Control of computer chip semi-conductor wafer fabs
|
|
- Barnard Reeves
- 5 years ago
- Views:
Transcription
1 Conrol of compuer chip semi-conducor wafer fabs Cos: x9 $ Reurn: years Cycle ime: 6 weeks WIP: 6, wafers, 8x6 $ Challenge: Conrol he queues a ~5 work seps Conrol of Manufacuring Sysems:! Sochasic racking of Fluid Soluion! and Robus Fluid Soluion! Simplex Algorihm o Solve Coninuous Linear Programs Eurandom Workshop on Robus Opimizaion in Applied Probabiliy Eindhoven, ovember 5 Gideon Weiss Universiy of Haifa Join work wih: Ana Kopzon, Yoni azarahy, Evgeny Shindin Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Conrol of compuer chip semi-conducor wafer fabs Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Conrol of compuer chip semi-conducor wafer fabs x9 Cos: $ Reurn: years Cycle ime: 6 weeks WIP: 6, wafers, 8x6 $ Challenge: Conrol he queues a ~5 work seps Conrol 6 jobs, sales value 8x6$, over 6 weeks Scheduling approach: Use combinaorial opimizaion o find opimal schedule Inracable because of size, Useless because of noise Queueing approach: Use Markov decision heory o opimize seady sae Inracable because of size, Useless because no seady sae Bridging approach: Queueing model, o accommodae noise Finie horizon, for scheduling he curren WIP Inracable, approximaion for large sysem. Opimize a Fluid Model wo approaches for he sochasics Model sochasic noise around exac fluid Solve a robus fluid model Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5
2 wo Fluid Approaches Modeling Sep : Muli class queueing neworks MCQ Manufacuring Plan Wafer Fab Model as Muliclass Queueing ework Finie Horizon Problem (Rolling Horizon model predicive) MCQ problem: conrol (Q(),()), over << Dynamics Q k () = Q k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k K k ' Fluid Model + Sochasic Deviaions Robus Fluid Model Bersimas, asrabadi, Paschalidis 5 Separaed Coninuous Linear Program Solved by Simplex ype Algorihm Weiss 8 rack Fluid Soluion Discree Sochasic azarahy, Weiss Asympoic Opimaliy? Maximal hroughpu? Objecive Capaciy min γ k k ( ) + c k Q k ()d k k () k (s) s k:s(k)=i re-enran line Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Modeling Sep : Fluid nework approximaion o MCQ SCLP Fluid problem is a separaed coninuous linear program V * = min ( γ u() + c q() )d q() fluid levels u() processing raes s.. q() = q() Ru(s)ds, R Rouing marix Mu(), M consiuen mean imes marix u(),q(), [, ] Robus fluid formulaion SCLP Robus fluid problem is a again an SCLP (Bersimas, asrabadi, Paschalidis 5) V Robus = min ( γ u() + c q() )d s.. q() = q() Ru(s)ds, u() M C Γ M! α(), I J β() u(),α(),β(),q(), [, ] Robus processing imes m k m k () = m k + z k ()!m k m k +!m k z k (), z k () Γ i machine i Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7
3 Derivaion of fluid formulaion (Bersimas, asrabadi, Paschalidis 5) Opimal fluid soluion (finie horizon) V Robus = min ( γ u() + c q() )d s.. q() = q() Ru(s)ds, u() M C Γ M! α(), I J β() u(),α(),β(),q(), [, ] Replace Mu() By max u U M ()u() consiuen mean ime marix Rouing R = m Sochasic: [ M ] i,k = k if s(k) = i + oise else Robus : m [ M ()] i,k = k + z k ()!m k if s(k) = i else!! Soluion of SCLP - Cenralized planning Primal Z i (u,) = max ( m k + z k ()!m k )u k () s(k)=i z k () Γ i s(k)=i z k () Dual Z i (u,) = minγ i β i () + α k () + m k u k () s(k)=i β i () + α k ()!m k u k (), k : s(k) = i β i (),α k () Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9 Opimal fluid soluion (finie horizon) Soluion of SCLP - Cenralized planning racking he fluid soluion IVQs Model deviaions from fluid soluion as MCQ wih infinie virual queues IVQ Pariions ime horizon o a finie number of inervals piecewise consan processing raes (conrols) coninuous piecewise linear fluid level (saes) Use disribued conrol o sabilize deviaions heorem: Policy is asympoically opimal - as sysem becomes large Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5
