When the value of \(a\) is close to one, the controller response will be
slower because the controller gain decreases, but the controller will be
more robust to system uncertainties and measurement errors, since it is
far from being in the instable region defined by Eq. (4). If \(a\) is
close to zero, the controller will respond more quickly, but there will
be a higher risk of oscillations and divergence. In the extreme case,
when \(a\) equals zero, called the deadbeat scenario, convergence is
achieved after one iteration, but an exact model of the system is
required to predict the transfer function, and there must be no external
disturbances or measurement errors (Xu & Tan, 2002). In the present
study, the parameter \(a\) was set to 0.3.
The transfer function was updated online for each cycle during the
experiment, i.e., the transfer function was calculated after each
iteration with the values obtained for \(u_{k-1}\) and \(y_{k-1}\),
using Eq. (3). The controller gain was also updated for each cycle as
this depends on the transfer function. This allows correction for
possible disturbances in the downstream processing, such as loss of
capacity of the columns or changes in the concentration or flow rate of
the feed. Therefore, the speed of the controller was adjusted depending
on the variation in the transfer function. Combining Eq. (5) with Eq.
(3) allows the learning control law in Eq. (2) to be re-written as
follows: