A complete analytical solution to the equationTA - FT = LCand its applications

In this note, a complete, analytical, and restriction-free solution with complete and explicit freedom of the matrix equation
TA - FT = LC
is proposed. Here
(A, C)
is given and is observable, and
F
is in the Jordan form with arbitrary given eigenvalues. This solution appears to be new because it can be applied directly to obtain significantly better solutions to the following three basic design problems: 1) 2-D system eigenvalue assignment; 2) function observer design; and 3) state feedback eigenstructure design, as shown in this note.