EVENT-DRIVEN CONTROL, COMMUNICATION, AND OPTIMIZATION

Transcription

1 EVENT-DRIVEN CONTROL COMMUNICATION AND OPTIMIZATION C. G. Cassandras Dvson of Sysems Engneerng and Dep. of Elecrcal and Compuer Engneerng and Cener for Informaon and Sysems Engneerng Boson Unversy Chrsos G. Cassandras CODES Lab. - Boson Unversy

2 TIME-DRIVEN v EVENT-DRIVEN CONTROL REFERENCE - ERROR CONTROLLER INPUT PLANT OUTPUT MEASURED OUTPUT SENSOR EVENT-DRIVEN CONTROL: Ac only when needed or on TIMEOUT - no based on a cloc REFERENCE - ERROR CONTROLLER INPUT PLANT OUTPUT MEASURED OUTPUT EVENT: gstate 0 SENSOR Chrsos G. Cassandras CODES Lab. - Boson Unversy

3 OUTLINE Reasons for EVENT-DRIVEN Conrol Communcaon and Opmzaon EVENT-DRIVEN Conrol n Dsrbued Wreless Sysems EVENT-DRIVEN Sensvy Analyss for Hybrd Sysems Chrsos G. Cassandras CODES Lab. - Boson Unversy

4 REASONS FOR EVENT-DRIVEN MODELS CONTROL OPTIMIZATION Many sysems are naurally Dscree Even Sysems DES e.g. Inerne all sae ransons are even-drven Mos of he res are Hybrd Sysems HS some sae ransons are even-drven Many sysems are dsrbued componens nerac asynchronously hrough evens Tme-drven samplng nherenly neffcen open loop samplng Chrsos G. Cassandras CODES Lab. - Boson Unversy

5 REASONS FOR EVENT-DRIVEN MODELS CONTROL OPTIMIZATION Many sysems are sochasc acons needed n response o random evens Even-drven mehods provde sgnfcan advanages n compuaon and esmaon qualy Sysem performance s ofen more sensve o even-drven componens han o me-drven componens Many sysems are wrelessly newored energy consraned me-drven communcaon consumes sgnfcan energy UNNECESSARILY! Chrsos G. Cassandras CODES Lab. - Boson Unversy

6 CYBER-PHYSICAL SYSTEMS INTERNET CYBER Daa collecon: relavely easy PHYSICAL Conrol: a challenge Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

7 TIME-DRIVEN v EVENT-DRIVEN SYSTEMS TIME-DRIVEN SYSTEM STATES STATE SPACE: X R DYNAMICS: f TIME EVENT-DRIVEN SYSTEM STATES s 4 s 3 s 2 s TIME STATE SPACE: { } X s s s s DYNAMICS: e ' f e e 2 e 3 e 4 e 5 EVENTS Chrsos G. Cassandras CODES Lab. - Boson Unversy

8 SYNCHRONOUS v ASYNCHRONOUS BEHAVIOR Indsngushable evens INCREASING TIME GRANULARITY Wased cloc cs More wased cloc cs Chrsos G. Cassandras Even more wased cloc cs CODES Lab. - Boson Unversy

9 SYNCHRONOUS v ASYNCHRONOUS COMPUTATION y TIME y y 2 TIME Tme-drven synchronous mplemenaon: - Sum repeaedly evaluaed unnecessarly - When evaluaon s acually needed s done a he wrong mes! Chrsos G. Cassandras CODES Lab. - Boson Unversy

