Externally Positive Systems of Order <i>k</i> and Their Application to Undershoot and Overshoot Analysis

A linear time-invariant system is said to be externally positive if it maps non-negative inputs to non-negative outputs, for zero initial conditions. Although this property has proven useful in many applications, in certain cases we may be interested only in specific classes of inputs. For instance, preventing overshoot for all possible non-negative input signals is often unnecessary. Instead, we may want to restrict our attention to, e.g., step inputs. Motivated by this, we generalize the notion of externally positive systems by introducing externally positive systems of order k, i.e., systems that map non-negative inputs to outputs whose kth-primitive is non-negative, for zero initial conditions. As a particular case, externally positive systems of order 0 coincide with externally positive systems. We show how this class of systems is a natural generalization of externally positive systems, by extending well-known necessary and sufficient conditions to the case k0. Finally, we leverage this new class of systems to derive necessary and sufficient conditions to determine the presence of overshoot and undershoot in linear time-invariant feedback control systems.