IN THE. C. G. Cassandras

Transcription

1 IN THE OF C. G. Cassandras Boston Unversty

2 OUTLINE COMPLEXITY FUNDAMENTAL LIMITS BARGAIN HUNTING IDEAS AND PITFALLS - SAMPLE PATH ANALYSIS - DECOMPOSITION - ABSTRACTION - SURROGATE PROBLEMS - HIGH PROBABILITY v CERTAINTY THOUGHTS ON MANAGING COMPLEXITY

3 COMPLEXITY PHYSICAL COMPLEXITY OPERATIONAL COMPLEXITY + STOCHASTIC COMPLEXITY NUMERICAL COMPLEXITY

4 THREE FUNDAMENTAL COMPLEXITY LIMITS 1/T 1/2 LIMIT NP-HARD LIMIT one order ncrease Tradeoff n estmaton between ACCURACY requres INFO. GENERALITY two orders STRATEGY and EFFICIENCY ncrease = n DECISION learnng effort SPACE of an algorthm SPACE (e.g., [Wolpert SIMULATION and Macready, LENGTH IEEE T) TEC, 1997] SPACE NO-FREE-LUNCH LIMIT

5 THREE FUNDAMENTAL COMPLEXITY LIMITS 1/T 1/2 LIMIT NP-HARD LIMIT Effect s MULTIPLICATIVE! NO-FREE-LUNCH LIMIT

6 A BARGAIN EXAMPLE USING SAMPLE PATH ANALYSIS

7 A COMPLEX SYSTEM IN [0, θ] x(t) θ x & = f ( u, t; θ ) f f UNKNOWN

8 A COMPLEX SYSTEM IN [0, θ] CONTINUED PROBLEM: Determne θ to mnmze: x(t) subject to J T ( θ ) L( x( t); θ ) dt x & = = T 0 f ( u, t; θ ) θ Random ff Random Events

9 A COMPLEX SYSTEM IN [0, θ] CONTINUED α(t) LOSS θ x(t) β(t) x& = 0 0 α ( t) β ( t) x( t) = 0, α( t) β ( t) 0 x( t) = θ, α( t) β ( t) 0 otherwse θ LOSS Increase θ to to keep keep low low WORKLOAD Decrease θ to to keep keep low low

10 A COMPLEX SYSTEM IN [0, θ] CONTINUED PROBLEM: Determne θ to trade off LOSS vs WORKLOAD: N N EL = T T dx = 1 F T [ L(θ ] = T ) (θ ) E dx EQ [ Q ] T (θ = ) T T (θ ) Exdt xdt 0 0 = 1 F = T LOSS F 1 F 2 WORKLOAD

11 SENSITIVITY ANALYSIS Try and get : LOSS Senstvty: dl T dθ WORKLOAD Senstvty: dq T dθ

12 SENSITIVITY ANALYSIS - RESULTS Φ(θ) = set of perods wth at least one overflow nterval LOSS Senstvty: dl T dθ = Φ(θ ) UNBIASED ESTIMATE dl dθ T E = d dt E [ L ] T τ k = tme between frst overflow and end of perod, k Φ(θ) WORKLOAD Senstvty: No knowledge of detaled dynamcs, stochastc characterstcs, or even model parameters dqt dθ = τ k k Φ (θ ) UNBIASED ESTIMATE F 1 F 2 WORKLOAD WORKLOAD dq dθ T E = LOSS LOSS d dt E [ Q ] T τ

13 THE MORAL Often, partal knowledge of system dynamcs s adequate to allow useful nferences from observed state trajectory data; n partcular: SENSITIVITY INFORMATION Ths bargan apples to a large (but not unversal) class of problems; otherwse, the NFL lmt gets you! x & = ( x, u, t; θ ) What about ths system wth? Smlar results, but more nfo. needed regardng model parameters f Feedback

14 DECOMPOSITION AND ABSTRACTION

15 HIERARCHICAL DECOMPOSITION??? PLANNING DISCRETE-EVENT PROCESSES FLIGHT PLAN COMMANDS, RANDOM EVENTS AIRCRAFT FLIGHT DYNAMICS PHYSICAL PROCESSES

