Robust Adaptive Multilevel Control of a Quadrotor

This study proposes a new multilevel control of a quadrotor with dynamic uncertainties and time-varying external disturbances. The quadrotor model is partitioned into three subsystems: the vertical position, the horizontal position and the rotational subsystems. First, a double loop integral fast terminal sliding mode control with an adaptive estimator for disturbances’ upper-bounds (ADIFTSMC) is proposed for the altitude subsystem to ensure that the quadrotor reaches the desired height. Secondly, a radial basis function neural network backstepping controller (RBFNNBC) is applied to the horizontal subsystem. Finally, by combining a finite time exact disturbance observer with backstepping nonsingular fast terminal sliding mode control (FDOBNFTSMC), the rotational angles converge to the reference angles in the presence of the time-varying disturbances. Furthermore, a Lyapunov stability analysis is used to prove that the tracking errors converge to a small neighborhood of the origin. Numerical simulations illustrate the feasibility of the compound control structure.


I. INTRODUCTION
The applications of unmanned aerial vehicles (UAV), quadrotors to be specific, have increased in many areas. The main attributes of quadrotors lie in hovering, very fast maneuvers, vertical take-off and landing (VTOL), low cost and small size [1]- [4]. Quadrotors have several applications such as military surveillance, exploration, wild fire surveillance, delivery, aerial photography, rescue missions, agriculture, mapping, e.t.c [5]- [7]. The quadrotor dynamic model is underactuated strongly coupled nonlinear system. Implementing robust control for such systems is challenging and have attracted tremendous interest from the field automatic control. In recent years, researchers have proposed various nonlinear control methods to cope with the varying operating conditions of the quadrotor [8], [9].
The dynamic model of a quadrotor is usually in strict feedback form which is appropriate for designing backstepping control [10]- [13]. In order to eliminate the steady state errors, an integral error term is included in the backstepping design [14], [15]. Nonetheless, these papers did not account for the uncertainties and external disturbances. An adaptive The associate editor coordinating the review of this manuscript and approving it for publication was Shihong Ding . backstepping controller was used to addressed the uncertainties [16]- [18]. However, adaptive control methods are limited to systems with constant or slow time varying disturbances. A sliding mode control (SMC) technique is essential for mitigating all kind of bounded uncertainties and disturbances [19]- [21]. A SMC was combined with adaptive backstepping approach for the altitude and the attitude tracking of an octorotor [22]. Furthermore, fuzzy logic system (FLS) and neural networks (NN) based controllers have recently caught the attention of researchers [23]- [28].
The singularity issue that may arise in the control of a quadrotor is avoided by using the unit quaternion representation of the quadrotor attitude [29]. In [30], a robust chattering-free SMC was studied for the quaternion model of a hovering quadrotor. A quaternion-based attitude tracking controller was proposed for a quadrotor subjected to external disturbances. Nevertheless, the quaternion representation of the quadrotor may bring ambiguities. A nonsigular fast terminal (NFTSMC) was suggested in [31], [32] to avoid singularities and achieve a tracking control of the quadrotor with actuator faults. A robust adaptive NTFSMC was designed in [33] for attitude and displacement tracking control of a quadrotor subjected to unknown modelling errors. An adaptive NFTSMC based on dynamic inversion has been designed for trajectory tracking of a quadrotor [34].
A cascade structure consisting of the orientation and the position dynamics of the quadrotor was proposed in [35]. The orientation part consists of the fully actuated subsystem while the position part consists of the underactuated subsystem, which were managed by independent controllers. In [36], the quadrotor model is not only divided into fully actuated and underactuated subsystems, but also included the propeller subsystem to tackle the actuator faults. In [37], a linear parameter varying controller and feedback linearization controller were designed for the attitude and the position of a quadrotor respectively. In [38], an active disturbance rejection control (ADRC) and backstepping SMC have been constructed for the quadrotor attitude and the position respectively. In [39], a backstepping SMC and an internal model controller have been implemented for the quadrotor rotational and translational subsystems. In [40], an adaptive backstepping controller was implemented for the position subsystem, and an adaptive backstepping FTSMC was developed for the attitude subsystem. In [41], a regular SMC and backstepping SMC with adaptive fault observer have been presented for quadrotor attitude and position respectively. In [42], a backstepping controller and a chattering-free backstepping SMC have been presented for the quadrotor rotational and translational subsystems respectively. In [43], a backstepping control method was designed for the displacement subsystem, while a SMC strategy was utilised to control the attitude. In [44], a RBFNN based SMC was developed for the position subsystem and a robust integral of the signum error (RISE) was designed for the orientation subsystem. In [45], an integral SMC has been designed for the position subsystem and a backstepping SMC has been developed for the rotation subsystem.
In order to obtain an improved control performance and disturbance attenuation, a disturbance observer (DO) can be integrated into the quadrotor control design. A nonlinear DO assimilated with SMC was used to control the hovering state of a quadrotor [46]. A DO based attitude tracking controller was developed in [47] to estimate the Coriolis terms and external disturbances. The actuator faults and the uncertainties hampering the performances of the quadrotor have been tackled using the DO based controllers [48]- [50]. Contrary to the traditional DO which guarantees asymptotic stability, the finite time DO (FDO) can attained a finite time stability [50]- [53].
In the light of the foregoing discussion, we present a new robust control scheme for a quadrotor under time-varying disturbances and model uncertainties. The proposed control strategy is divided into into three subcontrollers: ADIFTSMC for altitude control, RBFNNBC for horizontal position control, and FDOBNFTSMC for rotational angle control. The main contributions of the proposed method are presented as follows 1) Contrary to [37]- [43], [45] where the quadrotor control design is divided into two subcontrollers, we divided the proposed control scheme into three subcontrollers: the altitude, the horizontal position and the attitude controllers. 2) Unlike the adaptive NFTSMC [33], ADRC [38], backstepping FTSMC [40], RISE [44], backstepping SMC [45] implemented for the attitude control, in this paper, a backstepping is combined with NFTSMC and FDO (FDONFTSMC) to control the attitude angles. The advantage of this scheme is that the disturbances are exactly estimated and guarantees the robust trajectory following in finite time. 3) In contrast to Adaptive backstepping [40], backstepping [43], RBFNN based SMC [44], integral SMC [45] developed for the position subsystem, a RBFNN based backstepping is devised to stabilised the horizontal position subsystem and produce the desired Euler angles. The RBFNN identify the uncertain functions together with the external disturbances and the time derivative of the virtual control laws. 4) For the first time, an ADIFTSMC is proposed to control the altitude of the quadrotor in the face of environmental disturbances and dynamic uncertainties. This controller is free from singularities and integrates the advantages of integral and fast terminal SMC. Moreover, it requires no knowledge of the disturbances' upper-bounds. 5) The finite time stability of the multilevel control, ADIFTSMC-RBFNNBC-FDOBNFTSMC, has been established using the Lyapunov function. The remainder of this article is arranged as follows. Section II presents the quadrotor dynamic model. Section III displays the design of the proposed control strategy. Section IV demonstrates the simulation and comparison results. We conclude the paper in Section V.

