Set operations and order reductions for constrained zonotopes

Raghuraman, Vignesh; Koeln, Justin P.

doi:10.1016/j.automatica.2022.110204

Cited by 44 publications

(14 citation statements)

References 37 publications

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“…For example, if X safe = p H p is the union of finitely many polytopes H p , the collection C i k will consist of P roj x (Z i k ) ∩ H p for all p. Each P roj x (Z i k ) ∩ H p is a constrained zonotope, whose CG-Rep can be obtained by Lemma 1. For details, see [18]. Finally, if f is twice continuously differentiable, LE(z * , f , Z) will converge to a singleton set as Z does.…”

Section: B Splitting Methodsmentioning

confidence: 99%

“…Compared to other operations, the problem of Minkowskisubtracting a set from a zonotopic minuend (in G-Rep) receives less attention, and is first studied in [17], where the subtrahend is also assumed to be a zonotope (in G-Rep). The exact Minkowski difference is not necessarily a zonotope, but a zonotopic under-approximation can still be found efficiently [18], [19] using the encoding techniques developed in [20]. Based on these developments, a scalable backward reachability algorithm is obtained for linear systems with additive disturbances in [19].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Yang

Zhang

Jeannin

et al. 2022

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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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.

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Section: B Splitting Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Yang

Zhang

Jeannin

et al. 2022

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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show abstract

“…These objects are well known for representing reachable sets of dynamic systems (Althoff, 2015), which can enable formally-verified motion planning, fault detection, and navigation (Bhamidipati & Gao, 2020;Kousik et al, 2019;Shetty & Gao, 2020). These objects can also be extended to constrained zonotopes, which can represent arbitrary convex polytopes (Raghuraman & Koeln, 2022;Scott et al, 2016), avoiding the limitation of symmetry. While zonotopes are widely used in robotics for path planning and collision avoidance, they have not been applied in the field of GNSS localization for addressing multipath/NLOS effects.…”

Section: Related Workmentioning

confidence: 99%

Set-Valued Shadow Matching Using Zonotopes for 3D-Map-Aided GNSS Localization

Bhamidipati

Kousik

Gao

2022

navi

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GNSS-based localization is often unreliable in dense urban areas. As illustrated in Figure 1, direct line-of-sight (LOS) GNSS signals can be blocked or reflected by tall buildings (Hofmann-Wellenhof et al., 1992), creating non-line-of-sight (NLOS) and multipath effects, thereby lowering the number of visible GNSS satellites available for localization (Zhu et al., 2018). In particular, LOS satellite signals in the cross-street direction are more likely to be blocked or reflected by buildings than signals along the street. Consequently, positioning accuracy is degraded in urban areas, especially in the cross-street direction.Robustness to degraded accuracy is important for safety-critical GNSS applications such as autonomous driving and drone delivery. As detailed later in our discussion of related work, set-valued position estimates enable robustness guarantees by ensuring an entire set of possible positions lies outside of obstacles or inside

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“…This could cause a computational burden issue for a deep FNN. However, it's possible to utilize the order reduction methods proposed in [18], [19] to reduce the complexity of the approximated constrained zonotopes.…”

Section: B Over-approximation Output Analysismentioning

confidence: 99%

“…The price of accuracy, however, is that the number of constrained zonotopes and the order of constrained zonotopes will grow exponentially. Thus, order reduction techniques as proposed in [18], [19] are needed for analyzing deep neural networks. Nevertheless, as shown in Section VI, the computation time of our exact analysis algorithm is comparable with other state-of-the-art algorithms.…”

Section: Reachability Analysis and Safety Verification For Linear Sys...mentioning

confidence: 99%

Safety Verification of Neural Feedback Systems Based on Constrained Zonotopes

Zhang¹,

Xu²

2022

Preprint

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Artificial neural networks (ANNs) have been utilized in many feedback control systems and introduced new challenges regarding the safety of the system. This paper considers the problem of verifying whether the trajectories of a system with a feedforward neural network (FNN) controller can avoid unsafe regions, using a constrained zonotope-based reachability analysis approach. FNNs with the rectified linear unit activation function are considered in this work. A novel set-based method is proposed to compute both exact and overapproximated reachable sets for linear discrete-time systems with FNN controllers, and linear program-based sufficient conditions are presented to certify the safety of the neural feedback systems. Reachability analysis and safety verification for neural feedback systems with nonlinear models are also considered. The computational efficiency and accuracy of the proposed method are demonstrated by two numerical examples where a comparison with state-of-the-art methods is also provided.

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Set operations and order reductions for constrained zonotopes

Cited by 44 publications

References 37 publications

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Set-Valued Shadow Matching Using Zonotopes for 3D-Map-Aided GNSS Localization

Safety Verification of Neural Feedback Systems Based on Constrained Zonotopes

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