Optimal fractional control applied to a single link flexible robot with disturbances and payload changes

Abstract

This article explores and compares the use of the linear quadratic regulator (LQR) theory to control a single-degree-of-freedom flexible link robot ( FLR ) using an integer-order model and a fractional-order model. The fractional-order model is obtained from the integer-order one by introducing new “fictitious” fractional states. The LQR technique determines feedback gains to optimally control a system by considering its dynamics and cost function. Two LQR schemes are designed here: one for the integer-order model using a linear matrix inequality reformulation and the other for the fractional-order model involving analytical tuning and numerical optimization. A genetic algorithm is used to optimize the fractional-order LQR by minimizing the same cost function as the integer-order model. The resulting control system minimizes disturbances, ensuring accurate tracking, stability, and robustness in flexible robotic systems. The complete scheme integrates active disturbance rejection control ( ADRC) and Luenberger observers to estimate internal states. Simulations and experiments demonstrate that the fractional-order LQR outperforms the integer-order version in terms of energy efficiency, evaluated using total variation ( TV) and the integral absolute control signal ( IACS) . Then we have shown that feeding back fictitious fractional states besides the standard integer-order ones reduces the LQR cost index and improves the robot performance.