II. QUADROTOR DYNAMIC MODEL
The schematic representation of the quadrotor is depicted in Fig. 1. The quadrotor dynamic model is a six degrees of freedom underactuated system with four inputs. The dynamic model is given by [19] where x,y,z stand for the position of quadrotor centre of gravity, φ, θ, and ψ denote the pitch, roll and yaw angles respectively, r is the moment of inertia of the rotor, I x , I y , I z are the moment of inertia of the quadrotor, i (i = x, y, z, φ, θ, ψ) represent the time-varying disturbances, Schematic for the quadrotor system [14].
The relation between the thrusts, u i (i = 1, 2, 3, 4), and the angular velocities of the four propellers, ω i (i = 1, 2, 3, 4), is given by where C d and K p denote the drag and aerodynamic coefficients respectively, g stands for the acceleration due to gravity, V x , V y and V z are false inputs created to take care of the under-actuated subsystem. From (8), the total thrust u 1 and the desired angles (φ d , θ d ) can be derived as

III. CONTROL DESIGN
In this section, a nonlinear robust adaptive controllers are developed to steer the quadrotor states to track the reference trajectories in the premise of external disturbances and model uncertainties. The control block diagram is illustrated in Fig. 2. We designed ADIFTSMC, RBFNNBC, and FDOBNFTSMC for the quadrotor vertical position, horizontal position and attitude subsystems respectively.
Theorem 1: Consider the altitude subsystem (3), the DIFTSM surface (12), if the controller is constructed as α i (i = 1, 2, 3) are the estimate of α i (i = 1, 2, 3) and they are updated online by the following adaptation laws in which γ i > 0 (i = 0, 1, 2) are constant adaptation gains, then the altitude closed-loop system is ultimately bounded.

