一类欠驱动Euler Lagrange系统的近似模型镇定
Stabilizing of a Class of Underactuated Euler Lagrange System Using an Approximate Model
摘要：
能量成形方法，受控拉格朗日，是一种著名的方法来稳定欠驱动欧拉拉格朗日（EL）系统。在这种方法中，为了构造一个控制规则，一些非线性，非齐次偏微分方程（PDE），这被称为匹配条件，必须解决。本文针对一类欠驱动电致发光系统，提出了一种近似求解这些匹配条件的方法。为了发展的方法，势能匹配条件被转换成一组线性偏微分方程使用近似的惯性矩阵。因此，可分配的势能函数和受控惯性矩阵，都被构造为这些偏微分方程的公共解。然后，找到了陀螺力和耗散力作为动能匹配条件的解。最后，通过增加能量成形规则和附加耗散注入来构造控制规则，以提供渐近稳定性。最后给出了采用该方法得到的控制规则的闭环系统的稳定性分析。为了证明所提出的方法的成功，倒立摆的稳定性问题的推车被认为是。
英文摘要：
The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the under-actuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear, nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of under-actuated EL systems. To develop the method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. So the assignable potential energy function and the controlled inertia matrix, both are constructed as a common solution of these PDEs. Afterwards, the gyroscopic and dissipative forces are found as the solution of the kinetic energy matching condition. Finally, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed loop system which used the control rule derived with the proposed method is also given. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.
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