Line zonotopes: A tool for state estimation and fault diagnosis of unbounded and descriptor systems

TL;DR

Descriptor systems pose unique challenges for set-based state estimation and fault diagnosis due to static constraints and potential instability. The paper introduces line zonotopes (LZs), a CLG-rep based extension of constrained zonotopes that supports unbounded sets while preserving efficient set operations, and extends open-loop active fault diagnosis to reachable tubes. It provides a state estimator that works without a priori bounded initial sets and an open-loop fault diagnosis framework that uses the entire output sequence to reduce conservatism. Numerical experiments show LZs achieve tighter enclosures and robust fault isolation in unstable or unobservable LDS, with favorable computation relative to CZ-based methods.

Abstract

This paper proposes new methods for set-based state estimation and active fault diagnosis (AFD) of linear descriptor systems (LDS). Unlike intervals, ellipsoids, and zonotopes, constrained zonotopes (CZs) can directly incorporate linear static constraints on state variables - typical of descriptor systems - into their mathematical representation, leading to less conservative enclosures. However, for LDS that are unstable or not fully observable, a bounded representation cannot ensure a valid enclosure of the states over time. To address this limitation, we introduce line zonotopes, a new representation for unbounded sets that retains key properties of CZs, including polynomial time complexity reduction methods, while enabling the description of strips, hyperplanes, and the entire n-dimensional Euclidean space. This extension not only generalizes the use of CZs to unbounded settings but can also enhance set-based estimation and AFD in both stable and unstable scenarios. Additionally, we extend the AFD method for LDS from Rego et al. (2020) to operate over reachable tubes rather than solely on the reachable set at the final time of the considered horizon. This reduces conservatism in input separation and enables more accurate fault diagnosis based on the entire output sequence. The advantages of the proposed methods over existing CZ-based approaches are demonstrated through numerical examples.

Figures (6)

• Figure 1: Top: The feasible sets for the system \ref{['eq:desc_systemmotivational']}: the initial set $X_0$ (red), and the sets $S_0$ (yellow), $S_1$ (cyan), and $S_2$ (green). Bottom: The feasible sets for the system \ref{['eq:desc_systemmotivational']} with $X_0$ (red) unbounded along $x_2$ and $x_3$ (denoted by arrows).
• Figure 2: Examples of line zonotopes: a strip $S = \{ {\mathbf{x}} \in \mathbb{R}^2 : |[-1\,\; 1] {\mathbf{x}} - 1| \leq 0.5\} = \left( {\mathbf{I}}_{2}, \bm{0}_{2\times1}, \bm{0}_{2\times1}, [-1\,\; 1], -0.5, 1 \right)_\text{LZ}$ (left), and the set $( [1 \,\;5 \,\; 3]^T, {\mathbf{G}}_z, \bm{0}_{3\times1}, 0, [-2 \,\; 1 \,\; -1], 2)_\text{LZ}$, ${\mathbf{G}}_z \neq {\mathbf{0}}$ (right).
• Figure 3: The zonotope $Z$ (gray), the strip $S$ (blue), the zonotope obtained using the conservative intersection method in Bravo2006 (yellow), and the line zonotope $Z \cap S$ computed in CLG-rep (red)
• Figure 4: The strips $S_1$ (blue) and $S_2$ (red), and the intersection $S_1 \cap S_2$ computed computed in CLG-rep (magenta).
• Figure 5: The radii of the enclosures $\hat{X}_k$ obtained using the methods proposed in Puig2018, Rego2020b, and the line zonotope method proposed in Section \ref{['sec:desc_estimationLZ']} (top), as well as the projections of $\hat{X}_k$ onto $x_3$ centered at $x_3$ (bottom). The circles at the bottom red curves denote the instant $k$ in which the enclosure generated by LZ becomes bounded.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Proposition 1
• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• Remark 4
• Remark 5
• Definition 3