# Two-Phase Non-Singular Terminal Sliding Mode Control of Nonlinear Systems
## 2 Preliminaries  
Finite time systems usually can be categorized into these types:  
(i) the power rate type, described by \(\dot{x}=-kx^{\alpha},x(0)=x_{0}\), where \(0<\alpha<1\) is a fraction where both numerator and denominator are positive odd integers; the settling time function is obtained as \(t_{s}=\frac{x_{0}^{1-\alpha}}{k(1-\alpha)}\), in which the independent variable is the initial state \(x_{0}\);  
(ii) the power rate exponential type, given by \(\dot{x}=-px-kx^{\alpha},x(0)=x_{0}\), where \(p>0\), and \(0<\alpha<1\) is a fraction where both numerator and denominator are positive odd integers; the settling time function is derived as \(t_{s}=\frac{1}{p(1-\alpha)}\ln(1+\frac{p}{k}x_{0}^{1-\alpha})\), similarly depending on the initial state \(x_{0}\);  
(iii) the double power rate type, depicted as \(\dot{x}=-k_{1}x^{\alpha}-k_{2}x^{\beta},x(0)=x_{0}\), where \(0<\alpha<1\) and \(\beta>1\) are fractions where both numerator and denominator are positive odd integers; the settling time function is \(t_{s}=\frac{x_{0}^{1-\alpha}}{k_{1}(1-\alpha)}F(a;b;c;-\frac{k_{2}}{k_{1}}x_{0 }^{\beta-\alpha})\), where \(a=1\), \(b=\frac{1-\alpha}{\beta-\alpha},c=b+a,F(\cdot,\cdot;\cdot;\cdot)\) is the Gauss hypergeometric function, and initial state \(x_{0}\) is also an independent variable.  
It is observed that the convergence rate of the finite time systems is different, where the settling time, from an arbitrary initial state \(x_{0}\) to the equilibrium point \(x(t)=0\), seemingly varies with the initial state \(x_{0}\). In particular, for the double power rate one (iii), the accurate expression of the settling time is characterized by the hypergeometric function, for which is difficult to obtain the solution.  
An agreement is reached substantially that the convergence rate of the finite time system needs to be fast and the arriving time needs to be short. Recently, a fixed-time stability was put forward in [32], by which the influence of the initial state on the settling time is eliminated. It provides an upper bound of settling time which is independent of the initial state. Later, Ref. [36] devised a finite-duration attractor, which offers an alternative choice of settling time so that the impact of the initial state can be customized by the designer. Specifically, the upper bound of equation (iii) given by [32; 36] is presented as following  
\[\bar{t}_{s,d}=\frac{1}{k_{1}(1-\alpha)}+\frac{1}{k_{2}(\beta-1)} \tag{1}\]  
\[\bar{t}_{s,e} = \frac{x_{t}^{1-\alpha}}{k_{1}(1-\alpha)}+\frac{(x_{t}^{1-\beta}- x_{0}^{1-\beta})}{k_{2}(\beta-1)} \tag{2}\]  
where \(x_{t}\) is the transition state chosen by the designer.  
**Remark 1**.: _Obviously, the upper bound in (2) equals to (1) when \(x_{t}=1\) and the initial state \(x_{0}\) is naturally ignored. A tight bound is provided by [36] for the double power rate one (iii), which presents a general expression of the settling time, with an appropriate selection of the transition state._  
**Remark 2**.: _Facilitated by the main thought of [36], we aim to construct a new type of terminal sliding mode surface by which the convergence rate improves and the singularity is avoided and presents an effective analysis method where a tight bound of the settling time is offered._