Constrained Matrix Zonotopes

• Constrained matrix zonotopes are set-valued objects in matrix space defined by a center, generator matrices, and linear equality constraints that couple their parameters.
• They enable precise uncertainty representation and propagation in LTI systems by capturing non-convex dependencies without over-approximation.
• Their closed-form operations with constrained polynomial zonotopes support scalable, data-driven reachability analysis for systems with measurement noise.

A constrained matrix zonotope (CMZ) is a set-valued object in $\mathbb{R}^{m\times n}$ that generalizes the notion of zonotopes by allowing linear equalities on the generator-weight parameters. CMZs provide a compact, algebraically closed representation for parameterizing sets of matrices, including those arising as feasible linear system models in data-driven identification and as operators in reachability computations. CMZs underpin exact operations with constrained polynomial zonotopes (CPZs), making them central to non-convex reachable set propagation for linear time-invariant (LTI) systems in the presence of uncertainty and constraints (Zhang et al., 2 Apr 2025).

1. Formal Definition

A CMZ $\mathcal{N} \subset \mathbb{R}^{m \times n}$ is defined by a center matrix $C \in \mathbb{R}^{m \times n}$, generator matrices $$G_{(1)}, \ldots, G_{(\gamma)} \in \mathbb{R}^{m \times n}$$, a set of linear constraint matrices $A_{(k)}$ and a right-hand side $B$, all parameterized over real scalars $\alpha_{(k)}$. Its explicit representation is

$$\mathcal{N} =\Bigl\{\,C+\sum_{k=1}^\gamma\alpha_{(k)}\,G_{(k)} \;\Bigm|\; \sum_{k=1}^\gamma\alpha_{(k)}\,A_{(k)}=B,\; \alpha_{(k)}\in[-1,1]\Bigr\}.$$

By stacking generators and constraints, this becomes

$$\mathcal{N} = \{C+G\alpha \mid A\alpha = B,\; \|\alpha\|_{\infty} \leq 1\},$$

with $G \in \mathbb{R}^{m\times(n\gamma)}$ and $A$ concatenating all $A_{(k)}$. The coefficients $\alpha_{(k)}$ describe each generator’s amplitude and are coupled linearly.

2. Constraint Structure and Comparison to Zonotopes

CMZs generalize classic zonotopes by introducing algebraic coupling among the generator weights. For standard zonotopes, the set

$$\mathcal{Z} = \{c + G\xi \mid \|\xi\|_\infty \leq 1\}$$

has independent $\xi$ weights. In contrast, CMZs include constraint equations $A\alpha = B$, defining a (possibly non-convex) slice through the generator-weight hypercube. These constraints enforce dependency among parameters, which in the matrix case can themselves be matrix-valued, maintaining linearity with respect to $\alpha_{(k)}$. The resulting set may capture non-convex and highly structured uncertainty that cannot be represented with classical zonotopes.

3. Exact Multiplication with Constrained Polynomial Zonotopes

Given a CPZ $\mathcal{P} \subset \mathbb{R}^n$, the exact propagation of its image under all $N \in \mathcal{N}$ is defined as

$$\mathcal{N}\, \otimes\, \mathcal{P} = \{ Np \mid N \in \mathcal{N},\; p \in \mathcal{P}\} \subseteq \mathbb{R}^m.$$

The key result ensures that $\mathcal{N} \otimes \mathcal{P}$ is always a CPZ whose parameters can be constructed in closed form: $$\boxed{ \mathcal{N}\otimes\mathcal{P} \;=\; \langle\, C_{\mathcal N}c_{\mathcal P}, \, [G_{\mathcal N}c_{\mathcal P},\, C_{\mathcal N}G_{\mathcal P},\, G_f], \, E_{\mathcal{NP}}, \, A_{\mathcal{NP}}, \, B_{\mathcal{NP}}, \, R_{\mathcal{NP}}, \, id_{\mathcal{NP}} \rangle_{\mathrm{CPZ}} }$$ Here, $G_{\mathcal N}c_{\mathcal P}$ and $C_{\mathcal N}G_{\mathcal P}$ arise from expanding the matrix–vector multiplication, and $G_f$ comprises pairwise generator products ($$G_f^{((i-1)h_{\mathcal P}+j)} = G_{\mathcal N}^{(i)} G_{\mathcal P}^{(\cdot,j)}$$). The new constraint and exponent matrices, along with identifier merging, enforce that dependencies among $\alpha_{(k)}$ are maintained exactly—yielding a CPZ that encodes the precise image without relaxation or projection.

