Fractional derivative control designs by inhomogeneous sampling for systems with nonlinear elements

This paper presents an efficient numerical method to realize a discrete model of fractional order PID controllers for systems containing backlash nonlinearity. This model is based on a class of Stieltjes integrals transferred from the Riemann-Liouville definition. The discretization method is to calculate on inhomogeneous sampling periods which are getting longer as the operation points go back toward the initial time. It leads to the effective quality which has low computational costs and enough accuracy. A practical controller design is applied for an inverted pendulum system containing gear-backlash. The simulation and experimental results show that the fractional PID control system has superior performance against backlash nonlinearity and thus the backlash vibration can be suppressed.