分数阶微积分及其在粘弹性材料和控制理论中的应用
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摘要
“数学是给予万物高深莫测的别名的艺术。而‘分数阶微积分'这一美丽的-乍听起来有些神秘的一名称正是这些别名中的一个。它同时也是数学理论的精华.”——Igor Podlubny,1999。
到目前为止,分数阶微分算子和分数阶积分算子在粘弹性理论中得到了最广泛的应用。许多文献中提到应用分数阶微分作为粘弹性材料的数学模型是件很自然的事情。值得一提的是,分数阶理论之所以在粘弹性材料建模上得到如此大的发展的主要原因是分数阶材料在工程领域的广泛应用。并且,只要给予适当的假设,几乎所有的分数阶模型都能很好的归纳材料的变形特征,并对解释实际问题起着相当大的作用。基于这些理论和实践的研究,并且考虑到粘弹性材料的四大特征,松弛、蠕变、预调和滞后,徐明瑜教授和我提供了一种新的粘弹性材料特性的数学描述并将其作为本学术论文的第一大部分内容。这些理论可以直接指导各种材料试验并且丰富了现有的结果。
这篇博士论文的另一部分阐述的是分数阶微积分在控制理论中的应用。目前,受许多物理现象的启发,世界范围内的学者们应用含有分数阶导数和积分的分数阶本构关系和分数阶系统方程在不同领域做出了越来越多的贡献。这些新模型比起经典的整数阶模型更胜一筹。另外,分数阶导数和积分为描述不同事物的记忆性和遗传性提供了一个有力的工具。这是分数阶模型最明显的特征之一。同时,也是分数阶粘弹性模型在实际中得到如此成功实践的原因之一。尽管如此,由于缺少相应的数学工具,分数阶动力系统的研究只占据了控制系统理论和应用很小的一部分空间。尽管在分数阶控制领域也有一些成功的尝试,但总的来说在时域中的讨论几乎被避开了。为了填补这项空白,我们(我的导师,合作者和我自己)发表了一系列这方面的学术论文并希望这些有关基础理论的文章有助于分数阶微积分在理论和应用方面的发展。首先,我们导出了分数阶Lyapunov直接方法,它是系统渐进稳定的充分条件。同时,我们还提出了Mittag-Leffler稳定的概念,它是指数稳定的广义形式。另外,我们还研究了分数阶通用自适应稳定理论。基于这些理论,我们还做了一系列的有很好结果的试验,但基于本文的连贯性和可读性,这一部分被省略掉了。第三,在对分数阶通用自适应稳定问题的研究中,我们回答了“什么时候(广义)Mittag-Leffler函数是Nussbaum函数?”。这一理论提供了一类全新的Nussbaum函数。通过应用这一类新的函数,我们得到了许多成功的试验和仿真结果。最后,在对上述问题的研究中,我们得到了许多灵感和启示,丰富了我们下一步的研究内容。
总之,在未来的时间里将会涌现出更多的“分数阶”物理问题。我们引用下面的话来作为本文陈述部分的结束:
“…所有的系统需要一个含有分数阶导数的方程来描述…系统对所有之前发生的事情都有记忆性。用发生过的事情来预测未来是有必要的。…
这个结论是明显且不可避免的:‘逝去的事物仍有记忆'。换句话说,一切自然事物都是分数阶的。”——S.Westerlund,1991。
本学位论文的正文部分共分六章:
在第一章中,我们介绍了Riemann-Liouville和Caputo分数阶算子的基本概念和性质。另外,我们还列出了Mittag-Leffler函数,两参数Mittag-Leffler,广义Mittag-Leffler函数(三参数Mittag-Leffler函数)和四参数Mittag-Leffler函数的基本定义。这些函数是许多分数阶微分方程的基本解。此外,我们给出了Fox H-函数的定义和无穷项级数表示。这种函数包含了几乎所有在应用数学和统计学领域被广泛应用的特殊函数:Mittag-Leffler类型函数,广义超越几何函数,广义Bessel函数和Meijer's G-函数,等等。在本章的最后,我们对分数阶微积分及其在粘弹性材料和控制理论中应用的发展作了精要概括。
