NOTES¶

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

Why ERA Is Most Accurate¶

• Large matrix (Hankel 75×50) SVD processes lots of information
• NExT preprocessing improves SNR
• Balanced realization ensures numerical stability
• Jacobi SVD converges fast (~15 sweeps) on embedded platforms

Algorithm¶

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¶

• Highest overall accuracy (freq/damping/shapes)
• Data ≥ 20s (60s recommended)
• Final confirmation method

Design Deep Dive¶

1. Hankel Matrix (p,q) Tradeoffs¶

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.