16 HIEARARCHICAL DECOMPOSITION TIME TIME SCALE CONTINUED MODEL??? Weeks - Months PLANNING Dff. Eq s, Flows, LP Mnutes - Weeks DISCRETE-EVENT PROCESSES Automata, Petr nets, Queueng, Smulaton msec - Hours PHYSICAL PROCESSES Dff. Eq s, Detaled Smulaton

17 HYBRID CONTROL SYSTEM What exactly does that mean? DISCRETE-EVENT PROCESSES CONTROL CONTROL PHYSICAL PROCESSES

18 WHAT S A HYBRID SYSTEM? z & 1 = g1( z1, u1, t) z & 2 = g2( z2, u2, t) TIME-DRIVEN DYNAMICS TIME x 1 x 2 x 0 x 1 = f1( x0, z1, u1, t) x 1 x 2 = f2( x1, z2, u2, t) More More on on modelng modelng frameworks, frameworks, open open problems, problems, etc: etc: [Proc. [Proc. of of IEEE IEEE Specal Specal Issue Issue (Antsakls, (Antsakls, Ed.), Ed.), 2000] 2000] NEW MODE EVENT-DRIVEN DYNAMICS

19 HYBRID SYSTEMS IN COOPERATIVE CONTROL TARGET EVENT: threat sensed THREATS TIME-DRIVEN DYNAMICS BASE EVENT: nfo. communcated by team member

20 HYBRID SYSTEM IN MANUFACTURING Key questons facng manufacturng system ntegrators: How to ntegrate process control wth operatons control? How to mprove product QUALITY wthn reasonable TIME? PROCESS CONTROL Physcsts Materal Scentsts Chemcal Engneers... OPERATIONS CONTROL Industral Engneers, OR Schedulers Inventory Control...

21 HYBRID SYSTEM IN MANUFACTURING CONTINUED Throughout a manuf. process, each part s characterzed by A PHYSICAL state (e.g., sze, temperature, stran) A TEMPORAL state (e.g., total tme n system, total tme to due-date) PHYSICAL STATE Tme-drven Dynamcs NEW PHYSICAL STATE OPERATION TEMPORAL STATE Event-drven Dynamcs NEW TEMPORAL STATE

22 DECOMPOSITION

23 LESS COMPLEX MORE COMPLEX What exactly does that mean? TIME-DRIVEN SYSTEM HYBRID SYSTEM EVENT-DRIVEN SYSTEM LESS COMPLEX DECOMPOSITION

24 ABSTRACTION (AGGREGATION)

25 LESS COMPLEX MORE COMPLEX TIME-DRIVEN SYSTEM ZOOM OUT HYBRID SYSTEM EVENT-DRIVEN SYSTEM LESS COMPLEX ABSTRACTION (AGGREGATION)

26 LESS COMPLEX MORECOMPLEX TIME-DRIVEN SYSTEM HYBRID SYSTEM HYBRID SYSTEM ABSTRACTION (AGGREGATION) EVENT-DRIVEN SYSTEM DECOMPOSITION

27 WHAT IS THE RIGHT ABSTRACTION LEVEL? TOO FAR model not detaled enough JUST RIGHT good model TOO CLOSE too much undesrable detal CREDIT: W.B. Gong

28 DECOMPOSITION IN OPTIMAL CONTROL

29 OPTIMAL CONTROL PROBLEMS Physcal State, z z N Temporal State, x x N Get to desred fnal physcal state z N n mnmum tme x N, subject to N-1 swtchng events Mnmze: - devatons from N desred physcal states (z -q ) 2 x - devatons from 1 target x 2 desred tmes (x - τ ) 2

30 OPTIMAL CONTROL PROBLEMS Temporal state Tme-drven Dynamcs mn u N x x = 1 1 L ( z ( t), u ( t)) dt s.t. z & = g ( z, u, t) x +1 = f (x,u,t) Physcal state Event-drven Dynamcs

31 OPTIMAL CONTROL PROBLEMS CONTINUED mn u N = 1 x L ( z ( t), u ( t)) dt + x 1 ψ ( x ) Cost under u (t) over [x -1,x ] φ ( x, x 1) Cost of swtchng tme x mn u N [ φ ( x, x ) + ψ ( x )] 1 = 1 Let: s = x - x -1 Tme spent at th operatng mode Assume: φ x, x ) = φ ( s ) ( 1

