EVENT-DRIVEN CONTROL AND OPTIMIZATION:
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1 EVENT-DRIVEN CONTROL AND OPTIMIZATION: WHERE LESS IS OFTEN MORE 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 OUTLINE Reasons for EVENT-DRIVEN Conrol and Opmzaon EVENT-DRIVEN Conrol n Dsrbued Sysems EVENT-DRIVEN Conrol n Managng Uncerany EVENT-DRIVEN Sensvy Analyss Chrsos G. Cassandras CODES Lab. - Boson Unversy
3 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
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 TIME-DRIVEN v EVENT-DRIVEN SYSTEMS TIME-DRIVEN SYSTEM STATES STATE SPACE: X DYNAMICS: f TIME EVENT-DRIVEN SYSTEM STATES s 4 s 3 s 2 s TIME STATE SPACE: X s s s s DYNAMICS: ' f e e e 2 e 3 e 4 e 5 EVENTS Chrsos G. Cassandras CODES Lab. - Boson Unversy
7 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
8 SYNCHRONOUS v ASYNCHRONOUS COMPUTATION y y + y Asynchronous evens 2 TIME 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
9 SELECTED REFERENCES - EVENT-DRIVEN CONTROL - 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
10 EVENT-DRIVEN CONTROL IN DISTRIBUTED SYSTEMS
11 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 ???????? 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
12 OPTIMAL COVERAGE IN A MAZE hp://codescolor.bu.edu/coverage Chrsos G. Cassandras Zhong Zhong and and Cassandras Cassandras CODES Lab. - Boson Unversy
13 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 ??????? 4 2?? Chrsos G. Cassandras CODES Lab. - Boson Unversy
14 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 Perceved even densy Chrsos G. Cassandras CODES Lab. - Boson Unversy
15 CONTINUED DISTRIBUTED COOPERATIVE SCHEME Se d p R s s H N N 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 s H s s Desred dsplacemen = V Cassandras and L 2005 Zhong and Cassandras 20 Cassandras and L 2005 Zhong and Cassandras 20
16 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 replaced by node neghborhood Dscreze p usng a local grd Chrsos G. Cassandras CODES Lab. - Boson Unversy d d s d p p R s H N s H s s
17 DISTRIBUTED COOPERATIVE OPTIMIZATION N sysem componens processors agens vehcles nodes one common objecve: s mn H s s s s.. N N consrans on each s mn H s s s.. s N consrans on s mn H s s N s N s.. consrans on s N Chrsos G. Cassandras CODES Lab. - Boson Unversy
18 DISTRIBUTED COOPERATIVE OPTIMIZATION Conrollable sae s = n s s 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
19 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
20 ASYNCHRONOUS COOPERATION 2 3 Nodes no synchronzed delayed nformaon used Updae frequency for each node s bounded + echncal condons Chrsos G. Cassandras s s d s converges Berseas Berseas and and Tssls Tssls CODES Lab. - Boson Unversy
21 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
22 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: j j s j j j Mos recen value s j j j j j d j j j Lnear predcon Chrsos G. Cassandras CODES Lab. - Boson Unversy
23 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
24 WHEN SHOULD A NODE COMMUNICATE? Compare ERROR FUNCTION g j o THRESHOLD Node communcaes s sae o node j only when deecs ha s rue sae devaes from j esmae of so ha g j j j Even-Drven Conrol Chrsos G. Cassandras CODES Lab. - Boson Unversy
25 CONVERGENCE Asynchronous dsrbued sae updae process a each : s s d s Esmaes of oher nodes K d s f evaluaed by node THEOREM: Under ceran condons here es posve consans α and K δ such ha lm H s 0 sends oherwse updae INTERPRETATION: Even-drven cooperaon achevable wh mnmal communcaon requremens energy savngs Chrsos G. Cassandras Zhong Zhong and and Cassandras Cassandras IEEE IEEE TAC TAC CODES Lab. - Boson Unversy
26 COONVERGENCE WHEN DELAYS ARE PRESENT g j Error funcon rajecory wh NO DELAY 0 j 0 j j 2 j 3 Red curve: Blac curve: g g ~ j j DELAY Chrsos G. Cassandras 0 j 0 j j j 2 j 3 j j 2 3 j j 4 4 CODES Lab. - Boson Unversy
27 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 Zhong and and Cassandras Cassandras IEEE IEEE TAC TAC Chrsos G. Cassandras CODES Lab. - Boson Unversy
28 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
29 DEMO: OPTIMAL DISTRIBUTED DEPLOYMENT WITH OBSTACLES SIMULATED AND REAL Chrsos G. Cassandras CODES Lab. - Boson Unversy
30 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
31 EVENT-DRIVEN CONTROL IN MANAGING UNCERTAINTY
32 UNCERTAINTY: CONTRAST TWO APPROACHES ESTIMATE-AND-PLAN VS HEDGE-AND-REACT Decsons planned ahead Need accurae sochasc models Curse of dmensonaly Delay decsons unl las possble nsan No dealed sochasc model Smpler op. problems - Dynamc Programmng DP - Marov Decson Processes MDP - Recedng Horzon Conrol RHC - Model Predcve Conrol MPC Chrsos G. Cassandras CODES Lab. - Boson Unversy
33 UNCERTAINTY: CONTRAST TWO APPROACHES ESTIMATE-AND-PLAN VS HEDGE-AND-REACT Decson Tme Decson Tme Chrsos G. Cassandras CODES Lab. - Boson Unversy
