Solving the HGRAIL using NCProcess: Step 3
The starting polynomial equations for this step will be the
output from the first call to
NCProcess1
as well as the
four new equations
that we have just derived.
We ran NCProcess1 for two iterations.
Once again we go directly to the
spreadsheet which
NCProcess1 created.
There is no need to record all of it here, since at this
stage we shall
be concerned only with the undigested polynomial equations.
There are only
two undigested polynomial equations which are not banal.
The expressions with unknown variables
and knowns
The expressions with unknown variables
and knowns
Step 4
Now we analyze these two polynomial equations.
The first polynomial equation is a
(RicattiLyapunov) equation in
.
Numerical methods for solving Ricatti equations
are common.
For this reason assuming that a Ricatti equation has a solution is a
socially acceptable
necessary condition throughout control
engineering.
Thus we can consider
.
to be known.
A first glance at the second equation reveals
that the same products of unknowns appear over and over.
Also we can see that this equation is symmetric. It would not take an
experienced person long to realize that by multiplying this
equation on the left by
and on the right by
,
we will have an equation in one unknown.
Now we can replace
with a new
variable X. This yields
Observe that this is an equation in the one unknown X.
End game
Now let us compare what we have found to the well known
solution of (HGRAIL).
In that theory there are two Ricatti equations due to Doyle,
Glover, Khargonekar and Francis.
These are the DGKF X and Y equations.
One can read off that the
equation which we found is the DGKF
equation for
,
while the Ricatti equation which we just analyzed
is the DGKF
equation.
Indeed what we have proved is that if (HGRAIL) has a solution
with
invertible and if b and c
are given by formulas
derived in
Step 2,
then
 the DGKF
and
,
equations must have a solution

and
are selfadjoint

is invertible
All that remains is to prove the converse