L1 ⚛ L1-432 ⊙ Testnet

Optimal Control (LQR)

Control Theory · Linear quadratic control

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⬇ L1-432.md

# ⚛ L1 Principle — Optimal Control (LQR) **ID:** `L1-432` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Control Theory — *Linear quadratic control* > **🎯 Problem class:** parameter estimation · **🧮 Solution space:** control gain matrix > **📡 Carrier:** N/A · **🌫 Noise:** deterministic > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555257 --- ## 🧠 1. Introduction **Optimal Control (LQR)** is a **parameter-estimation problem** whose unknown lives in **control gain matrix** space, within the **Linear quadratic control** sub-domain of **Control Theory**. Measurements consist of N/A via a **full state feedback lqr** sensing mechanism. The forward operator applies, in order: computes eigen-pairs of a linear operator; S · lqr · gain computation operator; O · lyapunov · closed loop operator. Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered control_gain_matrix is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); model_uncertainty_A_B dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → S · lqr · gain computation → **y** (detector). ``` y = `S.lqr.gain_computation` x (deterministic) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.lqr.gain_computation` | S · lqr · gain computation operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `E.eigensolve` | Computes eigen-pairs of a linear operator | **🛡 Analytical properties** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.lyapunov.closed_loop` | O · lyapunov · closed loop operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Control Theory | | Sub domain | Linear quadratic control | | Carrier | N/A | | Problem class | parameter_estimation | | Solution space | control_gain_matrix | | Noise model | deterministic | | Integration axis | time_horizon | | Difficulty delta | 3 | | L dag | 2.5 | ## 📡 4. Measurement Model Existence of the recovered control_gain_matrix is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); model_uncertainty_A_B dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | closed_loop_performance_index | | Secondary | stability_margin_dB | ## 📏 5. Operating Range (Ω) **Center problem class:** `parameter_estimation` · **Forward operator:** `full_state_feedback_lqr` **Center point:** | Parameter | Unit | Value | |---|---|---| | Q r ratio | — | 1 | | N inputs m | m | 1 | | System order n | N | 4 | | Spectral radius a | — | 0.9 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Q r ratio | — | 0.01 – 100 | | N inputs m | m | 1 – 10 | | System order n | N | 1 – 50 | | Spectral radius a | — | 0.0 – 0.999 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 1.05 closed_loop_performance_index | Metric | Range | |---|---| | Closed loop performance index | 0.9 – 2.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **closed_loop_performance_index**, with κ = `500` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xec453bc6b303c57f8ed7c38e68dc69f7f37ac381200764f902b1645782b13e78` - **Chain tx hash:** `0x2a18c6e029f0d084e0113108cf9d25d0b316b60d89c51e8a94c126a64707b504` - **Chain block:** `41555257` --- ## File Mapping This bundle consists of: `L1-432.md`, `L1-432.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-432.md` | Source of truth — edit this | Human or LLM | | `L1-432.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-432`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-432-001 Optimal Control (LQR) — Nominal + Mismatch Specs

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📋 JSON metadata

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  "chain_tx_hash": "0x2a18c6e029f0d084e0113108cf9d25d0b316b60d89c51e8a94c126a64707b504",
  "domain": "Control Theory",
  "hardness_fn": {
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    "metric": "closed_loop_performance_index",
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  "physics_fingerprint": {
    "L_DAG": 2.5,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Control Theory",
    "integration_axis": "time_horizon",
    "noise_model": "deterministic",
    "primitives": [
      "E.eigensolve",
      "S.lqr.gain_computation",
      "O.lyapunov.closed_loop"
    ],
    "problem_class": "parameter_estimation",
    "sensing_mechanism": "full_state_feedback_lqr",
    "solution_space": "control_gain_matrix",
    "sub_domain": "Linear quadratic control",
    "title": "Optimal Control (LQR)"
  },
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      "forward_operator": "full_state_feedback_lqr",
      "input_format": "measurement_only",
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        "spectral_radius_A": 0.9,
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      "n_inputs_m": [
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      "spectral_radius_A": [
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      ],
      "system_order_n": [
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        50
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Linear quadratic control",
  "title": "Optimal Control (LQR)"
}

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