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Kalman Filter

Purpose and Usage

Framework and Model


$$x_k = Ax_{k-1} + Bu_{k-1} + w_{k-1}$$

Each current signal value $x^k$ is a combination of previous signal value $x_{k-1}$ times a constant, a control signal $u_{k}$ and a process noise and a process noise signal $w_{k-1}$ (which usually considered as zero).

$$ y_{k} = Hx_{k} + v_{k} $$

In reality, covariance matrix $Q$ and $R$ may change in every iteration. We assume they are constant here however.


Find the best (recursive) estimate of the state $x$ of the system.

Computational Origins

Define $e_{k}^{-}$ to be a priori state estimate at step k given knowledge of the process prior to step $k$, and define $e_{k}$ to be a posteriori state estimate at step $k$ given measurement $z_{k}$. Then a priori and a posteriori estimate errors can be defined as:

$$e_{k}^{-} \equiv x_{k} - \hat{x}_{k}^{-}$$

$$e_{k} \equiv x_{k} - \hat{x}_{k}$$

The a priori estimate error covariance is then

$$P_{k}^{-} = E[e_{k}^{-}e_{k}^{-T}]$$

The a posteriori estimate error covariance is then

$$P_{k} = E[e_{k}e_{k}^{T}]$$

Then How can we optimally (linearly) combine the estimate and measurement to obtain the best reconstruction of the true x?

The answer given in “The Probabilistic Origins of the Filter” found.

$$ \hat{x} = \hat{x}_{k}^{-} - K \times residual $$

Where residual is $z_k - H \hat{x}_{k}^{-}$. It also can be called as measurement innovation.

The Kalman filter gains are derived by minimizing the posterior error covariance, resulting in

$$K_k = \frac{P_k^-H^T}{(HP_k^-H^T + R)^{-1}}$$

If the a priori estimate of the process noise is zero, Then

$$K = 0$$

If the measurement noise is zero

$$K = H^{-1}$$

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