Supervisory Control under Partial Observation

Transcription

1 Supervisory Control under Prtil Observtion Dr Rong Su S1-B1b-59, Shool of EEE Nnyng Tehnologil University Tel: , Emil: EE6226 Disrete Event Dynmi Systems 1

2 Outline Motivtion The Conept of Observbility Supervisor Synthesis under Prtil Observtion Exmple Conlusions EE6226 Disrete Event Dynmi Systems 2

3 Three Min Conepts in Control Controllbility llows you to improve the dynmis of system by feedbk e.g. ontrollbility in the RW supervisory ontrol theory Observbility llows you to deploy suh feedbk by using the system's output Optimlity gives rise to forml methods of ontrol synthesis e.g. supremlity in the RW supervisory ontrol theory EE6226 Disrete Event Dynmi Systems 3

4 Exmple G d b b Σ={,b,,d} Σ ={b,d} S d b EE6226 Disrete Event Dynmi Systems 4

5 Exmple (ont.) G b d:unobservble b Σ={,b,,d} Σ ={b,d} S d b EE6226 Disrete Event Dynmi Systems 5

6 Some Intuitions Supervisor n only t upon reeiving observble events Prtil observtion fores supervisor to be onservtive We n enble or disble n unobservble event EE6226 Disrete Event Dynmi Systems 6

7 Outline Motivtion The Conept of Observbility Supervisor Synthesis under Prtil Observtion Exmple Conlusions EE6226 Disrete Event Dynmi Systems 7

8 Observbility Given G φ(σ), let Σ o Σ nd P:Σ * Σ o * be the nturl projetion. A lnguge K L(G) is (G,P)-observble, if ( s K)( σ Σ) sσ L(G) K P -1 P(s)σ K= b b Σ o = {b} EE6226 Disrete Event Dynmi Systems 8

9 Or equivlently K L(G) is (G,P)-observble, if for ny s K, sʹ Σ * nd σ Σ, sσ L(G) K sʹσ L(G) P(s)=P(sʹ) sʹσ L(G) K or equivlently, sσ K sʹσ L(G) P(s)=P(sʹ) sʹσ K (Think bout why they re equivlent) EE6226 Disrete Event Dynmi Systems 9

10 Exmple 1 d Σ = {,b,,d} Σ o = {} K ={, b} K b Question: is K (G,P)-observble? yes EE6226 Disrete Event Dynmi Systems 10

11 Exmple 2 d Σ = {,b,,d} Σ o = {} K ={, b} K b d Question: is K (G,P)-observble? no EE6226 Disrete Event Dynmi Systems 11

12 Exmple 3 d Σ = {,b,,d} Σ o = {,} K ={, b} K b d Question: is K (G,P)-observble? yes EE6226 Disrete Event Dynmi Systems 12

13 (G,P)-observbility is deidble. But how? EE6226 Disrete Event Dynmi Systems 13

14 Proedure of Cheking Observbility : Step 1 Let G = (X,Σ,ξ,x 0,X m ) Suppose K is reognized by A = (Y,Σ,η,y 0,Y m ), i.e. K=L m (A) Let A' = G A = (X Y,Σ,ξ η,(x 0,y 0 ),X m Y m ) Sine K=L(A) L(G), we hve L(G A)=L(A) A stte (x,y) X Y is boundry stte of A' w.r.t. G, if ( s L(A')) ξ η((x 0,y 0 ),s)=(x,y), i.e. (x,y) is rehble from (x 0,y 0 ) ( σ Σ) ξ(x,σ)! η(y,σ)!, where! denotes is defined Let B be the olletion of ll boundry sttes of A' w.r.t. G B is finite set. (Why?) EE6226 Disrete Event Dynmi Systems 14

15 Proedure of Cheking Observbility : Step 2 For eh boundry stte (x,y) B, we define two sets T(x,y) := {s L(A') ξ η((x 0,y 0 ),s)=(x,y)} (T(x,y) is regulr, why?) Σ(x,y) := {σ Σ ξ(x,σ)! η(y,σ)!} Theorem K is observble w.r.t. G nd P, iff for ny boundry stte (x,y) B, P -1 P(T(x,y))Σ(x,y) K = EE6226 Disrete Event Dynmi Systems 15

16 Exmple d 2 2 Σ = {,b,,d} Σ o = {} K ={, b} b b 1 G A 1 EE6226 Disrete Event Dynmi Systems 16

17 Exmple Step 1 d Σ = {,b,,d} Σ o = {} K ={, b} (0,0) b A'=G A (2,2) (1,1) (3,3) B={(1,1), (2,2)} EE6226 Disrete Event Dynmi Systems 17

