Limitations of feedback properties imposed by open-loop right half plane poles

An integral constraint on the closed-loop transfer function of unstable open-loop systems is used to quantify a tradeoff between feedback properties of the closed-loop system. This results in a lower bound on the peak of the complementary sensitivity function for unstable plants. The lower bound illustrates the nature of the tradeoff that is imposed, and provides insight into the difficulties imposed on the control of unstable systems. It can be used to quantitatively evaluate tradeoffs between the frequency of the open-loop unstable pole, the peak of the complementary sensitivity functions, and the bandwidth of the closed-loop system. Such a relationship has long been recognized, and is embodied in the classical design rule of thumb which states that the closed-loop bandwidth must be at least twice as large as the frequency of the unstable pole. Results which give insight into this rule and a theoretical basis for it are provided.< >