturbances, inertia parameters which are unknown but constant, input saturation, and actuator failures. In [18], a finite-time prescribed performance adaptive fuzzy tracking controller is constructed for the unknown nonlinear systems, then the error of output tracking reaches a stable state in finite time and is always constrained within a prescribed boundary. To solve the problem of parametric uncertainties and time-varying external perturbations in the closed-loop system, an adaptive neural finite-time controller integrated with prescribed performance is proposed in [19] to ensure the tracking error is constrained within the prescribed bounds.

It is worth emphasizing that the settling time of the above literatures is related to initial conditions. In the actual system, the settling time may become especially long when the initial conditions are large. To deal with this problem, Polyakov [20] propose the concept of fixed-time method. In [21], a fixed-time controller is proposed for the attitude tracking of reusable launch vehicle systems with uncertainties and external disturbances. According to [22], a nonsingular terminal sliding mode control method is investigated for the 6-DOF RLV attitude control with a mass of parameter uncertainties and atmospheric disturbances, and the excellent control performance is guaranteed. In [23], a fixed-time controller combined with prescribed performance is designed for the uncertain stochastic nonlinear systems affected by unknown measurement sensitivity and input quantization. Although many results have been achieved in fixed-time and prescribed performance, the application to quadrotor aircraft is still a problem worthy of further studying.

From the practical point of view, continuous communication may lead to large resource consumption and frequent updates of controllers, which is difficult to be implemented. The event-triggered strategy is an effective way to solve these problems. In [24], an adaptive event-triggered finite-time controller combined with prescribed performance is investigated for a kind of strict-feedback nonlinear systems with external disturbances, which can guarantee the tracking error reaches a small residual set in finite-time and sufficiently reduce the communication burden. In [25], an adaptive event-triggered fixed-time controller with prescribed performance is discussed for uncertain nonstrict-feedback nonlinear systems, then the signals involved are all bounded.

Despite of theoretical research for prescribed performance, fixed-time and event-triggered mechanisms have been studied in different systems and achieved satisfying control performance. However, there are few research results referring to safety, fast convergence and communication resources during the trajectory tracking of quadrotor aircraft. To achieve excellent tracking performance, a prescribed performance fixed-time event-triggered controller is proposed for the quadrotor aircraft in this paper. In comparison with the existing results, the primary contributions of this paper can be summarized as follows:

1) Compared with the existing constraint control strategies in Refs. [26], [27], hyperbolic tangent function is selected as index function to constrain the trajectory tracking errors, which can guarantee the error of trajectory tracking gets to an arbitrarily small residual set and improve the convergence accuracy. In addition, the transient performance such as overshoot, regulation time of the system, can be adjusted by the proposed prescribed performance strategy.

2) For the existing results on finite-time control in Refs. [28], [29], the convergence time is explicitly related to the initial condition of quadrotor. In the practical system, the convergence time will be extremely long if the initial position is too large, which limits its practical applications. As an improvement to finite-time control, the fixed-time strategy is implemented in this paper for both the position loop and the attitude loop of the quadrotor aircraft, where the tracking error can be guaranteed to converge zero in a fixed-time and the maximum convergence time can be obtained regardless of initial states.

3)Unlike the most existing event-triggered mechanisms in Refs. [30], [31], the measurement error in our research work is related to the controller, which can reduce difficulty of controller design in the actual system. Moreover, the attitude loop only needs to transmit information and update control laws at some discrete trigger time, while maintaining the fixed time stability of the quadrotor aircraft system, and the lower limit of the trigger interval is mathematically derived to ensure Zeno-free phenomenon.

The organizational structure of this essay is presented as follows. In Section 2, a dynamic model of the 6-DOF quadrotor aircraft is founded. Then, a fixed-time controller with prescribed performance is constructed in the position loop. In Section 3, an event-triggered fixed-time controller with prescribed performance is implemented in the attitude loop and the stability of this system is proved by Lyapunov function. After that, the simulation results demonstrate the validity and superiority of the proposed methods in Section 4. Finally, conclusions and future works are given in Section 5.