  We propose several continuous data assimilation (downscaling) algorithms
based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We
show that for sufficiently large choices of the control parameter and
resolution and assuming that the observed data is error-free, the solution of
the controlled system converges exponentially (in $L^2$ and $H^1$ norms) to the
reference solution independently of the initial data chosen for the controlled
system. Furthermore, we show that a similar result holds when controls are
placed only on the horizontal (or vertical) variables, or on a single
Els\"asser variable, under more restrictive conditions on the control parameter
and resolution. Finally, using the data assimilation system, we show the
existence of abridged determining modes, nodes and volume elements.
