In this article, adaptive backstepping sliding mode controller and adaptive fractional backstepping sliding mode controller methods are proposed to control an electrostatic microplate with a piezoelectric layer. Based on the modified couple stress theory, a size-dependent mathematical model is proposed, in which the microplate is modeled using the Kirchhoff plate theory. To take into account the geometric nonlinearities, the von Kármán nonlinear strains are considered in the mathematical model. The Hamilton’s principle is used to obtain the nonlinear equation of motion of the system, which is then converted into a nonlinear ordinary differential equation via the Galerkin technique. The validity of the results obtained from the proposed reduced-order model is checked through direct numerical simulation of the partial differential equation using the finite element method. Finally, two Lyapunov-based control approaches that are adaptive backstepping sliding mode and adaptive fractional backstepping sliding mode are applied to the system. In the adaptive fractional backstepping sliding mode controller method, the adaptive control law is used to evaluate the upper bound of uncertainties and random disturbances. The fractional-order form of sliding surface is used to reduce the amount of chattering and also to improve the tracking error. In the results section, first, numerical studies are conducted to study the effect of system parameters, such as material length scale parameter, aspect ratio, fractional order, and robustness of the controller, on the performance of the system. In addition, the performances of the two control methods are compared, and the merits and demerits of each method are discussed.

在本文中，提出了自适应反步滑模控制器和自适应分数反步滑模控制器方法来控制带压电层的静电微板。基于改进的耦合应力理论，提出了一种尺寸相关的数学模型，其中使用基尔霍夫板理论对微板进行建模。为了考虑几何非线性，在数学模型中考虑了vonKármán非线性应变。使用汉密尔顿原理获得系统的非线性运动方程，然后通过Galerkin技术将其转换为非线性常微分方程。通过使用有限元方法对偏微分方程进行直接数值模拟，可以验证从所提出的降阶模型获得的结果的有效性。最后，将两种基于李雅普诺夫的控制方法分别为自适应反推滑模和自适应分数反推滑模。在自适应分数阶倒推滑模控制器方法中，自适应控制律用于评估不确定性和随机干扰的上限。滑动表面的分数阶形式用于减少颤动量并改善跟踪误差。在结果部分中，首先进行数值研究以研究系统参数的影响，例如材料长度比例参数，长宽比，分数阶，控制器的鲁棒性和系统性能。此外，比较了两种控制方法的性能，并讨论了每种方法的优缺点。