宽带噪声激励下带有分数阶控制器的强非线性系统的随机平均技术

STOCHASTIC AVERAGING TECHNIQUE OF STRONGLY NONLINEAR SYSTEMS WITH FRACTIONAL-ORDER CONTROLLER UNDER THE WIDE-BAND NOISE EXCITATION

运用随机平均法研究了宽带噪声激励下带有分数阶PI~λD~μ控制器的强非线性系统.首先,应用广义谐波平衡技术,将分数阶PI~λD~μ控制力分解为幅值依赖的等效拟线性阻尼力和拟线性回复力,得到了受控整数阶等效非线性系统.然后,运用基于广义谐和函数的随机平均法得到关于幅值的平均伊藤微分方程.最后,建立并求解相应的简化Fokker-Planck-Kolmogorov(FPK)方程,得到稳态概率密度函数.作为算例,考察了Duffing-van de Pol振子.数值结果表明随机平均法能够达到较高的精度,分数阶PI~λD~μ控制器能够对系统响应进行有效的控制.此外,宽带噪声参数ξ_i、ω_i及D_i改变时,本文提出的方法仍具有较好的适用性,分数阶控制器仍同样具有非常好的控制效果.

The stochastic averaging technique is used to study the strongly nonlinear systems with fractional-order PIλDμcontroller under wide-band noise excitation.Firstly,the generalized harmonic balance technique is applied to decouple the fractional order PIλDμcontrol force into the amplitude-dependent equivalent quasi-linear damping force and restore force,and the original system is replaced by a controlled equivalent nonlinear system.Then,the averaged It■differential equation for the amplitude is derived by using the stochastic averaging method based on the generalized harmon⁃ic function.Finally,the stationary probability density function is obtained from the reduced Fokker-Planck-Kolmogorov(FPK)equation.As an illustrative example,Duffing-van de Pol oscillator is investigated.Numerical results display that stochastic averaging method can yield the high precision,and the fractional order PIλDμcontroller can control the system response effectively.Furthermore,the study finds that the proposed method can work well and the fractional order PIλDμcontroller still has very good control effect for the change of wide-band noise parametersξi,ωi,and Di.