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Step 4

Now we analyze these two polynomial equations.

The first polynomial equation is a (Ricatti-Lyapunov) equation in tex2html_wrap_inline5600 . 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 tex2html_wrap_inline5600 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 tex2html_wrap_inline5726 and on the right by tex2html_wrap_inline5728 , we will have an equation in one unknown.

Now we can replace tex2html_wrap_inline5730 with a new variable X. This yields

-X A - A X + X B C + C B X - X B B X + X B B X

Observe that this is an equation in the one unknown X.

Wed Jul 3 10:27:42 PDT 1996