Output Feedback Adaptive Dynamic Surface Sliding-Mode Control for Quadrotor UAVs with Tracking Error Constraints

In this paper, a fuzzy adaptive output feedback dynamic surface sliding-mode control scheme is presented for a class of quadrotor unmanned aerial vehicles (UAVs). ,e framework of the controller design process is divided into two stages: the attitude control process and the position control process. ,e main features of this work are (1) a nonlinear observer is employed to predict the motion velocities of the quadrotor UAV; therefore, only the position signals are needed for the position tracking controller design; (2) by using the minimum learning technology, there is only one parameter which needs to be updated online at each design step and the computational burden can be greatly reduced; (3) a performance function is introduced to transform the tracking error into a new variable which can make the tracking error of the system satisfy the prescribed performance indicators; (4) the slidingmode surface is introduced in the process of the controller design, and the robustness of the system is improved. Stability analysis proved that all signals of the closed-loop system are uniformly ultimately bounded. ,e results of the hardware-in-the-loop simulation validate the effectiveness of the proposed control scheme.


Introduction
Quadrotor unmanned aerial vehicles (UAVs), as a new product in the field of small UAVs, have become a research hotspot among research and scholars all over the word [1][2][3][4][5].
e main advantages of quadrotor UAVs, such as flying in any direction, take off and land vertically, and hover at an ideal attitude, make the quadrotor UAVs widely used in more important fields, such as providing medical assistance, transporting special resources, disaster monitoring, and agricultural mapping. However, quadrotor UAVs are a complex physical system with the following characteristics, such as multivariate, nonlinearity, underactuating, and strong coupling, which make it very difficult to design an effective adaptive robust flight controller.
With the development of intelligence control theory, different kinds of advanced nonlinear control methods, which combine the linear control methods with intelligence control theory, such as feedback linearization control [12], model predictive control [13,14], adaptive backstepping control [15,16], adaptive sliding-mode control (SMC) [17][18][19][20], fault-tolerant control [21], dynamic surface control (DSC) [22][23][24][25], and adaptive fuzzy control [26,27], have been proposed to achieve attitude and position trajectory tracking performance of quadrotor UAVs. In [28], a novel neural network-based output feedback controller is developed for a group of quadrotor UAVs. In [29], the prescribed performance backstepping dynamic surface control (DSC) scheme is proposed to solve the problem of trajectory tracking control for a quadrotor UAV with control input saturation. In [5], a fuzzy-based compound quantized control strategy is applied to the Quanser Qball-X4 quadrotor experimental platform, which achieved precise position control and tracking performance.
Among the above control schemes, the backstepping strategy has been widely used in the controller design for nonlinear systems. For instance, in [30], the trajectory tracking controller based on the backstepping approach was developed for the quadrotor model, while the PD control was used to attenuate the effects caused by system uncertainties. In [16], a nonlinear disturbance observer-based backstepping control method has been proposed to address the problem of loss of actuators' effectiveness. However, one drawback of backstepping is the "explosion of complexity" caused by the recurrent derivation of the virtual control law in each design step. To deal with this problem, the DSC control method has been proposed for a class of nonlinear systems, by introducing a first-order low-pass filter in each design step, and the shortcoming was overcome [22,[31][32][33].
An effective way to deal with the uncertainty of system parameters and unmodeled dynamics is to design an adaptive controller using the general approximation ability of the fuzzy logic system (FLS) and neural networks (NNs) [34][35][36]. e number of adaptive laws which depends on the fuzzy rules or the NN weights will be significantly increased as the number of fuzzy rules or the NN weights increase. To overcome this problem, a new method by estimating the norm rather than each item of the weight vector was proposed in [37,38].
us, the number of adaptive laws is reduced significantly. Actually, the quadrotor UAVs are not only time-varying coupled and nonlinear systems, but also suffer from perturbations such as payload variations and nonlinear friction. erefore, it is necessary to design a controller with adaptive capability, fast convergence, and robust performance for the quadrotor UAVs.
As a widely used nonlinear control algorithm, the sliding-mode control (SMC) is known for its excellent performance properties for complex high-order nonlinear systems in the presence of uncertain conditions [39,40]. In [18], the SMC trajectory tracking controller was proposed for quadrotor UAVs by considering the wind perturbations and external disturbance components. In [19], a hierarchical control strategy based on the double-loop integral slidingmode controller was designed for the position and attitude tracking of quadrotor UAVs with sustained disturbances and parameter uncertainties. Most of the existing literature focuses on using the SMC method to solve the attitude control of quadrotor UAVs instead of the position trajectory tracking control design because the transformed dynamic equation has a preferred form for the attitude control.
However, a major constraint in the controller design of quadrotor UAVs is that all the state variables of the system are required to be measurable. But in practical application, under some unpredictable factors, it will cause the measurement sensor to fail. [41][42][43][44][45]. In [41], an adaptive output feedback control scheme has been proposed for a class of uncertain SISO nonlinear systems under the constraint that only the system output can be obtained. In [43], a fuzzy state observer-based control method is designed for an uncertain MIMO nonlinear system, and by using the state observer, the problem of state immeasurability has been solved. Traditionally, the tracking performance in adaptive control schemes has been confined to ensure that the tracking error converges to a residual set, the size of which is determined by the explicit design parameters and some unknown bounded terms. e upper bounds of the tracking error are difficult to calculate, so it is a very practical work to make the prior selection of the tracking performance satisfy certain steady state behavior. In [46,47], a prescribed performance control scheme has been proposed for a class of nonlinear systems, and by constructing a prescribed performance function, the tracking error of the system was transformed into a new variable to ensure that the convergence rate was no less than a prespecified value, and the steady-state error remains within the prescribed range. However, limited attention has been paid to this issue for the controller design of quadrotor UAVs.
Inspired by the aforementioned discussions, an adaptive output feedback dynamic surface sliding-mode control for a class of quadrotor UAV system is presented where the fuzzy approximators are used to approximate the unknown items of the system. e main contributions of the proposed control scheme are as follows: Firstly, to our best knowledge, this is the first time to use the dynamic surface control techniques with the slidingmode method to design and test the robust controller of quadrotor UAVs in the platform of hardware-in-the-loop simulation, leading to a greatly simplified structure of the controller and improved robustness of the system. Secondly, by introducing performance and error transformed functions in the controller, the tracking error of the quadrotor UAVs is transformed into a new error constraint variable which can ensure the prescribed transient performance of the system. irdly, by estimating the norm of the FLS weights instead of estimating each variables of the weight vector, there is only one parameter needed to be updated at each step. us, the computing time is reduced.
Finally, the nonlinear state observer is introduced to predict the unmeasurable state of the quadrotor such as the angular velocity state of the quadrotor. en, only the measurable attitude and position information are required in the implementation of the controller of the quadrotor. e rest of this paper is organized as follows. In Section 2, problem statement and preliminaries including the mathematical model of the quadrotor, fuzzy logic systems (FLSs), and performance function are introduced. e process of the controller design and analysis of stability are given in Sections 3 and 4, respectively. Section 5 shows the results of the hardware-in-the-loop simulation to validate the effectiveness of the proposed control scheme.