34 TIME-DRIVEN v EVENT-DRIVEN RHC TIME-DRIVEN: mus be small Compuaonally nensve EVENT-DRIVEN: Compuaonal nensy depends on even frequency NEXT EVENT NEXT EVENT NEXT EVENT Chrsos G. Cassandras CODES Lab. - Boson Unversy
35 COOPERATIVE RECEDING HORIZON CRH CONTROL: MAIN IDEA Do no aemp o assgn nodes o arges Cooperavely seer nodes owards PLANNING hgh epeced reward regons HORIZON H Repea process when even occurs Worry abou fnal node-arge assgnmen a he las possble nsan ACTION HORIZON h u Turns ou nodes converge o arges on her own! Solve opmzaon problem by selecng all u o mamze oal epeced rewards over H u 2 u 3 REWARD-MAXIMIZATION MISSION Chrsos G. Cassandras CODES Lab. - Boson Unversy
36 TARGET ASSIGNMENT MAIN IDEA IN CRH APPROACH: Replace comple Dscree Sochasc Opmzaon problem by a sequence of smpler Connuous Opmzaon problems Solve each new problem whenever a PREDEFINED EVENT occurs e.g. some node ges o some arge or when a RANDOM EVENT or a TIMEOUT occurs Bu how do we guaranee ha nodes ulmaely head for he desred DISCRETE TARGET POINTS? Chrsos G. Cassandras CODES Lab. - Boson Unversy
37 STABILITY ANALYSIS TARGETS: y NODES: j DISTANCE: d j DEFINITION: Node rajecory M generaed by a conroller s saonary f here ess some such ha V j v y s for some N j M. QUESTION: Under wha condons s a CRH-generaed rajecory saonary? Targe Sze Chrsos G. Cassandras CODES Lab. - Boson Unversy
38 MAIN STABILITY RESULT Local mnma of objecve funcon J: l l l 2M... M R l L Vecor of node posons a h eraon of CRH conroller: Theorem: Suppose H mn d. l j If for all l = L = y for some = N j = M hen J J b b > 0 s a consan. j j If all local mnma concde wh arges he CRH-generaed rajecory s saonary Chrsos G. Cassandras CODES Lab. - Boson Unversy
39 MAIN STABILITY RESULT QUESTION: When do all local mnma concde wh arge pons? Node N arges If here ess a y s.. R N j j R j y y y y j j 0 2 Nodes arge 2 Nodes 2 arges L L and and Cassandras Cassandras IEEE IEEE TAC TAC Chrsos G. Cassandras CODES Lab. - Boson Unversy
40 BOSTON UNIVERSITY TEST BEDS Chrsos G. Cassandras SMARTS Kcoff Meeng CISE - CODES Lab. - Boson Unversy
41 REWARD MAXIMIZATION DEMO Chrsos G. Cassandras CODES Lab. - Boson Unversy
42 EVENT-DRIVEN SENSITIVITY ANALYSIS
43 REAL-TIME STOCHASTIC OPTIMIZATION GOAL: ma E[ L ] CONTROL/DECISION Parameerzed by SYSTEM PERFORMANCE E[ L ] NOISE 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
44 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 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
45 REAL-TIME STOCHASTIC OPTIMIZATION: HYBRID SYSTEMS Sample pah CONTROL/DECISION Parameerzed by HYBRID SYSTEM PERFORMANCE E[ L ] NOISE 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
46 PERFORMANCE OPTIMIZATION AND IPA Performance merc objecve funcon: ; 0 T EL ; 0 T J L N 0 L d IPA goal: dj ; 0 T - Oban unbased esmaes of normally d dl - Then: n n n n d dl d NOTATION: Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy
47 HYBRID AUTOMATA G h Q X E U f Inv guard 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 : dscree sae ranson funcon : Q X E Q Inv: se defnng an nvaran condon doman Inv Q X guard: se defnng a guard condon guard Q Q X : rese funcon : Q Q X E X q 0 : nal dscree sae 0 : nal connuous sae Chrsos G. Cassandras CODES Lab. - Boson Unversy
48 HYBRID AUTOMATA Unrelable machne wh meous T ' 0 IDLE BUSY ' 0 DOWN u 0 0 [ T ] Rese ' 0 ' 0 K Invaran Guard : physcal sae of par n machne : cloc : START : STOP : REPAIR Chrsos G. Cassandras CODES Lab. - Boson Unversy
49 THE IPA CALCULUS
50 IPA: THREE FUNDAMENTAL EQUATIONS. Connuy a evens: Tae d/d f f ' ] [ ' ' If no connuy use rese condon d q q d ' Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy Sysem dynamcs over + ]: f NOTATION:
51 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
52 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
53 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 Cassandras e al Europ. J. Conrol 200
54 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
55 IPA PROPERTIES 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]
56 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
57 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
58 Chrsos G. Cassandras CODES Lab. - Boson Unversy SOLVING PROBLEMS WITH LINEAR TIME-DRIVEN DYNAMICS ] s.. mn 0 K u u u b a d u L K Common o parameerze conrols usng bass funcons l : L l l l u 0
59 Chrsos G. Cassandras CODES Lab. - Boson Unversy Recall IPA equaons: f f ' ] [ ' ' du u f du u f v dv e v f e ' 0 g g f g or SOLVING PROBLEMS WITH LINEAR TIME-DRIVEN DYNAMICS When endogenous even [g = 0] occurs a : l l L l l l L l l l l b b a a l a l a l e a b e
60 CYBER-PHYSICAL SYSTEMS INTERNET CYBER Daa collecon: relavely easy PHYSICAL Conrol: a challenge Chrsos G. Cassandras CISE - CODES Lab. - Boson Unversy
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