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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)

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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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3 AA BB DD CC {x 1, x 2, x 3 } A B D C B (x1, x2, x3) = (2, -3.5, 1) dx 1 /dt = 2 x 1 (t) = 2t + x 1 (0) dx 2 /dt = -3.5 dx 3 /dt = 1 Piecewise Constant Rates g

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4 AA BB DD CC x c | x c | ’ g x2x2 x1x1 Rectangular Guards

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5 AA BB DD CC (x 1 2 x 1 1) (x 2 3.5 x 2 0.5) g x2x2 x1x1 Initial Regions

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Highly Expressive Piecewise constant rates; rectangular guards The control state (mode) reachability problem is undecidable. Given q f, Whether there exists a trajectory (q 0, x 0 ) (q 1, x 1 ) ……. (q m, x m ) such that q m = q f q 0 = initial mode; x 0 in initial region. HKPV’95

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7 (q 0, x 0 ) (q 1, x 1 ) g (q 0 ) (q 2, x 2 ) (q 3, x 3 ) (q 4, x 4 )

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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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9 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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10 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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11 Control Applications PLANT Digital Controller Sensors actuators The reset assumption is untenable.

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12 PLANT Digital Controller Sensors actuators [HK’97]: Discrete time assumption. The plant state is observed only at (periodic) discrete time points T 0 T 1 T 2 ….. T i+1 – T i =

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13 Discrete time behaviors The discrete time behavior of a hybrid automaton: Q : The set of modes q 0 q 1 …q m is a state sequence iff there exists a run (q 0, v 0, ) (q 1, v 1 ) … (q m, v m ) 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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16 g gg g + g k-1 k The value of x i 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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17 h h + h hh k+1k If a mode change takes place at t = k is then x i starts evolving at ’(x i ) at some t’ in [k+ h, k+h + h ] h and h are fixed rationals.

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18 Global Hybrid Automata PLANT Digital Controller Sensors Actuators ? x1 ? x2 ? x3 (x1) (x2) (x3)

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19 PLANT Sensors Actuators ? x1 ? x2 ? x3 (x1) (x2) (x3) p 1 p 2 p 3 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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22 [(s 0 1, v 0 1 ), (s 0 2, v 0 2 ), (s 0 3, v 0 3 )] [ (s 0 1 ), (s 0 2 ), (s 0 3 )]

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23 [(s 0 1, v 0 1 ), (s 0 2, v 0 2 ), (s 0 3, v 0 3 )] [(s 1 1, v 1 1 ), (s 1 2, v 1 2 ), (s 1 3, v 1 3 )] [(s 2 1, v 2 1 ), (s 2 2, v 2 2 ), (s 2 3, v 2 3 )] [ (s 0 1 ), (s 0 2 ), (s 0 3 )]

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24 [(s 0 1, v 0 1 ), (s 0 2, v 0 2 ), (s 0 3, v 0 3 )] [(s 1 1, v 1 1 ), (s 1 2, v 1 2 ), (s 1 3, v 1 3 )] [(s 2 1, v 2 1 ), (s 2 2, v 2 2 ), (s 2 3, v 2 3 )] [ (s 0 1 ), (s 0 2 ), (s 0 3 )] [(s 3 1, v 3 1 ), (s 3 2, v 3 2 ), (s 3 3, v 3 3 )]

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25 [(s 0 1, v 0 1 ), (s 0 2, v 0 2 ), (s 0 3, v 0 3 )] [(s 1 1, v 1 1 ), (s 1 2, v 1 2 ), (s 1 3, v 1 3 )] [(s 2 1, v 2 1 ), (s 2 2, v 2 2 ), (s 2 3, v 2 3 )] [ (s 0 1 ), (s 0 2 ), (s 0 3 )] [(s 4 1, v 4 1 ), (s 4 2, v 4 2 ), (s 4 3, v 4 3 )] [(s 3 1, v 3 1 ), (s 3 2, v 3 2 ), (s 3 3, v 3 3 )]

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26 Discrete time behavior: (Global) state sequences [s 0 1, s 0 2, s 0 3 ] [s 1 1,s 1 2,s 1 3 ] [s 2 1, s 2 2, s 2 3 ] [s 3 1, s 3 2, s 3 3 ]....

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27 Discrete time behaviors The discrete time behavior of DHA is L(DHA) (S p1 S p2 ….. S pn )* the set of global state sequences of DHA. L(DHA) is regular?

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28 Discrete time behaviors The discrete time behavior of DHA is L(DHA) (S p1 S p2 ….. S pn )* 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 Discretization Local 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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Aut x Aut x will keep track of the current value of x CTL(x) = p if x Ctl(p) A move of Aut x : read the current rate of x from Aut CTL(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 v min v max

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Quotienting the value space of x INIT x

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Quotienting the value space of x cc’ 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 AUT CTL(x) ||||| ’| Find the largest positive rational that evenly divides all these rationals. Use it to divide [v min, v max] 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 Aut x : If Aut x is in state I and CTL(x) = p and Aut p ’s state is then Aut x moves from I to I’ = (I)

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) ( ( ) I’ = 1 (I) I (s 1, 1 ) I’ Aut(x 1 ) (s 2, 2 ) )( I I’

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I1 (s 2, 2 ) I3 (s’ 2, ’ 2 ) g (s 2, 2 ) (s’ 2, ’ 2 ) (v1, v3) satisfies g for some (v1, v3) in I1 I3 I2 Aut (p 2 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 ) 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(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 )

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 ) The automata can ‘drift” in time steps.

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Aut(x 1 ) Aut(p 3 ) Aut(x 3 )Aut(x 2 ) Aut(p 2 ) Aut(p 1 ) Aut ------ The asynchronous product of { Aut(x 1 ), Aut(p 1 ), Aut(x 2 ), Aut(p 2 ), Aut(x 3 ), Aut(p 3 ) } 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]

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[s10, s20, s30] [ --, s21, -- ][ --, s22, -- ] [ --, s23, -- ] [s11, --, --] [s12, --, --] [s13, --, --] [---, ---, s31] [---, ---, 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) p 1 p 2 p 3 Communicating hybrid automata: Synchronize on common actions; message passing; shared memory …

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p 1 p 2 p 3 Time-triggered protocol; Each controller is implemented on an ECU Study 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.

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