User:Ramkakarala/sandbox

Asymptotic Form

The Kalman filter may be written:

${\hat {\mathbf {x} }}_{k\mid k}=\mathbf {F} _{k}{\hat {\mathbf {x} }}_{k-1\mid k-1}+\mathbf {K} _{k}[\mathbf {z} _{k}-\mathbf {H} _{k}\mathbf {F} _{k}{\hat {\mathbf {x} }}_{k-1\mid k-1}]$

The gain matrices $\mathbf {K} _{k}$ evolve independently of the measurements $\mathbf {z} _{k}$. From above, the four equations for updating the Kalman filter are as follows:

$\mathbf {P} _{k\mid k-1}=\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k}$

$\mathbf {S} _{k}=\mathbf {R} _{k}+\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}$

$\mathbf {K} _{k}=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}$

$\mathbf {P} _{k|k}=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}$

Since the gain matrices depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices $\mathbf {K} _{k}$ to an asymptotic matrix $\mathbf {K} _{\infty }$ holds under broad conditions established in Walrand and Dimakis . Simulations establish the number of steps to convergence. For the moving truck example described above, with $\Delta t=1$. and $\sigma _{a}^{2}=\sigma _{z}^{2}=\sigma _{x}^{2}=\sigma _{\dot {x}}^{2}=1$, simulation shows convergence in $10$ iterations.

Using the asymptotic gain, and assuming $\mathbf {H} _{k}$ and $\mathbf {F} _{k}$ are independent of $k$, the Kalman filter becomes a linear time-invariant filter:

${\hat {\mathbf {x} }}_{k}=\mathbf {F} {\hat {\mathbf {x} }}_{k-1}+\mathbf {K} _{\infty }[\mathbf {z} _{k}-\mathbf {H} \mathbf {F} {\hat {\mathbf {x} }}_{k-1}]$