Design of fractional MOIF and MOPIF controller using PSO algorithm for the stabilization of an inverted pendulum-cart system

The topic of this paper is the design of two fractional order schemes, based on a state feedback for linear integer order system. In the first one of the state feedback is associated with a fractional order integral ($I^{\alpha }$) controller. In the second structure the state feedback is associated with a fractional order proportional integral ($P{I}^{\alpha }$) controller. With such controllers, the closed loop system with state feedback described by the state equations splits in n-subsystems with different fractional orders derivatives of the state variable. In order to find the optimal parameters value of both controllers ($I^{\alpha }$) and ($P{I}^{\alpha }$), a multi-objective particle swarm optimization algorithm is used, with the integral of absolute error, the overshoot $M_p$, the Buslowicz stability criterion are considered as objective functions. The multi-objective integral fractional order controller and the multi-objective proportional integral fractional order controller are applied to stabilize the inverted pendulum-cart system (IP-C), and their performance is compared to the fractional order controller. The simulation results of these innovative controllers are also compared with those obtained by conventional proportional–integral–derivative and fractional order proportional–integral–derivative controllers. The robustness of the proposed controllers against disturbances is investigated through simulation runs, considering the non-linear model of the IP-C system. The obtained results demonstrate that our approach not only leads to high effectiveness but also showcases remarkable robustness, supported by both simulation and experimental results.