Data-driven Set-based Estimation of Polynomial Systems with Application to SIR Epidemics

The paper’s Proposition 2 claims that M_Θp = (M^+_{X|Z} − M_w) Ω(Z_{x|z}, U^−)† contains all matrices Θp consistent with the data, but it neither assumes nor proves a full-row-rank (persistency of excitation) condition for the monomial data matrix Ω. Without rank(Ω)=m_a, right-multiplication by a pseudoinverse only recovers the minimum-norm solution; the general solution includes an extra additive nullspace term Y(I−ΩΩ†). The paper’s proof sketch does not address this and relies on a terse substitution using an interval pseudoinverse, leaving a logical gap. By contrast, the candidate solution is correct: it derives the same inclusion and explicitly states the needed rank condition for exactness, while noting that without it the displayed set captures only the minimum-norm Θp. See the paper’s setup of f_p(ζ)=Θ_p h(ζ) and data model (3)–(5) , the construction of Zx|z(k) in Lemma 1 (6) from z(k)=Hx(k)+γ(k) , the stacked bound M^+_{X|Z} in Proposition 1 (8) , the interval/pseudoinverse conventions , and Proposition 2 (9) with its brief proof sketch and “contains all matrices” phrasing .