A Comparison of Set-Based Observers for Nonlinear Systems

Set-based state estimation computes sets of states consistent with a system model given bounded sets of disturbances and noise. Bounding the set of states is crucial for safety-critical applications so that one can ensure that all specifications are met. While numerous approaches have been proposed for nonlinear discrete-time systems, a unified evaluation under comparable conditions is lacking. This paper reviews and implements a representative selection of set-based observers within the CORA framework. To provide an objective comparison, the methods are evaluated on common benchmarks, and we examine computational effort, scalability, and the conservatism of the resulting state bounds. This study highlights characteristic trade-offs between observer categories and set representations, as well as practical considerations arising in their implementation. All implementations are made publicly available to support reproducibility and future development. This paper thereby offers the first broad, tool-supported comparison of guaranteed state estimators for nonlinear discrete-time systems.

💡 Research Summary

This paper presents the first comprehensive, tool‑supported comparison of guaranteed state estimators for nonlinear discrete‑time systems. The authors first motivate set‑based state estimation as a necessity for safety‑critical applications where probabilistic filters cannot provide hard inclusion guarantees. They then introduce the mathematical preliminaries, focusing on five set representations commonly used in the literature: intervals, ellipsoids, polytopes (specifically zonotopes, zonotope bundles, and constrained zonotopes). For each representation the authors discuss closure properties (e.g., under Minkowski sum, linear map, intersection), computational complexity, and typical sources of conservatism.

Building on the categorization introduced for linear set‑based observers (Althoff & Rath, 2021), the paper classifies nonlinear observers into three families: intersection‑based, propagation‑based, and interval‑based. Intersection‑based observers follow the classic predict‑then‑correct paradigm. In the prediction step the reachable set is over‑approximated using inclusion functions (e.g., mean‑value extension) or by expressing the dynamics as a difference of convex (DC) functions. The correction step intersects the predicted set with measurement strips or, for constrained zonotopes, with the exact measurement set. Representative algorithms include FRad‑A, VolMin‑A, CZMV, ZDC, CZDC, FRad‑B, VolMin‑B, CZN‑A, and CZN‑B. The DC‑based methods (ZDC, CZDC) are noteworthy because they can tightly bound a wide class of nonlinearities by decomposing them into two convex components and using supporting tangents for each.

Propagation‑based observers avoid explicit intersection by embedding a Luenberger correction matrix G into the set‑valued dynamics. The only nonlinear method found in the literature is FRad‑C, which computes G online using a Kalman‑filter‑like optimization and relies on conservative linearization (first‑order Taylor expansion plus a Hessian‑based bound on higher‑order terms).

Interval‑based observers are less developed for general nonlinear discrete‑time systems; the authors note that pure interval observers do not exist in this setting, and existing approaches essentially combine separate lower‑ and upper‑bound propagation with an intersection step.

The experimental section implements all selected observers within the open‑source CORA toolbox (MATLAB) and evaluates them on a set of benchmark problems ranging from 2‑ to 4‑dimensional nonlinear models (e.g., a two‑link robotic arm, a nonlinear vehicle dynamics model, and a nonlinear electrical circuit). For each observer the authors measure computational time, memory consumption, and conservatism, the latter quantified by the volume of the estimated set (or an equivalent hyper‑rectangle measure).

Key findings include:

• Zonotope‑based observers (FRad‑A, VolMin‑A, etc.) scale well with dimension, offering modest computational effort and acceptable conservatism, but they cannot directly handle intersections, which can increase over‑approximation.
• Constrained‑zonotope observers (CZMV, CZN‑A/B) achieve the smallest set volumes, especially when combined with DC decomposition, but suffer from rapid growth of generator and constraint matrices, leading to higher runtime and memory usage. Effective order‑reduction techniques are essential for practical use.
• Ellipsoid‑based observers (e.g., ESO‑E) are computationally cheap even in higher dimensions, yet the need to compute a minimum‑volume enclosing ellipsoid after each intersection introduces significant conservatism.
• Propagation‑based FRad‑C is the fastest because it avoids intersection entirely, but its reliance on conservative linearization makes it less accurate for strongly nonlinear dynamics.
• Pure interval‑based approaches are the most conservative and have limited applicability in the tested scenarios.

The authors conclude that the choice of observer and set representation involves a clear trade‑off: zonotopes are attractive for real‑time, higher‑dimensional applications; constrained zonotopes with DC decomposition are preferable when tight bounds are paramount; ellipsoids are suitable for low‑dimensional fast‑prototype analysis; and propagation‑based methods excel when computational budget is extremely limited.

Finally, the paper contributes all source code, benchmark definitions, and a CORA extension publicly, enabling reproducibility and providing a baseline for future research on set‑based estimation of nonlinear systems.