Adaptive Robust Dynamic Surface Integral Sliding Mode Control for Quadrotor UAVs under Parametric Uncertainties and External Disturbances

A robust adaptive fuzzy nonlinear controller based on dynamic surface and integral sliding mode control strategy (ADSISMC) is proposed to realize trajectory tracking for a class of quadrotor UAVs. In this study, the composite factors including parametric uncertainties and external disturbances are added to controller design, which make it more realistic. +e quadrotor model is divided into two subsystems of attitude and position that make the control design become feasible. +e main contributions of the proposed ADSISMC strategy are as follows: (1) +e combination of dynamic surface and integral sliding mode makes the system always in sliding stage by finding the appropriate initial position compared with the common sliding mode, and the complexity of explosion in backstepping method is eliminated. (2) By introducing the fuzzy system, the unknown functions and uncertainties can be approximated which significantly improves the robustness and the tracking performance. (3) +e switching control strategy is utilized to compensate for the errors between estimated and ideal inputs; the tracking performance of the whole system has been significantly improved. +e simulation results show the effectiveness of the proposed control method.


Introduction
As a newborn member of the small unmanned aerial vehicle (UVA) family, quadrotor has attracted much research interest due to its extensive utility in several important applicants, such as commercial photography, military surveillance, rescue mission, and agricultural investigation [1][2][3][4]. Compared with traditional unmanned fixed-wing flight vehicles and manned airplanes, the main advantages of the quadrotor lie in small size, low cost, stable hovering, vertical take-off and landing (VTOL), convenient portability, and versatile features [5]. However, trajectory tracking control of the quadrotor is a thorny problem because of its nonlinear, underactuated dynamics, and strong coupling [6,7]. Moreover, the quadrotor system is susceptible to external disturbances such as wind and nonlinear frictions. What is more, taking robustness of the trajectory tracking controller into consideration poses a bigger challenge [8].
In early quadrotor research stage, many studies used conventional linear control methods such as proportionalintegral-derivative (PID) [9,10] and linear quadratic regulator (LQR) [11] to design the quadrotor controller in order to improve the simpleness and practicability. e linear control technology was developed to stabilize the quadrotor by neglecting the unimportant factors and linearizing the dynamic model. erefore, it is poor and even not acceptable for the tracking accuracy and the robustness of the quadrotor. To overcome the drawbacks of the aforementioned linear control approaches, a large number of nonlinear control strategies, including backstepping control [12][13][14][15], sliding mode control (SMC) [16][17][18][19][20], and feedback linearization control, are utilized to improve the tracking performance of the flight control system.
Backstepping control method has received extensive attention, not only in the quadrotor control, but also in some mechanical systems. e main idea of backstepping technology is to select the appropriate state variable function as the recursive virtual control input. Once the final control input is received, the stability of the whole system is guaranteed. In [21][22][23][24], a backstepping controller has been designed to stabilize the attitude system of quadrotor. In [24], attitude control using hybrid backstepping methodology based on Frenet-Serret theory is studied in detail. e results show that the controller has good robustness under wind disturbance. To solve the problem of trajectory tracking, in [25], an adaptive controller combining parameter adaptive and backstepping control is designed. However, the obvious limitation of conventional backstepping design is the problem of "complexity of explosion" caused by the repeated differentiation of some nonlinear functions and the lack of robustness against uncertainties. To overcome this limitation of traditional backstepping control, dynamic surface control (DSC) is proposed as an effective alternative method [26][27][28][29][30]. In [27], a dynamic surface control method based on RBF neural network approximation is proposed for a class of nonlinear time-delay systems with state variables all measurable, which greatly simplifies the design process of the controller.
Moreover, to enhance the attitude performance robustness, disturbance observer (DOB), parameter estimation [31], and the approximation-based adaptive control are generally combined with DSC to handle external disturbances and parameterized uncertainties. For instance, a class of adaptive control methods using fuzzy logic systems or neural networks to approximate unknown functions in nonlinear systems have been proposed in [32][33][34][35][36][37][38]. In [33], a dynamic surface control-based adaptive fuzzy control method is proposed to overcome the "explosion of complexity" problem of classical backstepping. In [35], a robust dynamic surface controller based on extended state observer is presented for a quadrotor UAV subject to external disturbances and parametric uncertainties. In [36], both indirect and direct global neural controllers with the dynamic surface design are developed for the strict-feedback systems. e simulation results are presented to demonstrate the feasibility of the proposed global neural DSC design. A robot control scheme based on dynamic surface considering output error constraints, unknown dynamics, and bounded disturbance has been proposed in [39]; by introducing an improved virtual variable, the robustness of the control system was improved. However, the performance properties and robustness are not taken into account in these papers. As a commonly used nonlinear control method, the SMC is utilized as an effective method to design robust controllers for a specific class of nonlinear tracking problems in the presence of uncertain conditions [40][41][42][43][44][45]. Traditional SMC features the low sensitivity to the disturbances and parameter variations of the system [46][47][48][49][50][51]. In [50], a method based on second-order sliding mode control is used to avoid the chattering phenomenon for quadrotor UAVs. In [51], a robust backstepping sliding mode nonlinear controller for quadrotor UAVs is proposed to improve the robustness of the controller against model uncertainty and external disturbances. Compared with traditional sliding mode control method, the integral sliding mode (ISM) can guarantee that the system always meets the desired dynamic performance index during the whole arrival period which significantly improves the robustness of the control system.
Motivated by the aforementioned observations, a new control methodology combined with dynamic surface and ISMC is proposed for the quadrotor trajectory tracking problem under parametric uncertainties and external disturbances. e main contributions of this paper are summarized as follows: (1) By fusing the technique of DSC and the integral SMC, a new integral sliding mode robust dynamic surface trajectory tracking controller is designed, which eliminates the "explosion of complexity" in the backstepping and improves the robustness of the whole system. (2) e FLSs are introduced to approach the ideal control law. And the estimations of the weight vector norm are utilized in the FLSs to significantly reduce the number of online estimation parameters. erefore, the amount of calculation is obviously reduced, and the structure of the proposed controller is simplified. (3) e adaptive switching control is introduced to compensate the error between the real control law and the ideal control law, and the tracking performance of the whole system has been significantly improved. e rest of this paper is organized as follows. e modeling of a quadrotor and some preliminaries are introduced in Section 2. e control algorithms are introduced in Section 3. Section 4 gives the stability analysis of the control system. Extensive simulations under different operating scenarios are given in Section 5. is paper ends with the conclusions in Section 6.

