Observability and temporal properties of hybrid systems: analysis and verification

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Observability and temporal propertiesof hybrid
systems analysis and verification
Department of Electrical Engineering and Computer
Science Center of Excellence DEWS - University of
LAquila
Doctoral Dissertation
Co-Advisor Prof. S. Di Gennaro
Advisor Prof. M.D. Di Benedetto
Doctoral studies coordinator Prof. A. Germani

• PhD candidate
• A. DInnocenzo

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Outline

• Introduction
• Hybrid systems
• Discrete state observability
• Verification techniques by abstraction
• Conclusions

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Leit motif

• We discuss verification of observability and
  temporal properties of hybrid systems
• that are systems with discrete and continuous
  aspects in their dynamics
• Pro have huge expressive power...
• Con but are complex to deal with

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Problems addressed

• Observability

• Model checking

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Challenges in hybrid systems verification

• Because of generality of the model, these
  problems rarely have a closed solution.
• Some of them are even undecidable!
• Huge state space complexity analysis is a
  fundamental issue.
• A comprehensive theory for observability
  verification and model checking is still missing

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Outline

• Introduction
• Hybrid systems
• Definition
• Executions as formal languages
• Discrete state observability
• Verification techniques by abstraction
• Conclusions

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Hybrid system example
Gear 1
Gear 2
Gear 3
Gear 4
Gear 5
each gear is associated to different continuous
dynamics
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Hybrid system definition
Discrete Layer
Invariant Sets Guard Sets Reset Maps
q3
q1
q2
Continuous Layer
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Hybrid execution
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Hybrid time basis Lygeros et al. - IEEE TAC 03

• Finite or infinite collection of intervals

Q
X
q4
q3
q2
q1
q0
t
t0
t0 t1
t1 t2
t3 t4
t2 t3
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Language of executions
q2
q4
q3
q1
4 s
2 s
1 s
3 s
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Regular language of executions

• Consider observations without time
  delaysthen L, P, LQb, PQb are regular
  languages
• Regular languages are closed w.r.t. union,
  intersection, concatenation.

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Outline

• Introduction
• Hybrid systems
• Discrete state observability
• Motivation
• Non deterministic case
• Stochastic case
• Verification techniques by abstraction
• Conclusions

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Discrete state observability motivation
MED05
Qb unauth. crossing
Engines Running
Taxiing
Ask for crossing grant
Taxi on airport way
Taxiing
Unobs.
Waiting at stop-bar
Unauthorized crossing
Unobs.
Emergency Braking
Crossing
Unobs.
Unobs.
Authorized crossing
Taxi to hangar
Crossing completed
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Non deterministic caseobservability definition

• Definition Set Qb ? Q is observable for hybrid
  system H if observer of Qb exists.

LNCIS05, CDC06
Hybrid system
Observer of Qb
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Classical observability definition

• Proposition Classical discrete state
  observability is a special case of observability
  of Qb

Observer of q1
Observer of H

Observer of qN
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Observability condition

• Proposition Set Qb is observable for hybrid
  system H if and only if

Q0
Qb
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Observability verification

• Algorithm
• Compute regular languages PQb and PQ\Qb
• Compute intersection PQb ? PQ\Qb
• Check if PQb ? PQ\Qb is empty.

Algorithm terminates in polynomial time w.r.t.
dimension of discrete state space
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When the discrete output is not enough
H
H

• H embeds continuous information by means of extra
  discrete outputs and their generation time
• Observability definition with bounded delay.

• Proposition If H is observable,then H is
  observable with delay ?.

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Extra output design
Extra output set is neither unique nor free
minimize set of extra signals

• Algorithm to find minimal set of extra output
  information to satisfy observability (exponential
  time).
• Algorithm to find sub-optimal solution
  (polynomial time).

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Stochastic casehidden Markov hybrid system
Gear 1
Gear 2
Gear 3
Gear 4
Gear 5
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Discrete state estimator
Viterbi algorithm generates for each observation
the probability distribution of discrete state
Viterbi algorithm
H
Observer of Qb
Want to define observability such that, for any
observation, we are almost sure that the observer
estimate is true
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Observability with bounded reliability
Definition Set Qb is observable with
reliability ( ?1, ?2 ) if
ADHS06

• Proposition Observability with reliability (1,1)
  is equivalent to observability in the non
  deterministic setting.

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Observability verification decidability
Run Viterbi algorithm for any p?PThis might
require infinite iterations.

