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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

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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: • 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**• 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**• 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.**VF x 5 dx/dt = 3 dx/dt = -1.5 5 x 2.8 2.8 VD**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**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**• 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.**[AT’04]: The discrete time behavior of an hybrid**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**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**PLANT Sensors Actuators Digital Controller (x1) ? x1 (x2) ? x2 (x3) ? x3**Distributed Hybrid Automata**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**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)**[(s01), (s02), (s03)]**[(s01, v01), (s02, v02), (s03, v03)]**[(s21, v21), (s22, v22), (s23, v23)]**[(s11, v11), (s12, v12), (s13, v13)] [(s01), (s02), (s03)] [(s01, v01), (s02, v02), (s03, v03)]**[(s21, 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)]**[(s21, 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:**(Global) state sequences [s01, s02, s03] [s11,s12,s13] [s21, s22, s23] [s31, s32, s33] . . . .**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?**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…..**Global HA**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**Global HA Syntactic Product Local discretization Discretization Product Global FSA Network of FSAs**Network of HAs**Local discretization Global FSA Network of FSAs**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) }**Autx**• 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**vmax vmin**Quotienting the value space of x**c c’ c, c’ , ., . ., the constants that appear in some guard**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**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)**(**) ( ) ( ) I’ I (s2, 2) (s1, 1) I Aut(x1) I’ I’ = 1(I)**I1**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(x1)**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(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)**Aut(x1)**Aut(p3) Aut(p1) Aut(x3) Aut(x2) Aut(p2)

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