Based on robust sliding mode and linear active disturbance rejection control for attitude of quadrotor load UAV

In this paper, a mass adaptive control method combining with robust sliding mode control (SMC) and linear active disturbance rejection control (LADRC) is designed for the quadrotor load unmanned aerial vehicle (UAV) with mass variation. In detail, firstly, the mass variation in the quadrotor affects the position of its centroid, taking into account the changes of centroid position, which makes the established model more accurate. Moreover, a mass adaptive law is designed to eliminate the influence of mass variation. Secondly, SMC can enhance the robustness of the controller, improve the anti-disturbance performance and overcome the problem of low control precision caused by bandwidth limitation of LADRC. The linear extended state observer estimates the external disturbances of the system and the internal unmodeled dynamics caused by the SMC chattering in real time, and then, the total disturbance is compensated by the proportional–derivative controller. The proposed scheme combines the advantages of SMC and LADRC and complements each other. Thirdly, in order to simplify the parameter setting, the adaptive control is introduced in LADRC to adjust the controller parameters in real time, which is beneficial to the stability analysis of the control system. Then Lyapunov stability theory is used to prove the stability of the whole system. Finally, the simulation is compared with LADRC and dynamic surface active disturbance rejection control. The results show that the designed scheme has smaller overshoot and faster response speed, which proves its superiority. Moreover, the designed adaptive law is also effective, it can eliminate the influence of parameter deviation, so that the proposed scheme can track the reference signal stably even in the presence of disturbances.


