不确定分数阶非线性系统Mittag-Leffler自适应控制

Adaptive control of fractional order nonlinear system (FONS) is one of the research frontiers in nonlinear science. Based on different fractional order derivative definitons, valid power laws of fractional order derivative in both salar and vector versions for fractional order system (FOS) are proposed, to obtain a direct relation between fractional order derivative of energy function and differential equation of system, providing new theoretic results to the application of fractional order Lyapunov direct method. Based on Mittag-Leffler stability theory, energy function construction methods are proposed, with sufficient and necessary conditons for fractional Lypunov function of adaptive control obtained. Based on comparison principles, new fractional calculus properties are studied, with new methods for upper bound and convergence analysis of energy function of uncertain FOS proposed. Stabilization, tracking control and synchronization for uncertain FONS are studied, based on sliding mode control, robust control and feedback control. Classical backstepping and dynamic surface control are generalized to FOS, and fractional backstepping and dynamic surface control methods are proposed, with modelling error, external disturbance, unknown control direction and distributed parameter considered. This project provides new theoretical results for stability analysis and control of FOS.

分数阶非线性系统自适应控制是非线性科学研究前沿。本项目针对不确定分数阶非线性系统，基于不同分数阶微分定义，提出有效的标量及向量的分数阶导数幂律不等式，建立能量函数的分数阶导数与系统微分方程的直接联系，为分数阶李雅普诺夫直接法的应用提供理论新依据。基于分数阶李雅普诺夫理论(Mittag-Leffler Stability Theory)，提出不确定系统能量函数构造方法，给出自适应控制分数阶李雅普诺夫函数的充要条件。基于比较原理，研究分数阶微积分新性质，提出不确定分数阶系统能量函数上界和收敛性分析新方法。基于滑模控制、鲁棒控制、状态反馈等方法，研究不确定分数阶非线性系统镇定、跟踪、同步问题。综合考虑系统的建模误差、外部扰动、控制方向未知、分布参数等情况，将经典自适应反步控制和动态面控制推广到分数阶系统，提出分数阶自适应反步控制和动态面控制方法。本项目将丰富分数阶系统稳定性及控制理论。

研究了不确定分数阶非线性系统，基于不同分数阶微分定义，提出有效的标量及向量的分数阶导数幂律不等式，建立能量函数的分数阶导数与系统微分方程的直接联系，为分数阶李雅普诺夫直接法的应用提供理论新依据。基于分数阶李雅普诺夫理论，提出不确定系统能量函数构造方法，给出自适应控制分数阶李雅普诺夫函数的充要条件。基于比较原理，研究分数阶微积分新性质，提出不确定分数阶系统能量函数上界和收敛性分析新方法。..将无源性引入分数阶系统，提出分数阶无源滑模控制方法。针对含有未知参数和外部扰动的严格反馈分数阶非线性系统，将自适应反步控制推广到分数阶系统，提出分数阶系统控制器反步设计方法、分数阶未知参数估计更新律和未知上界估计分数阶更新律。针对含有模型误差的分数阶非线性系统，设计分数阶积分滑模面和趋近律，提出分数阶滑模控制方法。针对含有任意不确定性的严格反馈分数阶非线性系统，将自适应动态面控制推广到分数阶系统，提出分数阶动态面控制方法。针对不确定分数阶非线性系统，基于分数阶反步方法，提出分数阶反步滑模控制方法。基于分数阶微积分理论，提出分数阶反馈控制方法。 该研究丰富了分数阶系统稳定性及控制理论，还可以应用到信号处理和保密通信等实际工程领域。.