Discrete-Time Kalman Filter

Overview

Consider a linear, time-varying system with the following dynamics:

\[ \begin{align} \mathbf{x}_k &= \mathbf{F}_{k - 1} \mathbf{x}_{k - 1} + \mathbf{G}_{k - 1} + \mathbf{w}_{k - 1} \\ \mathbf{y}_k &= \mathbf{H}_{k} \mathbf{x}_k + \mathbf{v}_k, \end{align} \]

where $\mathbf{w}_{k - 1}$ and $\mathbf{v}_{k}$ are zero-mean,

Consider a linear time-varying system with the following dynamics:

$$ \begin{align}

\end{align} $$

Discrete-Time Systems

Consider the following linear discrete-time system:

\[ \mathbf{x}_k = \mathbf{F}_{k - 1} \mathbf{x}_{k - 1} + \mathbf{G}_{k - 1} \mathbf{u}_{k - 1} + \mathbf{w}_{k - 1}, \]

where $\mathbf{u}_{k - 1}$ is a known input and $\mathbf{w}_{k - 1}$ is a Gaussian zero-mean white noise with covariance $\mathbf{Q}_{k - 1}$.

The expetation of $\mathbf{x}_k$ is:

\[ \bar{\mathbf{x}}_k = \mathbb{E}\left[ \mathbf{x}_k \right] = \mathbf{F}_{k - 1} \bar{\mathbf{x}}_{k - 1} + \mathbf{G}_{k - 1} \mathbf{u}_{k - 1}. \]

The covariance of $\mathbf{x}_k$ is:

\[ \begin{align} \mathbf{P}_{k} &= \mathbb{E} \left[ \left( \mathbf{x}_k - \bar{\mathbf{x}}_k \right) \left( \cdots \right)^T \right] \\ &= \mathbb{E} \left[ \left(\mathbf{F}_{k - 1} \mathbf{x}_{k - 1} + \mathbf{G}_{k - 1} \mathbf{u}_{k - 1} + \mathbf{w}_{k - 1} - \bar{\mathbf{x}}_k \right) \left( \cdots \right)^T \right] \\ &= \mathbb{E} \left[ \left( \mathbf{F}_{k - 1} \left( \mathbf{x}_{k - 1} - \bar{\mathbf{x}}_{k - 1} \right) + \mathbf{w}_{k - 1} \right) \left( \cdots \right)^T \right] \\ &= \mathbb{E} \left[ \mathbf{F}_{k - 1} \left(\mathbf{x}_{k - 1} - \bar{\mathbf{x}}_{k - 1} \right) \left(\mathbf{x}_{k - 1} - \bar{\mathbf{x}}_{k - 1} \right)^T \mathbf{F}^T_{k - 1} \right]\\ &\quad + \mathbb{E} \left[ \mathbf{w}_{k - 1} \mathbf{w}^T_{k - 1} \right] \\ &\quad + \underbrace{\mathbb{E} \left[ \mathbf{F}_{k - 1} \left( \mathbf{x}_{k - 1} + \bar{\mathbf{x}}_{k - 1} \right) \mathbf{w}^T_{k - 1} \right]}_{\mathbf{0}} \\ &\quad + \underbrace{\mathbb{E} \left[ \mathbf{w}_{k - 1} \left( \mathbf{x}_{k - 1} - \bar{\mathbf{x}}_{k - 1} \right)^T \mathbf{F}^T_{k - 1} \right]}_{\mathbf{0}}. \end{align} \]

Hence, the covariance of $\mathbf{x}_k$ is:

\[ \begin{align} \mathbf{P}_k = \mathbf{F}_{k - 1} \mathbf{P}_{k - 1} \mathbf{F}^T_{k - 1} + \mathbf{Q}_{k - 1}, \end{align} \]

which is known as the discrete-time Lyapunov equation. The mean and the covariance completely characterizes $\mathbf{x}$ in a statistical sense since it is a Gaussian random variable:

\[ \mathbf{x}_{k} \sim \mathcal{N}(\bar{\mathbf{x}}_k, \mathbf{P}_k). \]