Research On Fractional-Order Stochastic Resonance And Its Application In Signal Detection

Stochastic resonance(SR)is a nonlinear phenomenon occurred in stochastic dynamic systems.By virtue of the cooperation of three factors,i.e.,the input signal,noise and the dynamic system,an appropriate amount of noise can improve the output signal of the dynamic system,which is similar to the resonance effect in physics.This nonlinear phenomenon is called SR.Traditional SR means the resonance peak phenomenon when the system output performance varies with the noise strength.The SR in a broad sense means the nonmonotonous behavior of a maximum value when the system output signal varies with the parameters of the input signal,with the parameters of the dynamic system,and with the other noise parameters.The occurrence of a resonance peak is also called bona fide SR when the system output signal varies with the frequency of the input signal.In many engineering fields,signals have the properties of long correlation or local correlation,which can be described by fractional-order mathematical model.Fractional derivative has excellent characteristics such as dynamic memory of historical state,therefore many mathematical models are more suitable to be described by fractional derivative ones.In this thesis,based on the properties of fractional-order derivative and the statistical characteristics of noises,the SR phenomena in linear and nonlinear fractional-order oscillators are studied,and signal detection in noisy environment is investigated by virtue of the SR phenomenon in fractional-order linear oscillators.The research contents of this thesis are as follows:(1)The SR for two linear oscillators with two kinds of fractional-order derivatives subject to dichotomous noise is studied,the conditions of steady-state solution for the oscillators are given.The linear oscillator consists of two models: the first model is an oscillator with viscous damping and random characteristic frequency,and the second one with random characteristic frequency and signal modulation noise.For the oscillator with viscous damping and random characteristic frequency,stochastic multi-resonance phenomenon can be observed on the curve of the output amplitude gain(OAG)versus the first fractional derivative.The SR phenomenon can be observed on the curves of the OAG versus the second fractional derivative,versus the viscous damping,versus the two friction coefficients,versus the frequency of the driving signal,as well as versus the system characteristic frequency.For the oscillator with random characteristic frequency and signal modulation noise,SR appears on the relationship curves of the OAG versus the fractional exponents,versus the friction coefficient,versus the frequency of the driving signal,as well as versus the correlation rate of the multiplicative noise.The influence of the coupling noise strength between the additive and multiplicative noise on the OAG is discussed.The SR phenomenon for fractional-order linear oscillator with random delay and random damping is studied.The multiplicative noise of the oscillator is modled as a dichotomous noise.Based on the property of dichotomous noise,by using linear system theory and the small delay approximation,the expression of the OAG is obtained.The results show that OAG presents a resonance effect with the variety of the delay time,with the variety of the intensity and correlation of random delay,with the variety of the strength and correlation of random damping,as well as with the variety of the system characteristic frequency and the frequency of driving signal.Traditional SR,the SR in a broad sense,the stochastic multi-resonance and the bona fide stochastic multi-resonance are observed.The nonmonotonic effects of fractional exponent,of friction coefficient,of damping coefficient and of the coupling noise strength on the OAG are discussed.(2)The stochastic multi-resonance for a fractional-order linear oscillator with trichotomous noise is investigated,in which the characteristic frequency of the system is disturbed by a trichotomous noise,and the kernel function of the system being in the form of Mittag-Leffler function.Based on linear system theory,the mathematical expression of the output signal amplitude(SPA)of the system is derived,the necessary conditions for the system stability are given.Research results show that the SR phenomenon is observed on the curve of the SPA versus the noise amplitude,versus the memory time and memory exponent.The nonmonotonous influence of the steady-state probability of trichotomous noise,the influence of the frequency of driving signal and of the system characteristic frequency,as well as that of the viscous damping coefficient on the SPA is discussed.(3)The SR for a fractional-order nonlinear oscillator with square-wave signal and dichotomous noise is studied.Under the adiabatic approximation condition,based on the nonlinear SR theory,the expression of the output signal-to-noise ratio(SNR)for the system is obtained.The results show that there exists a single-peak when the SNR varies with the additive noise intensity.Three resonance peaks can be observed on the curves of the SNR versus the fractional exponent.SNR shows a resonant peak with slowly changing hat shape as the system changes.Meanwhile,the SNR is a nonmonotomous function of the multiplicative noise intensity.The effect of the noise rate and that of the amplitude of the driving singal on the SNR is discussed.(4)Finally,a physical experiment investigation of signal detection in noisy environment is carried out by virtue of the SR phenomenon in a fractional-order linear oscillator.Firstly,the SR effect of a fractional-order linear oscillator with random characteristic frequencies subject to additive noise are studied theoretically.Then,a 0.5-order linear oscillator with random characteristic frequency is designed,and two physical experiments are carried out to to enhance the performance of a CMI data stream in a noise environment and to diagnose failure for ATX power supply.For the CMI data stream enhancement experiment,by tuning the multiplicative noise intensity and the characteristic frequency of the system,the waveform quality of the output data stream and the decision accuracy of the output stream can be optimized,and the CMI stream can be completely and accurately determined.For the fault diagnosis experiment for ATX power supply,by selecting the appropriate multiplicative noise intensity and system characteristic frequency,the characteristic frequency corresponding to fault state for the power supply can be extracted from the output signal of fractional-order linear oscillator,therefore,the fault state for power supply can be diagnosed.