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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 168 (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.
Approximate Reachability Analysis of PiecewiseLinear Dynamical Systems
, 2000
"... . In this paper we describe an experimental system called d=dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much small ..."
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Cited by 140 (31 self)
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. In this paper we describe an experimental system called d=dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previouslyused methods. The d=dt system can, so far, treat nontrivial continuous systems, hybrid systems, convex differential inclusions and controller synthesis problems. 1 Introduction The problem of calculating reachable states for continuous and hybrid systems has emerged as one of the major problems in hybrid systems research [G96,GM98,DM98,KV97,V98,GM99,CK99,PSK99,HHMW99]. It constitutes a prerequisite for exporting algorithmic verification methodology outside discrete systems or hybrid systems with piecewisetrivial dynamics. For computer scientists it poses new challenges in treating continuous functions and their approximations and in applying computational geometry...
Computational Techniques for Hybrid System Verification
 IEEE Trans. on Automatic Control
, 2003
"... Abstract—This paper concerns computational methods for verifying properties of polyhedral invariant hybrid automata (PIHA), which are hybrid automata with discrete transitions governed by polyhedral guards. To verify properties of the state trajectories for PIHA, the planar switching surfaces are p ..."
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Cited by 115 (5 self)
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Abstract—This paper concerns computational methods for verifying properties of polyhedral invariant hybrid automata (PIHA), which are hybrid automata with discrete transitions governed by polyhedral guards. To verify properties of the state trajectories for PIHA, the planar switching surfaces are partitioned to define a finite set of discrete states in an approximate quotient transition system (AQTS). State transitions in the AQTS are determined by the reachable states, or flow pipes, emitting from the switching surfaces according to the continuous dynamics. This paper presents a method for computing polyhedral approximations to flow pipes. It is shown that the flowpipe approximation error can be made arbitrarily small for general nonlinear dynamics and that the computations can be made more efficient for affine systems. The paper also describes CheckMate, a MATLABbased tool for modeling, simulating and verifying properties of hybrid systems based on the computational methods previously described. Index Terms—Hybrid systems, model checking, reachability, verification. I.
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 110 (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 78 (24 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
Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations
 In Hybrid Systems : Computation and Control
, 2000
"... Abstract. A general verification algorithm is described. It is then shown how ellipsoidal methods developed by A. B. Kurzhanski and P. Varaiya can be adapted to the algorithm. New numerical algorithms that compute approximations of unions of ellipsoids and intersections of ellipsoids and polyhedra ..."
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Cited by 73 (1 self)
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Abstract. A general verification algorithm is described. It is then shown how ellipsoidal methods developed by A. B. Kurzhanski and P. Varaiya can be adapted to the algorithm. New numerical algorithms that compute approximations of unions of ellipsoids and intersections of ellipsoids and polyhedra were developed. The presented techniques were implemented in the verification tool called VeriSHIFT and some practical results are discussed.
Computational techniques for the verification of hybrid systems
 Proceedings of the IEEE
, 2003
"... Hybrid system theory lies at the intersection of the fields of engineering control theory and computer science verification. It is defined as the modeling, analysis, and control of systems that involve the interaction of both discrete state systems, represented by finite automata, and continuous sta ..."
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Cited by 72 (9 self)
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Hybrid system theory lies at the intersection of the fields of engineering control theory and computer science verification. It is defined as the modeling, analysis, and control of systems that involve the interaction of both discrete state systems, represented by finite automata, and continuous state dynamics, represented by differential equations. The embedded autopilot of a modern commercial jet is a prime example of a hybrid system: the autopilot modes correspond to the application of different control laws, and the logic of mode switching is determined by the continuous state dynamics of the aircraft, as well as through interaction with the pilot. To understand the behavior of hybrid systems, to simulate, and to control these systems, theoretical advances, analyses, and numerical tools are needed. In this paper, we first present a general model for a hybrid system along with an overview of methods for verifying continuous and hybrid systems. We describe a particular verification
Beyond HYTECH: Hybrid systems analysis using interval numerical methods
 in HSCC
, 2000
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Reachability Analysis Using Polygonal Projections
 IN HYBRID SYSTEMS: COMPUTATION AND CONTROL
, 1999
"... Coho is a reachability analysis tool for systems modeled by nonlinear, ordinary differential equations. Coho represents highdimensional objects using projections onto planes corresponding to pairs of variables. This representation is compact and allows efficient algorithms from computational geome ..."
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Cited by 57 (5 self)
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Coho is a reachability analysis tool for systems modeled by nonlinear, ordinary differential equations. Coho represents highdimensional objects using projections onto planes corresponding to pairs of variables. This representation is compact and allows efficient algorithms from computational geometry to be exploited while also capturing dependencies in the behaviour of related variables. Reachability is performed by integration where methods from linear programming and linear systems theory are used to bound trajectories emanating from each face of the object. This paper has two contributions: first, we describe the implementation of Coho and, second, we present analysis results obtained by using Coho on several simple models.