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Decoupling of Filtering Equations

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Abstract

The limiting (or steady-state) Kalman filter provides a very efficient method for estimating the state vector in a time-invariant linear system in real-time.

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Fig. 9.1

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Correspondence to Charles K. Chui .

Exercises

Exercises

  1. 9.1

    Consider the two-dimensional real-time 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 zero-mean Gaussian white noise sequences. The \(\alpha -\beta \) tracker associated with this system is defined by

    $$\begin{aligned} {\left\{ \begin{array}{ll} \check{\mathbf {x}}_{k}=\left[ \begin{array}{c@{\quad }c} 1 &{} h\\ 0 &{} 1 \end{array}\right] \check{\mathbf {x}}_{k-1}+\left[ \begin{array}{c} \alpha \\ {\beta /h}\end{array}\right] (v_{k}-[1\quad 0]\left[ \begin{array}{c@{\quad }c} 1 &{} h\\ 0 &{} 1 \end{array}\right] \check{\mathbf {x}}_{k-1})\\ \check{\mathbf {x}}_{0}=E(\mathbf {x}_{0})\ . \end{array}\right. } \end{aligned}$$
    1. (a)

      Derive the decoupled Kalman filtering algorithm for this \(\alpha -\beta \) tracker .

    2. (b)

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

  2. 9.2

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

  3. 9.3

    Consider the three-dimensional radar-tracking 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

    $$\begin{aligned} w_{k}=sw_{k-1}+\eta _{k} \end{aligned}$$

    and \(\{\underline{\xi }_{k}\},\ \{\eta _{k}\}\) are both uncorrelated zero-mean Gaussian white noise sequences, as described in Chap. 5. The associated \(\alpha -\beta -\gamma -\theta \) tracker for this system is defined by the algorithm:

    $$\begin{aligned} {\left\{ \begin{array}{ll} \check{X}_{k}=\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] \check{X}_{k-1}+\left[ \begin{array}{c} \alpha \\ \beta /h\\ \gamma /h^2\\ \theta \end{array}\right] \{v_{k}-[1\quad 0\quad 0]\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] \check{X}_{k-1}\}\\ \check{X}_{0}=\left[ \begin{array}{c} E(\mathbf {x}_{0})\\ 0 \end{array}\right] , \end{array}\right. } \end{aligned}$$

    where \(\alpha ,\beta ,\gamma \), and \(\theta \) are constants (cf. Fig. 9.2).

    1. (a)

      Compute the matrix

      $$\begin{aligned} \mathrm {\Phi }=\{I-\left[ \begin{array}{c} \alpha \\ \beta /h\\ \gamma /h^{2}\\ \theta \end{array}\right] [1\quad 0\quad 0]\}\left[ \begin{array}{c@{\quad }c} A &{} 0\\ 0 &{} s \end{array}\right] . \end{aligned}$$
    2. (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 z-transform of the \(\alpha -\beta -\gamma -\theta \) filter is taken.)

    3. (c)

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

    4. (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 \(g_{3}=\gamma /h^{2}\).

  4. 9.4

    Verify equations (9.17)–(9.20).

  5. 9.5

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

  6. 9.6

    Verify the equations in (9.21).

  7. 9.7

    Verify the equation in (9.22).

Fig. 9.2
figure 2

Block diagram of the 4-parameter tracker

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Chui, C.K., Chen, G. (2017). Decoupling of Filtering Equations. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/978-3-319-47612-4_9

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