data-driven-reachability-matrix-perturbation

name	data-driven-reachability-matrix-perturbation
description	Data-driven reachability analysis framework using matrix perturbation theory. Provides Cai-Zhang bounds for matrix zonotopes and constrained matrix zonotopes. Enables efficient reachable-set propagation with coefficient-space approximation. Use for: safety verification of uncertain systems, robust control synthesis, formal verification of dynamical systems, computational reachability analysis.

Data-Driven Reachability Analysis Using Matrix Perturbation Theory

Overview

This skill provides a matrix zonotope perturbation framework that leverages matrix perturbation theory to characterize how noise-induced distortions alter the dynamics within sets of models. The framework derives interpretable Cai-Zhang bounds for matrix zonotopes (MZs) and extends them to constrained matrix zonotopes (CMZs).

Core Innovation

The framework introduces:

1. Cai-Zhang bounds: Interpretable error bounds for matrix zonotope operations
2. Coefficient-space approximation: Over-approximation of constrained coefficient space by unconstrained zonotope
3. Scalable reachable-set update: Replacing CMZ-constrained-zonotope products with unconstrained MZ-zonotope multiplication

When to Use This Skill

Use this methodology when:

• Analyzing safety of uncertain dynamical systems
• Computing reachable sets for data-driven models
• Verifying properties of systems with bounded noise
• Need computationally efficient reachability analysis
• Working with linear systems subject to disturbances

Mathematical Framework

Matrix Zonotope (MZ)

A matrix zonotope M is defined as:

M = { M_0 + sum(beta_i * M_i) | beta_i in [-1, 1] }

where:

• M_0 is the center matrix
• M_i are generator matrices
• beta_i are bounded coefficients

Constrained Matrix Zonotope (CMZ)

A constrained matrix zonotope extends MZ with linear constraints:

C = { M_0 + sum(beta_i * M_i) | A*beta = b, beta in [-1, 1] }

where A*beta = b represents linear equality constraints on coefficients.

Cai-Zhang Bounds

For matrix perturbations, the Cai-Zhang bound provides:

|| Delta M || <= sqrt(sum(||M_i||^2)) * epsilon

where epsilon characterizes the noise level.

Algorithm: Reachable-Set Propagation

Step 1: System Representation

Represent uncertain system as:

x_{k+1} = (A + Delta A)*x_k + (B + Delta B)*u_k + w_k

where:

• A, B are nominal matrices
• Delta A, Delta B are bounded uncertainties (matrix zonotopes)
• w_k is process noise

Step 2: Coefficient-Space Approximation

Convert CMZ to over-approximating unconstrained zonotope:

C ⊆ ~M

where ~M has more generators but no constraints.

Step 3: Scalable Reachable-Set Update

Replace CMZ-CZ products with MZ-zonotope multiplication:

// Traditional: CMZ × CZ (computationally expensive)
// Proposed: MZ × Zonotope (efficient)

R_{k+1} = A · R_k ⊕ B · U ⊕ W

where:

• · is matrix-zonotope multiplication
• ⊕ is Minkowski sum
• R_k, U, W are zonotopes

Implementation

def reachable_set_propagation(system, initial_set, num_steps):
    """
    Compute reachable sets using matrix perturbation theory.
    
    Args:
        system: Linear system with matrix zonotope uncertainties
        initial_set: Initial state zonotope
        num_steps: Number of propagation steps
    
    Returns:
        List of reachable sets at each time step
    """
    reachable_sets = [initial_set]
    
    for k in range(num_steps):
        # Get current reachable set
        R_k = reachable_sets[-1]
        
        # Compute next reachable set using coefficient-space approximation
        R_next = matrix_zonotope_multiplication(system.A_mz, R_k)
        R_next = minkowski_sum(
            R_next,
            matrix_zonotope_multiplication(system.B_mz, system.U)
        )
        R_next = minkowski_sum(R_next, system.W)
        
        reachable_sets.append(R_next)
    
    return reachable_sets

Key Features

Feature	Benefit
Cai-Zhang bounds	Interpretable error characterization
Coefficient-space approximation	Computational tractability
Unconstrained MZ operations	Efficient matrix operations
Formal guarantees	Safety verification capability

Performance

The proposed method demonstrates:

• Substantially faster computation vs traditional CMZ-based propagation
• Maintained accuracy through tight over-approximation
• Scalable to higher-dimensional systems

Applications

1. Safety Verification: Check if system trajectories stay within safe regions
2. Controller Synthesis: Design controllers that guarantee safety
3. Fault Detection: Identify anomalous behavior outside expected reachable sets
4. Robust Planning: Plan trajectories considering all possible uncertainties

References

• Paper: "Data-Driven Reachability Analysis Using Matrix Perturbation Theory"
• Authors: Peng Xie, Abdulla Fawzy, Zhen Zhang, Amr Alanwar
• arXiv: 2604.13862v1
• Published: April 15, 2026
• Category: Systems and Control (eess.SY)

Related Concepts

• Zonotope calculus
• Reachability analysis
• Uncertainty quantification
• Formal verification
• Robust control
• Set-based computing

Activation Keywords

• data-driven reachability analysis
• matrix perturbation theory
• matrix zonotope
• constrained matrix zonotope
• cai-zhang bounds
• reachable-set propagation
• formal verification uncertain systems
• safety verification zonotopes