Dynamic Model of Quadrotor UAVs.
e schematic configuration of the quadrotor in this paper is shown in Figure 1. e basic movements are vertical movement, front and back movement, left and right movement, pitch rotation, roll rotation, and yaw rotation. On changing the rotor speed altogether with the same quantity, the lift forces will change, in this case, affecting the attitude of the vehicle. e complicated motions of a quadrotor can be divided into two typical parts, and each part represents a subsystem with coupled terms. e first part is associated with the translational positions, and the second part is associated with the rotational angles. And in this section, we will deduce the mathematical model of a quadrotor UAV, including navigation equations and moment equations.
Define Λ � [ϕ, θ, ψ] T ∈ R 3 and w � [p, q, r] T , where ϕ, θ, and ψ represent the roll angle, pitch angle, and yaw angle with respect to the inertia frame and p, q, and r denote the angular velocity of the roll, pitch, and yaw with respect to the body-fixed frame. Let R BG denote the transformation matrix between the inertia frame and the body-fixed frame using Euler-Lagrange formulation, which can be expressed as where S (·) and C (·) denote the sin(·) and cos(·), respectively. Let P � [x, y, z] T ∈ R 3 denote the position with respect to the inertia frame. According to Newton's second law, the relationship between combined force F G and acceleration in the ground coordinate is and we can get the translational dynamic equations of the quadrotor where D x � d x _ x, D y � d y _ y, and D z � d z _ z, in which d x , d y , and d z are the air drag coefficients; is the lift force generated by the rotors with respect to the body coordinate system, in which Ω i , i � 1, . . . , 4 denote the rotary speed of the front, right, rear, and left rotors and k is the lift coefficient of the rotor.
According to the kinetic equation, the relationship between Λ and w can be described as where T (·) denotes tan(·) and the transformation matrix Q(Λ) is bounded according to ‖Q(Λ)‖ F < Q(Λ) max for a known constant Q(Λ) max provided − (π/2) < ϕ < (π/2) and − (π/2) < θ < (π/2) [23]. M B0 is defined as the torque provided by the rotors with respect to the body-fixed frame and is described as follows: where l is the distance between a rotor and the center of mass of the quadrotor, F i (i � 1, . . . , 4) � kΩ 2 i denotes the lift provided by the rotor, and we get τ i � k ψ kΩ 2 i � τΩ 2 i , in which τ represents the antitorque coefficient. Using the Newton-Euler equation, we can get the rotational dynamic equation of the quadrotor: where J B � diag(J xx , J yy , J zz ) is a symmetric positive definite constant matrix with J xx , J yy , and J zz being the rotary inertia with respect to the O b x b , O b y b , and O b z b axes, the signal × represents the cross multiplication, and M r and M d are the resultant torques due to the gyroscopic effects and the resultant of the aerodynamic frictions torque. ey are given as where J R represents the moment of inertia of each rotor and d ϕ , d θ , and d ψ are the corresponding aerodynamic drag coefficients. From (6), the following equation can be obtained: where With the help of (4), the following equations can be obtained:  Complexity where

