In this paper, a novel approach is proposed for adjusting the position of a magnetic levitation system using projection recurrent neural network-based adaptive backstepping control (PRNN-ABC). The principles of designing magnetic levitation systems have widespread applications in the industry, including in the production of magnetic bearings and in maglev trains. Levitating a ball in space is carried out via the surrounding attracting or repelling magnetic forces. In such systems, the permissible range of the actuator is significant, especially in practical applications. In the proposed scheme, the procedure of designing the backstepping control laws based on the nonlinear state-space model is carried out first. Then, a constrained optimization problem is formed by defining a performance index and taking into account the control limits. To formulate the recurrent neural network (RNN), the optimization problem is first converted into a constrained quadratic programming (QP). Then, the dynamic model of the RNN is derived based on the Karush-Kuhn-Tucker (KKT) optimization conditions and the variational inequality theory. The convergence analysis of the neural network and the stability analysis of the closed-loop system are performed using the Lyapunov stability theory. The performance of the closed-loop system is assessed with respect to tracking error and control feasibility.

中文翻译：

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基于递归神经网络的自适应最优反步策略控制磁悬浮系统

在本文中，提出了一种使用基于投影递归神经网络的自适应反步控制 (PRNN-ABC) 来调整磁悬浮系统位置的新方法。磁悬浮系统的设计原理在工业中有着广泛的应用，包括磁轴承的生产和磁悬浮列车。使球在空间中悬浮是通过周围的吸引或排斥磁力来实现的。在这样的系统中，致动器的允许范围很重要，尤其是在实际应用中。在所提出的方案中，首先进行了基于非线性状态空间模型的反步控制律的设计过程。然后，通过定义性能指标并考虑控制限来形成约束优化问题。为了制定循环神经网络 (RNN)，首先将优化问题转换为约束二次规划 (QP)。然后，基于Karush-Kuhn-Tucker (KKT) 优化条件和变分不等式理论推导出RNN 的动态模型。神经网络的收敛性分析和闭环系统的稳定性分析采用李雅普诺夫稳定性理论进行。闭环系统的性能根据跟踪误差和控制可行性进行评估。神经网络的收敛性分析和闭环系统的稳定性分析采用李雅普诺夫稳定性理论进行。闭环系统的性能根据跟踪误差和控制可行性进行评估。神经网络的收敛性分析和闭环系统的稳定性分析采用李雅普诺夫稳定性理论进行。闭环系统的性能根据跟踪误差和控制可行性进行评估。