Coprime Factorizations & Bézout Identity

• System coprime factorizations express transfer functions as products of stable matrices using a Bézout identity, ensuring no common unstable factors.
• They underpin robust controller synthesis, fault detection, and distributed control by leveraging state-space methods and algebraic solution techniques.
• Recent advances include algorithms for high-dimensional, sparsity-preserving factorizations and resilient control schemes via Riccati equations and LMIs.

A system coprime factorization expresses a transfer function (or polynomial) matrix as a product or quotient of matrix pairs without common unstable factors, underpinned by the existence of solutions to an associated Bézout identity. This framework is central to robust controller synthesis, system structural analysis, and fault/attack detection. The Bézout identity guarantees solvability of underlying Diophantine-type equations and underlies constructions of image and kernel representations that are crucial for both control and diagnostic functions. Recent developments span computational algebraic methods for arbitrary dimension, sparse networked control via structured coprime factorizations, and unified schemes for resilient control and detection in cyber-physical systems (Sabău et al., 2022, Li et al., 12 Jan 2026, Cayron, 2021).

1. Fundamental Definitions and Bézout Identities

Let $G(\lambda) \in \mathbb{R}(\lambda)^{p \times m}$ denote a linear time-invariant (LTI) plant transfer function, with $\lambda = s$ (continuous) or $z$ (discrete). The factorization domain $\mathcal{S}$ is the stability region: $\mathrm{Re}\, s < 0$ (continuous) or $|z|<1$ (discrete).

Right coprime factorization (RCF):

$G(\lambda) = N(\lambda) M(\lambda)^{-1}$ with $N(\lambda)\in\mathbb{R}(\lambda)^{p\times m}$, $M(\lambda)\in\mathbb{R}(\lambda)^{m\times m}$ both analytic and invertible on $\mathcal{S}$, and there exist $$X(\lambda), Y(\lambda)\in\mathbb{R}(\lambda)^{m\times m}$$, analytic in $\mathcal{S}$, such that

$$N(\lambda) Y(\lambda) + M(\lambda) X(\lambda) = I_m.$$

Left coprime factorization (LCF):

$$G(\lambda) = \tilde{M}(\lambda)^{-1} \tilde{N}(\lambda)$$ with $$\tilde{M}(\lambda)\in\mathbb{R}(\lambda)^{p\times p}$$, $$\tilde{N}(\lambda)\in\mathbb{R}(\lambda)^{p\times m}$$ analytic and full-row-rank on $\mathcal{S}$, and with $$\tilde{X}(\lambda), \tilde{Y}(\lambda)\in\mathbb{R}(\lambda)^{p\times p}$$ analytic on $\mathcal{S}$ such that

$$\tilde{Y}(\lambda)\tilde{N}(\lambda) + \tilde{X}(\lambda)\tilde{M}(\lambda) = I_p.$$

A pair $(M, N)$ (resp. $(\tilde{M},\tilde{N})$) is fully coprime if $M$ (resp. $\tilde{M}$) has full normal rank and no zeros outside $\mathcal{S}$ (Sabău et al., 2022, Li et al., 12 Jan 2026).

The double (block) Bézout identity combines right and left factor pairs:

$$\begin{bmatrix} X & Y \ -\tilde{N} & \tilde{M} \end{bmatrix} \begin{bmatrix} M & -\tilde{Y} \ N & \tilde{X} \end{bmatrix} = I_{p+m}$$

The upper-left and lower-right diagonal blocks yield the single (right or left) Bézout identities (Li et al., 12 Jan 2026).

2. Existence Results and State-Space Construction

Existence of stable coprime factorizations relies on standard system-theoretic properties: a minimal realization $x' = Ax + Bu$, $y = Cx + Du$ of $G(\lambda)$ is assumed stabilizable and detectable; under these, state-space approaches yield stable coprime factors in $\mathcal{RH}_\infty$.

A pivotal construction is the System-Response-Type Realization (SRTR). The state-space can be partitioned so that, for any $K\in\mathbb{R}^{(n-p)\times p}$,

$$W(\lambda)=A_{11}+A_{12}(\lambda I-A_{22})^{-1}A_{21} + K[\,A_{12}(\lambda I-A_{22})^{-1}A_{21} - A_{11}]$$

$$V(\lambda)=B_1+A_{12}(\lambda I-A_{22})^{-1}B_2 + K[\,A_{12}(\lambda I-A_{22})^{-1}B_2 - B_1]$$

then $$G(\lambda) = (\lambda I - W(\lambda))^{-1} V(\lambda)$$. The pair $(\lambda I - W, V)$ forms a left coprime factorization when the original realization is minimal. All SRTRs of specified degree are parameterized by $K$ (Sabău et al., 2022).

Passing from coprime factorizations to (and from) normalized form is realized via Riccati equations. Specifically, a minimal realization of an LCF can be used to define a non-symmetric algebraic Riccati equation, whose right-stabilizing solution yields an associated SRTR (Sabău et al., 2022, Li et al., 12 Jan 2026).

