倒立摆模型的三种时滞反馈控制的比较

Comparison of Three Time-Delayed Feedback Controllers for Inverted Pendulum

带有时滞的比例-微分(Proportional-derivative, PD)反馈控制、比例-积分-微分(Proportional-integral-derivative, PID)反馈控制和比例-导数-二阶导数(Proportional-derivative-acceleration, PDA)反馈控制的倒立摆控制模型是人体平衡控制问题的常用理论模型,此闭环系统的快速稳定等方面的研究成果对理解和提升具有明显反应时滞的老年人的人体平衡能力具有重要意义.本文首先考察时滞PID反馈控制作用下的倒立摆系统的稳定性,利用重特征根条件得到了闭环系统的最优增益解析表达式、最优衰减因子及相应的临界时滞,然后分别比较了三种时滞反馈作用下的倒立摆控制系统的最优指数衰减因子、临界时滞以及控制效果.结果表明,在相同参数条件下,时滞PDA反馈控制作用下的闭环系统具有比其他两种时滞反馈控制系统更快的衰减速率和更大的临界时滞,因此从快速稳定的角度看,时滞PDA反馈控制策略在人体平衡问题中比另外两种控制策略更有优势.

The inverted pendulum control model with delayed PD(proportional-derivative)feedback controller,delayed PID(proportional-integral-derivative)feedback controller,and delayed PDA(proportional-derivative-acceleration)feedback controller are three common theoretical models for the human balancing problem.From the viewpoint of fast-stablization of the closed-loop system,it is important to understand and to improve the balance ability of the elderly with obvious response delay.In this paper,we firstly study the stability of the inverted pendulum control system under a delayed PID controller,and then obtain the analytical expression of optimal gain,the optimal exponential decay factor,and the corresponding critical time delay of the closed-loop system utilizing the condition of multiple characteristic roots.Comparison between the three delayed control strategies is made for the optimal exponential decay factor,critical time-delay and the performance of the controlled inverted pendulum.It shows that the delayed PDA feedback has a faster decay rate and a greater critical time delay than the other two strategies under the same parameters.Therefore,the delayed PDA feedback controller is more suitable for the human balancing problem from the viewpoint offast-stabilization.