I'm just learning about Kalman filters, and I'm trying to understand some notation. The book that I am reading through sets up a system with the state-space realization:
$$\dot{x}(t) = A(t)x(t) + B(t)w(t)$$ $$y(t) = C(t)x(t)$$
where $w(t)$ is a white noise process. Then, we suppose that the observations $z(t) = y(t) + v(t)$ are available, where $v(t)$ is another white noise process, and $E\{w(t)v^T(t)\} = 0$. Then, it says "the objective is to design a linear system $H$ that operates on the measurements $z(t)$ and produces an estimate $\hat{y}(t|t) = C(t)\hat{x}(t|t)$ of $y(t) = C(t)x(t)$ given measurements at time $t$ so that the covariance $E\{e(t|t)e^T(t|t)\}$ is minimized, where $e(t|t) = x(t) - \hat{x}(t|t)$.
I guess I'm primarily confused on what the notation $x(t|t)$ (or $y(t|t)$, etc) means, and how to deal with it. I'm used to seeing the notation $E\{x|y\}$ to mean the expected value of $x$ given $y$, so is this a similar thing - the estimation given a time $t$? Later in this section, we look at $\dot{x}(t|t)$, and I'm not sure what that means, either...
$\hat{y}(t|t)$ is an estimate of $y(t)$ from observations up to time t. Similarly for $\hat{x}(t|t)$. If you keep reading you may see notation like $\hat{x}(t|t-1)$ (or something similar) which would be an estimate of $x(t)$ from observations up to time $t-1$. Kalman filter is an efficient way of incrementally updating new observations into the estimator.