Optimal design of mechanical controllers

机械控制器的优化设计

• 批准号: 2404310
• 金额: --
• 依托单位: UNIVERSITY OF EXETER
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2404310
• 关键词: Optimal; design; mechanical; controllers

Many engineering design problems can be specified as so-called H2 norm minimisation problems. Examples include the design of railway vehicles suspensions, where it is desired to minimise track wear, and the design of automotive vehicle suspensions, where it is desired to optimise road comfort. In these cases, the performance of the system (i.e., the railway or automotive vehicle) is controlled through interconnection with a network of mechanical and electrical components (the suspension), and the H2 norm of the controlled system will depend on the components' parameters such as the spring stiffnesses and damping rates. The design problem is to choose these parameters in order to minimise this H2 norm. As an added level of complexity, the optimisation must also be carried out over a range of different network structures corresponding to the possible arrangements of components within the network. This results in a mathematically challenging, nonlinear and computationally intensive optimisation problem at the intersection of linear algebra, dynamics, graph theory, real algebraic geometry and optimisation. The objective of this PhD is to develop mathematical and computational tools for the optimal design of such electromechanical networks. The project will couple together two recently obtained results: 1. A solution of the so-called minimal realisation problem for series-parallel mechanical networks whose dynamic degree does not exceed three (i.e., their behaviour is governed by a differential equation of third order or lower), 2. A recently discovered algorithm for computing a system's H2 norm, which allows for a more efficient and numerically stable calculation of the H2 norm than commercially available algorithms. A first activity in this PhD project is to exploit the newly discovered algorithm for H2 norm computation to maximise the efficiency of H2 norm calculations, and to quantify the benefits relative to commercially available algorithms. The most computationally intensive step in the algorithm involves computing the solution to a polynomial Diophantine equation. Solutions to said equation can be obtained by computing polynomial subresultant and subremainder sequences through a generalised version of the extended Euclidean algorithm. Moreover, the coefficients in these polynomial sequences correspond to subdeterminants of Hurwitz matrices, which also allow one to determine the stability of the system with little additional computational effort. The PhD will begin by exploring the gains in computational efficiency and robustness that amount from exploiting these connections. In the context of optimisation, the algorithm can be implemented symbolically to obtain explicit expressions for the H2 norm in terms of component parameters (the spring stiffnesses and damping rates in the aforementioned examples). This can improve the efficiency of so-called gradient-based optimisation methods to enable the optimal component parameters to be found for a given mechanical network. Another objective of this PhD project is to implement this by combining the symbolic algebra and optimisation capabilities of MATLAB, together with the theory of semi-algebraic geometry and the mathematical theory of electrical circuits, in order to determine the optimal series-parallel mechanical network of dynamic degree less than or equal to three for a given application. A potential subsequent activity is to investigate the possibility of extending these results to mechanical networks of higher dynamic degree. This would constitute a considerable extension to the existing research and a significant breakthrough in the field. Further work may also be undertaken under the broad remit of control systems design for passive and renewable energy systems. This project is closely aligned with a number of EPSRC research areas under the Mathematical Sciences theme, most notably Control engineering.

许多工程设计问题可以指定为所谓的 H2 范数最小化问题。例子包括铁路车辆悬架的设计，其中希望最大限度地减少轨道磨损，以及汽车悬架的设计，其中希望优化道路舒适度。在这些情况下，系统（即铁路或汽车）的性能通过与机械和电气组件（悬架）网络的互连进行控制，受控系统的 H2 范数将取决于组件的参数，例如弹簧刚度和阻尼率。设计问题是选择这些参数以最小化 H2 范数。由于增加了复杂性，还必须在与网络内组件的可能排列相对应的一系列不同网络结构上进行优化。这导致了线性代数、动力学、图论、实代数几何和优化交叉领域的数学挑战、非线性和计算密集型优化问题。该博士学位的目标是开发数学和计算工具来优化此类机电网络的设计。该项目将结合最近获得的两项成果：1. 串并联机械网络所谓的最小实现问题的解决方案，其动态度不超过三（即，它们的行为由三阶或更低的微分方程控制），2. 最近发现的计算系统 H2 范数的算法，与商业上可用的相比，该算法可以更有效且数值稳定地计算 H2 范数 算法。该博士项目的第一个活动是利用新发现的 H2 范数计算算法来最大限度地提高 H2 范数计算的效率，并量化相对于商用算法的优势。该算法中计算量最大的步骤涉及计算多项式丢番图方程的解。所述方程的解可以通过利用扩展欧几里得算法的广义版本计算多项式子结果和子余数序列来获得。此外，这些多项式序列中的系数对应于 Hurwitz 矩阵的子行列式，这也使得人们无需额外的计算工作就可以确定系统的稳定性。博士课程将首先探索利用这些连接所带来的计算效率和鲁棒性方面的收益。在优化的背景下，可以象征性地实现该算法，以获得组件参数（上述示例中的弹簧刚度和阻尼率）方面的 H2 范数的显式表达式。这可以提高所谓的基于梯度的优化方法的效率，从而能够为给定的机械网络找到最佳的组件参数。该博士项目的另一个目标是通过结合 MATLAB 的符号代数和优化功能、半代数几何理论和电路数学理论来实现这一目标，以确定给定应用的动态度小于或等于 3 的最佳串并联机械网络。一个潜在的后续活动是研究将这些结果扩展到更高动态程度的机械网络的可能性。这将构成现有研究的相当大的扩展和该领域的重大突破。进一步的工作也可以在被动和可再生能源系统的控制系统设计的广泛范围内进行。该项目与数学科学主题下的许多 EPSRC 研究领域密切相关，尤其是控制工程。