ABSTRACT

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).

摘要

三自由度（3-DOF）XZ倒立摆作为一种不稳定、多输入多输出、欠驱动的高阶非线性系统，其控制问题面临着许多挑战。本文利用多目标遗传算法（MOGA）针对 XZ 倒立摆系统优化设计了模糊自适应分层滑模控制器（FAHSMC）。为此，首先，通过五个步骤的正向变换过程将系统的状态变量转换为新的状态变量。在此转换中，XZ 倒立摆系统的动力学方程被重新组织为用于控制实现的适当形式。然后，设计了新颖的FAHSMC用于系统的稳定和跟踪控制，并通过Lyapunov稳定性理论证明了控制器的稳定性分析。之后，通过六个步骤的向后变换过程计算 XZ 倒立摆的状态和控制变量。最终，MOGA 用于所提出的控制方法的帕累托最优设计，该方法应用于所关注的 XZ 倒立摆系统。选择X方向小车位置跟踪误差、Z方向小车位置跟踪误差和摆锤角度误差作为三个不一致的目标函数进行多目标优化运算。