e Mathematical Model of UAV.
e quadrotor UAV is an underactuated system because it has six degrees of freedom, but only four actual inputs [1,2]. In this paper, the quadrotor UAV with four rotors is shown in Figure 1. e equations of the dynamic quadrotor UAV are basically a rotating rigid body with six degrees of freedom [4,5] which are usually derived by Newton-Euler formulas [8][9][10].
Define ξ � [ϕ, θ, φ] T and ω b � [p, q, r] T , with φ, θ, and ϕ being the angle of roll, pitch, and yaw with respect to the inertia frame. p, q, and r are the angular velocity of roll, pitch, and yaw with respect to the body-fixed frame. e rotation matrix from the rigid frame to the inertia frame can be expressed as where S (·) and C (·) denote sin(·) and cos(·), respectively. According to the rotation matrix R t , the relationship between _ ξ and ω b can be described as can be got easily online. It should be noted that to make the roll and pitch angles physically meaningful, they are both limited to (− π/2, π/2). In particular, the yaw angle is also limited to (− π/2, π/2) in this study, while P � [x, y, z] T ∈ R 3 is the position with respect to the inertial frame. e translational dynamic equations of the quadrotor are given as where d x , d y , and d z are the air drag coefficients which are added in (8) to model the drag force caused by translational motions; F is the lift force generated by rotors with respect to the body-fixed frame.
By combining (7) and (8), a compact affine nonlinear equation of the quadrotor UAV is given as 12 is the state variable; f(X) and g(X) are smooth functions on X. Equation (10) is expended as follows: where U i , i � 1, 2, 3, 4 represents the control inputs defined as follows: Jy , Rotor 3 Rotor 2 Rotor 1

Fuzzy Logic Systems (FLSs).
In this study, the Fuzzy Logic Systems (FLSs) are introduced to approximate the continuous unknown functions on a given compact set. e FLSs consist of three main parts: fuzzy rule base, fuzzification, and defuzzification operators. e form of the fuzzy rules of the fuzzy controller is Rule l: If x 1 is F l 1 and x 2 is F l 2 and . . .and x n is F l n . en y is G l , l � 1, 2, . . . , N. where x(t) � [x 1 , x 2 , . . . , x n ] T and y are the input and output of the whole fuzzy system, respectively. And N is the number of the rules. e fuzzy basis functions forms are defined as where y l � max y∈R μ G l (y). Denoting then equation of the fuzzy system can be rewritten as sup Inverse transformation x, y, z, ϕ, θ, φ  Complexity where α * is the optimal fuzzy parameter vector, ε(x) is the approximation error satisfies ‖ε(x)‖ ≤ ε ε > 0.

