L1 ⚛ L1-429 ⊙ Testnet

Unscented Kalman Filter (UKF)

Control Theory · Sigma-point filtering

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⬇ L1-429.md

# ⚛ L1 Principle — Unscented Kalman Filter (UKF) **ID:** `L1-429` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Control Theory — *Sigma-point filtering* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** nonlinear state estimate > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555240 --- ## 🧠 1. Introduction **Unscented Kalman Filter (UKF)** is a **nonlinear inverse problem** whose unknown lives in **nonlinear state estimate** space, within the **Sigma-point filtering** sub-domain of **Control Theory**. Measurements consist of N/A via a **sigma point state estimation** sensing mechanism. The forward operator applies, in order: S · ukf · sigma points UT operator; adds a prior term that biases the solution toward smoothness/sparsity; O · innov · sigma point update operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered nonlinear_state_estimate 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); heavy_tail_distribution 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 · ukf · sigma points UT → O · innov · sigma point update → **y** (detector). ``` y = `O.innov.sigma_point_update` `S.ukf.sigma_points_UT` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.ukf.sigma_points_UT` | S · ukf · sigma points ut operator | | `O.innov.sigma_point_update` | O · innov · sigma point update operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Control Theory | | Sub domain | Sigma-point filtering | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | nonlinear_state_estimate | | Noise model | gaussian | | Integration axis | discrete_time | | Difficulty delta | 3 | | L dag | 3 | ## 📡 4. Measurement Model Existence of the recovered nonlinear_state_estimate 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); heavy_tail_distribution dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | RMSE_vs_EKF_improvement | | Secondary | sigma_consistency_NEES | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `sigma_point_state_estimation` **Center point:** | Parameter | Unit | Value | |---|---|---| | N steps | — | 200 | | Q r ratio | — | 1 | | N state dim | — | 6 | | Nonlinearity degree | — | 1 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N steps | — | 20 – 5000 | | Q r ratio | — | 0.01 – 100 | | N state dim | — | 2 – 20 | | Nonlinearity degree | — | 0.1 – 3.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.06 RMSE_vs_EKF_improvement | Metric | Range | |---|---| | Rmse vs ekf improvement | 0.01 – 0.3 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **RMSE_vs_EKF_improvement**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xc35665859661d5427d608cee5f81e5216807c057c8a57e7308a243870d8d824b` - **Chain tx hash:** `0x6f06ba118e9d95beb7c8737e3f9b55bc338242074ab33c87c006731acdf9cb24` - **Chain block:** `41555240` --- ## File Mapping This bundle consists of: `L1-429.md`, `L1-429.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-429.md` | Source of truth — edit this | Human or LLM | | `L1-429.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 `L1-429`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-429-001 Unscented Kalman Filter (UKF) — Nominal + Mismatch Specs

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📋 JSON metadata

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  "chain_hash": "0xc35665859661d5427d608cee5f81e5216807c057c8a57e7308a243870d8d824b",
  "chain_tx_hash": "0x6f06ba118e9d95beb7c8737e3f9b55bc338242074ab33c87c006731acdf9cb24",
  "domain": "Control Theory",
  "hardness_fn": {
    "delta": 3,
    "kappa": 1000,
    "metric": "RMSE_vs_EKF_improvement",
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    "regime": "Existence of the recovered nonlinear_state_estimate 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); heavy_tail_distribution dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.",
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  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Control Theory",
    "integration_axis": "discrete_time",
    "noise_model": "gaussian",
    "primitives": [
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      "O.regularize",
      "O.innov.sigma_point_update"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "sigma_point_state_estimation",
    "solution_space": "nonlinear_state_estimate",
    "sub_domain": "Sigma-point filtering",
    "title": "Unscented Kalman Filter (UKF)"
  },
  "size_tiers": {
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      "N_steps"
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    ],
    "center_spec": {
      "epsilon_fn_center": "0.06 RMSE_vs_EKF_improvement",
      "forward_operator": "sigma_point_state_estimation",
      "input_format": "measurement_only",
      "omega": {
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        "Q_R_ratio": 1.0,
        "n_state_dim": 6,
        "nonlinearity_degree": 1.0
      },
      "problem_class": "nonlinear_inverse"
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    "epsilon_bounds": {
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        0.3
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    },
    "omega_bounds": {
      "N_steps": [
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        5000
      ],
      "Q_R_ratio": [
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        100
      ],
      "n_state_dim": [
        2,
        20
      ],
      "nonlinearity_degree": [
        0.1,
        3.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Sigma-point filtering",
  "title": "Unscented Kalman Filter (UKF)"
}

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