Kalman Filter

Original Paper

Let a general plant model be given by a random process

$$\begin{array}{rl}\dot{x}& =Ax+Bu+Gw\\ y& =Cx+v\end{array}$$

Where we have a noise $v$

• Autocovariance/autocorrelation ${\mathcal{A}}_v(t,\tau )=E[v(t)v(t{)}^T]=\delta (t-\tau )R_v$

And a disturbance, $w$

• Autocovariance/autocorrelation ${\mathcal{A}}_w(t,\tau )=E[w(t)w(t{)}^T]=\delta (t-\tau )Q_w$

Both are represented by a zero mean white Gaussian signal, and are uncorrelated.

• ${\mathcal{A}}_{vw}(t,\tau )=E[v(t)w(t{)}^{\mathbb{T}}]=0$

Kalman Gain Covariance Matrix of the Estimation Error

Combining these gives us the covariance update equation.

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Kalman Filter in Continuous Time Discrete Time Kalman Filter

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Covariance Matrix of the Estimation Error LQR and Kalman Filter Duality Colored Noise in Kalman Filter Kalman Filter for Time Varying Models

Covariance Update Equation for Kalman Filter

We therefore get the following covariance update equation (Covariance Matrix):

$$\dot{P}=(A-LC)P+P(A-LC{)}^T+E[e(w^T{G}^T-v^T{L}^T)]+E[e(w^T{G}^T-v^T{L}^T){]}^T$$

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Here, we encounter a problem: L is time-varying. We solve this by using the following Transition Matrix

$$\begin{array}{rl}\dot{\Phi }(t,\tau )& =(A-L(t)C)\Phi (t,\tau )\end{array}$$

Where $\Phi (t,t)=\mathbb{I}$ can be used to recover the solution

$$\begin{array}{rl}\dot{e}& =(A-LC)e+Gw-Lv\\ e(t)& =\Phi (t,0)e_0+{\int }_0^t\Phi (t,\tau )(Gw(\tau )-L(\tau )v(\tau ))d\tau \end{array}$$

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Giving us the final equation for $\dot{P}$

$$\dot{P}=(A-LC)P+P(A-LC{)}^T+G{Q}_w{G}^T+L{R}_v{L}^T$$

Where $Q_w$ and $R_v$ is defined at the top of this document.