Solution of the robust servomechanism problem subject to specified gain margin and time lag tolerance constraints

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. It is shown that a plant in state space description has a specified gain margin tolerance with respect to a given control configuration if and only if a certain augmented state space description of the plant remains stable when used in the same control configuration. Applications of these results are then made to obtain existence conditions to solve the stabilization problem and the robust servomechanism problem with gain margin and time lag tolerance constraints. In particular, it is shown that: (i) Generically, there exists no controller to stabilize an unstable multivariable system so that the resultant closed loop system possesses a non zero gain margin. (ii) 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.