摘要
本文论讨有速度反馈的继电器控制线性系统的周期振荡(极限环)问题。这类系统有两条与x轴平行的开关线,将相平面分成三部分。其轨线由 x=y y=-q(x±(r╱q))-py (1)确定,其中p^2<4q,r>0。令,并记x轴至开关线的距离为a。 我们用点变换法证明 1.当p>0,a>0时,存在一个正数r_1>0,r≥r_1,则在整个相平面只有一个不稳定环,r<r_1,无环。 2.当p<0,a>0时,存在r_2>0,如r>r_2,则存在两环Γ_1和Γ_2,Γ_1Γ_2,Γ_1是稳定环,Γ_2是不稳定环。r=r_2,只一个不稳定环;r<r_2时无环。
The paper discusses the Problem of Periodic oscillation(limi t cycle) of a relay-controlled linear system with rate-feedback.This system has two switching lines,parallel to the x-axis and equidistant from it,thus dividing the phase plane into three parts.Its trajectories are defined bywhere and let the distance from x-axis toeach of the switching lines be denoted by a.By use of the method of point-to-point transformation we prove1.If P>0,a>0,then there exists a positive number r1,such that when ≥r1i there will only be one unstable cycle on the entire Phase plane,and when r<r1,there will be no cycle2.If p<0,a>0 then there exists a positive number r2,such that there are two cycles Γ1 and Γ2 on the phase plane,Γ1 Γ2,when r>r2,Γ1 is stable and Γ2 unstable.But system(1) has only one unstable cycle when r=r2 and no cycle when r<r2
