Adaptive neural-fuzzy inference system to control dynamical systems with fractional order dampers

In this paper, an adaptive neural fuzzy inference system (ANFIS)-based control technique is proposed to stabilize dynamical systems with fractional order dampers. For this purpose, a linear quadratic regulator (LQR) is first designed for the analogous linearized integer order systems where the fractional damper is replaced by the combination of an integer spring and an integer damper. Next, the ANFIS-based controller is trained based on the responses of the closed-loop LQR-controlled system under different scenarios such as several initial conditions and/or inputs. Since the number of fuzzy rules increases exponentially by increasing the number of inputs, a fusion function proposed in the literature is used to reduce the number of inputs in the ANFIS-based controller. Hence the number of fuzzy rules is reduced as well. The result of this training is a trained ANFIS-LQR controller that can be used for stabilizing the fractional-order models with fractional order dampers. As an illustrative example, the proposed technique is employed to stabilize an under-actuated double inverted pendulum on the cart with fractional order dampers.