Fundamental theorem of linear state feedback for singular systems

The problem is solved of simultaneously assigning the finite and infinite eigenstructure of the controllable singular system Ex'(t)=Ax(t)+Bu(t) by the proportional state feedback law u(t)=Fx(t). Given m monic polynomials d/sub 1/(s), . . .,d/sub m/(s) of degrees d/sub 1/,. . ., d/sub m/ satisfying d/sub i+1/(s) mod d/sub i/(s), and eta ( eta =nullity of E) nonnegative integers p/sub 1/, . . .,p eta d/sub 1/+. . .+d/sub m/+p/sub 1/+p/sub eta /=rank E, necessary and sufficient conditions are established for the existence of a real feedback map F so that the d/sub i/(s)'s are the invariant polynomials and the p/sub i/'s are the infinite pole orders of the closed-loop system Ex'(t)=(A+BF)x(t).<>