L1 ⚛ L1-426 ⊙ Testnet

Linear State-Space System Analysis

Control Theory · Linear systems

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⬇ L1-426.md

# ⚛ L1 Principle — Linear State-Space System Analysis **ID:** `L1-426` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Control Theory — *Linear systems* > **🎯 Problem class:** parameter estimation · **🧮 Solution space:** state vector > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555240 --- ## 🧠 1. Introduction **Linear State-Space System Analysis** is a **parameter-estimation problem** whose unknown lives in **state vector** space, within the **Linear systems** sub-domain of **Control Theory**. Measurements consist of N/A via a **state space output measurement** sensing mechanism. The forward operator applies, in order: computes eigen-pairs of a linear operator; S · observability · gramian operator; O · lyapunov · stability operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered state_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 ~= 50); model_order_mismatch 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** → S · observability · gramian → **y** (detector). ``` y = `S.observability.gramian` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.observability.gramian` | S · observability · gramian 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.stability` | O · lyapunov · stability operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Control Theory | | Sub domain | Linear systems | | Carrier | N/A | | Problem class | parameter_estimation | | Solution space | state_vector | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 3 | | L dag | 2.5 | ## 📡 4. Measurement Model Existence of the recovered state_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 ~= 50); model_order_mismatch dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | state_RMSE_normalized | | Secondary | output_prediction_RMSE | ## 📏 5. Operating Range (Ω) **Center problem class:** `parameter_estimation` · **Forward operator:** `state_space_output_measurement` **Center point:** | Parameter | Unit | Value | |---|---|---| | Noise snr db | dB | 30 | | System order n | N | 5 | | Sampling rate hz | Hz | 1000 | | Condition number kappa | — | 100 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Noise snr db | dB | 0 – 60 | | System order n | N | 2 – 50 | | Sampling rate hz | Hz | 10 – 100000.0 | | Condition number kappa | — | 10 – 1000000.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.05 state_RMSE_normalized | Metric | Range | |---|---| | State rmse normalized | 0.001 – 0.5 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **state_RMSE_normalized**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x0f32999d9f39215efc5d2aa3148591be18774780792deff2c9d41328f738731f` - **Chain tx hash:** `0x4b41989773fb14f4ca09a6dfdcc5b2cbb43d2a59112bc58e84d1b3bf4441114e` - **Chain block:** `41555240` --- ## File Mapping This bundle consists of: `L1-426.md`, `L1-426.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-426.md` | Source of truth — edit this | Human or LLM | | `L1-426.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 `L1-426`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-426.md).
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L2-426-001 Linear State-Space System Analysis — Nominal + Mismatch Specs

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📋 JSON metadata

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    "integration_axis": "time",
    "noise_model": "gaussian",
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    "sub_domain": "Linear systems",
    "title": "Linear State-Space System Analysis"
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      "system_order_n": [
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Linear systems",
  "title": "Linear State-Space System Analysis"
}

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