[PDF] - Control from Computer Science Oded Maler CNRS-VERIMA G PDF Document

SLIDE 1 Control from Computer Science Oded Maler CNRS-VERIMA G Grenoble, F rance

SLIDE 2 Control from Computer Science Oded Maler Mo del-based System Design

World Formal Model Analysis Design Abstract Controller Experiments Thinking Implementation O I Controller

1

SLIDE 3 Control from Computer Science Oded Maler The Coee Machine Physics-Info rmation Info rmation Pro cessing Buttons Coins Coins Drinks M 1 5 4 6 M 2 drink-ready st-tea st-coffee 3 2 1 coin-in cancel coin-out 7 8 9 req-coffee req-tea reset

k

done P

rt

F rom!T

Event

t yp es Meaning 1 E ! M 1 coin-in a coin w as inserted 2 E ! M 1 cancel cancel button p ressed 3 M 1 ! E coin-out release the coin 4 M 1 ! M 2

k

sucient money inserted 5 M 1 ! M 2 reset money returned to user 6 M 2 ! M 1 done drink distribution ended 7 E ! M 2 req-coffee coee button p ressed req-tea tea button p ressed 8 E ! M 2 drink-ready drink p repa ration ended 9 M 2 ! E st-coffee sta rt p repa ring coee st-tea sta rt p repa ring tea 2

SLIDE 4 Control from Computer Science Oded Maler The Tw

Sub-Machines

done/ 1 coin-in/

k

cancel/coin-out, reset drink-ready/done drink-ready/done A C B D

k/

reset/ M 2 req-coffee/st-coffee req-tea/st-tea M 1 3

SLIDE 5 Control from Computer Science Oded Maler The Global Mo del 0A 1B drink-ready/ drink-ready/ 1C 1D 0C 0D cancel/coin-out cancel/coin-out req-tea/st-tea req-coffee/st-coffee cancel/coin-out coin-in/ No rmal b ehavio rs: 0A coin-in 1B cancel coin-out 0A 0A coin-in 1B req-coffee st-coffee 1C drink-ready 0A 4

SLIDE 6 Control from Computer Science Oded Maler An Unexp ected Behavio r 0A 1B drink-ready/ drink-ready/ 1C 1D 0C 0D cancel/coin-out cancel/coin-out req-tea/st-tea req-coffee/st-coffee cancel/coin-out coin-in/ 0A coin-in 1B req-coffee st-coffee 1C cancel coin-out 0C drink-ready 0A 5

SLIDE 7 Control from Computer Science Oded Maler Fixing the Bug drink-ready/done drink-ready/done A C B D reset/ req-coffee/st-coffee,lock req-tea/st-tea,lock M 2

k/

M 1 1 coin-in/

k

2 lock/ cancel/coin-out, reset done/ 6

SLIDE 8 Control from Computer Science Oded Maler Fixing the Bug { the Global Mo del 0A 1B drink-ready/ 2C 2D coin-in/ cancel/coin-out req-tea/st-tea req-coffee/st-coffee drink-ready/ 7

SLIDE 9 Control from Computer Science Oded Maler The Mo ral

f

the Sto ry 1) Many systems can b e mo deled as a comp

sition
f

interacting automata (transition systems, discrete event systems). 2) P

tential

b ehavio rs

f

the system co rresp

nd

to paths in the global transition graph

f

the system. 3) These paths a re lab eled b y input events. Each input sequence might generate a dierent b ehavio r. 4) W e w ant to mak e sure that a system resp

nds

co rrectly to all conceivable inputs. 5) F

r

every individual input sequence w e can simulate the reaction

f

the system. But w e cannot do it exhaustively due to the huge numb er

f

input sequences. 6) V erication is a collection

f

automatic and semi- automatic metho ds to analyze all the paths in the graph. 7) This is ha rd fo r humans to do and even fo r computers. 8

SLIDE 10 Control from Computer Science Oded Maler Mo del I: Closed Systems A transition system is S = (X ;

)

where X is nite and

:

X ! X is the transition function. The state-space X has no numerical meaning and no interesting structure. X k is the set

f

all sequences

f

length k ; X

the

set

f

all sequences. Behavio r: The b ehavio r

f

S sta rting from an initial state x 2 X , is

=
[0];
[1];

: : : 2 X

s.t.
[0]

= x and fo r every i,

[i

+ 1] =

(

[i]) Basic Reachabilit y Problem: Given x and a set P

X

, do es the b ehavio r

f

S sta rting at x reach P ? 9

SLIDE 11 Control from Computer Science Oded Maler Solution b y F

rw

a rd Simulation

[0]:=x

F := fx g rep eat

[k

+ 1]:= ( [k ]) F k +1 := F k [ f [i + 1]g until F k +1 = F k F

:=F

k x 1 x 2 x 3 x 4 x 5 fx 1 g; fx 1 ; x 2 g; fx 1 ; x 2 ; x 3 g; fx 1 ; x 2 ; x 3 ; x 5 g Ho w to do it fo r continuous system dened b y _ x = f (x) ? 10

SLIDE 12 Control from Computer Science Oded Maler Mo del I I: Systems with One Input A

ne-input

transition system is S = (X ; V ;

)

where X and V a re nite

:

X

V

! X is the transition function. Behavio r Induced b y Input: Given an input sequence 2 V

,

the b ehavio r

f

S sta rting from x 2 X in the p resence

f

is a sequence

(

) =

[0];
[1];

: : : 2 X

suc

h that

[i

+ 1] =

(

[i]; [i]): v 1 v 2 v 2 v 1 v 1 v 1 v 2 v 2 x 1 x 2 x 3 x 4 x 5 v 1 ; v 2 x 1 v 1

!

x 2 v 2

!

x 3 v 2

!

x 5 v 1

!

x 2 v 1

!

x 4 11

SLIDE 13 Control from Computer Science Oded Maler Reachabilit y fo r Op en Systems The reachabilit y p roblem: Is there some input sequence 2 V

such

that

(

) reaches P ? F

r

every given w e can use the p revious algo rithm, simulate and

btain

F

(

). F

r

an automaton with n states all states a re reachable b y sequences

f

length < n. F

=

[

2V

n F

(

) v 1 v 2 x 4 x 5 x 5 v 2 v 1 x 3 x 4 x 5 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 v 1 v 2 x 3 x 1 x 2 x 3 x 5 x 2 x 1 x 2 x 5 12

SLIDE 14 Control from Computer Science Oded Maler A Mo re Ecient W a y Many dierent inputs lead to the same state. Immediate successo rs:

(x)

= fx : 9u

(x;

u) = x g Successo rs

f

a set F :

(F

) = f (x) : x 2 F g F

rw

a rd reachabilit y algo rithm (b readth-rst): F := fx g rep eat F k +1 := F k [

(F

k ) until F k +1 = F k F

:=F

k v 1 v 2 v 1 v 2 v 1 v 2 x 3 x 5 x 2 x 1 x 2 x 5 v 1 v 2 x 4 x 5 x 5 x 3 x 1 v 1 v 2 Complexit y:

nly

O (n

log

n

jV

j) 13

SLIDE 15 Control from Computer Science Oded Maler V a riations: Depth-First and Backw a rds Depth-rst: v 1 v 1 x 3 x 2 v 1 v 2 x 4 x 5 x 5 v 2 v 1 x 3 x 4 x 5 v 1 v 2 x 5 x 2 x 5 x 1 v 2 Backw a rds: nd all states from which there is an input leading to P . Immediate p redecesso rs:

1

(x) = fx : 9u

(x

; u) = xg F := P rep eat F k +1 := F k [

1

(F k ) until F k +1 = F k F

:=F

k 14

SLIDE 16 Control from Computer Science Oded Maler Admissible Inputs So fa r w e have assumed that the external environment can generate all sequences in V

.

