Intermittent Constrained Predictive Control of Mechanical Systems

Frequency-domain identification of the human controller

System identification techniques applied to experimental human-in-the-loop data provide an objective test of three alternative control-theoretical models of the human control system: non-predictive control, predictive control, and intermittent predictive control. A two-stage approach to the identification of a single-input single-output control system is used: first, the closed-loop frequency response is derived using the periodic property of the experimental data, followed by the fitting of a parametric model. While this approach is well-established for non-predictive and predictive control, it is here used for the first time with intermittent predictive control. This technique is applied to data from experiments with human volunteers who use one of two control strategies, focusing either on position or on velocity, to manually control a virtual, unstable load which requires sustained feedback to maintain position or low velocity. The results show firstly that the non-predictive controller does not fit the data as well as the other two models, and secondly that the predictive and intermittent predictive controllers provide equally good models which cannot be distinguished using this approach. Importantly, the second observation implies that sustained visual manual control is compatible with intermittent control, and that previous results suggesting a continuous control model for the human control system do not rule out intermittent control as an alternative hypothesis. Thirdly, the parameters identified reflect the control strategy adopted by the human controller.

Intermittent control: a computational theory of human control

The paradigm of continuous control using internal models has advanced understanding of human motor control. However, this paradigm ignores some aspects of human control, including intermittent feedback, serial ballistic control, triggered responses and refractory periods. It is shown that event-driven intermittent control provides a framework to explain the behaviour of the human operator under a wider range of conditions than continuous control. Continuous control is included as a special case, but sampling, system matched hold, an intermittent predictor and an event trigger allow serial open-loop trajectories using intermittent feedback. The implementation here may be described as "continuous observation, intermittent action". Beyond explaining unimodal regulation distributions in common with continuous control, these features naturally explain refractoriness and bimodal stabilisation distributions observed in double stimulus tracking experiments and quiet standing, respectively. Moreover, given that human control systems contain significant time delays, a biological-cybernetic rationale favours intermittent over continuous control: intermittent predictive control is computationally less demanding than continuous predictive control. A standard continuous-time predictive control model of the human operator is used as the underlying design method for an event-driven intermittent controller. It is shown that when event thresholds are small and sampling is regular, the intermittent controller can masquerade as the underlying continuous-time controller and thus, under these conditions, the continuous-time and intermittent controller cannot be distinguished. This explains why the intermittent control hypothesis is consistent with the continuous control hypothesis for certain experimental conditions.

Predictive feedback control and Fitts' law

Fitts' law is a well established empirical formula, known for encapsulating the "speed-accuracy trade-off". For discrete, manual movements from a starting location to a target, Fitts' law relates movement duration to the distance moved and target size. The widespread empirical success of the formula is suggestive of underlying principles of human movement control. There have been previous attempts to relate Fitts' law to engineering-type control hypotheses and it has been shown that the law is exactly consistent with the closed-loop step-response of a time-delayed, first-order system. Assuming only the operation of closed-loop feedback, either continuous or intermittent, this paper asks whether such feedback should be predictive or not predictive to be consistent with Fitts law. Since Fitts' law is equivalent to a time delay separated from a first-order system, known control theory implies that the controller must be predictive. A predictive controller moves the time-delay outside the feedback loop such that the closed-loop response can be separated into a time delay and rational function whereas a non- predictive controller retains a state delay within feedback loop which is not consistent with Fitts' law. Using sufficient parameters, a high-order non-predictive controller could approximately reproduce Fitts' law. However, such high-order, "non-parametric" controllers are essentially empirical in nature, without physical meaning, and therefore are conceptually inferior to the predictive controller. It is a new insight that using closed-loop feedback, prediction is required to physically explain Fitts' law. The implication is that prediction is an inherent part of the "speed-accuracy trade-off".

The frequency of human, manual adjustments in balancing an inverted pendulum is constrained by intrinsic physiological factors

While standing naturally and when manually or pedally balancing an equivalent inverted pendulum, the load sways slowly (characteristic unidirectional duration approximately 1 s) and the controller, calf muscles or hand, makes more frequent adjustments (characteristic unidirectional duration 400 ms). Here we test the hypothesis that these durations reflect load properties rather than some intrinsic property of the human neuromuscular system. Using a specialized set-up mechanically analogous to real standing, subjects manually balanced inverted pendulums with different moments of inertia through a compliant spring representing the Achilles tendon. The spring bias was controlled by a sensitive joystick via a servo motor and accurate visual feedback was provided on an oscilloscope. As moment of inertia decreased, inverted pendulum sway size increased and it became difficult to sustain successful balance. The mean duration of unidirectional balance adjustments did not change. Moreover, the mean duration of unidirectional inverted pendulum sway reduced only slightly, remaining around 1 s. The simplest explanation is that balance was maintained by a process of manual adjustments intrinsically limited to a mean frequency of two to three unidirectional adjustments per second corresponding to intermittent control observed in manual tracking experiments. Consequently the inverted pendulum sway duration, mechanically related to the bias duration, reflects an intrinsic constraint of the neuromuscular control system. Given the similar durations of sway and muscle adjustments observed in real standing, we postulate that the characteristic duration of unidirectional standing sway reflects intrinsic intermittent control rather than the inertial properties of the body.