Inner approximated reachability analysis

Goubault, Éric; Mullier, Olivier; Putot, Sylvie; Kieffer, Michel

doi:10.1145/2562059.2562113

Cited by 34 publications

(21 citation statements)

References 27 publications

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“…Unlike that in step 1) in Subsection 3.2, ∪ t∈[0,5.0] O(t; ∂I 0 ) illustrated in Fig. 3 is equal to ∪ 4 i=1 ∪ t∈[0,5.0] O(t; I 0,i ) and is computed without the assumption that system (12) starting from I 0 evolves in the viable domain X within the time interval [0, 5.0]. According to Lemma 1 in [25], ∪ t∈[0,5.0] Ω 1 (t; I 0 ) ⊂ X in Assumption 1 is guaranteed.…”

Section: Examples and Discussionmentioning

confidence: 99%

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

Bai

Wang

Feng

et al. 2021

IEEE Trans. Automat. Contr.

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This note explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this note is a class of DDEs whose dynamics are subject to perturbations and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over-and under-approximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for perturbed ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.

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Section: Examples and Discussionmentioning

confidence: 99%

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

Bai

Wang

Feng

et al. 2021

IEEE Trans. Automat. Contr.

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“…In interval analysis, a computation often provides a proof of unique existence of a solution within a resulting interval. This technique also applies in interval-based reachability analysis [15,14], but it is not considered in most of the methods for hybrid systems. Our method enforces the use of the proof to verify more generic temporal properties.…”

Section: Related Workmentioning

confidence: 99%

Monitoring Bounded LTL Properties Using Interval Analysis

Ishii

Yonezaki

Goldsztejn³

2015

Electronic Notes in Theoretical Computer Science

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Verification of temporal logic properties plays a crucial role in proving the desired behaviors of hybrid systems. In this paper, we propose an interval method for verifying the properties described by a bounded linear temporal logic. We relax the problem to allow outputting an inconclusive result when verification process cannot succeed with a prescribed precision, and present an efficient and rigorous monitoring algorithm that demonstrates that the problem is decidable. This algorithm performs a forward simulation of a hybrid automaton, detects a set of time intervals in which the atomic propositions hold, and validates the property by propagating the time intervals. A continuous state at a certain time computed in each step is enclosed by an interval vector that is proven to contain a unique solution. In the experiments, we show that the proposed method provides a useful tool for formal analysis of nonlinear and complex hybrid systems.

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“…As a result, significant advances of outer-approximate reachability analysis techniques for both linear and nonlinear systems have been reported in the literature based on various representations of sets such as intervals [35], zonotopes [1], polyhedra and support functions for polyhedral sets [10], [15], ellipsoids [22], level sets [30], Taylor models [6] and semi-algebraic sets [41], [19]. Computational methods for inner-approximations have received increasing attention just recently, e.g., [41], [21], [17], [7], [44]. It nevertheless has a wide range of practical applications including collision avoidance and surveillance.…”

Section: Introductionmentioning

confidence: 99%

Inner-Approximating Reachable Sets for Polynomial Systems With Time-Varying Uncertainties

Bai

Fränzle

Zhan

2020

IEEE Trans. Automat. Contr.

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In this paper we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sub-level set of the unique Lipschitz viscosity solution to a Hamilton-Jacobi partial differential equation (HJE). We show that inner-approximations of the backward reachable set can be formed by zero sub-level sets of its viscosity super-solutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity super-solutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semi-definite program. We also prove that polynomial solutions to the formulated semi-definite program exist and can produce a convergent sequence of inner-approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this work. Several illustrative examples demonstrate the merits of our approach. Related WorkAs mentioned above, inner-approximate reachability analysis of ordinary differential equations subject to time-varying uncertainties and state constraints, is still in its infancy and thus provides an open area of research.For ordinary differential equations free of time-varying uncertainties and state constraints, [17], [18] proposed a method based on modal intervals with affine forms to inner-approximate reachable sets using intervals. By making use of the homeomorphism property of the solution mapping, a boundary based reachability analysis method was proposed to inner-approximate reachable sets with polytopes in [44], and it was extended to a class of delay differential equations in [43]. Since reachable sets of nonlinear systems tend to be non-convex, the above mentioned methods based on convex set representations may result in poor approximations. As accuracy is also an important factor in performing reachability analysis (e.g.,[37], [28]), more complex shapes of representations such as Taylor models and semi-algebraic sets are desirable.[7] proposed a Taylor model backward flowpipe method to compute inner-approximations. [41] proposed an iterative method, with each iteration relying on solving semi-definite programming problems, to compute semi-algebraic inner-approximations for polynomial systems using the advection map of the given dynamical system. [45] extended the method in [44] to compute semi-algebraic inner-approximations of reachable sets for polynomial systems and beyond by solving semi-definite programming problems. Recently, [42] formulated the problem of solving HJEs as a semi-definite program to compute inner-approximations for polynomial systems. For state-constrained polynomial systems ...

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Inner approximated reachability analysis

Cited by 34 publications

References 27 publications

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

Over- and Underapproximating Reach Sets for Perturbed Delay Differential Equations

Monitoring Bounded LTL Properties Using Interval Analysis

Inner-Approximating Reachable Sets for Polynomial Systems With Time-Varying Uncertainties

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