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

利用popov频率法,讨论了三阶直接控制系统dXdt=AX+bf(σ),σ=cTX零解的绝对稳定性,获得了A在cTb·trA2-cTA2b≤0的条件下,其零解绝对稳定的充分必要条件为cTb≤0,cTA-1b≥0;特别是当A=diag(-ρ1,-ρ1,-ρ2)时,零解绝对稳定的充分必要条件为cTb≤0,cTA-1b≥0。

In this paper, the absolute stability of 3-rd order direct control system dXdt=AX+(bf(σ),) σ=c^TX is discussed by means of Popov frequency method. The main results are: if A=(a_(ij))_(3×3), Reλ(A)<0, and c^Tb·trA^2-c^TA^2b≤0, the necessary and sufficient condition for absolute stability of the system is that c^Tb≤0, c^TA^(-1)b≥0; especially when A=(diag(-ρ_1, -ρ_1, -ρ_2),) the necessary and sufficient condition for absolute stability of the system is c^Tb≤0, (c^TA^(-1)b≥0.)