Introduction
The quadrotor UAV has the advantages of simple structure and strong maneuverability. It has become a research hotspot and has been widely used in many fields [1,2]. The quadrotor load UAV can carry out tasks such as object handling, rescue and data collection [3], so the research on the quadrotor cannot be ignored.
The quadrotor has a complicated mathematical model, which makes it particularly difficult to control.
In [4], a robust Proportion Integration Differentiation (PID) controller is proposed to realize the quadrotor trajectory tracking task, which is combined with the power reduction method to effectively reduce the power used. In order to establish a more accurate mathematical model, in [5], the acceleration and velocity vector of the quadrotor are considered, and a nonlinear PID controller with parameter tuning is designed using genetic algorithm, but the method design is more complicated. Aiming at the yaw and altitude channels of the quadrotor, in [6], a multiplemodel adaptive controller is designed to eliminate the influence of parameter uncertainty on the quadrotor system. For the trajectory tracking control of the quadrotor, in [7], a robust adaptive recursive sliding mode control (ARSMC) scheme is adopted to effectively eliminate the influence of model uncertainty and unknown disturbance on the quadrotor attitude control system. In [8], considering the attitude and altitude control of the quadrotor system with bounded disturbances, an adaptive non-singular terminal sliding mode control (ANTSMC) is proposed to track the reference input in a finite time and eliminate the chattering effect. In [9], a global sliding mode control algorithm is proposed to realize the trajectory tracking of the quadrotor, which is robust to model uncertainty and external disturbances. In [10], a multilayer neural dynamic controller is designed to improve the control precision of the quadrotor, which can directly control the motor input to ensure that the quadrotor can effectively track the desired trajectory.
The structure of active disturbance rejection control (ADRC) technology is simple, and there is no need of the accurate mathematical model, so it can be used in quadrotor system with nonlinear and strong couplings [11]. In [12], an obstacle avoidance method based on the rotational potential field is designed, wherein ADRC is used to eliminate disturbances and couplings. In [13], a model-based control scheme is designed to enhance the anti-disturbance performance of the quadrotor system by using ADRC technology. However, there are too many parameters in this scheme, which is not conducive to parameter setting. Thus, in order to improve the performance affected by the input delay and disturbances, ADRC method is adopted in [14], which significantly improves the flight control performance of the quadrotor. However, the design is complicated with many nonlinear functions. In [15], a dynamic surface active disturbance rejection controller (DSADRC) is designed to solve the trajectory tracking problem of the quadrotor, which uses dynamic surface control law to replace the PD controller. However, the algorithm has many parameters and the control precision need further improved. In [16], in view of the complex coupling and strong nonlinearity of the powered parafoil system, an ADRC scheme is used to compensate its coupling. This method is taken the coupling as disturbance, and the extended state observer is used to estimate the total disturbance, which improves the control precision of the system. In [17], the dynamics model of the airship system is established, and the ADRC scheme is adopted to control its horizontal trajectory tracking, which can effectively eliminate the influence of external disturbance and model uncertainty. However, the traditional ADRC contains nonlinear functions and many parameters, which makes parameter tuning complicated. Therefore, Gao introduces the concept of bandwidth and proposes a linear active disturbance rejection controller (LADRC) with fewer parameters, which consists of a PD controller and a linear extended state observer (LESO) [18].
Besides the advantages of strong anti-disturbance ability and simple structure, by using LADRC, there is less the adjustable parameters, which greatly improves its application in actual engineering. In [19], an improved LADRC strategy is proposed to address the strong coupling and nonlinear problems of the quadrotor system, which can track the reference trajectory quickly and effectively, however, without performing strict stability analysis. Variable speed micro-hydro plant is a complex nonlinear system; in [20], LADRC is used to estimate and compensate the disturbances in real time, which improves the robustness of the control system. In order to solve the oscillation system with time delay, an internal model controller (IMC) is used in [21], which compensates the time delay, and then LADRC is adopted for approximation. In view of the uncertain factors and significant disturbances in the boiler system, in [22], the LADRC control scheme is proposed to control the steam temperature, which improves the control precision of the system.
Despite the above advantages, LADRC has strong correlation with its parameters, which leads to the complexity of parameter tuning. Moreover, it is well known that although a higher bandwidth can better track the desired signal and suppress the disturbance, in the practical application, the bandwidth is limited by the dynamic uncertainties. Furthermore, high bandwidth may cause oscillations and even lead to system instability, as well as increase operating costs. Therefore, in practical applications, bandwidth is limited by the requirements of performance and dynamic uncertainties, which leads to the decrease in LADRC control precision.
In the existing researches, most mathematical model of the quadrotor is established by assumption of constant mass. However, this assumption may not hold true in practical applications. For example, the quadrotor is used for carrying objects, pesticide spraying, etc. Its mass is variable, which will lead to the position of its centroid to change and cause timevarying disturbances, thus affecting its control precision. Therefore, the time-varying mass problem should be considered in the process of quadrotor modeling.
In this paper, we have established a more accurate mathematical model and proposed a control scheme that combines robust SMC and LADRC. By comparing with DSADRC proposed in [15], strict stability analysis is carried out and the performance of the control system is improved by introduction of SMC. In order to make the established model more accurate, we have considered the mass change of the quadrotor. Moreover, the designed control scheme has fewer parameters and stronger anti-disturbance performance. The main contributions of this paper are as follows: 1. The mathematical model of the quadrotor load UAV is established, taking into account the air resistance and the moment acting on the quadrotor. Moreover, the centroid position of the quadrotor from the changing mass has been fully considered. Therefore, the centroid position is correlated with the mass, which can make the established model more accurate. Furthermore, the adaptive law is introduced to solve the problem of mass variation, which can effectively reduce the influence of mass change on the quadrotor attitude system. 2. A control scheme combining with robust SMC and LADRC is proposed. The scheme combines the advantages of both and complements each other. LADRC can reduce the dependency of the controller on precise model, wherein LESO is used to estimate the internal and external disturbances of the system and the unmodeled dynamics caused by the SMC chattering, and compensate them with the PD controller. SMC can further enhance the robustness of the controller and overcome the disadvantage of low control precision due to limited bandwidth by using LADRC method. 3. In view of the strong correlation of LADRC parameters, the introduction of adaptive control can adjust the parameters of controller in real time according to the state of the system, which greatly simplifies the parameter setting process and reduces the influence of parameter deviation on the system. Moreover, it provides a novel method to prove the stability of the controller, which is beneficial to the stability analysis of the whole system. 4. The stability of the system is strictly proved by the Lyapunov stability theory. In the case of external disturbance and mass variation, compared with LADRC and DSADRC, the proposed control scheme shows stronger anti-disturbance ability and faster response speed, which confirms its effectiveness.
The rest of this article is structured as follows: The mathematical model of the quadrotor is established in Sect. 2; the control scheme is given in Sect. 3; in Sect. 4, the stability analysis of the system is given; Sect. 5 is the simulation results and discussion; Sect. 6 summarizes the paper.

