Finite-time Attitude Control with Chattering Suppression for Quadrotors Based on High-order Extended State Observer

This article investigates a finite-time attitude control with chattering suppression for quadrotors based on a high-order extended state observer (HESO). To improve the anti-disturbance capacity of quadrotors, a HESO capable of estimating fast time-varying disturbances is presented to enhance system robustness by regarding higher-order disturbance derivatives as extended arguments. Then, with the estimates of HESO, a finite-time attitude control policy including a double hyperbolical reaching law (DHRL) is introduced, alleviating control chattering and guaranteeing rapid response. The most significant feature is a finite time regulation for quadrotors without incurring severe oscillations can be obtained in the event of fast time-varying uncertainties. Moreover, the stability analysis is proved by a Lyapunov theorem. Finally, the effectiveness of proposed control scheme is demonstrated by numerical and experimental results.


I. INTRODUCTION
During the past decades, quadrotors have received extensive attention [1]- [5] on account of the abundant applications in military and civilian fields such as monitoring [6], patrolling [7], ground mapping, searching and rescuing [8]. They have characteristics of prominent agility, low cost and small size. However, the quadrotors are typical under-actuated systems with modelling nonlinearities accompanied by highlycoupled states, parametric uncertainties and external disturbances. Therefore, the motion control for quadrotors is always a challenging research topic. Especially, it is necessary to point out that the attitude control is an essential precondition for quadrotor applications. Researches on attitude control schemes for quadrotors have conducted abundant of advanced approaches in recent years. For instance, backstepping control [9]- [10], active disturbance rejection control [11]- [12], sliding mode control (SMC) [13]- [14] and the like.
Ascribe to the excellent anti-interference ability and strong robustness, sliding mode control (SMC) has become one of the most popular control strategies for the quadrotor attitude control [13]- [16]. Nevertheless, it is undeniable that the impact in the course of engineering practices caused by the serious oscillations of SMC is considerable. However, it is undeniable that the severe oscillations in control torques of conventional SMC is considerable in practice. So as to solve abovementioned chattering matter, researchers dedicate themselves to proposing creative SMC policies, including but not be limited to terminal SMC [17]- [18], adaptive SMC [19], disturbance estimation-based SMC [20]. A free-model-based terminal SMC method is developed for the attitude and position control for quadrotors with model parameter variations [17]. An adaptive SMC scheme is proposed for mismatched uncertainties quadrotors [19] to achieve robust tracking. By introducing an adaptive finite-time extended state observer (FTESO) to estimate the lumped disturbances, a continuous fast nonsingular terminal SMC is elaborated in [18]. In [20], focusing on actuator faults and model uncertainties, a novel SMC in combination with a neural network is investigated. As mentioned above, the disturbance estimation-based SMC has been regarded as a preferable method to retard chattering problem. The principle behind estimation-based SMC lies in the setting of switching gain value, by adjusting it larger than the error of disturbance estimation rather than the max value of disturbance, such that control action is assured to be continuous and smooth [21]. Absolutely, the critical point of disturbance estimation-based SMC is the choice of disturbance estimator.
A great deal of disturbance estimators are employed and combined within SMC construction to alleviate the chattering of control inputs, such as sliding mode observer (SMO) [22], extended state observer (ESO) [23]- [25] and neural network (NN) [26]. However, one should notice that the SMO design not only requires the upper bound of disturbance to be known, but also there exists obvious chattering along with its outputs. Meanwhile, for conventional ESO design, which can only handle the fixed or slow time-varying disturbances, has poor ability in dealing with fast time-varying perturbations. In addition, with respect to NN, large calculational cost is inevitable because it is a very burdensome process to adjust the appropriate parameters to prompt the weight update to converge in limited time. However, even though stronger robustness and smoother control signals are available, it worth being suggested that rapid transient convergence performance is unattainable arising from the selection of tiny switching gains [22]- [26].
In order to address the slow transient convergence problem, adaptive SMC and SMC with improved reaching law are widely applied [27]- [30]. In [27], an adaptive SMC has been proposed, which could realize rapid convergence and faster adaption for quadrotor system. In [28], a novel exponential reaching law (ERL) is invented to overcome the convergence performance dilemma in SMC. However, the appearance of additional adaptive rule leads to a huge computational burden, which occurred in the process of online updating. Moreover, although modified reaching law avoids the deficiency of the former, it still suffers from chattering problem [29]. Recently, a neoteric double hyperbolical reaching law (DHRL) [30] is proposed by means of combining hyperbolic tangent and hyperbolic cosine functions. It prompts the system convergence and eliminates the discontinuity in control performance owing to it allows the sliding mode variable enclosing to zero infinitely rather than traversing it. Hence, it is a meaningful attempt to introduce DHRL into the quadrotor attitude control design to improve the tracking performance Inspired by aforementioned discussions, herein we present a HESO-based SMC with a double hyperbolical reaching law to stabilize quadrotor attitude tracking, remedy the sluggish convergence and further suppress oscillations. The contributions of this article can be listed as follows: 1) A HESO is devised by extending traditional ESO with higher extended order for quadrotor attitude control to estimate the rapid time-varying disturbances. Compare with previous SMO [22] or ESO [23]- [25], higher precision is obtained on the premise of overcoming the limitation of slow time-varying disturbances. Furthermore, unlike NN [26] suffers from complexity parameter selection and heavy computational burden, the HESO only needs to adjust the bandwidth, such that a reliable disturbance estimation could be attained.
2) Different from ecumenical disturbance estimation-based SMC for quadrotor attitude control subject to slower convergence. This article adopts a new DHRL which can guarantee rapid convergence and contain no extra oscillations contrast with orthodox RLs [28], [31].
The outline of this paper is organized as follows. Section II establishes the attitude model of quadrotors. Section III elaborates the design of the HESO-based controller with DHRL and stability analysis. Section IV discusses the simulation and experimental tests with corresponding results. At last, Section V concludes this paper.

