Trajectory tracking control for quadrotor based on time-varying threshold event-triggered extended state observer

Considering the problems of unknown external disturbances and parametric uncertainties in trajectory tracking of a quadrotor, a backstepping control method based on time-varying threshold event-triggered extended state observer (TETESO) and a nonlinear differentiator with lead compensation is proposed in this paper. First, to facilitate the controller design, the quadrotor system is divided into the position subsystem and the attitude subsystem, which are transformed into the original cascaded dynamics. Second, a TETESO which can observe the unmeasurable states and the total disturbance as well as reduce the unnecessary communication resources between sensor and observer is proposed. It is proved that the observation errors are bounded, and Zeno behavior is avoided. Third, a new controller synthesized by the proposed TETESO and the backstepping control method is designed for the position subsystem and the attitude subsystem. To address the “differential explosion” problem in the backstepping control method, a nonlinear differentiator with lead compensation is used to compute the derivative of desired attitudes in the controller. Moreover, the stability analysis of the closed-loop system is proved and the proposed controller can ensure the ultimate boundedness of all signals. Finally, the effectiveness and superiority of the proposed control method are verified by simulation.


Introduction
With the development of quadrotors in recent years, it has attracted considerable attention from many scholars. Compared with traditional manned aircraft and unmanned fixed-wing aircraft, the quadrotor has the advantages of vertical take-off and landing, hover stability, easy portability, small size, and so on (Bouabdallah and Siegwart, 2007;Hamel et al., 2002;Hoffmann et al., 2007;Huang et al., 2022aHuang et al., , 2022bLazim et al., 2019). Therefore, the quadrotor has been widely used in the civil and military fields.
To improve the trajectory tracking performance of a quadrotor, a lot of research has been done on the trajectory tracking control of a quadrotor. In Chen et al. (2016), a nonlinear elastic trajectory control method based on backstepping control method and a nonlinear disturbance observer were proposed for a quadrotor. In Xu and Ozguner (2007), a sliding mode control was developed to stabilize a quadrotor, and the stability of the closed-loop system is proved. In Zheng et al. (2014), to facilitate the controller design, a second-order sliding mode that unifies the position error and the attitude error into a sliding mode surface was proposed for a quadrotor. In Jia et al. (2017), a controller that combines integral backstepping control and sliding mode control was introduced to achieve the trajectory tracking of a quadrotor. In Xiong and Zheng (2014), the quadrotor system is divided into the fully actuated subsystem and the underactuated subsystem and then a novel robust terminal sliding mode control and a sliding mode control were designed for the fully actuated subsystem and the underactuated subsystem, respectively. However, the control methods proposed in the above references still have some shortcomings, which have a limitation of utilizing the full-state measurement including the position information, velocity, attitude information, angular rate states, and so on. It is difficult to design the controller without the above states of a quadrotor. In reality, to ensure the accurate and reliable flight of a quadrotor, the navigation board of a quadrotor usually integrates the global positioning system (GPS) module, the three-axis gyroscope, the accelerometer, and so on (Zhou et al., 2017). But in some cases, the use of gyroscopes and accelerometer sensors is often disturbed by a heavy measurement noise, which will greatly reduce the accuracy of the 1 School of Automation Engineering, University of Electronic Science and Technology of China, China 2 Yangtze Delta Region Institute, University of Electronic Science and Technology of China, China full-state feedback controller design and produce a negative impact on the flight performance of quadrotor. Therefore, the above control methods are difficult to be applied in reality.
In addition, there are still some problems such as unknown external disturbances and parametric uncertainties in trajectory tracking of a quadrotor, which will reduce the tracking accuracy of the nonlinear controller design. Some research has been done on the above problems. In Zhao et al. (2021), an adaptive external disturbance estimation algorithm was designed for a quadrotor. In Maqsood and Qu (2020), a sliding mode control method based on a new nonlinear disturbance observer was adopted to realize the trajectory tracking of a quadrotor. Although the disturbance observers adopted in Zhao et al. (2021) and Maqsood and Qu (2020) can estimate external disturbances, the disturbance observers are based on the assumption of slow variation of external disturbances, that is, it is assumed that the derivative of external disturbance is zero, which is a disadvantage since external disturbances always change in reality, and it may be difficult for the disturbance observer used in Zhao et al. (2021) and Maqsood and Qu (2020) to be carried out in engineering applications.