10 SELECTED REFERENCES - EVENT-DRIVEN CONTROL COMMUNICATION ESTIMATION OPTIMIZATION - Asrom K.J. and B. M. Bernhardsson Comparson of Remann and Lebesgue samplng for frs order sochasc sysems Proc. 4s Conf. Decson and Conrol pp T. Shma S. Rasmussen and P. Chandler UAV Team Decson and Conrol usng Effcen Collaborave Esmaon ASME J. of Dynamc Sysems Measuremen and Conrol vol. 29 no. 5 pp Heemels W. P. M. H. J. H. Sandee and P. P. J. van den Bosch Analyss of even-drven conrollers for lnear sysems Inl. J. Conrol 8 pp P. Tabuada Even-rggered real-me schedulng of sablzng conrol ass IEEE Trans. Auom. Conrol vol. 52 pp J. H. Sandee W. P. M. H. Heemels S. B. F. Hulsenboom and P. P. J. van den Bosch Analyss and epermenal valdaon of a sensor-based even-drven conroller Proc. Amercan Conrol Conf. pp J. Lunze and D. Lehmann A sae-feedbac approach o even-based conrol Auomaca 46 pp P. Wan and M. D. Lemmon Even rggered dsrbued opmzaon n sensor newors Proc. of 8h ACM/IEEE Inl. Conf. on Informaon Processng n Sensor Newors Zhong M. and Cassandras C.G. Asynchronous Dsrbued Opmzaon wh Even-Drven Communcaon IEEE Trans. on Auomac Conrol AC-55 2 pp Chrsos G. Cassandras CODES Lab. - Boson Unversy

11 EVENT-DRIVEN CONTROL IN DISTRIBUTED USUALLY WIRELESS SYSTEMS

12 MOTIVATIONAL PROBLEM: COVERAGE CONTROL Deploy sensors o mamze even deecon probably unnown even locaons even sources may be moble sensors may be moble R Hz/m ??? Ω???? 4 2 0? Meguerdchan e al IEEE INFOCOM 200 Cores e al IEEE Trans. on Robocs and Auomaon 2004 Cassandras and L Eur. J. of Conrol 2005 Gangul e al Amercan Conrol Conf Hussen and Spanovc Amercan Conrol Conf Hoayem e al Amercan Conrol Conf Perceved even densy daa sources over gven regon msson space Chrsos G. Cassandras CODES Lab. - Boson Unversy

13 OPTIMAL COVERAGE IN A MAZE hp:// Zhong and Cassandras 2008 Chrsos G. Cassandras CODES Lab. - Boson Unversy

14 COVERAGE: PROBLEM FORMULATION N moble sensors each locaed a s R 2 Daa source a ems sgnal wh energy E Sgnal observed by sensor node a s SENSING MODEL: p s P[Deeced by A s ] A daa source ems a Sensng aenuaon: p s monooncally decreasng n d - s R Hz/ 50 m ?????? Ω? Chrsos G. Cassandras CODES Lab. - Boson Unversy

15 COVERAGE: PROBLEM FORMULATION Jon deecon prob. assumng sensor ndependence s [s s N ] : node locaons P s N [ p ] s Even sensng probably OBJECTIVE: Deermne locaons s [s s N ] o mamze oal Deecon Probably: ma s R P s d Ω Perceved even densy Chrsos G. Cassandras CODES Lab. - Boson Unversy

16 CONTINUED DISTRIBUTED COOPERATIVE SCHEME Se [ ] d p R s s H N N Ω Chrsos G. Cassandras CODES Lab. - Boson Unversy Mamze Hs s N by forcng nodes o move usng graden nformaon: [ ] d d s d p p R s H N Ω s H s s β Desred dsplacemen V Cassandras and L EJC 2005 Zhong and Cassandras IEEE TAC 20

17 CONTINUED DISTRIBUTED COOPERATIVE SCHEME has o be auonomously evaluaed by each node so as o deermne how o move o ne poson: CONTINUED Use runcaed p Ω replaced by node neghborhood Ω Dscreze p usng a local grd Chrsos G. Cassandras CODES Lab. - Boson Unversy [ ] d d s d p p R s H N Ω s H s s β

18 DISTRIBUTED COOPERATIVE OPTIMIZATION N sysem componens processors agens vehcles nodes one common objecve: mn H s s s s s.. N N consrans on each s mn s s.. H s s N consrans on s mn s N H s s N s.. consrans on s N Chrsos G. Cassandras CODES Lab. - Boson Unversy

19 DISTRIBUTED COOPERATIVE OPTIMIZATION Conrollable sae s n s s α d s Sep Sze mn H s s s Chrsos G. Cassandras N s.. consrans on s Updae Drecon usually d s H s requres nowledge of all s s N Iner-node communcaon CODES Lab. - Boson Unversy

