On the solution of matrix inequalities in the Kalman-Yakubovich theorem via hidden parameters of positive rational functions Original Research Article

For minimal (and for controllable) SISO systems (A, B, C) with rational transfer function Z(p) = C(pI − A)− 1 B satisfying Re Z(p) ≥ 0 for Re p > 0, solutions H > 0 (H ≥ 0) of the inequality HA + A* H ≤ 0 satisfying C = B* H are constructed. The approach is based on the consideration of “hidden parameters” of the elementary-fraction decomposition of Z(p) and the construction of passive realizations of Z(p). In that way a new constructive proof of the Kalman-Yakubovich frequency theorem is obtained.