Benchmarking and Reasoning Distillation of Large Language Models for Feedback Controller Design in Complex Dynamical Systems

Although remarkable capabilities have been demonstrated by Large Language Models (LLMs) across scientific domains, feedback controller design remains underexplored. Existing benchmarks focus mainly on linear single-Degree-of-Freedom (DoF) systems and large API-hosted models, leaving performance on complex controller-design tasks and feasibility for edge deployment unclear. To address these limitations, we introduce the Complex Dynamics-to-Control Benchmark for Large Language Models (CoDyControlBench), comprising 132 system configurations across five evaluation dimensions: number of DoF, system type, coupling level, damping regime, and controller type. Six state-of-the-art LLMs were evaluated over three independent runs, including three commercial models (GPT, Gemini, and Claude) and three open-source models (GLM, DeepSeek, and Qwen). GPT achieved the highest design success rate at 94.8\%, whereas Qwen showed the lowest rate at 50.0\%. Across the benchmark dimensions, DoF and controller type exhibited the largest model-averaged variations in design success, with success-rate ranges of 36.3\% and 17.6\%, respectively, exceeding those associated with system type, coupling level, and damping regime. Comparison of GPT and Qwen showed that their performance gap arose mainly from the control-design knowledge, particularly gain selection and the use of transient-limiting mechanisms. For edge deployment, a specialized 1.5B-parameter model was developed through reasoning distillation. The reasoning-distilled model outperformed the answer-distilled and base model on CoDyControlBench, maintained stable performance across 1-6 DoFs, and achieved successful traget tracking in all three physical trials on a pneumatic-artificial-muscle-driven robotic arm. These results establish a benchmark baseline and highlight the potential of lightweight, edge-deployable controller-design models.