DSISMC Controller Design Procedure
In this section, the design process of dynamic surface integral sliding mode controller is proposed and the control system block diagram is shown in Figure 2. e controller design process is divided into two parts: the position tracking controller design and attitude tracking controller design, as shown in Tables 1 and 2, respectively. To make the presentation clear, the specific controller design process is given in Appendix A. In Table 1, e i , (i � 1, 2, 3) are the tracking error, and x i , (i � 2, 4, 6) are the virtual laws. e low-pass first-order filters (T1.3), (T1.9), and (T1.15) are presented in each step to get a new variable x id , (i � 2, 4, 6) with the time constants τ i , (i � 1, 2, 3). e integral sliding mode surfaces S i , (i � 1, 2, 3) are selected in each step which improves the robustness of the system against disturbances and parameters uncertainties. Due to the existence of unknown functions and parameters, the fuzzy system v ifs , (i � 1, 2, 3) are utilized to approximate v i , (i � 1, 2, 3) with α i , (i � 1, 2, 3) and ξ i , (i � 1, 2, 3) being the adjustable parameters and fuzzy basis vectors, respectively. en, the switching control law v ivs , (i � 1, 2, 3) are introduced to compensate the error of v i , (i � 1, 2, 3) and the ideal input. Step 1 (T1.18) Table 2: Attitude control algorithm of UAVs.

Stability Analysis of the Closed-Loop System
In this section, stability analysis of the proposed control system is established to confirm that all signals in the closed loop are ultimately bounded. e errors of the first-order filter are presented as follows: e derivative of y i (i � 2) in time can be obtained as then, one can obtain where is a continuous function. e following formula can also be obtained: en, the following inequalities hold: 4,6,8,10,12).
Consider the Lyapunov function candidate with and en, the following theorem can be obtained.    Proof.
e specific proof process is presented in Appendix B.

Simulations
In this section, the following simulations are given to validate the effectiveness and the performance of the proposed adaptive dynamic surface integral sliding mode control. e parameters for the quadrotor UAV adopted in this paper are presented in Table 3 Figure 3 shows the 3D tracking trajectory by using the proposed controller. e position and yaw angle trajectories are shown in Figure 4, and the tracking errors are given in Figure 5. Figure 6 shows the control signals. Figure 7 shows the change of roll and pitch angles. From the tracking performance, it can be concluded that the proposed control scheme can guarantee that all the variables are bounded and that the control system has strong robustness. Figures 8 to 10 illustrate the tracking performance under the different uncertainty cases. e proposed scheme has the better robustness against external disturbances and uncertainty parameters. Figures 11 to 13 show the tracking error comparisons between the ADSISMC, the ADSC, and the ASMC methods. Meanwhile, the maximum values (MVTE) and the root mean square values (RMSVTE) of tracking error in steady of the proposed scheme and the other two schemes are given in Table 4. e simulation results show that the proposed scheme has better tracking performance and robustness compared with the ADSC and the ASMC methods.

Conclusion
is paper proposed a dynamic surface integral sliding mode control scheme for a quadrotor UAV under the conditions of parameter uncertainty and external disturbances. Virtual control inputs are introduced in the robust controller design to guarantee the trajectory tracking performance, and the problem of "explosion of complexity" in the backstepping design has been greatly simplified. e fuzzy systems are utilized to approximate the ideal control inputs and the switch control is introduced to compensate for errors between estimated and ideal inputs which improves the control performance and robustness of the whole system. In addition, the stability analysis of the overall system through Lyapunov stability theory is presented, and all signals of the closed loop are ultimately bounded. Finally, the simulation results show that robustness and improved tracking performance can be achieved with the proposed control scheme.