4 Muli class queueing neworks (Harrison 88, Dai 94, ) MCQ wih infinie virual queues (Massey 84, W 6) Muli-class queueing neworks wih infinie virual queues! IVQ Q k () = Q k () + A k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k ' Κ Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Muli class queueing neworks (Harrison 88, Dai 94, ) MCQ wih infinie virual queues (Massey 84, W 6) IVQ in he fluid soluion Opimal Fluid Soluion In each inerval, pariion buffers 6 6 K = {5, 7, 9} k K : Q k () = Q k () + A k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k ' Κ k K : Q k () = Q k () + α k S k ( k ()) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 K = {,,,, 4, 6, 8} 5,7,9 are empy 7 fully uilizes machine 9 and 6 fully uilize machine Machine pumps ou of 8 bu is no fully uilized. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5
5 Maximum pressure policy (assiulas, Solyar, Dai & Lin) Disribued conrol policy for MCQ: Observe queue of each class and is downsream classes Calculae pressure of class Serve Max Pressure class a each machine racking he fluid soluion wih IVQ and Max Pressure! racking heorem (Dai and Lin 5): Max pressure achieves sabiliy for any sysem wih offered load < rae sabiliy if offered load = heorem: Same resul holds for MCQ wih IVQs We model deviaions from fluid soluion as MCQ wih IVQs Sabiliy implies: deviaions are negligible if sysem large Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7 Asympoic opimaliy (azarahy, W 9) heorem: Le Q() be he queue lengh process of a finie horizon MCQ. Le Q () be scalings of Q(), wih Q ()=Q(), and wih -fold speed Of processing. Le q() be he opimal fluid soluion and le V f be is Objecive value. (i) Le V denoe he objecive values of Q () for any general policy. hen: (ii) Under max pressure racking of he fluid soluion and lim inf V V f Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 a.s. lim Q() = q() a.s. uniformly on < < lim V = V f a.s. Finie horizon conrol of a processing sysem! Model as a muli-class queueing nework MCQ! Fluid nework approximaion o MCQ Sochasic vs. Robus! Soluion of fluid problem! separaed coninuous linear program SCLP! racking he opimal fluid soluion! MCQ wih infinie virual queues IVQ! - minimal daa raes and roues,! - cenralized planning for fluid,! - disribued execuion for racking! - rolling horizon implemenaion! Sochasic: hroughpu Opimal for large! Robus: Lose some hroughpu sable for smaller Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9
6 Separaed coninuous linear programming! Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() u(), x() b < < primal saes x k () primal conrols u j () SCLP Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () u(), x() < < u(), x() < < min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) Dual runs in reversed ime p(),q() < < p(),q() < < Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5
7 Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () Piecewise consan conrols and linear saes Soluion feaures: pariion of ime horizon = < < < = piecewise consan conrols u() coninuous piecewise linear x() eed o know (primal and dual): u(), x() < < Inerval lenghs: au n = n - n- Linear Inerval Equaions min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) Dual runs in reversed ime Values of primal and dual conrols, u(), p(-): u n j, p n k Slopes of primal and dual saes,!x(),!q( ) :!x k n,!q j n Raes LP p(),q() < < x k ()p k ( )d = q j ( )u j ()d = primal dual feasible complemenary slack soluions are opimal Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Inial sae values x() q(): x k, q j Boundary LP Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Complemenary slack opimal soluions!x k p k x!x k k Boundary and Rae LPs max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() u(), x() b < < min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d p(),q() < < = = = = Boundary max d x s.. Fx α min b q s.. H 'q γ u j!q j!q j q j x q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7
8 Boundary and Rae LPs d max (( γ + ( ) c )u() + d x() ) d d d s.. G u(s)ds d + Fx() α + a H u() b d min (( α + ( ) a ) p() + b q() ) d d d s.. G p(s)ds + H q() γ + c d F p() d Equaions for inerval lengh!x k x!x k k p k x k ( ) = u(), x() < < p(),q() < < Boundary max d x s.. Fx α min b q s.. H 'q γ = = = = Raes x max c'u + d!x s.. Gu + F!x a Hu b min a' p + b!q s.. G ' p + H '!q c F ' p q d u j q j ( ) =!q j!q j q j Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9 Soluion of a simple SCLP example Solving he example 6 max (8 )u () x () ( ) d s.. u (s)ds + x () + u () u, x Primal and dual problems max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Parameric soluion Insead of =6 we solve for parameric, saring a and increasing o We solve simulaneously he primal problem and is dual. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5
