Gain margin and time lag tolerance constraints applied to the stabilization problem and robust servomechanism problem

Definitions of gain margin tolerance and time lag tolerance for a multivariable linear time-invariant system are given which correspond to a generalization of the notions of gain and phase margin of classical control system theory. Existence conditions are then given to solve the stabilization problem and the robust servomechanism problem with gain margin and time lag tolerance constraints. In particular, the following is shown for a system described in terms of a state-state model: 1) Generically, there exists a controller to stabilize an unstable
m
-input,
r
-output system so that the resultant closed-loop system possesses a nonzero gain margin, if and only if min
(r, m) = 1
. 2) For stable plants, there always exists a controller to solve the robust servomechanism problem so that it has an arbitrary specified gain margin/time lag tolerance constraint, if and only if there exists a solution to the robust servomechanism problem for the plant. A controller synthesis method is then given which solves the robust servomechanism problem so that it has a specified gain margin/time lag tolerance. Various examples of the proposed algorithm are given to illustrate the type of results that may be obtained.