TinySysid · ERA — Design notes¶
Current implementation and parameter meaning¶
ERA_SVD_THRESH is currently 0.002f: retain components above 0.2% of the leading singular value. This is an amplitude ratio, not cumulative energy (which involves squared singular values). The 0.5% in historical diagrams and code comments below is retained for comparison and is not the current default.
ERA depends on correlation estimates, Hankel dimensions, signal-to-noise ratio and selected order. Claims of highest accuracy or fixed error bounds require common inputs, parameters and observed runs; SVD alone does not establish them.
ERA — Eigensystem Realization Algorithm: State-space OMA
Based on NExT (Natural Excitation Technique) + Hankel matrix + SVD + balanced realization. Extracts a state-space model from time-domain response, then identifies modal parameters from it. Validate results against input quality, model order and residuals.
Intuition¶
What ERA Does¶
ERA's recipe: 1. NExT: Convert random responses into "free decay"-like signals (cross-correlation) 2. Hankel matrix: Arrange time series into a block matrix—rows = "past", columns = "future" 3. SVD: Decompose this matrix, keep largest singular values → remove noise, keep system dynamics 4. Balanced realization: Reconstruct system matrix \(A\) from SVD results; its eigenvalues are the system's poles
Numerical design choices¶
- Large matrix (Hankel 75×50) SVD processes lots of information
- NExT preprocessing improves SNR
- Balanced realization helps manage scaling; numerical stability still needs validation
- Jacobi SVD converges fast (~15 sweeps) on embedded platforms
Algorithm¶
Historical threshold sketch (current value: 0.2%)
1. NExT: cross-correlation → free-decay-like signals
2. Build Hankel matrices
├─ H₀: [p·n_ch × q·n_ch] (default p=15, q=10)
└─ H₁: shifted by 1 lag
3. SVD: H₀ = U·Σ·Vᵀ
└─ Select order n (σ[k] > 0.5%·σ[0])
4. Balanced realization
├─ A = Σ⁻¹/²·Uᵀ·H₁·V·Σ⁻¹/²
└─ λ = eig(A) → discrete poles
5. Discrete → continuous modal parameters
When to Use ERA¶
- Joint frequency, damping and mode-shape estimation with a Hankel/SVD memory budget
- Data ≥ 20s (60s recommended)
- Final confirmation method
Design Deep Dive¶
1. Hankel Matrix (p,q) Tradeoffs¶
The table retains earlier estimates for five channels. Memory mainly describes a single Hankel matrix, not the entire algorithm workspace. Timing and accuracy lack complete build and run conditions and are not current performance guarantees.
| p,q | Hankel size | Memory | Accuracy | Time |
|---|---|---|---|---|
| (10,5) | 50×25 | 5 KB | Fair | ~1s |
| (15,10) | 75×50 | 15 KB | Good | ~2s |
| (20,15) | 100×75 | 30 KB | Better | ~3s |
| (30,20) | 150×100 | 60 KB | Best | ~5s (PSRAM) |
2. Jacobi SVD Choice¶
ERA needs full SVD (all SVs/vectors). Jacobi over QR/Householder because: no extra memory allocation, natural rank determination, float-friendly.
3. Balanced Realization¶
ERA simplifies: truncate Hankel SVD → construct A from H₁. The balancing step (Σ⁻¹/²·Uᵀ·H₁·V·Σ⁻¹/²) is included but the full balanced transform is omitted—ERA assumes the system is already controllable/observable.