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Differential Dynamic Logic for Hybrid Systems
, 2007
"... Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, ..."
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Cited by 78 (46 self)
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Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations. With the goal of developing a theoretical and practical foundation for deductive verification of hybrid systems, we introduce a dynamic logic for hybrid programs, which is a program notation for hybrid systems. As a verification technique that is suitable for automation, we introduce a free variable proof calculus with a novel combination of realvalued free variables and Skolemisation for lifting quantifier elimination for real arithmetic to dynamic logic. The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts. Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations. In a case study with cooperating traffic agents of the European Train Control System, we further show that our calculus is wellsuited for verifying realistic hybrid systems with parametric system dynamics.
Safety verification of hybrid systems by constraint propagation based abstraction refinement
, 2005
"... This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and impr ..."
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Cited by 75 (11 self)
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This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and improve it by developing an additional refinement step that employs interval constraint propagation to add information to the abstraction without introducing new grid elements. Moreover, the resulting method allows switching conditions, initial states and unsafe states to be described by complex constraints instead of sets that correspond to grid elements. Nevertheless, the method can be easily implemented since it is based on a welldefined set of constraints, on which one can run any constraint propagation based solver. Tests of such an implementation are promising.
Computing differential invariants of hybrid systems as fixedpoints
, 2008
"... Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose righthand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, ..."
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Cited by 58 (21 self)
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Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose righthand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control.
KeYmaera: A hybrid theorem prover for hybrid systems
 IJCAR. VOLUME 5195 OF LNCS
, 2008
"... KeYmaera is a hybrid verification tool for hybrid systems that combines deductive, real algebraic, and computer algebraic prover technologies. It is an automated and interactive theorem prover for a natural specification and verification logic for hybrid systems. KeYmaera supports differential dyn ..."
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Cited by 56 (24 self)
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KeYmaera is a hybrid verification tool for hybrid systems that combines deductive, real algebraic, and computer algebraic prover technologies. It is an automated and interactive theorem prover for a natural specification and verification logic for hybrid systems. KeYmaera supports differential dynamic logic, which is a realvalued firstorder dynamic logic for hybrid programs, a program notation for hybrid automata. For automating the verification process, KeYmaera implements a generalized freevariable sequent calculus and automatic proof strategies that decompose the hybrid system specification symbolically. To overcome the complexity of real arithmetic, we integrate real quantifier elimination following an iterative background closure strategy. Our tool is particularly suitable for verifying parametric hybrid systems and has been used successfully for verifying collision avoidance in case studies from train control and air traffic management.
Verifying Analog Oscillator Circuits Using Forward/Backward Abstraction Refinement
 In DATE 2006: Design, Automation and Test in Europe
, 2006
"... Properties of analog circuits can be verified formally by partitioning the continuous state space and applying hybrid system verification techniques to the resulting abstraction. To verify properties of oscillator circuits, cyclic invariants need to be computed. Methods based on forward reachability ..."
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Cited by 42 (1 self)
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Properties of analog circuits can be verified formally by partitioning the continuous state space and applying hybrid system verification techniques to the resulting abstraction. To verify properties of oscillator circuits, cyclic invariants need to be computed. Methods based on forward reachability have proven to be inefficient and in some cases inadequate in constructing these invariant sets. In this paper we propose a novel approach combining forward and backwardreachability while iteratively refining partitions at each step. The technique can yield dramatic memory and runtime reductions. We illustrate the effectiveness by verifying, for the first time, the limit cycle oscillation behavior of a thirdorder model of a differential VCO circuit. 1.
Robust Test Generation and Coverage for Hybrid Systems
, 2007
"... Testing is an important tool for validation of the system design and its implementation. Modelbased test generation allows to systematically ascertain whether the system meets its design requirements, particularly the safety and correctness requirements of the system. In this paper, we develop a fr ..."
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Cited by 42 (13 self)
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Testing is an important tool for validation of the system design and its implementation. Modelbased test generation allows to systematically ascertain whether the system meets its design requirements, particularly the safety and correctness requirements of the system. In this paper, we develop a framework for generating tests from hybrid systems’ models. The core idea of the framework is to develop a notion of robust test, where one nominal test can be guaranteed to yield the same qualitative behavior with any other test that is close to it. Our approach offers three distinct advantages. 1) It allows for computing and formally quantifying the robustness of some properties, 2) it establishes a method to quantify the test coverage for every test case, and 3) the procedure is parallelizable and therefore, very scalable. We demonstrate our framework by generating tests for a navigation benchmark application.
Symbolic analysis for improving simulation coverage of simulink/stateflow models
 in EMSOFT ’08: Proceedings of the 8th ACM international conference on Embedded software, 2008
"... Aimed at verifying safety properties and improving simulation coverage for hybrid systems models of embedded control software, we propose a technique that combines numerical simulation and symbolic methods for computing statesets. We consider systems with linear dynamics described in the commercial ..."
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Cited by 37 (4 self)
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Aimed at verifying safety properties and improving simulation coverage for hybrid systems models of embedded control software, we propose a technique that combines numerical simulation and symbolic methods for computing statesets. We consider systems with linear dynamics described in the commercial modeling tool Simulink/Stateflow. Given an initial state x, and a discretetime simulation trajectory, our method computes a set of initial states that are guaranteed to be equivalent to x, where two initial states are considered to be equivalent if the resulting simulation trajectories contain the same discrete components at each step of the simulation. We illustrate the benefits of our method on two case studies. One case study is a benchmark proposed in the literature for hybrid systems verification and another is a Simulink demo model from Mathworks.
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.