Mathematical model
According to Fig. 1, the quadrotor can be modeled using six coordinates, namely x, y and z for the position coordinates, and /, h and w for the attitude coordinates [23]. When the quadrotor is loaded, its center of mass moves down from the origin of the body fixed frame B. The dynamics model of the quadrotor load UAV is established with reference to the earth fixed frame E and the body fixed frame B [24].
According to the matrix transformation relationship between the earth fixed frame E and the body fixed frame B [25,26], the following formula can be given as With the consideration of Newton's second law, the position dynamics model of the quadrotor load UAV can be given as where V E ¼ ½ x y z T ; V is the linear velocity of the quadrotor in the earth fixed frame E; G E is the total lift provided by the four motors; M is the mass of the quadrotor; m is the mass of the load; Considering the body fixed frame B, the total lift of UAV is as follows according to the transformation formula between the earth fixed frame E and the body fixed frame B, it can be given as According to formulas (2) and (4), the position dynamics model of the quadrotor load UAV is as follows When the quadrotor UAV is loaded, its center of mass shifts downward, resulting in an increase in the distance from the motor to the centroid, which affects the pitch and roll angle movement of the quadrotor. The quadrotor carries the load through the mount platform, and the two can be regarded as a whole. The centroid coordinate of the quadrotor is ðx 1 ; y 1 ; z 1 Þ; the centroid coordinate of the load is ðx 2 ; y 2 ; z 2 Þ. The following formula gives the mathematical method of determining the position of the centroid.
where ðx a ; y a ; z a Þ is the centroid coordinate of the quadrotor load UAV. According to Fig. 1, the centroid coordinates of the quadrotor and the load are ð0; 0; 0Þ and ð0; 0; z c Þ, respectively. Substituting into the above equation, the centroid coordinates of the quadrotor load UAV can be obtained as ð0; 0; z a Þ. Considering the Newton-Euler equation [27,28], the attitude dynamics model is written as [29,30] where is the angular velocity of the quadrotor rotating about each axis; L is the resultant moment acting on the quadrotor; N and O are the pulling moment and reaction torque of the quadrotor, respectively.
where b 1 is the lift coefficient; b 2 is the anti-torque coefficient; l is the length of the quadrotor arm. According to formulas (8) (9) and (10), the attitude dynamics of the quadrotor is obtained as where /, h and w are, respectively, the roll, pitch and yaw angles of the quadrotor UAV. Use the virtual control variables (U 1 , U 2 , U 3 , U 4 ) to simplify the dynamic model, as shown in the following formula According to the above formula, the dynamic model of the quadrotor load UAV is written in the following form The quadrotor dynamics model in formula (13) can also be expressed in the following form [31]. where

Control scheme design
Aiming at the problems of strong coupling, nonlinear and mass variation in the quadrotor load UAV, this section proposes a control scheme combining robust sliding mode control and LADRC and establishes an adaptive control of mass. Figure 2 shows the block diagram of the control scheme. The attitude channel and altitude channel of the quadrotor are controlled by SMC and LADRC, which can effectively reduce the influence of external disturbances on the quadrotor, and LADRC can also eliminate the coupling between each channel and improve the performance of the overall control scheme. Regarding parameter tuning problems, the gains of the controller are generally obtained by experiential methods or optimization algorithms. In this paper, in order to simplify the parameter setting and eliminate the problem of time-varying mass, adaptive control is adopted to adjust the parameters and mass variation in real time. Figure 3 shows the structure diagram of the controller.

Design of sliding mode controller
The SMC has the characteristics of easy implementation and strong robustness [32]. The introduction of SMC can further improve the anti-disturbance performance of the controller and solve the problem of lower control precision of LADRC.
The sliding surface can be designed as where w 1 is the adjustable parameter, e 1 ¼ z 1 À u d is the tracking error, z 1 is the estimation of the system state u by the LESO, u d is the input signal of the system.
where z 3 is LESO's estimate of the total disturbance.
With the consideration of (16), the sliding mode control law is designed in the following form where w 2 is a positive parameter.

