## 5 Improved Tracking Differentiator (SYSTD)
### Phase-Plane Analysis of the Tracking Differentiator System  
By studying the types of singular points and the characteristics of phase-trajectories in the system phase-plane, the convergence characteristics of the second-order nonlinear time-invariant system can be analyzed, and, thus, the constraint conditions of the parameters can be obtained.  
First, perform a Taylor expansion of the function \(sys(x,a_{4},a_{5})\) at the zero point. Its expansion with remainder is as follows:  
\[y=\frac{1}{\sqrt{a_{4}}}(x-\frac{a_{5}*x^{3}}{2*a_{4}}+\frac{3*a_{5}^{2}*x^{5} }{8*a_{4}^{2}}-\frac{5*a_{5}^{3}*x^{7}}{16*a_{4}^{3}}+\frac{35*a_{5}^{4}*x^{9} }{128*a_{4}^{4}}\cdots)+R_{n}(x) \tag{35}\]  
The absolute-value ratios of the first-order term coefficient to other multi-order term coefficients are \(\frac{2a_{4}}{a_{5}},\frac{8a_{4}^{2}}{3a_{5}^{2}},\frac{16a_{4}^{3}}{5a_{5}^ {3}},\frac{128a_{4}^{4}}{35a_{5}^{4}}\), etc. If the linear part of the function \(sys()\) is to play a dominant role in the extremely small neighborhood of \(x=0\) and reduce the chattering of the TD output, it is necessary to set, which can increase the weight of the linear part. Therefore, in the extremely small neighborhood of \(x=0\), there is the following:  
\[sys(x,a_{4},a_{5})\approx\frac{x}{\sqrt{a_{4}}} \tag{36}\]  
At this time, system \(\mathrm{S}_{0}\) can be approximately expressed as a linear system \(\mathrm{S}_{3}\), which is as follows:  
\[\left\{\begin{array}{l}\dot{z}_{1}(t)=z_{2}(t)\\ \dot{z}_{2}(t)=-\frac{z_{1}(t)}{\sqrt{a_{41}}}-\frac{z_{2}(t)}{\sqrt{a_{42}}} \end{array}\right. \tag{37}\]  
Its Jacobian matrix at the origin is as follows:  
\[X=\begin{bmatrix}0&1\\ -\frac{1}{\sqrt{a_{41}}}&-\frac{1}{\sqrt{a_{42}}}\end{bmatrix} \tag{38}\]  
It can be seen that the matrix has no zero eigenvalue, so the origin of the phase-plane is an elementary singular point of system \(\mathrm{S}_{3}\). Let \(i_{1}=\frac{1}{\sqrt{a_{41}}}\), \(i_{2}=\frac{1}{\sqrt{a_{42}}}\) the eigenvalues of matrix \(X\) are as follows:\[\lambda_{1,2}=\frac{-i_{2}\pm\sqrt{i_{2}^{2}-4i_{1}}}{2} \tag{39}\]  
When \(i_{2}\geq 2\sqrt{i_{1}}\), the eigenvalues of matrix \(X\) are all real numbers, and the singular point (0, 0) is a stable node of system S\({}_{3}\). In this paper, eight groups of phase-plane points are selected to analyze the phase-trajectory characteristics under this condition. Set \(i_{1}=6\) and \(i_{2}=16\) and draw its phase-trajectory diagram as follows:  
It can be seen, from Figure 6, that the phase-trajectory of the system is smooth and finally directly approaches the origin, and the system state converges stably. Similarly, when, the eigenvalues of matrix \(X\) are all imaginary numbers, and the singular point (0, 0) is a stable focus of system S\({}_{3}\). Additionally, we can select eight groups of phase-plane points, set \(i_{1}=16\) and \(i_{2}=6\), and draw their phase-trajectory diagram as shown is Figure 7.  
Obviously, differently from the simple and fixed approximation method in the neighborhood of the node, the phase-trajectory in this figure approaches the origin in a spiral form, and the oscillation of the system during the transition process is relatively large. In summary, in order to reduce the overshoot oscillation during the system transition and increase the weight of the linear component in the neighborhood range, in the selection of system parameters, it can be considered to satisfy the constraint.  
\[\left\{\begin{array}{l}a_{41}\geq 16a_{42}^{2}\\ a_{4i}\geq 0.6a_{5i}\;i=1,2\end{array}\right. \tag{40}\]