L1 ⚛ L1-434 ⊙ Testnet

PID Controller Tuning

Control Theory · Classical control

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⬇ L1-434.md

# ⚛ L1 Principle — PID Controller Tuning **ID:** `L1-434` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Control Theory — *Classical control* > **🎯 Problem class:** parameter estimation · **🧮 Solution space:** PID parameter vector > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555257 --- ## 🧠 1. Introduction **PID Controller Tuning** is a **parameter-estimation problem** whose unknown lives in **PID parameter vector** space, within the **Classical control** sub-domain of **Control Theory**. Measurements consist of N/A via a **closed loop step response** sensing mechanism. The forward operator applies, in order: applies a smooth nonlinear function element-wise; S · zn · tuning rule operator; O · iae · integral criterion operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered PID_parameter_vector 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 ~= 10); plant_gain_variation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → Pointwise nonlinearity → S · zn · tuning rule → O · iae · integral criterion → **y** (detector). ``` y = `O.iae.integral_criterion` `S.zn.tuning_rule` f(·) x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `N.pointwise` | Applies a smooth nonlinear function element-wise | | `S.zn.tuning_rule` | S · zn · tuning rule operator | | `O.iae.integral_criterion` | O · iae · integral criterion operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Control Theory | | Sub domain | Classical control | | Carrier | N/A | | Problem class | parameter_estimation | | Solution space | PID_parameter_vector | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 3 | | L dag | 2 | ## 📡 4. Measurement Model Existence of the recovered PID_parameter_vector 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 ~= 10); plant_gain_variation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | IAE_integral_absolute_error | | Secondary | overshoot_percent | ## 📏 5. Operating Range (Ω) **Center problem class:** `parameter_estimation` · **Forward operator:** `closed_loop_step_response` **Center point:** | Parameter | Unit | Value | |---|---|---| | Plant k | — | 1 | | Plant l | — | 0.1 | | Plant tau | — | 1 | | Desired pm deg | deg | 45 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Plant k | — | 0.1 – 10.0 | | Plant l | — | 0.001 – 10.0 | | Plant tau | — | 0.01 – 100.0 | | Desired pm deg | deg | 30 – 70 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.3 IAE_integral_absolute_error | Metric | Range | |---|---| | Iae integral absolute error | 0.05 – 2.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **IAE_integral_absolute_error**, with κ = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x4c6477238f4388535011959bfca70d1a63e1ca808d7addb600173e33dce8bb79` - **Chain tx hash:** `0x4f8b46faf1b793eb6437471a64e3238a1392f3e209b75afea7b58207e93f0525` - **Chain block:** `41555257` --- ## File Mapping This bundle consists of: `L1-434.md`, `L1-434.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-434.md` | Source of truth — edit this | Human or LLM | | `L1-434.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 `L1-434`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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📋 JSON metadata

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    "sub_domain": "Classical control",
    "title": "PID Controller Tuning"
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      "plant_tau": [
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        100.0
      ]
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Classical control",
  "title": "PID Controller Tuning"
}

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