倾斜轨道上的三级倒立摆数学模型的建立与可控性研究

Mathematical modelling and study of controllability of a triple inverted pendulum on an inclined rail

给出倾斜轨道上的三级倒立摆机构的动能、势能和损耗能,利用拉格朗日方程推导出倾斜轨道上的单电机控制的三级倒立摆理论上的数学模型。给出了线性化状态方程、可观性矩阵的秩和可控性矩阵的秩及条件数,并进行原理性数学仿真。仿真结果表明,直接利用系统的全部状态进行线性反馈,三级倒立摆在有些初始条件下是稳定的,而在另一些初始条件下又是不稳定的。这表明三级倒立摆是可观、可控但又是难控制的。

Based on Lagrange's equations of motion, the dynamic equations of a triple inverted pendulum on an inclined rail controlled by a single motor are studied. After simplifying the model and linearizing around the equilibrium state, the rank of the observability matrix, the rank and the condition value of the controllability matrixes are obtained. Simulation on principle are given. The simulation indicates that, using state variable linear feedback, the triple inverted pendulum is stable under some initial conditions, and is unstable under other initial conditions. The triple inverted pendulum is observable and controllable, but has lower controllability.