A high order system is controlled with the aid of a fast model.
The simulations here consist of three and four cascaded integrators.
At each step, the state of the fast model is set to correspond to that
of the system.
The fast model is run with maximum and minimum value of the input u.
At some point the error and all the state derivatives of the model will
have the same sign as the input.
It can be said that they are then all 'onside'.
At this point the iteration ceases.
While the model runs, tests are made to determine the drive to be
applied to the plant.
In the first example, the drive is
for which the iteration lasts longest - taking longest for all
variables to come 'onside'. The model is run with a coarser step
length than the plant.
The second example has the original
strategy of choosing the drive for which the position error is
'offside' furthest ahead in time. It can be seen that a sequence
of overshoots will occur.
In the third example, 'slugging' has
been added. The values of derivatives are exaggerated when
applied to the model initial conditions.
The fourth example stretches the 'all
offside' strategy to control a fourth-order system. There are
The fifth example adds 'slugging' to
the 'all offside' control of a fourth order system to avoid the
overshoots, at some expense in settling time.
The next example shows that even a
five-integrator system can be controlled!
Try tuning a simple nonlinear controller to
match the performance!