# ⚛  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`

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## 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._