A generalized stochastic optimal control formulation for heart rate regulation during treadmill exercise

Abstract

The stochastic optimal control formulation for heart rate (HR) regulation during treadmill exercise is extended here to encompass low-pass characteristics in the compensator and in the input sensitivity function. The latter governs the response of the treadmill speed command to disturbances arising from physiological heart rate variability (HRV), and it must be shaped to give appropriate control loop behaviour. In a comparative test series involving 20 healthy male subjects, the low-pass compensator was found to give substantially and significantly lower mean intensity of changes in treadmill speed (1.0 vs. $ 12.2 \times 10^{-4} $ m2/s2, low-pass vs. non-low-pass compensators, $ p = 5.1 \times 10^{-8} $ ), but at the cost of a significant increase in mean root-mean-square tracking error (2.46 vs. 1.74 bpm, $ p = 6.5 \times 10^{-6} $ ). The experimental results demonstrated accurate and robust control of HR across the 20 subjects tested, despite a simple pre-existing nominal model having been used for controller design. The results also provide strong evidence that the magnitude of HRV decreases naturally over time. The principal design issue in this application is that of suppression of disturbances arising from HRV, but changes in treadmill speed must remain acceptable to the runner. The theoretical extension of the stochastic optimal control problem formulation derived here allows this to be addressed by appropriate shaping of the key sensitivity functions using two tuning parameters, thus facilitating the tradeoff between tracking accuracy and control signal intensity.

Keywords

• Optimal control
• heart rate control
• physiological control
• treadmills
• polynomial approach
• sensitivity functions