New Conditions for the Stability and Instability of Feedback Systems

In this paper we derive some new and useful conditions on the open loop transfer function, necessary for closed loop stability. The derivation is based on imposing the sign-invariance of the closed loop characteristic polynomial evaluated on the real axis. Although this is a simple necessary condition for closed loop stability, it leads to fairly stringent requirements on the open loop transfer function. Also, the results can be stated only in terms of the numbers of real poles and zeros of the open loop system even though the system may have complex poles and zeros. Using the same approach new necessary conditions for stabilizability by P, PI and PID controllers as well as by arbitrary controllers are also presented.