含有执行器饱和约束的非齐次分数阶系统的分析和控制

The application of fractional order controllers in practical engineering systems encounters theoretical challenges, due in part to ubiquitous actuator saturation in engineering and in part to the incommensurate form of closed-loop systems. Most of the exsting research on the stability issues of fractional order nonlinear systems are still at preliminary stage, and motivated by and extended from integer order references. One of the important challenges in the field of fractional order control theory is extremely difficult to find a uniform structural fractional order Lyapunov function candidates which demonstrate the characteristic of fractional order operator. The incommensurate orders and difficulties of finding proper Lyapnuov function candidate limit the practical applications of fractional order controllers. In this project, we will focus on the incommensurate fractional order linear systems subject to actuator saturation, and investigate the stability problem utilizing fractional order Lyapunov approach. In order to estimate the domain of attraction and design stabilization controller, the commensurate fractional order approximation model of incommensurate fractional order systems will be firstly proposed. Two kinds of fractional order Lyapunov function candidate including order-dependent Lyapunov function and that composed of fractional order potential energy and kinetic energy, will be built up to deal with the problem of the asymptotic stability of incommensurate fractional order systems subject to actuator saturation. The proposed stability condition is expected to reveal the order coupling mechanism and energy decay law of incommensurate fractional order systems. The algorithms for the estimation and enlargement of domain of attraction, and the designing approach for state-feedback stabilizing controller and fractional order dynamic output-feedback stabilizing controller will be advanced. Additionally, we will design hardware-in-loop platform to verify the proposed stabilization approach and our theoretical results. It is expected that this project would be beneficial to the solutions of applying fractional order controllers to practical engineering systems. The implementation of this project will provide theoretical guidance and engineering experience for the practical applications of fractional order controllers.

执行器饱和现象广泛存在于实际工程系统中。分数阶控制器在工程系统中的应用不可避免遇到两个挑战：闭环非齐次、执行器饱和。现有的分数阶非线性系统的稳定性研究还处于初步阶段，构造结构统一的、体现分数阶特征的李雅普诺夫候选函数依然是分数阶领域的一个开放问题，极大制约了分数阶控制的工程应用。本项目以含执行器饱和的非齐次分数阶线性系统为研究对象，以体现分数阶特征的李雅普诺夫函数构造技术为主线，以提出吸引域估计和镇定方法为目标，以非齐次分数阶系统的同维齐次等价模型的建立为切入点，以构造阶次相关的李雅普诺夫函数和分数阶动势能形式的李雅普诺夫函数为手段，系统研究非齐次分数阶系统在执行器饱和约束下的渐近稳定问题，揭示其阶次耦合机制，探索其能量衰减规律，提出一套吸引域估计及其扩大算法、状态反馈和分数阶动态输出反馈镇定方法，并在硬件仿真平台验证理论结果，为分数阶方法在实际工程系统的应用提供理论和技术支撑。

执行器饱和现象广泛存在于实际工程系统中。分数阶控制器在工程系统中的应用不可避免遇到两个挑战：闭环非齐次、执行器饱和。现有的分数阶非线性系统的稳定性研究还处于初步阶段，构造结构统一的、体现分数阶特征的李雅普诺夫候选函数依然是分数阶领域的一个开放问题，极大制约了分数阶控制的工程应用。本项目针对含有执行器饱和的分数阶系统，获得了等价的增广非齐次分数阶系统描述，提出了增广系统的吸引域估计算法；针对现有分数阶控制器整定方法无法在时域分析与频域分析方法之间建立联系的问题，以一阶惯性加滞后系统和通用的高阶系统为对象，研究了分数阶PID控制器、分数阶TID控制器的整定问题，给出了参数的边界条件，通过期望相位裕度确定可取值的最大频率，保证未知参数方程组的解的存在性，在保证期望增益穿越频率可取值的前提下，给出完备的稳定控制器集合，并根据平坦相位约束和时域指标整定最优控制器，获得满意的控制性能；针对含执行器饱和的整数阶系统，研究了分数阶PI控制器的整定问题方法，提出了一种基于描述函数法的分析方法，基于开环Nyquist曲线分析了含执行器饱和的系统在分数阶PI控制器作用下的稳定域和输出特点，提出了无稳态误差的最优分数阶PI控制器整定方法。.但是由于现有分数阶李雅普诺夫方法的限制，本项目在含有执行器饱的分数阶系统吸引域扩大估计和通用李雅普诺夫函数构造方面，尚未取得突破性进展。本项目的结果丰富了分数阶系统稳定性分析方法和控制器设计方法。