In this post, we explain how the expected value of the state vector and state covariance matrix of linear dynamical systems are propagated in discrete-time. These equations, together with the equations describing the recursive least squares method which are derived in our previous post, are important for deriving the Kalman filter equations.
Linear Dynamical System Affected by the Process Noise (Disturbances)
We are considering the following state equation
(1) 
where
![Q_{k}=E[\mathbf{w}_{k}\mathbf{w}_{k}^{T}]](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-266e7fa5fa9769934ad70b1a27d1a31b_l3.png "Rendered by QuickLaTeX.com")
![E[\cdot]](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-3941b329206b0baa01fad23b819cc5f7_l3.png "Rendered by QuickLaTeX.com")
A few comments are in order. If the system is time-invariant, then the matrices 
State Vector Expectation Propagation
Let us apply the expectation operator to the equation (1), as the result, we obtain
(2) ![\begin{align*}E[\mathbf{x}_{k}]=E[A_{k-1}\mathbf{x}_{k-1}]+E[B_{k-1}\mathbf{u}_{k-1}]+E[\mathbf{w}_{k-1}] \end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-c8cdeac4faab74cceb6c500935fff1ad_l3.png "Rendered by QuickLaTeX.com")
For notation brevity, we introduce the following notation
(3) ![\begin{align*}E[\mathbf{x}_{k}]=\overline{\mathbf{x}}_{k}\end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-d06195533e5a2184859a2c4f74873a33_l3.png "Rendered by QuickLaTeX.com")
Taking into account that ![E[\mathbf{w}_{k-1}] =0](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-a7ed027dca1e69e99e0c873d4d7f01e6_l3.png "Rendered by QuickLaTeX.com")
(4) 
State Covariance Matrix Propagation
We define the state covariance matrix as follows
(5) ![\begin{align*}P_{k}=E[(\mathbf{x}_{k}-\overline{\mathbf{x}}_{k})(\mathbf{x}_{k}-\overline{\mathbf{x}}_{k})^{T}]\end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-417f70d13177299d2165a6e57a3374b4_l3.png "Rendered by QuickLaTeX.com")
Consequently, we first need to expand this expression
(6) 
and then compute the expectation of every term in that expression. By substituting the equations (1) and (4) in (10), we obtain
(7) 
Since 
(8) ![\begin{align*}E[A_{k-1}(\mathbf{x}_{k-1}-\overline{\mathbf{x}}_{k-1})\mathbf{w}_{k-1}^{T}]=0 \\E[\mathbf{w}_{k-1}(\mathbf{x}_{k-1}-\overline{\mathbf{x}}_{k-1})^{T}A_{k-1}^{T}]=0\end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-718487f6340e86c492ddc9856a12d4f9_l3.png "Rendered by QuickLaTeX.com")
Also, we have
(9) ![\begin{align*}E[\mathbf{w}_{k-1}\mathbf{w}_{k-1}^{T}]=Q_{k-1}\end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-971507cefd2ddffcaa13ca9dada3219d_l3.png "Rendered by QuickLaTeX.com")
(10) ![\begin{align*}P_{k-1}=E[(\mathbf{x}_{k-1}-\overline{\mathbf{x}}_{k-1})(\mathbf{x}_{k-1}-\overline{\mathbf{x}}_{k-1})^{T}]\end{align*}](https://aleksandarhaber.com/wp-content/ql-cache/quicklatex.com-8f67b069d73bd06e87bed34045985dae_l3.png "Rendered by QuickLaTeX.com")
By using these expressions, and by taking the expectation of (7), we obtain the final equation for the propagation of the state covariance matrix
(11) 