Remark 1.
It is worth noting to point out that the roll, pitch, and yaw angles are limited to (− π/2, π/2) which is physically meaningful. By combing (3) and (10), a nonlinear equation of the quadrotor UAV is given as follows: _ where T is the state vector and f(X) and g(X) are the smooth functions. e dynamic of quadrotor UAVs can be described as follows: where g is the gravity acceleration and U i , i � 1, . . . , 4 are the control inputs which can be expressed as Dividing the unknown parameters a i (i � 4, . . . , 11) into two parts, known part a iN and unknown part Δa i , it is expressed as follows:

Fuzzy Logic Systems (FLSs).
e FLS is composed of three main components: fuzzy rule base, fuzzification, and defuzzification operators. e fuzzy rule base comprises a collection of fuzzy "IF-THEN" rules of the following form: . , x n ] T ∈ U and y are the FLS input and output, respectively, N is the number of rules, and fuzzy sets F l i and G l are associated with the fuzzy membership functions μ F l i (x i ) and μ G l (y). rough the singleton function, center average defuzzification, and product inference, the FLS can be expressed as where y l � max y∈R μ G l (y). e fuzzy basis function is defined as 4 Complexity Denoting (15) can be rewritten as Lemma 1. According to [34], FLSs can effectively approximate any continuous nonlinear function with any small approximated error in a compact set. It can be expressed as where the continuous nonlinear function (17), and ε > 0 is the approximated error. erefore, F(x) can be described as where the minimum approximated error |σ * | ≤ ε and W * is an ideal weight vector and can be defined as

Performance and the Error Transformation Functions.
Similar to [46], the mathematical expression of the prescribed tracking performance is given by where e i (t) � y i − x id , i � 1, . . . , 6, are the tracking errors, the performance function p i (t) is defined as a smooth and decreasing positive function, and κ i and β i are the given positive constants. Moreover, κ i p i (0) and β i p i (0) represent the lower and upper bounds of the undershoot of e i (t) and − κ i p i (∞) and β i p i (∞) are the maximum allowable size of e i (t). e error transformation function is chosen as where S i is the transformed error variable and Υ i (S i ) is a smooth strictly increasing function with the following properties: Note that if S i is kept bounded, we have − κ i < Υ i (S i ) < β i , and thus (21) holds. e inverse transformation of Υ i (S i ) can be expressed as where S i , e i (t), and p i (t), are the transformed errors, the output tracking errors, and their corresponding performance functions.
In this paper, we choose and differentiating (25) yields where x id are the reference signals and e i (t) are the output tracking errors. From the properties of the transformation, it is clear that η i and v i are bounded and 0 < η i0 ≤ η i .

