Abstract
The limiting (or steadystate) Kalman filter provides a very efficient method for estimating the state vector in a timeinvariant linear system in realtime.
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Exercises
Exercises

9.1
Consider the twodimensional realtime tracking system
$$\begin{aligned} \left\{ \begin{array}{rl}\mathbf {x}_{k+1}&{}=\left[ \begin{array}{c@{\quad }c} 1 &{} h\\ 0 &{} 1 \end{array}\right] \mathbf {x}_{k}+\underline{\xi }_{k}\\ \quad v_{k}&{}=[1\quad 0]\mathbf {x}_{k}+\eta _{k},\end{array}\right. \end{aligned}$$where \(h>0\), and \(\{\underline{\xi }_{k}\},\ \{\eta _{k}\}\) are both uncorrelated zeromean Gaussian white noise sequences. The \(\alpha \beta \) tracker associated with this system is defined by
$$\begin{aligned} {\left\{ \begin{array}{ll} \check{\mathbf {x}}_{k}=\left[ \begin{array}{c@{\quad }c} 1 &{} h\\ 0 &{} 1 \end{array}\right] \check{\mathbf {x}}_{k1}+\left[ \begin{array}{c} \alpha \\ {\beta /h}\end{array}\right] (v_{k}[1\quad 0]\left[ \begin{array}{c@{\quad }c} 1 &{} h\\ 0 &{} 1 \end{array}\right] \check{\mathbf {x}}_{k1})\\ \check{\mathbf {x}}_{0}=E(\mathbf {x}_{0})\ . \end{array}\right. } \end{aligned}$$
(a)
Derive the decoupled Kalman filtering algorithm for this \(\alpha \beta \) tracker .

(b)
Give the conditions under which this \(\alpha \beta \) tracker is a limiting Kalman filter.

(a)

9.2
Verify the decoupled formulas of \(x_{k},\dot{x}_{k}\), and \(\ddot{x}_{k}\) given in Sect. 9.2 for the realtime tracking system (9.11).

9.3
Consider the threedimensional radartracking system
$$\begin{aligned} \left\{ \begin{array}{rl} \mathbf {x}_{k+1}&{}=\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} h &{} h^2/2 \\ 0 &{} 1 &{} h\\ 0 &{} 0 &{} 1 \end{array}\right] \mathbf {x}_{k}+\underline{\xi }_{k}\\ \quad v_{k}&{}=[1\quad 0\quad 0]\mathbf {x}_{k}+w_{k},\end{array}\right. \end{aligned}$$where \(\{w_{k}\}\) is a sequence of colored noise defined by
$$\begin{aligned} w_{k}=sw_{k1}+\eta _{k} \end{aligned}$$and \(\{\underline{\xi }_{k}\},\ \{\eta _{k}\}\) are both uncorrelated zeromean Gaussian white noise sequences, as described in Chap. 5. The associated \(\alpha \beta \gamma \theta \) tracker for this system is defined by the algorithm:
$$\begin{aligned} {\left\{ \begin{array}{ll} \check{X}_{k}=\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] \check{X}_{k1}+\left[ \begin{array}{c} \alpha \\ \beta /h\\ \gamma /h^2\\ \theta \end{array}\right] \{v_{k}[1\quad 0\quad 0]\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] \check{X}_{k1}\}\\ \check{X}_{0}=\left[ \begin{array}{c} E(\mathbf {x}_{0})\\ 0 \end{array}\right] , \end{array}\right. } \end{aligned}$$where \(\alpha ,\beta ,\gamma \), and \(\theta \) are constants (cf. Fig. 9.2).

(a)
Compute the matrix
$$\begin{aligned} \mathrm {\Phi }=\{I\left[ \begin{array}{c} \alpha \\ \beta /h\\ \gamma /h^{2}\\ \theta \end{array}\right] [1\quad 0\quad 0]\}\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] . \end{aligned}$$ 
(b)
Use Cramer’s rule to solve the system
$$\begin{aligned}{}[zI\mathrm {\Phi }]\left[ \begin{array}{c} \tilde{X}_{1}\\ \tilde{X}_{2}\\ \tilde{X}_{3}\\ W \end{array}\right] =z\left[ \begin{array}{c} \alpha \\ \beta /h\\ \gamma /h^{2}\\ \theta \end{array}\right] V \end{aligned}$$for \(\tilde{X}_{1},\tilde{X}_{2},\tilde{X}_{3}\) and W. (The above system is obtained when the ztransform of the \(\alpha \beta \gamma \theta \) filter is taken.)

(c)
By taking the inverse ztransforms of \(\tilde{X}_{1},\tilde{X}_{2},\tilde{X}_{3},\) and W, give the decoupled filtering equations for the \(\alpha \beta \gamma \theta \) filter.

(d)
Verify that when the colored noise sequence \(\{\eta _{k}\}\) becomes white; namely, \(s=0\) and \(\theta \) is chosen to be zero, the decoupled filtering equations obtained in part (c) reduce to those obtained in Sect. 9.2 with \(g_{1}=\alpha ,\ g_{2}=\beta /h\), and \(g_{3}=\gamma /h^{2}\).

(a)
 9.4

9.5
Prove Theorem 9.1 and observe that conditions (i)–(iii) are independent of the sampling time h.

9.6
Verify the equations in (9.21).

9.7
Verify the equation in (9.22).
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Chui, C.K., Chen, G. (2017). Decoupling of Filtering Equations. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_9
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DOI: https://doi.org/10.1007/9783319476124_9
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