H∞ and H2 optimal controllers for periodic and multi-rate systems

The solutions to the optimal l² to l ² disturbance rejection problem (H^∞) as well as to the LQG (linear quadratic Gaussian) (H²) problem in periodic systems using the lifting technique are presented. Both problems involve a causality condition on the optimal LTI (linear time invariant) compensator when viewed in the lifted domain. The H^∞ problem is solved using the Nehari´s theorem, whereas in the H² problem the solution is obtained using the projection theorem in Hilbert spaces. The authors demonstrate that exactly the same method of solution to the H^∞ and H² problems in periodic systems applies when considering the same problems in multirate sampled data systems