Simulation, optimisation, and control of multirate dynamics

Thesis

Multirate systems, which exhibit dynamics across various time scales, are common in both scientific and real-world applications. Predicting and prescribing their behaviour, however, is challenging, as resolving fast dynamics requires fine-grid treatment, affecting the accuracy and efficiency of computational methods. This thesis comprehensively tackles this challenge, focusing on simulation, open-loop control, and robust feedback strategies. The proposed methods build upon each other, leveraging the properties of the system's model to design bespoke methods for high-fidelity simulation and optimal control at reduced computational cost.

We begin by employing a multirate variational approach for uncontrolled simulations of multirate systems, achieving great reductions in the number of intermediate variables for negligible penalties in accuracy. We prove that the resulting discrete model is structure-preserving and derive its order of accuracy for a class of dissipative systems. Implementing this model for optimal control, we demonstrate the adjoint system inherits the structure-preservation and accuracy of the variational model, while the multirate formulation significantly reduces the number of optimisation variables and constraints. These findings are showcased through two theoretical and two practical aerospace examples: simultaneous slewing and vibration control of flexible satellites, and control of satellites in three-body gravitational environment using a six degree-of-freedom model.

This variational open-loop control strategy can not account for modelling errors or disturbances and often results in an intractible nonconvex problem. Therefore, we formulate a novel linearisation strategy for variational models and implement the multirate variational approach within a tube-based nonlinear receding horizon (RH) strategy. This results in a successive convex approximation approach that reduces the size of the optimal control problem, and without loss of feasibility or stability guarantees, increases the receding horizon time step compared to the time step used to discretise the fast dynamics.

Although providing feedback, this technique does not account for disturbances and does not provide guarantees for obtaining a local optimum of the original nonlinear problem. For this purpose, we develop a novel successive linearisation RH strategy for systems with explicit and implicit models.~We prove that unless terminated early, the technique obtains a local optimum of the original nonlinear problem and demonstrate how variational models can be used to reduce the size of the resulting convex problems while preserving some structures and the convexity of the continuous model. For systems with explicit models, this technique is extended to consider the presence of additive disturbances. Recursive feasibility and stability are proved and demonstrated through examples.

Authors: Lishkova, Y

• DOI: 10.5287/ora-qygvdzzyg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: convex programming; multirate systems; spacecraft control; variational calculus; structure-preserving integrators; convex trajectory design; model predictive control; optimization-based control; optimal control
• Subjects: Numerical methods; Aerospace engineering; Mathematics; Control Engineering