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 (Ricatti-Lyapunov) 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

  1. the DGKF and , equations must have a solution
  2. and are self-adjoint
  3. is invertible

All that remains is to prove the converse

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