20 SYNCHRONIZED TIME-DRIVEN COOPERATION COMMUNICATE UPDATE 2 3 Drawbacs: Ecessve communcaon crcal n wreless sengs! Faser nodes have o wa for slower ones Cloc synchronzaon nfeasble Bandwdh lmaons Secury rss Chrsos G. Cassandras CODES Lab. - Boson Unversy

21 ASYNCHRONOUS COOPERATION 2 3 Nodes no synchronzed delayed nformaon used Updae frequency for each node s bounded echncal condons Chrsos G. Cassandras s s α d s converges Berseas and Tssls 997 CODES Lab. - Boson Unversy

22 ASYNCHRONOUS EVENT-DRIVEN COOPERATION UPDATE COMMUNICATE 2 3 UPDATE a : locally deermned arbrary possbly perodc COMMUNICATE from : only when absoluely necessary Chrsos G. Cassandras CODES Lab. - Boson Unversy

23 WHEN SHOULD A NODE COMMUNICATE? Node sae a any me : s Node sae a : s AT UPDATE TIME : s j : node j sae esmaed by node Esmae eamples: s j j j Mos recen value j j s j j j j α d j j j j Lnear predcon Chrsos G. Cassandras CODES Lab. - Boson Unversy

24 WHEN SHOULD A NODE COMMUNICATE? AT ANY TIME : j : node sae esmaed by node j If node nows how j esmaes s sae hen can evaluae j Node uses s own rue sae he esmae ha j uses j and evaluaes an ERROR FUNCTION j g Error Funcon eamples: j j 2 Chrsos G. Cassandras CODES Lab. - Boson Unversy

25 WHEN SHOULD A NODE COMMUNICATE? Compare ERROR FUNCTION g j o THRESHOLD δ Node communcaes s sae o node j only when deecs ha s rue sae devaes from j esmae of so ha g j δ j j δ Even-Drven Conrol Chrsos G. Cassandras CODES Lab. - Boson Unversy

26 CONVERGENCE Asynchronous dsrbued sae updae process a each : s s α d s Esmaes of oher nodes Kδ d s δ δ f evaluaed by node THEOREM: Under ceran condons here es posve consans α and K δ such ha lm H s sends updae oherwse 0 INTERPRETATION: Even-drven cooperaon achevable wh mnmal communcaon requremens energy savngs Chrsos G. Cassandras Zhong and Cassandras IEEE TAC 200 CODES Lab. - Boson Unversy

27 COONVERGENCE WHEN DELAYS ARE PRESENT j g δ Error funcon rajecory wh NO DELAY 0 j 0 j j 2 j 3 Red curve: Blac curve: j j g g ~ DELAY δ Chrsos G. Cassandras 0 j 0 j j j 2 σ j 3 j j σ 2 σ 3 j j 4 σ 4 CODES Lab. - Boson Unversy

28 COONVERGENCE WHEN DELAYS ARE PRESENT Add a boundedness assumpon: ASSUMPTION: There ess a non-negave neger D such ha f a message s sen before -D from node o node j wll be receved before. INTERPRETATION: a mos D sae updae evens can occur beween a node sendng a message and all desnaon nodes recevng hs message. THEOREM: Under ceran condons here es posve consans α and K δ such ha lm H s 0 NOTE: The requremens on α and K δ depend on D and hey are gher. Zhong and Cassandras IEEE TAC 200 Chrsos G. Cassandras CODES Lab. - Boson Unversy

29 SYNCHRONOUS v ASYNCHRONOUS OPTIMAL COVERAGE PERFORMANCE Energy savngs Eended lfeme SYNCHRONOUS v ASYNCHRONOUS: No. of communcaon evens for a deploymen problem wh obsacles Chrsos G. Cassandras SYNCHRONOUS v ASYNCHRONOUS: Achevng opmaly n a problem wh obsacles CODES Lab. - Boson Unversy

30 DEMO: OPTIMAL DISTRIBUTED DEPLOYMENT WITH OBSTACLES SIMULATED AND REAL Chrsos G. Cassandras CODES Lab. - Boson Unversy

31 DEMO: REACTING TO EVENT DETECTION Imporan o noe: There s no eernal conrol causng hs behavor. Algorhm ncludes racng funconaly auomacally Chrsos G. Cassandras CODES Lab. - Boson Unversy

