A novel Lyapunov-based methodology for both stability analysis and stabilisation of second-order sliding mode algorithms in the presence of exogenous disturbances is presented. Nowadays, second-order sliding modes such as the well-known super-twisting algorithm are widely recognised for their usefulness in the design of controllers, observers, and exact differentiators. However, a theoretical proof for an optimal and systematical selection of parameters for the algorithm to be stable is of particular interest. The solution proposed in this paper is based on combining exact convex representations of the nonlinear algorithms with a strict Lyapunov function, which leads to stability or design conditions in terms of linear matrix inequalities that can be efficiently solved via convex optimisation techniques, allowing us to provide an enlarged and optimal parameter setting in contrast with existing results; moreover, the systematic nature of the proposal is able to deal with different scenarios avoiding ad-hoc solutions of previous works. Effectiveness of the results are verified via simulation examples.

提出了一种新颖的基于Lyapunov的方法，用于在存在外源干扰的情况下进行稳定性分析和二阶滑模算法的稳定性。如今，二阶滑动模式（例如众所周知的超扭曲算法）因其在控制器，观察者和精确微分器设计中的有用性而广为人知。然而，对于使算法稳定的最优且系统地选择参数的理论证明尤其令人关注。本文提出的解决方案是基于将非线性算法的精确凸表示与严格的Lyapunov函数相结合的结果，这可以通过线性优化来解决线性矩阵不等式的稳定性或设计条件，通过凸优化技术可以有效地解决这些问题，与现有结果相比，允许我们提供更大且最佳的参数设置；此外，该建议书的系统性使其能够处理各种情况，从而避免了先前工作的临时解决方案。通过仿真实例验证了结果的有效性。