![Page 1: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/1.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE: modelling stochastic hybridsystems
Vashti GalpinLaboratory for Foundations of Computer Science
University of Edinburgh
Joint work with Jane Hillston (University of Edinburgh)and Luca Bortolussi (University of Trieste)
20 April 2011
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 2: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/2.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Introduction
I stochastic hybrid process algebraI discrete behaviourI continuous behaviour, expressed as ODEsI stochastic behaviour, here using exponential distribution
I why use a process algebraI compositionalI language allows for abstract reasoning
I outlineI HYPE and stochastic HYPEI train gate exampleI checking for infinite discrete behaviourI I-graphs and acyclicity
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 3: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/3.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Introduction
I stochastic hybrid process algebraI discrete behaviourI continuous behaviour, expressed as ODEsI stochastic behaviour, here using exponential distribution
I why use a process algebraI compositionalI language allows for abstract reasoning
I outlineI HYPE and stochastic HYPEI train gate exampleI checking for infinite discrete behaviourI I-graphs and acyclicity
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 4: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/4.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Introduction
I stochastic hybrid process algebraI discrete behaviourI continuous behaviour, expressed as ODEsI stochastic behaviour, here using exponential distribution
I why use a process algebraI compositionalI language allows for abstract reasoning
I outlineI HYPE and stochastic HYPEI train gate exampleI checking for infinite discrete behaviourI I-graphs and acyclicity
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 5: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/5.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 6: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/6.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components
controllers
(C1(V) BC∗ · · · BC∗ Cn(V)
)
BC∗(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 7: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/7.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components
controllers
(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 8: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/8.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)
well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 9: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/9.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 10: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/10.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
components are parameterised by variables
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 11: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/11.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 12: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/12.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
events have event conditions: guards and resets
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 13: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/13.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
events have event conditions: guards and resets
ec(aj) = (f (V),V ′ = f ′(V)) discrete events
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 14: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/14.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
events have event conditions: guards and resets
ec(aj) = (f (V),V ′ = f ′(V)) discrete events
ec(aj) = (r ,V = f ′(V)) stochastic events
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 15: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/15.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 16: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/16.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
influences are defined by a triple
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 17: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/17.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
influences are defined by a triple
αj = (ιj , rj , I (V))
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 18: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/18.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)well-defined component
C (V)def=∑j
aj : αj .C (V) + init : α .C (V)
influences are defined by a triple
αj = (ιj , rj , I (V))
influence names are mapped to variables
iv(ιj) ∈ V
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 19: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/19.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)
controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 20: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/20.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)
controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 21: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/21.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 22: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/22.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 23: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/23.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 24: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/24.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Stochastic HYPE model
components controllers(C1(V) BC∗ · · · BC∗ Cn(V)
)BC∗
(Con1 BC∗ · · · BC∗ Conm
)controller grammar
M ::= a.M | 0 | M + M
Con ::= M | Con BC∗ Con
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 25: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/25.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 26: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/26.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 27: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/27.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 28: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/28.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 29: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/29.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 30: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/30.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 31: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/31.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 32: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/32.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Semantics
I HYPE semantics
I no stochastic events, only a
I SOS generates labelled transition system (LTS)
〈H, σ〉 a−−→ 〈H ′, σ′〉
I simple mapping of LTS to hybrid automaton (HA)
I stochastic HYPE semantics
I mapping to transition driven stochastic HA (TDSHA)
I compositional mapping using TDSHA product on shared events
I subset of piecewise deterministic Markov processes (PDMP)
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 33: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/33.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 34: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/34.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 35: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/35.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 36: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/36.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 37: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/37.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 38: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/38.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 39: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/39.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 40: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/40.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 41: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/41.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
