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22
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.
Computation of an overapproximation of the backward reachable set using subsystem level set functions
 DYNAMICS OF CONTINUOUS, DISCRETE & IMPULSIVE SYSTEMS SERIES A: MATHEMATICAL ANALYSIS
, 2004
"... In this paper, we present a method to decompose the problem of computing the backward reachable set for a dynamic system in a space of a given dimension, into a set of computational problems involving level set functions, each defined in a lower dimensional (subsystem) space. This allows the potent ..."
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Cited by 18 (3 self)
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In this paper, we present a method to decompose the problem of computing the backward reachable set for a dynamic system in a space of a given dimension, into a set of computational problems involving level set functions, each defined in a lower dimensional (subsystem) space. This allows the potential for great reduction in computation time. The overall system is considered as an interconnection of either disjoint or overlapping subsystems. The projection of the backward reachable set into the subsystem spaces is overapproximated by a level set of the corresponding subsystem level set function. It is shown how this method can be applied to twoplayer differential games. Finally, results of the computation of polytopic overapproximations of the unsafe set for the two aircraft conflict resolution problem are presented.
Reachability Under Uncertainty
"... The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances, which may also be interpreted as the action of an adversary in a gametheoretic setting. It defines possible notions of reachability under uncertainty emphasizing th ..."
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Cited by 14 (2 self)
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The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances, which may also be interpreted as the action of an adversary in a gametheoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between openloop and closedloop control. Solution schemes for calculating reachability sets are indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) and nonanticipative (minmax) piecewise openloop control with corrections and to corresponding notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closedloop reach sets under uncertainty may be found through a solution of the forward HamiltonJacobiBellmanIsaacs (HJBI) equation.The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.
Ellipsoidal techniques for hybrid dynamics: the reachability problem
 in New Directions and Applications in Control Theory
"... This report deals with the dynamics of hybrid systems under piecewise openloop controls restricted by hard bounds. It is assumed that the system equations may be reset when crossing some prespecified domains (“the guards”) in the state space. Therefore the continuous dynamics which govern the motion ..."
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Cited by 11 (0 self)
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This report deals with the dynamics of hybrid systems under piecewise openloop controls restricted by hard bounds. It is assumed that the system equations may be reset when crossing some prespecified domains (“the guards”) in the state space. Therefore the continuous dynamics which govern the motion between the guards are complemented by dicrete transitions which govern the resets. A state space model for such systems is proposed and reachability sets for such models are described. The computational side of the problem is treated through ellipsoidal techniques which indicate routes for numerical algorithms.
Polytopic Approximations of Reachable Sets applied to Linear Dynamic Games and to a Class of Nonlinear Systems
, 2005
"... This paper presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a HamiltonJacobi partial differential equation (PDE). Numerical solutions which provide converg ..."
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Cited by 9 (0 self)
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This paper presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a HamiltonJacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems [1]. In this paper, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and normbounded nonlinear systems. Three illustrative examples are presented.
Ellipsoidal techniques for reachability under state constraints
 SIAM JOURNAL OF CONTROL AND OPTIMIZATION
, 2003
"... The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with timevarying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoidalvalued tubes. These tubes touch the reach ..."
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Cited by 7 (1 self)
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The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with timevarying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoidalvalued tubes. These tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating elliposids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.
Applications of polytopic approximations of reachable sets to linear dynamic games and a class of nonlinear systems
 Proceedings of the American Control Conference
"... This paper presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a HamiltonJacobi partial differential equation (PDE). Numerical solutions which provide convergen ..."
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Cited by 5 (1 self)
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This paper presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a HamiltonJacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems [1]. In this paper, we show that these can be extended to feedback linearizable nonlinear systems, linear dynamic games, and normbounded nonlinear systems. Three illustrative examples are presented. 1
Optimization Methods for Target Problems of Control
"... The present report indicates an array of reachability problems relevant for nonstandard target problems of control. The problems are solved through dynamic optimization techniques for systems with nonintegral costs. This leads to new types of generalized HamiltonJacobiBellmantype equations in the ..."
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Cited by 3 (0 self)
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The present report indicates an array of reachability problems relevant for nonstandard target problems of control. The problems are solved through dynamic optimization techniques for systems with nonintegral costs. This leads to new types of generalized HamiltonJacobiBellmantype equations in the general case and allows treatment through duality methods of convex analysis and minmax theory in the linear case.
A Verified Hierarchical Control Architecture for Coordinated MultiVehicle Operation
 INT. J. ADAPT. CONTROL SIGNAL PROCESS.
, 2005
"... A layered control architecture for executing multivehicle team coordination algorithms is presented along with the specifications for team behavior. The control architecture consists of three layers: team control, vehicle supervision and maneuver control. It is shown that the controller implementat ..."
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Cited by 3 (1 self)
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A layered control architecture for executing multivehicle team coordination algorithms is presented along with the specifications for team behavior. The control architecture consists of three layers: team control, vehicle supervision and maneuver control. It is shown that the controller implementation is consistent with the system specification on the desired team behavior. Computer simulations with accurate models of autonomous underwater vehicles illustrate the overall approach in the coordinated search for the minimum of a scalar field. The coordinated search is based on the simplex optimization algorithm.
APPROXIMATION OF REACHABLE SETS USING OPTIMAL CONTROL ALGORITHMS
"... (Communicated by Peng Shi) Abstract. We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence a ..."
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(Communicated by Peng Shi) Abstract. We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented. 1. Introduction. The