Excitation Richness Monitor — System Identification/Validation

Excitation Richness Monitor

System Identification / Validation

Watches whether the data flowing past is rich enough to identify a model with, by accumulating the regressor information matrix and reducing it to two scalars:

S = Σ φφ′,   outputs det(S) and trace(S)

It is the missing half of every recursive estimator's warm-up caveat. An estimator reports what it fitted; this reports whether the data supported fitting anything, so a diagram can gate “trust the estimate” on “the data could actually identify it”.

How to read the two outputs. det(S) is the product of the eigenvalues of S, so it collapses toward zero as soon as any direction of the regressor space goes unexcited – which is exactly the condition that makes a parameter unidentifiable. It is not scale-free, though: it grows like the n-th power of the signal amplitude. trace(S) is the sum of the same eigenvalues and supplies that scale, so the quantity worth thresholding is det against a power of trace rather than det alone.

Stateful and discrete by nature: it declares itself discrete-only and always steps at its own rate.

Ports

• phi – the regressor, a column [n, 1]. n is taken from this port, not from any parameter, so nothing can disagree with the signal wired in. The usual source is a Tapped Delay of the input, or the same regressor already feeding a Recursive Least Squares block.
• det – det(S), a scalar. Zero means the data is rank-deficient in at least one direction.
• trace – trace(S), a scalar, always positive once any signal has arrived. The scale to judge det against.

Parameters

• Accumulation Method – how the sum is aged. The two are different computations, not two tunings of one:
  • Forgetting Factor – an exponential tail, S ← λ·S + φφ′. Cheap: the only state is S itself. Responds smoothly, and never fully forgets.
  • Sliding Window – a rectangular window of the last N regressors, held in a shift register and evicted exactly. Answers “was the data rich over this stretch” with a definite boundary, at the cost of carrying N×n past values.
• Forgetting Factor (λ) – in (0, 1], used by the first method only. It scales the accumulator every step, so an unexcited direction decays as λ^k while an excited one settles near 1/(1−λ). Unlike a recursive estimator's forgetting factor, this one cannot wind up: the accumulator is multiplied by λ rather than divided by it, so the recursion is a contraction and λ near 1 is safe here even though it is dangerous in Recursive Least Squares.
• Window Length (N) – a positive whole number, used by the second method only. Until N regressors have arrived the window is still filling, and det is legitimately small – that is the block reporting a real warm-up, not an artefact.
• ⚠ λ and N are bounded by the EXPORT format, not by stability. det is the n-th power of the accumulator's scale, and the scale is roughly 1/(1−λ) or N, so det grows very fast in both. The Q16.16 fixed-point format the three HDL targets use saturates near 32768: measured on a 3-tap regressor of a unit-amplitude signal, λ = 0.997 reaches det ≈ 1.3×10⁶ and overflows, λ = 0.93 peaks near 287, and a window of N = 32 near 3400. Software targets carry the full double range and are unaffected. If a model is destined for HDL, choose λ and N so that (scale)ⁿ stays inside the format, or read the block's trace output instead, which grows only linearly.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets, and the three HDL targets are genuinely synthesizable Q16.16 rather than this family's usual simulation-only real. That is the whole reason the measure is det rather than the textbook smallest eigenvalue: the accumulation is a rank-1 update, the determinant is a sum of products and the trace is a sum, so no division, no square root and no iteration reaches the emitted datapath. The order n and the method are both fixed at export time, so each core carries one accumulation path and a determinant unrolled into n! terms with their signs baked in – a flat sum of products in the seven software targets, and a chained product in the three HDL ones, where each multiply has to be brought back down by the fraction width.

The order is capped at 6, and the limit is n! rather than n³: a divide-free determinant means cofactor expansion, because every O(n³) route (LU, LDL′, Cholesky) needs a division and Cholesky a square root as well. The expansion costs n! products – 6 at n=3, 720 at n=6, 40,320 at n=8 – so an order above 6 is refused with that reason rather than emitted as a megabyte of unrolled arithmetic. trace is O(n) and has no such limit.

A normalized richness scalar is deliberately not an output. det / (trace/n)ⁿ is dimensionless and equals 1 exactly when all eigenvalues are equal, but it needs a runtime division and an n-th power, which would carry all three HDL targets into simulation-only real for a quantity that is already buildable downstream: n is known at design time, so Gain, Product and Divide express it from the two ports this block emits, and only the diagrams that ask for it pay for it.

Simulink bridge

None. The System Identification Toolbox has no block with this port shape, and it is not installed on the machine this library is verified against, so a bridge could not be parity-verified even if one were written. Every block in this family registers Support::None for the same reason. It therefore has no parity testbench, which is the documented consequence; code export verification still covers it across all ten languages.

Notes

• Not a state space: the reduction to det is nonlinear in the state, so no A/B/C/D describes the block and model reduction correctly refuses it.
• Feed it the SAME regressor the estimator sees. The measure is only meaningful about the model being fitted – a richness verdict computed from a different vector answers a different question.
• A rank-deficient regressor gives det = 0 exactly, in every backend. If phi is built by scaling one signal into a column, S is rank 1 and det is identically zero for any n ≥ 2: that is the correct answer, and it is also the commonest way to wire the block up meaninglessly.