Designing an extended state observer (ESO), Gao (2003), Han (2009) is an effective method to address the aforementioned problems. Moreover, by constructing ESO, all system states can be estimated. More importantly, the ''total disturbance'' (including internal disturbance and external disturbance) acting on the quadrotor system can also be estimated and compensated. Utilizing these observed values could make the controller have stronger robustness. It is worth noting that the ESO has the characteristic of model-free, that is, its form does not depend on the system model. In the research of model-free, a novel exploration-exploitation-based adaptive law for model-free control method was proposed (Tutsoy et al., 2023). Based on the above advantages, the ESOs have been widely used in various fields including the electrohydraulic system (Guo et al., 2016), the ship (Liu et al., 2017), the quadrotor (Dun et al., 2022;Li et al., 2020;Shao et al., 2018;Wang et al., 2020;Zhang et al., 2018Zhang et al., , 2020, the missile (Shao and Wang, 2015), and so on. However, the ESOs used in previous references rely on continuous sampling; sometimes, it may consume a lot of network resources and calculation resources. Aiming at network transmission problems, a set of asynchronous fault detection (FD) filters is proposed (Long et al., 2021). The event-triggered mechanism can avoid the waste of network resources and calculation resources. In Zhang et al. (2022), a fuzzy adaptive event-triggered control protocol is proposed by the backstepping procedure. To avoid consuming a lot of network resources and calculation resources between sensor and observer, some event-triggered extended state observers have been proposed. In Huang et al. (2017), an event-triggered extended state observer was proposed, and the triggering condition did not rely on the states. In Yu et al. (2019), a novel event-triggered extended state observer that adopts an output predictor was proposed. In Liu et al. (2020), an event-triggered extended state observer and an event-triggered finite-time convergent extended state observer were proposed for dynamic positioning vessels, respectively. However, the event-triggered extended state observers proposed in the above references have some drawbacks. For example, Huang et al. (2017) and Liu et al. (2020) are fixed threshold event-triggered extended state observers. The fixed event-triggered mechanism will produce a continuous triggering effect when the amplitude of the transmission signal is too large. On the contrary, it may maintain a very long update interval when the amplitude of the transmission signal is too small, which is opposite to the original intention of designing the event-triggered mechanism.
Motivated by the above discussions, a time-varying threshold event-triggered extended state observer (TETESO) is proposed for a quadrotor in this paper. Different from the general event-triggered mechanism, the threshold of the proposed TETESO is time-varying. When the amplitude of the transmission signal is large, the TETESO can meet the requirement that still maintaining a long update interval to better reduce the number of events and save the communication resources between sensor and observer. While the amplitude of the transmission signal is small, a smaller threshold makes more precise transmission signals applied to the observer such that better performance of the observer could be obtained. The TETESO can keep a balance between saving communication resources and maintaining the performance of the observer. Furthermore, a time-varying threshold is added with a saturation limitation, which avoids an excessively large magnitude of transmission signal jumping suddenly so that a large impulse will be applied to the observer and the performance of the observer will be degraded. Compared with the existing references, the main contributions of this paper are as follows: 1. Different from the feedback controllers designed in Chen et al. (2016), Jia et al. (2017), Xu and Ozguner (2007), Xiong and Zheng (2014), and Zheng et al. (2014) that use the velocity information of a quadrotor, the proposed controller only uses the measurable position and attitude states of a quadrotor and eliminates the negative effects of unknown external disturbances and parametric uncertainties. 2. In contrast to the traditional ESO (Gao, 2003), a TETESO is proposed in this paper. Such an improved ESO can greatly reduce the unnecessary communication consumption between sensor and observer and ensures an accurate observation performance. Zeno behavior of the TETESO is excluded. 3. To eliminate the ''differential explosion'' problem in the traditional backstepping control method, a novel nonlinear differentiator is developed in the paper, which not only shows the potential to address the problem of differential explosion but also can reduce the chattering phenomenon. 4. It is proved that observation errors of the TETESO are bounded, and the relationship between the observer performance and the observer parameters is obtained. Besides, the stability of the closed-loop system is analyzed using the Lyapunov theory.
