system model

Let $t=1,2,\mathrm{\dots }$ denote discrete time instants. By a system model we mean a mathematical model defined by a conditional probability density function $f(y_t|u_t,d(t-1))$ where

$y_t$

is the system output in time $t$,

$u_t$

is the system input and

$d(t-1)$

denotes the sequence of data $d_0,\mathrm{\dots },d_{t-1}$ where $d_t=(u_t,y_t)$.

Such a system has time-invariant (constant) parameters.If the model parameters are unknown (uncertain, variable), we introduce the definition in the form $f(y_t|u_t,d(t-1),\theta )$. Here, $\theta$ is a (possibly multi-dimensional) parameter.