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Succinct Approximations of Distributed Hybrid Behaviors
P.S. ThiagarajanSchool of Computing, National University of Singapore
Joint Work with: Yang Shaofa
IIST, UNU, Macau
(To be presented at HSCC 2010)
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Hybrid Automata
Hybrid behaviors: Mode-specific continuous dynamics +
discrete mode changes Standard model: Hybrid Automata
• Piecewise constant rates• Rectangular guards
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A
B D
C
{x1, x2, x3}
A
B
D
CB (x1, x2, x3) = (2, -3.5, 1)
dx1/dt = 2 x1(t) = 2t + x1(0)
dx2/dt = -3.5 dx3/dt = 1
Piecewise Constant Rates
g
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A
B D
C
x c | x c | ’
g
x2
x1
Rectangular Guards
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5
A
B D
C
(x1 2 x1 1)
(x2 3.5 x2 0.5)
g
x2
x1
Initial Regions
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Highly Expressive
Piecewise constant rates; rectangular guards The control state (mode) reachability
problem is undecidable. Given qf, Whether there exists a trajectory
(q0, x0) (q1, x1) ……. (qm, xm) such that qm = qf
q0 = initial mode; x0 in initial region.
HKPV’95
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(q0, x0)
(q1, x1)
g(q0)
(q2, x2)
(q3, x3)
(q4, x4)
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Two main ways to circumvent undecidability If its rate changes as the result of a mode
change, Reset the value of a variable to a pre-
determined region
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Hybrid Automata with resets.
dx/dt = 3 dx/dt = -1.5
x 5
x 2.8
VF
VD
5
2.8
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Hybrid Automata with resets.
dx/dt = 3 dx/dt = -1.5
x 5
x 2.8
VF
VD
5
2.8
x [2, 4]
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Control Applications
PLANT
Digital Controller
Sensors actuators
The reset assumption is untenable.
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PLANT
Digital Controller
Sensors actuators
[HK’97]: Discrete time assumption.
The plant state is observed only at (periodic) discrete time points T0 T1 T2 …..
T i+1 – Ti =
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Discrete time behaviors
The discrete time behavior of a hybrid automaton: Q : The set of modesq0 q1 …qm is a state sequence iff there exists a
run (q0, v0, ) (q1, v1) … (qm, vm) of the automaton. The discrete time behavior of Aut is
L(Aut) Q* the set of state sequences of Aut.
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[HK’97]: The discrete time behavior of (piecewise
constant + rectangular guards) an hybrid automaton is regular.
A finite state automaton representing this language can be effectively constructed.
Discrete time behavior is an approximation. With fast enough sampling, it is a good approximation.
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[AT’04]: The discrete time behavior of an hybrid automaton is regular even with delays in sensing and actuating (laziness)
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g
g
g + gk-1
k
The value of xi reported at t = k is the value at some t’ in [(k-1)+ g, (k-1)+g +g]
g and g are fixed rationals
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h h + h
h
k+1k
If a mode change takes place at t = k is then xi starts evolving at ’(xi) at some t’ in [k+ h, k+h + h]
h and h are fixed rationals.
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Global Hybrid Automata
PLANT
Digital Controller
Sensors
Actuators
? x1
? x2
? x3
(x1)
(x2)
(x3)
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PLANTSensors
Actuators
? x1
? x2
? x3
(x1)
(x2)
(x3)
p1
p2
p3
Distributed Hybrid Automata
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No explicit communication between the automata.. However, coordination through the shared memory of the plant’s state space.
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The Communictaion graph of DHA
Obs(p) --- The set of variables observed by p
Ctl(p) --- The set of variables controlled by p
Ctl(p) Ctl(q) =
Nbr(p) = Obs(p) Ctl(p)
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[(s0
1, v01), (s0
2, v02), (s0
3, v03)]
[(s01), (s0
2), (s03)]
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[(s0
1, v01), (s0
2, v02), (s0
3, v03)]
[(s11, v1
1), (s12, v1
2), (s13, v1
3)]
[(s21, v2
1), (s22, v2
2), (s23, v2
3)]
[(s01), (s0
2), (s03)]
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[(s0
1, v01), (s0
2, v02), (s0
3, v03)]
[(s11, v1
1), (s12, v1
2), (s13, v1
3)]
[(s21, v2
1), (s22, v2
2), (s23, v2
3)]
[(s01), (s0
2), (s03)]
[(s3
1, v31), (s3
2, v32), (s3
3, v33)]
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[(s0
1, v01), (s0
2, v02), (s0
3, v03)]
[(s11, v1
1), (s12, v1
2), (s13, v1
3)]
[(s21, v2
1), (s22, v2
2), (s23, v2
3)]
[(s01), (s0
2), (s03)]
[(s41, v4
1), (s42, v4
2), (s43, v4
3)]
[(s31, v3
1), (s32, v3
2), (s33, v3
3)]
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Discrete time behavior:(Global) state sequences
[s01, s0
2, s03] [s1
1,s12,s1
3] [s21, s2
2, s23] [s3
1, s32, s3
3] . . . .
