# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## 8 Posterior Cramer-Rao Lower Bound (PCRLB)  
Consider the discrete-time dynamic and measurement models given, respectively, by  
\[\mathbf{x}_{k} =\mathbf{f}_{k-1}(\mathbf{x}_{k-1},\mathbf{w}_{k-1}), \tag{97}\] \[\mathbf{z}_{k} =\mathbf{h}_{k}(\mathbf{x}_{k},\mathbf{n}_{k}). \tag{98}\]  
Let \(\mathbf{\hat{x}}_{k|k}\) be an unbiased estimator of the state vector \(\mathbf{x}_{k}\), based on measurement sequence \(\mathbf{Z}_{k}=\{\mathbf{z}_{1},...,\mathbf{z}_{k}\}\) and prior density \(p(\mathbf{x}_{0})\). Then, the mean square error matrix (MSEM) of \(\mathbf{\hat{x}}_{k|k}\), denoted as \(\mathbf{\Sigma}_{k|k}\), has a lower bound, called the PCRLB \(\mathbf{J}_{k}^{-1}\)[PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018):  
\[\mathbf{\Sigma}_{k|k}:=E((\mathbf{\hat{x}}_{k|k}-\mathbf{x}_{k})(\mathbf{ \hat{x}}_{k|k}-\mathbf{x}_{k})^{\prime})\geq\mathbf{J}_{k}^{-1}, \tag{99}\]  
where \(\mathbf{J}_{k}\) represents the Bayesian information matrix [PERSON] and [PERSON] (2018). [PERSON] et al. [PERSON] et al. (2019) derived an elegant recursive expression to compute \(\mathbf{J}_{k}\) recursively by using  
\[\mathbf{J}_{k}=\mathbf{D}_{k-1}^{33}-(\mathbf{D}_{k-1}^{12})^{\prime}(\mathbf{ J}_{k-1}+\mathbf{D}_{k-1}^{11})^{-1}\mathbf{D}_{k-1}^{12}+\mathbf{J}_{k,Z}, \tag{100}\]  
Figure 4: The IMM filter based on CKF.  
where, for additive Gaussian process noise and measurement noise (valid for our current problem) [5; 20; 61],  
\[\mathbf{D}_{k-1}^{11}=E\{\mathbf{\hat{F}}_{k-1}^{\prime}\mathbf{Q}_{k-1}^{-1} \mathbf{\hat{F}}_{k-1}\}, \tag{101}\]  
\[\mathbf{F}_{k-1}=\left[\
abla_{\mathbf{x}_{k-1}}\mathbf{f}_{k-1}^{\prime}( \mathbf{x}_{k-1})\right]^{\prime}, \tag{102}\]  
\[\mathbf{D}_{k-1}^{12}=-E\{\mathbf{\hat{F}}_{k-1}^{\prime}\}\mathbf{Q}_{k-1}^{-1}, \tag{103}\]  
\[\mathbf{D}_{k-1}^{33}=\mathbf{Q}_{k-1}^{-1}, \tag{104}\]  
\[\mathbf{J}_{k,Z}=E\{\mathbf{\hat{H}}_{k}^{\prime}\mathbf{R}_{k}^{-1}\mathbf{ \hat{H}}_{k}\}, \tag{105}\]  
\[\mathbf{\hat{H}}_{k}=\left[\
abla_{\mathbf{x}_{k}}\mathbf{h}_{k}^{\prime}( \mathbf{x}_{k})\right]^{\prime}. \tag{106}\]  
The Jacobian matrices \(\mathbf{\hat{F}}_{k-1}\) in (102) and \(\mathbf{\hat{H}}_{k}\) in (106) are nonlinear functions of the true states. Therefore, it is not possible to calculate the expected values in equations (101), (103), and (105) by using analytical methods. We evaluated these expected values using Monte Carlo runs. For the NCV motion,  
\[\mathbf{D}_{k-1}^{11}=(\mathbf{F}_{k-1}^{1})^{\prime}\mathbf{Q}_{k-1}^{-1} \mathbf{F}_{k-1}^{1}, \tag{107}\]  
\[\mathbf{D}_{k-1}^{12}=-\mathbf{F}_{k-1}^{\prime}\mathbf{Q}_{k-1}^{-1}. \tag{108}\]  
A detailed derivation of the Jacobian \(\mathbf{\hat{F}}_{k}\) for the time evolution function is presented in Appendix A. A detailed derivation of the Jacobian \(\mathbf{\hat{H}}_{k}\) for the AMTI measurement function is presented in Appendix B and is given by  
\[\mathbf{\hat{H}}_{k}=\begin{bmatrix}\frac{u_{y,k}}{r_{k}}&\mathbf{0}_{1\times 2 }\\ \frac{u_{y,k}}{r_{k}}&-\frac{u_{y,k}}{r_{k}}&\mathbf{0}_{1\times 2}\\ \frac{\mathbf{v}_{k}^{\prime}}{r_{k}}\big{[}\mathbf{I}_{2}-\mathbf{u}_{k} \mathbf{u}_{k}^{\prime}\big{]}&\mathbf{u}_{k}^{\prime}\end{bmatrix} \tag{109}\]