A. The Controller Design Procedures
Step 1. Define the position error: where x 1 d is the desired x position command, and the derivative of e 1 with respect to time is Define the virtual control x 2 , where c 1 is a positive constant. To solve the problem of "complexity of explosion" caused by the repeated differentiation, a new state variable x 2 d is introduced and let x 2 pass through the following first-order filter with constant τ 1 (T1.3) to obtain x 2 d where x 2 d is the output of the first-order filter; the filter error is y 1 � x 2 d − x 2 . A proper integral sliding mode manifold is chosen (T1.4): where k 1 and k 2 are both the positive constant. If the sliding mode control is in an ideal state, the derivative of S 1 with respect to time is Assuming that the perturbations and parameters in the equation are known, the control law for x motion in an ideal state is designed as follows: In most cases, the system parameters are uncertain and there are also unknown external disturbances, which makes it difficult to obtain the ideal control signalv * 1 . erefore, the fuzzy system is used to approximate the ideal control signalv * 1 and obtain v * where ε 1 is the approximation error, and |ε 1 | < E 1 . Introducing switching control law v 1vs (T1.5) to compensate for v * where E 1 is the estimation of E 1 . en, the actual control law (T1.6) is obtained as Consider the Lyapunov function where η 1 and ρ 1 are positive constant. e derivative of Γ 1 with respect to time can be presented below: (A.14) So Substituting (A.15) into (A.13), then (A. 16) e adaptive law and the switching control (T1.5) are chosen below: And formula (A.16) becomes To make the Lyapunov function _ Γ 1 ≤ 0, the adaptive law of switching control is updated below: en, e similar design processes are presented to design the trajectory tracking of y-axis position x 3 and z-axis position x 5 using the dynamic surface integral sliding mode control. Introduce the variables v 2 � (C x7 S x9 S x11 − S x7 S x11 )U 1 and v 3 � (C x7 C x9 )U 1 , the specific procedures are presented in Step 2 and Step 3.
Step 2. Define the position error: where x 3 d is the desired y-position command, and the derivative of e 3 with respect to time is Define the virtual control x 4 , where c 2 is a positive constant. A new state variable x 4 d is introduced and let x 4 pass through the following first-order filter with constant τ 2 (T1.9) to obtain x 4 d where x 4 d is the output of the first-order filter; the filter error is y 4 � x 4 d − x 4 . A proper integral sliding mode manifold is chosen (T1.10): where k 3 and k 4 are the positive constant. If the sliding mode control is in an ideal state, and the derivative of S 2 with respect to time is en, a variable v 2 � (C x7 S x9 S x11 − S x7 S x11 )U 1 is introduced to be a new control input. Assuming that the perturbations and parameters in the equation are known, the control law for y-motion in an ideal state is designed as follows: In most cases, the system parameters are uncertain and there are also unknown external disturbances, which makes it difficult to obtain the ideal control signal v * 2 . erefore, the fuzzy system is used to approximate the ideal control signal v * 2 and obtain v * where ε 2 is the approximation error, and |ε 2 | < E 2 . Introducing switching control law v 2vs to compensate for v * 2 and v 2fz 14 Complexity where E 2 is the estimation of E 2 . en, the actual control law (T1.12) is obtained Consider the Lyapunov function where η 2 and ρ 2 are positive constant. e derivative of Γ 2 with respect to time can be presented as below: (A. 35) e adaptive law (T1.11) is chosen below: And formula (A.35) becomes To make the Lyapunov function _ Γ 2 ≤ 0, the adaptive law of switching control is updated below: Step 3. Define the position error: where x 5 d is the desired z-position command, and the derivative of e 5 with respect to time is Define the virtual control x 6 , where c 3 is a positive constant. A new state variable x 6 d is introduced and let x 6 pass through the following first-order filter with constant τ 3 (T1.15) to obtain where x 6 d is the output of the first-order filter; the filter error is y 6 � x 6 d − x 6 . A proper integral sliding mode manifold is chosen (T1.16): where k 5 and k 6 are the positive constant. If the sliding mode control is in an ideal state, and the derivative of S 3 with respect to time is en, a variable v 3 � (C x7 C x9 )U 1 is introduced to be a new control input. Assuming that the perturbations and parameters in the equation are known, the control