a new verification algorithm for planar differential inclusions

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Synchronous Workshop 2003

A New Verification Algorithm for

Planar Differential Inclusions

Gordon Pace

University of Malta

December 2003

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Scientific Models

• Discrete systems– CSs’ favourite domain– What I should be talking about here …

• Continuous systems– Engineers’ domain– Differential equations

• Hybrid Systems

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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.

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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.

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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?

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Hybrid Automata

On Off

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On Off

Dynamics

Invariant

Guard Reset

Hybrid Automata

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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.

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Swimmer in a whirlpool

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Polygonal Differential Inclusion Systems (SPDIs)

• A partition of the plane into convex polygons

• Constant differential inclusion for each region describing allowable dynamics

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Swimmer SPDI

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Swimmer SPDI

Arrows:System dynamics

Polygons:Discrete states

(Transformed) coordinates:

two continuous states

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Swimmer SPDI

Arrows:System dynamics

Polygons:Discrete states

Position on line:one continuous state

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Swimmer SPDI

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Swimmer SPDI

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Some undecidable extensions

• Three or more dimensions

• Variant differential inclusions

• SPDIs with arbitrary resets

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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.

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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

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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.

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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.

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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.

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Summary

We have an algorithm to

decide SPDI reachability

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Summary

But it does not guarantee shortest

counter-example

unless exhaustive search is performed

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Forward model checking

Termination Condition:

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SPDI model checking

Termination Condition:[ [

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SPDI model checking

This follows loops (non-iteratively) in one

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SPDI model checking

This is the invariance kernel of the SPDI

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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.

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BFS algorithm which guarantees

shortest abstract counter-example

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Allows us to apply standard

model-checking verification optimisations

to SPDI verification

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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?

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x 2 [min{c1, 1 x + 2}, max{c1, 1 s + 2}]

a new verification algorithm for planar differential inclusions

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