# Two-Phase Non-Singular Terminal Sliding Mode Control of Nonlinear Systems
## 1 Introduction  
Sliding mode control (SMC) has been widely studied due to the simplicity of its structure and computation, and strong robustness to external disturbances and the uncertainties of systems. It has been applied in many industrial systems such as robots, converter systems, motor positioners and so on [1; 2; 3; 4]. SMC is essentially a two-fold nonlinear control method, which consists of (i) the selection of a sliding-mode surface, which determines the expected performance of systems when the system stays in its neighborhood, and (ii) the design of the control strategy, which steers the system to reach and reside on the sliding-mode surface strictly. The conventional SMC schemes usually adopt the linear sliding-mode surface (LSM), which is inherently a nonlinear control scheme. It suffers from a chattering phenomenon and fails to satisfy the finite time convergence of the system dynamics. To suppress the chattering phenomenon, plenty of methods have been proposed, including but not limited to higher-order sliding modes (HOSM) [5; 6; 7; 8] and super-twisting algorithms (STA) [9; 10; 11], and disturbance observers [12], and extended state observers [13]. To guarantee the finite time convergence, the terminal sliding mode (TSM) control approach has been developed [14; 15; 16; 17], which applies the nonlinear sliding-mode surface to offer a fast and finite time convergences property and steady tracking performance. However, the singularity issue exists in the TSM control scheme, which is undesirable and may deteriorate the performance to a large extent in practical applications. Substantial research has been undertaken to overcome the singularity problem [18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. An indirect approach can be found in [18; 20; 21; 22; 23]. Refs [18; 20] propose a modified TSM controller for second-order systems, where the power of the sliding surface is different from the common one. In [21], the terminal sliding mode is defined by inserting the arctangent function, suppressing the singularity. Ref. [22] proposes a new sliding variables to achieve the non-singularity. These indirect approaches aim at preventing the occurrence of singularity from the beginning, whereas the direct approach restricts the singularity by the limitation techniques straight [24; 25]. The authorsof [24; 25] utilize the saturation and sinusoidal function, respectively, to restrain the singular term. Furthermore, a derivative and integral TSM control is presented in [28; 29]. The sliding mode control can also be combined with other control schemes, such as adaptive control [30; 31]. It is noted that the study of the settling time of system dynamics are not involved in the aforementioned literature, which excites more attention recently. The fixed-time stability (FTS), which investigates the estimation of the upper bound of the settling time, is first presented in [32], where the intrinsic feature of the fixed-time convergence is revealed and the application to the design of the SMC controller is also discussed. The most attractive feature of the FTS is that the estimation of the bound of the settling time has nothing to do with the initial condition, which is given as a constant and determined by the system parameters only. There are some studies on the FTS [25; 33; 34; 35]. The authors of [25; 33] offer a fast convergence solution for the design of TSM schemes. In [35], a state-dependent exponential coefficient is used in the sliding surface design. On the basis of the FTS, there is a doubt as to whether the bound of the settling time independent of the initial state is optimal or not and whether the selection of the segmentation of the system state is optimal and unique or not. Given this consideration, it is our great passion to improve the convergence property by finding the possibility of a less conservative bound of the settling time through other alternative system states.