We consider sliding-mode control systems subject to unmatched perturbations. Classical first-order sliding-mode techniques are capable to compensate unmatched perturbations if differentiations of the output of sufficiently high order are included in the sliding variable. For such perturbations it is commonly assumed that they do not affect the relative degree of the system. In this contribution we consider perturbations that alter the relative degree of the process and study their impact on the closed-loop control system with a classical first-order sliding-mode design. In particular we consider systems with full (nominal) relative degree subject to a perturbation reducing the relative degree by one and analyse the resulting closed-loop system. It turns out that the sliding-manifold is not of reduced dimension and the uniqueness of the solution may be lost. Also attractivity of the sliding-manifold and global stability of the origin may be lost whereas the disturbance rejection properties of the sliding-mode control are not impaired. We present a necessary and sufficient condition for the existence of unique solutions for the closed-loop system. The second-order case is studied in great detail and allows to parametrically specify the conditions obtained before. We derive a necessary condition for the global asymptotic stability of the closed-loop system. Further we present a constructive condition for the global asymptotic stability of the closed-loop system using a piece-wise linear Lyapunov function. Each of the prominent results is illustrated by a numerical example.

我们考虑滑模控制系统受到无与伦比的扰动。如果滑动变量中包含足够高阶输出的微分，则经典的一阶滑动模式技术能够补偿不匹配的扰动。对于此类扰动，通常假设它们不会影响系统的相对程度。在这篇文章中，我们考虑了改变过程相对程度的扰动，并研究了它们对具有经典一阶滑模设计的闭环控制系统的影响。特别是，我们考虑具有完整（标称）相对度数的系统受到扰动，将相对度数减少 1，并分析由此产生的闭环系统。事实证明，滑动流形不是降维的，解的唯一性可能会丢失。滑动流形的吸引力和原点的全局稳定性也可能会丢失，而滑动模式控制的干扰抑制特性不会受到损害。我们提出了闭环系统唯一解存在的充分必要条件。对二阶情况进行了非常详细的研究，并允许参数化地指定之前获得的条件。我们推导出闭环系统全局渐近稳定的必要条件。此外，我们使用分段线性 Lyapunov 函数为闭环系统的全局渐近稳定性提出了建设性条件。每个突出的结果都用一个数值例子来说明。滑动流形的吸引力和原点的全局稳定性也可能会丢失，而滑动模式控制的干扰抑制特性不会受到损害。我们提出了闭环系统唯一解存在的充分必要条件。对二阶情况进行了非常详细的研究，并允许参数化地指定之前获得的条件。我们推导出闭环系统全局渐近稳定的必要条件。此外，我们使用分段线性 Lyapunov 函数为闭环系统的全局渐近稳定性提出了建设性条件。每个突出的结果都用一个数值例子来说明。滑动流形的吸引力和原点的全局稳定性也可能会丢失，而滑动模式控制的干扰抑制特性不会受到损害。我们提出了闭环系统唯一解存在的充分必要条件。对二阶情况进行了非常详细的研究，并允许参数化地指定之前获得的条件。我们推导出闭环系统全局渐近稳定的必要条件。此外，我们使用分段线性 Lyapunov 函数为闭环系统的全局渐近稳定性提出了建设性条件。每个突出的结果都用一个数值例子来说明。