磁浮列车的动力稳定性分析与Liapunov指数

ANALYSIS OF DYNAMIC STABILITY FOR MAGNETICLEVITATION VEHICLES BY LIAPUNOVCHARACTERISTIC NUMBER

针对弹性轨道上的磁悬浮列车线性动力控制系统，给出了采用Ｌｉａｐｕｎｏｖ特性指数判别动力系统稳定性的判据：当动力系统的全部Ｌｉａｐｕｎｏｖ特性指数小于零时，动力控制系统是稳定的；而当动力系统中有一Ｌｉａｐｕｎｏｖ特性指数大于零时，动力系统就成为不稳定的．由于Ｌｉａｐｕｎｏｖ特性指数可以由数值积分方式方便地给出，这一判断方法对于不便搜索参数稳定区域的高维动力系统，与寻求周期变系数线性常微分方程动力系统的Ｆｌｏｑｕｅｔ理论的变换矩阵特征值相比，具有计算量小和搜索方法简单等优点．数值算例表明本文方法是有效的．

This paper is to give an analysis of dynamic stability of the dynamic control system ofa magnetic levitation vehicle moving over flexible guideways by means of the Liapunov characteristic numbers. In the dynamic control system considered here, the Bernoulli-Euler theory of beamsis employed for behaving deformation of guideways, and the negative feedback of displacement andits velocity of the gap bewteen the 'maglev' and the guideways is employed. After the modaltechnique is taken in analysis of deform4tion of guideways, and small disturbance of the relativedisplacement in vertical direction is assumed in the study of dynamic stability, a system of ordinarydifferential equations with periodically variable coefficients are obtained to characterize the behavior of dynamic stability of the dynamic control system. Then, a conclusion on the dynamic stabilityof the dynamic control system is gained by means of the Liapunov characteristic numbers. Theresulted criterion for the dynamic control system is that when the Liapunov characteristic numbersof the system all are less than zero, the dynamic system is stablly controlled; otherwise, if there isone Liapunov characteristic number which is greater than zero in the system, the controlled system becomes unstable. Since the Liapunov characteristic numbers can be obtained by a numericalintegral to the dynamic system, this method for judging stability of the controlled system has themerit of small computation and simple of searching method to compare with the method based onthe Floquet theory to which one should search all characteristic values of matrix of basic solutionsof the dynamic system when it is of high dimension. Finally, some numerical examples are givento show that the method of Liapunov characteristic number is efficient.