Transcript Hybrid Systems

Hybrid Systems
Presented by:
Arnab De
Anand S
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.

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.
Formal Model of Hybrid Systems
Model Hybrid Systems as graphs:
 Vertices represent continuous activities.
 Edges represent transition.
Formal Model cont’d…
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 є Loc, v є V.
Formal Model cont’d…




Lab: finite set of synchronization labels,
containing the stutter label τ
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.
Time-deterministic hybrid
system
There is at most one activity for each
location and each valuation such that
f(0) = v
Denoted by φl[v].
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:
Thermostat example revisited
Hybrid Systems as
Transition Systems
Composition of Hybrid Systems
Linear Hybrid System
A time-deterministic hybrid system is
linear if:
1.
The activity functions are of the form
2.
The invariant for each location is
defined by a linear formula over Var.
Linear Hybrid System cont’d…
3.
For all transitions, the transition
relation µ is defined by a guarded set
of non-deterministic assignments
If αx = βx, we write
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.
Special cases cont’d…
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 x#c or
x-y#c (c non-negative integer).

Special cases cont’d…

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.
Example of LHS:
Leaking Gas Burner
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.
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
Forward analysis

Forward time closure




Set of valuations reachable from some v єP 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 є P by
executing transition e

.

(l,v) a (l’,v’)
Forward Analysis (contd…)



Region: A set of states
Define (l,P) = {(l,v) | v є P }
Extension to regions: for R=UlєLoc(l,Rl)
Forward Analysis (contd…)

A symbolic run on H is (in)infinite sequence





ρ: (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 Σ), the reachable region
(I*) is the set of states reachable from I

.
Forward Analysis (contd…)

Reachable region is least fixed point of


Or Rl of valuations for l є Loc if lfp of


.
[ψ] = set of valuations that satisfy ψ


.
Ψ is a linear formula
Pv is linear if P=[ψ] for some ψ
Forward Analysis (contd…)






For linear H, if P is linear, then so is <P>l
and poste[P]
pc  Var is a control var with range Loc
A region R is linear of all Rl([ψl]) are linear
Region R is defined by
Do successive approx.
Terminate for simple mutirated timed systems
Example : leaking gas burner
.
.

Backward Analysis

Backward time closure


Precondition


.
.
Extension
Backward Analysis (contd…)

Initial region


Equations Initial region if lfp




.
.
.
<P>l and pree[P] are linear
In example, we find set of states from which
ψR=y≥60 20z ≤y is reachable. We get null set
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




.
Ψ is state predicate and zєC
Formulas of TCTL are interpreted over state
space of H
Timed CTL (contd…)

Clocks can be used to express timing
constraints



A run ρ=σ0 t0 σ1 t1
For a state ρi=(li,vi), position =(i,t)


.
(0≤t ≤ti)
Positions are lexicographically ordered

.
TCTL (contd…)




For all positions =(i,t)
Clock valuation ξ: CR≥0
ξ+t and ξ[z=0]
Extended state (σ, ξ)
Model Checking (contd…)

(σ, ξ) ╞ Φ, if
Model Checking algorithm



σ╞ Φ, of (σ,ξ) Φ for all ξ evaluations
Computes Characteristic set [Φ]
(l,v) є (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
Examples

ΦU Φ’ computed as UiRi with

◊≤c Φ computed as ¬UiRi[z=0] with
Thank you