## 4 Experimental Results and Discussion
### Numerical Simulations and Analysis  
To design the simulation scenarios, suppose that \(\mathbf{X}=[x,y,\dot{x},\dot{y}]\) represents the state of the maneuvering target, including the position and velocity. The sensor is the radar, and its position is fixed at the coordinate origin. Thus, the measurement equation is  
\[\mathbf{Z}_{k}=\left[\begin{array}{c}r_{k}\\ \theta_{k}\end{array}\right]=\left[\begin{array}{c}\sqrt{({x_{k}}^{2}+{y_{k }}^{2})}\\ \tan^{-1}({y_{k}}/{x_{k}})\end{array}\right]+\mathbf{V}_{k} \tag{40}\]  
where \(r_{k}\) and \(\theta_{k}\) are the distance and angle measurement components of the radar, respectively.  
The high-speed maneuvering target moves in a two-dimensional scenario with 100 s; the initial position is [0 m, 400 m], and the initial velocity is [10 m/s, 40 m/s]; the target moves at a constant velocity in a straight line from 0 s-20 s, with a constant right turn rate being \(-0.1\) rad/s from 20 s-40 s, constant velocity in a straight line at 40 s-60 s, constant left turn with the turn rate being 0.1 rad/s at 60 s-80 s, and a constant velocity in a straight line at 80 s-100 s. In a Cartesian coordinate system, the true trajectory of the maneuvering target is depicted in Figure 4.  
Both the IMM and CPIMM include three models: one Constant Velocity (CV) model [44] and two Coordinated Turn (CT) models [44].  
For the CV model, the state transition matrix is  
\[\mathbf{F}_{CV}=\left[\begin{array}{cccc}1&T&0&0\\ 0&1&0&0\\ 0&0&1&T\\ 0&0&0&1\end{array}\right] \tag{41}\]  
where \(T\) is the sampling period, and \(T=1\) s.  
The process noise is calculated with  
\[\mathbf{V}_{CV}=N(.;0;{\sigma_{CV}}^{2}) \tag{42}\]  
where \({\sigma_{CV}}^{2}\) denotes the covariance of the process noise, and \({\sigma_{CV}}^{2}=5\).  
For the CT model, the state transition matrix is  
\[\mathbf{F}_{CT}=\begin{bmatrix}1&\frac{\sin(\omega T)}{\omega}&0&-\frac{1-\cos( \omega T)}{\omega}\\ 0&\cos(\omega T)&0&-\sin(\omega T)\\ 0&\frac{1-\cos(\omega T)}{\omega}&1&\frac{\sin(\omega T)}{\omega}\\ 0&\sin(\omega T)&0&\cos(\omega T)\end{bmatrix} \tag{43}\]  
where \(\omega\) is the turn rate.  
The process noise meets with  
\[\mathbf{V}_{CT}=N(.;0;{\sigma_{CT}}^{2}) \tag{44}\]  
where \({\sigma_{CT}}^{2}\) denotes the covariance of the process noise, and \({\sigma_{CT}}^{2}=20\).  
The two models use the following transition and mode probabilities.  
The transition probability matrix is  
Figure 4: True trajectory of the maneuvering target.  
\[\left[\begin{array}{ccc}0.99&0.005&0.005\\ 0.005&0.99&0.005\\ 0.005&0.005&0.99\end{array}\right].\]  
The initial mode probability matrix is  
\[[\begin{array}{ccc}0.99&0.005&0.005].\end{array}\]  
The measurement covariance is \(\boldsymbol{R}=diag\big{(}\sigma_{w}^{2},\sigma_{w}^{2}\big{)}=diag(15,15)\).  
Case 1 is where the sensor has no faults and is in a normal work situation. Case 2 is when the sensor has faults, and \(\delta\boldsymbol{h}_{out,k}=\left[\delta r_{k}\;\delta\theta_{k}\right]^{ \mathrm{T}}\), where \(\delta r_{k}\) and \(\delta\theta_{k}\) represents the distance deviation within \([-5\;\mathrm{m},\,5\;\mathrm{m}]\) and the angle drift rate within \([0\;1.0\times 10^{-2}\;\mathrm{rad}/\mathrm{s}]\), respectively.  
Every simulation experiment is tested 100 times using Monte Carlo, and algorithms of two sets of comparison simulations are evaluated by root mean square error (RMSE), which is computed through  
\[\boldsymbol{E}=\sqrt{\left(\frac{1}{N}\sum_{i=1}^{N}\left(\hat{\boldsymbol{X} }^{i}-\boldsymbol{X}^{i}\right)\left(\hat{\boldsymbol{X}}^{i}-\boldsymbol{X} ^{i}\right)^{T}\right)} \tag{45}\]  
where \(\hat{\boldsymbol{X}}^{i}\) and \(\boldsymbol{X}^{i}\) are the estimated and true state in the \(i^{\mathrm{th}}\) simulation, respectively; \(N\) is the number of simulations; \(N=100\).