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Orthogonal Projection and Kalman Filter

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Abstract

The elementary approach to the derivation of the optimal Kalman filtering process discussed in Chap. 2 has the advantage that the optimal estimate \(\hat{\mathbf {x}}_{k}=\hat{\mathbf {x}}_{k|k}\) of the state vector \(\mathbf {x}_{k}\) is easily understood to be a least-squares estimate of \(\mathbf {x}_{k}\) with the properties that (i) the transformation that yields \(\hat{\mathbf {x}}_{k}\) from the data \(\overline{\mathbf {v}}_{k}=[\mathbf {v}_{0}^{\top }\cdots \mathbf {v}_{k}^{\top }]^{\top }\) is linear, (ii) \(\hat{\mathbf {x}}_{k}\) is unbiased in the sense that \(E(\hat{\mathbf {x}}_{k})=E(\mathbf {x}_{k})\), and (iii) it yields a minimum variance estimate with \((Var(\overline{\underline{\epsilon }}_{k, k}))^{-1}\) as the optimal weight.

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Fig. 3.1
Fig. 3.2
Fig. 3.3

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Correspondence to Charles K. Chui .

Exercises

Exercises

  1. 3.1.

    Let \(A\ne 0\) be a non-negative definite and symmetric constant matrix. Show that \(\mathrm {tr}\, A>0\). (Hint: Decompose A as \(A=BB^{\top }\) with \(B\ne 0\).)

  2. 3.2.

    Let

    $$\begin{aligned} \hat{\mathbf {e}}_{j}=C_{j}(\mathbf {x}_{j}-\hat{\mathbf {y}}_{j-1})=C_{j}\left( \mathbf {x}_{j}-\sum _{i=0}^{j-1}\hat{P}_{j-1, i}\mathbf {v}_{i}\right) , \end{aligned}$$

    where \(\hat{P}_{j-1, i}\) are some constant matrices. Use Assumption 2.1 to show that

    $$\begin{aligned} \langle \underline{\eta }_{\ell },\ \hat{\mathbf {e}}_{j}\rangle =O_{q\times q} \end{aligned}$$

    for all \(\ell \ge j\).

  3. 3.3.

    For random vectors \(\mathbf {w}_{0},\cdots , \mathbf {w}_{r}\), define

    $$\begin{aligned}&Y(\mathbf {w}_{0}, \cdots , \mathbf {w}_{r})\\ =&\left\{ \mathbf {y}:\mathbf {y}=\sum _{i=0}^{r}P_{i}\mathbf {w}_{i},\quad P_{0}, \cdots , P_{r},\ constant\ matrices\right\} . \end{aligned}$$

    Let

    $$\begin{aligned} \mathbf {z}_{j}=\mathbf {v}_{j}-C_{j}\sum _{i=0}^{j-1}\hat{P}_{j-1, i}\mathbf {v}_i \end{aligned}$$

    be defined as in (3.4) and \(\mathbf {e}_{j}=\Vert \mathbf {z}_{j}\Vert ^{-1}\mathbf {z}_{j}\). Show that

    $$\begin{aligned} Y(\mathbf {e}_{0}, \cdots ,\mathbf {e}_{k})=Y(\mathbf {v}_{0}, \cdots ,\mathbf {v}_{k}). \end{aligned}$$
  4. 3.4.

    Let

    $$\begin{aligned} \hat{\mathbf {y}}_{j-1}=\sum _{i=0}^{j-1}\hat{P}_{j-1, i}\mathbf {v}_i \end{aligned}$$

    and

    $$\begin{aligned} \mathbf {z}_{j}=\mathbf {v}_{j}-C_{j}\sum _{i=0}^{j-1}\hat{P}_{j-1, i}\mathbf {v}_i. \end{aligned}$$

    Show that

    $$\begin{aligned} \langle \hat{\mathbf {y}}_{j},\ \mathbf {z}_{k}\rangle =O_{n\times q},\qquad j=0, 1, \cdots , k-1. \end{aligned}$$
  5. 3.5.

