## 2 The UKF Detection of PQDs  
### Unscented Kalman Filter  
The state space equations of nonlinear discrete systems are described as follows:  
\[X_{k+1}=f(X_{k})+W_{k} \tag{1}\]  
\[Z_{k}=\Psi(\overline{X}_{k})+V_{k} \tag{2}\]  
where \(X_{k+1}\) is state vector, \(Z_{k}\) is measurement vector, \(f(\cdot),\Psi(\cdot)\) is a nonlinear transformation and a nonlinear function satisfied by the system. \(W_{k}\) is a process noise, \(V_{k}\) is measurement noise. \(W_{k}\sim N(0,Q),V_{k}\sim N(0,R)\), where \(N\) represents the normal distribution, \(W_{k}\) and \(V_{k}\) follow normal distributions with means of 0 and covariances of \(Q\) and \(R\), respectively. The mathematical modeling of the UKF is divided into the following six steps:  
Step 1: Define initialization state  
\(\overline{X}_{0}\) and Variance matrix \(P_{0}\) as follows:  
\[\overline{X}_{0}=E(X_{0}) \tag{3}\]  
\[P_{0}=E[(X_{0}-\overline{X}_{0})(X_{0}-\overline{X}_{0})^{T}] \tag{4}\]  
where \(E(\cdot)\) is the expectation operator used to describe the mean of a random variable. Step 2: Sigma Point Sampling for State Variables  
Sigma points are used in the UKF to represent the distribution of the state vector. They are generated by adding and subtracting the square root of the weighted covariance matrix from the current state mean. Sigma points effectively capture the state distribution under nonlinear transformations. The calculation process is shown in Equation (5), which generates the first sigma point, sigma points from \(1\) to \(n\), and sigma points from \(n+1\) to \(2n\). \[\chi_{i}=\left\{ \begin{array}{rl}\overline{x}&i=0\\ \overline{x}+[\sqrt{(n+\lambda)P_{x}}]_{i}&i=1,2,\ldots,n\\ \overline{x}-[\sqrt{(n+\lambda)P_{x}}]_{i}&i=n+1,n+2,\ldots,2n\end{array} \right. \tag{5}\]where \(\lambda\) is the scale factor, \({(\sqrt{P})}^{T}(\sqrt{P})=P\), \(P\) is the covariance matrix, \({(\sqrt{P})}_{\rm i}\) is the i-th column of the root of the matrix. Step 3: Complete the sigma sampling point nonlinear conversion of state variables using the following equation:  
\[\chi_{\rm k+1}=f(\chi_{\rm k}) \tag{6}\]  
\[\overline{\rm X}_{\rm k+1}=\sum_{i=0}^{2n}W_{\rm i,m}\chi_{\rm k+1} \tag{7}\]  
\[P_{\rm k+1}=\sum_{i=0}^{2n}W_{\rm i,c}(\chi_{\rm k+1}-\overline{\rm X}_{\rm k+ 1})(\chi_{\rm k+1}-\overline{\rm X}_{\rm k+1})^{T}+Q_{\rm k+1} \tag{8}\]  
where \(Q_{\rm k+1}\) is the covariance matrix of process noise \(W_{\rm k}\) at time k + 1; \(W_{i,c}\) is the weight coefficient of the ith Sigma point when the state is updated; \(W_{\rm i,m}\) is the weight coefficient of the ith Sigma point when the measurement is updated. Step 4: Complete nonlinear conversion of sigma sampling points for measurement using the following equation:  
\[\chi_{\rm i}=\left\{ \begin{array}{cc}\overline{x}&i=0\\ \overline{x}+[\sqrt{(n+\lambda)P_{\rm x}}]_{\rm i}&i=1,2,\ldots,n\\ \overline{x}-[\sqrt{(n+\lambda)P_{\rm x}}]_{\rm i}&i=n+1,n+2,\ldots,2n\\ \\ Z_{\rm k+1}=\Psi(\chi_{\rm k+1})\end{array} \right. \tag{9}\]  
\[\overline{\rm Z}_{\rm k+1}=\sum_{i=0}^{2n}W_{\rm i,m}Z_{\rm k+1} \tag{10}\]  
\[P_{\rm zz}=\sum_{i=0}^{2n}W_{\rm i,c}(Z_{\rm k+1}-\overline{\rm Z}_{\rm k+1}) (Z_{\rm k+1}-\overline{\rm Z}_{\rm k+1})^{T}+R_{\rm k+1} \tag{11}\]  
where \(R_{\rm k+1}\) is the covariance matrix of measurement noise \(V_{\rm k}\) at time k + 1. Step 5: The covariance between the state vector and the measurement vector is described as follows:  
\[P_{\rm zz}=\sum_{i=0}^{2n}W_{\rm i,c}(\chi_{\rm k+1}-\overline{\rm X}_{\rm k+ 1})(Z_{\rm k+1}-\overline{\rm Z}_{\rm k+1})^{T} \tag{12}\]  
Step 6: The UKF gain is calculated, the state vector and variance are updated by the following:  
\[k_{\rm k+1}=P_{\rm zz}P_{\rm zz}^{-1} \tag{13}\]  
\[\overline{\rm X}_{\rm k+1}=\overline{\rm X}_{\rm k}+k_{\rm k+1}(Z_{\rm k+1}- \overline{\rm Z}_{\rm k+1}) \tag{14}\]  
\[R_{\rm k+1}=P_{\rm k}+k_{\rm k+1}P_{\rm zz}k_{\rm k+1}^{T} \tag{15}\]  
where \(P_{\rm zz}\) is the covariance matrix between the state and the measurement; \(P_{\rm zz}\) is the covariance matrix of the observation vector itself.