This paper presents a backstepping sliding mode control (BSMC) scheme for uncertain underactuated nonlinear systems. Since, underactuated electro-mechanical nonlinear systems (UEMNS) operate under the fewer number of inputs than the degree of freedom, therefore, the control of such systems remains a complex task. In this work, a class of UEMNS is considered which can be transformed into the so-called regular form. In the design process, these UEMNS are first transformed into regular form. In this form, the systems are properly subdivided into series cascaded blocks in which the one of the blocks is indirectly driven by the applied control input whereas the second block is directly controlled by the control input. This structure is apparently suitable for backstepping design. Hence, all the controller steps are designed via the proposed backstepping sliding mode technique. The step by step stability is proved rigorously by considering the Lyapunov approach. In term of benefits, this designed control law provides low-frequency vibration as compared to conventional SMC along with robust performance in the presence of matched uncertainties. This claim is verified via the numerical simulation results of the cart-pendulum system. For further confirmation, these results are also compared with the standard literature results to demonstrate the benefits and effectiveness of the aforesaid control scheme for such class.

中文翻译：

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一类欠驱动机电非线性系统的鲁棒反步滑模控制设计

本文提出了一种用于不确定欠驱动非线性系统的反推滑模控制（BSMC）方案。由于欠驱动机电非线性系统 (UEMNS) 在输入数量少于自由度的情况下运行，因此，对此类系统的控制仍然是一项复杂的任务。在这项工作中，考虑了一类可以转化为所谓正则形式的 UEMNS。在设计过程中，首先将这些 UEMNS 转化为规则形式。在这种形式中，系统被适当地细分为串联级联块，其中一个块由应用的控制输入间接驱动，而第二个块由控制输入直接控制。这种结构显然适合后步设计。因此，所有控制器步骤都是通过所提出的反推滑模技术设计的。通过考虑Lyapunov方法严格证明了逐步稳定性。在好处方面，与传统 SMC 相比，这种设计的控制律提供低频振动，并在存在匹配不确定性的情况下提供稳健的性能。这一说法通过推车-摆系统的数值模拟结果得到验证。为了进一步确认，这些结果还与标准文献结果进行了比较，以证明上述控制方案对该类的好处和有效性。与传统 SMC 相比，这种设计的控制律提供低频振动，并在存在匹配不确定性的情况下提供稳健的性能。这一说法通过推车-摆系统的数值模拟结果得到验证。为了进一步确认，这些结果还与标准文献结果进行了比较，以证明上述控制方案对该类的好处和有效性。与传统 SMC 相比，这种设计的控制律提供低频振动，并在存在匹配不确定性的情况下提供稳健的性能。这一说法通过推车-摆系统的数值模拟结果得到验证。为了进一步确认，这些结果还与标准文献结果进行了比较，以证明上述控制方案对该类的好处和有效性。