• Theorem Observability with reliability
• ( ?1, ?2 ) is decidable if the regular
• language P has a special structure

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Asymptotic observability
Definition Set Qb is observable in K steps if
observer generates correct estimate after K
discrete transitions.

• Previous results hold, with slight differences in
  definitions, conditions and verification
  algorithms

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Outline

• Introduction
• Hybrid systems
• Discrete state observability
• Verification techniques by abstraction
• Translating hybrid to timed automata
• Diagnosability (observability) analysis
• Verification of temporal properties
• Conclusions

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Verification by abstraction
Hybrid system H
Abstraction T
Propertytrue on T
Propertytrue on H

• Construct abstraction T to preserve properties of
  interest
• Verification procedure on T

Find conditions to construct an abstraction T
such thatproperty true for Hif and only if
true for T
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Abstraction methods
Hybrid system H
Discrete event system D
safety
Untimed
temporal properties
Hybrid system H
Timedautomaton T
Durationalgraph G
Timed
Timed abstraction Pro preserve time
information! Con more complex algorithms
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Abstraction algorithm

• Algorithm to construct G from H
• Split discrete states of H depending on incoming
  edges
• Define invariants and guards of G using
  continuous dynamics of H
• Define function ? that associates to any discrete
  state of H a set of discrete states of G. These
  sets partition the state space of G.

HSCC06
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Diagnosability definition
HSCC07

• Definition Set Qb is ?-diagnosable for H if it
  is possible to detect after a delay ? that Qb has
  been visited.

Proposition Qb is ?-diagnosable iff
Proposition Set Qb is observable if and only if
it is??-diagnosable with ?0.
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Diagnosability is preserved by G

• Proposition Set Qb is ?-diagnosable for H if
  ?(Qb) is ?-diagnosable for G.

Proposition With the assumption of memoryless
reset, Qb is ?-diagnosable for H if and only if
?(Qb) is?-diagnosable for G.
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Diagnosability verification complexity
Complexity
Timed automata
PSPACE
?
Durational graphs
PTIME
?
Discrete event systems
PTIME
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Diagnoser
Proposition A diagnoser for a hybrid automaton
H is a durational graph syncronized with the
output of H.
faulty
Diagnoser
Hybrid automaton H
nonfaulty
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Diagnosability verification of a camless valve
system

• Hybrid model
• Abstraction
• Diagnosabilityverification

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Model checking on hybrid automata hybrid Kripke
structure
Gear 1
Gear 2
Gear 3
Gear 4
Gear 5
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Temporal logic

• A temporal logic is a formalism for defining
  sequences of transitions

Computation Tree Logic (CTL), Linear Time Logic
(LTL) along every path, the atomicproposition
A?? will eventually hold
Temporal property on executions
Yes
Hybrid Kripke structure K
Model checking
Timed abstraction of K
No counterexample
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Transition systems

• Q set of states
• Q0 initials states
• ? labels
• E Q x ? x Q transition relation
• ? observations
• ? Q ? ? observation map

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Simulation relation

• Definition Given T1, T2, a relation? ? Q1 x Q2
  is a simulation relationof T1 by T2 if for all
  (q1, q2) ? ?
• ?(q1) ?(q2)
• for all (q1, ?, q1 ) ? E1, there exists (q2, ?,
  q2 ) ? E2 such that (q1, q2) ? ?

Definition T2 simulates T1 if simulation
relation ? exists, and any initial state of T1
related to initial state of T2
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Temporal properties are preserved by G

• Theorem The abstraction G is a simulation of
  the hybrid automaton H.

Proposition The universal fragment of temporal
logics is preserved by G.
Remark Diagnosability cannot be expressed by a
temporal logic formula.
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Landing scheduling problem
A/C 2
A/C 1
R
W
Y
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Landing scheduling verification

• Timed-CTL (TCTL) formula

Aircraft 2 will start landing after aircraft 1,
with a time delay in tm,tM
Hybrid model H of the aircraft
Timed Abstraction T
Verifyformula on T
Formulatrue on H
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Conclusion

• Discrete state observability definition
• Polynomial time verification, extra output design
• Observability with reliability, verification
  decidability
• Timed abstraction procedure
• Diagnosability verification complexity
• Model checking by timed simulation relations

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Future work

• Use all continuous output information
• Extend stochastic model
• Approximate time abstractions given H and
  maximum error ?, construct T such that d(H,T) ? ?.