Since
We achievė where Considering the Lyapunov function (18), we get According to the above proof, all the error signals, S z , α i , (i = 0, 1, 2) in the closed loop system are guaranteed to be ultimately bounded in the compact set defined by ϑ z ≡

B. HORIZONTAL POSITION CONTROL
The horizontal subsystem can be rewritten as = [ x y ] T . Define the tracking error variable e χ = χ − χ d . The Lyapunov function is selected as L 1 = 1 2 e 2 χ . The derivative of L 1 with respect to time is thuṡ The virtual controller is designed as α χ =χ d − χ e χ , with χ = T χ > 0 is a constant diagonal matrix. The error betweenχ and α χ is computed as ε χ =χ − α χ . Substitutinġ (27) yieldṡ The time derivative of χ yieldṡ By utilising the approximation property of RBFNN, we have where W = [W T x W T y ] T is the optimal weight vector, ξ = diag{ξ x ξ y } is the basis function matrix with h i as the width, c i is the center, ϕ = [ϕ x ϕ y ] T is the RBFNN approximation error vector satisfying ϕ ≤ ϕ * , with ϕ * being a positive constant. The second Lyapunov function is define as follows whereŴ is the estimate of W , τ > 0 is a constant. Proof 2: Substituting (33) and (34) into (32), one getṡ The following inequalities exist By integrating (36) and considering the definition of the Lyapunov function (31), one can have Therefore, the error signals e χ , ε χ andW are bounded in the compact set ϑ χ ≡ {ζ χ : The attitude subsystem is as follows where η = [φ, θ, ψ] T is the orientation of the quadrotor, A finite time disturbance observer [55] can be designed for (38) as follows where µ 0 , µ 1 and µ 2 are the estimate of , η and˙ η , respectively, sig ρ (χ) = |χ| ρ sgn(χ). Subsequently, η can be exactly estimated in finite time.
Theorem 3: Consider the attitude subsystem (38) with control law (44)-(47) for reaching and staying on the surface (43), and supposed that the parameters of the FDO (39) are properly selected so that the disturbances can be exactly estimated, then the attitude signals track the desired angles in finite time.

IV. SIMULATION
In this section, a comparison of the simulations is carried out to demonstrate the effectiveness of the proposed control scheme. The efficacy of the proposed ADIFTSMC-RBFNNBC-FDOBNFTSMC has been compared with the controllers in [40] where an adaptive backstepping (AB) and adaptive backstepping FTSMC (ABFTSMC) were designed for the position and the attitude subsystems respectively, and in [45] where integral SMC (ISMC) and Backstepping SMC (BSMC) were utilised for the attitude and the position subsystems respectively.
The quadrotor parameters are obtained from [19]. The initial condition of each of the quadrotor states is 0.001. The time varying external disturbances are modelled as sinusoidal functions i = 2sin(0.8t) (i = x, y, z, φ, θ, ψ). The control design parameters are given in Table 1. The desired signals are given by   Fig. 3, it can be inferred that the ADIFTSMC/RBFNNBC provide the highest trajectory tracking precision and the quickest dynamical responses compared to both AB and ISMC. Moreover, the tracking error responses in Fig. 4 show that the ADIFTSMC/RBFNNBC converges to zero in the shortest time.
In Fig. 5, the comparison between the FDOBNFTSMC, BSMC and ABFTSMC has been presented. It can be observed that in the presence of the disturbances, the attitude angles followed the reference angles with greater accuracy under the FDOBNFTSMC. Contrary to BSMC and ABFTSMC, the FDOBNFTSMC can exactly estimate and compensate the time varying disturbances in finite time. The tracking responses of the yaw angle are demonstrated in Fig. 6. This figure illustrates that the tracking errors converge to zero in the shortest time under FDOBNFTSMC. The control inputs are presented in Fig. 7.

V. CONCLUSION
In this article, a novel multilevel control of a quadrotor has been developed to guarantees the robust trajectory following in the presence of model uncertainties, system coupling and environmental disturbances. The nonlinear control method is divided into three subcontrollers. Firstly, an ADIFTSMC is developed for the altitude subsystem to keep the quadrotor flying at the desired height irrespective of the external disturbances. Then, the horizontal position subsystem is controlled with a RBFNNBC to stabilised it and generate the reference Euler angles for the attitude subsystem. The RBFNN approximates the uncertain dynamics, external disturbances and the derivatives of the virtual control inputs. Subsequently, a FDOBNFTSMC has been developed for the attitude subsystem. Numerical simulations have shown the superiority of the proposed multilevel control structure, ADIFTSMC-RBFNNBC-FDOBNFTSMC, over some existing methods. The finite time stability of the overall system is presented based on the Lyapunov theory. In future work, the effectiveness of the proposed scheme will be investigated in real time control of a quadrotor.