Results 1  10
of
38
Approximation metrics for discrete and continuous systems
 IEEE Transactions on Automatic Control
, 2005
"... Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In thi ..."
Abstract

Cited by 105 (16 self)
 Add to MetaCart
(Show Context)
Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudometrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudometrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the socalled branching distance between the systems. Approximations for the pseudometrics can be obtained by considering Lyapunovlike functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems.
Approximate simulation relations for hybrid systems
, 2006
"... Approximate simulation relations have recently been introduced as a powerful tool for the approximation of discrete and continuous systems. In this paper, we extend this notion to hybrid systems. Using the socalled simulation functions, we develop a computationally effective characterization of ap ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
Approximate simulation relations have recently been introduced as a powerful tool for the approximation of discrete and continuous systems. In this paper, we extend this notion to hybrid systems. Using the socalled simulation functions, we develop a computationally effective characterization of approximate simulation relations which can be used for hybrid systems approximation. An example of application in the context of safety verification is shown.
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 ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
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.
Reachability analysis of multiaffine systems
 In Hybrid Systems: Computation and Control, LNCS 3927
, 2006
"... Abstract We present a computationally attractive technique to study the reachability of rectangular regions by trajectories of continuous multiaffine systems. The method is iterative. At each step, finer partitions and finite quotients that overapproximate the reachability properties of the initi ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
(Show Context)
Abstract We present a computationally attractive technique to study the reachability of rectangular regions by trajectories of continuous multiaffine systems. The method is iterative. At each step, finer partitions and finite quotients that overapproximate the reachability properties of the initial system are produced. We exploit some convexity properties of multiaffine functions on rectangles to show that the construction of the quotient at each step requires only the evaluation of the vector field at the set of all vertices of all rectangles in the partition and finding the roots of a finite set of scalar affine functions. This methodology can be used for formal analysis of biochemical networks, aircraft and underwater vehicles, where multiaffine models are widely used.
Bisimulation for general stochastic hybrid systems
 In HSCC 2005
, 2005
"... Abstract. In this paper we define a bisimulation concept for some very general models for stochastic hybrid systems (general stochastic hybrid systems). The definition of bisimulation builds on the ideas of Edalat and of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result is that this ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we define a bisimulation concept for some very general models for stochastic hybrid systems (general stochastic hybrid systems). The definition of bisimulation builds on the ideas of Edalat and of Larsen and Skou and of Joyal, Nielsen and Winskel. The main result is that this bisimulation for GSHS is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same kind of bisimulation for their continuous parts and respectively for their jumping structures.
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 ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
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.
A homology theory for hybrid systems: Hybrid homology
 Lect. Notes in Computer Science 3414
, 2005
"... Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or Hspace. The homotopy colimit can be appl ..."
Abstract

Cited by 18 (9 self)
 Add to MetaCart
(Show Context)
Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or Hspace. The homotopy colimit can be applied to this Hspace to obtain a single topological space; the hybrid homology of an Hspace is the homology of this space. The result is a spectral sequence converging to the hybrid homology of an Hspace, providing a concrete way to compute this homology. Moreover, the hybrid homology of the Hspace underlying a hybrid system gives useful information about the behavior of this system: the vanishing of the first hybrid homology of this Hspace—when it is contractible and finite—implies that this hybrid system is not Zeno. 1
Controller synthesis for bisimulation equivalence
 Systems and Control Letters
"... Abstract. The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discreteevent systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that th ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discreteevent systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality. 1.
Approximate bisimulations for constrained linear systems
, 2005
"... In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for transition ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
In this paper, inspired by exact notions of bisimulation equivalence for discreteevent and continuoustime systems, we establish approximate bisimulation equivalence for linear systems with internal but bounded disturbances. This is achieved by developing a theory of approximation for transition systems with observation metrics, which require that the distance between system observations is and remains arbitrarily close in the presence of nondeterministic evolution. Our notion of approximate bisimulation naturally reduces to exact bisimulation when the distance between the observations is zero. Approximate bisimulation relations are then characterized by a class of Lyapunovlike functions which are called bisimulation functions. For the class of linear systems with constrained disturbances, we obtain computable characterizations of bisimulation functions in terms of linear matrix inequalities, set inclusions, and optimal values of static games. We illustrate our framework in the context of safety verification.
Robust ReachAvoid Controller Synthesis for Switched Nonlinear Systems
"... Abstract — In this paper, we describe a method to automatically synthesize controllers that provide hard guarantees of safety and target reachability for sampleddata switched systems under bounded continuous disturbances. Techniques from hybrid system verification are used to perform continuous tim ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Abstract — In this paper, we describe a method to automatically synthesize controllers that provide hard guarantees of safety and target reachability for sampleddata switched systems under bounded continuous disturbances. Techniques from hybrid system verification are used to perform continuous time differential game calculations on each sampling interval. Iterative procedures are given for computing the set of states for which there exists an admissible control policy so that the closedloop system satisfies the properties of safety and reachability over a finite time horizon. From this computation, we show how to obtain an explicit state feedback policy in the form of multiple reachable sets, and an algorithm is given for using this feedback law in closedloop control of the switched system. A simulation example of automated aerial refueling is used to illustrate the application of our approach. I.