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Succinct Approximations of Distributed Hybrid Behaviors. P.S. Thiagarajan School of Computing, National University of Singapore Joint Work with: Yang Shaofa IIST, UNU, Macau (To be presented at HSCC 2010). Hybrid Automata. Hybrid behaviors:

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Succinct approximations of distributed hybrid behaviors

Succinct Approximations of Distributed Hybrid Behaviors

P.S. Thiagarajan

School of Computing, National University of Singapore

Joint Work with: Yang Shaofa

IIST, UNU, Macau

(To be presented at HSCC 2010)


Hybrid automata
Hybrid Automata

  • Hybrid behaviors:

  • Mode-specific continuous dynamics + discrete mode changes

  • Standard model: Hybrid Automata

    • Piecewise constant rates

    • Rectangular guards


Piecewise Constant Rates

A

A

{x1, x2, x3}

g

B

B

D

D

B (x1, x2, x3) = (2, -3.5, 1)

dx1/dt = 2 x1(t) = 2t + x1(0)

dx2/dt = -3.5 dx3/dt = 1

C

C


Rectangular Guards

A

g

B

D

C

x2

x  c | x  c |   ’

x1


Initial Regions

(x1  2  x1  1) 

(x2  3.5  x2  0.5)

A

x2

g

x1

B

D

C


Highly expressive
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; x0in initial region.

  • HKPV’95


(q4, x4)

(q1, x1)

g

(q0)

(q2, x2)

(q0, x0)

(q3, x3)


Two main ways to circumvent undecidability
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


Hybrid automata with resets
Hybrid Automata with resets.

VF

x  5

dx/dt = 3

dx/dt = -1.5

5

x  2.8

2.8

VD


Hybrid automata with resets1
Hybrid Automata with resets.

VF

x  5

x  [2, 4]

dx/dt = 3

dx/dt = -1.5

5

x  2.8

2.8

VD


Control applications
Control Applications

PLANT

actuators

Sensors

Digital Controller

The reset assumption is untenable.


PLANT

actuators

Sensors

Digital Controller

[HK’97]: Discrete time assumption.

The plant state is observed only at (periodic) discrete time points T0 T1 T2 …..

T i+1 – Ti = 


Discrete time behaviors
Discrete time behaviors

  • The discrete time behavior of a hybrid automaton:

    • Q : The set of modes

    • q0 q1 …qm is a state sequenceiff 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.


  • [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.



automaton is regular even with delays in sensing and actuating (laziness) g

k

k-1

g

g + g

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


h automaton is regular even with delays in sensing and actuating (laziness)

h + h

k

k+1

h

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.


Global hybrid automata
Global Hybrid Automata automaton is regular even with delays in sensing and actuating (laziness)

PLANT

Sensors

Actuators

Digital Controller

(x1) 

? x1

(x2) 

? x2

(x3) 

? x3


Distributed Hybrid Automata automaton is regular even with delays in sensing and actuating (laziness)

PLANT

Sensors

Actuators

? x1

p1

(x1) 

? x2

p2

(x2) 

p3

(x3) 

? x3


No explicit communication between the automata.. However, coordination through the shared memory of the plant’s state space.


The Communictaion graph of DHA coordination through the shared memory of the plant’s state space.

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)


[ coordination through the shared memory of the plant’s state space.(s01), (s02), (s03)]

[(s01, v01), (s02, v02), (s03, v03)]


[(s coordination through the shared memory of the plant’s state space.21, v21), (s22, v22), (s23, v23)]

[(s11, v11), (s12, v12), (s13, v13)]

[(s01), (s02), (s03)]

[(s01, v01), (s02, v02), (s03, v03)]


[(s coordination through the shared memory of the plant’s state space.21, v21), (s22, v22), (s23, v23)]

[(s11, v11), (s12, v12), (s13, v13)]

[(s01), (s02), (s03)]

[(s31, v31), (s32, v32), (s33, v33)]

[(s01, v01), (s02, v02), (s03, v03)]


[(s coordination through the shared memory of the plant’s state space.21, v21), (s22, v22), (s23, v23)]

[(s41, v41), (s42, v42), (s43, v43)]

[(s11, v11), (s12, v12), (s13, v13)]

[(s01), (s02), (s03)]

[(s31, v31), (s32, v32), (s33, v33)]

[(s01, v01), (s02, v02), (s03, v03)]


Discrete time behavior: coordination through the shared memory of the plant’s state space.

(Global) state sequences

[s01, s02, s03] [s11,s12,s13] [s21, s22, s23] [s31, s32, s33] . . . .


