Control of Hybrid Behavioral Automata by Interconnection

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Control of Hybrid Behavioral Automata by
Interconnection

• A.A. Julius S.N. Strubbe A.J. van der Schaft
• Dept. Applied Mathematics
• University of Twente
• The Netherlands
• http//www.math.utwente.nl/juliusaa

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Presentation Overview

• Motivating example
• Active passive automata
• Hybrid behavioral automata (HBA)
• Composition of hybrid behavioral automata
• About control by composition
• Conclusions

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Motivating example

• An engine with possible overheating is modeled as
  an automaton.
• A controller is to be designed such that the
  overheating is not harmful.
• A naive solution use a controller that does not
  offer the event overheat, hence blocking the
  overheating from happening.
• Compatibility problem?

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Active passive automata (1)

• Define two kinds of transitions, active and
  passive.
• Active events can occur without synchronizing
  partners.
• Passive events only occur when induced by active
  events.
• Compatibility is embedded in the composition
  rules.

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Active passive automata (2)
Composition rules/semantics

• The composition operator is commutative and
  associative.

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Hybrid Behavioral Automata (1)

• Use active-passive model for discrete dynamics.
• Use behavioral approach for continuous dynamics.
• What is the behavioral approach? It is a
  semantical approach to systems theory.
• A behavior is a collection of possible
  trajectories.
• Why behavioral approach? It is general. It allows
  for general treatment of some classical control
  problem.

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Hybrid Behavioral Automata (2)

• In every location/discrete state, we define a
  behavior.
• Dynamic maps map trajectories and time to a
  space.
• Example
• The combination of dynamic maps and target areas
  are used to express guards and invariants.
• Example a trajectory satisfies the guard
  at time if
• Such couples are called dynamic predicates.

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Hybrid Behavioral Automata (3)

• To every transition (i.e. active or passive) we
  define a reset map.
• The reset map take trajectories and time and give
  a subset of the behavior in the target location.
• Formally, an HBA
• L the set of locations, W the set of continuous
  variables, A the set of transition labels, T and
  P set of active and passive transitions, Inv the
  invariant, and is the continuous behavior.

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Hybrid Behavioral Automata (4)

• The active transitions are expressed as
• is the initial location, the label, the
  target location, the guard of the transition,
  and the reset map.
• The passive transitions are expressed as
• Thus, passive transitions are not guarded.

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Composition of HBA (1)

• Composition of the discrete dynamics is done a la
  active passive automata.
• Composition of the continuous dynamics is done a
  la behavioral approach, i.e. taking intersection
  of behaviors.
• Define
• The HBA is characterized by the
  7-tuple
• The dynamic predicate is satisfied iff
  both and are
  satisfied.

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Composition of HBA (2)

• The set of transitions are determined through

• The composition operation is also commutative and
  associative.

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About control by composition (1)

• For this purpose we use rooted HBA, i.e. HBA with
  initial location.
• Given a plant expressed as a rooted HBA, we
  are to design a controller to achieve the
  specification
• In the composition, we allow not using all
  continuous variables.
• The equivalence is understood in the context of
  behavioral/language equivalence. We match the
  collection of possible hybrid trajectories.

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About control by composition (2)

• The solution to the control problem is presented
  for a special class of HBA.
• The controller used is a passive controller. It
  only has passive transitions.
• The idea is that the controller follows the
  discrete dynamics of the plant. Control is mainly
  due to interaction of continuous dynamics.
• Some conditions are given for solvability using
  this idea.

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Conclusions

• There is a need for asymmetric synchronization.
• In the paper we use passive active approach to
  realize it.
• The idea of HBA is built on it, adding to it
  continuous dynamics a la behavioral approach.
• Control is seen as composition of automata.
• A special class of control problems is solved
  with passive controller.
• More general methodology for solving control
  problems is desired.