本章内容是一个分数阶微积分及其应用的基本介绍和精要概括。所有的这些定义和内容都在随后的各个章节中得到了应用,并且,这些定义和特殊函数的性质使本文更易于读者理解。
在第二章中,我们首先证明了整数阶Kelvin模型在锯齿波加载下的滞后和预调。随后,我们证明了产生滞后和预调的模型和输入的充分条件。我们还讨论了滞后和预调的单调和极限性质。我们证明了滞后回环的宽度单调递减趋于非负值。同时我们还证明了,应用分数阶Kelvin模型和准线性理论以上结果仍然成立。
我们是第一个给出分数阶粘弹性固体材料滞后和预调的数学描述的科研小组。这一工作给许多材料学试验提供了理论依据和支持。另外,基于各式各样真实的材料试验,我们可以通过调整模型参数来更好的描述和指导分数阶模型在更大范围内的应用和实践。
在第三章中,我们讨论了粘弹性固体材料模型滞后回环和能量损耗的变化和趋势。我们的结论之一是:(3.4.15)式的正负是判断第(n+1)周期和第n个周期能量损耗变化的充要条件。我们还证明了Boltzmann叠加原理对于周期输入的适用性。应用分数阶Kelvin模型和准线性理论,本章中上述结论仍成立。
在真实材料的运动中,能量损耗几乎是不可避免的现象。尤其是在周期加载下能量损耗的趋势和极限性质是一项非常有意义的工作,这也是本章的主要内容。基于整数阶模型和分数阶模型,叠加原理和拟线性原理,这一问题得到了较好的回答并希望其结论有助于指导实际材料试验。
在第四章中,我们提出了Mittag-Leffler稳定的定义和分数阶里亚普诺夫直接方法。我们还证明了分数阶比较原理。另外,应用Caputo分数阶微积分算子,我们扩展了Riemann-Liouville分数阶系统的应用。为了支持这些理论,我们还提供了一个实例。
Mittag-Leffler稳定这一新定义是经典的指数衰减的扩展。它还可以在更广泛的意义下来描述整数阶和分数阶系统的稳定性理论。
在第五章中,应用分数阶通用自适应稳定理论,我们研究了三种分数阶标量系统的稳定性问题。这三种系统是:(Ⅰ)分数阶动力,整数阶控制;(Ⅱ)分数阶动力,分数阶控制;(Ⅲ)含有分数阶粘弹性材料和分数阶电磁力学模型的通用自适应稳定问题。文章证明了这三种情况下控制量的有界性。另外,我们还首次证明了对于Mittag-Leffler函数E_α(—λt~α)当α∈(2,3]和λ＞0时是Nussbaum函数。几个仿真结果验证了以上的理论。
通用自适应稳定理论看似一件神奇的事情,因为,它可以在不知道系统参数的情况下保证系统的稳定性.这一理论在整数阶系统中已经得到了完善的发展。但是,它在分数阶系统中的应用还是一个新课题。另外,我们已经证明了应用当Mittag-Leffler函数是Nussbaum函数时这一理论在理论和实验中都是非常有意义的工作。
在第六章中,我们建立了广义Mittag-Leffler函数和Nussbaum函数之间的关系。另外,当广义Mittag-Leffler函数是Nussbaum函数时,这些广义Mittag-Leffler函数的导数仍然是Nussbaum函数。其中的Matlab图像应用了YangQuan Chen老师编辑的M-文件。最后,我们讨论了Hessenberg矩阵的行列式问题。
这一工作意义非常,因为,当我们把“(广义)Mittag-Leffler函数是Nussbaum函数”应用到直流永磁电动机的通用自适应稳定实验中时,得到了非常好的结论。出于连贯性的考虑,这一部分并没有出现在本文中。
"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.
