分数阶系统的最优控制

Fractional-order systems are dynamical systems whose orders of derivative are positive real numbers and have recently been a hot spot in the field of automatic control. How to obtain optimal solution has always been the core problem in the field of optimal control and pseudospectral methods are an effective approach to deal with this issue. However, unlike conventional dynamical systems whose orders of derivative are positive integers, the solutions of fractional-order systems are usually singular near the boundaries (hereafter referred to as boundary singularity), and this brings great challenge on the theory and method to existing pseudospectral methods for solving fractional optimal control problems (FOCPs). Based on our previous work and existing results, we will in this project develop differential/integral fractional pseudospectral methods with the ability of capturing the boudary singularity, establish the equivalence between them and their costate mapping principles, and perform analysis and verification for the proposed theories and methods. It is expected that the present project may fundamentally solve the boundary singularity of fractional-order systems and bring breakthroughs in both theory and method as follows: 1) In theory, establishing the optimality conditions for FOCPs, revealing the motion laws of optimal solution of FOCPs, and expounding the costate mapping principles for fractional pseudospectral methods; 2) In method, acquiring a suite of equivalent fractional pseudospectral methods with the ability of capturing the boudary singularity and estimating costate, and providing powerful means for the numerical solution of FOCPs in the field of science and engineering.

分数阶系统是指导数阶次为正实数的动态系统，近年来成为了控制领域的研究热点。如何获取最优解一直是最优控制领域的核心问题，伪谱法是解决此问题的有效方法。然而，与常规动态系统不同，分数阶系统的解在端点附近常表现出奇异性(简称端点奇异性)，这给利用现有伪谱法来求解其优化控制问题(FOCP)带来了理论和方法上的巨大挑战。本项目拟在申请人前期研究的基础上，充分借鉴已有成果，研究具有端点奇异逼近能力的微分型/积分型分数阶伪谱法，建立他们之间的等价性和相应的协态映射原理，并对前述理论和方法进行分析验证。通过本项目的研究，有望从根本上解决分数阶系统的端点奇异性问题，并带来如下理论和方法上的突破：1)理论上，建立FOCP的最优性条件，揭示其最优解的运动规律，阐明分数阶伪谱法的协态映射原理；2)方法上，获得一套具有端点奇异逼近和协态估计能力的等价分数阶伪谱法，为科学与工程领域的FOCP提供强有力的数值求解手段。

常规动态系统的阶次为整数，而分数阶系统的阶次为实数，与前者不同的是，后者的解在端点附近常表现出奇异性，这给现有的最优控制理论和方法带来了严峻挑战。考虑到伪谱法在求解经典最优控制问题中的显著优势，项目主持人与合作者将常规伪谱法推广到了分数阶最优控制问题中，主要开展了以下工作：(1)针对分数阶动态系统的类型，研究了具有端点奇异逼近能力的微分型/积分型分数阶伪谱法，给出了相应分数阶伪谱矩阵的定义及其稳定的计算方法，并阐明了适用于端点奇异逼近的参数整定机制；(2)从分数阶Birkhoff插值的独特角度建立了微分型与积分型分数阶伪谱法之间的等价性，为理解分数阶伪谱法提供了一个新的视角；(3)建立了分数阶最优控制问题的最优性条件，研究了分数阶伪谱法的协态映射原理，获得了相应的协态估计公式，解决了分数阶伪谱法所得解的最优性问题。上述这些研究成果进一步完善了最优控制的理论框架，同时也为深入认识和发展伪谱法提供了新的视角，并为最优控制理论和方法在科学与工程中的深入应用奠定了坚实基础。