Remark 1
In order to ensure that the designed sliding mode control is stable and effective, an appropriate parameter w 2 is selected to satisfy w 2 [ 0, which is proved by formula (46).

Design of linear active disturbance rejection control
The quadrotor is subject to various disturbances in flight. The LADRC can effectively eliminate the influence of disturbances on the quadrotor and improve the anti-disturbance performance of the controller [33]. The dynamics of the quadrotor in formula (14) is written into the nonlinear system in the following form.
where u ¼ u 1 u 2 ½ T is the measurable system state; Q and P are the nonlinear functions of the system; b is the external disturbances.
The total disturbance of the system is designed in the following form In order to estimate the total disturbance, z 1 ¼ u, z 2 ¼ _ u and z 3 ¼ Hðu; vðnÞÞ are defined as extended state space representations of the system, where z ¼ ½ z 1 z 2 z 3 T .
The nonlinear system (18) can be given as vÞ. z can be obtained in real time through the LESO.
where g ¼ g 1 g 2 g 3 ½ T is the gain vector of LESO.

Remark 2
The LESO can treat the combined effect of the unknown external and internal disturbances of the system as extended state and observe this extended state through output feedback and then use the PD controller to compensate. In addition, the LESO does not depend on the specific mathematical model of the system; it is only related to the order of the plant.
The gain of the observer is parameterized by the characteristic equation; it can be given as where g ¼ 3x 3x 2 x 3 Â Ã T , x is the bandwidth of LESO.

Remark 3
The increase in the bandwidth x of LESO is helpful to improve the control precision of LADRC, but increasing x will cause the output of the controller to increase, which is not conducive to the practical application of engineering. In addition, the high bandwidth will also cause oscillation and lead to system instability. Therefore, in practical applications, the control precision of LADRC is reduced due to the limitation of bandwidth. In order to solve this problem, this paper introduces SMC to improve the overall performance of the controller.
It is defined that the estimated values of the extended state space representation of the system arê z 1 ¼û,ẑ 2 ¼_ u andẑ 3 ¼Ĥðu; vðnÞÞ. The control law is designed as where u p is the output of PD controller.
Remark 4 In order to simplify the parameter setting, adaptive control is introduced to adjust k p and k d in real time, which is beneficial to the stability analysis of the whole quadrotor system and provides a novel method to enhance the stability of the controller. Formulas (36) and (37) give the adaptive law of k p and k d , and the adaptive adjustment curves of k p and k d are given in Fig. 7.

Adaptive law design for parameters adjustment
According to the tracking error e 2 ¼ u À u d , the filter tracking error is obtained as where a ¼ t 1 is the appropriately chosen coefficient, so that when d ! 0, it satisfies e 2 ! 0. Take the derivative of d to get where The feedback linearization method is used to define the tracking control of the input signal to achieve the approximate purpose. As shown in the following formula n ¼Ĥ À1 ðu; cÞ ð 28Þ where c ¼Ĥðu; nÞ is pseudo-control [34,35],Ĥðu; nÞ is any approximation of Hðu; vÞ. By adding and subtractingĤðu; nÞ to the right of formula (27), we can get: _ u 2 ¼Hðu; v; nÞ þĤðu; nÞ ð 29Þ Assumption 1Ĥðu; nÞ is an arbitrary approximation of Hðu; vÞ, soHðu; v; nÞ is close to zero.
The pseudo-control design is as follows where a is any positive parameter.

Adaptive law design for mass variation
In this article, we mainly study the attitude control of the quadrotor load UAV, so the adaptive law of mass is established according to the altitude channel. It can be obtained from formulas (15) and (26).
According to the certainty equivalence principle can be given as where r [ 0 is an adjustable parameter;m is an estimate of the mass of the load. Substituting formula (33) into formula (32) can obtain LetM c ¼ 1=ðM þmÞ, M c ¼ 1=ðM þ mÞ. The above formula is rewritten as follows 4 Stability analysis Theorem 1 Consider the nonlinear system in formula (15), the following adaptive law is applied to make the assumption valid.
where #, f and e are the appropriately selected parameters.