18 Exmple Step 2 For the boundry stte (1,1) we hve T(1,1) = {b} Σ(1,1) = {} P -1 P(T(1,1))Σ(1,1) K = {b,} {,b} = {} For the boundry stte (2,2) we hve T(2,2) = {} Σ(2,2) = {d} P -1 P(T(2,2))Σ(2,2) K = {d} {,b} = K is not observble w.r.t. G nd P (0,0) b d (2,2) (1,1) (3,3) EE6226 Disrete Event Dynmi Systems 18

19 Properties of Observble Lnguges Suppose K 1 nd K 2 re losed, observble w.r.t. G nd P. Then K 1 K 2 is observble w.r.t. G nd P K 1 K 2 my not be observble w.r.t. G nd P Given plnt G, let O(G):={K L(G) K is losed nd observble w.r.t. G nd P} The prtilly ordered set (poset) (O(G), ) is meet-semi-lttie The gretest element my not exist (i.e. no supreml observble sublnguge) EE6226 Disrete Event Dynmi Systems 19

20 Exmple Σ={,b,,d,e} b e e Σ o ={} d d e G K 1 K 2 K 1 K 2 is observble, but K 1 K 2 is not. (Why?) EE6226 Disrete Event Dynmi Systems 20

21 Outline Motivtion The Conept of Observbility Supervisor Synthesis under Prtil Observtion Exmple Conlusions EE6226 Disrete Event Dynmi Systems 21

22 Min Existene Result Theorem 1 Let K L m (G) nd K. There exists proper supervisor iff K is ontrollble with respet to G K is observble with respet to G nd P K is L m (G)-losed, i.e. K = K L m (G) EE6226 Disrete Event Dynmi Systems 22

23 Supervision under Prtil Observtion Suppose K is ontrollble, observble nd L m (G)-losed. Let A=(Y,Σ o,η,y 0,Y m ) be the nonil reognizer of P(K). We onstrut new utomton S=(Y,Σ,λ,y 0,Y m ) s follow: For ny y Y, n event σ Σ Σ o is ontrol-relevnt w.r.t. y nd K, if ( s K) η(y 0,P(s))=y sσ K Let Σ(y) be the olletion of ll events in Σ Σ o ontrol-relevnt w.r.t. y, K We define the trnsition mp λ:y Σ Y s follows: λ is the sme s η over Y Σ o For ny y Y nd σ Σ(y), define λ(y,σ):=y (i.e. selfloop ll events of Σ(y) t y) For ll other (y,σ) pirs, λ(y,σ) is undefined S is proper supervisor of G under PO suh tht L m (S/G)=K EE6226 Disrete Event Dynmi Systems 23

24 Exmple d Σ = {,b,,d} Σ o = {} K ={, b} A, b b b S G K L m (S/G)=K? EE6226 Disrete Event Dynmi Systems 24

25 Diffiulty of Synthesis Given plnt G nd speifition SPEC, let O(G,SPEC):={K L m (G) L m (SPEC) K is ontrollble nd observble} Unfortuntely, there is no supreml element in O(G,SPEC). EE6226 Disrete Event Dynmi Systems 25

26 Solution 1: A New Supervisory Control Problem Given G, suppose we hve A E L(G) nd Σ=Σ o Σ. To synthesize supervisor S under prtil observtion suh tht A L(S/G) E (*) Let O(A) := {K A K is losed nd observble w.r.t. G nd P} Let C(E) := {K E K is losed nd ontrollble w.r.t. G} Theorem (Feng Lin) Assume A. The (*) problem hs solution S iff inf O(A) sup C(E) EE6226 Disrete Event Dynmi Systems 26

27 Solution 2 : The Conept of Normlity Given N M Σ *, we sy N is (M,P)-norml if N = M P -1 P(N) In prtiulr, tke N=M P -1 (K) for ny K Σ o*. Then N is (M,P)-norml. ( s 1,s 2 M) (s 1,s 2 ) ker P P(s 1 )=P(s 2 ) N/ker P M/ker P M EE6226 Disrete Event Dynmi Systems 27

28 Properties of Normlity Let N(E ; M) := {N E N is (M,P)-norml} for some E Σ * The poset (N(E ; M), ) is omplete lttie The union of (M,P)-norml sublnguges is norml (intuitive explntion?) The intersetion of (M,P)-norml sublnguges is norml (intuitive explntion?) Lin-Brndt formul : sup N(E ; M) = E P -1 P(M E) In TCT : N = Supnorm(E,M,Null/Imge) Let E L m (G), nd N(E ; L(G)):={N E N is (L(G),P)-norml} N(E ; M) is losed under rbitrry unions, but not under intersetions EE6226 Disrete Event Dynmi Systems 28

