On the Maximally-Permissive Range Control Problem in Partially-Observed Discrete Event Systems

Transcription

1 On the Mximlly-Permissie Rnge Control Prolem in Prtilly-Osered Disrete Eent Systems Xing Yin nd Stéphne Lfortune EECS Deprtment, Uniersity of Mihign 55th IEEE CDC, De 2-4, 206, Ls Vegs, USA X.Yin & S.Lfortune (UMih) CDC 206 /4

2 Introdution S(s) Plnt G S: E o Γ Superisor s P P(s) E = E E u = E o E uo Superisor: S: E o 2 E ; Disle eents in E sed on its osertions Closed-loop Behior: L(S/G) X.Yin & S.Lfortune (UMih) CDC 206 2/4

3 Disrete Eent Systems: Logil Properties Sfety: Regulr sulnguge L L(G) Mximl Permissieness: Optimlity riterion is set inlusion. Only disle n eent if solutely neessry to gurntee sfety When Σ Σ o, there exists unique mximlly-permissie superisor, i.e., the supreml ontrollle & norml superisor [Lin nd Wonhm, 988] Mximl superisor is NOT unique in generl X.Yin & S.Lfortune (UMih) CDC 206 /4

4 Non-Uniqueness of Lolly Mximl Solutions Non-Uniqueness of Lolly Mximl Solutions 0 o 2 o 4 B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 4/4

5 Non-Uniqueness of Lolly Mximl Solutions Non-Uniqueness of Lolly Mximl Solutions 0 o 2 o 4 B E = Two inomprle solutions, E o = *o+ o o o 4 X.Yin & S.Lfortune (UMih) CDC 206 4/4

6 Non-Uniqueness of Lolly Mximl Solutions Non-Uniqueness of Lolly Mximl Solutions 0 o 2 o 4 B E = Two inomprle solutions, E o = *o+ o o o 4 Computtion of Solutions: [Cho & Mrus, 989], [Ben Hdj-Aloune, Lfortune & Lin, 996], [Tki & Ushio, 200], [Komend & n Shuppen, 2005], [Ci, Zhng & Wonhm, 205], [Yin & Lfortune, 206] X.Yin & S.Lfortune (UMih) CDC 206 4/4

7 Non-Uniqueness of Lolly Mximl Solutions Non-Uniqueness of Lolly Mximl Solutions 0 o 2 o 4 B E = Two inomprle solutions, E o = *o+ o o o 4 Computtion of Solutions: [Cho & Mrus, 989], [Ben Hdj-Aloune, Lfortune & Lin, 996], [Tki & Ushio, 200], [Komend & n Shuppen, 2005], [Ci, Zhng & Wonhm, 205], [Yin & Lfortune, 206] How to hoose mong lolly mximl solutions? X.Yin & S.Lfortune (UMih) CDC 206 4/4

8 Rnge Control Prolem Using Lower Bound Lnguge s Criterion Forml Gurntees: Sfety + Mximlity + Minimlly Required Behior X.Yin & S.Lfortune (UMih) CDC 206 5/4

9 Rnge Control Prolem Using Lower Bound Lnguge s Criterion Forml Gurntees: Sfety + Mximlity + Minimlly Required Behior L(G) L X.Yin & S.Lfortune (UMih) CDC 206 5/4

10 Rnge Control Prolem Using Lower Bound Lnguge s Criterion Forml Gurntees: Sfety + Mximlity + Minimlly Required Behior L(G) L L r X.Yin & S.Lfortune (UMih) CDC 206 5/4

11 Rnge Control Prolem Using Lower Bound Lnguge s Criterion Forml Gurntees: Sfety + Mximlity + Minimlly Required Behior L(G) L L r Prolem (Rnge Control Prolem for Sfety nd Mximl-Permissieness) Let G e the plnt nd L r nd L e two prefix-losed lnguges. Find superisor S: E o Γ suh tht C. L r L S/G L C2. For ny S stisfying C, we he L S/G L S /G X.Yin & S.Lfortune (UMih) CDC 206 5/4

12 Rnge Control Prolem Using Lower Bound Lnguge s Criterion Forml Gurntees: Sfety + Mximlity + Minimlly Required Behior L(G) L Mx L r Mx 2 Mx Prolem (Rnge Control Prolem for Sfety nd Mximl-Permissieness) Let G e the plnt nd L r nd L e two prefix-losed lnguges. Find superisor S: E o Γ suh tht C. L r L S/G L C2. For ny S stisfying C, we he L S/G L S /G X.Yin & S.Lfortune (UMih) CDC 206 5/4

