论文标题
使用近似模型稳定一类不足的Euler Lagrange系统
Stabilizing of a Class of Underactuated Euler Lagrange System Using an Approximate Model
论文作者
论文摘要
能量塑形方法是控制的拉格朗日，是稳定稳定不足的Euler Lagrange（EL）系统的众所周知的方法。在这种方法中，为了构建控制规则，必须解决某些非线性的，非均匀的偏微分方程（PDE），这些方程称为匹配条件。在本文中，提出了一种方法，以获取一类未经局部EL系统的匹配条件的近似解决方案。为了开发该方法，使用惯性矩阵的近似值将势能匹配条件转化为一组线性PDE。因此，可分配的势能函数和受控的惯性矩阵，都是这些PDE的常见解决方案。之后，发现陀螺和耗散力是动能匹配条件的解。最后，控制规则是通过添加能量塑造规则和额外的耗散注射来构建的，以提供渐近稳定性。还给出了使用建议方法得出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，考虑了推车上倒摆的稳定性问题。
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.
