Computation of parametric barrier functions for dynamical systems using interval analysis

Bouissou, Olivier; Chapoutot, Alexandre; Djaballah, Adel; Kieffer, Michel

doi:10.1109/cdc.2014.7039472

Cited by 24 publications

(15 citation statements)

References 25 publications

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“…If f is not polynomial but also contains transcendental function symbols, then the constraints can be solved when being robust [16,Theorem 6]. However, even under the restriction to robust constraints, solving the constraint is often prohibitively costly, although sometimes possible [3]. …”

Section: Notation and Problem Descriptionmentioning

confidence: 99%

Computing ODE-barriers in hyper-rectangles

Ratschan

2016

2016 International Workshop on Symbolic and Numerical Methods for Reachability Analysis (SNR)

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In this paper we develop a computationally cheap, but incomplete method for computing barriers of ordinary differential equations. We consider a barrier of an ODE to be a subset of the state-space such that no trajectory of the ODE can start inside of the barrier and end outside. We compute barriers that may serve as a certificate that no trajectory may lead from a given face of a hyper-rectangle to the opposite face. Such a method can be useful for more involved techniques for analyzing dynamical systems, especially for techniques based on abstraction.

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Section: Notation and Problem Descriptionmentioning

confidence: 99%

Computing ODE-barriers in hyper-rectangles

Ratschan

2016

2016 International Workshop on Symbolic and Numerical Methods for Reachability Analysis (SNR)

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“…Brauer [7], Habets and Peiffer [18, §2]). 5 The comparison principle hinges on an appropriate comparison theorem, which establishes the relationship between the solutions of the one-dimensional system to the solutions of the differential inequality. Below we state a particularly useful comparison theorem (a corollary to the comparison theorem in Walter [49, Ch.…”

Section: Remarkmentioning

confidence: 99%

“…For example, constraint programming techniques for solving inequality constraints over the reals were studied by Ratschan [39] and applied to search for Lyapunovlike functions [40]. Computation of strict barrier certificates using interval constraint satisfaction techniques was later investigated by Bouissou [5], Djaballah, et al [12]. Another intriguing alternative studied by Sankaranarayanan et al [44] (and later Yang et al [50]) is the linear relaxation approach based on so-called Handelman representations [21] (which allow the use of linear programming to establish the positive semi-definite property of a polynomial over a compact convex polyhedron); this technique was observed to be much less prone to numerical errors than methods based on interior-point solvers.…”

Section: Generating Vector Barrier Certificates Using Sdpmentioning

confidence: 99%

Vector Barrier Certificates and Comparison Systems

Sogokon¹,

Ghorbal²,

Tan³

et al. 2018

Formal Methods

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Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems of non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.

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“…For the numerical resolution some methods grid the state space [7,27] which makes them computationally expensive. Lyapunov-based methods [24], level-set methods [20], or barrier functions [4] are attractive since they do not perform any integration through time. Now, these methods generally require a parametric expression for candidate Lyapunov-like functions [23].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Sliding Surfaces of Cyber-Physical Systems

Jaulin

Bars

2020

Acta Cybern

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When implementing a non-continuous controller for a cyber-physical system, it may happen that the evolution function of the closed-loop system is not anymore piecewise continuous along the trajectory, mainly due to if statements inside the control algorithm. As a consequence, an unwanted chattering effect may occur. This behavior is often difficult to observe even in simulation. We propose here a set-membership method based on interval analysis to detect different types of discontinuities. One of them is the sliding surface where the state trajectory jumps indefinitely between two distinct behaviors. As an application, we consider the validation of a sailboat controller. We show that our approach is able to detect and explain some unwanted sliding effects that may be observed in rare and specific situations on our actual sailboat robots.

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Computation of parametric barrier functions for dynamical systems using interval analysis

Cited by 24 publications

References 25 publications

Computing ODE-barriers in hyper-rectangles

Computing ODE-barriers in hyper-rectangles

Vector Barrier Certificates and Comparison Systems

Characterizing Sliding Surfaces of Cyber-Physical Systems

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