PhD Research Project in Simulation, optimisation and control of multirate dynamics

本研究项目的主要目的是为机械系统的仿真、优化和控制开发有效的数值方法。该方法将利用保持结构的多速率积分方案，从而以更低的计算成本在不同的空间和时间尺度上对系统进行高度精确的处理。然后将这些方案应用于航天器和飞行器动力学背景下的最优控制问题的解决，并研究它们的精度、收敛性和稳定性。因此，该项目属于EPSRC的工程和数学科学研究领域，有助于促进控制工程和数值分析之间的跨学科联系。在机械系统的模拟中，我们的目标是以最小的计算量以最准确的方式再现它们的行为。然而，大多数问题的非线性性质使得精确的解是不可能的，并且需要使用离散化方法来模拟手头系统的行为和性质。为此，以往的研究主要集中于开发保辛能量动量积分器。在强迫或耗散系统的背景下，特别感兴趣的是使用变分积分器，由离散朗朗日-达朗贝尔原理导出。它们保留了辛结构以及系统的动量和能量，因此可以提高精度并减少保守或弱耗散问题的计算成本。为了在最优控制的框架中捕捉这些特性，一种新的直接方法被称为离散力学和最优控制（DMOC）。通过拉格朗日-达朗贝尔原理的离散化，导出了机械系统的描述和最优控制问题的必要最优性条件。将保持结构的时间步进方程作为非线性优化问题的等式约束，然后用适当的非线性优化算法求解。采用多速率变分积分器实现了DMOC方案的进一步改进，该方案允许在不同时间尺度上的动力学在辛和动量保持方案中有效地集成。通过在拉格朗日函数的离散逼近中选择正交，可以减少必要的函数求值的次数，从而推导出纯显式或部分显式的方案。到目前为止，DMOC的多速率版本只在弹簧摆的情况下进行了检查，根据微观-宏观阶跃比例，与单速率DMOC相比，显示出显著的计算节省。3 .项目起点：DMOC的多速率版本的唯一实现的结果是有希望的，但是该方案需要针对更多的测试用例进行验证，这将是这个项目的第一个重点。一旦对具有不同时间尺度动力学的简单系统完成了对该方法的精度和收敛性的深入研究，该项目将转向将该方法应用于卫星编队飞行问题。通过航天器的合作来完成任务对它们的相对运动和定位提出了非常严格的要求。由于来自其他行星的引力吸引，在不同的时间尺度上存在动力学，使它们的控制进一步复杂化。采用小步集成的方法可以保证快速动力学的稳定集成，但计算量大。因此，多速率DMOC方法有望呈现出巨大的计算节省和精度的提高。

1 Introduction The main aim of this research project will be the development of efficient numerical methods for the simulation, optimisation and control of mechanical systems. The methods will make use of structure-preserving multirate integration schemes and thus offer highly accurate treatment of systems on different space and time scales at a decreased computational cost. These schemes will then be applied for the solution of optimal control problems in the context of spacecraft and vehicle dynamics and their accuracy, convergence and stability will be investigated. Thus, this project falls within both the Engineering and the Mathematical sciences EPSRC research areas and helps to facilitate the cross-disciplinary connection between Control Engineering and Numerical analysis. 2 Background In the simulation of mechanical systems, we aim to reproduce their behaviour in the most accurate way with the smallest computational effort. The nonlinear nature of most problems, however, renders an exact solution impossible and requires the use of discretization methods to model the behaviour and properties of the system at hand. For this purpose, previous research has focused on the development of symplectic-energymomentum preserving integrators. Of particular interest in the context of forced or dissipative systems is the use of variational integrators, derived by discretizing Langrange-d'Alembert principle. They preserve the symplectic structure as well as the momentum and energy of the system and thus allow for improvement in accuracy and reduction in computational cost for conservative or weakly dissipative problems.To capture these properties in the framework of optimal control, a new direct approach called Discrete Mechanics and Optimal Control (DMOC) was developed. Within it both the description of the mechanical system and the necessary optimality conditions for the optimal control problem are derived through the discretisation of the Lagrange-d'Alembert principle. The structure preserving time-stepping equations serve as equality constraints for the nonlinear optimisation problem, which is then solved by an appropriate nonlinear optimisation algorithm. A further advancement in the DMOC scheme was achieved by the use of multirate variational integrators, which allows for dynamics at different time scales to be integrated efficiently in a symplectic and momentum-preserving scheme. Through a choice of quadrature in the discrete approximation of the Lagrangian function one can reduce the number of necessary function evaluations, deriving purely or partly explicit schemes. Thus far, the multirate version of DMOC has been examined only in the case of a spring pendulum, showing significant computational savings in respect to the single rate DMOC depending on the micro-macro step proportionality. 3 Project starting point: The results from the sole implementation of the multirate version of DMOC are promising, however the scheme needs to be validated against more test cases and this will be the first focus of this project. Once a thorough investigation of the accuracy and convergence properties of this method is completed for simpler systems with dynamics of different time scales, the project will turn toward applying the method to the problem of satellite formation flying. Achieving tasks through cooperation of the spacecrafts places very strict requirements on their relative motion and positioning. Their control is further complicated by the presence of dynamics on different time scales due to the gravity attractions from other planets. Integrating the whole system with small steps would assure stable integration of the fast dynamics but lead to large computational effort. Thus, the multirate DMOC method is expected to present great computational savings and accuracy improvements.