13 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

14 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

15 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

16 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ The effet of enling n eent depends on future informtion y o o o o 0 n x z o o o o 4 m B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

17 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ The effet of enling n eent depends on future informtion y o o o o 0 n x z o o o o 4 m B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

18 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ The effet of enling n eent depends on future informtion y o o o o 0 n x z o o o o 4 m B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

19 Diffiulties in the Rnge Control Prolem Enling less now my result in lrger future ehior 5 o o 4 B E =, E o = *o+ The effet of enling n eent depends on future informtion y o o o o 0 n x z o o o o 4 m B E =, E o = *o+ X.Yin & S.Lfortune (UMih) CDC 206 6/4

20 Solution Oeriew System Model G Sfety Speifition H s.t. L H = L Lower Bound Speifition R s.t. L R = L r All Inlusie Controller (AIC) Strit Su-Automton Control Simultion Reltion (CSR) Mximlly-Permissie Sfe Superisor Ahieing the Miniml Required Behior X.Yin & S.Lfortune (UMih) CDC 206 7/4

21 Solution Oeriew System Model G Sfety Speifition H s.t. L H = L Lower Bound Speifition R s.t. L R = L r All Inlusie Controller (AIC) Strit Su-Automton Control Simultion Reltion (CSR) Mximlly-Permissie Sfe Superisor Ahieing the Miniml Required Behior Assumptions A There exists superisor hieing L R (W.l.o.g.; infiml losed ontrollle & osererle super-lnguge [Rudie & Wonhm, 992]) A2 R H G (W.l.o.g.; [Cho & Mrus, 989]) X.Yin & S.Lfortune (UMih) CDC 206 7/4

22 Biprtite Trnsition System Definition. (BTS). [Yin & Lfortune, IEEE TAC, 206] A iprtite trnsition system T w.r.t. G is 7-tuple T = (Q Y, Q Z, h YZ, h ZY, E, Γ, y 0 ) where Q Y I is the set of Y-sttes; Q Z I Γ is the set of Z-sttes so tht z = (I z, Γ z ); h YZ : Q Y Γ Q Z represents the unoserle reh; h ZY : Q Z E Q Y represents the osertion trnsition; 5 w w w , * + * + *, w+,5, *, w+ 5,6, 6 G: E =, w, E o = *,, + A BTS X.Yin & S.Lfortune (UMih) CDC 206 8/4

23 All Inlusie Controller Definition. (AIC). The All Inlusie Controller AIC(G) = (Q AIC Y, Q AIC Z, h AIC YZ, h AIC ZY, E, Γ, y 0 ), is defined s the lrgest BTS suh. For ny y Q Y AIC, there exists t lest one ontrol deision 2. For ny z Q Z AIC, we he 2.. ll fesile oserle eents re defined 2.2. I z only ontins legl sttes 5 w w w G , * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, X.Yin & S.Lfortune (UMih) CDC 206 9/4

24 Control Simultion Reltion Definition. (Control Simultion Reltion) Let T nd T 2 e two BTSs. A reltion Φ = Φ Y Φ Z Q Y T Q Y T 2 (QZ T Q Z T 2 ) is sid to e ontrol simultion reltion from T to T 2 if the following onditions hold:. y 0, y 0 Φ Y ; 2. For eery y, y 2 γ γ 2 Φ Y, we he tht: for ny y z in T, there exists y 2 z 2 suh tht z, z 2 Φ Z nd γ γ 2.. For eery z, z 2 Φ z, we he tht: for ny z σ y in T, there exists z 2 σ y2 suh tht y, y 2 Φ Y. X.Yin & S.Lfortune (UMih) CDC 206 0/4

25 Control Simultion Reltion Definition. (Control Simultion Reltion) Let T nd T 2 e two BTSs. A reltion Φ = Φ Y Φ Z Q Y T Q Y T 2 (QZ T Q Z T 2 ) is sid to e ontrol simultion reltion from T to T 2 if the following onditions hold:. y 0, y 0 Φ Y ; 2. For eery y, y 2 γ γ 2 Φ Y, we he tht: for ny y z in T, there exists y 2 z 2 suh tht z, z 2 Φ Z nd γ γ 2.. For eery z, z 2 Φ z, we he tht: for ny z σ y in T, there exists z 2 σ y2 suh tht y, y 2 Φ Y. There exists unique mximl CSR from T to T 2 Φ T, T 2 = lim k F k ( Q Y T Q Y T 2 QZ T Q Z T 2 ) At most Q Y T Q Y T 2 + Q Z T Q Z T 2 itertions X.Yin & S.Lfortune (UMih) CDC 206 0/4

