# Anomaly Detection and Identification in Satellite Telemetry Data Based on Pseudo-Period
## 3 Anomaly Detection and Identification of Satellite Telemetry Data Based on Phase-Plane Trajectory  
Phase-plane analysis is a graphical analysis method for solving low-dimensional dynamic systems, which is usually applied to system analysis, for a second-order system, such as:  
\[x^{\prime\prime}+f(x^{\prime},x)=0 \tag{11}\]  
where \(f(x^{\prime},x)\) is a linear function or nonlinear function of \(x^{\prime}\) and \(x\). The phase-plane of data is a two-dimensional plane in which the data value \(x\) is taken as the horizontal axis and the derivative value \(x^{\prime}\) as the vertical axis. The phase-plane trajectory shows the motion of the system under different conditions. Suppose the second-order system differential equation is:  
\[x^{\prime\prime}+\omega_{n}{}^{2}x=0 \tag{12}\]  
The solution to the equation is:  
\[\frac{dx^{\prime}}{dx}=-\frac{\omega_{n}{}^{2}x}{x^{\prime}} \tag{13}\]  
And the phase-plane trajectory equation:  
\[x^{2}+\left(\frac{x^{\prime}}{\omega_{n}}\right)^{2}=A^{2} \tag{14}\]  
where \(A=\sqrt{x^{\prime}_{0}/\omega_{n}^{2}+x_{0}^{2}}\). Equation (14) is an elliptic equation. The parameter A depends on the initial value of \(x^{\prime}\) and \(x\). If the value of parameter A takes different values, the phase-plane trajectory of elliptic will be different. As the most real satellite telemetry data is nonstationary, the differential equation corresponding to the certain problem will be more complicated under different circumstances. So, there are more significant differences in the different temporal shapes of telemetry data. The phase-plane trajectory could be applied in abnormal data detection and identification for satellite telemetry data. When the phase-plane was used to map the satellite telemetry data, the actual telemetry data were taken as the mapping function. The phase-plane trajectory map of the satellite telemetry data could be obtained by mapping the satellite telemetry data on the phase-plane. Figures 6-8 present the time-domain diagrams of three different shape data and their corresponding phase-plane trajectories, respectively. Figure 6: (**a**) Standard sinusoidal data; (**b**) Phase-plane trajectory of the standard sinusoidal data. Figure 6b shows the phase-plane trajectory of sinusoidal data for which the trajectory shape was approximately circular. The phase-plane trajectory points were mainly distributed in the boundary area of the phase-plane plots. Figure 7b shows the phase-plane trajectory of the sinusoidal data with hopping noise from time 12 to 16 in Figure 7a, where the trajectory was close to an ellipse. The phase trajectory points were mainly distributed in the horizontal axis area of the phase-plane plots, and the corresponding phase trajectory of the hopping noise points was located in the boundary area of the phase-plane plots. Figure 8b shows the phase-plane trajectory of the impulse data containing the noise, where the trajectory was mainly concentrated in the central area of the phase-plane. The phase trajectory of the pulse points was located in the boundary area of phase-plane plots. It was observed from the distribution characteristics of the above data that the phase-plane trajectories were highly sensitive to the change in the data shape. The data with different shapes had different phase-plane trajectories. A weak change in the data shape was clearly reflected in the phase-plane trajectories. Therefore, by comparing the phase-plane trajectories of the normal satellite telemetry data with the measured telemetry data, the identification of the anomaly data could be achieved. To detect and identify the anomalous satellite telemetry data based on the phase-plane trajectory, the phase-plane trajectory characteristics of the data should be further extracted in order to quantify the difference in the phase-plane trajectory. Regarding the phase-plane trajectory as the distribution of data  
Figure 8: (**a**) Pulse data with noise; (**b**) Phase-plane trajectory of the pulse data with noise.