How to discretize continuous state-action spaces in Q-learning: A symbolic control approach

Q-learning is widely recognized as an effective approach for synthesizing controllers to achieve specific goals. However, handling challenges posed by continuous state-action spaces remains an ongoing research focus. This paper presents a systematic analysis that highlights a major drawback in space discretization methods. To address this challenge, the paper proposes a symbolic model that represents behavioral relations, such as alternating simulation from abstraction to the controlled system. This relation allows for seamless application of the synthesized controller based on abstraction to the original system. Introducing a novel Q-learning technique for symbolic models, the algorithm yields two Q-tables encoding optimal policies. Theoretical analysis demonstrates that these Q-tables serve as both upper and lower bounds on the Q-values of the original system with continuous spaces. Additionally, the paper explores the correlation between the parameters of the space abstraction and the loss in Q-values. The resulting algorithm facilitates achieving optimality within an arbitrary accuracy, providing control over the trade-off between accuracy and computational complexity. The obtained results provide valuable insights for selecting appropriate learning parameters and refining the controller. The engineering relevance of the proposed Q-learning based symbolic model is illustrated through two case studies.