Remark 5
In order to eliminate the singularity problem that may exist in the adaptive law (36) and (37) whenk p andk d are zero. Integrating _k p and _k d can write the adaptive law in the following form.
wherek p andk d are the estimated values of the controller parameters. As long as the appropriate constants k p0 and k d0 are selected, it can be ensured thatk p ¼ k p Àk p andk d ¼ k d Àk d are not zero.

Assumption 2
The signals of the whole system are bounded, and the tracking error converges to the neighborhood of zero.
Proof of Theorem 1 The positive definite Lyapunov function is designed as follows.
Take the derivative of the above formula Substituting formulas (22), (31) and (35) into the above formula can obtain

Simulation results and discussions
In this part, MATLAB is used for simulation test. The initial value of attitude angles are [0,0,0] rad, and the initial value of altitude is 0 m. The parameters of the quadrotor and the controller are given in Tables 1 and  2, respectively.
Example 1 In this test, the mass of the load is kept unchanged, and m = 5 kg. The input signals of the system are The simulation results are compared with LADRC and DSADRC. Figures 4 and 5 show the output curves and the error curves of the system.

Remark 6
The DSADRC in this simulation is proposed in [15]. In order to make the comparison convincing, in the simulation experiment, the relevant parameters of DSADRC have been adjusted many times to ensure that DSADRC is close to the optimal control state.
It can be seen from the simulation results that in the altitude channel, the response speed of SMC-ADRC is faster than that of DSADRC. SMC-LADRC can track the input signal in 2 s, while DSADRC and LADRC can achieve stable tracking in 2.5 s and 3.5 s, respectively. In the pitch angle and roll angle channels, the response speed of SMC-LADRC and DSADRC is faster than that of LADRC. The overshoot of the proposed scheme is smaller than that of DSADRC and LADRC, and stable tracking is achieved in 0.5 s, while that of DSADRC and LADRC are 0.8 s and 1.5 s, respectively. In the yaw angle channel, SMC-LADRC has a faster response speed, achieving stable tracking in 0.3 s, while DSADRC and LADRC achieve stable tracking in 0.5 s and 1.3 s, respectively. Based on the above analysis, it can be concluded that the designed control scheme is better than LADRC and DSADRC. It has a faster response speed and smaller overshoot and can track the reference signal quickly and stably, which proves that the SMC-LADRC designed in this paper is effective.
Example 2 In this test, in order to test the control performance of the designed method and LADRC under the same bandwidth, keeping the parameters and input signals unchanged. Moreover, the SMC-LADRC without PD adaptive is introduced in the simulation to verify the effectiveness of the proposed adaptive law. Gaussian noise with mean value of 0 and variance of 2 is introduced to simulate the noise disturbance in the process of quadrotor flight. Figure 6 shows simulation diagram of Gaussian noise. Figure 7 shows the adaptive curves of k p and k d . The output curves and error curves under Gaussian noise are given in Figs. 8 and 9.
Remark 7 In order to prove the effectiveness of parameter adaptive law, the comparison with SMC-LADRC without PD adaptive is added in the simulation. Figures 8 and 9 show that the response speed of SMC-LADRC without PD parameter adaptive is slightly faster than that of SMC-LADRC. In the roll angle and pitch angle channels, the overshoot of SMC-LADRC with PD parameter adaptive is smaller than that of SMC-LADRC without PD parameter adaptive. Figure 7 shows that in the roll angle and pitch angle channels, the PD parameter adaptive curves have obvious changes in the beginning stage. This is the reason for the smaller overshoot of the SMC-LADRC, Through the analysis of Fig. 7, it can be clearly seen that the values of k p and k d are adjusted in real time, indicating that the introduction of adaptive law is effective, which can adjust the parameters according to the state of the system. In addition, it can also eliminate the parameter deviations and improve the control precision. Figures 8 and 9 show that in all channels, the SMC-LADRC tracking curves have small fluctuation, while that of LADRC are large. The SMC-LADRC without PD adaptive has smaller fluctuation in the altitude channel. In the roll angle and pitch angle channels, it has a relatively high overshoot within 0.3 s. However, in the yaw angle channel, the control precision of SMC-LADRC without PD adaptive has little difference from that of SMC-LADRC. In summary, this indicates that the introduction of SMC is effective; it can well overcome the problem of low control precision caused by the limited bandwidth of LADRC, which improves the anti-disturbance performance of the system. In addition, the introduction of adaptive law can eliminate the influence of parameter deviation and reduce the overshoot of the system, thus improving the control precision of the controller.

