Overapproximating Reachable Tubes of Linear Time-Varying Systems

Serry, Mohamed; Reißig, Gunther

doi:10.1109/tac.2021.3057504

Cited by 13 publications

(6 citation statements)

References 39 publications

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“…Given a non-empty subset X ⊆ R n and a measurable set S ⊆ R, X S denotes the set of Lebesgue measurable maps with domain S and values in X. Given a non-empty compact subset W ⊆ R m , a compact interval Let Ω, Ω ′ , Γ, Γ ′ ⊆ R n be nonempty and bounded, and let A, B ∈ R m×n , then the following hold (see [16,Lemma A.2]):…”

Section: Preliminariesmentioning

confidence: 99%

“…Herein, we consider the 5-D linear system given in [2, Equation 3.11, p. 39], with [0, T ] = [0, 2], and under-approximate the reachable tube R([0, T ]) using the proposed method with two values of the discretization parameter N (N ∈ {10, 100}), where we set ǫ h = 1−1/N 3 and ǫu = 1−1/N 2 . As the exact reachable tube is not known, we use the convergent over-approximation method of Serry and Reissig [16], with a refined time discretization (400 steps) and an accurate approximation of the matrix exponential (L(•, 10)) to produce an accurate representation of the reachable tube and compare it with the computed under-approximations from the proposed method. Figure 2 displays two projections of the over-approximation and the under-approximations.…”

Section: B 5-d Systemmentioning

confidence: 99%

“…These optimal control-based approaches assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general. In addition, the mentioned approaches use non-convex representations for under-approximations of reachable tubes, which are challenging to analyze in the contexts of verification and synthesis (see the introduction of [16]). Serry [17] proposed a set-propagation technique that yields convergent under-approximations of forward reachable sets and tubes for a general class of linear systems, where the under-approximations of reachable sets and tubes are given as convex sets and finite unions of convex sets, respectively, which can be analysed with relative ease.…”

Section: Introductionmentioning

confidence: 99%

“…Fig.2. x 2 − x 3 (top) and x 4 − x 5 (bottom) projections of the overapproximation (grey), using the method of Serry and Reissig[16], and the under-approximations (black), using the proposed method with N = 10 (left) and N = 100 (right), of the reachable tube R([0, T ]) of the 5-D system in Section VII-B.…”

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confidence: 99%

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Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

Serry¹,

Li²

2021

Preprint

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Under-approximations of reachable sets and tubes have received recent research attention due to their important roles in control synthesis and verification. Available under-approximation methods designed for continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general. In this note, we attempt to overcome this drawback for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes utilizing approximations of the matrix exponential. In particular, we consider the class of continuoustime LTI systems with an identity input matrix and uncertain initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes with first order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, the proposed method is implemented in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.

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Section: Preliminariesmentioning

confidence: 99%

Section: B 5-d Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

Serry¹,

Li²

2021

Preprint

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“…The Hausdorff distance and the topology induced by it, have found widespread applications in mathematical economics [32], stochastic geometry [53], set-valued analysis [48], image processing [35] and pattern recognition [36]. The distance δ has several useful properties with respect to set operations, see e.g., [17,Lemma 2.2], [49,Lemma A2].…”

mentioning

confidence: 99%

Hausdorff Distance between Norm Balls and their Linear Maps

Haddad¹,

Halder²

2022

Preprint

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We consider the problem of computing the (two-sided) Hausdorff distance between the unit p 1 and p 2 norm balls in finite dimensional Euclidean space for 1 < p 1 < p 2 ≤ ∞, and derive a closed-form formula for the same. When the two different 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 the 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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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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Overapproximating Reachable Tubes of Linear Time-Varying Systems

Cited by 13 publications

References 39 publications

Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

Hausdorff Distance between Norm Balls and their Linear Maps

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

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