32 HIERARCHICAL DECOMPOSITION mn u N [ φ + ] ( s ) ψ ( x ) = 1 s.t. z& = g ( z, u, t) x +1 = f (x,u,t) HIGHER LEVEL PROBLEM: mn s [ N + ] * φ ( s ) ψ ( x ) = 1 s.t. x +1 = f (x,s,t) LOWER LEVEL PROBLEMS: mnφ ( s u ) = s 0 L ( z ( t), u ( t)) dt s.t. z& = g ( z, u, t) FIXED s

33 HIERARCHICAL DECOMPOSITION f z CONTINUED f z z 0 z +1 s s +1 0 z +2 x -1 x x +1 u * ( z 0, z u, s mn 0 f θ ( z, z, s ) = f x 1 ux 2 φ ( s ( 0 f z, z, s ) ) mnφ ( z ), u, s ) + mn φ+ 1( s 1) 0 f + u 1 z 1, z, s ) ( ROUTINE OPTIMAL CONTROL PROBLEM! REALLY CHALLENGING PROBLEM! z mn N [ 0 f θ + ] ( z, z, s ) ( x ) ψ 0 f, z, s = 1 Typcal example: s.t. x +1 = f (x,u,t) x + = max( x, a + 1 ) + ( z, u 1 s )

34 HIERARCHICAL DECOMPOSITION CONTINUED Decomposton works well n ths case but we stll have to solve all LOWER-LEVEL problems and a HIGHER-LEVEL problem [Gokbayrak [Gokbayrak and and Cassandras, Cassandras, 2000] [Xu 2000] [Xu and and Antsakls, Antsakls, 2000] 2000] x 1 x 2 and there s also the ssue of selectng among many possble modes to swtch to [Bemporad [Bemporad et et al, al, 2000] 2000]

35 ABSTRACTION

36 ABSTRACTION OF A DISCRETE-EVENT SYSTEM DISCRETE-EVENT SYSTEM

37 ABSTRACTION OF A DISCRETE-EVENT SYSTEM DISCRETE-EVENT SYSTEM TIME-DRIVEN FLOW RATE DYNAMICS EVENTS HYBRID SYSTEM

38 STOCHASTIC FLOW MODELS (SFM) CONTROLLER θ α(t) α(t) p(x) γ(t) x(t) β(t) α(t), α(t), β(t): β(t): arbtrary stochastc processes (pecewse contnuously dfferentable) dx dt = 0 0 λ( t) p( x( t)) x( t) = 0, x( t) = θ, λ( t) p(0) 0 λ( t) p( θ ) 0 otherwse λ(t) = α(t) - β(t) λ(t) = α(t) - β(t) feedback

39 WHY SFM? Lower resoluton model of real system ntended to capture just enough nfo. on system dynamcs Aggregates many events nto smple contnuous dynamcs, preserves only events that cause drastc change computatonally effcent (e.g., orders of magntude faster smulaton) If used for CONTROL purposes, another BARGAIN opportunty arses

40 AN EXAMPLE OF A BARGAIN USING A SURROGATE PROBLEM

41 THRESHOLD BASED BUFFER CONTROL REAL SYSTEM ARRIVAL PROCESS LOSS K x(t) L(K): Loss Rate Q(K): Mean Queue Lenth PROBLEM: Determne Κ to mnmze [Q(K) + R L(K)] SFM α(t) RANDOM PROCESS γ(t) θ x(t) β(t) RANDOM PROCESS SURROGATE PROBLEM: Determne θ to mnmze [Q SFM (θ) + R L SFM (θ)]

42 THRESHOLD BASED BUFFER CONTROL CONTINUED Real System 21 J(K) DES SFM Opt. Algo K SFM Optm. Algorthm usng SFM-based gradent estmates

43 SURROGATE PROBLEM IDEA DISCRETE SET Observe and estmate but but 2. ths ths Observe s s real and estmate Ths Ths s s not not real (e.g., gradent) H(r n ) SURROGATE CONTINUOUS SET ρ n Update r n rn = f(ρn ) ρ n r 1. Transform ρ = ρ + η H ( r n+ 1 n n n ρ ) ρ n+1 r * = f(ρ * )???

