分数阶系统关键参数迭代学习辨识的基础理论及应用研究

In this project, the iterative learning identification of order and initialization response of fractional order systems are studied based on the fractional adaptive iterative learning control and order sensitivity function, which can be applied to the fractional system theory and iterative learning control. To the fractional adaptive iterative learning control, the differential type, filtering type and nonlinear learning laws are designed to the adaptive study for three kinds of fractional orders and types so that the finite time and infinite time perfect tracking control problems are realized as well. The analyticity and numerical implementation of order sensitivity function are analyzed. The construction of parameter learning law and its convergence analysis are applied to the iterative learning identification of constant order and variable order systems. The idea of fractional order element networks provides a practical way to the experiments. The iterative learning identifications of distributed order and initialization response are unified the identification of kernel functions, and the convergence analysis can be associated with a number of integral transforms. The identified results are applied to optimize the control strategies, reference learning law, and fractional nonlinear robust iterative learning control. The main aim of this project is to build the fundamental theoretical system of fractional order iterative learning identification, to extend the research fields of fractional order iterative learning control, and to combine the theory and experiments of the above conclusions such that both can be benefited from each other.

本项目从分数阶自适应迭代学习控制和阶灵敏度函数入手，研究分数阶系统阶数和初始化响应的迭代学习辨识方法，实现其在分数阶系统理论和迭代学习控制中的应用。针对分数阶自适应迭代学习控制方法，设计微分型、滤波器型和非线性学习律，研究其对分数阶系统三类阶数及类型的自适应性，实现有限时间和无限时间上的精确控制；研究阶灵敏度函数的解析性和数值实现方法；针对定常阶和变分数阶系统阶数的迭代学习辨识，构造参数型学习律分析其收敛性，并通过分数阶单元网络思想提出可行的实现方案；针对分布阶系统阶数和分数阶系统初始化响应，统一采用核函数迭代学习辨识的方法，并将各式积分变换引入该方法的收敛性分析；将辨识的结果用于控制策略的优化，并实现其在参考值型学习律和分数阶非线性鲁棒迭代学习控制中的应用。初步建立起分数阶迭代学习辨识的基础理论体系，扩大分数阶迭代学习控制的研究范围，通过理论指导实验并在实践中寻找理论创新。

随着科学技术的发展，诸多领域的专家指出其研究对象具有明显的非整数阶特性，并因此提出了大量分数阶系统辨识的需求。然而，现有的分数阶系统辨识方法均属于整数阶系统辨识方法的推广，要么无法体现分数阶系统本身特有的性质，要么存在精度欠佳等问题。因此，本项目从分数阶全参数自适应迭代学习控制和阶灵敏度函数入手，针对分数阶系统特有的阶次和初始化响应对系统控制和辨识所带来的挑战，借鉴了鲁棒自适应控制的现有结论，基于灰箱系统、黑箱系统和实际系统设计了微分型、滤波器型和非线性迭代学习辨识策略，实现了定常阶、变分数阶和分布阶三大分数阶系统关键参数的辨识，特别是通过分数阶阶次学习律和分数阶最小二乘等方法避免了局部陷阱问题，实现了分数阶和变分数阶的精确辨识。将阶灵敏度函数用于分布阶系统阶次和初始化函数的辨识，并提出基于预调的分数阶初始化系统辨识策略，且由此引出了分数阶系统可辨识性的概念。上述理论结果被成功应用于分数阶迭代学习控制方法的优化，轮式机器人多故障诊断，数控精密磨床，动力电池建模及模拟等领域。目前，该项目已初步建立起分数阶迭代学习辨识的基础理论体系，扩大了分数阶迭代学习控制的研究范围，通过理论指导实验并在实践中寻找理论创新。

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