The Research On Self-oscillation Of Closed-loop Control System Based On Е.П.Попов Harmonic Linearization

Closed-loop control system is nonlinear system in practical engineeringapplication and self-oscillation is the important characteristic. Therefore it is veryimportant to ensure the system operating regularly that put self-oscillation undercontrol. Phase plane method and describing function method are two kinds of classicalresearch methods for self-oscillation. Both of them have limitation for the research ofself-oscillation. Based on the principles of singularity， the phase plane method dealswith differential equation of second order to solve in most cases qualitative question.While the describing function method could be applied to solve the higher orderdifferential equation and deal with quantitative problem by linearization of nonlinearelement with only one sine wave signal as input. Therefore the describing functionmethod is more suitable for self-oscillation research of closed-loop control system. Atthe same time it is noticeable that external input (referring mostly to slowly varyinginput) apart from sine wave signal as input of nonlinear element always brings aboutsignificant benefits to self-oscillation control of closed-loop system, so we shouldlook for a certain kind of method satisfied with the self-oscillation research with orwithout external input. Е.П.Попов harmonic linearization proposed by Е.П.Поповfrom former Soviet Union meets the requirement, which combines the constantcomponent coming from slowly varying input and periodic component coming fromsine wave signal as self-oscillation value.Firstly this paper discussed the research of describing function method astheoretical foundation of Е.П.Попов harmonic linearization. Especially the criterionsarising from characteristic root criterion of linear system used by describing functionmethod for stabilities of self-oscillation and nonlinear system were presented. Phasestability criterion, Mihajlovic’s criterion and Hurwitz’s criterion were included.Secondly the practical examples were given to illustrate the concrete usage ofЕ.П.Попов harmonic linearization. This section mainly included two parts accordingto the reasons for generating constant component of self-oscillation. One was externalinput; the other was non-odd symmetric nonlinear characteristic. Thirdly the paperstudied the typical self-oscillation parameters of both symmetric nonlinearcharacteristic and asymmetric nonlinear characteristic, which provided detailedcalculation and diagramming process of amplifying factor equation and constant component equation. It could be used as a reference in the engineering self-oscillationcomputation. In the following sections the paper introduced DC mine hoist systems asclosed-loop control system with self-oscillation solved by Е.П.Попов harmoniclinearization. The V-M system (thryristor-motor mine hoist system) was chosen astypical DC mine hoist system because thyristor and amplifying element of speed-loopcoexisted in the V-M system as the oscillation elements which rarely discussed in theformer research. Calculation of self-oscillation parameters by Е.П.Попов harmoniclinearization, stability determination with phase stability criterion and Mihajlovic’scriterion and adjustment with system parameters to self-oscillation were core contentsin the discussion. MATLAB, Mathematica and MAPLE were used for solution andtaking deviation of high-order differential equation, which made up for theover-complication of computation of Е.П.Попов harmonic linearization. The codeand procedure were proved to be helpful in the simulation and calculation.To develop further research toward self-oscillation of more than one nonlinearelement in a certain system, an electrical motorcycle model with permanent magnetbrushless DC motor (PMBLDCM) was constructed in an upcoming chapter. Becausethe current loop and speed loop undertake different roles in the operation ofPMBLDCM at different stages, they would generate self-oscillation at different timeinstead of the same time. Therefore Е.П.Попов harmonic linearization was applied tosolve nonlinear oscillation issues for each of them. These contents were be helpful forthe self-oscillation research of multi-components in the future. Finally the paper madean expansion of Е.П.Попов harmonic linearization to analysis the absolute stability ofan aircraft yaw angle control system. The Lyapunov direct method was introduced toobtain absolute stable condition. Considering the self-oscillation represents thecritical stable condition of nonlinear system, the popov stability criterion was appliedto simulate the absolute stability with variants of system parameter settingsconsidering the maximum value of amplifying factor provided by Е.П.Поповharmonic linearization. Simulation results indicated that the absolute stable conditioncould guarantee the stability of the system.Based on the contents above this paper proposed that the research on theself-oscillation caused by multiple nonlinear components at the same time should beexpanded using Е.П.Попов harmonic linearization. Although the process would bemore diffcult the results will be more consistent with the complexity of nonlinearsystem. To promote the Е.П.Попов harmonic linearization in engineering research weshould introduce more advanced algorithm into this method. Especially the data analyzing and mining for the prediction of self-oscillation should be brought intoimportant consideration.