State-space → transfer-function drops the feedthrough contribution

[gabrielfrasantos]

Severity: high
Domain: control_analysis
Status: VERIFIED — read + hand-traced against a210d34 on 2026-08-10
Suggested labels: bug, numerical-correctness, control-analysis

Summary

ToTransferFunction() applies the feedthrough matrix D only to the leading numerator coefficient.
The terms D · charPoly[k] for k ≥ 1 are omitted, so any system with D ≠ 0 converts to the
wrong transfer function.

Location

numerical/control_analysis/TransferFunctionStateSpace.hpp

Evidence

The Faddeev–LeVerrier recursion for the characteristic polynomial and the N_k matrices is
correct. The numerator assembly is not:

std::array<T, n + 1> numCoeffs{};
numCoeffs[0] = sys.D.at(0, 0);

SqMat N{ SqMat::Identity() };
for (std::size_t k{ 1 }; k <= n; ++k)
{
    auto CB{ sys.C * N * sys.B };
    numCoeffs[k] = CB.at(0, 0);          // missing + D * charPoly[k]
    N = sys.A * N;
    for (std::size_t i{ 0 }; i < n; ++i)
        N.at(i, i) += charPoly[k];
}

Expected

Since G(s) = C(sI−A)⁻¹B + D:

numerator(s) = D · charPoly(s) + Σ_k (C N_{k−1} B) s^{n−k}

so numCoeffs[k] = C·N_{k−1}·B + D·charPoly[k] for all k ≥ 1.

Trace

A = −1, B = C = 1, D = 2, n = 1:

charPoly = [1, 1]                      (denominator s + 1)
true G(s) = 1/(s+1) + 2 = (2s + 3)/(s + 1)

code: numCoeffs[0] = 2
      numCoeffs[1] = C·I·B = 1
      => (2s + 1)/(s + 1)              wrong DC gain: 1 instead of 3

Why tests do not catch it

Existing round-trip tests use strictly proper systems (D = 0), where the missing term vanishes
identically. The one feedthrough test does not convert back.

Suggested fix

numCoeffs[k] = CB.at(0, 0) + sys.D.at(0, 0) * charPoly[k];

Add a round-trip test with D ≠ 0 and an analytic DC-gain assertion.