A. NOTATION
In this article, the standard definitions are presented below. mn × is a set of real numbers with m rows and n columns.
T [] denotes the transposed matrix.
() diag stands for the diagonal matrix. and refer to the absolute value and the Euclidian norm, respectively. As depicted in Fig.1, the construction of quadrotor attitude system is consisted of a rigid cross frame and four rotors. Two reference frames are established to express the attitude kinematics of quadrotors, i.e., inertia frame

B. MODELING OF A QUADRTOR
Remark 1: All the necessary data of quadrotor attitude system can be measured readily by the inertial measurement units (IMUs) embedded onboard. Particularly, in our experimental rig, it needs to be pointed out that quadrotors are restricted by a universal joint, which connects the quadrotor and the fixed iron shelf, so that the transformation matrix R is always nonsingular.
The control objective of this paper is to constructure a HESO-based SMC for quadrotors to accomplishing: 1) The lumped unknow disturbances could be recovered by HESO and the error of estimation is ensured to be exponentially convergent.
2) All the errors considered in this paper can accomplish ultimately uniformly bounded (UUB) results and a fast attitude tracking behavior can be achieved with a smooth control action.

III. CONTROLLER DESIGN
In this part, a HESO-based SMC will be design within a double hyperbolical reaching law for quadrotor attitude dynamics (2). The architecture of the proposed controller is clarified in Fig.2.

A. DHRL DESIGN
As the fundamental part of SMC, the reaching law plays a significant role in sliding mode surface design. Different RLs will cause various degrees of effects on system convergence behavior. For quadrotor attitude system (2), here a double hyperbolical reaching law (DHRL) including cosh( ) and tanh( ) is applied to realize fast convergence and chattering suppression of the preceding system. The form of the DHRL is as follows: 12   For the sake of demonstrating, the effectiveness of the two hyperbolic functions of proposed DHRL are shown in Fig.3. Evidently, the hyperbolical cosine function term has a bigger value prompting i s to converge rapidly in a considerable degree while distances between i s and the convergency value is far. The hyperbolic tangent function term takes over control when i s closes to the convergency value to guarantee that i s could infinitely approach zero but never arrive at or go across it. Benefiting from the above characteristics, additional chattering cropped up in traditionary RLs, will be totally averted.
To sum up, subsisting (6) and (7) into (5) obtains The analysis for the condition 0 0 s  is similar to aforementioned procedure, so the process is omitted in this paper. The above analysis manifests that i s can converge to a bounded non-zero set in finite time by a proper selection of related parameters. ■

Remark 2:
Although compared with traditional RLs using sgn( ) , DHRL effectively overcomes the extra chattering by employing the hyperbolic tangent function, it is difficult to guarantee finite-time stability for the quadrotor attitude system because of the asymptotic property of the convergence boundary [32]- [33]. Therefore, a small set is introduced here to assist the rapid dynamic convergence of system.

B. HESO DESIGN
The fundamental mechanism of established ESO is that to think of the interior uncertainties and exterior unmodeled dynamics as lumped disturbances and augment the lumped disturbances as an extended item in system. In the process of conventional ESO construction, only the liner first derivative of lumped disturbances is involved, which leads to an assumption that the lumped disturbances are constant or slow changing. However, such conditions are unavailable in reality, because actual interference is always complex and time-varying.
To cope with this nonnegligible problem, inspired by the higher-order expansion of Taylor polynomials, herein a HESO is devised by extending the derivative of lumped disturbances to the th n order and regarding them as extended states of system. For the sake of subsequent exposition, define the following notations as Then the attitude dynamics (1) is transformed into Assuming j M is continuously differentiable, obeying the principle of high-gain observer design [34], the particular HESO with the mission of acquiring the estimations of lumped disturbances can be designed as   , where i  is the gain coefficient. Then (9) can be directly transformed into a form with high gains as following: And the observation error dynamics can be described in the following form: 1, 1, Then constructing a Lyapunov function This completes the proof. When 3 i = , 3, j e denotes the estimation error of the lumped disturbances j M . ■ Remark 3: By using above proposed HESO, disturbance estimation with high precision can be obtained through a proper selection of the extension order n . Different from traditional ESO, whose estimation performance is limited to single parameter regulation, proposed HESO has a higher degree of freedom due to the selection of extension order n can provide an assistance when bandwidth adjusting is insufficient.

