The Convex Geometry of Integrator Reach Sets

Haddad, Shadi; Halder, Abhishek

doi:10.23919/acc45564.2020.9147611

Cited by 13 publications

(19 citation statements)

References 26 publications

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“…Our objective is to study the geometric aspects of (4) in detail. This paper significantly expands our preliminary work [4]: here we consider multi-input integrators as opposed to the single input case considered in [4]. Even for the single input case, while [4,Thm.…”

Section: Introductionmentioning

confidence: 81%

“…This paper significantly expands our preliminary work [4]: here we consider multi-input integrators as opposed to the single input case considered in [4]. Even for the single input case, while [4,Thm. 1] derived an exact formula for the volume of the reach set, that formula involved limit and nested sums, and in that sense, was not really a closed-form formula [6] -certainly not amenable for numerical computation.…”

Section: Introductionmentioning

confidence: 81%

“…2) and the single integrator reach set R 2 = {x 0 (3)} [−µ 2 t, µ 2 t], is a cylinder † . In [4], we explicitly derived that vol (R 1 ) = 2 3 µ 2 1 t 3 , and therefore, the volume of this cylindrical set must be equal to height of the cylinder × cross sectional area…”

Section: A Volumementioning

confidence: 99%

“…Step 2: Motivated by ( 59), we focus on deriving the r jdimensional volume of R j ({x 0 }, t). For this purpose, we proceed as in [4] by uniformly discretizing the interval [0, t] into n subintervals (i − 1)t n , it n , i = 1, 2, . .…”

Section: Appendixmentioning

confidence: 99%

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The Curious Case of Integrator Reach Sets, Part I: Basic Theory

Haddad¹,

Halder²

2021

Preprint

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This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, we establish that this compact convex set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.

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Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 81%

Section: A Volumementioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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The Curious Case of Integrator Reach Sets, Part I: Basic Theory

Haddad¹,

Halder²

2021

Preprint

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“…the Löwner-John ellipsoid[19,20],[15, p. 69] containing R 𝑁 , i.e., to solve Ch. 3.7.2]; one of them is based on the S procedure[29,34,35] that works well in practice, see e.g.,[17, Sec. V].…”

mentioning

confidence: 99%

Anytime ellipsoidal over-approximation of forward reach sets of uncertain linear systems

Haddad

Halder

2021

Proceedings of the Workshop on Computation-Aware Algorithmic Design for Cyber-Physical Systems

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Computing tight over-approximation of reach sets of a controlled uncertain dynamical system is a common practice in verification of safety-critical cyber-physical systems (CPS). While several algorithms are available for this purpose, they tend to be computationally demanding in CPS applications since here, the computational resources such as processor availability tend to be scarce, time-varying and difficult to model. A natural idea then is to design "computation-aware" algorithms that can dynamically adapt with respect to the processor availability in a provably safe manner. Even though this idea should be applicable in broader context, here we focus on ellipsoidal over-approximations. We demonstrate that the algorithms for ellipsoidal over-approximation of reach sets of uncertain linear systems, are well-suited for anytime implementation in the sense the quality of the over-approximation can be dynamically traded off depending on the computational time available, all the while guaranteeing safety. We give a numerical example to illustrate the idea, and point out possible future directions.

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A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps

Haddad,

Halder

2023

Set-Valued Var. Anal

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We consider the problem of computing the (two-sided) Hausdorff distance between the unit $\ell _{p_{1}}$ ℓ p 1 and $\ell _{p_{2}}$ ℓ p 2 norm balls in finite dimensional Euclidean space for $1 \leq p_{1} < p_{2} \leq \infty $ 1 ≤ p 1 < p 2 ≤ ∞ , and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the $k_{1}$ k 1 and $k_{2}$ k 2 unit $D$ D -norm balls, which are certain polyhedral norm balls in $d$ d dimensions for $1 \leq k_{1} < k_{2} \leq d$ 1 ≤ k 1 < k 2 ≤ d . When two different $\ell _{p}$ ℓ p norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different $\ell _{p}$ ℓ p unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.

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The Convex Geometry of Integrator Reach Sets

Cited by 13 publications

References 26 publications

The Curious Case of Integrator Reach Sets, Part I: Basic Theory

The Curious Case of Integrator Reach Sets, Part I: Basic Theory

Anytime ellipsoidal over-approximation of forward reach sets of uncertain linear systems

A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps

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