Dynamic Analysis And Control Of Fractional-order Nonlinear Systems

With the rapid development of fractional calculus theory,its application in the field of engineering technology has been widely concerned by experts at home and abroad.The organic combination of fractional calculus and the classical nonlinear dynamic system can not only promote the in-depth study of fractional calculus theory but also promote the engineering application of the fractional-order nonlinear system.There are many kinds of vibration control devices in the suspension and traction drive system of high-speed vehicles,all of which have certain fractional-order characteristics,and the suspension and traction drive system often contains many nonlinear factors such as segmentation,friction,lag and so on.Therefore,the research on the dynamics and control of the fractional-order nonlinear system of the high-speed vehicles is of great theoretical significance and application value for the analysis of the operation quality and performance safety of the high-speed vehicle system.In this thesis,the dynamics and control of several kinds of fractional-order nonlinear systems are discussed based on previous research.The main work contents are as follows:(1)The research status of fractional-order nonlinear systems at home and abroad is described,and the related knowledge of fractional calculus and the main innovation points of this thesis are given.(2)The dynamic characteristics of the van der Pol system under fractional time-delayed feedback control are studied.The approximate analytical solution of the main resonance of the system is obtained based on the averaging method,and the amplitude-frequency equation for the steady-state solution is established.Then,the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed.It could be found that time delay periodically affects the amplitude-frequency curve of the system.(3)The subharmonic resonance of Mathieu-Duffing oscillator with fractional derivative subjected to external harmonic excitation is investigated.Based on the KBM asymptotic method,the approximate analytical solution for the subharmonic resonance under parametric-forced jointed excitation is obtained,and the stability criterion of the steady-state solution is presented.The influence of the fractional-order term and excitation amplitude on the subharmonic resonance response of the system is analyzed.(4)Based on the stability theory of the fractional dynamical system,the stability of fractional van der Pol system and Rayleigh system with time-delayed feedback are analyzed respectively.Regarding time delay as a bifurcation parameter,the existence conditions of Hopf bifurcation of these two kinds of systems are obtained.Under some conditions,the critical value of time delay is calculated.(5)The chaotic threshold of the Duffing system with fractional-order derivative subjected to external harmonic excitation is investigated by the Melnikov method.An approximate integer-order system equivalent to the fractional-order Duffing system is obtained via the harmonic balance method,and they have the same amplitude-frequency response equation.The chaotic boundary conditions of the integer-order system are obtained by constructing the Melnikov function to predict the chaotic behavior of the fractional-order Duffing system.Finally,the main conclusions of the thesis are summarized,and some existing problems and future research are pointed out.