26 Synthesis Steps Synthesis Steps:. Construt AIC(G) tht ontins ll sfe superisors 2. Construt BTS T R tht relizes the superisor hieing L r. Compute the mximl CSR Φ from T R to AIC(G) 4. Construt BTS T tht relizes mximl superisor hieing L r y using T R, AIC(G) nd Φ - For eh Y-stte, enle s mny eents s possile without iolting Φ X.Yin & S.Lfortune (UMih) CDC 206 /4

27 Exmple G 2 5 w w w R X.Yin & S.Lfortune (UMih) CDC 206 2/4

28 Exmple G 2 5 w w w R Step : Construt AIC(G), * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, X.Yin & S.Lfortune (UMih) CDC 206 2/4

29 Exmple G 2 5 w w w R Step 2: Construt the BTS T R, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, * +, * +,{} T R X.Yin & S.Lfortune (UMih) CDC 206 2/4

30 Exmple G 2 5 w w w R, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + Φ, * + * + 5,6, T R Step : Compute the mximl CSR Φ,{} X.Yin & S.Lfortune (UMih) CDC 206 2/4

31 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T X.Yin & S.Lfortune (UMih) CDC 206 2/4

32 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T X.Yin & S.Lfortune (UMih) CDC 206 2/4

33 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T * +, * + X.Yin & S.Lfortune (UMih) CDC 206 2/4

34 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T * +, * + X.Yin & S.Lfortune (UMih) CDC 206 2/4

35 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T * +, * + X.Yin & S.Lfortune (UMih) CDC 206 2/4

36 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T, * + * + *, w+,5, *, w+ X.Yin & S.Lfortune (UMih) CDC 206 2/4

37 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T * +, * + *, w+,5, *, w+ 6 X.Yin & S.Lfortune (UMih) CDC 206 2/4

38 Exmple, * +,5, *, w+ *, w+ * +,2,,,,5, 6 AIC(G),4,5,,4,4, * + 5,6, Φ * +, * +,{} T R Step 4: Construt mximl BTS T * +, * + *, w+,5, *, w+ 5,6, 6 X.Yin & S.Lfortune (UMih) CDC 206 2/4

39 Synthesis Steps Reisited Synthesis Steps:. Construt AIC(G) tht ontins ll sfe superisors 2. Construt BTS T R tht relizes the superisor hieing L r. Compute the mximl CSR Φ from T R to AIC(G) 4. Construt BTS T tht relizes mximl superisor hieing L r y using T R, AIC(G) nd Φ - For eh Y-stte, enle s mny eents s possile without iolting Φ X.Yin & S.Lfortune (UMih) CDC 206 /4

40 Synthesis Steps Reisited Synthesis Steps:. Construt AIC(G) tht ontins ll sfe superisors 2. Construt BTS T R tht relizes the superisor hieing L r. Compute the mximl CSR Φ from T R to AIC(G) 4. Construt BTS T tht relizes mximl superisor hieing L r y using T R, AIC(G) nd Φ - For eh Y-stte, enle s mny eents s possile without iolting Φ The strtegy is not ompletely greedy. Future informtion hs lredy een pre-proessed nd onsidered! X.Yin & S.Lfortune (UMih) CDC 206 /4

41 Synthesis Steps Reisited Synthesis Steps:. Construt AIC(G) tht ontins ll sfe superisors 2. Construt BTS T R tht relizes the superisor hieing L r. Compute the mximl CSR Φ from T R to AIC(G) 4. Construt BTS T tht relizes mximl superisor hieing L r y using T R, AIC(G) nd Φ - For eh Y-stte, enle s mny eents s possile without iolting Φ The strtegy is not ompletely greedy. Future informtion hs lredy een pre-proessed nd onsidered! X.Yin & S.Lfortune (UMih) CDC 206 /4

42 Synthesis Steps Reisited Synthesis Steps:. Construt AIC(G) tht ontins ll sfe superisors 2. Construt BTS T R tht relizes the superisor hieing L r. Compute the mximl CSR Φ from T R to AIC(G) 4. Construt BTS T tht relizes mximl superisor hieing L r y using T R, AIC(G) nd Φ - For eh Y-stte, enle s mny eents s possile without iolting Φ Complexity O 2 2 X +2 E = O AIC 2 Mximl CSR Φ + O AIC Depth First Serh X.Yin & S.Lfortune (UMih) CDC 206 /4

43 Summry Contriution Sole the Mximlly-Permissie Rnge Control Prolem New pproh using AIC nd CSR Gurntees oth mximl-permissieness nd miniml requirement Softwre DPO-SYNT: Future Work Consider non-lokingness in ddition to sfety X.Yin & S.Lfortune (UMih) CDC 206 4/4