Remark 2.
It can be seen that a new variable S i is introduced through the above transformation process (21)- (25). If the designed control system can guarantee that S i is bounded, then the tracking error e i is bounded and meets formula (21).
is means that the tracking error is always kept within the e control objective is to design an adaptive controller so that S i is bounded.

Nonlinear Observer.
For a class of nonlinear systems with (A 0 , C) in the observer canonical form is given by with with x ∈ R n , y ∈ R, u ∈ R, b i ∈ R n (i ≥ 2), and f(x) is the unknown smooth function. e vector b is general and not in a restricted form. Only the output y is assumed to be Complexity 5 measurable [48]. For the uncertain system (27), the nonlinear state observer is established as where x is the estimation of the state x and K is the observer gain vector; K is chosen so that the characteristic polynomial of A − KC T is strictly Hurwitz. us, for a given a matrix e function f(x) is the estimation of f(x). In the next section, we choose FLSs to approximate f(x). According to (27) and (29), the observer error can be expressed as

The Process of the Controller Design
In this section, an adaptive FLS dynamic surface slidingmode control scheme is proposed for position and attitude trajectory tracking control. e structure of the proposed control scheme is shown in Figure 2. e recursive design procedure contains two parts. Part 1 is the position trajectory tracking control and part 2 is the attitude trajectory tracking control. Each part contains three design steps, which are shown in Tables 1 and 2. e details of the controller design process are shown in Appendix A.
It should be note that θ i , (i � 1, . . . , 6) are the estimations of θ i with θ i � ‖W * i ‖ 2 , and ‖W * i ‖ 2 and ξ i (X i ) are the ideal weight vector and fuzzy basis function vector of FLSs which are used to approximate the unknown continuous nonlinear function at each design step.

Remark 3.
For the attitude trajectory tracking control sysytem, χ 1 , χ 2 , and χ 3 can be regarded as known and the input U 1 can be solved. e denominator of U 1 will not cause singularity since the yaw angle is limited to (− π/2, π/2). Remark 4. In the traditional sliding-mode control method, the existence of the signum function will cause chattering in the control system. In practical applications, the saturation function sat(·) [49] or the hyperbolic tangent function tanh(·) [50] are generally used to eliminate the chatting phenomenon.
Proof. Please see Appendix B for details.

Hardware in the Loop Simulation Results
In this paper, the hardware-in-loop testing platform is used to verify the effectiveness of the proposed control scheme. e experiment environment and the experimental system architecture are shown in Figures 3 and 4, where the following components are included: Table 1: e proposed DSCSM design for position trajectory control.

Complexity
Case 1: normal case: we assume that there are no uncertainties in the model, and all parameters of the quadrotor are normal. e initial state vector is set to be x(0) � [0.02, 0, 0.02, 0, 0, 0, 0, 0, 0, 0, 0.1, 0] T . And, the fuzzy membership function is adopted as where l � 1, . . . , 5 and k � 2, 6, 8, 10, 12. Cases 2, 3, and 4: uncertainty (15%, 30%, and 50% added) in rotary inertia: in these cases, we consider three different model uncertainties 15%, 30%, and 50% separately added in the yaw axis. e experimental results are shown in Figures 5-15. Figures 5-9 illustrate the comparison experimental results of the tracking trajectory in Case1 between the proposed control method and the traditional PID control method. From Figures 6-9, it can be seen clearly that all the tracking errors of the position and yaw angles of the proposed control scheme are always kept within the performance function curves. at is, the control method proposed in this paper obtains much better control performance by comparing with the traditional PID control scheme. Figure 10 shows the control signals. Figure 11 shows the response curves of roll and pitch angles. Figures 12 and 13 show the change of six adaptive parameters. Figures 14 and 15 show the trajectories of x i and x i , (i � 2, 4, 6, 8, 10, 12). We can see that the proposed state observer can quickly approximate the output of the system. Figure 16 shows the 3D tracking trajectory with uncertainties 15%, 30%, and 50% added in the yaw rotating axes. Figure 17 illustrates the results of the tracking error under cases 2, 3, and 4. Also, the maximum value of the tracking error (MVTE) and the root mean square value of the tracking error (RMSVTE) in the steady state (t > 5 s) are        18 20 x 8 x 8 Figure 15: Actual value and estimated value.
Reference signal 15% 30% Normal case Complexity hown in Table 4. From Table 4, we can see that the proposed control scheme has strong robustness, and even the uncertainty in yaw rotary inertia is up to 50%.

