A New Verification Algorithm for Planar Differential Inclusions
-
Upload
nakeisha-johnson -
Category
Documents
-
view
24 -
download
2
description
Transcript of A New Verification Algorithm for Planar Differential Inclusions
![Page 1: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/1.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
A New Verification Algorithm for
Planar Differential Inclusions
Gordon Pace
University of Malta
December 2003
![Page 2: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/2.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Scientific Models
• Discrete systems– CSs’ favourite domain– What I should be talking about here …
• Continuous systems– Engineers’ domain– Differential equations
• Hybrid Systems
![Page 3: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/3.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
A Hybrid System
• Typical example: A heated room with a a thermostat.
• Room temperature T continuous variable,• State of heater (on or off) is a discrete
variable,• Different (continuous/differential) equations
regulate room temperature depending whether heater is on or off.
![Page 4: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/4.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
The Heated Room: Required Parameters
• Dynamics in different (discrete) states;
• When to switch from one state to another;
• Whether any continuous variables are reset discontinuously when switching from one state to another.
![Page 5: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/5.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
The Heated Room:Typical questions
• Reachability questions: Can the room temperature rise over 5% above the thermostat setting?
• ‘Qualitative’ system behaviour: Given a loop (a sequence of discrete states) what continuous behaviour is possible within that loop?
![Page 6: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/6.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Hybrid Automata
On Off
![Page 7: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/7.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
On Off
Label
Dynamics
Invariant
Guard Reset
Hybrid Automata
![Page 8: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/8.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Verification of Hybrid Automata
• Undecidable in general.
• Even (good) testing is difficult!
• Most complete approaches look at sub-problems eg limiting differential equations, limiting number of continuous variables.
![Page 9: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/9.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 10: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/10.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 11: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/11.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 12: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/12.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 13: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/13.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 14: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/14.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 15: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/15.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 16: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/16.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 17: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/17.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 18: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/18.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 19: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/19.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer in a whirlpool
![Page 20: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/20.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Polygonal Differential Inclusion Systems (SPDIs)
• A partition of the plane into convex polygons
• Constant differential inclusion for each region describing allowable dynamics
![Page 21: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/21.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer SPDI
![Page 22: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/22.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer SPDI
Arrows:System dynamics
Polygons:Discrete states
(Transformed) coordinates:
two continuous states
![Page 23: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/23.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer SPDI
Arrows:System dynamics
Polygons:Discrete states
Position on line:one continuous state
![Page 24: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/24.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer SPDI
![Page 25: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/25.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Swimmer SPDI
![Page 26: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/26.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Some undecidable extensions
• Three or more dimensions
• Variant differential inclusions
• SPDIs with arbitrary resets
![Page 27: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/27.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Some observations (1)
• Position on edges can be described as a single real number.
• Starting from a position s on an edge and ending at t on another edge, the linear inclusion limits guarantees:
t 2 [1 s + 2, 1 s + 2]• Similarly if we went through a number of
edges in between.
![Page 28: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/28.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Result:
• Given a loop of region edges, we can compute the reachable polygon without iterating.
• We can compute the effect of following an abstract trace:
e1…ei(ei+1…ej)*ej+1…ek(ek+1…el)* … en
![Page 29: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/29.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Some observations (2)
• For any self-crossing path through an SPDI, there exists a non-self-crossing one with the same start and end points.
• A path which follows a loop (a number of times), leaves it and goes through the loop again, can be replaced by one which enters the loop only once.
![Page 30: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/30.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Result:
• Any path through an abstract trace which is ‘too long’ also belongs to a shorter abstract path:
e1…ei(ei+1…ej)*ej+1…ek(ek+1…el)* … en
• Only a finite number of paths need be explored to check reachability.
![Page 31: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/31.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Summary
• We can (non-iteratively) calculate the effect of following an abstract path.
• A finite number of abstract paths cover all possible concrete paths from one edge to another.
• These abstract paths can be calculated.
![Page 32: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/32.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Summary
• We can (non-iteratively) calculate the effect of following an abstract path.
• A finite number of abstract paths cover all possible concrete paths from one edge to another.
• These abstract paths can be calculated.
We have an algorithm to
decide SPDI reachability
![Page 33: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/33.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Summary
• We can (non-iteratively) calculate the effect of following an abstract path.
• A finite number of abstract paths cover all possible concrete paths from one edge to another.
• These abstract paths can be calculated.
But it does not guarantee shortest
counter-example
unless exhaustive search is performed
![Page 34: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/34.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Forward model checking
[
Termination Condition:
![Page 35: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/35.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
SPDI model checking
[ [
Termination Condition:[ [
![Page 36: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/36.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
SPDI model checking
[ [
Termination Condition:[ [
This follows loops (non-iteratively) in one
step
![Page 37: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/37.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
SPDI model checking
[ [
Termination Condition:[ [
This is the invariance kernel of the SPDI
![Page 38: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/38.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Invariance kernel of a loop
• The greatest set of points such that every trajectory starting in such points must remain in the set forever.
• Can be calculated using a non-iterative algorithm.
• The set is the union of all invariance kernels.
![Page 39: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/39.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Invariance kernel of a loop
• The greatest set of points such that every trajectory starting in such points must remain in the set forever.
• Can be calculated using a non-iterative algorithm.
• The set is the union of all invariance kernels.
BFS algorithm which guarantees
shortest abstract counter-example
![Page 40: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/40.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Invariance kernel of a loop
• The greatest set of points such that every trajectory starting in such points must remain in the set forever.
• Can be calculated using a non-iterative algorithm.
• The set is the union of all invariance kernels.
Allows us to apply standard
model-checking verification optimisations
to SPDI verification
![Page 41: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/41.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
Future work
• Implementation of the new algorithm and standard optimisations
• Case studies and safe approximation generators
• How can this be applied to discrete systems with one continuous variable and differential inclusion transitions?
![Page 42: A New Verification Algorithm for Planar Differential Inclusions](https://reader031.fdocuments.net/reader031/viewer/2022032708/56812b26550346895d8f2b2c/html5/thumbnails/42.jpg)
Hyb
rid
Syste
m V
eri
ficati
on
Synchronous Workshop 2003
x 2 [min{c1, 1 x + 2}, max{c1, 1 s + 2}]