I standard example for hybrid systems modelling
I railway track with guarded crossing
I model track and train from 1500m before to 100m after gate
I random delay between trains
I two sensorsI first: 1000m before crossing sends approach signal to controller
I second: 100m after crossing sends exit signal to controller
I gate controller takes time to respond to signals
I gate opens and closes at fixed speed
I safety propertyI when the train is 100m before the gate, the gate is closed
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 42: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/42.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
400
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Time
train distancegate position
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 43: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/43.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Reasoning about stochastic hybrid systems
I want to ensure certain types of behaviour do not occur
I Zeno behaviourI infinite number of events in a finite time periodI very hard to check
I instantaneous Zeno behaviourI infinite number of events in a time instantI no continuous behaviour between eventsI not representative of realityI not permitted in piecewise deterministic Markov processes
I want to check for this behaviour abstractlyI without investigating the full behaviour of a modelI consider controller and event conditionsI combine these and ignore continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 44: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/44.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Reasoning about stochastic hybrid systems
I want to ensure certain types of behaviour do not occur
I Zeno behaviourI infinite number of events in a finite time periodI very hard to check
I instantaneous Zeno behaviourI infinite number of events in a time instantI no continuous behaviour between eventsI not representative of realityI not permitted in piecewise deterministic Markov processes
I want to check for this behaviour abstractlyI without investigating the full behaviour of a modelI consider controller and event conditionsI combine these and ignore continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 45: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/45.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Reasoning about stochastic hybrid systems
I want to ensure certain types of behaviour do not occur
I Zeno behaviourI infinite number of events in a finite time periodI very hard to check
I instantaneous Zeno behaviourI infinite number of events in a time instantI no continuous behaviour between eventsI not representative of realityI not permitted in piecewise deterministic Markov processes
I want to check for this behaviour abstractlyI without investigating the full behaviour of a modelI consider controller and event conditionsI combine these and ignore continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 46: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/46.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Reasoning about stochastic hybrid systems
I want to ensure certain types of behaviour do not occur
I Zeno behaviourI infinite number of events in a finite time periodI very hard to check
I instantaneous Zeno behaviourI infinite number of events in a time instantI no continuous behaviour between eventsI not representative of realityI not permitted in piecewise deterministic Markov processes
I want to check for this behaviour abstractlyI without investigating the full behaviour of a modelI consider controller and event conditionsI combine these and ignore continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 47: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/47.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Reasoning about stochastic hybrid systems
I want to ensure certain types of behaviour do not occur
I Zeno behaviourI infinite number of events in a finite time periodI very hard to check
I instantaneous Zeno behaviourI infinite number of events in a time instantI no continuous behaviour between eventsI not representative of realityI not permitted in piecewise deterministic Markov processes
I want to check for this behaviour abstractlyI without investigating the full behaviour of a modelI consider controller and event conditionsI combine these and ignore continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 48: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/48.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic idea
I how does the guard and reset of one event affect whetheranother event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph construction
I node: controller state, event, binary activation vector forevents, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 49: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/49.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph construction
I node: controller state, event, binary activation vector forevents, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 50: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/50.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph construction
I node: controller state, event, binary activation vector forevents, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 51: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/51.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph construction
I node: controller state, event, binary activation vector forevents, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 52: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/52.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph construction
I node: controller state, event, binary activation vector forevents, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 53: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/53.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph constructionI node: controller state, event, binary activation vector for
events, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 54: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/54.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph constructionI node: controller state, event, binary activation vector for
events, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 55: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/55.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph constructionI node: controller state, event, binary activation vector for
events, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 56: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/56.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph constructionI node: controller state, event, binary activation vector for
events, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditions
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 57: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/57.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Avoiding instantaneous Zeno behaviour
I basic ideaI how does the guard and reset of one event affect whether
another event can occur immediately that event
I capture this information in a finite graph using controller info
I if acyclic then no infinite sequences of events
