# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer  
[PERSON]  
1X'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi'an 710119, China; [EMAIL_ADDRESS] (F.J)  
1X'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi'an 710119, China; jingfeng@opt. sliding mode is designed to reduce the chattering of sliding mode control and achieve better tracking performance and anti-disturbance performance, as shown in [12]. In [13], a scheme based on a neural network and sliding mode control is proposed. A double-layer neural network is used to estimate the unmodeled dynamics of the system, which reduces the dependence of sliding mode control on the system model and improves the shortcomings of sliding mode control. The design of a control algorithm based on an adaptive fuzzy and neural network depends on experience, and the convergence speed is relatively slow, which is suitable for slow-varying external interference application scenarios. In recent years, the extended state observer (ESO) has been widely used in various mechanical systems because it can effectively estimate system uncertainties and external disturbances. In [14], a sampled-data extended state observer was developed for a two-link flexible manipulator, which was used to reconstruct the uncertain parameters of the system and effectively estimate the non-minimum phase behavior of the flexible link and the uncertainty of the model. In [15], the extended state observer is used to estimate the total disturbance of the FLM online, including external disturbances, changes in control gain, and rigid-flexible coupling dynamics. The above method shows that the ESO can effectively estimate the system uncertainty, external disturbance, and flexible dynamics of FLM. Therefore, combining the advantages of sliding mode control and ESO, this paper proposes a sliding mode control scheme based on ESO. Furthermore, in order to achieve the finite-time stability of FLM and overcome the shortcoming that sliding mode control requires the upper bound information of the total dis-turbance of the system, an adaptive nonsingular terminal sliding mode (ANTSM) controller based on ESO is proposed in this paper. However, both ESO and SMC require prior knowledge of the system, such as input gain, model parameters, etc., and are assumed to be known in advance. From an engineering point of view, the above information is often unknown. Several adaptive observers are proposed in [16; 17; 18; 19] so that the system controller design does not need to know the control gain in advance. However, the above adaptive observer requires the system input to be measurable, which cannot be satisfied in many applications. In order to overcome this difficulty, in [20], data-driven adaptive ESO is developed into disturbance and parameter estimation, and an adaptive ESO based on position and acceleration information is proposed to estimate the unknown input gain, unmeasured velocity, and concentrated disturbance of marine autonomous vehicles. This method requires additional acceleration sensor information. In [21], data-driven adaptive ESOs are applied to the group control of multiple autonomous vehicles with unknown input gains. An adaptive ESO [22] is proposed to estimate the system state and time-varying control gain of the marine navigation ship system. In [23], an adaptive ESO is applied to an electro-hydraulic loading system with unknown input gain. These adaptive ESOs are online estimations of system parameters, external disturbances, and system states, which can easily cause system instability. However, the above research results provide a theoretical basis for the estimation of unknown system parameters of a non-integral chain essential system. As a typical non-integral series system, the FLM system can use an adaptive ESO to estimate the unknown model parameters of FLM.