The Impact of Nonminimum-Phase Zeros on Human-in-the-Loop Control Systems

We present results from an experiment in which 33 human subjects interact with a dynamic system 40 times over a one-week period. The subjects are divided into three groups. For each interaction, a subject performs a command-following task, where the reference command is the same for all trials and all subjects. However, each group interacts with a different dynamic system, which is represented by a transfer function. The transfer functions have the same poles but different zeros. One has a minimum-phase zero
$z_{\textrm {m}} < 0$
, another has a nonminimum-phase zero
$z_{\textrm {n}} = - z_{\textrm {m}} > 0$
, and the last has a slower (i.e., closer to the imaginary axis) nonminimum-phase zero
$z_{\textrm {sn}} \in (0,z_{\textrm {n}})$
. The experimental results show that nonminimum-phase zeros tend to make dynamic systems more difficult for humans to learn to control. We use a subsystem identification algorithm to identify the control strategy that each subject uses on each trial. The identification results show that the identified feedforward controllers approximate the inverse dynamics of the system with which the subjects interact better on the last trial than on the first trial. However, the subjects interacting with the minimum-phase system are able to approximate the inverse dynamics in feedforward more accurately than the subjects interacting with the nonminimum-phase system. This observation suggests that nonminimum-phase zeros are an impediment to approximating inverse dynamics in feedforward. Finally, we provide evidence that humans rely on feedforward-step-like-control strategies with systems (e.g., nonminimum-phase systems) for which it is difficult to approximate the inverse dynamics in feedforward.