In this article, two boundary feedback controllers are designed via the backstepping approach for a class of gantry crane systems. To provide an accurate and concise representation of the dynamic behavior, the gantry crane with a flexible cable is described by a hybrid system. The hybrid system is formed by an ordinary differential equation coupled with a partial differential equation. In the first control strategy, a backstepping-based boundary state-feedback controller is proposed for the gantry crane to transport a payload to an expected position with less shaking. In the second control strategy, a boundary output-feedback controller is explored with an observer estimating the inaccessible states. By using the backstepping technique and kernel functions, the original systems with different control strategies are transformed into target systems. By using the operator semigroup and Lyapunov stability theories, the target system is proven to be well-posed and exponentially stable, respectively. Finally, numerical simulations and comparisons are provided to illustrate the efficiency and the advantages of the proposed methods.

中文翻译：

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一类龙门起重机系统的反步边界控制


在本文中，通过反步法为一类龙门起重机系统设计了两个边界反馈控制器。为了提供动态行为的准确和简洁的表示，具有柔性电缆的龙门起重机由混合系统描述。混合系统由常微分方程与偏微分方程耦合形成。在第一种控制策略中，提出了一种基于反步的边界状态反馈控制器，用于龙门起重机将有效负载运输到震动较小的预期位置。在第二种控制策略中，通过观察者估计不可访问的状态来探索边界输出反馈控制器。通过使用反步技术和核函数，将具有不同控制策略的原始系统转化为目标系统。通过使用算子半群和Lyapunov稳定性理论，分别证明了目标系统是适定的和指数稳定的。最后，通过数值模拟和比较来说明所提出方法的效率和优点。