Stability analysis and robust control of heart beat rate during treadmill exercise

We investigate a nonlinear dynamical model of a human's heart beat rate (HBR) during a treadmill exercise. We begin with a rigorous analysis of the stability of the model that extends significantly the results available in the literature. In particular, we first identify a simple set of necessary and sufficient conditions for both input-state stability and Lyapunov stability of the system, and then prove that the same conditions also hold when the model parameters are subject to unknown but bounded perturbations. The second part of the paper is devoted to the design and analysis of a control structure for this model, where the treadmill speed plays the role of the control input and the output is the subject's HBR, which is intended to follow a prescribed pattern. We propose a simple control scheme, suitable for a practical implementation, and then analyze its performance. Specifically, we prove (i) that the same conditions that guarantee the stability of the system also ensure that the controller attains a desired level of performance (quantified in terms of the admissible deviation of the HBR from the prescribed profile) and (ii) that the controller is robust to bounded perturbations both in the system parameters and the control input. Numerical simulations are also presented in order to illustrate some of the theoretical results.