32 BOSTON UNIVERSITY TEST BEDS Chrsos G. Cassandras SMARTS Kcoff Meeng CISE - CODES Lab. - Boson Unversy

33 EVENT-DRIVEN SENSITIVITY ANALYSIS

34 REAL-TIME STOCHASTIC OPTIMIZATION GOAL: ma E[ L ] Θ CONTROL/DECISION Parameerzed by SYSTEM PERFORMANCE E[ L ] NOISE η L n n n n L GRADIENT ESTIMATOR L DIFFICULTIES: - E[L] NOT avalable n closed form - L no easy o evaluae - L may no be a good esmae of E[ L ] Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

35 REAL-TIME STOCHASTIC OPTIMIZATION FOR DES: INFINITESIMAL PERTURBATION ANALYSIS IPA Sample pah Model CONTROL/DECISION Parameerzed by Dscree Even Sysem DES PERFORMANCE E[ L ] NOISE η L n n n n IPA L Chrsos G. Cassandras L For many bu NOT all DES: - Unbased esmaors - General dsrbuons - Smple on-lne mplemenaon [Ho and Cao 99] [Glasserman 99] [Cassandras ] CISE - CODES Lab. - Boson Unversy

36 REAL-TIME STOCHASTIC OPTIMIZATION: HYBRID SYSTEMS Sample pah CONTROL/DECISION Parameerzed by HYBRID SYSTEM PERFORMANCE E[ L ] NOISE η L n n n n L IPA L A general framewor for an IPA heory n Hybrd Sysems? Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

37 PERFORMANCE OPTIMIZATION AND IPA Performance merc objecve funcon: ; 0 T E[ L ; 0 T ] J L N 0 L d IPA goal: dj ; 0 T - Oban unbased esmaes of normally d - Then: dl η n n n n d dl d NOTATION: Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

38 HYBRID AUTOMATA G h Q X E U f φ Inv guard ρ q 0 0 Q: se of dscree saes modes X: se of connuous saes normally R n E: se of evens U: se of admssble conrols f : vecor feld f : Q X U X φ : dscree sae ranson funcon φ : Q X E Q Inv: se defnng an nvaran condon doman Inv Q guard: se defnng a guard condon guard Q Q X ρ : rese funcon ρ : Q Q X E X X q 0 : nal dscree sae 0 : Chrsos G. Cassandras nal connuous sae CODES Lab. - Boson Unversy

39 HYBRID AUTOMATA Unrelable machne wh meous T ' 0 IDLE α BUSY ' 0 DOWN u [ < T ] Rese ' 0 ' 0 β K γ Invaran Guard : physcal sae of par n machne : cloc α : START β : STOP γ : REPAIR Chrsos G. Cassandras CODES Lab. - Boson Unversy

40 THE IPA CALCULUS

41 IPA: THREE FUNDAMENTAL EQUATIONS. Connuy a evens: Tae d/d : f f ' ] [ ' ' If no connuy use rese condon δ υ ρ d q q d ' Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy Sysem dynamcs over ]: f NOTATION:

42 IPA: THREE FUNDAMENTAL EQUATIONS Solve over ]: ' ' f f d d du u f du u f v dv e v f e nal condon from above Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy 2. Tae d/d of sysem dynamcs over ]: f ' ' f f d d NOTE: If here are no evens pure me-drven sysem IPA reduces o hs equaon

43 3. Ge dependng on he even ype: IPA: THREE FUNDAMENTAL EQUATIONS - Eogenous even: By defnon 0 0 g - Endogenous even: occurs when g g f g - Induced evens: y y Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

44 Ignorng reses and nduced evens: IPA: THREE FUNDAMENTAL EQUATIONS f f ' ] [ ' ' du u f du u f v dv e v f e ' 0 g g f g or Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy 2 3 ' Recall: Cassandras e al Europ. J. Conrol 200

45 IPA PROPERTIES Bac o performance merc: N d L L 0 L L NOTATION: Then: N d L L L d dl 0 Wha happens a even mes Wha happens beween even mes Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