到目前为止,分数阶微分算子和分数阶积分算子在粘弹性理论中得到了最广泛的应用。许多文献中提到应用分数阶微分作为粘弹性材料的数学模型是件很自然的事情。值得一提的是,分数阶理论之所以在粘弹性材料建模上得到如此大的发展的主要原因是分数阶材料在工程领域的广泛应用。并且,只要给予适当的假设,几乎所有的分数阶模型都能很好的归纳材料的变形特征,并对解释实际问题起着相当大的作用。基于这些理论和实践的研究,并且考虑到粘弹性材料的四大特征,松弛、蠕变、预调和滞后,徐明瑜教授和我提供了一种新的粘弹性材料特性的数学描述并将其作为本学术论文的第一大部分内容。这些理论可以直接指导各种材料试验并且丰富了现有的结果。
这篇博士论文的另一部分阐述的是分数阶微积分在控制理论中的应用。目前,受许多物理现象的启发,世界范围内的学者们应用含有分数阶导数和积分的分数阶本构关系和分数阶系统方程在不同领域做出了越来越多的贡献。这些新模型比起经典的整数阶模型更胜一筹。另外,分数阶导数和积分为描述不同事物的记忆性和遗传性提供了一个有力的工具。这是分数阶模型最明显的特征之一。同时,也是分数阶粘弹性模型在实际中得到如此成功实践的原因之一。尽管如此,由于缺少相应的数学工具,分数阶动力系统的研究只占据了控制系统理论和应用很小的一部分空间。尽管在分数阶控制领域也有一些成功的尝试,但总的来说在时域中的讨论几乎被避开了。为了填补这项空白,我们(我的导师,合作者和我自己)发表了一系列这方面的学术论文并希望这些有关基础理论的文章有助于分数阶微积分在理论和应用方面的发展。首先,我们导出了分数阶Lyapunov直接方法,它是系统渐进稳定的充分条件。同时,我们还提出了Mittag-Leffler稳定的概念,它是指数稳定的广义形式。另外,我们还研究了分数阶通用自适应稳定理论。基于这些理论,我们还做了一系列的有很好结果的试验,但基于本文的连贯性和可读性,这一部分被省略掉了。第三,在对分数阶通用自适应稳定问题的研究中,我们回答了“什么时候(广义)Mittag-Leffler函数是Nussbaum函数?”。这一理论提供了一类全新的Nussbaum函数。通过应用这一类新的函数,我们得到了许多成功的试验和仿真结果。最后,在对上述问题的研究中,我们得到了许多灵感和启示,丰富了我们下一步的研究内容。
总之,在未来的时间里将会涌现出更多的“分数阶”物理问题。我们引用下面的话来作为本文陈述部分的结束:
“…所有的系统需要一个含有分数阶导数的方程来描述…系统对所有之前发生的事情都有记忆性。用发生过的事情来预测未来是有必要的。…
这个结论是明显且不可避免的:‘逝去的事物仍有记忆'。换句话说,一切自然事物都是分数阶的。”——S.Westerlund,1991。
本学位论文的正文部分共分六章:
在第一章中,我们介绍了Riemann-Liouville和Caputo分数阶算子的基本概念和性质。另外,我们还列出了Mittag-Leffler函数,两参数Mittag-Leffler,广义Mittag-Leffler函数(三参数Mittag-Leffler函数)和四参数Mittag-Leffler函数的基本定义。这些函数是许多分数阶微分方程的基本解。此外,我们给出了Fox H-函数的定义和无穷项级数表示。这种函数包含了几乎所有在应用数学和统计学领域被广泛应用的特殊函数:Mittag-Leffler类型函数,广义超越几何函数,广义Bessel函数和Meijer's G-函数,等等。在本章的最后,我们对分数阶微积分及其在粘弹性材料和控制理论中应用的发展作了精要概括。
