NOTES¶
SSI — Stochastic Subspace Identification: Cross-Order Voting for Best Damping
Build block Toeplitz matrices, perform SVD, use shift invariance to extract system matrices, scan across multiple model orders with voting. Best damping accuracy when data is sufficient.
Intuition¶
Core Idea: Big Matrix from Correlation¶
Like ERA, SSI builds a large matrix and does SVD, but SSI uses a Toeplitz matrix (symmetric block matrix from cross-correlation).
The key innovation is cross-order voting:
Order 2 → n candidate modes
Order 4 → n candidate modes
Order 6 → ...
Order 16 → ...
All candidates → cluster → stable modes are real
This avoids manual model order selection—modes that keep appearing across orders are the real physical ones.
Pairwise Variant (SSI-PW)¶
Process 2 channels at a time, build 2×2 block Toeplitz, run SSI on each pair, then vote across all results. More robust for weak modes.
SSI vs ERA¶
| ERA | SSI | |
|---|---|---|
| Matrix type | Hankel | Toeplitz |
| SVD | Once | Multiple (cross-order) |
| Damping | High | Highest (with enough data) |
| Data needed | Moderate | More (≥ 60s recommended) |
Design Deep Dive¶
1. Toeplitz vs Hankel¶
| ERA (Hankel) | SSI (Toeplitz) | |
|---|---|---|
| Structure | rows=past, cols=future | rows=future, cols=past |
| Post-SVD | Truncate → balanced realization → A | Truncate → observability → shift invariance → A |
| Noise type | Impulsive | Stationary random |
| Damping | High | Highest (with sufficient data) |
Toeplitz columns are cross-correlation functions themselves, closer to "statistical averages" and more natural for stationary random excitation. Hankel processes a single realization and is more sensitive to transient components.
2. Cross-Order Voting Design¶
SSI's key advantage: no manual model order selection. All orders n=2,4,...,16 pool candidates → 10% frequency tolerance clustering → majority vote.
Why SSI-PW (pairwise) is more robust: global SSI builds nch×blockrows Toeplitz (60×60 for 5ch). SSI-PW processes 2ch at a time (24×24). Smaller matrices are more stable; weak modes aren't drowned by strong modes on other channels.
3. SSI vs ERA Selection¶
| Condition | Recommended | Why |
|---|---|---|
| Short data (< 30s) | ERA | Hankel works better with limited data |
| Abundant data (> 60s) | SSI | Toeplitz statistical averaging wins |
| Damping critical | SSI | Cross-order voting + averaging |
| Frequency critical | ERA | Direct output from balanced realization |
| ## When to Use SSI |
- Plenty of data (≥ 60s, recommended 120s)
- Damping accuracy is critical
- Need automatic model order selection