## 5 Optimal Gain Design Strategy
### Relationship with Steady State PVM Kalman Filters  
As described in Section 1, one of the reasons for considering the \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filters in practical use is the analytical performance predictions of the Kalman filter (and its variants). However, the proposed gain design strategy does not use the Kalman filter, and the relationship between the designed \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) and Kalman filters is therefore unclear. This section clarifies this relationship analytically. Indeed, the proposed strategy corresponds to an optimization of the elements of the covariance matrix of the process noise with respect to the RMS index. To prove this, we now derive the relationship between steady state Kalman gains and the arbitrary covariance matrix of process noise. The covariance matrix is expressed as [4]:  
\[\mathbf{Q}_{\text{gen}}=\begin{pmatrix}a&b\\ b&c\end{pmatrix}, \tag{33}\]  
where \(a>0\), \(b>0\), \(c>0\) and the dimensions of \(a\), \(b\) and \(c\) are m\({}^{2}\), m\({}^{2}\)/s and m\({}^{2}\)/s\({}^{2}\), respectively. For example, substituting \((a,b,c)=(qT^{4}/4,qT^{3}/2,\,qT^{2})\) into Equation (33) gives the \(\mathbf{Q}_{\text{ra}}\) of (16); substituting \((a,b,c)=(q_{\text{v}}T^{2},q_{\text{v}}T,q_{\text{v}})\) (\(q_{\text{v}}\) is the variance of the velocity noise) yields the random-velocity model [9]; and \(b=0\) leads to a diagonal \(\mathbf{Q}\), which is also a well-used setting in real applications [1; 2].  
The relationship between steady state Kalman gains and \(\mathbf{Q}_{\text{gen}}\) is derived as:  
\[a =\frac{T^{2}B_{\text{v}}}{1-R_{\text{xv}}\beta^{2}-(1-\theta) \alpha-\theta}\{(\beta^{2}\alpha+2\beta^{3}+\beta^{2}+(\theta-1)\theta\beta)R _{\text{xv}}^{2}\] \[+\alpha^{2}((1-\theta)+2\alpha\beta(1-\theta)+\beta^{2}\theta+ \beta(-\theta^{2}+3\theta-2))R_{\text{xv}}\] \[+\alpha\theta(1-\theta)+\theta(\theta-1)\}, \tag{34}\] \[b =\frac{\beta^{3}R_{\text{xv}}^{2}+(\alpha(\beta(1-\theta)-\theta ^{2}+\theta)+\beta^{2}\theta+\beta\theta+\theta^{2}-\theta)R_{\text{xv}}}{1- R_{\text{xv}}\beta^{2}-(1-\theta)\alpha-\theta}TB_{\text{v}},\] (35) \[c =\frac{(\alpha\beta\theta+\beta^{2}(\theta+1)-\beta\theta)R_{ \text{xv}}+\alpha\theta(-\beta-\theta)+\beta\theta+\theta^{2}}{1-R_{\text{xv }}\beta^{2}-(1-\theta)\alpha-\theta}B_{\text{v}}. \tag{36}\]  
The derivation of these is given in Appendix C.  
Equations (34)-(36) transform the optimal gains of the \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filter as elements of the covariance matrix of the process noise of the PVM Kalman filter and hence are useful in their design. In substituting these optimal designed gains into Equations (34)-(36), we obtain a Kalman filter having the same steady state performance. Similarly, solving Equations (34)-(36) with respect to \((\alpha,\beta,\theta)\) and using a set \(\mathbf{Q}_{\text{gen}}\), we find the steady state gains of the designed PVM Kalman filter and its performance using Equations (20), (26) and (28). Moreover, optimization of the proposed strategy of Equation (32) with respect to \((\alpha,\beta,\theta)\) is equivalent to optimization of the RMS index with respect to \((a,b,c)\) using Equations (34)-(36).  
The tracking filters and their design strategies that this paper considers are summarized in Table 2. In the following subsections, the performance of the \(\alpha\)-\(\beta\)-\(\eta\)-\(\theta\) filter realized with the proposed strategy is investigated and is compared with that of the conventional \(\alpha\)-\(\beta\) filter and RA model-based filter design.