3. Algorithms and Computational Procedures

State-space and algebraic methods are complemented by algorithms for structured or sparse coprime factorizations and for high-dimensional polynomial/vector settings. For analytic transfer matrices or polynomial vectors:

• N-dimensional Bézout identity algorithms: Given $p=(p_1,\ldots,p_N)\in\mathbb{Z}^N$, all integer solutions $x$ to $p\cdot x=d=\gcd(p)$ are computed by constructing a unimodular matrix $U$ whose columns describe the solution space as $x = b_1 + z_2 b_2 + \cdots + z_N b_N$, with $p\cdot b_1=1$ and $p\cdot b_j=0$ for $j\geq2$. Cayron's algorithm recursively builds such cells, then applies lattice reduction for optimal basis vectors (Cayron, 2021).
• Sparsity-preserving coprime factorizations: Zero-patterns for distributed or networked control are achieved by imposing linear constraints on $K$ during SRTR construction, with stability imposed via LMIs or spectral constraints. The resultant convex feasibility problem can be solved efficiently using SDP solvers (Sabău et al., 2022).

The table summarizes these methodologies and their scope:

Approach	Domain	Main Computational Steps
State-space factorizations	LTI $\mathcal{RH}_\infty$	Riccati equations, state-space partitioning
Polynomial Bézout/Diophantine	$\mathbb{Z}[x]^N$	Euclid/Lattice reduction via Cayron's alg.
Sparsity-preserving SRTR	Structured/Networked LTI	Linear + LMI constraints on SRTR $K$

4. Bézout Identity: Algebraic and Analytical Details

The algebraic approach interprets coprimeness as matrix polynomials with no common left/right divisors, allowing the multi-dimensional Bézout identity to be constructed via a generalization of the Euclidean algorithm. The parametric solution, as generated by unit-cell methods, mirrors the construction of left/right coprime factor matrices.

The analytical (state-space) approach stabilizes the realization through Riccati-based state feedback and observer gains, resulting in factors that are both stable and satisfy the Bézout identity. Normalized coprime factorizations further enforce co-inner or inner structures optimal for $\mathcal{H}_2$ or $\mathcal{H}_\infty$ performance (Sabău et al., 2022, Cayron, 2021).

5. Applications in Control, Fault/Attack Detection, and Distributed Systems

Coprime factorizations with the Bézout identity are foundational for several advanced applications:

• Youla–Kučera parameterization: The family of all stabilizing controllers is parameterized by stable free transfer $Q$, with the controller constructed as

$$K_Q(s) = (Y - QM)^{-1} (X - QN), \quad Q \in \mathcal{RH}_\infty,$$

under the Bézout identity $NX + MY=I$. This enables systematic controller redesign and robustification (Sabău et al., 2022, Li et al., 12 Jan 2026).

• Fault and attack detection: The image and kernel representations induced by $(M, N)$ and $(-\tilde{N}, \tilde{M})$ span the nominal input-output behaviors. Residual generators based on the kernel subspace detect faults/attacks: deviations from expected behavior are mapped to orthogonal subspaces, ensuring high sensitivity and robustness for diagnosis (Li et al., 12 Jan 2026).
• Structure-preserving distributed control: For networks (e.g., ring, mesh, star), imposing structural constraints through SRTR and associated LP/LMI-based algorithms yields controllers that respect interconnection sparsity, enabling scalable distributed implementation (Sabău et al., 2022).

6. Illustrative Examples and Comparative Algorithms

In (Sabău et al., 2022), a 6-node ring network is used to demonstrate the construction of distributed sparse stabilizing controllers. Minimal realizations, coprime-based convex design, and subsequent structure-imposing algorithms yield low-order, nearest-neighbor controllers with explicit transfer-function forms, implementing efficient distributed feedback.

Fast Bézout identity computation in the integer case is shown for $p = (2,5,3)$, with the algorithm yielding short Bézout vectors and an explicit parametric representation of all solutions. Coprime factorization is immediate, with the unimodular matrix capturing coprimeness without need for time-consuming Smith normal form computations (Cayron, 2021).

7. Key Assumptions and Unimodularity Considerations

The existence and utility of system coprime factorizations with Bézout identity rest on:

• Plant stabilizability/detectability, ensuring stable factorizations.
• Factor (and Bézout solution) properness and internal stability.
• Unimodularity of the Bézout matrix, guaranteeing invertibility of mappings and robustness in residual-based filtering and kernel/image representations.
• For analytic domains, normalized (co-inner/inner) factors ensure optimality for $\mathcal{H}_2$/$\mathcal{H}_\infty$ design (Li et al., 12 Jan 2026).

These conditions are essential for stability, uniqueness (modulo unimodular equivalence), and the analytical tractability of controller synthesis, distributed implementation, and system monitoring.