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

Abstract

The critical length that limits stabilizability for delayed proportionalderivative-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.

Special issue on PID control in the information age: Theoretical advances and applications

“…Kovacs and Insperger 19 studied the PID control of an inverted pendulum, which is intended for modeling human balancing task such as stick balancing on the fingertip. They analyzed a particular variant of PID control, that is, the delayed proportional–derivative–acceleration (PDA) feedback, together with predictor feedback control.…”

An observer‐based periodic delayed feedback for fixed‐time stabilization of input‐delay systems

“…1 However, the challenge lies in utilizing finite-dimensional instantaneous state feedback to transfer these infinite poles to the left half-plane. A prevalent solution to this problem is the predictor-based scheme, where the predictor compensates for the time delay, and the controller is developed based on the resulting delay-free system to stabilize the system [2][3][4] and other related references.…”

Digital stability of the Furuta pendulum based on angle detection

“…In case of the Furuta pendulum, these effects are usually overlooked or considered to be negligible due to the low computational costs of the controllers based on few input and output signals in the presence of available high sampling rates. However, when the angle detection of the pendulum is based on noisy digital image processing, similarly to the human-delayed visual processing (Stepan 2009; Zhang et al, 2018; Kovacs and Insperger 2021), only low sampling rates can be achieved, and sampling delay may become critical especially in case of short pendulums. Similar difficulties are discussed and studied, when camera-based image processing is used for modal testing (Wang et al, 2022) or for lane-keeping of automated vehicles (Rossetter and Gerdes 2005; Talvala et al, 2011).…”