CONDITIONS FOR ROBUSTNESS AND NONROBUSTNESS OF THE STABILITY OF FEEDBACK-SYSTEMS WITH RESPECT TO SMALL DELAYS IN THE FEEDBACK LOOP

It has been observed that for many stable feedback control systems, th e introduction of arbitrarily small time delays into the loop causes i nstability. In this paper we present a systematic frequency domain tre atment of this phenomenon for distributed parameter systems. We consid er the class of all matrix-valued transfer functions which are bounded on some right half-plane and which have a limit at +infinity along th e real axis. Such transfer functions are called regular. Under the ass umption that a regular transfer function is stabilized by unity output feedback, we give sufficient conditions for the robustness and for th e nonrobustness of the stability with respect to small time delays in the loop. These conditions are given in terms of the high-frequency be havior of the open-loop system. Moreover, we discuss robustness of sta bility with respect to small delays for feedback systems with dynamic compensators. In particular, we show that if a plant with infinitely m any poles in the closed right half-plane is stabilized by a controller , then the stability is not robust with respect to delays. We show tha t the instability created by small delays is itself robust to small de lays. Three examples are given to illustrate these results.