Trajectory tracking for uncertainty time delayed-state self-balancing train vehicles using observer-based adaptive fuzzy control.

Complex systems, such as the interconnected self-balancing vehicles system, are known to be highly nonlinear, under-actuated, and challenging to control. Their complexity is further increased by the presence of time delays and external disturbances. Therefore, all attempts to control the interconnected self-balancing vehicles system were based on accurate determination of its states. However, it is well known that modelling of interconnected self-balancing vehicles systems is not an easy task. In this paper, trajectory tracking of a series of two-wheeled self-balancing vehicles, named B2-train system, is addressed. The highly nonlinear under-actuated system is analyzed and a nonlinear dynamic model of the B2-train system is derived using the Lagrangian method. The adaptive fuzzy controller is designed to approximate the unknown system parameters under the constraint that only the system output is available for measurement. To address this the constraint, a nonlinear state observer is employed to estimate the states including time delays. The aim is to design a state observer-based adaptive fuzzy controller using variable structure (VS) technique and a time delayed compensator which ensures the robust asymptotic stability of the closed-loop system and guarantees an H∞ norm bound constraint on disturbance attenuation for all admissible uncertainties based on Lyapunov criterion. Finally, a time-delayed B2-train system with 2 vehicles is used to demonstrate the performance and robustness of the proposed control scheme.