44 WHEN DOES THIS PROVABLY WORK? Need some structural propertes; otherwse, the NFL lmt gets you! Smlartes to Ordnal Optmzaton [Ho [Ho et et al, al, JDEDS JDEDS 1992] 1992] Resource allocaton problems [Gokbayrak [Gokbayrak and and Cassandras, Cassandras, JOTA JOTA 2002] 2002] Cooperatve control problems see Sesson FrA06

45 BYPASSING COMPLEXITY IN COOPERATIVE CONTROL CONTINUED V 2 V 3 V 1 V 4 V 5

46 BYPASSING COMPLEXITY IN COOPERATIVE CONTROL V 2 Current Control Horzon CONTINUED V 3 V 1 V 4 V 6 Optmal headng Over Event-Drven Recedng Horzon V 5

47 BYPASSING COMPLEXITY IN COOPERATIVE CONTROL MAIN IDEA: Replace complex Dscrete Stochastc Optmzaton problem CONTINUED by a sequence of smpler Contnuous Optmzaton problems But how do we guarantee that vehcles actually head for desred DISCRETE POINTS? It turns out they do! Can replace HARD problem by several SIMPLER ones

48 DANGERS OF DECOMPOSITION, ABSTRACTION What exactly does that mean? TIME-DRIVEN SYSTEM EVENT-DRIVEN SYSTEM

49 DANGERS OF DECOMPOSITION, ABSTRACTION a LOW-RESOLUTION. MODEL e.g., x = f(x,a) OUTPUT: x(a) AVERAGING a = E[A(u,ω)] a = E[A(u,ω)] A(u,ω N ) A(u,ω N )... A(u,ω 1 ) A(u,ω 1 ) INPUT u RANDOMNESS ω HIGH-RESOLUTION SIMULATOR OUTPUT: Data from N smulated scenaros

50 WHY THIS FAILS... a LOW-RESOLUTION. MODEL e.g., x = f(x,a) OUTPUT: x(a) Average corresponds to to unlkely scenaro x(a) s s way off... MEAN a Prob. Prob. Densty Functon of of A obtaned from from Hgh Hgh Resoluton model x

51 WHY THIS FAILS... SIMPLE AVERAGING: f(e[a]) SAMPLE, THEN AVERAGE: E[f(A)] If ultmate OUTPUT s x(a) = 0 or 1 ths can result n 0 nstead of 1 completely wrong concluson!

52 WHAT S THE WAY AROUND THIS? QUESTION: To average or not to average? ANSWER: Average just enough Replace AVERAGE by CONDITIONAL AVERAGES, one for each CLUSTER CLUSTER = group of smlar scenaros from Hgh Resoluton model CLUSTER ANALYSIS

53 CLUSTERING a m LOW-RESOLUTION MODEL a 1 LOW-RESOLUTION MODEL... AVERAGING x = E[f(a)] x = E[f(a)]... CLUSTERING a 1a 1... a ma m A(u,ω N ) A(u,ω N )... A(u,ω 1 ) A(u,ω 1 ) INPUT u RANOMNESS ω HIGH-RESOLUTION SIMULATOR

54 HIGH PROBABLILITY v CERTAINTY

55 HIGH PROBABILITY v CERTAINTY Prob. of Success Brthday Paradox CERTAINTY v 99% CONFIDENCE 366 v No. of samples

56 A BRAVE NEW COMPLEX WORLD

57 A BRAVE NEW COMPLEX WORLD

58 MANAGING COMPLEXITY Better HARDWARE and SOFTWARE help System ARCHITECTURE v OPERATIONAL CONTROL HIGH v LOW RESOLUTION models (Too much detal can hurt) Know what PROBLEM needs to be solved, then develop METHODOLOGIES (otherwse, NFL lmt gets you!) MODEL-DRIVEN v DATA-DRIVEN approaches (Embrace DATA -- and the NETWORK that gets data to you)

59 ACKNOWLEDGEMENTS Thanks to several current and former students whose contrbutons are reflected n ths talk + Colleagues Y.C. (Larry) Ho We-Bo Gong Yora Ward Ben Melamed Young Cho Ly Da Xren Cao Davd Castanon Jerry Wohletz Yanns Paschalds + Sponsors NSF, AFOSR, AFRL, DARPA, NASA, EPRI, ARO, UT, ALCOA, NOKIA, GE