Fractional Variational Integration and Optimal Control

大规模最优控制问题可以通过将原始(连续时间)问题近似为一个非线性规划问题(NLP)来数值求解。这种近似(或转录)过程或多或少是准确的，这取决于用于离散化系统动力学和成本的积分算法，以及所采用的相关网格密度(积分步长)。虽然这一程序在一般意义上已经确立，但仍有一些问题限制了这一方法的范围和效用。这些因素包括所得到的NLP的大小、稀疏性和条件性，以及用于离散化问题的积分算法的精度。我们将特别关注与机电系统有关的最优控制问题，并将寻求利用这些问题的特殊结构来提高它们的可解性。解决这些问题的标准方法是使用单独的建模和最优控制求解阶段，然后用通用软件求解最优控制问题。在(电子)机械系统的情况下，这种方法有一些我们试图弥补的缺点：(I)机械建模和最优控制基于密切相关的变分原理，但这一共同的遗产没有被标准的求解过程所利用；(Ii)通用的数值积分算法可以破坏(电子)机械系统模型的几何结构。在保守系统的背景下，不适当的积分方案可能产生数值耗散并破坏其他守恒量，如系统的动量。我们建议通过使用变分或辛积分格式来确保这种情况不会发生。(3)由于标准方法没有利用力学模型的特殊结构，决策变量的数量不必要地增加了一倍。这源于广义速度和系统状态向量中广义位置的出现。在离散力学模型(基于离散欧拉-拉格朗日方程的模型)中，广义速度可以用广义位置上的时间差来表示，因此可以从问题中消除。对于机电系统，我们的建议是使用保持对称性的积分方案将建模和优化阶段合并为一个整体。使用经典变分方法(如定常作用)推导运动方程仅限于无损系统。这一建议的主要思想是将现有的变分建模和积分方法扩展到具有耗散的机电系统。耗散包括阻力损失、阻尼、空气动力损失、磁滞损失和摩擦。为了确保一个纯粹的变分公式，我们将利用分数阶微积分来模拟系统中的耗散项。20世纪90年代末引入了用分数导数模拟耗散的概念，如线性摩擦或阻力。然而，将其用于发展耗散机电系统的保结构变分积分和最优控制方法是一个全新的研究领域，也是本方案的主要目的。开发的方法和算法将用于解决汽车行业的“工业强度”应用问题。机电系统出现在汽车工业的许多领域，最优控制问题出现在混合动力系统控制等领域。一旦开发出可行的理论和软件，它们将在两个汽车项目中进行评估。第一次将与一级方程式车队进行，第二次将与一家领先的发动机和动力总成制造商进行。

Large-scale optimal control problems can be solved numerically by approximating the original (continuous-time) problem with a nonlinear programming problem (NLP). This approximation (or transcription) process is more or less accurate, depending on the integration algorithm used to discretise the system dynamics and cost, and the associated mesh density (integration step size) employed. While this procedure is well established in general terms, there are a number of issues that limit the scope and utility of this approach. These include the size, sparsity and conditioning of the resulting NLP, the accuracy of the integration algorithm used to discretise the problem. We will focus particularly on optimal control problems relating to electro-mechanical systems, and will seek to exploit the special structure of these problems in order to improve their solubility. The standard approach to the solution of these problems is to use separate modelling and optimal control solution phases, with the optimal control problem then solved with general-purpose software. In the case of (electro-)mechanical systems this approach has a number of drawbacks that we seek to remedy: (i) mechanical modelling and optimal control are based on closely related variational principles, but this common heritage is not exploited by the standard solution process; (ii) general-purpose numerical integration algorithms can destroy the geometric structure of (electro-)mechanical system models. In the context of conservative systems inappropriate integration schemes may produce numerical dissipation and destroy other conserved quantities such the system's momentum. We propose to ensure that this does not occur through the use of variational, or symplectic integration, schemes. (iii) Since the special structure of mechanical models is not exploited in the standard approach, the number of decision variables is needlessly doubled. This follows from the appearance of the generalised velocities and the generalised positions in the system's state vector. In discrete mechanical models (models based on a discrete Euler-Lagrange equation) the generalised velocities are expressible in terms of time differences in the generalised positions and thus can be eliminated from the problem. For electro-mechanical systems our proposal is to combine the modelling and optimisation phases into a single whole using symmetry-preserving integration schemes.The derivation of the equations of motion using classical variational methods, such as stationary action, is limited to lossless systems. The main idea in this proposal is to extend existing variational modelling and integration approaches to electro-mechanical systems with dissipation. Dissipation includes such things as resistive losses, damping, aerodynamic losses, hysteresis losses and friction. To ensure a purely variational formulation we will make use of fractional calculus to model the dissipative terms in the system. The concept of modelling dissipation such as linear friction, or resistance by means of fractional derivatives was introduced in the late 1990s. However, its use to develop structure-preserving variational integration and optimal control methods for dissipative electro-mechanical systems is a completely new field of research and the main objective in this proposal. The developed methods and algorithms will be used to solve 'industrial strength' application problems from the automotive sector. Electro-mechanical systems appear in many areas of the automotive industry with optimal control problems arise in areas such as hybrid powertrain control. Once workable theory and software has been developed, they will be evaluated in two automotive projects. The first will be conducted with a Formula One team, while the second will be conducted with a leading engine and powertrain manufacturer.