非线性非保守系统的稳定性判据

Stability Criterion of Nonlinear Non-conservative System

对于用Lienard方程描写的非线性自治电路,采用基波分析法,在适当端口施加正弦电压源uS,求得注入网络电流的基波分量IS1=Um(Gi+jBi).令基波输入导纳(Gi,Bi)=(0,0);如果求出有一组合理的实数解(ωS,Um)∈R2,说明网络存在有周期振荡,相图显示有稳定极限环。根据等效推力理论,可以求出变阻尼力在一周期中贡献能量的等效平均值Df。可以证明Df的符号值代表iS1实功成份的流向,成为判定网络稳定性的依据。Df是振幅值Um的函数,它在零值平衡点邻域随Um的变化趋势,可以确定系统极限环的稳定性。振荡的自激保持说明一周期内注入网络的能量为零。极限环包围的面积代表网络内的总储能E,它在一周期内每一瞬间都在发生变化,但经历一周期后E保持不变。结论的普遍性可推广到三阶非线性方程。其正确性可用SIMULINK仿真验证。

For the nonlinear autonomous electronic circuit described by Lienard equation, we can find the fundamental harmonic component is1 = Um （Gi＋jBi） of the current of injected into networks by fundamental--wave analysi, when sinusoidal voltage source Us is applied to the prop- er port. Let first harmonic input admittance（Gi ,Bi）= （0,0） ,if a real field solution （ws ,Um ）∈R2 is obtained, it is illustrated that there exist the periodic oscillations in networks; and stable limit cycle can be shown in the phase portrait. The equivalent average value Df of energy contributed by the variational damping force in a cycle can be derived according to equivalent thrust theory. It can be proved that the sign of Df represents the flow direction of action component of is1. It will be- come the gist which decides the stability of network. Df is a function of Um, the stability of the limit cycle can be judged by the Df change trend with Um in a little zone near zero equilibrium point. When self-excited oscillation is maintained , it is illustrated that the energy injected into network in a cycle is equal to zero. The area encircled by limit cycle represents overall storage en- ergy E inside the network. It incessantly changes inside a cycle; but always maintains invariable- ness after experiencing each cycle. The universality of the conclusion can be spread to third-order nonlinear differential equation. Its correctness can be verified by the simulink.