Limiting Kalman Filter

Abstract

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

Exercises

• 6.1. Prove that the estimate $\tilde{\mathbf {x}}_{k-1}$ in (6.7) is an unbiased estimate of $\mathbf {x}_{k-1}$ in the sense that $E(\tilde{\mathbf {x}}_{k-1})=E(\mathbf {x}_{k-1})$.

• 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 zero-mean Gaussian white system and measurement noise processes, respectively, and $\vec {\mathbf {x}}_{k}$ be defined by (6.4). Show that

  $$\begin{aligned} \langle \mathbf {x}_{k-1}-\vec {\mathbf {x}}_{k-1},\ \underline{\xi }_{k-1}\rangle =0 \end{aligned}$$

  and

  $$\begin{aligned} \langle \mathbf {x}_{k-1}-\vec {\mathbf {x}}_{k-1},\ \underline{\eta }_{k}\rangle =0. \end{aligned}$$

• 6.7. Verify that for the Kalman gain $G_{k}$, we have

  $$\begin{aligned} -(I-G_{k}C)P_{k, k-}{}_{1}C^{\top }G_{k}^{\top }+G_{k}R_{k}G_{k}^{\top }=0. \end{aligned}$$

  Using this formula, show that

  $$\begin{aligned} P_{k, k}&=(I-G_{k}C)AP_{k-1, k-1}A^{\top }(I-G_{k}C)^{\top }\\&\quad +(I-G_{k}C){{\Gamma }} Q_{k}{{\Gamma }}^{\top }(I-G_{k}C)^{\top }+G_{k}RG_{k}^{\top }. \end{aligned}$$

• 6.8. By imitating the proof of Lemma 6.8, show that all the eigenvalues of $(I-GC)A$ 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

  $$\begin{aligned}&\langle \underline{\epsilon }_{k-1}, \underline{\xi }_{k-1}\rangle =0,&\quad \langle \underline{\epsilon }_{k-1}, \underline{\eta }_{k}\rangle =0,\\&\langle \underline{\delta }_{k-1},\ \underline{\xi }_{k-1}\rangle =0,&\quad \langle \underline{\delta }_{k-1},\ \underline{\eta }_{k}\rangle =0, \end{aligned}$$

  where $\{\underline{\xi }_{k}\}$ and $\{\underline{\eta }_{k}\}$ are zero-mean 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 steady-state) 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 zero-mean Gaussian white noise sequences with variances q and r, respectively.