Abstract
In this chapter, we consider the special case where all known constant matrices are independent of time.
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Exercises
Exercises

6.1.
Prove that the estimate \(\tilde{\mathbf {x}}_{k1}\) in (6.7) is an unbiased estimate of \(\mathbf {x}_{k1}\) in the sense that \(E(\tilde{\mathbf {x}}_{k1})=E(\mathbf {x}_{k1})\).

6.2.
Verify that
$$\begin{aligned} \frac{d}{ds}A^{1}(s)=A^{1}(s)\left[ \frac{d}{ds}A(s)\right] A^{1}(s) . \end{aligned}$$ 
6.3.
Show that if \(\lambda _{min}\) is the smallest eigenvalue of P, then \( P\ge \lambda _{min}I\). Similarly, if \(\lambda _{max}\) is the largest eigenvalue of P then \(P\le \lambda _{max}I\).

6.4.
Let F be an \(n\times n\) matrix. Suppose that all the eigenvalues of F are of absolute value less than 1. Show that \(F^{k}\rightarrow 0\) as \( k\rightarrow \infty \).

6.5.
Prove that for any \(n\times n\) matrices A and B,
$$\begin{aligned} (A+B)(A+B)^{\top }\le 2(AA^{\top }+BB^{\top })\ . \end{aligned}$$ 
6.6.
Let \(\{\underline{\xi }_{k}\}\) and \(\{\underline{\eta }_{k}\}\) be sequences of zeromean Gaussian white system and measurement noise processes, respectively, and \(\vec {\mathbf {x}}_{k}\) be defined by (6.4). Show that
$$\begin{aligned} \langle \mathbf {x}_{k1}\vec {\mathbf {x}}_{k1},\ \underline{\xi }_{k1}\rangle =0 \end{aligned}$$and
$$\begin{aligned} \langle \mathbf {x}_{k1}\vec {\mathbf {x}}_{k1},\ \underline{\eta }_{k}\rangle =0. \end{aligned}$$ 
6.7.
Verify that for the Kalman gain \(G_{k}\), we have
$$\begin{aligned} (IG_{k}C)P_{k, k}{}_{1}C^{\top }G_{k}^{\top }+G_{k}R_{k}G_{k}^{\top }=0. \end{aligned}$$Using this formula, show that
$$\begin{aligned} P_{k, k}&=(IG_{k}C)AP_{k1, k1}A^{\top }(IG_{k}C)^{\top }\\&\quad +(IG_{k}C){{\Gamma }} Q_{k}{{\Gamma }}^{\top }(IG_{k}C)^{\top }+G_{k}RG_{k}^{\top }. \end{aligned}$$ 
6.8.
By imitating the proof of Lemma 6.8, show that all the eigenvalues of \((IGC)A\) are of absolute value less than 1.

6.9.
Let \(\underline{\epsilon }_{k}=\hat{\mathbf {x}}_{k}\vec {\mathbf {x}}_{k}\) where \(\vec {\mathbf {x}}_{k}\) is defined by (6.4), and let \(\underline{\delta }_{k}= \mathbf {x}_{k}\hat{\mathbf {x}}_{k}\). Show that
$$\begin{aligned}&\langle \underline{\epsilon }_{k1}, \underline{\xi }_{k1}\rangle =0,&\quad \langle \underline{\epsilon }_{k1}, \underline{\eta }_{k}\rangle =0,\\&\langle \underline{\delta }_{k1},\ \underline{\xi }_{k1}\rangle =0,&\quad \langle \underline{\delta }_{k1},\ \underline{\eta }_{k}\rangle =0, \end{aligned}$$where \(\{\underline{\xi }_{k}\}\) and \(\{\underline{\eta }_{k}\}\) are zeromean Gaussian white system and measurement noise processes, respectively.

6.10.
Let
$$\begin{aligned} B_{j}=\langle \underline{\epsilon }_{j},\ \underline{\delta }_{j}\rangle A^{\top }C^{\top },\qquad j=0, 1,\ \cdots , \end{aligned}$$where \(\underline{\epsilon }_{j}=\hat{\mathbf {x}}_{j}\vec {\mathbf {x}}_{j}\), \(\underline{\delta }_{j}=\mathbf {x}_{j}\hat{\mathbf {x}}_{j}\), and \(\vec {\mathbf {x}}_{j}\) is defined by (6.4). Prove that \(B_{j}\) are componentwise uniformly bounded.

6.11.
Derive formula (6.41).

6.12.
Derive the limiting (or steadystate) Kalman filtering algorithm for the scalar system:
$$\begin{aligned} \left\{ \begin{array}{rl} x_{k+1}&{}=ax_{k}+\gamma \xi _{k}\\ v_{k}&{}=cx_{k}+\eta _{k}, \end{array}\right. \end{aligned}$$where \(a,\ \gamma \), and c are constants and \(\{\xi _{k}\}\) and \(\{\eta _{k}\}\) are zeromean Gaussian white noise sequences with variances q and r, respectively.
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Chui, C.K., Chen, G. (2017). Limiting Kalman Filter. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_6
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DOI: https://doi.org/10.1007/9783319476124_6
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