C. HESO-based SMC DESIGN
According to above analysis, in this part, a HESO-based SMC containing the DHRL will be stated for quadrotor attitude dynamics.
Here, the attitude command is given as Theorem 3: For the relevant quadrotor attitude dynamics (2), along with HESO design (9), sliding mode surface (13) and proposed control law (15), supposing all assumptions are met, then the estimation error 3 2  2  3  2  3,  2  1  2  3  1   2  2  3  3  3,  3,  2  2  3  3  1  1   2  33  3,  2  3 11 Up to now, estimation error and sliding mode variable are both proved UUB. Besides, referring to (13), one can derive that the attitude tracking error i e is also ultimately bounded. In conclusion, all errors in the researched quadrotor attitude system are proved bounded. ■ Remark 4: Some experience about parameter tuning is given here to help the selection of the controller parameters for acquiring satisfactory control performance, which are extracted as follows: 1) For controller module, the control gains 1i  , 2i  and 3i  should be selected larger enough to guarantee fast convergence of control signals and achievement of tracking mission. Nevertheless, it should be noted that overlarge gains may cause unexpected chattering into control inputs. Therefore, suitable control parameters cannot be more significant for controller to realize an admirable attitude tracking performance.
2) In terms of proposed HESO, a higher extended order n results to an estimation with higher precision, but the sensitivity to measurement noise of HESO will inevitably increase and unanticipated oscillations may appear. In consequence, a compromise between estimation accuracy and system robustness should be carefully considered.  disturbances, the performance of attitude tracking is comparatively perfect. Furthermore, owing to the concerned DHRL, the convergence process of attitude tracking errors has been shortened and chattering in the control inputs can be suppressed effectively. The precise estimations of lumped disturbances are guaranteed by selecting a proper bandwidth of HESO. In summary, by implementing DHRL and HESO in attitude tracking control of quadrotors, satisfactory tracking property, rapid dynamic response along with precise disturbance estimation can be concurrently realized.

B. COMPARISON SIMULATIONS
To show up the advantages of proposed control scheme, a comparison is taken here between HESO-based SMC and continuous nonlinear terminal SMC (CNTSMC) [22]. In order to ensure the fairness of comparison, the convergence speed of two SMCs is demanded to be identical by tuning the related parameters advisedly. Figs.7-8 show the comparative results of attitude tracking with same disturbances and quantized in TABLE 2. By comparing the quantitative comparative performance indices, it can be found clearly that under the premise of compensating the lumped disturbances and tracking the desired commands accurately, one can obtain a more accurate tracking performance contributed by proposed method. Additionally, the control oscillations, which are occurred in CNTSMC severely, are effectively alleviated with the aid of devised HESO and DHRL whether in transient or steady state. Hence, the control consumption is reduced naturally.

Index
HESO-based SMC CNTSMC [22] Convergence time   This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.   , which are offered by the Links-UAV Testbench. The attitude command is selected the same as before. Other parameter selection of experimental validation is identical with that in simulation showed in TABLE 1. Experimental results are demonstrated in Figs.13-14. One can find an accurate and fast attitude tracking performance produced by the proposed control method due to the merit of HESO and DHRL. According to Fig.14, it is obviously to see that continuous control actions with a slight oscillation can be obtained by proposed control method, which is acceptable because of the measurement noises involved in attitude information measured by a low cost IMU. Definitely, a costly IMU will effectively improve measurement accuracy and reduce measurement noises, as a consequence, one should make a compromise between control performance and equipment cost.

D. EXPERIMENTAL COMPARISONS
To emphasize the excellence of the proposed SMC, a comparative experiment is executed with CNTSMC devised in [22]. The relevant parameters stay the same with previous simulation. As a matter of convenience, the pitch angle is chose as a typical example for comparison and analysis.  Experiment comparative performances are displayed in Fig.15 and quantized TABLE 4. By the picture one can see that both controllers could produce a continuous attitude tracking performance under the specified time-vary command. Evidently, the CNTSMC has nonnegligible deficiencies in tracking accuracy and control stability. The proposed HESO-based SMC provides a more accurate tracking behavior and a smoother control action.

V. CONCLUSION
In this article, a HESO-based SMC with a DHRL is presented for quadrotor attitude tracking under unknown disturbances. Focusing on improving control accuracy, a HESO with single parameter regulation is devised via a higher-order expansion with respect to the derivatives of lumped disturbances to obtain a more accurate disturbance estimation. To avoid extra chattering bought by traditional reaching law, an original DHRL consisting of two different hyperbolical functions is employed, which produces neither additional oscillation nor sluggish dynamic response. The analysis of the total system stability is specified based on the Lyapunov theorem. Simulations along with experiments have been implemented and adequate results have been acquired to indicate the prominent characteristics of proposed control scheme.