Optimal design of mechanical controllers

许多工程设计问题可以归结为所谓的H2范数最小化问题。例如，轨道车辆悬架的设计希望将轨道磨损降至最低，而汽车悬架的设计则希望优化道路舒适性。在这种情况下，系统(即铁路或汽车)的性能是通过与机械和电气部件(悬架)网络的互连来控制的，而受控系统的H2标准将取决于部件的参数，如弹簧刚度和阻尼率。设计问题是选择这些参数以最小化这个H2范数。作为增加的复杂性级别，还必须在对应于网络内组件的可能布置的一系列不同网络结构上执行优化。这导致了在线性代数、动力学、图论、实代数几何和优化的交叉点上的数学挑战、非线性和计算密集型的优化问题。该博士学位的目标是开发数学和计算工具，以优化此类机电网络的设计。该项目将把最近获得的两个结果结合在一起：1.解决动态度不超过3的串并联力学网络的所谓最小实现问题(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的H2范数的算法，它允许比商业上可用的算法更有效和更稳定地计算H2范数。这个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化H2范数计算的效率，并量化相对于商业可用算法的好处。该算法中计算量最大的步骤是计算一个多项式丢番图方程的解。该方程的解可以通过使用扩展欧几里德算法的推广版本计算多项式、次结果和次余数序列来获得。此外，这些多项式序列中的系数对应于Hurwitz矩阵的子行列式，这也允许人们在几乎不需要额外计算的情况下确定系统的稳定性。PHD将首先探索利用这些联系在计算效率和健壮性方面的收益。在优化的背景下，该算法可以象征性地实现，以获得关于部件参数(前述例子中的弹簧刚度和阻尼率)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优的部件参数。这个PHD项目的另一个目标是通过结合MatLab的符号代数和优化能力，以及半代数几何理论和电路数学理论来实现这一点，以便为给定的应用确定动态度小于或等于3的最优串并联机械网络。一个潜在的后续活动是研究将这些结果扩展到动态度更高的机械网络的可能性。这将是对现有研究的相当大的扩展，也是该领域的一项重大突破。还可以在无源和可再生能源系统的控制系统设计的广泛范围内开展进一步的工作。这个项目与EPSRC在数学科学主题下的一些研究领域密切相关，最引人注目的是控制工程。

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.