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Discrete time behaviors
The discrete time behavior of DHA is L(DHA) (Sp1 Sp2 ….. Spn)* the set of global state sequences of DHA.
L(DHA) is regular?
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Discrete time behaviors
The discrete time behavior of DHA is L(DHA) (Sp1 Sp2 ….. Spn)* the set of global state sequences of DHA.
L(DHA) is regular? Yes. Construct the (syntactic product) AUT of DHA. AUT will have piecewise constant rates and
rectangular guards. Hence…..
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Network of HAs Global HA
m ---- the number of component automata in DHA
The size of DHA will be linear in m
The size of AUT will be exponential in m.
Can we do better?Global FSA
Syntactic product
Discretization
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Global FSA
Syntactic Product
DiscretizationLocal discretization
Network of FSAs
Product
Network of HAs Global HA
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Global FSA
Local discretization
Network of FSAs
Network of HAs
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Location node Variable node
For each node, construct an FSA
Each FSA will “read” from all its neighbor FSAs to make its moves.
Nbr(p) = Ctl(p) Obs(p) Nbr(x) = {p | x Ctl(p) Obs(p) }
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Autx
Autx will keep track of the current value of x CTL(x) = p if x Ctl(p) A move of Autx:
read the current rate of x from AutCTL(x) and update the current value of x
Can only keep bounded information Quotient the value space of x
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Quotienting the value space of x
vminvmax
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Quotienting the value space of x
INITx
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Quotienting the value space of x
c c’
c, c’ , ., . ., the constants that appear in some guard
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Quotienting the value space of x
, ’ ….. rates of x associated with modes in AUTCTL(x)
|| |’|
Find the largest positive rational that evenly divides all these rationals.Use it to divide [vmin, vmax] into uniform intervals
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Quotienting the value space of x
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Quotionting the value space of x
) ( ()
A move of Autx:
If Autx is in state I and CTL(x) = p and Autp’s state is then Autx moves from I to I’ = (I)
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) ( ()
I’ = 1(I)
I (s1, 1)
I’
Aut(x1)
(s2, 2)
)( I I’
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I1
(s2, 2) I3
(s’2, ’2)
g
(s2, 2)
(s’2, ’2)
(v1, v3) satisfies g for some (v1, v3) in I1 I3
I2
Aut (p2 )
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
Each automaton will have a parity bit. This bit flips every time the automaton makes a move. Initially all the parities are 0.
A variable node automaton makes a move only when its parity is the same as all its neighbors’
A location node automaton makes a move only when its parity is different from all its neighbors.
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
The automata can ‘drift” in time steps.
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Aut(x1)Aut(p3)
Aut(x3) Aut(x2)Aut(p2)
Aut(p1)
Aut ------ The asynchronous product of { Aut(x1), Aut(p1), Aut(x2), Aut(p2), Aut(x3), Aut(p3) }
Each global state of Aut will induce a global state of DHA.
[(I1, (s1, 1), I2, (s2, 2), I3, (s3, 3)] [s1, s2, s3]
In fact, each complete state sequence of Aut will induce a global state sequence of DHA.
f
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[s10, s20, s30]
[ -- , s21, -- ] [ -- , s22, -- ] [ -- , s23, -- ]
A state sequence of Aut is complete iff all the FSA make an equal number of moves along .
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[s10, s20, s30][s11, --, --] [s12, --, --] [s13, --, --]
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[s10, s20, s30]
[---, ---, s31][---, ---, s32]
[---, ---, s32]
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[s10, s20, s30]
[ -- , s21, -- ] [ -- , s22, -- ] [ -- , s23, -- ]
[s11, --, --] [s12, --, --] [s13, --, --]
[---, ---, s31][---, ---, s32]
[---, ---, s32]
[s10, s20, s30] [s11, s21, s31] [s12, s22, s32] [s13, s23, s33]
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The main results Suppose is a complete state sequence of
Aut. Then f() is a global state sequence of DHA.
If is global state sequence of DHA then there exists a complete sequence of Aut such that f() = .
In the absence of deadlocks, every state sequence of Aut can be extended to a complete state sequence.
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Extensions
Laziness Delays in communictions between the
plant and controllers. Different granularities of time for the
controllers …….
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The marked graph connection
Can be used to derive partial order reduction verification algorithms.
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61
PLANT
Sensors
? x1
? x2
? x3
(x1)
(x2)
(x3)
p1
p2
p3
Communicating hybrid automata:Synchronize on common actions; message passing; shared memory …
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p1
p2
p3
Time-triggered protocol; Each controller is implemented on an ECUStudy interplay between plant dynamics and the performance of the computational platform
Plant
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Summary
The discrete time behavior of distributed hybrid automata can be succinctly represented. as a network of FSA (communicating as in
asynchronous cellular automata). many extensions possible. Finite precision assumption can yield
stronger results.