A model of postural control in quiet standing: robust compensation of delay-Induced Instability using intermittent activation of feedback control.

摘要

The main purpose of this study is to compare two different feedback controllers for the stabilization of quiet standing in\udhumans, taking into account that the intrinsic ankle stiffness is insufficient and that there is a large delay inducing instability\udin the feedback loop: 1) a standard linear, continuous-time PD controller and 2) an intermittent PD controller characterized\udby a switching function defined in the phase plane, with or without a dead zone around the nominal equilibrium state. The\udstability analysis of the first controller is carried out by using the standard tools of linear control systems, whereas the\udanalysis of the intermittent controllers is based on the use of Poincare´ maps defined in the phase plane. When the PDcontrol\udis off, the dynamics of the system is characterized by a saddle-like equilibrium, with a stable and an unstable\udmanifold. The switching function of the intermittent controller is implemented in such a way that PD-control is ‘off’ when\udthe state vector is near the stable manifold of the saddle and is ‘on’ otherwise. A theoretical analysis and a related simulation\udstudy show that the intermittent control model is much more robust than the standard model because the size of the\udregion in the parameter space of the feedback control gains (P vs. D) that characterizes stable behavior is much larger in the\udlatter case than in the former one. Moreover, the intermittent controller can use feedback parameters that are much smaller\udthan the standard model. Typical sway patterns generated by the intermittent controller are the result of an alternation\udbetween slow motion along the stable manifold of the saddle, when the PD-control is off, and spiral motion away from the\udupright equilibrium determined by the activation of the PD-control with low feedback gains. Remarkably, overall dynamic\udstability can be achieved by combining in a smart way two unstable regimes: a saddle and an unstable spiral. The\udintermittent controller exploits the stabilizing effect of one part of the saddle, letting the system evolve by alone when it\udslides on or near the stable manifold; when the state vector enters the strongly unstable part of the saddle it switches on a\udmild feedback which is not supposed to impose a strict stable regime but rather to mitigate the impending fall. The\udpresence of a dead zone in the intermittent controller does not alter the stability properties but improves the similarity with\udbiological sway patterns. The two types of controllers are also compared in the frequency domain by considering the power\udspectral density (PSD) of the sway sequences generated by the models with additive noise. Different from the standard\udcontinuous model, whose PSD function is similar to an over-damped second order system without a resonance, the\udintermittent control model is capable to exhibit the two power law scaling regimes that are typical of physiological sway\udmovements in humans.