Lipschitz continuous sliding-mode controllers (LCSMC) are developed as the integral of discontinuous SMC, producing control signals of finite slope. Nevertheless, LCSMC still generate chattering in the presence of fast parasitic dynamics. In this paper, an analysis of chattering in systems driven by LCSMC is performed using the Harmonic Balance (HB) approach. Two kinds of LCSMC are considered: the first one is based on a linear sliding variable (LSV) and the second one on a terminal switching variable (TSV). Predictions of the amplitude and frequency of self-excited oscillations allowed to compute the average power consumed by the controller, in order to maintain the trajectories into the real sliding mode. A comparison of LCSMC with the Super-Twisting controller (STC), which produce a continuous control signal with infinite slope, is performed. Theoretical predictions and simulation results confirm that LCSMC may induce fast-oscillations (chattering) of smaller amplitude and average power than those ones caused by the STC. But, surprisingly, the chattering generated by LSV-LCSMC could be smaller than that one caused by TSV-LCSMC, when the actuators are fast enough. On the other hand, it tuns that if the sliding dynamics of the LSV-LCSMC closed-loop is of similar speed as the actuators dynamics, the system can loose even practical stability.

中文翻译：

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Lipschitz 连续滑模控制器的颤振分析

Lipschitz 连续滑模控制器 (LCSMC) 是作为不连续 SMC 的积分而开发的，可产生有限斜率的控制信号。尽管如此，LCSMC 在存在快速寄生动态的情况下仍然会产生颤振。在本文中，使用谐波平衡 (HB) 方法对由 LCSMC 驱动的系统中的颤振进行了分析。考虑了两种 LCSMC：第一种基于线性滑动变量 (LSV)，第二种基于终端开关变量 (TSV)。自激振荡的幅度和频率的预测允许计算控制器消耗的平均功率，以便将轨迹保持在真实的滑动模式中。LCSMC 与 Super-Twisting 控制器 (STC) 进行了比较，后者产生具有无限斜率的连续控制信号。理论预测和仿真结果证实，与由 STC 引起的那些相比，LCSMC 可能会引起幅度和平均功率更小的快速振荡（颤动）。但是，令人惊讶的是，当执行器足够快时，LSV-LCSMC 产生的颤动可能比 TSV-LCSMC 产生的颤动小。另一方面，如果 LSV-LCSMC 闭环的滑动动力学与执行器动力学的速度相似，则系统甚至可能失去实际稳定性。