Algorithm 1. Cognitive Radar Tracking Recursion Based on PF

Initialization

1.    ${\widehat{x}}_0^{\left(i\right)}\sim p\left(x_0\right)$, ${\mathsf{\omega }}_k^{(i)}=1/N_s$, $i=1,\dots ,N_s$

Controller Optimization

2.    ${\widehat{x}}_k^{\left(i\right)}\sim q\left(x_k^{(i)}|{\widehat{x}}_{k-1}^{(i)},{\mathbf{z}}_k\right)$, the mean $E_{p_k\left(x_k^{}|{\widehat{x}}_{k-1}^{\left(i\right)}\right)}\left[{\widehat{x}}_k^{}\right]=f_x\left({\widehat{x}}_{k-1}^{\left(i\right)}\right)$, $i=1,\dots ,N_s$

3.    ${\mathsf{\omega }}_k^{(i)}\left(\mathbf{\theta }\right)\propto p\left({\mathbf{z}}_k|{\widehat{x}}_k^{(i)}\right)p\left({\widehat{x}}_k^{(i)}|x_{k-1}^{(i)}\right)/q\left(x_k^{(i)}|x_{0:k}^{(i)},{\mathbf{z}}_{1:k}\right)$ $, i=1,\dots ,N_s$ $, \mathrm{normalize} {\tilde{\mathsf{\omega }}}_k^{(i)}\left(\mathbf{\theta }\right)={\mathsf{\omega }}_k\left(x_{0:k}^{\left(i\right)}\right)/{\displaystyle \sum _{i=1}^{N_s}{\mathsf{\omega }}_k\left(x_{0:k}^{\left(i\right)}\right)}$

4.    $\mathrm{If} N_{eff}=1/{\displaystyle \sum _{i=1}^{N_s}{\left({\tilde{\mathsf{\omega }}}_k^{\left(i\right)}\right)}^2}<N_{th}\left(\mathrm{empirical}\right)$ $, \mathrm{then} \left[{\left\{{\widehat{x}}_k^{\left(i\right)},{\widehat{\mathsf{\omega }}}_k^{(i)}\left(\mathbf{\theta }\right)\right\}}_{i=1}^{N_s}\right]=\mathrm{RESAMPLE}\left[{\left\{x_k^{\left(i\right)},{\tilde{\mathsf{\omega }}}_k^{(i)}\left(\mathbf{\theta }\right)\right\}}_{i=1}^{N_s}\right]$

5.    ${\mathbf{P}}_k\left(\mathbf{\theta }\right)={\displaystyle \sum _{i=1}^{N_s}{\mathsf{\omega }}_k^{(i)}\left(\mathbf{\theta }\right)\left({\widehat{x}}_k^{\left(i\right)}-{\widehat{x}}_k^{}\right){\left({\widehat{x}}_k^{\left(i\right)}-{\widehat{x}}_k^{}\right)}^T}$

6.    ${\mathbf{\theta }}_k^{*}=\arg \underset{{\mathbf{\theta }}_k^{}\in \mathcal{P}}{\min }\left[\mathrm{Tr}\left({\overline{\mathbf{P}}}_{k+1|k+1}^{}\left({\mathbf{\theta }}_k\right)\right)\right]$ $. \mathrm{Select} \mathrm{the} \mathrm{optimal} {\mathbf{\theta }}_k={\mathbf{\theta }}_k^{*}$

Motion Update and Measurement

7.    $f^{-}\left({\widehat{x}}_{k-1}^{(i)}\right)={\displaystyle \int q\left(x_k^{(i)}|x_{k-1}^{(i)};{\mathbf{\theta }}_k\right)f\left({\widehat{x}}_{k-1}^{(i)}\right)d{x}_{k-1}^{}}$$, {\widehat{x}}_{k-1}^{\left(i\right)}={\mu }_k^{-}=E_k^{-}\left[{\widehat{x}}_k^{}\right]$

8.    ${\widehat{y}}_k=\arg \underset{y}{\min }\ln f\left({\mathbf{z}}_k^{}|y;{\mathbf{\theta }}_k\right)$

Information Update and State Estimation

9.    $\left\{{\widehat{x}}_k^{\left(i\right)},{\mathsf{\omega }}_k^{(i)}\right\}=\left\{x_k^{\left(i\right)},{\tilde{\mathsf{\omega }}}_k^{(i)};{\mathbf{\theta }}_k\right\}$ $, i=1,\dots ,N_s$ $, {\widehat{x}}_k\approx {\displaystyle \sum _{i=1}^{N_s}{\mathsf{\omega }}_k^{\left(i\right)}{\widehat{x}}_k^{\left(i\right)}}$