三阶直接控制系统绝对稳定的充分必要条件

NECESSARY AND SUFFICIENT CONDITIONS FOR 3-RD ORDER ABSOLUTE STABILITY OF SOME DIRECT CONTROL SYSTEMS

本文利用Popov频率法，讨论了三阶直接控制系统 σ＝cTX零解的绝对稳定性，主要获得如下结果：1°．假设A＝（aij）3×3，Reλ（A）＜0，且cTb·trA2＋cTA2b≤0，cT（A-1）2b≤0，则其零解绝对稳定的充分必要条件是cTb≤0；cTA-1b≥0．2°．假设A＝（aij）3ｘ3，A的特征根均为负实数，cTb＝0，则其零解绝对稳定的充分必要条件为cTA2b≥0；cTA-1b≥0．

In this paper,the absolute stability of 3rd-order direct control system dX/dt-AX+bf(σ),σ=cTX is studied. The Popov frequency method is applied and the following main results are obtained:1°If A= (aij)3ｘ3,Reλ(A) <0, and cTb. trA2-cTA2b≤0,cT(A-1)2b≤0, then the necessary and sufficient condition for obsolute stability of the system is that cTb≤0,cTA-1b≥0. 2° If A = (aij)3×3,the characteristic roots of the matrix A are all negatire,and cTb=0, then the necessary and sufficient condition for obsolute stability of the system is that cTA2b≥ 0, cTAb≥0.