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42
Spaceex: Scalable verification of hybrid systems
 In Proceedings of the International Conference on Computer Aided Verification
, 2011
"... Abstract. We present a scalable reachability algorithm for hybrid systems with piecewise affine, nondeterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an overapproximation of the reachable states. The algorithm improves over previous wo ..."
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Cited by 88 (5 self)
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Abstract. We present a scalable reachability algorithm for hybrid systems with piecewise affine, nondeterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an overapproximation of the reachable states. The algorithm improves over previous work by using variable time steps to guarantee a given local error bound. In addition, we propose an improved approximation model, which drastically improves the accuracy of the algorithm. The algorithm is implemented as part of SpaceEx, a new verification platform for hybrid systems, available at spaceex.imag.fr. Experimental results of full fixedpoint computations with hybrid systems with more than 100 variables illustrate the scalability of the approach. 1
Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization
"... Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These errors are adde ..."
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Cited by 33 (15 self)
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Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These errors are added as uncertain inputs, such that the reachable set of the locally linearized system encloses the one of the original system. The linearization error is controlled by splitting of reachable sets. Reachable sets are represented by zonotopes, allowing an efficient computation in relatively highdimensional space.
Exact state set representations in the verification of linear hybrid systems with large discrete state space
 In Automated Technology for Verification and Analysis, ATVA’07, volume 4762 of LNCS
, 2007
"... Copyright c © June 2007 by the author(s) ..."
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Computing Reachable States for Nonlinear Biological Models
, 2010
"... In this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models. We summarize the stateoftheart for linear systems and then develop a novel algorithm for computing reachable states for n ..."
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Cited by 14 (5 self)
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In this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models. We summarize the stateoftheart for linear systems and then develop a novel algorithm for computing reachable states for nonlinear systems. We report experimental results obtained using a prototype implementation applied to several biological models. We believe these results constitute a promising contribution to the analysis of complex models of biological systems.
Relational Abstractions For Continuous and Hybrid Systems
"... Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any s ..."
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Cited by 13 (5 self)
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Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any state of the system to a state that can potentially be reached at some future time instant. Such relations are then interpreted as discrete transitions that model the continuous evolution of states over time. We adapt templatebased invariant generation techniques for continuous dynamics to derive relational abstractions for continuous systems with linear as well as nonlinear dynamics. Once a relational abstraction hasbeen derived,theresultingsystemis apurelydiscrete, infinitestatesystem. Therefore, techniquessuchas kinductioncan be directly applied to this abstraction to prove properties, and bounded modelchecking techniques applied to find potential falsifications. We present the basic underpinnings of our approach and demonstrate its use on many benchmark systems to derive simple and usable abstractions. 1
A hybrid bounding method for computing an overapproximation for the reachable set of uncertain nonlinear systems
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2009
"... In this paper, we show how to compute an overapproximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relie ..."
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Cited by 10 (2 self)
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In this paper, we show how to compute an overapproximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relies on stateoftheart validated numerical integration methods. The second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty. Since the derived bracketing systems are piecewisedifferentiable functions, validated numerical integration methods cannot be used directly. Hence, our contribution resides in the use of hybrid automata to model the bounding systems. We give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set. The computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques. Sufficient conditions are given which ensure thepractical stability of the enclosures given by our hybrid bounding method. Two examples are also given which show that the performance of our method is very promising.
Reachability of Uncertain Nonlinear Systems Using a Nonlinear Hybridization
 of Lecture Notes in Computer Science
"... Abstract. In this paper, we investigate nonlinear reachability computation in presence of model uncertainty, via guaranteed set integration. We show how this can be done by using the classical Müller’s existence theorem. The core idea developed is to no longer deal with whole sets but to derive in ..."