Example 3
The load mass of the quadrotor load UAV in practical application may be constantly changing. In this test, keeping the input and Fig. 4 Tracking output curves parameters of the system unchanged, and make the mass variation as an external disturbance, the simulation results are compared with those of SMC-LADRC without the adaptive of mass. Figure 10 shows the variation in load mass m. The output curves of the system with the mass variation are given in Fig. 11. Figure 12 shows the error curves.
According to the analysis of Figs. 11 and 12, it can be seen that the SMC-LADRC with mass adaptive law can effectively reduce the influence of mass variation, while the SMC-LADRC without mass adaptive has obvious overshoot. In the roll angle and pitch angle channels, the SMC-LADRC without mass adaptive has a large overshoot within 0.3 s. However, in the yaw angle channel, the curves of the two are almost the  same. Based on the above analysis of the simulation results, it can be concluded that the designed mass adaptive law is effective, which can reduce the influence of mass changes on the quadrotor.
Remark 8 It can be seen from the mathematical model of the quadrotor in formula (13) that the mass change mainly affects the altitude, roll angle and pitch angle channels of the quadrotor and has no direct Example 4 In different locations on the earth, the gravitational acceleration g is different. This test gives the gravitational acceleration at the equator and the North Pole, which are g ¼ 9:780 and g ¼ 9:832, respectively. According to the quadrotor dynamic model in formula (13), g only affects the altitude channel of the quadrotor. Figures 13 and 14 show the output curves and error curves of the altitude channel with different gravitational accelerations. It can be seen from the figures that the three curves are basically consistent, indicating that the change of gravitational acceleration has very little impact on the quadrotor system, which shows that the proposed scheme has strong robustness.

Conclusion
In this paper, considering the mass variation in the quadrotor in practical application, we have established the dynamic model of the quadrotor load UAV, and a composite control scheme combining with robust SMC and LADRC is proposed to solve this problem. The LADRC is used to estimate and compensate the disturbances, and it can also eliminate the chattering effect of SMC, but it is limited by the bandwidth, which leads to the decrease in its control precision. The introduction of SMC can further improve the robustness of the system and overcome the problem of low control precision caused by limited bandwidth. Compared to [32] and [34], the introduction of neural networks makes the controller design too complicated. However, the control scheme combining SMC and LADRC designed in this paper has a simpler controller structure. Furthermore, in order to simplify the parameter setting process of the controller, the adaptive law is introduced to adjust the adjustable parameters in real time, and it also eliminates tracking errors caused by parameter deviations, which is beneficial to the stability analysis of the whole system. Moreover, an adaptive law is also designed to control the mass variation in the quadrotor in real time. The Lyapunov  stability theory proves that the system is stable. Finally, the simulation results are compared with LADRC and DSADRC to verify that the designed scheme has better stability and anti-disturbance ability. It can effectively reduce the influence of disturbances and overcome the shortcomings of low control precision of LADRC, which improves the control performance of the quadrotor system. Moreover, the proposed mass adaptive law is also effective.
However, the limitation of this paper is that we only conduct simulation research and have not verified it through actual experiments. In the future work, we will study the feasibility of the designed method in the practical application and make improvements based   on experiments. At the same time, the experience of [32] and [34] is considered to continue the research and explore whether SMC and neural network can be combined.
Authors' contribution The first draft of the manuscript was written by Zhaoji Wang, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding No funding was received to assist with the preparation of this manuscript.

Declarations
Conflicts of interest The authors declare that they have no conflict of interest.
Availability of data and material All data generated or analyzed during this study are included in this article.
Consent for publication This article has not been published previously; it is not under consideration for publication elsewhere.