46 IPA PROPERTIES: ROBUSTNESS THEOREM : If eher 2 holds hen dl/d depends only on nformaon avalable a even mes :. L s ndependen of over [ ] for all 2. L s only a funcon of and for all over [ ]: 0 f d d f d d L d d IMPLICATION: - Performance sensves can be obaned from nformaon lmed o even mes whch s easly observed - No need o rac sysem n beween evens! N d L L L d dl 0 Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy [Yao and Cassandras 200]

47 IPA PROPERTIES : ROBUSTNESS EXAMPLE WHERE THEOREM APPLIES smple racng problem: Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy d du f a f N w u a d g E T s.. ] [ mn 0 φ φ L NOTE: THEOREM provdes suffcen condons only. IPA sll depends on nfo. lmed o even mes f for nce funcons u e.g. b N w u a

48 IPA PROPERTIES: DECOMPOSABILITY THEOREM 2: Suppose an endogenous even occurs a wh swchng funcon g. If f 0 hen s ndependen of f. If n addon dg 0 d hen 0 IMPLICATION: Performance sensves are ofen rese o 0 sample pah can be convenenly decomposed Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

49 IPA PROPERTIES EVENTS f u w ; Evaluang However ; d ; d requres full nowledge of w and f values obvous may be ndependen of w and f values NOT obvous I ofen depends only on: - even mes - possbly f Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

50 IPA PROPERTIES In many cases: - No need for a dealed model capured by f o descrbe sae behavor n beween evens - Ths eplans why smple absracons of a comple sochasc sysem can be adequae o perform sensvy analyss and opmzaon as long as even mes are accuraely observed and local sysem behavor a hese even mes can also be measured. - Ths s rue n absracons of DES as HS snce: Common performance mercs e.g. worload sasfy THEOREM Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy

51 THE CLASSIC SCHEDULING PROBLEM: cµ-rule SERVICE RATES µ ARRIVAL PROCESSES 2 µ 2 µ N CONTROLLER: u { N} N Problem: mn u { N} T E T N 0 c d c > 0 N. cµ-rule: Always serve he non-empy queue wh hghes c µ value NOTE: cµ rule s an almos sac conrol polcy! Chrsos G. Cassandras CODES Lab. - Boson Unversy

52 OPTIMALITY OF cµ-rule Deermnsc model Smh 956 Classcal Queueng Theory: - M/G/ sysem - Co and Smh96 - Dscree me general arrvals geomercally dsrbued servce - Baras e al. 985; Buyuoc e al Dscree me servce mes wh ncreasng/decreasng falure raes Hrayama e al. 989 Flud models: - Deermnsc - Chen and Yao 993; Avram e al Flud lms heavy raffc Kngman 96; Wh 968; Harrson 968; Meghem 995 Chrsos G. Cassandras CODES Lab. - Boson Unversy

53 STOCHASTIC FLOW MODEL FOR SCHEDULING Chrsos G. Cassandras CODES Lab. - Boson Unversy ; ; 2 2 µ µ u u Sae dynamcs: oherwse ; 0 0 ; α α u u d d f n n n n n n n > 0 0 mn u u u µ µ µ µ α > 0 0 } mn{ u µ µ α [0] ; ; 2 α α 2 µ µ 2 ; u ; 2 u Capacy Consran: n n u µ ; 0 RANDOM PROCESSES

54 IPA FOR LINEAR HOLDING COSTS Sample funcon: T Q [ c c22 T 0 ] d THEOREM: If c µ > c 2 µ 2 hen Q < 0 * α µ > 0 0 cµ-rule s opmal Proof: Use IPA CALCULUS o deermne Q and show s < 0 NOTE: Resul ndependen of nflow rae process α n Unversaly of cµ-rule! Kebarghob and Cassandras J. DEDS 20 Chrsos G. Cassandras CODES Lab. - Boson Unversy

55 CONCLUSIONS See o combne TIME-DRIVEN wh EVENT-DRIVEN Conrol Communcaon and Opmzaon and eplo her relave advanages and dsadvanages EVENT-DRIVEN Conrol n Dsrbued Wreless Sysems: - Ac only when necessary when specfc evens occur EVENT-DRIVEN Sensvy Analyss for Hybrd Sysem - Sensves depend mosly on evens and are robus wh respec o nose Chrsos G. Cassandras CODES Lab. - Boson Unversy

56 THANK YOU