Conclusion
In this paper, an adaptive dynamic surface sliding-mode output feedback controller has been proposed for attitude and position control of a class of quadrotor UAVs with consideration of parametric uncertainties and disturbances.
By using the norm estimation approach, there is only one parameter which needs to be updated online at each design step regardless of the plant order and input-output dimension. Also, by introducing an error transformed function, the tracking performance of the quadrotor UAV has been achieved. e proposed control scheme can not only eliminate the problem of "explosion of complexity" existing in the backstepping control scheme but also improve the robustness of the system. e results of the hardware-inloop simulation validate the effectiveness of the proposed (A.9) e stabilization of Γ 1 can be obtained by designing the virtual control (T1.5) and the adaptation law (T1.6), where c 2 , μ 1 , and λ 1 are the positive constants and ε 1 is an arbitrarily small positive constant. For the external disturbance d t1 encountered in the quadrotor flight process, the sliding surface is added to maintain system stability with μ 1 ≥ |d t1 |. Substituting (T1.5) and (T1.6) into (A.9), we get Similar design procedures can be used to design adaptive DSC sliding-mode laws for trajectory tracking control of y axis position (x 3 ) and z axis position (x 5 ). Introduce two variables χ 2 � (C x 7 S x 9 S x 11 − S x 7 C x 11 )U 1 and χ 3 � (C x 7 C x 9 )U 1 . e corresponding control laws and adaptive laws are designed as follows.
Step 2. Let S 3 given by (26) be the second error variable. en, the derivative of S 3 can be expressed as According to (29) and (T1.8), (A.11) can be rewritten as where c 3 is a positive constant. Introduce a new state variable x 4 d , which can be obtained by the following first-order filter: Define the error surface (T1.10), and the time derivative of S 4 is (A.14) where F 2 (X 2 ) � f(x 4 ), X 2 � x 4 . FLSs are used to approximate the unknown function F 2 (X 2 ): with respect to the unknown optimal weight vector in (A.15), define θ 2 � ‖W * 2 ‖ 2 , and since θ 2 is unknown, let θ 2 be the estimation of θ 2 and θ 2 � θ 2 − θ 2 . Choosing the following proper sliding surface σ s2 � S 4 , consider the second Lyapunov function: Differentiating Γ 2 , we obtain

. (A.17)
Using Young's inequality, it can be verified that (A. 18) en, (A.17) can be rewritten as (A. 19) e stabilization of Γ 2 can be obtained by designing the virtual control (T1.11) and the adaptation law (T1.12), where c 4 , μ 2 , and λ 2 are the positive constants and ε 2 is an arbitrarily small positive constant. For the external disturbance d t encountered in the quadrotor flight process, the sliding surface is added to maintain system stability with μ 1 ≥ |d t2 |. Substituting (T1.11) and (T1.12) into (A.20), we get (A.20) Step 3. Let S 5 given by (26) be the third error variable. en, the derivative of S 5 can be expressed as According to (29) and adaptive laws (T1.14), (A.21) can be rewritten as _ S 5 � η 3 x 6 + k 1 x 5 − x 5 − η 3 v 3 , (A. 22) and the virtual control signal can be chosen as x 6 d as x 6 d � − k 1 (x 5 − x 5 ) + v 3 − S 5 c 5 /η 3 with c 5 being a positive constant. Introduce a new state variable x 6 d , which can be obtained by the following first-order filter: Define the error surface (T1.16), and the time derivative of S 6 is _ where F 3 (X 3 ) � f(x 6 ), X 3 � x 6 . Utilizing FLSs to approximate the unknown function F 3 (X 3 ), we obtain Define θ 3 � ‖W * 3 ‖ 2 , and let θ 3 be the estimation of θ 3 and θ 3 � θ 3 − θ 3 . Choosing the following proper sliding surface σ s3 � S 6 , consider the third Lyapunov function: e differentiation of Γ 3 is as follows: