Exact block-Hankel realization

Fix integers $d\ge 0$, $m,p\ge 1$, and $q\ge 1$. The input is a list of exact Markov matrices $M_0,\dots ,M_{2q-1}\in M_{p\times m}(\mathbb{R})\mathrm{.}$ Use the exact-real linear-algebra model: an exact rank test returns the rank of a finite real matrix; a thin rank-$d$ singular-value decomposition of a rank-$d$ matrix $H_0$ returns $H_0=U_d{\mathrm{\Sigma }}_d{V}_d^{\mathsf{T}}$ with ${\mathrm{\Sigma }}_d$ diagonal and strictly positive, and an exact Moore--Penrose routine returns a matrix satisfying the four Penrose equations. A routine failure or a returned object that fails its stated exact identities is reported explicitly. This specification makes no finite-bit computability or complexity claim for arbitrary real inputs.

Form the $q$-by-$q$ block Hankel matrices $H_0=[M_{i+j}{]}_{0\le i,j<q},H_1=[M_{i+j+1}{]}_{0\le i,j<q},$ each of shape $qp$ by $qm$. Execute the following steps.

1. Apply the exact rank test to $H_0$. If it fails, return failed-solve; if its value is not $d$, return inconsistent-rank-data.

2. If $d=0$, use the empty factorization $H_0=\widehat{O}\widehat{R}$ with $\widehat{O}$ of shape $qp$ by $0$ and $\widehat{R}$ of shape $0$ by $qm$. If $d>0$, obtain the thin rank-$d$ SVD in the declared exact model and set $\widehat{O}:=U_d{\mathrm{\Sigma }}_d^{1\mathrm{/}2},\widehat{R}:={\mathrm{\Sigma }}_d^{1\mathrm{/}2}{V}_d^{\mathsf{T}}\mathrm{.}$ Verify $H_0=\widehat{O}\widehat{R}$, $\mathrm{rank}\widehat{O}=d$, and $\mathrm{rank}\widehat{R}=d$; if any operation or verification fails, return failed-solve.

3. Compute the exact Moore--Penrose inverses ${\widehat{O}}^{+}$ and ${\widehat{R}}^{+}$ and verify their four Penrose equations. If a computation or verification fails, return failed-solve.

4. Set $\widehat{A}:={\widehat{O}}^{+}{H}_1{\widehat{R}}^{+}$, let $\widehat{B}$ be the first $m$ columns of $\widehat{R}$, and let $\widehat{C}$ be the first $p$ rows of $\widehat{O}$. Return success with $(\widehat{A},\widehat{B},\widehat{C})$. For $d=0$, these are the corresponding empty matrices.

The routine can report failed-solve for a failed or invalid exact-operation backend. The existence theorems for finite-matrix SVDs and Moore--Penrose inverses ensure that the exact mathematical operations requested in steps 2–3 exist; this definition alone asserts no noisy rank selection or stable parameter recovery.

Markov parameter convention $M_k=C{A}^{k}B$

Rank of a finite-dimensional linear map

Finite-dimensional singular-value decomposition

Four Penrose equations

Existence and uniqueness of the finite-matrix pseudoinverse

Finite matrix product laws

Dahleh, Dahleh, and Verghese, Lectures on Dynamic Systems and Control, Chapter 25

Lall, State-Space Realization, Engr207c lecture notes