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 , another has a nonminimum-phase zero , and the last has a slower (i.e., closer to the imaginary axis) nonminimum-phase zero zsn ∈ (0,zn) . 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.

The Impact of Command-Following Task on Human-in-the-Loop Control Behavior.

This article presents results from an experiment in which 44 human subjects interact with a dynamic system 40 times over a one-week period. The subjects are divided into four groups. All groups interact with the same dynamic system, but each group performs a different sequence of command-following tasks. All reference commands have frequency content between 0 and 0.5 Hz. We use a subsystem identification algorithm to estimate the control strategy (feedback and feedforward) that each subject uses on each trial. The experimental and identification results are used to examine the impact of the command-following tasks on the subjects' performance and the control strategies that the subjects learn. Results demonstrate that certain reference commands (e.g., a sum of sinusoids) are more difficult for subjects to learn to follow than others (e.g., a chirp), and the difference in difficulty is related to the subjects' ability to match the phase of the reference command. In addition, the identification results show that differences in command-following performance for different tasks can be attributed to three aspects of the subjects' identified controllers: 1) compensating for time delay in feedforward; 2) using a comparatively accurate approximation of the inverse dynamics in feedforward; and 3) using a feedback controller with comparatively high gain. Results also demonstrate that subjects generalize their control strategy when the command changes. Specifically, when the command changes, subjects maintain relatively high gain in feedback and retain their feedforward internal model of the inverse dynamics. Finally, we provide evidence that subjects use prediction of the command (if possible) to improve performance but that subjects can learn to improve performance without prediction. Specifically, subjects learn to use feedback controllers with comparatively high gain to improve performance even though the command is unpredictable.

The Roles of Feedback and Feedforward as Humans Learn to Control Unknown Dynamic Systems.

We present results from an experiment in which human subjects interact with an unknown dynamic system 40 times during a two-week period. During each interaction, subjects are asked to perform a command-following (i.e., pursuit tracking) task. Each subject's performance at that task improves from the first trial to the last trial. For each trial, we use subsystem identification to estimate each subject's feedforward (or anticipatory) control, feedback (or reactive) control, and feedback time delay. Over the 40 trials, the magnitudes of the identified feedback controllers and the identified feedback time delays do not change significantly. In contrast, the identified feedforward controllers do change significantly. By the last trial, the average identified feedforward controller approximates the inverse of the dynamic system. This observation provides evidence that a fundamental component of human learning is updating the anticipatory control until it models the inverse dynamics.