Hybrid Systems

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Hybrid Systems

• Presented by
• Arnab De
• Anand S

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An Intuitive Introduction to Hybrid Systems

• Discrete program with an analog environment.
• What does it mean?
• Sequence of discrete steps in each step the
  system evolves continuously according to some
  dynamical law until a transition occurs.
  Transitions are instantaneous.

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A Motivating Example Thermostat

• The heater can be on or off.
• When the heater is on, the temperature increases
  continuously according to some formula.
• When the heater is off, the temperature
  decreases.
• Thermostat keeps the temperature within some
  limit by putting the heater on or off.

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Formal Model of Hybrid Systems

• Model Hybrid Systems as graphs
• Vertices represent continuous activities.
• Edges represent transition.

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Formal Model contd

• H (Loc, Var, Lab, Edg, Act, Inv)
• Loc finite set of vertices (locations)
• Var finite set of real-valued variables.
• A valuation v(x) assignes a real value to each
  variable. V is the set of valuations.
• A state is a pair (l, v), l ? Loc, v ? V.

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Formal Model contd

• Lab finite set of synchronization labels,
  containing the stutter label t
• Edg finite set of edges (transitions). e
  (l, a, µ, l)
• Stutter transition (l, µ, IdCon, l).
• Act set of activities, maps non-negative reals
  to valuations.
• Inv set of invariants at a location.

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Time-deterministic hybrid system

• There is at most one activity for each location
  and each valuation such that
• f(0) v
• Denoted by flv.

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Runs of a Hybrid System

• A state can change in two ways
• Discrete and Instantaneous transition that
  changes both l and v.
• Time delay that changes only v according to
  activities of the location.
• Some transition must be taken before the
  invariant becomes false.
• Run

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Thermostat example revisited
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Hybrid Systems as Transition Systems
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Composition of Hybrid Systems
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Linear Hybrid System

• A time-deterministic hybrid system is linear if
• The activity functions are of the form
• The invariant for each location is defined by a
  linear formula over Var.

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Linear Hybrid System contd

• For all transitions, the transition relation µ is
  defined by a guarded set of non-deterministic
  assignments
• If ax ßx, we write

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Special Cases of Linear Hybrid Systems

• If Act(l,x) 0 for all locations, then x is a
  discrete variable.
• A discrete variable x is a proposition if
• for all transitions.
• A finite-state system is a linear hybrid system
  all of whose variables are propositions.

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Special cases contd

• If Act(l,x) 1 for each location and
• for each transition, then x is a clock.
• A timed automaton is a LHS all of whose variables
  are either propositions or clocks and the linear
  expressions are boolean combination of
  inequalities of the form xc or x-yc (c
  non-negative integer).

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Special cases contd

• If for each location and
• for each edge, then x is an
  integrator. An integrator system is a LHS all of
  whose variables are propositions or integrators.

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Example of LHS Leaking Gas Burner
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Reachability problem

• Given two states, does there exist any run that
  starts at first state and ends at another.
• Verification of some invariant property is
  equivalent to the reachability question.
• Reachability is undecidable in general but
  decidable for some special cases.

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Verification of Linear Hybrid Systems

• H(Loc,Var,lab,Edg,Act,Inv)
• Do a reachability analysis
• Iteratively find out the reachable states
• Forward analysis computes step successors of a
  given set of states
• Backward analysis

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Forward analysis

• Forward time closure
• Set of valuations reachable from some v ?P by
  letting time progress
• .
• (l,v) ?t (l,v)
• Post condition of P w.r.t an edge e,
• The set of valuations reachable from v ? P by
  executing transition e
• .
• (l,v) ?a (l,v)

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Forward Analysis (contd)

• Region A set of states
• Define (l,P) (l,v) v ? P
• Extension to regions for RUl?Loc(l,Rl)

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Forward Analysis (contd)

• A symbolic run on H is (in)infinite sequence
• ? (l0,P0)(l1,P1),(li,Pi)
• .
• The region (li,Pi) is the set of states reachable
  from (l0,v0) after executing e0,.ei-1
• Every run of H can be represented by some
  symbolic run of H
• Given I (subset of S), the reachable region (I?)
  is the set of states reachable from I
• .

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Forward Analysis (contd)

• Reachable region is least fixed point of
• .
• Or Rl of valuations for l ? Loc if lfp of
• .
• ? set of valuations that satisfy ?
• ? is a linear formula
• PÍv is linear if P? for some ?

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Forward Analysis (contd)

• For linear H, if P is linear, then so is ltPgtl?
  and posteP
• pc ? Var is a control var with range Loc
• A region R is linear of all Rl(?l) are linear
• Region R is defined by
• Do successive approx.
• Terminate for simple mutirated timed systems

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Example leaking gas burner

• .
• .

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Backward Analysis

• Backward time closure
• .
• Precondition
• .
• Extension

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Backward Analysis (contd)

• Initial region
• .
• Equations Initial region if lfp
• .
• .
• ltPgtl? and preeP are linear
• In example, we find set of states from which
  ?Ry60? 20z y is reachable. We get null set

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Model Checking (Timed CTL)

• Check if H satisfies a requirement expressed in
  real-time temporal logic
• Define C (disjoint with Var)
• State predicate is a linear formula over Var U C
• The grammer
• .
• ? is state predicate and z?C
• Formulas of TCTL are interpreted over state space
  of H

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Timed CTL (contd)

• Clocks can be used to express timing constraints
• .
• A run ?s0 ?t0 s1 ?t1
• For a state ?i(li,vi), position ?(i,t)
• (0t ti)
• Positions are lexicographically ordered
• .

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TCTL (contd)

• For all positions ?(i,t)
• Clock valuation ? C?R0
• ?t and ?z0
• Extended state (s, ?)

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Model Checking (contd)

• (s, ?) F, if

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Model Checking algorithm

• s F, of (s,?) F for all ? evaluations
• Computes Characteristic set F
• (l,v) ? (R ? R) iff
• Single step until operator
• If R and R are linear so is R ? R
• Thus the modalities can be computed iteratively
  using ?
• Will terminate in simple multirate timed system

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Examples

• F?U F computed as UiRi with
• ??c F computed as UiRiz0 with

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Thank you