Zonotopic Tubes in Reachability Analysis

• Zonotopic tubes are time-parametrized sequences of zonotopes that rigorously enclose trajectory sets in dynamical systems, aiding in reachability and safety analysis.
• They employ a computationally efficient center-plus-generator representation, facilitating linear mappings, Minkowski sums, and order reduction to manage complexity.
• Applications span tube-based predictive control and safe navigation in robotics, supported by guarantees on error bounds, stability, and scalable computations.

A zonotopic tube is a time-parametrized sequence of zonotopes, typically constructed to over- or under-approximate the trajectory sets (reachable, backward-reachable, or error sets) of dynamical systems under bounded uncertainty. Zonotopes are convex polytopes that can be represented in a computationally advantageous center-plus-generator form, enabling efficient linear-algebraic propagation of set-valued dynamics. Zonotopic tubes provide rigorous, set-based enclosures for the evolution of system states or errors in both model-based and data-driven control, reachability, and safety analysis for a wide range of linear and certain classes of nonlinear systems.

1. Zonotopes: Formal Definitions and Algebra

A zonotope $Z \subset \mathbb{R}^n$ is defined as

$Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$

where $c \in \mathbb{R}^n$ is the center and $G = [g_1, \ldots, g_p] \in \mathbb{R}^{n \times p}$ is the generator matrix, whose columns are the generators of the zonotope (Serry et al., 2021, Yang et al., 2021). The family generalizes to higher-order objects, such as matrix zonotopes,

$$\mathcal{M} = \{C + \sum_{i=1}^s G^{(i)} \beta_i : \|\beta\|_\infty \le 1 \}$$

where $C$ is a center matrix and $G^{(i)}$ are matrix generators (Ghiasi et al., 24 Dec 2025).

Constrained zonotopes extend this form by imposing linear constraints $A \xi = b$ on the generator coefficients, which can capture intersections and non-centrally symmetric geometry (Zhang et al., 2 Apr 2026).

Algebraic operations on zonotopes include linear maps,

$$L Z = \{Lc + L G \xi : \|\xi\|_\infty \le 1 \} = \langle Lc, L G \rangle$$

and Minkowski sums,

$$Z_1 \oplus Z_2 = \langle c_1 + c_2, [G_1, G_2] \rangle$$

Both are closed in the space of zonotopes, supporting recursive set-valued propagation (Akhormeh et al., 21 Sep 2025).

2. Zonotopic Tubes for Reachability Analysis

In continuous-time linear time-varying (LTV) systems,

$Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$0

with zonotopic uncertainty on initial state and inputs, the reachable tube on interval $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$1 is constructed as a recursive sequence of zonotopes $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$2 over time steps $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$3 (Serry et al., 2021). At each step,

$Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$4

where $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$5 is a numerically approximated transition matrix, $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$6 is the input convolution-approximation (a zonotope), and $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$7 is a rigorously constructed remainder zonotope to enclose truncation and quadrature error. The tube provides a first-order convergent, guaranteed over-approximation of all possible system evolutions.

For discrete-time systems and backward reachability, the primary challenge arises from the Minkowski difference, which is not closed under zonotopes. The approach of (Yang et al., 2021) uses LP-based inner approximation to zonotope differences, controlling the conservatism and enabling scalable computation of backward-reachable zonotopic tubes by repeated application of template-based set subtraction, order reduction, and linear inversion.

Zonotopic tubes can be combined forward and backward to yield safe tubes for MPC feasibility and planning, enforcing the intersection or containment of the forward tube within a precomputed safe backward tube.

3. Data-Driven and Adaptive Zonotopic Tube Synthesis

Zonotopic tubes are foundational in data-driven reachability and predictive control frameworks. Instead of presuming exact model knowledge, methods such as Exponentially Forgetting Zonotopic Recursive Least Squares (EF-ZRLS) construct a time-varying zonotope of admissible models $Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$8, which is updated online from measurement data corrupted by bounded noise (Akhormeh et al., 21 Sep 2025). At each step, tube propagation is performed under the entire model set,

$Z = \{c + G \xi : \|\xi\|_\infty \le 1 \}$9

where $c \in \mathbb{R}^n$0 is a matrix zonotope encoding allowable dynamics, and generator-reduction ensures scalable complexity.