Discrete time behaviors1
Discrete time behaviors coordination through the shared memory of the plant’s state space.

  • The discrete time behavior of DHA is

    • L(DHA)  (Sp1  Sp2  ….. Spn)*

    • the set of global state sequences of DHA.

  • L(DHA) is regular?


  • Discrete time behaviors2
    Discrete time behaviors coordination through the shared memory of the plant’s state space.

    • 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…..


  • Global HA coordination through the shared memory of the plant’s state space.

    Network of HAs

    Syntactic product

    Discretization

    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


    Network of HAs coordination through the shared memory of the plant’s state space.

    Global HA

    Syntactic Product

    Local discretization

    Discretization

    Product

    Global FSA

    Network of FSAs


    Network of HAs coordination through the shared memory of the plant’s state space.

    Local discretization

    Global FSA

    Network of FSAs


    Location node coordination through the shared memory of the plant’s state space.

    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) }


    Aut x
    Aut coordination through the shared memory of the plant’s state space.x

    • Autxwill 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


    Quotienting the value space of x
    Quotienting the value space of x coordination through the shared memory of the plant’s state space.

    vmax

    vmin


    Quotienting the value space of x1
    Quotienting the value space of x coordination through the shared memory of the plant’s state space.

    INITx


    Quotienting the value space of x2
    Quotienting the value space of x coordination through the shared memory of the plant’s state space.

    c

    c’

    c, c’ , ., . ., the constants that appear in some guard


    Quotienting the value space of x3
    Quotienting the value space of x coordination through the shared memory of the plant’s state space.

    , ’ ….. 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


    Quotienting the value space of x4
    Quotienting the value space of x coordination through the shared memory of the plant’s state space.


    Quotionting the value space of x
    Quotionting the value space of x coordination through the shared memory of the plant’s state space.

    (

    )

    (

    )

    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)


    ( coordination through the shared memory of the plant’s state space.

    )

    (

    )

    (

    )

    I’

    I

    (s2, 2)

    (s1, 1)

    I

    Aut(x1)

    I’

    I’ = 1(I)


    I1 coordination through the shared memory of the plant’s state space.

    I2

    (s2, 2)

    I3

    (s2, 2)

    Aut (p2 )

    (v1, v3) satisfies g for some (v1, v3) in I1  I3

    g

    (s’2, ’2)

    (s’2, ’2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)

    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.


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)

    The automata can ‘drift” in time steps.


    Aut(x coordination through the shared memory of the plant’s state space.1)

    Aut(p3)

    Aut(p1)

    Aut(x3)

    Aut(x2)

    Aut(p2)

    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


    A state sequence coordination through the shared memory of the plant’s state space. of Aut is complete iff all the FSA make an equal number of moves along .

    [ -- , s23, -- ]

    [ -- , s21, -- ]

    [ -- , s22, -- ]

    [s10, s20, s30]


    [s10, s20, s30] coordination through the shared memory of the plant’s state space.

    [s13, --, --]

    [s12, --, --]

    [s11, --, --]


    [s10, s20, s30] coordination through the shared memory of the plant’s state space.

    [---, ---, s32]

    [---, ---, s31]

    [---, ---, s32]


    [ -- , s23, -- ] coordination through the shared memory of the plant’s state space.

    [ -- , s21, -- ]

    [ -- , s22, -- ]

    [s10, s20, s30]

    [s13, --, --]

    [s12, --, --]

    [s11, --, --]

    [---, ---, s32]

    [---, ---, s31]

    [---, ---, s32]

    [s10, s20, s30] [s11, s21, s31] [s12, s22, s32] [s13, s23, s33]


    The main results
    The main results coordination through the shared memory of the plant’s state space.

    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.


    Extensions
    Extensions coordination through the shared memory of the plant’s state space.

    Laziness

    Delays in communictions between the plant and controllers.

    Different granularities of time for the controllers

    …….


    The marked graph connection
    The marked graph connection coordination through the shared memory of the plant’s state space.

    Can be used to derive partial order reduction verification algorithms.


    Communicating hybrid automata: coordination through the shared memory of the plant’s state space.

    Synchronize on common actions; message passing; shared memory …

    Sensors

    PLANT

    ? x1

    p1

    (x1) 

    ? x2

    p2

    (x2) 

    p3

    (x3) 

    ? x3


    Plant coordination through the shared memory of the plant’s state space.

    p1

    p2

    p3

    Time-triggered protocol;

    Each controller is implemented on an ECU

    Study interplay between plant dynamics and the performance of the computational platform


    Summary
    Summary coordination through the shared memory of the plant’s state space.

    • 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.


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