# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## 2 Single-Model Nonlinear Filtering Algorithms
### Single-Model Filtering  
Let \(\mathbf{x}_{k}\) and \(\mathbf{z}_{k}\) denote the \(n\)-dimensional and \(m\)-dimensional target state and sensor measurement at time \(t_{k}\). The discrete-time dynamic and measurement models for a nonlinear filtering problem are described by [8; 20]  
\[\mathbf{x}_{k}=\mathbf{f}_{k-1}(\mathbf{x}_{k-1})+\mathbf{w}_{k-1}, \tag{1}\]  
\[\mathbf{z}_{k}=\mathbf{h}_{k}(\mathbf{x}_{k})+\mathbf{n}_{k}, \tag{2}\]  
where \(\mathbf{f}_{k-1}\) and \(\mathbf{h}_{k}\) are known functions for time evolution and measurement, respectively. We assume that the process noise \(\mathbf{w}_{k-1}\) is zero-mean white Gaussian for the time interval \((t_{k-1},t_{k}]\) with a covariance of \(\mathbf{Q}_{k-1}\) and that the measurement noise \(\mathbf{n}_{k}\) is zero-mean white Gaussian with a covariance of \(\mathbf{R}_{k}\) at time \(t_{k}\)[8; 20]. Our goal is to process the noisy measurements sequentially and estimate the target state. Next, we present the formal recursive Bayesian optimal estimation approach [20; 50]. Let \(\mathbf{Z}_{k}\) represent the complete collection of observations accumulated up to the time \(t_{k}\),  
\[\mathbf{Z}_{k}:=\{\mathbf{z}_{1},\mathbf{z}_{2},\ldots,\mathbf{z}_{k}\}. \tag{3}\]  
The conditional mean represents the optimal estimate in the MMSE sense and is given by [7; 8; 20; 50]  
\[\mathbf{\hat{x}}_{k|k}=E\{\mathbf{x}_{k}|\mathbf{Z}_{k}\}=\int\mathbf{x}_{k}p (\mathbf{x}_{k}|\mathbf{Z}_{k})d\mathbf{x}_{k}, \tag{4}\]  
where \(p(\mathbf{x}_{k}|\mathbf{Z}_{k})\) represents the conditional probability density function (pdf). Suppose we know the pdf \(p(\mathbf{x}_{k-1}|\mathbf{Z}_{k-1})\) at time \(t_{k-1}\). Our objective is to determine the posterior pdf \(p(\mathbf{x}_{k}|\mathbf{Z}_{k})\) after processing the measurement \(\mathbf{z}_{k}\). At time \(t_{k}\), the prediction density or prior density is given by the Chapman-Kolmogorov equation [20; 50]. \[p(\mathbf{x}_{k}|\mathbf{Z}_{k-1})=\int p(\mathbf{x}_{k}|\mathbf{x}_{k-1})p( \mathbf{x}_{k-1}|\mathbf{Z}_{k-1})d\mathbf{x}_{k-1}, \tag{5}\]  
where \(p(\mathbf{x}_{k}|\mathbf{x}_{k-1})\) is the transition density. By using [PERSON] rule, we obtain the posterior density:  
\[p(\mathbf{x}_{k}|\mathbf{Z}_{k})=\frac{p(\mathbf{z}_{k}|  
The quantity \(p(\mathbf{z}_{k}|\mathbf{x}_{k})\) is known as the likelihood of the state \(\mathbf{x}_{k}\) given the measurement \(\mathbf{z}_{k}\)[8]. From the dynamic model (1) with additive zero-mean white Gaussian noise, \(\mathbf{w}_{k-1}\), with a covariance of \(\mathbf{Q}_{k-1}\), we obtain the transition density as  
\[p(\mathbf{x}_{k}|\mathbf{x}_{k-1})=\mathcal{N}(\mathbf{x}_{k};\mathbf{f}_{k-1 }(\mathbf{x}_{k-1}),\mathbf{Q}_{k-1}). \tag{8}\]  
Similarly, from the measurement model (2) with additive zero-mean Gaussian noise, \(\mathbf{n}_{k}\), with a covariance of \(\mathbf{R}_{k}\), the likelihood can be expressed as  
\[p(\mathbf{z}_{k}|\mathbf{x}_{k})=\mathcal{N}(\mathbf{z}_{k};\mathbf{h}_{k}( \mathbf{x}_{k}),\mathbf{R}_{k}). \tag{9}\]  
Given the prior distribution \(p_{0}(\mathbf{x}_{0})\) of the target state, in principle, we can recursively determine the posterior density \(p(\mathbf{x}_{k}|\mathbf{Z}_{k})\) using (5)-(9). However, obtaining closed-form solutions is intractable in most cases due to multi-dimensional integrals in (5) and (7), and in most real-world problems, approximate or sub-optimal solutions are obtained. The EKF, UKF, CKF, GHF, CDF, DDF, and PF represent examples of approximate nonlinear filters. If we assume that all the densities are Gaussian, then the Bayesian recursion Equations (5)-(7) can be evaluated approximately, and the resulting class of filters is called Gaussian filters [16; 20; 23].