Sometimes w e have a mo re restricted environment, e.g. it will never p ro duce v 1 v 1 . W e can build an automaton which mo dels the environment and comp

se

it with the mo del

f

the system. v 1 v 2 v 2 v 1 v 1 v 1 v 2 v 2 x 1 x 2 x 3 x 4 x 5 v 1 ; v 2 v 1 v 2 v 2 v 1 v 1 v 2 v 2 x 1 x 2 x 3 x 5 v 1 v 2 v 2 15

SLIDE 17 Control from Computer Science Oded Maler V erication: The State-of-the-Art There a re algo rithms that tak e a description

f

any

p

en system and verify whether any

f

the admissible inputs drives the system into a set P . Such algo rithms alw a ys terminate after a nite numb er

f

steps. This is essentially what verication is all ab

ut.

The result is general: it is valid fo r every discrete nite-state system. Of course, nite systems can b e very la rge and sp ecial tricks a re needed to verify them. The analogue fo r continuous systems: do the same fo r a system dened b y _ x = f (x; u). 16

SLIDE 18 Control from Computer Science Oded Maler Systems with t w

Inputs

A t w

-input

transition system is S = (X ; U ; V ;

)

where X , U and V a re nite sets and

:

X

U
V

! X is the transition function. u 1 u 1 u 1 u 2 u 2 u 2 v 1 v 2 v 2 v 1 v 2 v 2 v 1 v 2 v 1 v 2 v 2 v 1 u 2 v 1 u 1 v 1 ; v 2 v 1 x 2 x 3 x 5 x 4 x 1

(x

1 ; u 1 ; v 1 ) = x 1

(x

1 ; u 1 ; v 2 ) = x 2

(x

1 ; u 2 ; v 1 ) = x 2

(x

1 ; u 2 ; v 2 ) = x 4 The b ehavio r in the p resence

f

t w

inputs,
2

U

and

2 V

:

a sequence

(

; ) s.t.

[i

+ 1] =

(

[i];

[i];

[i]) 17

SLIDE 19 Control from Computer Science Oded Maler Games and Strategies Interp retation

f

inputs: U : w e, the go

d

guys, the controller. V : they , the bad guys, disturbances. An antagonist game situation. Our goal is to cho

se

each time an element

f

U such that the b ehavio rs induces b y all p

ssible

disturbances a re go

d.

Strategy: a function c : X

!

U State strategy: a function c : X ! U . Each strategy c converts a t yp e I I I system into a t yp e I I system S c = (X ; V ;

c

) s.t.

c

(x; v ) =

(x;

c(x); v ). Synthesis fo r Reachabilit y: Let S = (X ; U ; V ;

)

let P

X

b e a set

f

\bad" states. The controller synthesis p roblem is: nd a strategy c such that all the b ehavio rs

f

the derived system S c = (X ; V ;

c

) never reach P . 18

SLIDE 20 Control from Computer Science Oded Maler Finding Winning States and Strategies Controllable Predecesso rs: F

r

S = (X ; U ; V ;

)

and F

X

, the set

f

controllable p redecesso rs

f

F is

(F

) = fx : 9u 2 U 8v 2 V

(x;

u; v ) 2 F g The states from which the controller, b y p rop erly selecting u, can fo rce the system into P in the next step. The follo wing backw a rd algo rithm nds the set F

f

\winning states" from which P can b e avoided fo rever. F := X

P

rep eat F k +1 := F k \

(F

k ) until F k +1 = F k F

:=F

k Rema rk: this is simila r to the Ramadge-W

nham

theo ry

f

discrete event control. 19

SLIDE 21 Control from Computer Science Oded Maler Synthesis Example u 1 u 1 u 1 u 2 u 2 v 1 v 2 v 2 v 1 v 2 v 2 v 1 v 2 v 1 v 2 v 2 v 1 u 2 v 1 u 1 v 1 ; v 2 v 1 x 2 x 3 x 5 x 4 x 1 u 2 W e w ant to avoid x 5 . F = fx 1 ; x 2 ; x 3 ; x 4 g F 1 = fx 1 ; x 2 ; x 3 g = F