    Let \(\mathbf {e}_{j}\) be defined as in Exercise 3.3. Also define

    $$\begin{aligned} \check{\mathbf {x}}_{k}=\sum _{i=0}^{k}\langle \mathbf {x}_{k},\ \mathbf {e}_{i}\rangle \mathbf {e}_{i} \end{aligned}$$

    as in (3.10). Show that

    $$\begin{aligned}\begin{gathered} \langle \mathbf {x}_{k},\ \underline{\xi }_{k}\rangle =O_{n\times n},\qquad \langle \check{\mathbf {x}}_{k|k},\ \underline{\xi }_{j}\rangle =O_{n\times n},\\ \langle \mathbf {x}_{k},\ \underline{\eta }_{j}\rangle =O_{n\times q},\qquad \langle \check{\mathbf {x}}_{k-1|k-1},\ \underline{\eta }_{k}\rangle =O_{n\times q}, \end{gathered}\end{aligned}$$

    for \(j=0, 1, \cdots , k\).

  6. 3.6.

    Consider the linear deterministic/stochastic system

    $$\begin{aligned} \left\{ \begin{array}{rcl} \mathbf {x}_{k+1}&{}=&{}A_{k}\mathbf {x}_k+B_{k}\mathbf {u}_{k}+\mathrm {\Gamma }_{k}\underline{\xi }_{k}\\ \mathbf {v}_{k}&{}=&{}C_{k}\mathbf {x}_{k}+D_{k}\mathbf {u}_{k}+\underline{\eta }_{k}, \end{array}\right. \end{aligned}$$

    where \(\{\mathbf {u}_{k}\}\) is a given sequence of deterministic control input m-vectors, \(1\le m\le n\). Suppose that Assumption 2.1 is satisfied. Derive the Kalman filtering algorithm for this model.

  7. 3.7.

    Consider a simplified radar tracking model where a large-amplitude and narrow-width impulse signal is transmitted by an antenna. The impulse signal propagates at the speed of light c, and is reflected by a flying object being tracked. The radar antenna receives the reflected signal so that a time-difference \(\mathrm {\Delta } t\) is obtained. The range (or distance) d from the radar to the object is then given by \(d=c\mathrm {\Delta } t/2\). The impulse signal is transmitted periodically with period h. Assume that the object is traveling at a constant velocity w with random disturbance \(\xi \sim N(0, q)\), so that the range d satisfies the difference equation

    $$\begin{aligned} d_{k+1}=d_{k}+h(w_{k}+\xi _{k}). \end{aligned}$$

    Suppose also that the measured range using the formula \(d= c\mathrm {\Delta } t/2\) has an inherent error \(\mathrm {\Delta } d\) and is contaminated with noise \(\eta \) where \(\eta \sim N(0, r)\), so that

    $$\begin{aligned} v_{k}=d_{k}+\mathrm {\Delta } d_{k}+\eta _{k}. \end{aligned}$$

    Assume that the initial target range is \(d_{0}\) which is independent of \(\xi _{k}\) and \(\eta _{k}\), and that \(\{\xi _{k}\}\) and \(\{\eta _{k}\}\) are also independent (cf. Fig. 3.2). Derive a Kalman filtering algorithm as a range-estimator for this radar tracking system.

  8. 3.8.

    A linear stochastic system for radar tracking can be described as follows. Let \(\mathrm {\Sigma }\), \(\mathrm {\Delta } A\), \(\mathrm {\Delta } E\) be the range, the azimuthal angular error , and the elevational angular error , respectively, of the target, with the radar being located at the origin (cf. Fig. 3.3). Consider \(\mathrm {\Sigma },\mathrm {\Delta } A\), and \(\mathrm {\Delta } E\) as functions of time with first and second derivatives denoted by \(\dot{\mathrm {\Sigma }}\), \(\mathrm {\Delta }\dot{A}\), \(\mathrm {\Delta }\dot{E}\), \(\ddot{\mathrm {\Sigma }}\), \(\mathrm {\Delta }\ddot{A}\), \(\mathrm {\Delta }\ddot{E}\), respectively. Let \(h>0\) be the sampling time unit and set \(\mathrm {\Sigma }_{k}=\mathrm {\Sigma }(kh)\), \(\dot{\mathrm {\Sigma }}_{k}=\dot{\mathrm {\Sigma }}(kh)\), \(\ddot{\mathrm {\Sigma }}_{k}=\ddot{\mathrm {\Sigma }}(kh)\), etc. Then, using the second degree Taylor polynomial approximation , the radar tracking model takes on the following linear stochastic state-space description:

    $$\begin{aligned} \left\{ \begin{array}{rcl} \mathbf {x}_{k+1}&{}=&{}\tilde{A}\mathbf {x}_{k}+\mathrm {\Gamma }_{k}\underline{\xi }_{k}\\ \mathbf {v}_{k}&{}=&{}\tilde{C}\mathbf {x}_{k}+\underline{\eta }_{k}, \end{array}\right. \end{aligned}$$

    where

    $$\begin{aligned} \mathbf {x}_{k}=&[\mathrm {\Sigma }_{k}\ \dot{\mathrm {\Sigma }}_{k}\ \ddot{\mathrm {\Sigma }}_{k}\ \mathrm {\Delta } A_{k}\ \mathrm {\Delta }\dot{A}_{k}\ \mathrm {\Delta }\ddot{A}_{k}\ \mathrm {\Delta } E_{k}\ \mathrm {\Delta }\dot{E}_{k}\ \mathrm {\Delta }\ddot{E}_{k}]^{\top },\\ \tilde{A}=&\left[ \begin{array}{ccccccccc} 1 &{} h &{} h^{2}/2 &{} &{} &{} &{} &{} &{} \\ 0 &{} 1 &{} h &{} &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} 1 &{} &{} &{} &{} &{} &{} \\ &{} &{} &{} 1 &{} h &{} h^{2}/2 &{} &{} &{} \\ &{} &{} &{} 0 &{} 1 &{} h &{} &{} &{} \\ &{} &{} &{} 0 &{} 0 &{} 1 &{} &{} &{} \\ &{} &{} &{} &{} &{} &{} 1 &{} h &{} h^{2}/2\\ &{} &{} &{} &{} &{} &{} 0 &{} 1 &{} h\\ &{} &{} &{} &{} &{} &{} 0 &{} 0 &{} 1 \end{array}\right] ,\\&\tilde{C}= \left[ \begin{array}{ccccccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \end{array}\right] , \end{aligned}$$

    and \(\{\underline{\xi }_{k}\}\) and \(\{\underline{\eta }_{k}\}\) are independent zero-mean Gaussian white noise sequences with \(Var(\underline{\xi }_{k})=Q_{k}\) and \(Var(\underline{\eta }_{k})=R_{k}\). Assume that

    $$\begin{aligned}\begin{gathered} \mathrm {\Gamma }_{k}=\left[ \begin{array}{ccc} \mathrm {\Gamma }_{k}^{1} &{} &{} \\ &{} \mathrm {\Gamma }_{k}^{2} &{} \\ &{} &{} \mathrm {\Gamma }_{k}^{3} \end{array}\right] ,\\ Q_{k}=\left[ \begin{array}{ccc} Q_{k}^{1} &{} &{} \\ &{} Q_{k}^{2} &{} \\ &{} &{} Q_{k}^{3} \end{array}\right] ,\qquad R_{k}=\left[ \begin{array}{ccc} R_{k}^{1} &{} &{} \\ &{} R_{k}^{2} &{} \\ &{} &{} R_{k}^{3} \end{array}\right] , \end{gathered}\end{aligned}$$

    where \(\mathrm {\Gamma }_{k}^{i}\) are \(3\times 3\) submatrices, \(Q_{k}^{i}\), \(3\times 3\) non-negative definite symmetric submatrices, and \(R_{k}^{i}\), \(3\times 3\) positive definite symmetric submatrices, for \(i=1, 2, 3\). Show that this system can be decoupled into three subsystems with analogous state-space descriptions.

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Chui, C.K., Chen, G. (2017). Orthogonal Projection and Kalman Filter. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/978-3-319-47612-4_3

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