Dual Control of Linear Systems from Bilinear Observations with Belief Space Model Predictive Control

We study finite-horizon quadratic control of linear systems with bilinear observations, in which the control input affects not only the state dynamics but also the partial observations of the state. In this setting, the separation principle can fail because control inputs influence the future quality of state estimates. State estimation requires an input-dependent Kalman filter whose gain and error covariance evolve as functions of the control inputs. To address this challenge, we propose a belief-space model predictive control ($\texttt{B-MPC}$) method that plans directly over both the estimated state and its error covariance. In particular, $\texttt{B-MPC}$ plans with a deterministic surrogate of the belief evolution defined by the input-dependent Kalman filter. Through numerical experiments in two synthetic settings, we show that $\texttt{B-MPC}$ can outperform both the separation-principle controller and its MPC variant in favorable regimes, and that these gains are accompanied by lower estimation covariance and more uncertainty-aware action choices.

Automating Thought of Search: A Journey Towards Soundness and Completeness

Planning remains one of the last standing bastions for large language models (LLMs), which now turn their attention to search. Most of the literature uses the language models as world models to define the search space, forgoing soundness for the sake of flexibility. A recent work, Thought of Search (ToS), proposed defining the search space with code, having the language models produce that code. ToS requires a human in the loop, collaboratively producing a sound successor function and goal test. The result, however, is worth the effort: all the tested datasets were solved with 100% accuracy. At the same time LLMs have demonstrated significant progress in code generation and refinement for complex reasoning tasks. In this work, we automate ToS (AutoToS), completely taking the human out of the loop of solving planning problems. AutoToS guides the language model step by step towards the generation of sound and complete search components, through feedback from both generic and domain specific unit tests. We achieve 100% accuracy, with minimal feedback iterations, using LLMs of various sizes on all evaluated domains.