基于Liapunov-Schmidt方法的局部静态分叉控制

Local static bifurcation control based on Liapunov-Schmidt method

针对某一类非线性系统 (在临界情况下 ,其线性化系统具有单个的零特征根 ) ,讨论了局部静态分叉控制问题 .首先 ,证明了临界非线性系统的局部静态分叉控制和局部稳定性之间的等价性 ;然后 ,利用 LS方法对原非线性系统进行约化 ,导出一个低阶的系统 ,在一定的条件下 ,此低阶系统与原系统的稳定性一致 ;最后 ,对约化后的低阶系统实施合适的局部状态反馈 ,使其局部渐近稳定 ,易知原非线性系统也局部渐近稳定 .从而实现了非线性系统的局部静态分叉控制 .

The local static bifurcation control problem in the bifurcation point neighbourhood of a class of nonlinear system was discussed. First, it prove that the local static bifurcation control is equal to the local stability of the critical nonlinear systems, so the local static bifurcation control problem can be transformed to the local stability problem; then the system can be simplified through Liapunov-Schmidt Method to be a low dimension system, whose stability is as same as that of the original system; and last the low dimension system was stabilized and so the local static bifurcation control of the original nonlinear system was realized.