I I-graph constructionI node: controller state, event, binary activation vector for
events, b1 . . . bn for n events
I edge: if transition in LTS with that event and event is enabledand after vector is an update by event of before vector
I update of vector by event determines which other events havebeen activated or disabled by this event
I include all controller states, all events, activation vector of 1’s
I gives an overapproximation, ignores initial conditionsVashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 58: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/58.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Controllers for train gate system
I system controller
Conadef= appr.Conl + exit.Cona
Conldef= appr.Conl + exit.Conl + lower.Cone
Conedef= appr.Cone + exit.Conr
Conrdef= appr.Conl + exit.Conr + raise.Cona
I gate internal controller
GC odef= raise.GC o + lower.GC l
GC ldef= raise.GC r + lower.GC l + closed.GC c
GC cdef= raise.GC r + lower.GC c
GC rdef= raise.GC r + lower.GC l + open.GC o
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 59: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/59.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Controllers for train gate system
I system controller
Conadef= appr.Conl + exit.Cona
Conldef= appr.Conl + exit.Conl + lower.Cone
Conedef= appr.Cone + exit.Conr
Conrdef= appr.Conl + exit.Conr + raise.Cona
I gate internal controller
GC odef= raise.GC o + lower.GC l
GC ldef= raise.GC r + lower.GC l + closed.GC c
GC cdef= raise.GC r + lower.GC c
GC rdef= raise.GC r + lower.GC l + open.GC o
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 60: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/60.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Controllers for train gate system
I sequencing of train travel
Seqadef= appr.Seqp Seqp
def= pass.Seqe
Seqedef= exit.Seqf Seqf
def= wait.Seqa
I controller states
Conx BC∗ GC y BC∗ Seqz will be written as CxGySz
I continuous part of the system is
Gate BC∗ Train BC∗ TimerL BC∗ TimerR
and remains unchanged by the occurrence of eventsbecause of form of well-defined components
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 61: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/61.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Controllers for train gate system
I sequencing of train travel
Seqadef= appr.Seqp Seqp
def= pass.Seqe
Seqedef= exit.Seqf Seqf
def= wait.Seqa
I controller states
Conx BC∗ GC y BC∗ Seqz will be written as CxGySz
I continuous part of the system is
Gate BC∗ Train BC∗ TimerL BC∗ TimerR
and remains unchanged by the occurrence of eventsbecause of form of well-defined components
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 62: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/62.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Controllers for train gate system
I sequencing of train travel
Seqadef= appr.Seqp Seqp
def= pass.Seqe
Seqedef= exit.Seqf Seqf
def= wait.Seqa
I controller states
Conx BC∗ GC y BC∗ Seqz will be written as CxGySz
I continuous part of the system is
Gate BC∗ Train BC∗ TimerL BC∗ TimerR
and remains unchanged by the occurrence of eventsbecause of form of well-defined components
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 63: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/63.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Event conditions
I 8 events (other then init)
ec(appr) = (D = −1000, T ′L = 0)
ec(pass) = (D = 0, true)
ec(exit) = (D = 100, T ′R = 0 ∧ D ′ = −1500)
ec(wait) = (delay , true)
ec(lower) = (TL = 5, T ′L = 0)
ec(raise) = (TR = 5 T ′R = 0)
ec(closed) = (G = 0, true)
ec(open) = (G = 90, true)
I let b1 . . . b8 be a binary vector where each element isassociated with an event using the above ordering
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 64: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/64.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Event conditions
I 8 events (other then init)
ec(appr) = (D = −1000, T ′L = 0)
ec(pass) = (D = 0, true)
ec(exit) = (D = 100, T ′R = 0 ∧ D ′ = −1500)
ec(wait) = (delay , true)
ec(lower) = (TL = 5, T ′L = 0)
ec(raise) = (TR = 5 T ′R = 0)
ec(closed) = (G = 0, true)
ec(open) = (G = 90, true)
I let b1 . . . b8 be a binary vector where each element isassociated with an event using the above ordering
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 65: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/65.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 66: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/66.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 67: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/67.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 68: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/68.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 69: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/69.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 70: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/70.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Instantaneous activation graph
I (CeGlSp, closed, 11110111)→ (CeGcSp, pass, 11110110)
I transition in controller, CeGlSpclosed−−−→ CeGcSp
I in source node, all events are enabled except lowerI in target node, lower and open are disabled
I closed disables openI closed occurs when G = 0 and G is not reset by closedI hence immediately after closed, G = 0I open occurs when G = 90 so disabled immediately after closed
I closed does not enable lower
I lower disables itself: ec(lower) = (TL = 5,T ′L = 0)
I wait disables all events because it has duration
I straightforward to determine event updates
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 71: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/71.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 72: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/72.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 73: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/73.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 74: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/74.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 75: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/75.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 76: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/76.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 77: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/77.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Well-behaved stochastic HYPE models
I well-behaved: no infinite sequence of simultaneous events
I theorem: HYPE model with an acyclic I-graph is well-behaved
I results for simple sequential controllers
I two well-behaved controllers with independent events havewell-behaved cooperation
I two well-behaved controllers whose unshared events do notactivate events of the other controller have well-behavedcooperation
I assuming all shared events appear in the cooperation set
I apply results to train gate controller
I use compositional results rather than working with allcontrollers
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 78: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/78.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved?
I Seqa is well-behavedI simple sequential controller
I one event is stochastic and inhibits all other events
I also exit inhibits itself and nothing activates it
I Cona is not well-behaved
I Conldef= appr.Conl + exit.Conr
I appr does not inhibit itself
I (Conl , appr, 10001)→ (Conl , appr, 10001)
I consider I-graphs of Cona BC∗ Seqa and Gatea
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 79: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/79.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved?