This paper is organized as follows: the mathematical model of a quadrotor is introduced in section ''Preliminaries and problem formulation.'' In section ''Time-varying threshold event-triggered extended state observer,'' a TETESO is proposed and analyzed. In section ''Control design and analysis,'' a backstepping control method based on the TETESO and a nonlinear differentiator is proposed for the position subsystem and the attitude subsystem, and the stability of the closed-loop system is analyzed. Simulation results are given to prove the validity of the proposed TETESO and the proposed control method in section ''Simulation results.'' Conclusions are drawn in section ''Conclusion.''

Preliminaries and problem formulation
The motion of a quadrotor can be regarded as a rigid body with 6 degrees of freedom including rotation and linear motion, and the model of a quadrotor is established by the earth-frame E and the body-frame B as shown in Figure 1, where (x, y, z) represent the position coordinates of the quadrotor; (f, u, c) represent the Euler angles of the quadrotor; and f, u, and c represent roll angle, pitch angle, and yaw angle, respectively.
To successfully derive the mathematical model of quadrotor, the following assumptions are made.
Assumption 1. The quadrotor is a rigid body, and its structure is symmetrical.
Assumption 2. The Euler angles of the quadrotor are bounded: roll angle f 2 À p 2 , p Remark 1. In reality, it is very difficult to measure the inertia coefficient I x , I y , I z , and air resistance coefficient k f of a quadrotor, and it is impossible to obtain their accurate values. We assume , and k f 0 are nominal parts, and DI x , DI y , DI z , and Dk f denote unknown and uncertain parts.
c are defined as unmeasurable states in this paper.
To facilitate the following controller design, the mathematical model of the position subsystem (1) is transformed into the original cascaded dynamics similarly, the mathematical model of the attitude subsystem (2) could be rewritten as where Remark 2. It is noted that according to Assumption 3 and Assumption 5, we have H 1 and H 2 are bounded and suppose that H 1 and H 2 satisfy H 1 k k+ H 2 k kł H m with H m . 0.
To realize the trajectory tracking of the quadrotor, the virtual control inputs are defined as follows According to equation (5) and the desired yaw angle c d , the control input u 1 , the desired pitch angle f d , and the desired roll angle u d are obtained as follows To sum up, the four control inputs of the quadrotor are assigned as follows: U 1 are the control inputs for the position (x, y, z), and U 2 are the control inputs for the attitude (f, u, c).

Time-varying threshold event-triggered extended state observer
To design TETESO for the quadrotor, the linear extended state observer (LESO) is first introduced. For a second-order nonlinear system with unknown external disturbances and parametric uncertainties where x 1 and x 2 are states which satisfies jx i j ł B with B . 0, i = 1, 2; b 0 is the known control gain; u is the control input.
f (x 1 , x 2 , t, w(t)) is an unknown function including the unknown external disturbances and parametric uncertainties, which is assumed to be derivable. Let f (x 1 , x 2 , t, w(t)) be a new state x 3 , then equation (7) could be rewritten as In system (8) where z 1 , z 2 , and z 3 are the observed values of x 1 , x 2 , and x 3 , respectively; w 0 . 0 is defined as the bandwidth of the observer. By selecting an appropriate w 0 , the observed values of LESO can be guaranteed to converge to their true values asymptotically, that is, lim Based on the LESO (9), a TETESO is proposed as follows with where t 2 ½t k , t k + 1 ), t k is the previous triggered events time instant, and k denotes the total number of event-triggering.
is the trigger condition of the event-triggering mechanism. n . 0, 0\s\1, and C . 0 are design parameters. The advantages of the designed time-varying threshold event-triggering mechanism are as follows: (1) when the value of the transmission signal becomes larger, the trigger threshold will also increase, which will not reduce the trigger times of the TETESO too much. (2) When the value of the transmission signal becomes small, a smaller threshold makes more precise transmission signals applied to the observer such that better performance of the TETESO could be obtained. (3) In addition, the trigger threshold is limited by an upper bound, which avoids a large signal pulse from being transmitted to the observer due to the sudden increase of transmission signal, thus reducing the observation effect of the TETESO.
Remark 3. When the system states are large or change rapidly, the parameters of the event-triggering mechanism including n, s, and C can be appropriately increased. On the contrary, when the system states are small, the parameters of the eventtriggering mechanism can be appropriately reduced, so that the TETESO can achieve a balance between the observation performance and reducing unnecessary communication resources.
