filters are how beliefs are updated from observation

discrete state filter

\begin{equation} b’(s’) = P(s’|b,a,o) \end{equation}

\(b’\) is what state we think we are in next, and its a probability distribution over all states, calculated given from \(b,a,o\) our current belief about our state, our action, and our observation.

We can perform this belief update by performing Bayes Theorem over \(o\):

\begin{align} b’(s’) &= P(s’|b,a,o) \\ &\propto P(o|b,a,s’) P(s’ | b,a) \end{align}

We have a system of differential equations:

\begin{equation} \begin{cases} \dv{I}{t} = -0.73 U(t) + 0.0438 + 0.4 \dv{M}{t} \\ \dv{U}{t} = 0.4I-0.012 \\ \dv{G}{t} = \dv{M}{t} - I(t) \end{cases} \end{equation}

where, \(M\) is a sinusoidal function which we can control.

We hope for this system to be as stable as possible.

First, let’s try to get a general solution of the system. The linearized(ish) solution takes the shape of:

\begin{equation} \dv t \mqty(I \\ U \\ G) = \mqty(0 & -x_1 & 0 \\ x_4 & 0 & 0 \\ -1 & 0 & 0 ) \mqty(I \\ U \\ G)+ \dv{M}{t}\mqty(x_3 \\ 0 \\ 1) + \mqty(x_2 \\ x_5 \\ 0) \end{equation}