Fixed-Time Perturbation-Based Extremum Seeking Control Method

Perturbation-based extremum seeking control(PESC)is a model-free,real-time online adaptive optimization control method.The basic principle of PESC is that to obtain the approximate gradient information of the plant to be optimized by adding a perturbation signal on the input of the plant and multiplying the same frequency perturbation signal to the output of the plant.Then,the adaptive optimization control of the unknown function or the steady state input-to-output mapping of the system can be completed by the adaptive law formed by the obtained approximate gradient information.PESC is used in many fields including anti-lock braking system,aircraft formation flight,maximum power point tracking of new energy generation such as solar,wind,and fuel cells,because of its advantages of simple principle,lower computational complexity,and model-free.With the wide application of PESC,the better dynamics,steady state,and robust performance are required.However,the classical stability analysis method of PESC requires the system to have three time scales,and the convergence speed of the system depends on the slowest time scale.In addition,the gradient estimation method of PESC is difficult to be designed as a non-asymptotically convergent form,which greatly limits the performance improvement of PESC.Moreover,the problems of existing PESC such as steady state oscillations and falling into local extreme points also limit its scope of application.According to the above mentioned problems,the PESC is studied in this paper from the aspects of improved gradient estimation methods,design of fixed-time(FX)convergence,steady state oscillation elimination,etc.First,a gradient estimator that estimates the gradient of an unknown function based on a perturbed signal,perturbation-based gradient estimator(PGE),is proposed.After that,a FX design of the proposed PGE is carried out.Furthermore,a FX adaptive law is designed and combined with the FX-PGE,a FX-PESC is proposed.Finally,for the steady state oscillation problem of FX-PESC,a design of non-oscillating steady state for FX-PESC is performed.The detail content is as follows:From the basic principle of the classical PESC and Newton-based extremum seeking(ES),a perturbation-based gradient estimator(PGE)is proposed by analyzing the mechanism of classical PESC for gradient information estimation and generalized design of the gradient estimation method.Based on the PGE,an ES based on gradient estimatoris designed,and its stability is analyzed by applying small-gain theorem and singular perturbation technology.Different from the traditional PESC,the ES based on gradient estimator gives a quantitative relationship between the time scale of the perturbation signal and the adaptive gain,which provides a basis for the selection of parameters.Then,a Newton ES based on the gradient estimator is designed based on the above foundation,and the stability analysis is also given.In addition,the above-mentioned improved extremum seeking is applied to the maximum power point tracking of fuel cells.Finite-time(FT)and FX(non-asymptotic)convergence design for the proposed PGE is performed by generalized homogeneous approximation technique,and FT-PGE and FX-PGE are obtained respectively.The FT and FX convergence characteristics of the nonasymptotically convergent PGEs are analyzed by implicit Lyapunov function approach.For the existence of bounded disturbance and measurement noise,the non-asymptotic robustness of FT-PGE and FX-PGE is analyzed.The sufficient conditions of FT and FX input-to-state stability are first given for the FT-PGE and FX-PGE respectively.The sufficient conditions are also applicable to existing FT and FX state observers.A FX adaptive law is designed,and the FX input-to-output stability of it is analyzed.Furthermore,the existing FT small-gain theorem is extended to the case of FX,and combined with the FX-PGE proposed above,a FX-PESC method is designed.Based on the above FX small-gain theorem,the stability of the proposed FX-PESC is analyzed,and the effectiveness of the proposed FX-PESC method is verified by simulation examples and its simulation in anti-lock braking system.According to the problems of falling into local extreme points and steady state oscillation,an ES without steady state oscillation is proposed.The rigorous stability proof of the proposed ES is given using singular perturbation theory,averaging method,the center manifold theorem,and Lyapunov method.Based on the same design ideas,the proposed FX-PESC is designed as non-oscillating steady state.Numerical simulations and the simulation in anti-lock braking system validate the characteristics of non-oscillating steady state and the ability to overcome local extreme points of the proposed ES without steady state oscillation.Besides,the simulation results also illustrate the effectiveness of the non-oscillating steady state design for the fixed-time PESC.