In this article, an implicit Euler algorithm for digital implementation of constrained stabilization is studied for the second-order systems. For that, a switching controller is designed in a discrete-time framework such that the system's position output converges to some predefined range, that is, ϱ ∈ (−ε, ε) in finite-time while the velocity output converges to the origin, that is, , in finite-time. The switching controller is switched to the implicit Euler implementation of twisting algorithm when ϱ ∉ (−ε, ε) and to an implicit Euler implementation of first-order sliding mode control when ϱ ∈ (−ε, ε). The combination of the two implicit Euler implementations achieves discrete-time constrained stabilization of second-order systems, avoiding the chattering caused by conventional explicit integration schemes. The usefulness of the proposed algorithm for constrained stabilization is illustrated by considering the container-slosh coupled dynamical system.

中文翻译：

---

用于二阶系统约束镇定的基于隐式欧拉的数字实现

在本文中，研究了用于二阶系统的约束镇定数字实现的隐式欧拉算法。为此，在离散时间框架中设计了一个开关控制器，使得系统的位置输出收敛到某个预定义的范围，即在有限时间内ϱ  ∈ (− ε ,  ε ) 而速度输出收敛到原点，也就是说，在有限时间内。当ϱ  ∉ (− ε ,  ε )时，切换控制器切换到扭曲算法的隐式欧拉实现，当ϱ  ∈ (− ε ,  ε )时切换到一阶滑模控制的隐式欧拉实现）。两种隐式欧拉实现的组合实现了二阶系统的离散时间约束稳定，避免了传统显式积分方案引起的抖动。通过考虑容器-晃动耦合动力系统，说明了所提出的算法对约束稳定的有用性。