Consider the following system.$\DeclareMathOperator{\sign}{sign}$ $$ \begin{aligned} &\dot{e_1}=c_1e_1+c_2e_2+\eta\\ &\dot{e_2}=c_3e_1+c_4e_2-c_5 \sign(e_1)+a \eta \end{aligned} $$ which $\eta$ is a gaussian random variable and {$a,c_1,c_2,c_3,c_4 \in \mathbb{R}$}, such that: \begin{equation} A=\begin{bmatrix}{c_1\ \ \ c_2\\c_3\ \ \ c_4}\end{bmatrix} \end{equation}
is an stable matrix, and $c_5> 0$. The output of this system is the following discontinuous signal: $$ u=c_5 \sign(e_1) $$ In addition as a result of sliding on the sliding surface, we conclude that, this signal is with probability of 0.5, $c_5$ and possibility of 0.5, $-c_5$.

Unfortunately this random signal is not i.i.d and this brings up this question:

If you consider the following LTI system. $$ \begin{aligned} &\dot{x}=Fx+Gu\\ &y_f=Cx \end{aligned} $$ which $x \in \mathbb{R}^2$. and $F$ is an stable matrix.

WHAT IS THE CO-VARIANCE OF ----> $y_f$? In addition is there any body who know any source that has dealt with the discontinuous signal that I have presented, I even don't know what is its specific name to search for any source on google, thanks, please post me some useful key-words


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