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Cited by 10 (2 self)
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Abstract. In this paper, we investigate nonlinear reachability computation in presence of model uncertainty, via guaranteed set integration. We show how this can be done by using the classical Müller’s existence theorem. The core idea developed is to no longer deal with whole sets but to derive instead two nonlinear dynamical systems which involve no model uncertainty and which bracket in a guaranteed way the space reachable by the original uncertain system. We give a rule for building the bracketing systems. In the general case, the bracketing systems obtained are only piecewise Ckcontinuously differential nonlinear systems and hence can naturally be modeled with hybrid automata. We show how to derive the hybrid model and how to address mode switching. An example is given with a biological process. 1 Introducion Computing reachable sets for hybrid systems is an important step when one addresses verification or synthesis taks. A key issue then lays in the calculation of the reachable space for continuous dynamics with nonlinear models. In this paper, we will also emphasize the presence of parameter uncertainty in the nonlinear dynamical models used for characterizing the continuous dynamics. Consider an uncertain dynamical system described by nonautonomous differential equations with the following form:{ ẋ(t) = f(x,p, t), x(t0) ∈ X0 ⊆ D, p ∈ P (1) where function f: D ×P × IR+ 7 → IRn is possibly nonlinear, D ⊆ IRn, X0 is the initial domain for state vector x at time t0 ≥ 0 and P is an uncertainty domain for parameter vector p. The reachable space of system (1) is then defined as follows R([t0, t];X0) = x(τ), t0 ≤ τ ≤ t  (ẋ(τ) = f(x,p, τ)) ∧ (x(t0) ∈ X0) ∧ (p ∈ P)
Computing reachable sets for uncertain nonlinear monotone systems
, 2009
"... We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work [1], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the ..."
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Cited by 10 (1 self)
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We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work [1], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of offdiagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields componentwise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, componentwise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.
Enclosing temporal evolution of dynamical systems using numerical methods. under submission
 In RSP. IEEE
, 2012
"... Abstract. Numerical methods are necessary to understand the behaviors of complex hybrid systems used to design controlcommand systems. Especially, numerical integration methods are heavily used in simulation to compute approximations of the solution of differential equations, including nonlinear a ..."
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Cited by 7 (2 self)
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Abstract. Numerical methods are necessary to understand the behaviors of complex hybrid systems used to design controlcommand systems. Especially, numerical integration methods are heavily used in simulation to compute approximations of the solution of differential equations, including nonlinear and stiff solutions. Nevertheless, these methods only produce approximate results and they should not be used in formal verification methods as is. We propose a systematic way to make explicit RungeKutta integration method safe with respect to the mathematical solution. As side effect, we can hence compare different integration schemes in order to pick the right one in different situations. 1
Hybridization Domain Construction using Curvature Estimation ∗ ABSTRACT
"... This paper is concerned with the reachability computation for nonlinear systems using hybridization. The main idea of hybridization is to approximate a nonlinear vector field by a piecewiseaffine one. The piecewiseaffine vector field is defined by building around the set of current states of the ..."
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Cited by 7 (2 self)
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This paper is concerned with the reachability computation for nonlinear systems using hybridization. The main idea of hybridization is to approximate a nonlinear vector field by a piecewiseaffine one. The piecewiseaffine vector field is defined by building around the set of current states of the system a simplicial domain and using linear interpolation over its vertices. To achieve a good timeefficiency and accuracy of the reachability computation on the approximate system, it is important to find a simplicial domain which, on one hand, is as large as possible and, on the other hand, guarantees a small interpolation error. In our previous work [8], we proposed a method for constructing hybridization domains based on the curvature of the dynamics and showed how the method can be applied to quadratic systems. In this paper we pursue this work further and present two main results. First, we prove an optimality property of the domain construction method for a class of quadratic systems. Second, we propose an algorithm of curvature estimation for more general nonlinear systems with nonconstant Hessian matrices. This estimation can then be used to determine efficient hybridization domains. We also describe some experimental results to illustrate the main ideas of the algorithm as well as its performance. 1.