## 3 Control System Design
### Finite-Time-Convergent Second-Order Differentiator  
So far, many differentiators have been presented to obtain the differential estimation of the system signals in recent decades, such as relatively simple linear differentiator [35], high-order sliding mode based differentiator [36] and sigmoid function-based nonlinear tracking differentiator [37], to name a few. There exists time-lagging phenomenon and chattering problems in the high-order sliding mode-based differentiator, while the nonlinear differentiator has a complicated structure which is  
Figure 2: Structure diagram of the proposed control scheme (detailed definition of the symbols in the figure can be found in Section 3.3).  
hard for parameters to regulate. To address the drawbacks mentioned above, a finite-time convergent differentiator [38] is introduced to the control strategy.  
**Lemma 1:**[38]_The finite-time-convergent second-order differentiator is designed as_  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2},\\ \varepsilon^{2}\dot{x}_{2}=-\text{sat}_{\varepsilon_{b}}\bigg{\{}\text{sign}( \phi(x))\cdot\big{|}\phi(x)\big{|}^{\frac{\beta}{2-\beta}}\bigg{\}}-\text{sat} _{\varepsilon_{b}}\big{\{}\text{sign}(x_{2})\cdot|\varepsilon x_{2}|^{\beta} \big{\}},\\ x_{out}=x_{2},\end{array}\right. \tag{3}\]  
_where \(x_{1},x_{2}\) and \(x_{out}\) denote the system state and the output of the differentiator, respectively; \(\varepsilon_{b}>0\) and \(\beta\in(0,1)\) are constant; \(\varepsilon>0\) is a perturbation parameter; \(r(t)\) is the input signal of the differentiator; and functions \(\phi(x)\) and \(\text{sat}_{\varepsilon_{b}}(x)\) are defined as follows:_  
\[\begin{array}{l}\phi(x)=x_{1}-r(t)+\frac{\text{sign}(x_{2})\cdot|\varepsilon x _{2}|^{2-\beta}}{2-\beta},\\ \text{sat}_{\varepsilon_{b}}(x)=\left\{\begin{array}{c}x,\quad|x|<\varepsilon _{b}\\ \varepsilon_{b}\text{sign}(x),\quad|x|\geq\varepsilon_{b}\end{array}\right..\end{array} \tag{4}\]  
_For a continuous and piecewise two-order derivable input signal \(r(t)\), there exists \(\gamma>0\) (where \(\rho\gamma>2\) and \(\rho=\beta/(2-\beta)\)) and a function \(\Gamma>0\), such that_  
\[e_{i}=x_{i}-r^{(i-1)}(t)=O\Big{(}\varepsilon^{\rho\gamma-i+1}\Big{)}. \tag{5}\]  
_For \(t>\varepsilon\Gamma(\Xi(\varepsilon)\cdot e(0))\), \(i=1\), and \(t_{j}>t\geq t_{j-1}+\varepsilon\Gamma\Big{(}\Xi(\varepsilon)\cdot e_{+}(t_{j-1 })\Big{)}\), \(j=\{1,\cdots,k+1\}\), \(i=2\), \(O(\varepsilon^{\rho\gamma-i+1})\) denotes the approximation of \(\varepsilon^{\rho\gamma-i+1}\) order between \(x_{i}\) and \(r^{(i-1)}(t)\), where \(\Xi(\varepsilon)=\text{diag}\{1,\varepsilon\},e(0)=[e_{1}(0),e_{2}(0)]^{T}\), \(e_{+}\Big{(}t_{j-1}\Big{)}=\Big{[}e_{1}\Big{(}t_{j-1}\Big{)},e_{2}^{+}\Big{(} t_{j-1}\Big{)}\Big{]}^{T}\), \(e_{+}^{+}\Big{(}t_{j-1}\Big{)}=x_{2}-\dot{r}^{+}\Big{(}t_{j-1}\Big{)}\) and \(\dot{r}^{+}\Big{(}t_{j-1}\Big{)}\) is the right derivative._  
**Remark 2:**_According to the proof of [PERSON]'s work [38], system (3) is finite-time-convergent. The smaller \(\varepsilon\) we choose, the shorter the convergent time is and the smaller the tracking errors are. It should be also noted that, if the selected \(x_{1}(0)\) is close to \(r(0)\), the peaking phenomenon can be eliminated. Otherwise, we can first select a relatively larger value of \(\varepsilon\) and then a relatively smaller one after a while._