Adaptive tube methods further alternate between tube shaping and feedback law updates, leveraging prior knowledge and online data to refine both disturbance and model zonotopes recursively (Ghiasi et al., 24 Dec 2025). This enables the co-design of tube geometry and feedback gains to achieve $c \in \mathbb{R}^n$1-contractivity (positively invariant shrinking tubes), jointly guaranteeing feasibility, disturbance attenuation, and robustness to model drift.

4. Zonotopic Tubes in Tube-Based Predictive and Robust Control

Tube-based MPC employs zonotopic tubes to enforce robust satisfaction of state and input constraints under bounded uncertainty. The standard paradigm uses a nominal trajectory $c \in \mathbb{R}^n$2 and an error variable $c \in \mathbb{R}^n$3, with feedback $c \in \mathbb{R}^n$4. The error evolves as

$c \in \mathbb{R}^n$5

A sequence $c \in \mathbb{R}^n$6 of zonotopes is constructed such that $c \in \mathbb{R}^n$7, computed via recursive set-inclusion (Russo et al., 2022, Farjadnia et al., 2024). The tightened constraints $c \in \mathbb{R}^n$8, $c \in \mathbb{R}^n$9 ensure robust satisfaction for all possible realizations. Stability and recursive feasibility are established through the existence of a common Lyapunov function over the zonotopic tube, and the tube evolution is independent of the MPC horizon length, scaling linearly in the number of generators.

In recent approaches, nonconvex or multi-layered zonotopic terminal sets are used to decrease conservatism and enlarge MPC feasible regions (Zhang et al., 2 Apr 2026). These schemes construct contractive constrained zonotopes and compute inner and outer approximations to the maximal robust positive invariant (MRPI) set, enabling layered certification of stability, feasibility, and screening regions in control synthesis.

5. Order Reduction and Computational Properties

Unmitigated propagation of generator matrices leads to linear or quadratic growth in generator dimension with time. Order-reduction techniques—including PCA-style bundling, support-function matching, and generator merging heuristics—are applied after each step to restrict the generator-count, typically trading minimal additional conservatism for substantial memory and CPU efficiency (Serry et al., 2021, Yang et al., 2021). For instance, volume loss below $G = [g_1, \ldots, g_p] \in \mathbb{R}^{n \times p}$0 per merge is typical, and backward reachability tubes with order capped at 5–10 maintain solution quality and scalability up to 100 steps in seconds.

Matrix zonotope reduction applies to the set-valued parameter estimators in the data-driven context, where reductions operate in vectorized space and maintain outer-containment of the model set (Akhormeh et al., 21 Sep 2025).

6. Theoretical Guarantees and Performance

Zonotopic tubes provide rigorous guarantees on over- or under-approximation error, feasibility margins, and (under suitable conditions) exponential stability. For reachable set over-approximation, the local discretization and quadrature error is absorbed as a remainder zonotope of order $G = [g_1, \ldots, g_p] \in \mathbb{R}^{n \times p}$1 in stepsize, with global convergence scaling as $G = [g_1, \ldots, g_p] \in \mathbb{R}^{n \times p}$2 and explicit conservatism control (Serry et al., 2021). In the scalable backward-reachability framework, volume ratios of the zonotopic inner approximation to the true Minkowski difference are typically $G = [g_1, \ldots, g_p] \in \mathbb{R}^{n \times p}$3, and generator-count control enables linear scaling in dimensionality and time-horizon (Yang et al., 2021).

Data-driven and adaptive zonotopic tubes maintain feasibility and robust constraint satisfaction with minimal samples and without reliance on conservative prior bounds. Probabilistic guarantees for recursive feasibility and closed-loop exponential stability are attained with high confidence for a wide range of system and disturbance realizations (Russo et al., 2022, Farjadnia et al., 2024, Ghiasi et al., 24 Dec 2025, Zhang et al., 2 Apr 2026).

7. Applications and Extensions

Zonotopic tubes underpin advanced methodologies in safe navigation, robotics, and cyber-physical systems verification, particularly under bounded data and structural uncertainty. They facilitate the construction of safe-motion corridors, elastic tubes for navigation, and robust feasible MPC strategies in high-dimensional, partially known systems (Ghiasi et al., 24 Dec 2025). The modularity of zonotopic tubes supports continual incorporation of online data, prior physical knowledge, and layered terminal-set designs, enabling both safety-critical control and formal verification tasks across discrete and continuous-time domains.

A plausible implication is continued expansion of zonotopic tube frameworks into nonlinear, real-time, and distributed contexts where scalability, compositionality, and online adaptability are critical.