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LLM + Control

Apply the LLM patterns from earlier sections — plain prompting, tool use, and multi-agent loops — to concrete control-theory problems. The pages in this section walk through representative tasks in rough order of difficulty. Early problems (Laplace transforms) a capable model handles fine with a plain prompt; later problems (plotting, PID tuning, real-world deployments) force you up the staircase.

The goal isn't to replace SymPy or python-control with a language model — they're better at the math than any model is. The goal is to see, problem by problem, where each layer of LLM machinery starts to become necessary, and to build an intuition for when to reach for which tool.

The staircase

Every page follows the same recipe: state the problem, pick a deterministic oracle for ground truth, run all three approaches, and compare.

flowchart LR
    P[Control problem] --> A1[Plain prompt]
    P --> A2[With deterministic tool]
    P --> A3[Writer and checker loop]
    A1 --> G[Ground-truth oracle]
    A2 --> G
    A3 --> G
    G --> R[Accuracy vs. cost]

Each rung reuses a pattern from earlier: plain prompting from Calling the API, deterministic tools from Tool Use, and writer/checker from Multi-agent. The staircase is the unifying story — every additional rung costs more tokens and more latency, but buys you something specific.

Pages

  • Laplace Transforms — Homework 2 Problem 3.3 and Homework 3 Problems 3.7 / 3.9. A plain prompt already matches the textbook for forward transforms, inverse via partial fractions, and ODE solving — no tool needed.
  • Control Plots — give the model three python-control-backed tools (root locus, Bode, Nyquist) and let it parse a natural-language transfer function into coefficient lists and call the matching tool. First page where a tool is genuinely required (plain prompts cannot emit plots).
  • PID Tuning — an agent with a deterministic step-response evaluator in the loop. Compares a single chat-style call (open-loop) against the full iterative loop (closed-loop) on a first-order PI tuning problem. The analogy to open-loop vs. closed-loop control is the pedagogical spine of the page.
  • Real-World Applications — three deployed projects (drone swarm choreography, autonomous spacecraft in KSPDG, long-horizon robotic manipulation) that put an LLM in the control stack, each with a different role: planner, fine-tuned controller, reflective agent. Videos and architecture diagrams, no code.