## 5 Improved Tracking Differentiator (SYSTD)
### Parameter Tuning Method of SYSTD  
Set the system input signal as \(A\)sin(\(\omega t\)), and define \(M(A)\) to describe the nonlinear function \(f(t)=sys(t)\):  
\[M(A)=\frac{f_{1}+je_{1}}{A} \tag{41}\]  
Figure 6: Phase-trajectories of node-type singular points. Figure 7: Phase-trajectories of focus-type singular points. where \(e_{1}\) and \(f_{1}\) are the first-order Fourier coefficients:  
\[ \begin{split} e_{1}&=\frac{1}{\eta}\int_{0}^{2\pi}f(t) \cos(\omega t)\cdot d(\omega t)\\ f_{1}&=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\sin(\omega t )\cdot d(\omega t)\end{split}  \tag{42}\]  
Because \(f(t)\) is an odd function, \(e_{1}\), let \(x_{1}(t)\) - \(r(t)\) = \(A\)sin(\(\omega t\)), and combined with Equation (42); the function \(sys(x_{1}(t)\) - \(r(t)\),\(a_{41}\),\(a_{51}\)) can be transformed into the following:  
\[M_{1}(A,a_{41},a_{51})=\frac{1}{\pi}\int_{0}^{\pi}sys(A\sin(\omega t),a_{41},a_ {51})\sin(\omega t)\cdot d(\omega t) \tag{43}\]  
Similarly, the function \(sys(x_{2}(t)/R\),\(a_{41}\),\(a_{51}\)) can also be transformed into the following:  
\[M_{2}(A,a_{42},a_{52})=\frac{1}{\pi}\int_{0}^{\pi}sys(A\sin(\omega t),a_{42},a _{52})\sin(\omega t)\cdot d(\omega t) \tag{44}\]  
Therefore, the linearized form of the nonlinear system S\({}_{2}\) can be expressed as follows:  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=-R^{2}[M_{1}(A,a_{41},a_{51})\cdot(x_{1}(t)-r(t))+M_{2}(A,a_{42 },a_{52})\cdot\frac{x_{2}(t)}{R}]\end{array} \right. \tag{45}\]  
From this, the transfer function between the input \(r(t)\) and the tracking \(x_{1}(t)\) can be derived as follows:  
\[G_{\frac{x_{1}}{T}}=\frac{X_{1}(s)}{r(s)}=\frac{R^{2}\cdot M_{1}(A,a_{41},a_{ 51})}{s^{2}+R\cdot M_{2}(A,a_{42},a_{52})s+R^{2}\cdot M_{1}(A,a_{41},a_{51})} \tag{46}\]  
Let \(T=\frac{1}{R\sqrt{M_{1}}}\), \(\xi=\frac{M_{2}}{2\sqrt{M_{1}}}\), \(\omega_{r}=\frac{1}{T}\) then Equation (46) can be simplified as:  
\[G_{\frac{x_{1}}{T}}=\frac{1}{T^{2}s^{2}+2\xi Ts+1} \tag{47}\]  
where \(T\) is the signal period, \(\xi\) is the damping coefficient, and \(\omega_{T}\) is the cut-off frequency. It should be noted that, when \(0<\xi<1\), system Equation (47) is actually an under-damped system. When the damping is not large enough, chattering may occur. Therefore, \(\xi\) must be greater than 1:  
\[M_{2}\geq 2\sqrt{M_{1}} \tag{48}\]  
where \(M_{1}>0\) and \(M_{2}>0\) to ensure the stability of the system. The frequency characteristics of the system can be described as follows:  
\[G_{\frac{x_{1}}{T}}(j\omega)=\frac{R^{2}M_{1}}{R^{2}M_{1}+RM_{2}\cdot(j\omega )-\omega^{2}} \tag{49}\]  
The amplitude-frequency and phase-frequency characteristics are as follows:  
\[L(\omega)=20\log(\left|G_{\frac{x_{1}}{T}}(\omega)\right|)=-20\log\sqrt{\frac {\omega^{4}+2(M_{2}^{2}-M_{1}^{2})R^{2}\omega^{2}+R^{4}M_{1}^{2}}{R^{4}M_{1}^{ 2}}} \tag{50}\]  
\[\varphi(\omega)=\angle\left|G_{\frac{x_{1}}{T}}(\omega)\right|=-\arctan(\frac {2 RM_{2}\omega}{RM_{1}-\omega^{2}}) \tag{51}\]  
Through simulation, as shown in Figure 8, we observe the influence of the comprehensive parameters of SYSTD on its dynamic characteristics. The results are as follows:  
1. [label=(0)]
2. Influence of the speed factor \(R\) on the frequency characteristics. First, fix \(M_{1}\) and \(M_{2}\), change \(R\), and then draw the Bode diagram of SYSTD as shown in Figure 8a. It can be seen from the figure that an increase in the value of \(R\) can increase the cut-off frequency of the system, which is conducive to the estimation of high-frequency components of the signal and shortens the system response time. However, if the value of \(R\) is too large, high-frequency noise will be introduced, reducing the filtering performance of SYSTD. Therefore, the selection of the value of \(R\) needs to balance the response speed and filtering performance. 2. Influence of parameter \(M_{1}\) on the frequency characteristics. First, fix \(M_{2}\) and \(R\), change \(M_{1}\), and then draw the Bode diagram of SYSTD as shown in Figure 8b.