I Seqa is well-behavedI simple sequential controller
I one event is stochastic and inhibits all other events
I also exit inhibits itself and nothing activates it
I Cona is not well-behaved
I Conldef= appr.Conl + exit.Conr
I appr does not inhibit itself
I (Conl , appr, 10001)→ (Conl , appr, 10001)
I consider I-graphs of Cona BC∗ Seqa and Gatea
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 80: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/80.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved?
I Seqa is well-behavedI simple sequential controller
I one event is stochastic and inhibits all other events
I also exit inhibits itself and nothing activates it
I Cona is not well-behaved
I Conldef= appr.Conl + exit.Conr
I appr does not inhibit itself
I (Conl , appr, 10001)→ (Conl , appr, 10001)
I consider I-graphs of Cona BC∗ Seqa and Gatea
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 81: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/81.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
I-graph of system controller and sequencer
ClSp p 111111
ClSe l 010111
ClSp l 111111
CeSp p 111101
ClSe l 111111
CeSe e 111101
ClSe e 111111
ClSa l 000110
ClSa l 111111
CeSa a 111101 CaSa a 111110
CrSa r 111111 ClSf l 111111
CeSf f 111101 CaSf f 111110
CrSf r 111111
CaSa a 111111 ClSa a 111111
CeSp p 111111 CeSe e 111111
CrSa a 111111 CeSa a 111111
CaSf f 111111 CeSf f 111111
CrSf f 111111 ClSf f 111111
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 82: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/82.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
I-graph of gate controller
Go ra 1111
Go lo 1011
Gl cl 0011
Go lo 1111
Gl cl 0111 Gl ra 0111
Gc ra 0110 Gr op 0011
Gl lo 1111Gl ra 1111
Gr op 1011Gr lo 1011
Go lo 1001
Gc lo 1111
Gc ra 0111
Gc ra 1111 Gr ra 1111 Gr lo 1111
Gc lo 1110 Gc ra 1110
Gr lo 1010
Gl cl 0010
Go lo 1101
Gl ra 0101
Go ra 1101
Gr op 0001
Gl cl 1111 Gr op 1111
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 83: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/83.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 84: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/84.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 85: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/85.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 86: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/86.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 87: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/87.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 88: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/88.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Train gate system: well-behaved
I Cona BC∗ Seqa is well-behaved
I GC o is well-behaved
I no event activates another event
I all shared events are synchronised on
I hence Cona BC∗ Seqa BC∗ GCo is well-behaved
I hence train gate model can be mapped successfully to apiecewise deterministic Markov process
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 89: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/89.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Conclusions
I stochastic HYPEI process algebra for stochastic hybrid systemsI extension of HYPEI different underlying semantic model
I illustrated through a railway gate system
I exclusion of infinite behaviour at a time instantI instantaneous Zeno behaviourI well-behaved stochastic HYPE modelsI construct I-graph from controller and event conditionsI overapproximation of behaviourI check for acyclicityI abstract from continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 90: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/90.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Conclusions
I stochastic HYPEI process algebra for stochastic hybrid systemsI extension of HYPEI different underlying semantic model
I illustrated through a railway gate system
I exclusion of infinite behaviour at a time instantI instantaneous Zeno behaviourI well-behaved stochastic HYPE modelsI construct I-graph from controller and event conditionsI overapproximation of behaviourI check for acyclicityI abstract from continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 91: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/91.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Conclusions
I stochastic HYPEI process algebra for stochastic hybrid systemsI extension of HYPEI different underlying semantic model
I illustrated through a railway gate system
I exclusion of infinite behaviour at a time instantI instantaneous Zeno behaviourI well-behaved stochastic HYPE modelsI construct I-graph from controller and event conditionsI overapproximation of behaviourI check for acyclicityI abstract from continuous behaviour
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011
![Page 92: Stochastic HYPE: modelling stochastic hybrid systemshomepages.inf.ed.ac.uk/vgalpin1/ps/Gal11a-slides.pdf · IntroductionStochastic HYPETrain gateIn nite instantaneous behaviourConclusions](https://reader036.vdocument.in/reader036/viewer/2022070821/5f22857c0fd639682f41fe98/html5/thumbnails/92.jpg)
Introduction Stochastic HYPE Train gate Infinite instantaneous behaviour Conclusions
Thank you
Vashti Galpin
Stochastic HYPE: modelling stochastic hybrid systems BCTCS 2011