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97
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.
Approximate Bisimulations for Nonlinear Dynamical Systems
, 2005
"... The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic n ..."
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Cited by 31 (6 self)
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The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunovlike inequalities, which for deterministic systems can be computed using recent sumofsquares techniques. Our framework is illustrated on a safety verification example.
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.
Symbolic model checking of hybrid systems using template polyhedra
 In TACAS’08  Tools and Algorithms for
, 2008
"... Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs o ..."
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Cited by 20 (7 self)
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Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs overapproximations of the reachable states using template polyhedra. Therefore, operations used in symbolic model checking such as intersection, union and postcondition across discrete transitions over template polyhedra can be computed efficiently using template polyhedra without requiring expensive vertex enumeration. Additionally, the verification of hybrid systems requires techniques to handle the continuous dynamics inside discrete modes. We propose a new flowpipe construction algorithm using template polyhedra. Our technique uses higherorder Taylor series expansion to approximate the time trajectories. The terms occurring in the Taylor series expansion are bounded using repeated optimization queries. The location invariant is used to enclose the remainder term of the Taylor series, and thus truncate the expansion. Finally, we have implemented our technique as a part of the tool TimePass for the analysis of affine hybrid automata. 1
Approximate bisimulation relations for constrained linear systems
 AUTOMATICA
, 2007
"... In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisi ..."
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Cited by 18 (5 self)
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In this paper, we define the notion of approximate bisimulation relation between two systems, extending the well established exact bisimulation relations for discrete and continuous systems. Exact bisimulation requires that the observations of two systems are and remain identical, approximate bisimulation allows the observation to be different provided they are and remain arbitrarily close. Approximate bisimulation relations are conveniently defined as level sets of a function called bisimulation function. For the class of linear systems with constrained initial states and constrained inputs, we develop effective characterizations for bisimulation functions that can be interpreted in terms of linear matrix inequalities, set inclusion and games. We derive a computationally effective algorithm to evaluate the precision of the approximate bisimulation between a constrained linear system and its projection. This algorithm has been implemented in a MATLAB toolbox: MATISSE. Two examples of use of the toolbox in the context of safety verification are shown.
Generating and Analyzing Symbolic Traces of Simulink/Stateflow Models
"... Abstract. We present a methodology and a toolkit for improving simulation coverage of Simulink/Stateflow models of hybrid systems using symbolic analysis of simulation traces. We propose a novel instrumentation scheme that allows the simulation engine of Simulink/Stateflow to output, along with the ..."
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Cited by 18 (3 self)
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Abstract. We present a methodology and a toolkit for improving simulation coverage of Simulink/Stateflow models of hybrid systems using symbolic analysis of simulation traces. We propose a novel instrumentation scheme that allows the simulation engine of Simulink/Stateflow to output, along with the concrete simulation trace, the symbolic transformers needed for our analysis. Given a simulation trace, along with the symbolic transformers, our analysis computes a set of initial states that would lead to traces with the same sequence of discrete components at each step of the simulation. Such an analysis relies critically on the use of convex polyhedra to represent sets of states. However, the exponential complexity of the polyhedral operations implies that the performance of the analysis would degrade rapidly with the increasing size of the model and the simulation traces. We propose a new representation, called the bounded vertex representation, which allows us to perform underapproximate computations while fixing the complexity of the representation a priori. Using this representation we achieve a tradeoff between the complexity of the symbolic computation and the quality of the underapproximation. We demonstrate the benefits of our approach over existing simulation and verification methods with case studies. 1
Reachability Analysis of Linear Systems using Support Functions
, 2009
"... This work is concerned with the algorithmic reachability analysis of continuoustime linear systems with constrained initial states and inputs. We propose an approach for computing an overapproximation of the set of states reachable on a bounded timeinterval. The main contribution over previous wor ..."
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Cited by 17 (2 self)
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This work is concerned with the algorithmic reachability analysis of continuoustime linear systems with constrained initial states and inputs. We propose an approach for computing an overapproximation of the set of states reachable on a bounded timeinterval. The main contribution over previous works is that it allows us to consider systems whose sets of initial states and inputs are given by arbitrary compact convex sets represented by their support functions. We actually compute two overapproximations of the reachable set. The first one is given by the union of convex sets with computable support functions. As the representation of convex sets by their support function is not suitable for some tasks, we derive from this first overapproximation a second one given by the union of polyhedrons. The overall computational complexity of our approach is comparable to the complexity of the most competitive available specialized algorithms for reachability analysis of linear systems using zonotopes or ellipsoids. The effectiveness of our approach is demonstrated on several examples.
Temporal Logic Verification Using Simulation
 In Proc. FORMATS’06
, 2006
"... Abstract. In this paper, we consider a novel approach to the temporal logic verification problem of continuous dynamical systems. Our methodology has the distinctive feature that enables the verification of the temporal properties of a continuous system by verifying only a finite number of its (simu ..."
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Cited by 16 (7 self)
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Abstract. In this paper, we consider a novel approach to the temporal logic verification problem of continuous dynamical systems. Our methodology has the distinctive feature that enables the verification of the temporal properties of a continuous system by verifying only a finite number of its (simulated) trajectories. The proposed framework comprises two main ideas. First, we take advantage of the fact that in metric spaces we can quantify how close are two different states. Based on that, we define robust, multivalued semantics for MTL (and LTL) formulas. These capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance from unsatisfiability. Second, we use the recently developed notion of bisimulation functions to infer the behavior of a set of trajectories that lie in the neighborhood of the simulated one. If the latter set of trajectories is bounded by the tube of robustness, then we can infer that all the trajectories in the neighborhood of the simulated one satisfy the same temporal specification as the simulated trajectory. The interesting and promising feature of our approach is that the more robust the system is with respect to the temporal logic specification, the less is the number of simulations that are required in order to verify the system. 1
Approximate reachability computation for polynomial systems
 in HSCC’06, vol. 3927 in LNCS
, 2006
"... Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time ste ..."
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Cited by 16 (9 self)
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Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system. 1