On the Nyquist envelope of an interval plant family

In this note, we study the envelope of the Nyquist plots generated by an interval plant family and show that this boundary is not always contained in the Nyquist plots of the Kharitonov plants. With this motivation in mind, we give a sufficient condition for an envelope point to be contained in the Nyquist plot of a Kharitonov plant and use it to generate large and critical portions of the Nyquist envelope; e.g., we show that the outer Nyquist envelope of a stable interval plant is generated by the Nyquist plots of the Kharitonov plants. Another by-product of this sufficient condition is a framework for developing new extreme point results for interval feedback systems. This framework is useful in computing the phase margin and the maximal peaking in the sensitivity and complementary sensitivity functions and in stating a robust version of the circle criterion. We also use this framework to easily explain existing extreme point results for the gain margin, the H _∞ norm and the positive realness of interval plants. One conclusion which emerges is this: Seemingly, all important properties of an interval feedback system are deducible from the Kharitonov plants