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

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.