## 5 Improved Tracking Differentiator (SYSTD)
### Design and Stability Judgment of SYSTD
\[\dot{V}(z_{1},z_{2})=sys(z_{1},a_{41},a_{51})\cdot z_{2}+z_{2}\dot{z}_{2}=-z_{2} \cdot sys(z_{2},a_{42},a_{52})\leq 0 \tag{26}\]  
It can be seen from Equation (26) that the equal sign holds only when \(z_{2}=0\). When \(z_{2}\
eq 0\), the function always holds. Therefore, in the entire phase space, the total derivative of \(V(z_{1},z_{2})\) along the solution of system \(S_{0}\) with respect to time \(t\) is always negative. Thirdly, it is known that the equal sign of Equation (26) holds only when \(z_{2}=0\). At this time, this occurs only when \(z_{1}=0\), \(\dot{z}_{2}=0\). Therefore, the set of points that satisfy \(\dot{V}(z_{1},z_{2})=0\) only contains the origin (0, 0). Finally,  
When \(|z_{1}|\rightarrow+\infty\), \(\int_{0}^{z_{1}}sys(\xi,a_{41},a_{51})d\xi\rightarrow+\infty\);  
When \(|z_{2}|\rightarrow+\infty\), \(\frac{1}{2}z_{2}^{2}\rightarrow+\infty\). Therefore, \(V(z_{1},z_{2})\) tends to positive infinity in the real domain \(R^{2}\). It can be seen from Lemma 1 that the zero-solution of system \(S_{0}\) is globally asymptotically stable. The proof is completed. \(\Box\)  
Next, this paper designs TD based on the function \(sys()\). Lemma 2 needs to be introduced first. Lemma 2 (20). : _In the sense of [PERSON], if the second-order differential equation is as follows:_  
\[\left\{ \begin{array}{l}\dot{z}_{1}(t)=z_{2}(t)\\ \dot{z}_{2}(t)=f(z_{1}(t),z_{2}(t))\end{array} \right. \tag{27}\]  
_Additionally, if it has all solutions bounded and satisfies the following:_  
\[\left\{ \begin{array}{l}\underset{t\rightarrow\infty}{\lim}z_{1}(t)=0\\ \underset{t\rightarrow\infty}{\lim}z_{2}(t)=0\end{array} \right. \tag{28}\]  
_Then, for any bounded measurable signal r(t) (\(t\in[0\), \(+\infty\))] and time T(T \(>0\)), there is a differential equation \(S_{1}\):_  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=R^{2}f(x_{1}(t)-r(t),\frac{x_{2}(t)}{R})\end{array} \right. \tag{29}\]  
_The solution component \(x_{1}(t)\) of \(S_{1}\) satisfies the following:_  
\[\underset{R\rightarrow\infty}{\lim}\int_{0}^{T}|x_{1}(t)-r(t)|dt=0 \tag{30}\]  
It can be seen from Lemma 2 that, within any finite time \(T\), as \(R\) increases, \(x_{1}(t)\) can fully approximate the input signal \(r(t)\). Combining this with Equation (21), the system \(S_{2}\) is constructed as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=R^{2}[-sys(x_{1}-r,a_{41},a_{51})-sys(\frac{x_{2}}{R},a_{42},a_{52 })]\end{array} \right. \tag{31}\]  
Theorem 2.: _If \(a_{41}\), \(a_{51}\), \(a_{42}\), and \(a_{52}\) are all positive numbers, and the \(\dot{r}(t)\) and \(\ddot{r}(t)\) of the input signal r(t) (\(t\in[0\), \(+\infty\))] are bounded, then, for any finite time T \(>0\), the system \(S_{2}\) satisfies:_  
_(1) The solutions of system \(S_{2}\) satisfy the following:_  
\[\underset{R\rightarrow\infty}{\lim}\int_{0}^{T}|x_{1}(t)-r(t)|dt=0 \tag{32}\]  
_Furthermore, \(x_{1}(t)\) converges to \(r(t)\), and \(x_{2}(t)\) converges to the generalized derivative of \(r(t)\)._  
_(2) System \(S_{2}\) is a perturbed form of system \(S_{1}\)._Proof. : First, because \(\mathrm{S}_{2}\) and \(\mathrm{S}_{1}\) are equivalent in form, it can be seen from Lemma 2 that conclusion (1) holds. Secondly, let the following be true:  
\[ \begin{array}{l}\eta=Rt\\ w_{1}(\eta)=x_{1}(t)-r(t)\\ w_{2}(\eta)=\frac{x_{2}(t)}{K}\end{array}  \tag{33}\]  
Take the derivatives of \(w_{1}\) and \(w_{2}\) to obtain the following:  
\[ \begin{array}{l}\frac{dw_{1}(\eta)}{dt}=w_{2}(\eta)-\frac{i(\frac{\eta}{2} )}{R}\\ \frac{dw_{2}(\eta)}{dt}=-sys(w_{1}(\eta),a_{41},a_{51})-sys(w_{2}(\eta),a_{42},a_{52})\end{array}  \tag{34}\]  
When \(R\rightarrow\infty\), system Equation (34) is equivalent to system \(\mathrm{S}_{0}\). At this time, \(z_{1}\) and \(z_{2}\) are not only the solutions of system \(\mathrm{S}_{0}\) but also the solutions of system Equation (33). Therefore, system \(\mathrm{S}_{2}\) is a perturbed form of system \(\mathrm{S}_{0}\) and has global zero-value asymptotic stability. The conclusion (2) holds and the proof is completed.