Sequential and Square-Root Algorithms

Abstract

It is now clear that the only time-consuming operation in the Kalman filtering process is the computation of the Kalman gain matrices.

Exercises

• 7.1. Give a proof of Lemma 7.1.

• 7.2. Find the lower triangular matrix L that satisfies:

  • (a) $LL^{\top }=\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} 2 &{} 3\\ 2 &{} 8 &{} 2\\ 3 &{} 2 &{} 14 \end{array}\right] .$

  • (b) $LL^{\top }=\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} 1 &{} 1\\ 1 &{} 3 &{} 2\\ 1 &{} 2 &{} 4 \end{array}\right] .$

• 7.3.

  • (a) Derive a formula to find the inverse of the matrix

    $$\begin{aligned} L=\left[ \begin{array}{c@{\quad }c@{\quad }c} \ell _{11} &{} 0 &{} 0\\ \ell _{21} &{} \ell _{22} &{} 0\\ \ell _{31} &{} \ell _{32} &{} \ell _{33} \end{array}\right] , \end{aligned}$$

    where $\ell _{11},\ \ell _{22}$, and $\ell _{33}$ are nonzero.

  • (b) Formulate the inverse of

    $$\begin{aligned} L=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \ell _{11} &{} 0 &{} 0 &{} \cdots &{} 0\\ \ell _{21} &{} \ell _{22} &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \vdots &{} &{} \ddots &{} 0\\ \ell _{n1} &{} \ell _{n2} &{} \cdots &{} \cdots &{} \ell _{nn} \end{array}\right] , \end{aligned}$$

    where $\ell _{11},\ \cdots ,\ \ell _{nn}$ are nonzero.

• 7.4. Consider the following computer simulation of the Kalman filtering process. Let $\epsilon \ll 1$ be a small positive number such that

  $$\begin{aligned}\begin{gathered} 1-\epsilon \not \simeq 1 \\ 1-\epsilon ^{2}\simeq 1 \end{gathered}\end{aligned}$$

  where “$\simeq$” denotes equality after rounding in the computer. Suppose that we have

  $$\begin{aligned} P_{k, k}=\left[ \begin{array}{c@{\quad }c} \frac{\epsilon ^{2}}{1\epsilon ^{2}} &{} 0\\ 0 &{} 1\end{array}\right] . \end{aligned}$$

  Compare the standard Kalman filter with the square-root filter for this example. Note that this example illustrates the improved numerical characteristics of the square-root filter.

• 7.5. Prove that to any positive definite symmetric matrix A, there is a unique upper triangular matrix $A^{u}$ such that $A=A^{u}(A^{u})^{\top }$.

• 7.6. Using the upper triangular decompositions instead of the lower triangular ones, derive a new square-root Kalman filter.

• 7.7. Combine the sequential algorithm and the square-root scheme with upper triangular decompositions to derive a new filtering algorithm.