Critical parameters for the robust stabilization of the inverted pendulum with reaction delay: State feedback versus predictor feedback

The critical length that limits stabilizability for delayed proportional-derivative-acceleration (PDA) feedback and for predictor feedback (PF) is analyzed for the inverted pendulum paradigm. The aim of this work is to improve the understanding of human balancing tasks such as stick balancing on the fingertip, which can be modeled as a pendulum cart system. The relation between the critical length of the balanced stick and the reaction time delay in the presence of sensory uncertainties, which are modeled as static parameter perturbations in the control gains, is investigated rigorously. Robust stabilizability analysis is performed using the real structured stability radius. Performance is assessed by the length of the shortest pendulum (critical length) that can still be balanced for a fixed reaction delay. For both PDA feedback and PF control with delay mismatch, it is observed that the relation between the critical length and the reaction delay remains quadratic in the presence of perturbations on the control gains (of fixed size). Numerical comparison shows that predictor feedback is superior over PDA feedback in terms of critical length: shorter pendulum can be balanced by PF than by PDA feedback for the same reaction delay and for the same static parameter perturbation. Furthermore, it is found that both control concepts are more sensitive to the change in the feedback delay than on the same relative change in the parameter uncertainties. Interpretation to human balancing suggests that it is more challenging for the nervous system to cope with reaction delay than with sensory uncertainties.