The

resulting \closed-lo

p"

system alw a ys remains in fx 1 ; x 2 ; x 3 g. u 1 v 1 v 2 v 1 v 2 u 1 v 1 ; v 2 x 2 x 3 x 1 u 2 20

SLIDE 22 Control from Computer Science Oded Maler Discrete Innite-State Systems Computer p rogram a re syntactic rep resentation

f

dynamical systems with innite state-space. rep eat y :=y + 1 until y = 4 State space: fx 1 ; x 2 g

Z

x 1 x 2 4 6 : : : 5 4 3 2 y 6= 4/y :=y + 1 x 1 y = 4 x 2 : : : F

rw

a rd reachabilit y algo rithm will terminate if sta rted from (x 1 ; 2) but not from (x 1 ; 5). The reachabilit y p roblem is unsolvable: there is no general algo rithm that solves every instance

f

it. \Deductive" app roach: p rove p rop erties \analytically". \Symb

lic"

app roach: reachabilit y using fo rmulae to rep resent sets

f

states, e.g. x = x 1 ^ y

5.

21

SLIDE 23 Control from Computer Science Oded Maler Continuous (and Hyb rid) Systems Why? ... Problems: state space R n , innite even when b

unded,

time domain R . Mathematical R vs. numerical R in the computer. Reachabilit y fo r _ x = f (x): When w e have a closed- fo rm solution, e.g. fo r _ x = Ax, the reachable set can b e written as F

=

fx e At : t

0g

but ho w to test whether F

\

P = ;? F

rw

a rd simulation: discretize time and replace the system with

[(n

+ 1)] =

[n]

+ h( [n]; ): P x

This

is not the \real" thing and it is not gua ranteed to converge but that's life. 22

SLIDE 24 Control from Computer Science Oded Maler Continuous Systems with Input Systems

f

the fo rm _ x = f (x; v ). Admissible inputs a re signals

f

the fo rm : T ! V . Problem: sho w that no admissible input drives the system into a set P . F

r

every w e can simulate and \compute" F

(

), but there is no nite subset

f

inputs that covers all reachable states. The set

f

all inputs is a doubly-dense tree, b

th

vertically (time) and ho rizontally (V ). x x 23

SLIDE 25 Control from Computer Science Oded Maler Incremental Reachabilit y Computation Breadth-rst computation

f

reachable states. x t

!

x denotes the existence

f

an input signal : [0; t] ! V that drives the system from x to x in t time. Let F b e a subset

f

X and let I b e a time interval. The I

successo

rs

f

F a re all the states that can b e reached from F within that time interval, i.e.

I

(F ) = fx : 9x 2 F 9t 2 I x t

!

x g: Semigroup p rop ert y:

[0;r

2 ] ( [0;r 1 ] (F )) =

[0;r

1 +r 2 ] (F ): F := fx g rep eat F k +1 := F k [

[0;r

] (F k ) until F k +1 = F k F

:=F

k 24

SLIDE 26 Control from Computer Science Oded Maler App ro ximate Reachabilit y Computation But

[0;r

] (F ) cannot b e computed exactly . W e can

ver-app

ro ximate it b y

such

that fo r every F

[0;r

] (F )

[0;r

] (F ) and

[0;r

] (F ) b elongs to some eective sub-class

f

R n , e.g. pp

lyhedra.

The result

f

the algo rithm is a set F

s.t.

F

F
and

hence F

\

P = ; implies the co rrectness

f

the system. x x 25

SLIDE 27 Control from Computer Science Oded Maler Conclusion W e have develop ed a system called d/dt which accepts as input a description

f

a continuous

r

a hyb rid system and computes automatically an

ver-

app ro ximation

f

the reachable states. Mo re ab

ut

it in the sp ecial session

n

reachabilit y . Challenge: use mo re kno wledge

n

the system dynamics in

rder

to increase the p erfo rmance and treat systems with higher dimensions. Challenge: develop algo rithms fo r automatic synthesis

f

strategies fo r systems with t w

inputs,

_ x = f (x; u; v ). 26