Study On Fractional Order System Approximation And Estimation Algorithm

The appearance of fractional calculus has greatly extended and expanded the field of calculus theory.In recent years,the useful properties of fractional calculus have been discovered in many scientific problems and engineering phenomena,which has promoted more and more researchers to use this theory to analyze and study various scientific problems and apply them in various fields,such as system modeling,con-troller design,biomedical and signal processing field.Although the theory of fractional calculus has been born for more than 300 years,the in-depth study of fractional systems needs further development.At present,the scientific research community is far from enough understanding of the basic units of fractional order systems.The properties of fractional order systems constructed by using such basic units are very unstable,and it is difficult to use them to study specific problems.Therefore,how to obtain a sta-ble,easy-to-build and simulated reference-level fractional order system is a key issue in the research.Some studies have shown that an integer order system can be used to approximate the basic unit of a fractional order system,but related researches are far from sufficient and can still be improved and has high research value.In addition to the construction of fractional order systems,the research on complex fractional order systems also needs further development.It is also one of the important directions in the research of fractional order control theory to estimate the unknowns of the system by using the noisy output of the system.The estimated results can be used to analyze systems performance and controller synthesis.However,fractional-order systems have complex dynamic characteristics and infinite-dimensional system charac-teristics.At the same time,because the development of the basic theory of fractional calculus is not comprehensive enough,in actual calculations,problems that singular integrals cannot be processed and high computational complexity are often encoun-tered.This adds a lot of difficulties to system.Therefore,this work has very important research value,but also a great challenge.In summary,this paper takes the solution to the above problems as the starting point,proposes a high-precision approximation scheme for fractional order systems,and improves some calculation methods in the basic theory of fractional calculus and applies them to a class of complex fractional system input and estimation of fractional derivative of the output.Firstly,this dissertation has made an in-depth study on the approximation of the fractional diferential operator s?,and specifically analyzed the relationship between the composition of the approximation system and the size of ?,and obtained a more generalized approximation form.The phase of the approximation system is analyzed,and the approximation accuracy is improved by optimizing the phase,and on this basis,an approximation scheme with a simpler calculation is proposed.Secondly,this dissertation further explores some basic theorems in the theory of fractional calculus,and puts forward a new calculation formula of fractional calculus,which avoids the defects of traditional definition in application.Thirdly,for a class of complex fractional-order linear system input estimation and the estimation of nonlinear terms in nonlinear systems,this dissertation proposes a fractional-order sliding window method.Different from the integer order system,frac-tional order system has the characteristic of long memory,that is,the value at each time point is closely related to the historical value,so the traditional sliding window method can not be directly applied to the estimation of fractional order system.This disser-tation uses modulation functions to construct a new sliding window method,which can effectively cope with the long memory characteristics of fractional-order systems,thereby accurately estimating the required input.Since the use of modulation func-tion to construct the recursive algorithm can avoid the influence of truncation error on the final result,this paper uses the modulation function method to first estimate a se-ries of commensurate derivatives of output of the fractional linear system,derives an effective estimation algorithm,and uses Chebyshev's inequality analyzes the param-eter optimization problem under noise interference and improves the accuracy of the method.Fourthly,estimating the fractional derivative of the output of such complex sys-tems is also one of the focuses of this dissertation.Finally,on the basis of completing the above-mentioned derivative estimation,this dissertation continues to expand the estimation range,derives the estimation method of the output derivative of any order,and extends the estimated system from a linear system to a nonlinear system,and obtains a good approximation result.To sum up,this dissertation firstly optimizes the simulation method of the frac-tional order system.Secondly,it optimizes some basic theories of fractional calculus.Based on these two works,the application of the modulation function method in the complex fractional system is improved.This paper makes innovations in three aspect-s:basic tools,basic theories and basic methods in the field of fractional order system research.