From Natural Language to Certified H-infinity Controllers: Integrating LLM Agents with LMI-Based Synthesis

We present $\text{textsc}S2C$ (Specification-to-Certified-Controller), a multi-agent framework that maps natural-language requirements to certified ${\mathcal{H}}_{\mathrm{\infty }}$ state-feedback controllers via LMI synthesis. $\text{textsc}S2C$ coordinates five roles – $\mathit{\text{SpecInt}}$ (spec extraction), $\mathit{\text{Solv}}$ (bounded-real lemma (BRL) LMI), $\mathit{\text{Tester}}$ (Monte Carlo and frequency-domain checks), $\mathit{\text{Adapt}}$ (spec refinement), and $\mathit{\text{CodeGen}}$ (deployable code). The loop is stabilized by a severity- and iteration-aware $\gamma$-floor guardrail and a decay-rate region constraint enforcing $\mathrm{\Re }\lambda (A+BK)<-\alpha$ with $\alpha =3.9\mathrm{/}{T}_s$ derived from settling-time targets. For state feedback, verification reports disturbance rejection $\parallel C(sI-(A+BK){)}^{-1}E{\parallel }_{\mathrm{\infty }}$ alongside time-domain statistics; discrete benchmarks are converted to continuous time via a Tustin (bilinear) transform when needed. On 14 COMPleib problems, $\text{textsc}S2C$ attains $\mathbf{\text{100\%}}$ synthesis success and $\mathbf{\text{100\%}}$ convergence within six iterations, with strong decay-rate satisfaction and near-target certified ${\mathcal{H}}_{\mathrm{\infty }}$ levels; it improves robustness metrics relative to single-shot BRL and BRL+$\alpha$ baselines. An ablation over LLM backbones (GPT-5, GPT-5 mini, DeepSeek-V3, Qwen-2.5-72B, Llama-4 Maverick) shows the pipeline is robust across models, while stronger models yield the highest effectiveness. These results indicate that LLM agents can integrate certificate-bearing control synthesis from high-level intent, enabling rapid end-to-end prototyping without sacrificing formal guarantees.