4. Closure Property and Proof Outline

The closure of the CPZ family under multiplication by CMZs is established by:

• Merging the $\alpha_{(k)}$ parameters and their identifiers across both operands, ensuring correct tracking of dependencies.
• Expressing generic elements as

$$N = C_{\mathcal N} + \sum_{i=1}^\gamma \left(\prod_{r=1}^a \alpha_{(r)}^{\overline E_{\mathcal N}^{(r,i)}}\right) G_{\mathcal N}^{(i)}, \quad p = c_{\mathcal P} + \sum_{j=1}^{h_{\mathcal P}} \left(\prod_{r=1}^a \alpha_{(r)}^{\overline E_{\mathcal P}^{(r,j)}}\right) G_{\mathcal P}^{(\cdot,j)}$$

and expanding $N p$.

• Aggregating all new constraint equations: each $\alpha$ must satisfy both sets of original equalities.
• As the construction requires no projection and preserves all algebraic structure, the resulting CPZ is exact: the image set contains no superfluous points.

5. Role in Data-Driven Reachability of LTI Systems

In data-driven LTI reachability, CMZs provide a non-parametric description of all consistent models given observed input/state/perturbation trajectories subject to noise, formulated as

$$x_{k+1} = \Phi_{\rm tr} x_k + \Gamma_{\rm tr} u_k + w_k.$$

Offline, feasible $(\Phi_{\rm tr}, \Gamma_{\rm tr})$ pairs are described by a CMZ $\mathcal{M}^{\Sigma}$ constructed from measurement data. Online, new trajectories intersect and refine this CMZ, reducing feasible model uncertainty.

The reachable state set at step $k$ is managed as a CPZ $\mathcal{R}_k$. Propagation over one step involves

$$\mathcal{R}_{k+1} = \mathcal{M}^\Sigma \otimes (\mathcal{R}_k \times \mathcal{U}_k) \boxplus \mathcal{Z}_w,$$

with

• $\otimes$: exact CMZ–CPZ multiplication,
• $\times$: Cartesian product (CPZ),
• $\boxplus$: CPZ addition (exact).

All operations preserve non-convexity and inter-generator dependencies, culminating in tight, non-conservative set approximations. The algorithms remain polynomial in generator and constraint counts, supporting scalable computation.

6. Computational Properties and Significance

CMZs are algorithmically tractable: their manipulation—intersection, exact multiplication, constraint integration—can be performed in closed form by matrix algebra, requiring no relaxation or convexification. This retains the structural richness of the original uncertainty sets, avoiding over-approximation that plagues polyhedral or ellipsoidal methods. For non-convex cases especially, the expressive power and exact closure properties of CMZ/CPZ analysis enable sharp, data-driven reachable set envelopes (Zhang et al., 2 Apr 2025).

7. Tabular Summary of Core Components

Element	Symbol(s)	Role
Center (Offset)	$C$	Mean matrix of the set
Generator Matrices	$G_{(k)}$, $G$	Directions for variability
Coefficient Vector	$\alpha = (\alpha_{(1)},\ldots)$	Free parameters (coupled/boxed)
Constraint Matrices	$A_{(k)}$, $A$	Coupling among $\alpha_{(k)}$
RHS of Constraints	$B$	Specifies the affine slice

These components are assembled algebraically to define and manipulate constrained matrix zonotopes in both theoretical and applied non-convex reachability analyses.