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38
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
Formal verification of hybrid systems
, 2011
"... In formal verification, a designer first constructs a model, with mathematically precise semantics, of the system under design, and performs extensive analysis with respect to correctness requirements. The appropriate mathematical model for embedded control systems is hybrid systems that combines th ..."
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Cited by 34 (0 self)
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In formal verification, a designer first constructs a model, with mathematically precise semantics, of the system under design, and performs extensive analysis with respect to correctness requirements. The appropriate mathematical model for embedded control systems is hybrid systems that combines the traditional statemachine based models for discrete control with classical differentialequations based models for continuously evolving physical activities. In this article, we briefly review selected existing approaches to formal verification of hybrid systems, along with directions for future research.
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.
Recent progress in continuous and hybrid reachability analysis
 In Proc. IEEE International Symposium on ComputerAided Control Systems Design. IEEE Computer
, 2006
"... Abstract — Setbased reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachabilit ..."
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Cited by 30 (1 self)
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Abstract — Setbased reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachability analysis has been limited so far to simple systems, both in the continuous and the hybrid domain. In this paper we present recent advances that, in combination, significantly improve this applicability, and allow us to find better balance between computational cost and accuracy. The presentation covers, in a unified manner, a variety of methods handling increasingly complex types of continuous dynamics (constant derivative, linear, nonlinear). The improvements include new geometrical objects for representing sets, new approximation schemes, and more flexible combinations of graphsearch algorithm and partition refinement. We report briefly some preliminary experiments that have enabled the analysis of systems previously beyond reach. I.
Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis
 HSCC’08, to appear
, 2008
"... In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algo ..."
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Cited by 24 (1 self)
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In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algorithm based on zonotopes. In order to use this technique for the analysis of hybrid systems, we must also deal with the discrete transitions in a satisfactory (i.e. scalable and accurate) way. For that purpose, we need to approximate the intersection of the continuous reachable sets with the guards enabling the discrete transitions. The main contribution of this paper is a novel algorithm for computing efficiently a tight overapproximation of the intersection of (possibly highorder) zonotopes with a hyperplane. We show the accuracy and the scalability of our approach by considering two examples of reachability analysis of hybrid systems.
Reachable set computation for uncertain timevarying linear systems
 IN: HYBRID SYSTEMS: COMPUTATION AND CONTROL
, 2011
"... This paper presents a method for using setbased approximations to the PeanoBaker series to compute overapproximations of reachable sets for linear systems with uncertain, timevarying parameters and inputs. Alternative representations for sets of uncertain system matrices are considered, including ..."
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Cited by 16 (10 self)
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This paper presents a method for using setbased approximations to the PeanoBaker series to compute overapproximations of reachable sets for linear systems with uncertain, timevarying parameters and inputs. Alternative representations for sets of uncertain system matrices are considered, including matrix polytopes, matrix zonotopes, and interval matrices. For each representation, the computational efficiency and resulting approximation error for reachable set computations are evaluated analytically and empirically. As an application, reachable sets are computed for a truck with hybrid dynamics due to a gainscheduled yaw controller. As an alternative to computing reachable sets for the hybrid model, for which switching introduces an additional overapproximation error, the gainscheduled controller is approximated with uncertain timevarying parameters, which leads to more efficient and more accurate reachable set computations.
Analysis of lactose metabolism in E.coli using reachability analysis of hybrid systems
 IEE PROCEEDINGS  SYSTEMS BIOLOGY
, 2007
"... We propose an abstraction method for medium scale biomolecular networks, based on hybrid dynamical systems with continuous multiaffine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easil ..."
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Cited by 14 (3 self)
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We propose an abstraction method for medium scale biomolecular networks, based on hybrid dynamical systems with continuous multiaffine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easily automated. An efficient reachability algorithm is possible for the resulting class of hybrid systems. We construct an approximation for an ordinary differential equation model of the lac operon, and show that our abstraction passes the same experimental tests that were used to validate the original model. The wellstudied biological system exhibits bistability and switching behavior, arising from positive feedback in the expression mechanism of the lac operon. The switching property of the lac system is an example of the major qualitative features that are the building blocks of higher level, more coarsegrained descriptions. Our approach is useful in helping correctly identify such properties and in connecting them to the underlying molecular dynamical details. We use reachability analysis together with the knowledge of the steady state structure to identify ranges of parameter values for which the system maintains the bistable switching property.
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.
Avoiding geometric intersection operations in reachability analysis of hybrid systems
 In Hybrid Systems: Computation and Control
, 2012
"... Although a growing number of dynamical systems studied in various fields are hybrid in nature, the verification of properties, such as stability, safety, etc., is still a challenging problem. Reachability analysis is one of the promising methods for hybrid system verification, which together with ..."
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Cited by 11 (6 self)
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Although a growing number of dynamical systems studied in various fields are hybrid in nature, the verification of properties, such as stability, safety, etc., is still a challenging problem. Reachability analysis is one of the promising methods for hybrid system verification, which together with all other verification techniques faces the challenge of making the analysis scale with respect to the number of continuous state variables. The bottleneck of many reachability analysis techniques for hybrid systems is the geometrically computed intersection with guard sets. In this work, we replace the intersection operation by a nonlinear mapping onto the guard, which is not only numerically stable, but also scalable, making it possible to verify systems which were previously out of reach. The approach can be applied to the fairly common class of hybrid systems with piecewise continuous solutions, guard sets modeled as halfspaces, and urgent semantics, i.e. discrete transitions are immediately taken when enabled by guard sets. We demonstrate the usefulness of the new approach by a mechanical system with backlash which has 101 continuous state variables.
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.