Demystifying Enigmatic Undershoot in Setpoint Command Following [Focus on Education]

It often occurs in practice that the response to a step input or setpoint command moves initially in a direction that is opposite to the direction of the asymptotic response. In many real-world applications, this phenomenon—called initial undershoot—presents a fundamental limitation on control system performance. Although the basic mechanism responsible for initial undershoot, namely, an odd number of real, positive zeros, is well understood, it is surprising that, as the setpoint changes, initial undershoot may appear or disappear for the same plant dynamics. The goal of this tutorial note is to investigate the causes of this puzzling phenomenon. In particular, for setpoint command following with a changing setpoint, this article shows (spoiler alert) that the internal state when the setpoint changes determines the presence or absence of initial undershoot. Complete proofs for both initial and delayed undershoot in both continuous time and discrete-time systems are given to make the article self-contained and useful for students and instructors of systems and control theory