29 Reltionship between Normlity nd Observbility Let K L m (G). Then K is (L(G), P)-norml K is observble w.r.t. G nd P Let Σ(K) := {σ Σ ( s K) sσ L(G) K} Σ(K) is the olletion of ll boundry events of K w.r.t. G K is observble w.r.t. G, P Σ(K) Σ o K is (L(G),P)-norml EE6226 Disrete Event Dynmi Systems 29

30 Supervisory Control under Normlity Given plnt G nd speifition E, let C(G,E) := {K L m (G) L m (E) K is ontrollble w.r.t. G} We define new set S(G,E) := {K Σ * K C(G,E) N(L m (E),L(G)) L m (G)-losed} S(G,E) is nonempty nd losed under rbitrry unions. sup S(G,E) exists Supervisory Control nd Observtion Problem (SCOP) to ompute proper supervisor S under prtil observtion suh tht L m (S/G) = sup S(G,E) EE6226 Disrete Event Dynmi Systems 30

31 The TCT Proedure for SCOP Given plnt G nd speifition E, let A = Supsop(E,G,Null/Imge) L m (A) = sup S(G,E) Bsed on A, we onstrut proper supervisor S under prtil observtion Why n we do tht? Beuse sup S(G,E) is ontrollble nd observble EE6226 Disrete Event Dynmi Systems 31

32 Outline Motivtion The Conept of Observbility Supervisor Synthesis under Prtil Observtion Exmple Conlusions EE6226 Disrete Event Dynmi Systems 32

33 Wrehouse Collision Control Cr 1 Reeiving Dok Cr 2 Trffi Light Trk 1 Trk 2 Trk 3 Trk 4 Sensor Dispthing Dok EE6226 Disrete Event Dynmi Systems 33

34 Plnt Model Enter Trk 1 Enter Trk 2 Enter Trk 3 Enter Trk 4 C 1 C Σ 1 = {11, 12, 13, 15}, Σ 1, = {11, 13, 15}, Σ 1,o = {11, 15} Σ 2 = {21, 22, 23, 25}, Σ 2, = {21, 23, 25}, Σ 2,o = {21, 25} EE6226 Disrete Event Dynmi Systems 34

35 Speifition To void ollision, C 1 nd C 2 n t reh the sme stte together Sttes (1,1), (2,2), (3,3) should be voided in C 1 C 2 EE6226 Disrete Event Dynmi Systems 35

36 Synthesis Proedure in TCT Crete the plnt G = Syn(C 1,C 2 ) (25 ; 40) Crete the speifition E = mutex(c 1,C 2,[(1,1),(2,2),(3,3)]) (20 ; 24) Supervisor Synthesis K = Supsop(E,G,[12,13,22,23]) (16 ; 16) EE6226 Disrete Event Dynmi Systems 36

37 Trnsition Struture of K EE6226 Disrete Event Dynmi Systems 37

38 A Proper Supervisor S under Prtil Observtion 11 12, ,23 22,23 K = L m (S/G) ,13 EE6226 Disrete Event Dynmi Systems 38

39 Some Ft Perform the following TCT opertion W = Condt(G,K) Only events 11 nd 21 re required to be disbled. Therefore, we only need one trffi light t Trk 1. EE6226 Disrete Event Dynmi Systems 39

40 A Slight Modifition Cr 1 Reeiving Dok Cr 2 Trffi Light Trk 1 Trk 2 Trk 3 Trk 4 Sensor Dispthing Dok Σ 1,o = {11, 15} Σ 2,o = {21, 25} Σ 1,o = {11, 13} Σ 2,o = {21, 23} EE6226 Disrete Event Dynmi Systems 40

41 Synthesis Result Crete the plnt G = Syn(C 1,C 2 ) (25 ; 40) Crete the speifition E = Mutex(C 1,C 2,[(1,1),(2,2),(3,3)]) (20 ; 24) Supervisor Synthesis K = Supsop(E,G,[12,15,22,25]) (empty) Explin intuitively why this n hppen (homework) EE6226 Disrete Event Dynmi Systems 41

42 Conlusions Prtil observtion is importnt for implementtion. A supervisor n mke move only bsed on observtions. The urrent observbility is not losed under set union. Thus, there is no supreml observble sublnguge (unfortuntely). Normlity is losed under set union. Thus, the supreml norml sublnguge exists. But the onept of normlity is too onservtive. EE6226 Disrete Event Dynmi Systems 42