Some necessary and sufficient conditions of stability on the approximation of a distributed delay control law

Implementation of delayed output feedback for linear systems with multiple input delays

“…Though the DOF approach is theoretically appealing, the implementation problem of the resulting controllers has been neglected in the previous studies. In the study of predictor feedback (or finite spectrum assignment) problem, it was pointed out that the implementation of controllers involving distributed delay terms can be a problem: if the integration in the controller is implemented by numerical integration approaches, such as the backwards rectangular rule, the trapezoidal rule and so on, the resulting closed‐loop system is unstable no mater how precise the numerical integration is [21–25]. This instability phenomenon was well explained in [26] where it was shown that the closed‐loop characteristic quasi‐polynomial may be of neutral type if the integration in the controller is implemented by approximated numerical methods.…”