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On ellipsoidal techniques for reachability analysis. Part II. Internal Approximations Boxvalued constraints
, 2001
"... Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with timevarying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constr ..."
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Cited by 120 (8 self)
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Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with timevarying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with boxvalued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, boxvalued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.
Effective Synthesis of Switching Controllers for Linear Systems
, 2000
"... In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the sys ..."
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Cited by 76 (8 self)
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In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the system from one "mode" to another in order to avoid a set of bad states, and propose an abstract algorithm which solves the problem by an iterative computation of reachable states. We have implemented a concrete version of the algorithm, which uses a new approximation scheme for reachability analysis of linear systems.
Hierarchical Modeling and Analysis of Embedded Systems
, 2003
"... This paper describes the modeling language CHARON for modular design of interacting hybrid systems. The language allows specification of architectural as well as behavioral hierarchy and discrete as well as continuous activities. The modular structure of the language is not merely syntactic, but is ..."
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Cited by 56 (20 self)
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This paper describes the modeling language CHARON for modular design of interacting hybrid systems. The language allows specification of architectural as well as behavioral hierarchy and discrete as well as continuous activities. The modular structure of the language is not merely syntactic, but is exploited by analysis tools and is supported by a formal semantics with an accompanying compositional theory of refinement. We illustrate the benefits of CHARON in the design of embedded control software using examples from automated highways concerning vehicle coordination
Reachability Analysis of Hybrid Systems via Predicate Abstraction
 Hybrid Systems: Computation and Control, Fifth International Workshop, LNCS 2289
, 2002
"... Predicate abstraction has emerged to be a powerful technique for extracting finitestate models from infinitestate discrete programs. This paper presents algorithms and tools for reachability analysis of hybrid systems by combining the notion of predicate abstraction with recent techniques for appr ..."
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Cited by 44 (7 self)
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Predicate abstraction has emerged to be a powerful technique for extracting finitestate models from infinitestate discrete programs. This paper presents algorithms and tools for reachability analysis of hybrid systems by combining the notion of predicate abstraction with recent techniques for approximating the set of reachable states of linear systems using polyhedra. Given a hybrid system and a set of userdefined predicates, we consider the finite discrete quotient whose states correspond to all possible truth assignments to the input predicates. The tool performs an onthefly exploration of the abstract system by using weakest preconditions to compute abstract transitions corresponding to the discrete switches and conservative polyhedral approximations to compute abstract transitions corresponding to continuous flows. Compared to tools such as Checkmate and d/dt, this approach requires significantly less computational resources as the emphasis is shifted from computing the reachable set to searching in the abstract quotient. We demonstrate the feasibility of the proposed technique by analyzing a parametric timingbased mutual exclusion protocol and safety of a simple controller for vehicle coordination.
Reachability analysis of nonlinear systems using conservative approximation
 In Oded Maler and Amir Pnueli, editors, Hybrid Systems: Computation and Control, LNCS 2623
, 2003
"... ..."
Abstraction and CounterexampleGuided Refinement in Model Checking of Hybrid Systems
, 2003
"... Hybrid dynamic systems include both continuous and discrete state variables. Properties of hybrid systems, which have an infinite state space, can often be verified using ordinary model checking together with a finitestate abstraction. Model checking can be inconclusive, however, in which case t ..."
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Cited by 38 (4 self)
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Hybrid dynamic systems include both continuous and discrete state variables. Properties of hybrid systems, which have an infinite state space, can often be verified using ordinary model checking together with a finitestate abstraction. Model checking can be inconclusive, however, in which case the abstraction must be refined. This paper presents a new procedure to perform this refinement operation for abstractions of hybrid systems. Following an approach originally developed for finitestate systems [11, 25], the refinement procedure constructs a new abstraction that eliminates a counterexample generated by the model checker. For hybrid systems, analysis of the counterexample requires the computation of sets of reachable states in the continuous state space. We show how such reachability computations with varying degrees of complexity can be used to refine hybrid system abstractions efficiently.
On the decidability of the reachability problem for planar differential inclusions
 In HSCC’2001, number 2034 in LNCS
, 2001
"... Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is ..."
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Cited by 34 (13 self)
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Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is the use of onedimensional affine Poincaré maps for which we can easily compute the fixpoints. As a first step, we show that between any two points linked by an arbitrary trajectory there always exists a trajectory without selfcrossings. Thus, solving the reachability problem requires considering only those. We prove that, indeed, there are only finitely many “qualitative types ” of those trajectories. The last step consists in giving a decision procedure for each of them. These procedures are essentially based on the analysis of the limits of extreme trajectories. We illustrate our algorithm on a simple model of a swimmer spinning around a whirlpool. 1
On efficient representation and computation of reachable sets for hybrid systems
 In HSCC’2003, LNCS 2289
, 2003
"... Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approxim ..."
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Cited by 28 (6 self)
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Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the overapproximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.
Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets
 in Hybrid Systems: Computation and Control
, 2001
"... We develop a general framework for solving the hybrid system reachability problem, and indicate how several published techniques fit into this framework. The key unresolved need of any hybrid system reachability algorithm is the computation of continuous reachable sets; consequently, we present ..."
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Cited by 26 (14 self)
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We develop a general framework for solving the hybrid system reachability problem, and indicate how several published techniques fit into this framework. The key unresolved need of any hybrid system reachability algorithm is the computation of continuous reachable sets; consequently, we present new results on techniques for calculating numerical approximations of such sets evolving under general nonlinear dynamics with inputs. Our tool is based on a local level set procedure for boundary propagation in continuous state space, and has been implemented using numerical schemes of varying orders of accuracy. We demonstrate the numerical convergence of these schemes to the viscosity solution of the HamiltonJacobi equation, which was shown in earlier work to be the exact representation of the boundary of the reachable set. We then describe and solve a new benchmark example in nonlinear hybrid systems: an autolander for a commercial aircraft in which the switching logic and continuous control laws are designed to maximize the safe operating region across the hybrid state space.