Guaranteed Margins for LQG Regulators

ity of the desired transfer-function matrix T,(s) is that the matrix equation (9) is consistent for m > 3 m (i.e., for the case where there are more equations than unknowns). However, as it is seen below, this condition is not sufficient. Suppose that it is required to test the admissibility of a desired transfer-function matrix TAs). Also consider another transfer-function matrix Tu@), for which Td(s) and T&) give rise to the same values for the matrices Jp and 4. Now, if these matrices make (9) consistent, only one of the transfer-function matrices need be admissible. And thus the condition that (9) is consistent for r = p is not sufficient for admissibility of Td(s). However, since the elements of Tds) are ratios of finite-order poly-nomials in s, there is an upper limit on the value of " p " for which the above equality (11) holds for distinct Td(s) and T,(s). The closed-loop system with the PID controller is of order (n + mXsee [ lOD. The numerator of the elements of TAs) are polynomials of maximal order (n + m-l) and denominators are polynomials of maximal order (n+m). Hence, taking the gains into account, (1 1) can hold for distinct T , s) and T,(s) if