系统增益变化时状态变量反馈系统稳定性的鲁棒性问题

On the Robustness of Stability of State Variable Feedback System with the Original Systems Gain Varied

状态空间法是自动控制系统分析与设计的主要方法之一．状态变量反馈系统的突出优点是，对于完全可控与完全可观系统来说，它的闭环极点可以任意配置．同时，它与传统系统一样，当系统参数变化时，也存在着稳定性的鲁棒性问题．笔者按着矩阵理论与根轨迹法，探讨了ＳＩＳＯ状态变量反馈系统当原系统增益发生变化时的稳定性鲁棒性问题．记αｉ为原系统特征方程诸系数，ａｉ为状态变量反馈系统期望闭环特征方程诸系数，对于二阶系统，原系统增益变化时，状态变量反馈系统保持稳定的充要条件是α１＜ａ１，α０＜ａ２．对于高阶系统，保持稳定的必要条件是αｉ＜ａｉ（ｉ＝０，１，…，ｎ－１）．对于状态变量反馈系统的设计，如果这些条件不能得到满足，那么一个实际系统就有可能是不稳定的．

According to the matrix theory and root locus method,the paper studied the robustness problem for SISD state variable feedback system when the original systems gain varied.Supposing the sign a i (i=0,1,…,n-1)denotes the coefficients of the characteristic equation of the original system,and d i  denotes the coefficients of the expected characteristic equation of the closed loop system,a1<α 1 and a o<α o are the necessary and sufficient conditions for the second order system to keep stable when the gain of the original system varies,and a i<α i are the necessary conditions for the high order system to keep stable.When designing a state variable feedback system,the above mentioned conditions must be satisfied,otherwise the actual system will be unstable.