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

多速率动力学仿真、优化和控制博士研究项目

• 批准号: 2280382
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2280382
• 关键词: PhD; Research; Project; Simulation; optimisation

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.

1引言本研究项目的主要目的将是开发有效的数值方法的模拟，优化和控制的机械系统。该方法将利用结构保持多速率积分计划，从而提供了高精度的处理系统在不同的空间和时间尺度上，以降低计算成本。然后将这些方案应用于解决航天器和飞行器动力学方面的最佳控制问题，并将研究其准确性、收敛性和稳定性。因此，该项目福尔斯工程和数学科学EPSRC研究领域，并有助于促进控制工程和数值分析之间的跨学科联系。2背景在机械系统的模拟中，我们的目标是以最小的计算工作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不可能得到精确的解，并且需要使用离散化方法来模拟系统的行为和属性。为此，以前的研究集中在辛能量动量保持积分器的发展。特别感兴趣的强迫或耗散系统的情况下，是使用变分积分，来自离散化的拉格朗日-达朗贝尔原理。他们保持辛结构以及动量和能量的系统，从而允许提高精度和减少计算成本的保守或弱dissipative problems.To捕捉这些属性的最优控制的框架中，一个新的直接方法称为离散力学和最优控制（DMOC）的开发。在它的机械系统的描述和必要的最优性条件的最优控制问题是通过拉格朗日-达朗贝尔原理的离散化。结构保持时间步进方程作为等式约束的非线性优化问题，然后解决了一个适当的非线性优化算法。DMOC方案的进一步发展是通过使用多速率变分积分器实现的，该积分器允许在不同时间尺度上的动态有效地集成在辛和动量保持方案中。通过在拉格朗日函数的离散近似中选择正交，可以减少必要的函数评估的数量，从而导出纯显式或部分显式的方案。到目前为止，DMOC的多速率版本已被检查，只有在弹簧摆的情况下，显示出显着的计算节省相对于单速率DMOC取决于微观-宏观步骤的比例。3项目起点：DMOC的多速率版本的单独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，这将是该项目的第一个重点。一旦完成对具有不同时间尺度动态的简单系统的这种方法的精度和收敛特性的彻底调查，该项目将转向将该方法应用于卫星编队飞行问题。通过航天器之间的协作完成任务，对航天器之间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，它们的控制由于不同时间尺度上的动力学而进一步复杂化。用小步长积分整个系统将确保快速动态的稳定积分，但会导致大的计算工作量。因此，多速率DMOC方法预计将带来巨大的计算节省和精度提高。