Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Abstract

Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closedloop systems. This paper is concerned with developing scalable algorithms that under-approximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics, and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to under-approximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-sized representation for it in polynomial time. Our algorithm finds a polynomial-sized under-approximation in polynomial time. We further analyze the conservatism of this under-approximation technique, and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms.

Koopman-Inspired Implicit Backward Reachable Sets for Unknown Nonlinear Systems

“…which proves relation (12). Combining (11) and (12) with the definition of εi , we have |E i Ã, B (x, u)−c * i | ≤ εi for all (x, u) ∈ S, which concludes the proof.…”

“…Our future work will aim to overcome this limitation by investigating how to obtain lifting functions incrementally in a way to yield monotonically better performance. From an algorithmic point of view, we plan to replace the polytopic reachability tools used in this paper to compute the BRS of the Koopman over-approximation with more efficient zonotopic ones [12] to improve scalability. Finally, it is also interesting to generalize the results to handle uncertainty in dynamics and measurements.…”

“…Backward reachability analysis has also been extensively studied as a useful tool to solve constrained control problems. Existing techniques include set-based methods [1], [10]- [12], discrete-abstractions [13], [14], Hamilton-Jacobi (HJ) reachability [15], [16], and more recent work for unknown dynamics [17]. However, most of the existing methods cannot be extended to general nonlinear systems, especially when the state space dimension is large [15].…”

Backward Reachability Analysis of Neural Feedback Systems Using Hybrid Zonotopes

“…Compared to forward reachability analysis, backward reachability analysis can be more efficient in finding safety violations for a system, especially in cases where the unsafe states are rare or hard to reach from many initial states. Although various techniques have been developed for backward reachability analysis on systems without NNs [15], [16], [17], they are not directly applicable to neural feedback systems due to the highly nonlinear and nonconvex nature of NNs. This letter aims to compute the exact BRS of a neural feedback system where the controller is a Feedforward Neural Network (FNN) with Rectified Linear Unit (ReLU) activation functions.…”

Backward Reachability Analysis of Neural Feedback Systems Using Hybrid Zonotopes

“…When the target region is chosen as the unsafe set for safety verification problems, the computed BRS will bound the back-propagated trajectories that enter the unsafe set. Although there exist various methods for computing the BRSs of systems without NNs [12], [13], they are not directly applicable to NFSs due to the inherently complex and nonlinear nature of NNs. For isolated ReLU-activated NNs, [9] proposed a method to compute the exact BRS of NNs by representing the NNs as piecewise linear functions via the activation patterns of the ReLU functions.…”