本章内容是一个分数阶微积分及其应用的基本介绍和精要概括。所有的这些定义和内容都在随后的各个章节中得到了应用,并且,这些定义和特殊函数的性质使本文更易于读者理解。
在第二章中,我们首先证明了整数阶Kelvin模型在锯齿波加载下的滞后和预调。随后,我们证明了产生滞后和预调的模型和输入的充分条件。我们还讨论了滞后和预调的单调和极限性质。我们证明了滞后回环的宽度单调递减趋于非负值。同时我们还证明了,应用分数阶Kelvin模型和准线性理论以上结果仍然成立。
我们是第一个给出分数阶粘弹性固体材料滞后和预调的数学描述的科研小组。这一工作给许多材料学试验提供了理论依据和支持。另外,基于各式各样真实的材料试验,我们可以通过调整模型参数来更好的描述和指导分数阶模型在更大范围内的应用和实践。
在第三章中,我们讨论了粘弹性固体材料模型滞后回环和能量损耗的变化和趋势。我们的结论之一是:(3.4.15)式的正负是判断第(n+1)周期和第n个周期能量损耗变化的充要条件。我们还证明了Boltzmann叠加原理对于周期输入的适用性。应用分数阶Kelvin模型和准线性理论,本章中上述结论仍成立。
在真实材料的运动中,能量损耗几乎是不可避免的现象。尤其是在周期加载下能量损耗的趋势和极限性质是一项非常有意义的工作,这也是本章的主要内容。基于整数阶模型和分数阶模型,叠加原理和拟线性原理,这一问题得到了较好的回答并希望其结论有助于指导实际材料试验。
在第四章中,我们提出了Mittag-Leffler稳定的定义和分数阶里亚普诺夫直接方法。我们还证明了分数阶比较原理。另外,应用Caputo分数阶微积分算子,我们扩展了Riemann-Liouville分数阶系统的应用。为了支持这些理论,我们还提供了一个实例。
Mittag-Leffler稳定这一新定义是经典的指数衰减的扩展。它还可以在更广泛的意义下来描述整数阶和分数阶系统的稳定性理论。
在第五章中,应用分数阶通用自适应稳定理论,我们研究了三种分数阶标量系统的稳定性问题。这三种系统是:(Ⅰ)分数阶动力,整数阶控制;(Ⅱ)分数阶动力,分数阶控制;(Ⅲ)含有分数阶粘弹性材料和分数阶电磁力学模型的通用自适应稳定问题。文章证明了这三种情况下控制量的有界性。另外,我们还首次证明了对于Mittag-Leffler函数E_α(—λt~α)当α∈(2,3]和λ＞0时是Nussbaum函数。几个仿真结果验证了以上的理论。
通用自适应稳定理论看似一件神奇的事情,因为,它可以在不知道系统参数的情况下保证系统的稳定性.这一理论在整数阶系统中已经得到了完善的发展。但是,它在分数阶系统中的应用还是一个新课题。另外,我们已经证明了应用当Mittag-Leffler函数是Nussbaum函数时这一理论在理论和实验中都是非常有意义的工作。
在第六章中,我们建立了广义Mittag-Leffler函数和Nussbaum函数之间的关系。另外,当广义Mittag-Leffler函数是Nussbaum函数时,这些广义Mittag-Leffler函数的导数仍然是Nussbaum函数。其中的Matlab图像应用了YangQuan Chen老师编辑的M-文件。最后,我们讨论了Hessenberg矩阵的行列式问题。
这一工作意义非常,因为,当我们把“(广义)Mittag-Leffler函数是Nussbaum函数”应用到直流永磁电动机的通用自适应稳定实验中时,得到了非常好的结论。出于连贯性的考虑,这一部分并没有出现在本文中。
"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.
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