We have found in our experimental work that there is often a rather large range in the values of the RPF proportionality constants K and R that will still be effective in controlling the system. This is fortunate since it allows us to make estimates of K and R from just a few experimental points in the neighborhood of the desired fixed point during the precontrol phase of the experiment. Here, we briefly derive the RPF equations for K and R using methods of control theory[6,7] in order to to shed light on the theoretical robustness of the RPF method.
We start with the general 1D return-map equation (Eq. 2 from reference [3])
where the notation is that used in reference [3],
is the value of the measured variable at the start of the
nth Poincaré cycle and 
is the value of the control
parameter during
the nth cycle. For control on a period-1 orbit, we are interested
in the natural dynamics of the system described by
Eq. 3 near the fixed point, 
, of the period-1 orbit for
; 
. To first
order in 
, 
, and
, we have
where 
,
, and
.
In reference [3], the RPF algorithm is obtained by using
the optimum possible strategy of choosing
such that the system is
brought to the fixed point as quickly as possible, namely two
Poincaré cycles.
Taking
and
, then the first and second iterate of
Eq. (4) gives the recursive control algorithm Eq. 1
with R and K given by
As shown in reference [3], 
and 
,
and Eqs. 2 are equivalent to Eqs. 5.
Here we take a different approach. We assume there is a bilinear
relationship between 
and 
where H and G are constants yet to be determined.
Equations 4 and 6 form a two-dimensional discrete
map describing the full control system including both the dynamics of the
nonlinear system and the recursive feedback control strategy.
The two-dimensional map is put in more conventional
form if we shift the n labels on
the 
by 1. This can be accomplished by defining a new parameter,
. Substitution into Eqs. 4 and
6 gives
The conventional 2D map is formed by substituting Eq. 8 into the last term of Eq. 7 giving
where
The full control system defined by Eq. 9 has a fixed
point at 
. The control system will produce
the desired control if this fixed point is stable.
The parameters 
, v, and w are determined by the
dynamics of the system near the fixed point and we are free to adjust
H and G. For successful feedback
control, the values of H and G must be chosen so that the fixed
point of Eq. 9 is stable.
The stability of the fixed point is determined by the magnitude of the
eigenvalues of 
:
where Tr
and Det
.
The fixed point will be stable if
Equations 11 and 12 put conditions on H and G such that the control strategy will work.
The best values of H and G would make 
and
the fixed point of the control system becomes superstable. The superstable
condition is satisfied if 
and 
where
The values of 
and 
that satisfy Eqs. 13 and 14
are
Equations 15 are identical to Eqs. 5 for the
RPF algorithm with 
and 
and we have shown
that the RPF algorithm forms a superstable control system. Of course, it
is not necessary to have the optimum values for K and R for the control
to be successful and there is a range of values satisfying the
stability conditions of Eqs. 11 and 12.
Finally, if we substitute 
and 
into 
, then
the map describing the superstable control system becomes
Iterating these equations once shows explicitly that, for the superstable
RPF control conditions, 
and
for any starting values of
.
Of course, 
and 
must be
small enough so that the linearization of the dynamics about 
is valid.
