Radius, Diameter, Incenter, Circumcenter, Width and Minimum Enclosing Cylinder for Some Polyhedral Distance Functions

Das, Sandip; Nandy, Ayan; Sarvottamananda, Swami

doi:10.1007/978-3-319-74180-2_24

Cited by 2 publications

(2 citation statements)

References 8 publications

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“…Remark 1: It is more complicated to calculate b P min than its counterpart b P max since the maximum of B ū P is reached on a vertex of Ū, while its minimum is not. An algorithm for determining b P min is given in [35]. Now that we have bounded the nominal reach time T * N , we can investigate the malfunctioning reach time T * M .…”

Section: A Nominal Reach Timementioning

confidence: 99%

Quantitative Resilience of Linear Systems

Bouvier¹,

Ornik²

2022

Preprint

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Actuator malfunctions may have disastrous consequences for systems not designed to mitigate them. We focus on the loss of control authority over actuators, where some actuators are uncontrolled but remain fully capable. To counteract the undesirable outputs of these malfunctioning actuators, we use real-time measurements and redundant actuators. In this setting, a system that can still reach its target is deemed resilient. To quantify the resilience of a system, we compare the shortest time for the undamaged system to reach the target with the worst-case shortest time for the malfunctioning system to reach the same target, i.e., when the malfunction makes that time the longest. Contrary to prior work on driftless linear systems, the absence of analytical expression for time-optimal controls of general linear systems prevents an exact calculation of quantitative resilience. Instead, relying on Lyapunov theory we derive analytical bounds on the nominal and malfunctioning reach times in order to bound quantitative resilience. We illustrate our work on a temperature control system.

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Section: A Nominal Reach Timementioning

confidence: 99%

Quantitative Resilience of Linear Systems

Bouvier¹,

Ornik²

2022

Preprint

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“…The abundant literature from late nineteenth century on Minkowski sum corroborates the importance and applicability of the concept. Another recent use of Minkowski sums, that is not highlighted much in the literature, is to compute distances for several types of polyhedral distance functions [4,5], especially when distances from polytopes are involved. Das et al [5] used Minkowski sums implicitly to compute diameter, width, minimum enclosing/stabbing sphere, maximum inscribed sphere and minimum enclosing/stabbing cylinder for several types of input and problem variations involving convex polygon/polytopes.…”

Section: Introductionmentioning

confidence: 99%

Computing the Minkowski Sum of Convex Polytopes in $\Re^d$

Das,

Sarvottamananda

2018

Preprint

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We propose a method to efficiently compute the Minkowski sum, denoted by binary operator ⊕ in the paper, of convex polytopes in d using their face lattice structures as input. In plane, the Minkowski sum of convex polygons can be computed in linear time of the total number of vertices of the polygons. In d , we first show how to compute the Minkowski sum, P ⊕ Q, of two convex polytopes P and Q of input size n and m respectively in time O(nm). Then we generalize the method to compute the Minkowski sum of n convex polytopes,, where P1, P2, . . . , Pn are n input convex polytopes and for each i, Ni is size of the face lattice structure of Pi. Our algorithm for Minkowski sum of two convex polytopes is optimal in the worst case since the output face lattice structure of P ⊕ Q for convex polytopes in d can be O(nm) in worst case.

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Radius, Diameter, Incenter, Circumcenter, Width and Minimum Enclosing Cylinder for Some Polyhedral Distance Functions

Cited by 2 publications

References 8 publications

Quantitative Resilience of Linear Systems

Quantitative Resilience of Linear Systems

Computing the Minkowski Sum of Convex Polytopes in $\Re^d$

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