Fractional Calculus And Its Applications To Viscoelastic Material And Control Theory

"Mathematics is the art of giving things misleading names.The beautiful-and at first look mysterious-name 'the fractional calculus' is just one of the those misnomers which are the essence of mathematics."——Igor Podlubny,1999.To date,the most extensive applications of fractional differential and integral operators is its applications to the viscoelastic theory.It is mentioned in many literatures that the use of fractional derivatives for the mathematical modeling of viscoelastic materials is quite natural. It should be mentioned that the main reasons for the theoretical development are mainly the wide use of fractional materials in various fields of engineering.Almost all of the fractional approaches to generalizations of the laws of deformation have been found useful for solving practical problems of viscoelasticity,if the results are properly interpreted[1].Based on these theoretical and practical researches and motivated by the four properties of viscoelastic materials,relaxation,creep,precondition and hysteresis,Prof.Mingyu Xu and I developed some new kinds of mathematical descriptions of viscoelastic phenomena,which can guide directly the material experiments and enrich the current results.The other part of this dissertation is the application of fractional calculus to the control theory.At present,motivated by some fractional physical phenomena,a growing number of works by many authors from various fields deal with fractional constitutive relationships and fractional systems which means equations involving derivatives and integrals of non-integer order.These new models are more adequate than the previously used integer-order models. Moreover,fractional-order derivatives and integrals provide a powerful instrument for the description of memory and hereditary properties of different substances.This is the most significant advantage of the fractional-order models and why fractional viscoelastic models get so many successful applications in reality.However,because of the absence of appropriate mathematical methods,fractional-order dynamical systems were studied only marginally in the theory and practice of control systems.There do have some successful attempts,but generally the study in the time domain has been almost avoided[1].For filling this gap, we(my advisors,cooperators and I) published a series of fundamental theoretical papers, which improved directly and potentially the theory and applications of fractional calculus. First,we derive the fractional Lyapunov direct method,which is a sufficient condition of asymptotic stability of systems.The definition of Mittag-Leffer stability is proposed that extends the classic exponential stability.Second,the theory of fractional Universal Adaptive Stabilization(UAS) is studied.Fractional viscoelastic theory is added to the fractional UAS.Accompany with the theory of fractional UAS,a number of experimental results we did indicate the advantages of fractional UAS,which are omitted in this thesis for the continuity and readability.Third,during our research on fractional UAS,we answered "When (generalized) Mittag-Leffler function is Nussbaum function?",which provides a new type of Nussbaum function.By using these new Nussbaum functions,we obtained many good experimental and simulation results.Lastly,the above researches give us lots of sparks and hints for our future works.Finally,it is possible that in the future there will appear more "fractional order" physical theories.We would like to end the statement part of this thesis with the following quotation:"...all systems need a fractional time derivative in the equations that describe them... systems have memory of all earlier events.It is necessary to include this record of earlier events to predict the furore...The conclusion is obvious and unavoidable:'Dead matter has memory'.Expressed differently, we may say that Nature works with fractional time derivatives."——S.Westerlund, 1991.The dissertation is divided into six Chapters.In Chapter 1,we introduce the basic definitions and properties of Riemann-Liouville and Caputo fractional operators.Moreover,we also present the Mittag-Leffler function, Mittag-Leffler function in two parameters,Generalized Mittag-Leffler function and Mittag-Leffler function in four parameters,which are the elementary solutions of many fractional differential equations.At the end of this chapter,we give the definition and infinite summations of the Fox H-function,which includes nearly all the special functions occurring in applied mathematics and statistics as its special cases,such as Mittag-Leffler type function, generalized hypergeomatric function,generalized Bessel function,Meijer's G-function and so on.The contents of this part can be a concise introduction and collection related to the basic knowledge of fractional calculus and its applications.Moreover,all the definitions and properties of this section are used in the following parts,which give the readers a better understanding of this dissertation.In Chapter 2,we first study the hysteresis and precondition of Kelvin models under the condition of loading and unloading of saw-tooth wave.Then we prove out a series of sufficient conditions of models and inputted strains for producing hysteresis and precondition in this paper.We discuss the monotonic and limit property of hysteresis and precondition.We prove that the hysteresis loop's width decreases with increase of period and is greater or equal to zero.At the same time we prove that for the fractional-order Kelvin model and under the condition of quasi-linear theory the above conclusions also hold.In this part,we are the first group who mathematically describe the hysteresis and precondition of fractional order viscoelastic solid models.This work gives us a theoretical guidance to the material experiments.Moreover,based on the experimental data of different materials,we can also adjust our model parameters,which extends the applications of our results.In Chapter 3,we discuss the process of changing and the tendency of hysteresis loop and energy dissipation of viscoelastic solid models.One of our conclusions is that under certain conditions,the sign of(3.4.15) is a sufficient and necessary condition for judging the sign of the difference between dissipated energy in the(n+1)th period and nth period.We have proved that the Boltzmann superposition principle also holds for inputted strain being constant on some domains.We prove that for the fractional-order Kelvin model and under the condition of quasi-linear theory the above conclusions also hold.Energy dissipation is an unavoidable phenomenon during the motion of the real materials. Especially under the periodic load,the tendency and limit properties of the energy dissipation becomes a meaningful work,which is also the main content of this section.Based on both of the integer order and fractional order material models,superposition principle and quasi-linear theory,this problem is well solved in this section,which is in hope of guiding the real experiments.In Chapter 4,we propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method.Then we propose the fractional comparison principle. Third,we extend the application of Riernann-Liouville fractional systems by using Caputo fractional systems.Finally,an illustrative example is provided as a proof of concept.The new definition,Mittag-Leffler stability,is an extension of the classical exponential decay.It can also describe the stability of both integer order and fractional order systems in a more generalized way.In Chapter 5,we study the asymptotic stability of three fractional scalar systems by using the method of universal adaptive stabilization:(Ⅰ) Fractional dynamics with integer-order control strategy,(Ⅱ) Fractional dynamics with fractional control strategy,and(Ⅲ) Application of fractional viscoelastic and electromagnetic theories in universal adaptive stabilization. It is shown that the control efforts for these three cases are bounded.Then we show that Mittag-Leffler function E_α(—λt~α) forα∈(2,3]andλ＞O is a Nussbaum function. Finally,several illustrated simulation results are provided as a proof of concept.The method of universal adaptive stabilization seems like a miracle that it can guarantee the stability of system without knowing exactly the system parameters.This theory is well developed in the integer order systems.But,its application to the fractional order system is still a new topic.Moreover,the Mittag-Leffler function as the Nussbaum fimction has already been proved to be a worthy work in both theory and experiment.In Chapter 6,we establish the relationships between Generalized Mittag-Leffler function and Nussbaum function.Moreover,the derivatives of Generalized Mittag-Leffler functions, which are Nussbaum functions are also Nussbaum functions.The Matlab figures are plotted by the M-file compiled by Dr.YangQuan Chen.Lastly,the determinant of Hessenberg matrix is also discussed.This work is meaningful because we get many very good results by applying the Generalized Mittag-Leffler function as the Nussbaum function to the universal adaptive stabilization experiments on DC motor,which is not included in this dissertation for continuity. Moreover,some of the classical results are included in this part.