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The control problem of a three-degree-of-freedom (3-DOF) X-Z inverted pendulum as an unstable, multi-input-multi-output, under-actuated and high-order nonlinear system is facing many challenges. In this paper, a Fuzzy Adaptive Hierarchical Sliding-Mode Controller (FAHSMC) is optimally designed for the X-Z inverted pendulum system utilizing the Multi-Objective Genetic Algorithm (MOGA). To this end, at first, the state variables of this system are converted to the new state variables via five steps of a forward transformation process. In this transformation, the dynamical equations of the X-Z inverted pendulum system are reorganized to an appropriate form for control implementations. Then, the novel FAHSMC is designed for stabilization and tracking control of the system, and the stability analysis of the controller is proved via the Lyapunov stability theory. After that, the state and control variables of the X-Z inverted pendulum are computed through six steps of a backward transformation process. Ultimately, the MOGA is exerted for Pareto optimal design of the proposed control approach applied on the regarded X-Z inverted pendulum system. The tracking error of the cart position in the X-direction, the tracking error of the cart position in the Z-direction and the angle error of the pendulum are selected as three inconsistent objective functions for multi-objective optimization operations. Numerical results and diagrams illustrate that the proposed approaches can accurately converge the system states to the desirable trajectories in a short time and yield superior Pareto optimal fronts in comparison with the Hierarchical Sliding-Mode Controller (HSMC) and the Adaptive Hierarchical Sliding-Mode Controller (AHSMC).

Pareto Optimal Design of a Fuzzy Adaptive Hierarchical Sliding-mode Controller for an X-Z Inverted Pendulum System