## 1 Introduction  
Sliding mode control (SMC) is a popular control algorithm and possesses several desirable characteristics, such as design simplicity, ease of implementation, and robustness to external disturbances, model uncertainties, and parameter variations. Thus, SMC has received a significant amount of attention and is widely found in diverse practical applications, including robotics [1, 2, 3, 4, 5, 6], electrical and mechanical systems [7, 8, 9, 10, 11, 12], and aerospace engineering [13, 14, 15, 16, 17]. Nonetheless, the conventional SMC still exhibits several drawbacks, namely chattering, asymptotic convergence, and susceptibility to mismatched disturbances. Chattering phenomena occur in the conventional SMC due to the use of a switching function, which leads to high oscillations in the control signal [18]. Significant efforts have been made in the literature to attenuate this problem, including the boundary layer [19], second and higher-order SMC [20, 21, 22], low-pass filtering [23, 24], and the disturbance observer [25]. Another problem found in the conventional SMC is that system states can only asymptotically converge to the equilibrium point. By comparison, terminal sliding mode control (TSMC) can drive system states both to the sliding mode surface and to the equilibrium point in finite time [26]. Although the conventional TSMC inherits the robustness characteristic from its predecessor, it still suffers from the chattering phenomenon and is prone to the singularity problem as the control signal instantly jumps to infinity. A large number of studies have been devoted to solving this problem, e.g., [27, 28, 29]. Recently, a chattering-free full-order TSMC was developed in [30] that satisfactorily resolves both the chattering and singularity problems mentioned above. A rather less common problem that SMC faces is mismatched disturbances, which act in a different channel than the control input. Some examples in which mismatched disturbances exist are magnetic levitation (MAGLEV) suspension systems, permanent magnet synchronous motor (PMSM), systems and flight control systems [31]. The various methods that contribute to tackling this problem include integral SMC [32; 33; 34] and adaptive SMC [35; 36; 37]. However, it is well known that the integral action may produce several undesirable effects, such as a large overshoot and a long settling time for the control system. Moreover, the nominal performance is normally sacrificed for robustness against disturbances in these methods. Disturbance observer-based SMC, in contrast, is a more promising approach to deal with mismatched disturbances [38; 39; 40; 41; 31; 42]. In [38], a nonlinear disturbance observer is introduced to develop a new sliding mode surface in an attempt to attenuate mismatched disturbances. However, the mismatched disturbance considered in [38] must be constant rather than time varying, which is a highly restrictive constraint, because the disturbance can continuously change over time in practical applications; that is, unless the mismatched disturbance is time-invariant, the algorithm in [38] fails to stabilize the control system. In addition, the nonlinear disturbance observer utilized in [38] has been proven to be exponentially stable only in the case of slow time-varying mismatched disturbances [43]. A remedy for this constant restriction on the mismatched disturbance formulation is provided in [39], wherein the proposed extended disturbance observer can estimate nonlinear and time-varying mismatched disturbances. However, the problem of being Lyapunov stable without convergence remains, thus limiting the potential of the control algorithm. Another effort to stabilize nonlinear systems is found in [41], where mismatched disturbances are estimated and then suppressed by an extended-state observer-based sliding mode control. Both matched and mismatched disturbances are considered in the paper.