On the Fokker-Planck Equation for Stochastic Hybrid Systems: Application to a Wind Turbine Model

Bect, Julien; Phulpin, Yannick; Baili, Hana; Fleury, Gilles

doi:10.1109/pmaps.2006.360298

Cited by 6 publications

(2 citation statements)

References 24 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is especially useful for the computation of the stationary distribution, as shown in [32] a nontrivial three-dimensional model of a wind turbine. This type of application of the FPK equation to the analysis of GSHSs relies on the availability of software components allowing an easy implementation of efficient numerical methods, in the spirit of Mitchell's Level Set Toolbox [33,34] for Hamilton-Jacobi equations.…”

Section: Discussionmentioning

confidence: 99%

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

Bect

2010

Nonlinear Analysis: Hybrid Systems

Self Cite

View full text Add to dashboard Cite

a b s t r a c tA general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHSs). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.

show abstract

Section: Discussionmentioning

confidence: 99%

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

Bect

2010

Nonlinear Analysis: Hybrid Systems

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the continuous-time case, a theoretical upper bound on the reachability probability is derived in Bujorianu (2004) using Dirichlet forms. The temporal evolution of the probability density function of the hybrid state has been characterized through generalized Fokker-Planck equations (Bect et al, 2006). Optimal control of stochastic hybrid systems is considered in Bensoussan and Menaldi (2000) and quasi-variational inequalities based on dynamic programming are derived for the optimal trajectory.…”

Section: Overview and Related Workmentioning

confidence: 99%

Methods for Reachability-based Hybrid Controller Design

Ding¹

2012

View full text Add to dashboard Cite

With the increasing complexity of systems found in practical applications, the problem of controller design is often approached in a hierarchical fashion, with discrete abstractions and design methods used to satisfy high level task specifications, and continuous abstractions and design techniques used to satisfy low level control objectives. Although such a separation allows the application of mature theoretical and computational tools from the realms of computer science and control theory, the task of ensuring desired closed-loop behaviors, which results from the composition between discrete and continuous designs, often requires costly and time consuming verification and validation. This problem becomes especially acute in safety-critical applications, in which design specifications are often subject to rigorous industry standards and government regulations. Hybrid systems, which feature state trajectories evolving on a combination of discrete and continuous state spaces, have been proposed as a possible approach to reconcile the analysis and design techniques from the discrete and continuous domains under a rigorous theoretical framework. However, designing controllers for general classes of hybrid systems is a highly nontrivial task, as such a design problem inherits both the difficulty of nonlinear control, as well as the range of theoretical and computational issues introduced by the consideration of discrete switching.This dissertation describes several efforts aimed towards the development of theoretical analysis tools and computational synthesis techniques to facilitate the systematic design of feedback control policies satisfying safety and target attainability specifications with respect to subclasses of hybrid system models. The main types of problems we consider are safety/invariance problems, which involve keeping the closed-loop state trajectory within a safe set in the hybrid state space, and reach-avoid problems, which involve driving the state trajectory into a target set subject to a safety constraint. These problems are addressed within the context of continuous time switched nonlinear systems and discrete time stochastic hybrid systems, as motivated by application scenarios arising in autonomous vehicle control and air traffic management.First, we provide several design techniques and synthesis algorithms for deterministic reachability problems formulated in the setting of switched nonlinear systems, with controlled switches between discrete modes, and bounded continuous disturbances. For scenarios in which the mode transitions proceed in a known sequence, a method is discussed for designing controllers to satisfy 1 sequential reachability specifications, consisting of a temporally ordered sequence of invariance and reach-avoid objectives. In particular, we use continuous time reachable sets to inform choices of feedback control policies within each discrete mode to satisfy both individual reachability objectives and compatibility conditions between successive modes. This technique is illus...

show abstract

Interdisciplinary Modeling of Autonomous Systems Deployed in Uncertain Dynamic Environments

Bujorianu

2011

Lecture Notes in Electrical Engineering

View full text Add to dashboard Cite

On the Fokker-Planck Equation for Stochastic Hybrid Systems: Application to a Wind Turbine Model

Cited by 6 publications

References 24 publications

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

Methods for Reachability-based Hybrid Controller Design

Interdisciplinary Modeling of Autonomous Systems Deployed in Uncertain Dynamic Environments

Contact Info

Product

Resources

About