Control Of Bifurcation And Chaos Based On Equivalent-input-Disturbance Approach

Nonlinearities widely exist in many practical systems.On one hand,nonlinearities have bad influences on some practical systems(such as electromechanical system)and degrade the control performance.Thus,we reject and compensate for nonlinearities in such practical systems,and study on how to avoid nonlinear behaviors.On the other hand,nonlinearities are important for some practical systems(such as bifurcation in cell division)to accomplish certain missions.Thus,we retain and control nonlinearities in such practical systems,and study on how to control nonlinear behaviors.Furthermore,disturbances degrade the control performance.And big disturbances even destroy system structure.Thus,we have to consider the influences of disturbances on nonlinear system control and study how to control nonlinear systems with disturbances.However,most researchers focused on studying on rejecting nonlinear behaviors with disturbances and few focused on controlling nonlinear behaviors with disturbances.So,this dissertation not only studies on rejecting nonlinear behaviors with disturbances,but also originally studies on controlling nonlinear behaviors with disturbances.For chaos rejection,we used the equivalent-input-disturbance approach to rejecting the effects of disturbances and nonlinearities,and thus avoid the chaotic motion.For bifurcation and chaos control,we transformed the problem into three problems: 1.How to reject disturbances and retain nonlinearities;2.How to reconstruct and control bifurcation characteristics;3.How to reconstruct and control chaos characteristics.In this dissertation,bifurcation and chaos control under disturbance are studied in four steps:(1)Chaos rejection.This dissertation considers chaos in a PI control system for the first time and presents an equivalent-input-disturbance(EID)-based control method to suppress the chaotic phenomenon.Since the chaos is caused by two nonlinear terms,we take them to be disturbances and use two EID estimators to separately compensate for them,and thus suppress any possible chaos.Simulation results show that the method is effective not only to suppress chaos for the PI control system,but also improve the control performance.A comparison of the method with nonlinear feedback control,backstepping control,and impulsive control shows the superiority of our method.(2)Disturbance rejection in nonlinear systems.If nonlinearities are necessary and have to be retained,we cannot take them to be disturbances and have to retain them while reject exogenous disturbances.However,most researches cannot reject exogenous disturbances only.Thus,this dissertation extends the EID approach and presents a nonlinear equivalent-input-disturbance(NEID)approach to rejecting an unknown exogenous disturbance in a nonlinear system.(3)Bifurcation reconstruction and control with disturbances.If we want to control the pitchfork bifurcation destroyed by disturbances,we have to reconstruct the bifurcation first.Since nonlinearities and the bifurcation parameter are necessary parts for bifurcation,this dissertation designs a nonlinear term and a bifurcation parameter in the EID approach to reconstruct the pitchfork bifurcation.(4)Chaos control with disturbances.Compared with the chaos control without disturbances,the chaos control under disturbances is difficult.The difficulty of this problem lies in that it is necessary to preserve the chaotic property of the system while suppressing the disturbance.And the influence of initial value sensitivity and pseudo-randomness of the chaotic property on the control performance of the system should also be considered.This dissertation presents the parallel equivalent-input-disturbance(PEID)approach to rejecting an unsolved disturbance in the conventional EID approach.Thus,the disturbancerejection performance is improved.And then,we use the PEID approach to replace the conventional EID approach in the NEID approach for reconstructing and controlling chaos.Finally,the phase diagram is used to verify the effectiveness of the presented method.