succinct approximations of distributed hybrid behaviors
Download
Skip this Video
Download Presentation
Succinct Approximations of Distributed Hybrid Behaviors

Loading in 2 Seconds...

play fullscreen
1 / 63

Succinct Approximations of Distributed Hybrid Behaviors - PowerPoint PPT Presentation


  • 104 Views
  • Uploaded on

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:

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Succinct Approximations of Distributed Hybrid Behaviors' - fala


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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
slide3
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

slide4
Rectangular Guards

A

g

B

D

C

x2

x  c | x  c |   ’

x1

slide5
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
slide7
(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.

slide12
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.
slide14
[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.
slide15
[AT’04]: The discrete time behavior of an hybrid automaton is regular even with delays in sensing and actuating (laziness)
slide16
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

slide17
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
Global Hybrid Automata

PLANT

Sensors

Actuators

Digital Controller

(x1) 

? x1

(x2) 

? x2

(x3) 

? x3

slide19
Distributed Hybrid Automata

PLANT

Sensors

Actuators

? x1

p1

(x1) 

? x2

p2

(x2) 

p3

(x3) 

? x3

slide20
No explicit communication between the automata.. However, coordination through the shared memory of the plant’s state space.
slide21
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)

slide22
[(s01), (s02), (s03)]

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

slide23
[(s21, v21), (s22, v22), (s23, v23)]

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

[(s01), (s02), (s03)]

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

slide24
[(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)]

slide25
[(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)]

slide26
Discrete time behavior:

(Global) state sequences

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

discrete time behaviors1
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 behaviors2
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…..
slide29
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

slide30
Network of HAs

Global HA

Syntactic Product

Local discretization

Discretization

Product

Global FSA

Network of FSAs

slide31
Network of HAs

Local discretization

Global FSA

Network of FSAs

slide32
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) }

aut x
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 x2
Quotienting the value space of x

c

c’

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

quotienting the value space of x3
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
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)

slide40
(

)

(

)

(

)

I’

I

(s2, 2)

(s1, 1)

I

Aut(x1)

I’

I’ = 1(I)

slide41
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)

slide42
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.

slide43
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide44
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide45
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide46
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide47
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide48
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide49
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide50
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide51
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

slide52
Aut(x1)

Aut(p3)

Aut(p1)

Aut(x3)

Aut(x2)

Aut(p2)

The automata can ‘drift” in time steps.

slide53
Aut(x1)

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

slide54
A state sequence  of Aut is complete iff all the FSA make an equal number of moves along .

[ -- , s23, -- ]

[ -- , s21, -- ]

[ -- , s22, -- ]

[s10, s20, s30]

slide55
[s10, s20, s30]

[s13, --, --]

[s12, --, --]

[s11, --, --]

slide56
[s10, s20, s30]

[---, ---, s32]

[---, ---, s31]

[---, ---, s32]

slide57
[ -- , s23, -- ]

[ -- , 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

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

Laziness

Delays in communictions between the plant and controllers.

Different granularities of time for the controllers

…….

the marked graph connection
The marked graph connection

Can be used to derive partial order reduction verification algorithms.

slide61
Communicating hybrid automata:

Synchronize on common actions; message passing; shared memory …

Sensors

PLANT

? x1

p1

(x1) 

? x2

p2

(x2) 

p3

(x3) 

? x3

slide62
Plant

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
  • 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.
ad