## 1 Introduction
Nonetheless, the estimation errors and the system state can only reach and vary in a small region around the equilibrium point, indicating that the observer and the closed-loop system are only Lyapunov stable without convergence. Consequently, the mismatched disturbance cannot be estimated precisely, and the system state cannot be stabilized in finite time. Motivated by previous works, to accurately estimate and eventually counteract the effect of mismatched disturbances for a class of nonlinear systems, a new design of disturbance observer-based TSMC is introduced in this paper. The finite-time disturbance observer employed in this work originates from an arbitrary-order exact robust differentiator developed in [21]. It forms the basis for the development of this disturbance observer, which has been used by a large number of researchers in various applications. Some of the works are discussed as follows. In [44], the above-mentioned observer is implemented to tackle the tracking control problem of robotic manipulators. In [45], the influence of mismatched disturbances on a class of nonlinear systems is attenuated based on this observer. Simulation of a PMSM then proved the effectiveness of the proposed controller. The same disturbance observer is employed in [46] to address the problem of airgap control of a MA-GLEV suspension vehicle. Another application of this disturbance observer can be found in [31], where the authors utilize it to control a DC-DC buck converter. Simulation and experiments on the converter have proven its disturbance rejection capability. These references convincingly demonstrated the effectiveness and the popularity of the disturbance observer considered in the current paper, for estimating and suppressing the influences of matched and mismatched disturbances in different highly nonlinear dynamic systems. Major aspects of the proposed control algorithm are presented as follows. First, mismatched disturbances are accurately estimated by the nonlinear finite-time disturbance observer. Second, the estimation results are integrated into a new terminal sliding mode surface to enhance system robustness against external disturbances, uncertainties, and unmodeled dynamics. Third, the utilization of a chattering-attenuated full-order terminal sliding mode surface structure in controller design effectively suppresses chattering and singularity phenomena in the control signal. Fourth, Lyapunov stability analysis alongside the design step theoretically increases the reliability of the proposed control scheme. In addition, although the control algorithm is developed for a class of nonlinear second-order systems, the obtained results are extendable to the nth-order formulation. Finally, comparative simulation studies, including a numerical example and an application to an Electro Hydrostatic Actuator (EHA) system, demonstrate the effectiveness of the proposed algorithm. This work can be perceived to be an extension and improvement of the references [30; 41] for several reasons. First, in [30], rigorous research on chattering-free full-order terminal sliding mode control was presented, thus forming the backbone of the development of our work. However, because the control algorithm developed in [30] was only designated to tackle the problem of matched disturbances, it cannot cope with mismatched disturbances, which are more challenging. Second, in [41], an extended-state observer-based sliding mode control (ESOSMC) was introduced in an attempt to address the problem of mismatched disturbances. However, this approach nonetheless exhibited some drawbacks. Although it inherited the chattering attenuation characteristic from the work in [30], the algorithm proposed in [41] was only able to drive the disturbance estimation and system state to a small, bounded region around the equilibrium without actually converging to it. Moreover, the mismatched disturbance used in [41] gradually converged to a constant rather than being time varying throughout the system operation.