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4.1 Brief Theoretical Discussion

Kalman Filter theory typically begins with a set of coupled linear equations expressed in state space form. Consider rectilinear motion covered most undergraduate physic courses:

Let's start with $F=ma$ and assume that acceleration is constant.


\begin{displaymath}
a = {dv \over dt} = {F\over m}
\end{displaymath} (1)

Therefor we can write the above relationship in a different form:

\begin{displaymath}
\begin{array}{rcl}
dv &=& {F\over m} dt \\
&&\\
\int_{v_0}...
...over m} t \\
&&\\
v & = & v_0 + {F \over m} t \\
\end{array}\end{displaymath} (2)

Finally we arrive at:

\begin{displaymath}
v = v_0 + {F \over m} t
\end{displaymath} (3)

Note that

\begin{displaymath}
v = {dx \over dt} = v_0 + {F \over m} t
\end{displaymath} (4)

so we can write:

\begin{displaymath}
dx = \left( v_0 + {F \over m} t \right)
\end{displaymath} (5)


\begin{displaymath}
x(t_1) = x(t_0) + v(t_0) t + {1 \over 2} a t^2
\end{displaymath} (6)

If we let $x_1$ be velocity and $x_2$ be position we can write the following set of coupled differential equations:

\begin{displaymath}
{d \over dt} \left[ \begin{array}{c} x_1  x_2 \end{array}...
...
+ \left[ \begin{array}{c} {t^2}/2  t \end{array} \right]
\end{displaymath} (7)

State Equation:

\begin{displaymath}
x_{k+1} = Hx_k + Bu_k + w_k
\end{displaymath} (8)

Observation Equation:

\begin{displaymath}
y_k = Cx_k + z_k
\end{displaymath} (9)


next up previous contents
Next: 5 Linear Regression Up: 4 Kalman Filters Previous: 4 Kalman Filters   Contents
Andrew Douglas 2005-03-15