The minimum principle for hybrid systems with partitioned state space and unspecified discrete state sequence

2010 ◽  
Author(s):  
Benjamin Passenberg ◽  
Peter E. Caines ◽  
Marion Sobotka ◽  
Olaf Stursberg ◽  
Martin Buss
Keyword(s):  
State Space ◽  
Hybrid Systems ◽  
Minimum Principle ◽  
Discrete State ◽  
State Sequence

Abstraction and Counterexample-Guided Refinement in Model Checking of Hybrid Systems

2003 ◽  
Vol 14 (04) ◽  
pp. 583-604 ◽  
Author(s):  
Edmund Clarke ◽  
Ansgar Fehnker ◽  
Zhi Han ◽  
Bruce Krogh ◽  
Joël Ouaknine ◽  
...  
Keyword(s):  
Model Checking ◽  
State Space ◽  
Hybrid Systems ◽  
State Variables ◽  
Discrete State ◽  
Continuous State Space ◽  
Continuous State ◽  
Finite State ◽  
Refinement Procedure ◽  
State Abstraction

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 finite-state 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 finite-state 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. Examples illustrate our counterexample-guided refinement procedure. Experimental results for a prototype implementation indicate significant advantages over existing methods.

scholarly journals Uncertainty quantification for quantum chemical models of complex reaction networks

2016 ◽  
Vol 195 ◽  
pp. 497-520 ◽  
Author(s):  
Jonny Proppe ◽  
Tamara Husch ◽  
Gregor N. Simm ◽  
Markus Reiher
Keyword(s):  
Free Energy ◽  
Electronic Structure ◽  
State Space ◽  
Time Scales ◽  
Reaction Network ◽  
Reaction Networks ◽  
Multiple Time Scales ◽  
Complex Reaction ◽  
Discrete State ◽  
Continuous State

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

scholarly journals Improved estimations of stochastic chemical kinetics by finite-state expansion

2021 ◽  
Vol 477 (2251) ◽  
pp. 20200964
Author(s):  
Tabea Waizmann ◽  
Luca Bortolussi ◽  
Andrea Vandin ◽  
Mirco Tribastone
Keyword(s):  
Master Equation ◽  
State Space ◽  
Stochastic Dynamics ◽  
Reaction Network ◽  
Discrete State ◽  
State Expansion ◽  
Analytical Approximations ◽  
Finite State ◽  
Stochastic Reaction ◽  
Discrete State Space

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

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