Proposition 1. Considering the system (8), the TETESO (10), and the triggering condition in (11), there exists a constant t . 0 such that for any k . 0 Proof: In the interval ½t k , t k + 1 ), considering the triggering condition Then, for the sample error According to x 2 j j\B with B . 0, we obtain that for B . 0. Thus, there exists such that It is shown that there is no Zeno phenomenon in the TETESO based on the triggering condition in equation (11). This completes the proof. According to equations (8) and (10), the observer estimation error system is given as where e i = z i À x i , i = 1, 2, 3. Equation (15) could be rewritten as To facilitate the following analysis, we define h i = ei w0 iÀ1 , i = 1, 2, 3, then it follows (16) where h= e 1 , e2 w0 , e3 Since A h is a Hurwitz matrix, there exists a positive definite matrix P satisfying the following equation where I is an identity matrix.
Theorem 2. Considering the system (8), the TETESO (10), and the event-triggering condition (11), suppose that the Assumption 5 holds, then the proposed TETESO is stable. Estimation errors are ultimately bounded and can approach the origin asymptotically.
Proof: Choose the Lyapunov function candidate as follows Defining k i = PB i k k with i = 1, 2, and applying Young's inequality to 2w 0 k 1 (sC + n) + 2hm w0 2 k 2 h T k k, we have from equation (20) is represented as the maximum eigenvalue of P. By setting the suitable parameters including w 0 , s, n and C such that a . 0. Finally, it can be easily inferred that Remark 4. From the above analysis, it can be seen that the observation errors of the TETESO are ultimately bounded and strictly restricted by O a . It is also concluded that the upper bound of observation error decreases with the increase of the bandwidth w 0 , while the upper bound of observation error increases with the increase of event-triggering coefficients including s, n, and C. Because when the eventtriggering coefficients are increased, the number of state transmissions is reduced between sensor and observer, thereby reducing the performance of the observer. Therefore, we need to make a balance between the performance of the observer and reducing the consumption of communication resources between sensor and observer.

Control design and analysis
In this section, the quadrotor control is divided into position subsystem control and attitude subsystem control. The block diagram of a quadrotor control scheme is shown in Figure 2.

Position subsystem control design
In this subsection, backstepping control method is designed for the position subsystem. Define position error E 1 as where X 1d = (x d , y d , z d ) T , X 1 = (x, y, z) T . Choose the Lyapunov function candidate V 1 as follows and time derivative of V 1 is to make _ V 1 =À k 1 E 1 T E 1 ł 0, let _ X 1d À X 2 =À k 1 E 1 with k 1 . 0, and X 2 = _ X 1d + k 1 E 1 . Define a virtual control law X 2d = _ X 1d + k 1 E 1 , On the basis of the above, define error E 2 as follows choose the Lyapunov function candidate V 2 as and time derivative of V 2 is ; therefore, the position subsystem controller U 1 is designed as where E 1 = X 1d À X 1 , _ E 1 = _ X 1d À X 2 , E 2 = X 2d À X 2 = _ X 1d + k 1 E 1 À X 2 . A TETESO is utilized for the position subsystem, which is given as where j 1 (t) = (j 11 , j 12 , j 13 ) T , and the triggering condition is as follows where s i , n i , C i , i = 1, 2, 3 are the design parameters and w 1 is the bandwidth. Z 1 , Z 2 , and Z 3 are estimated values of X 1 , X 2 , and X 3 , respectively. The mathematical expression of the controller (29) is as U 1 = € X 1d + E 1 + k 1 ( _ X 1d À X 2 ) + k 2 (X 2d À X 2 ) À X 3 . X 2 , X 3 in the controller (29) are replaced by Z 2 , Z 3 , and the final expression of the position subsystem controller is rewritten as Attitude subsystem control design In the following, the controller design of attitude subsystem will be introduced. Define attitude error E 4 as where X 4d = (f d , u d , c d ) T , X 4 = (f, u, c) T . Choose the Lyapunov function candidate V 3 as follows and time derivative of V 3 is to obtain _ V 3 =À k 3 E 4 T E 4 ł 0, let _ X 4d À X 5 =À k 3 E 4 with k 3 . 0, and X 5 = _ X 4d + k 3 E 4 . Select a virtual control law X 5d = _ X 4d + k 3 E 4 . Define error E 5 as follows Choose the Lyapunov function candidate V 4 as and time derivative of V 4 is The above equation can be rewritten as € X 4d + k 3 _ E 4 À U 2 À X 6 + E 4 =À k 4 E 5 , so the expression of the attitude subsystem controller U 2 is given by where Remark 5. Due to the coupling between the position subsystem and attitude subsystem in the roll and pitch channels, the calculation of the desired roll and pitch angles becomes complicated. As a result, it is difficult to directly obtain their differential signals. To address this problem, a thirdorder nonlinear differentiator with lead compensation is proposed.
and € X 4d in the attitude subsystem controller (39) are replaced by v 2 and v 3 .
