Optimal design of mechanical controllers

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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.