Delay-Dependent Stability Analysis of Self-Balancing Robot Control Systems with Time-Varying Delays

This study investigated the stability of a self-balancing robot control system under a delayed-feedback proportional-integral-derivative (PID) control scheme. To enhance the effectiveness of the analysis, a control method that explicitly considers time delays within the PID feedback loop was developed. First, a dynamic model of the self-balancing robot was established using a Lagrangian formulation, incorporating the time-delay effects from feedback control. The controller output was defined in terms of the velocity and steering angle of the robot, and a closed-loop control system with time-delayed feedback was derived. The Lyapunov–Krasovskii functional approach was used to analyze the stability of the system. In particular, an augmented Lyapunov functional was constructed and combined with inequality techniques based on auxiliary functions and matrix injection methods to achieve a delay-dependent stability criterion. This criterion enables the quantitative estimation of the admissible delay bounds under system stability. Finally, numerical simulations were conducted to validate the theoretical results. The case study results demonstrated that the proposed method provides a reliable, accurate, and superior stability analysis for PID-controlled self-balancing robots with time-varying delays.