Reachability Analysis for Linear Systems with Uncertain Parameters using Polynomial Zonotopes

TL;DR

This work presents a polynomial zonotope-based reachability framework for linear systems with uncertain parameters and inputs, enabling tight non-convex enclosures of time-targeted reachable sets. The method extends naturally to time-varying parameters, linear time-varying and nonlinear dynamics, as well as hybrid systems, while preserving dependencies across time to achieve higher accuracy than traditional convex over-approximations. A key contribution is the development of scalable optimization techniques for multi-affine zonotopes, leveraging problem structure to reduce complexity and improve plotting, intersection testing, and support-function calculations. Experimental results on benchmarks including a 9‑D vehicle platoon and Dubins car demonstrate superior tightness and competitive runtime compared with state-of-the-art tools such as CORA, highlighting practical impact for safety-critical cyber-physical systems.

Abstract

In real world applications, uncertain parameters are the rule rather than the exception. We present a reachability algorithm for linear systems with uncertain parameters and inputs using set propagation of polynomial zonotopes. In contrast to previous methods, our approach is able to tightly capture the non-convexity of the reachable set. Building up on our main result, we show how our reachability algorithm can be extended to handle linear time-varying systems as well as linear systems with time-varying parameters. Moreover, our approach opens up new possibilities for reachability analysis of linear time-invariant systems, nonlinear systems, and hybrid systems. We compare our approach to other state of the art methods, with superior tightness on two benchmarks including a 9-dimensional vehicle platooning system. Moreover, as part of the journal extension, we investigate through a polynomial zonotope with special structure named multi-affine zonotopes and its optimization problem. We provide the corresponding optimization algorithm and experiment over the examples obatined from two benchmark systems, showing the efficiency and scalability comparing to the state of the art method for handling such type of set representation.

Figures (14)

• Figure 1: Different classes of linear systems explained on the example of the Dubins car model, where $s_x$, $s_y$ represent the x- and y-position of the car and $v_x$, $v_y$ are the corresponding velocities. The final reachable set is shown in red and the reachable set for the whole time horizon in gray.
• Figure 2: Visualization of the polynomial zonotope in Example \ref{['example: single PZ']}.
• Figure 3: Comparison from Example \ref{['example: compare minkowski sum and direct sum']} between Minkowski sum $A~\mathcal{PZ} \oplus \mathcal{PZ}$, exact sum $A~ \mathcal{PZ} \boxplus \mathcal{PZ}$, and a mixture of both obtained by destroying the dependencies for the second dependent factor.
• Figure 4: Comparison between combining the homogeneous solution $\mathcal{H}(\tau_0)$ and particular solution $\mathcal{P}(\tau_0)$ via Minkowski sum (left) and via exact sum (right).
• Figure 5: Reachable set for the linear system from Example \ref{['example:linear']} (red), where the initial set is depicted in white with a black border.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Example 1
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Example 2
• Definition 2.6
• Proposition 1
• proof