ZETA: a library for Zonotope-based EsTimation and fAult diagnosis of discrete-time systems

TL;DR

ZETA provides a comprehensive MATLAB library for zonotopes, constrained zonotopes, line zonotopes, and interval arithmetic to enable guaranteed, set-based estimation and fault diagnosis in discrete-time systems. It unifies linear, descriptor, and nonlinear dynamics with multiple propagation and estimation strategies (MV, FO, DC, PR) and introduces polyhedral relaxations via a Polyrelax framework to automate nonlinear enclosures. Key contributions include efficient CZ complexity reduction, robust state estimation for linear and descriptor models, active fault diagnosis using MIQP/MILP, and scalable nonlinear propagation and estimation methods. The work offers an accessible, extensible toolset for reachability, estimation, and fault diagnosis, with demonstrations and comparisons to existing tools to highlight efficiency and reduced conservatism.

Abstract

This paper introduces ZETA, a new MATLAB library for Zonotope-based EsTimation and fAult diagnosis of discrete-time systems. It features user-friendly implementations of set representations based on zonotopes, namely zonotopes, constrained zonotopes, and line zonotopes, in addition to a basic implementation of interval arithmetic. This library has capabilities starting from the basic set operations with these sets, including propagations through nonlinear functions using various approximation methods. The features of ZETA allow for reachability analysis and state estimation of discrete-time linear, nonlinear, and descriptor systems, in addition to active fault diagnosis of linear systems. Efficient order reduction methods are also implemented for the respective set representations. Some examples are presented in order to illustrate the functionalities of the new library.

Figures (6)

• Figure 1: Conversions between set representations implemented in ZETA.
• Figure 2: Constrained zonotope $Z$ to be reduced (solid green), and the reduced $\bar{Z}$ obtained by ZETA (solid blue) and CORA (dot-dashed red). The CZonotope objects have been plotted using plot.
• Figure 3: Separated output reachable tubes using the MIQP fault diagnosis approach proposed in Scott2014, implemented in ZETA. The Zonotope objects have been plotted using plot.
• Figure 4: CZ enclosures of ${\mathbf{f}}(X)$ obtained using ZETA, along with propagated samples (black dots). The sets were plotted using plot.
• Figure 5: Estimated CZs $\hat{X}_k$ using different methods in ZETA, along with ${\mathbf{x}}_k$ ($\times$), for a few selected instants steps. The CZs were plotted using plot.

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Remark 1