A model of postural control in quiet standing: robust compensation of delay-induced instability using intermittent activation of feedback control

Abstract

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

Figure 1. Characterization of the 4 control models in the phase plane ().

In Models 1 and 2 the control is active in the whole plane. The shaded areas in Models 3 and 4 identify the areas where the control is switched off.

Figure 2. In the plane of proportional and derivative parameters (P and D, respectively) of the model 1 and model 2 feedback controllers, the figure identifies the region of stability (shaded triangle).

Body parameters: m (mass); I (moment of inertia); h (distance of the center of mass from the ankle); K (intrinsic stiffness); B (intrinsic viscosity); mgh (gravity toppling rate). Controller parameters: P, D, Δ (delay of the feedback loop). As Δ decreases, the triangle increases its area and tends to fill the whole first quadrant to the right of the critical value mgh-K. As Δ increases the triangle decreases its area and vanishes when it reaches the value .

Figure 3. Typical solutions of Model 3 in the phase plane.

In each plane, the initial state at t = 0 is represented by the thick curve segment labeled “1”. This state segment moves in the phase plane according to the DDE of eq. 3 in number order as labeled. A state of the system at time t is represented by the corresponding curve segment whose leading edge is and the tail-end is . The PD controller is turned on and off, respectively, if the tail-end of the segment is located in the on (white) and off (gray-shaded) regions in the phase plane. Dotted lines are the stable manifold (arrow heads directing the equilibrium) and the unstable manifold (arrow heads departing away from the equilibrium). A: A typical orbit of eq. 3 when the proportional gain P of the PD controller is small. B: A typical orbit of eq. 3 when the gain P is large.

Figure 4. Stability analysis of control model 3 (in the absence of noise) by means of the Poincaré map.

Alternation of PD-on and PD-off flows. The lines Σ and Π in the phase plane are related to the switching mechanism of the controller. (The shaded areas indicate that the PD-control is switched off.) Σ is also used as the section for the computation of the map. Two typical orbits from Σ to Σ are shown (thick curves) for two different values of the proportional controller gain P: s→p ₁ →p ₂ →p ₃ →s' and s→q ₁ →q ₂ →q ₃ →s'. The thin lines display the PD-on flows (unstable spiral) and the PD-off flow (saddle with a stable manifold).

Figure 5. Poincaré map θ' = F(θ) and its dynamics.

A: Two examples of numerically obtained Poincaré map for two different values of P. Representation of the return map was restricted to the angular values: θ to θ'. For each map, an initial tilt angle of a leading edge placed on Σ is given, and the subsequent transverse angles of the leading edge across Σ are obtained by and . B and C: A sequence of the tilt angles when the state of the system passes through the section Σ obtained by iterative use of the map in the panel A (filled points) and by the DDE simulation (open circles) for Model 3 with a = −0.4 s⁻¹, and they showed a good agreement. The sequence toward the equilibrium of the sway angle is monotonic in B (P/mhg = 0.54) and oscillatory in C (P/mhg = 0.64).

Figure 6. Comparison of the stability region in the P–D plane for the control Models 2 and 3.

The horizontal axis is normalized with respect to the critical stiffness (mgh) considering that the intrinsic stiffness is 80% of that value. The stability region of Model 2 is the striped triangle. The stability region of Model 3 is the grey-shaded area, with a gray intensity which is a function of the absolute slope of the Poincaré map: |dF/dθ|_θ = 0 : the darker the shade the quicker the recovery of upright equilibrium. |dF/dθ|_θ = 0 = 0 is maximal stability; |dF/dθ|_θ = 0 = 1 is neutral stability. Dotted areas correspond to instability (|dF/dθ|_θ = 0>1). The four panels show how the stability of Model 3 depends upon parameter a which identifies the switching mechanism of Fig. 1. In panels C and D, a small white thin region at the left upper edge of the gray region corresponds to parameter sets in which the equilibrium point is a stable node. Hence in this white region, the Poincaré map cannot be defined though the equilibrium point is stable.

Figure 7. Simulation of the four control models with and without noise: Model 1 (panel A); Model 2 (panel B); Model 3 (panel C); Model 4 (panel D).

Each panel shows: 1) trajectories in the phase plane (left-upper part without noise, left-lower part with noise); 2) corresponding angular sway sequences (middle-upper part without noise, middle-lower part with noise); 3) power spectral density for the model with noise (right part). In the shaded areas of the phase planes, PD control is switched off. For Models 1 and 2, the following parameters were used: P/mgh = 0.8, D = 270 Nms/rad, σ  = 2 Nm. For Models 3 and 4 the parameters in the PD-on regions were as follows: P/mgh = 0.25, D = 10 Nms/rad, σ  = 0.2 Nm, and a = −0.4 s⁻¹.

Figure 8. Sample of human postural sway, collected from a subject in quiet standing for 120 s.

Left upper panel: Angular sway sequence; Left lower panel: trajectory in the phase plane; Right panel: power spectral density of the angular sway.

Figure 9. Power spectral density functions (PSDs) of sway data for Model 4 with two different parameter values.

Left panel: P = 176 Nm/rad (), D = 10 Nms/rad, a = −0.4 s⁻¹. Right panel: P = 470 Nm/rad (), D = 10 Nms/rad, a = −0.4 s⁻¹. For Models 3 and 4 with a = −0.4 s⁻¹, the optimal the optimal value of P for the stability is about 60% of mgh (i.e. ) regardless the value of D. The values of P in the left and right panels are smaller and larger than the optimal value of P, respectively.

Figure 10. For the feedback control Model 4 (intermittent with dead zone) the figure shows the dependence of the scaling factor α of the PSD function upon the following parameters: 1) noise intensity; 2) proportional feedback gain P (normalized with respect to the critical stiffness mgh); 3) the slope a of the switching function.

The derivative feedback gain D is fixed at the value of 10 Nms/rad. The values of α, with appropriate choice of σ, P and a, are comparable with the physiological value which is about 1.5. A: a = 0 s⁻¹. B:a = −0.1 s⁻¹. C:a = −0.4 s⁻¹. D: a = −0.7 s⁻¹.