SI&DA - Lecture 25 'Asymptotic Behaviour of the Kalman Filter'

Summary

Discrete Riccati Equation: an equation that governs the covariance $p(t+1|t)$ given $p(t|t-1)$.

If we “compress” the whole KF algorithm of the following model:

$$\begin{array}{rl}& x(t+1)=ax(t)\\ & y(t)=x(t)+v(t)\end{array}$$

where: $v(t)\sim WP\left(0,{\sigma }_v^2\right)$ into 2 equation, we obtain:

$$\begin{array}{rl}\hat{x}(t+1|t)& =a\hat{x}(t|t-1)+aK(t)(y(t)-\hat{x}(t|t-1))\\ p(t+1|t)& =a^2\left(p(t|t-1)-\frac{p^2(t|t-1)}{p(t|t-1)+{\sigma }_v^2}\right)\\ & =\frac{a^2\cdot p(t|t-1)\cdot {\sigma }_v^2}{p(t|t-1)+{\sigma }_v^2}\end{array}$$

We defined this equation:

$$p(t+1|t)=\frac{a^2\cdot p(t|t-1)\cdot {\sigma }_v^2}{p(t|t-1)+{\sigma }_v^2}$$

as the Discrete Riccati Equation (for this particular model) and we analyzed what would happen for $t\to \infty$, we defined the following as the “Algebric Riccati Equation”:

$$\underset{t\to \infty }{\lim }p(\infty )=\frac{a^2\cdot p(\infty )\cdot {\sigma }_v^2}{p(\infty )+{\sigma }_v^2}$$

and we found 2 solution.

1. $p(\infty )=0$ (which it will assume if $|a|<1$, the system is asymptotically stable, given by a theorem)
2. $p(\infty )={\sigma }_v^2(a^2-1)$ (which it will assume if $|a|>1$, the system is unstable)

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TODO What does detectable mean ? #TODO What does stabilizable mean ?

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Prediction using an LTI Filter (not good):

LTI Filters using 2 different constant $K$ (which is bad, while in the KF, $K$ varies with each iteration, this graph does not show a KF) #TODO If there is time describe the graphs.