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:


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