Finally, the expression of the attitude subsystem controller U 2 is given by Remark 6. It is known that X 2 and X 5 are unmeasurable; besides, X 3 and X 6 are also unknown functions including unknown external disturbances and parametric uncertainties, so X 2 , X 3 , X 5 , and X 6 cannot be used directly in the controller (29) and (39). Furthermore, we need to reduce the loss of unnecessary communication resources between sensor and observer. The proposed TETESO can be used to solve the above problems.
It is noted that the matrix L is a positive definite by selecting suitable parameters including k 1 , k 2 , k 3 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , hence we can conclude that where a 1 =2l min (L) min 1, l min and l max denote the minimum eigenvalue of matrix L and maximum eigenvalue of matrix P 1 , respectively. It is concluded from equation (50) that Equation (51) means that V 3 is eventually restricted by m=a 1 , which can guarantee the ultimate boundedness of all states in the closed-loop system of a quadrotor.

Simulation results
The physical parameters of the quadrotor are selected, as shown in Table 1 (Zhang et al., 2018). The desired trajectories and desired yaw angle are chosen as x d = 6 cos (t), y d = 6 sin (t), z d = 6 sin (t), and c d = cos (t).
To verify the effectiveness and superiority of the proposed method, comparative simulations with the following control method are performed: 1. Backstepping based on the LESO control method (BS-LESO). 2. Backstepping based on the fixed threshold eventtriggered extended state observer (FETESO) (Liu et al., 2020) control method (BS-FETESO).
It is worth noting that the above comparative control methods are only different in the used extended state observer, and the other controller coefficients are the same.
It can be seen from Figure 3 that all three control methods can achieve the trajectory tracking of a quadrotor. Figures 4-6 and Figures 7-9 show that all three control methods can make Table 1. Parameters of the quadrotor.
the quadrotor track the desired trajectory and desired attitudes, and all tracking errors can be within a certain range. Moreover, the position tracking and attitude tracking performance of the BS-LESO control method is slightly better than the other control methods from Table 2. The reason is that the LESO continuously samples the transmission signal, which can ensure the performance of the LESO. The position tracking the performance of the BS-TETESO control method is slightly better than that of the BS-FETESO, and it can better realize the trajectory tracking of a quadrotor. As depicted in Figures 10-15 and Tables 3 and 4, the performance of the LESO is better than that of the FETESO and the TETESO. Under the event-triggered mechanism, the performance of the FETESO and the TETESO does not drop too much, which is an acceptable and tolerable result. Although the performance of the FETESO is slightly better than the TETESO, it is based on more communication     network resource consumption. Compared with the FETESO, the TETESO can maintain the better performance of the observer similar to the FETESO while reducing the most communication resources between observer and sensor.

Conclusion
Aiming at the problems of unknown external disturbances and parametric uncertainties in trajectory tracking of a quadrotor, a backstepping control method based on the designed TETESO and a novel nonlinear differentiator is proposed for the position subsystem and the attitude subsystem. The TETESO can not only observe the unmeasurable states and the total disturbance but also can reduce the unnecessary communication resources between sensor and observer by introducing a time-varying threshold event-triggered mechanism. It is proved that the observation errors are bounded, and Zeno behavior is avoided. To address the ''differential explosion'' problem in the backstepping control method, a nonlinear differentiator with lead compensation is used. The proposed control algorithm can guarantee the ultimate boundedness of all states in the closed-loop system. Finally, the effectiveness and superiority of the proposed control algorithm are verified by simulation. In future research, we will focus on the following two scopes. On the one hand, we will study how to construct a novel extended state observer to eliminate the negative impact of measurement noise on the observer's observation performance. On the other hand, we will study the design mechanism of velocity and input constraints.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.