law for z motion in an ideal state is designed as follows: In most cases, the system parameters are uncertain and there are also unknown external disturbances, which makes it difficult to obtain the ideal control signalv * 3 . erefore, the fuzzy system is used to approximate the ideal control signal v * 3 and obtain v * where ε 3 is the approximation error, and |ε 3 | < E 3 . Introducing switching control law v 3vs (T1.17) to compensate for v * 3 and v 3fz where E 3 is the estimation of E 3 . en, the actual control law (T1.18) is obtained as Consider the Lyapunov function where η 3 and ρ 3 are the positive constant. e derivative of Γ 3 with respect to time can be presented as below: (A.54) e adaptive law (T1.17) is chosen below: And formula (A.54) becomes (A.56) To make the Lyapunov function _ Γ 3 ≤ 0, the adaptive law of switching control is updated below: (A.57) en, (A.58) By associating v 1 ,v 2 , andv 3 , a group of virtual controls is obtained as follows: Remark 2. v 1 ,v 2 , andv 3 are combinations of available terms that can be given directly or measured above. erefore, the control input U 1 can be solved by regarding them as known in the controlled system (A.59). Apparently, (A.59) has four unknown variables, namely, x 7 , x 9 , x 11 , and U 1 . However, x 11 d is usually given as an extra reference signal in advance and the integral SMC controller is designed above to ensure the rapid convergence of x 11 to x 11 d . us, x 11 is regarded as known and can be replaced by x 11 d in this situation, and the unknown variables are reduced. So, we can obtain the unknown variables as follows: where x 7 d and x 9 d are desired the roll and pitch angle trajectory, and U 1 is part of the ultimate control laws, a � cos(x 11 d ) and b � sin(x 11d ).
In the attitude tracking system, x 7 d , x 9 d , and x 11 d are taken as the desired attitude trajectory, and the design procedure of attitude tracking contains three steps.
Step 4. Define the roll error: where x 7 d is the desired roll command, and the derivative of e 7 with respect to time is: Define the virtual controlx 8 , where c 4 is a positive constant. A new state variable x 8 d is introduced and let x 8 pass through the following first-order filter with constant τ 4 (T2.3) to obtain x 8 d where x 8 d is the output of the first-order filter; the filter error is y 8 � x 8 d − x 8 . A proper integral sliding mode manifold is chosen (T2.4): where k 7 and k 8 are the positive constant. If the sliding mode control is in an ideal state, the derivative of S 4 with respect to time is Assuming that the perturbations and parameters in the equation are known, the control law for roll motion in an ideal state is designed as follows: (A.67) In most cases, the system parameters are uncertain and there are also unknown external disturbances, which makes it difficult to obtain the ideal control signalU * 2 . erefore, the fuzzy system is used to approximate the ideal control sig-nalU * 2 and obtain where ε 4 is the approximation error, and |ε 4 | < E 4 . Introducing switching control law U 2vs (T2.5) to compensate for where E 4 is the estimation of E 4 . en, the actual control law (T2.6) is obtained as � − a 4 x 10 x 12 − a 5 ϖx 10 + a 6 x 8 − d 4 + _ Substituting (A.74) into (A.72), then And formula (A.75) becomes (A.77) To make the Lyapunov function _ Γ 4 ≤ 0, the adaptive law of switching control is updated below: (A.78) en, (A.79) e similar design processes are presented to design the trajectory tracking of pitch x 9 and yaw x 11 using the dynamic surface integral sliding mode control. e integral sliding mode manifolds (T2.10), (T2.16) and the first-order filter (T2.9), (T2.15) are chosen properly. e adaptive law and the switching control (T2.11), (T2.18) of the pitch and yaw are presented in Table 2. And (T2.12), (T2.18) are the actual control inputs of the pitch and yaw equation. c 5 , c 6 , τ 5 , τ 6 , k 9 , k 10 , k 11 , k 12 , η 5 , η 6 , ρ 5 , and ρ 6 are positive constants which need to be assigned to meet the performance requirements of the pitch and yaw angle system. e derivatives of Lyapunov function of pitch and yaw angle are presented as follows: (A.80)

B. The Proof of Theorem
Proof. Consider the Lyapunov function candidate (B.1) e derivative of V 1 with respect to time can be obtained as follows: Substitute the adaptive law and the switching control law into (B.2): Noting that, for any positive number λ,|y 2i B 2i | ≤ (y 2 2i B 2 2i /2λ) + (λ/2). Assume that |B 2i | < M 2i , where M 2i is a positive constant.
where α 0 is a positive constant.