On the general functional matrix for a linear system

Macfarlane [1] has shown that for any asymptotically stable linear system with constant coefficients there exists a general functional matrix that can be used to evaluate a very wide class of system functionals. A disadvantage is that the algebra needed to determine this matrix, while straightforward, is lengthy and tedious. It is shown that the general functional matrix can be obtained by a completely systematic procedure that involves little or no algebraic manipulation. The relationship with some recent work by Bass and Webber [4] on optimal linear systems with quartic and higher-order performance criteria is investigated, and a method based on the solution of the Liapunov matrix equation is suggested for obtaining the optimal nonlinear feedback control. An explicit form for the inverse of a related Kronecker sum is also given.