9 Solving he example Boundary LP max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example Raes LP > max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q max x s.. + x = x min s.. q + q = 4 q max u!x s.. u +! +!x = u + u = min p +!q s.. p! +!q = p p! U, u Z,!x,u P! U, p Z, p,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u Z,!x,u P! U, p Z, p,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5
10 Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u Z,!x,u P! U, p Z, p,!q P!x U, u,!x,u P p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9
11 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P 4! U, p Z, p,!q P!x 4 U, u,!x,u P p Z, p,!,!q P! U, u Z,!x,u P 4! U, p Z, p,!q P!x 4 U, u,!x,u P p Z, p,!,!q P q q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4
12 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P 5! U, p Z, p,!q P!x 5 U, u,!x,u P p Z, p,!,!q P! U, u Z,!x,u P 5 6! U, p Z, p,!q P!x 5 U, u,!x,u P 6 p Z, p,!,!q P!,!x,u,u P p,!, p,!q P q q q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 44 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 45 Complee soluion: p max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Analog from parameric LP Lemke algorihm Solving LP by parameric Lemke algorihm: max ( ( λ) + λc)' x ( ) x Ax ( λ) + λb max c' x Ax b x (b,c) 6 p u p u u p u 6 (, ) q (! b,!c) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 46 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 47
13 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 48 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 49 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5
14 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 54 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 55
15 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 56 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 57 Solving he fluid problem SCLP algorihm demo Summary: We discuss a mehod for finie horizon conrol of large manufacuring plans We large scale sochasic vs. Robus approach Sochasic approach is asympoically opimal, for large scale operaions Robus approach sub-opimal bu sable on smaller scale Calculaes cenralized fluid soluion for overall conrol Uses de-cenralized local conrol o rack opimal fluid soluion Uses minimum amoun of daa, readily available Rolling horizon implemenaion Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 58 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 59
16 References () A Simplex based algorihm o solve Separaed Coninuous Linear Programs Gideon Weiss, Mah Programming Series A, 5:5-98, 8. () ear opimal conrol of queueing neworks over a finie ime horizon Yoni azarahy and Gideon Weiss, Annals of OR 7:-49, 9 () A Push-Pull nework wih infinie supply of work Ana Kopzon, Yoni azarahy and Gideon Weiss, QUESA 6:75-, 9 (4) Posiive Harris recurrence and diffusion scale analysis of a Push-Pull queueing nework wih infinie supply of work Yoni azarahy and Gideon Weiss, Performance Evaluaion, 67:-7, (5) Robus Fluid Processing eworks Dimiris Bersimas, Ebrahim asrabadi and Ioannis Paschalidis, IEEE rans Auomaic Conrol 6():75-78, 5. hank you! (6) A new algorihm for sae-consrained separaed coninuous linear programs, X. Luo and Dimiris Bersimas, SIAM J Conrol & Opimizaion, 7:77, 998. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Where are we now? racking he fluid soluion Example nework µ Sacked Queue level represenaion: µ µ Sacked Queue Levels Q Q Finished Jobs Q rajecory of a single job Resource is he boleneck µ > µ + µ How o conrol queues, Schedule resource, for opimal draining ime Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6
17 Fluid formulaion Fluid policies: LBFS, min makespan, opimal min () + q () + q ()d s. : () = () µ u (s)ds q () = q () + µ u (s)ds µ u (s)ds q () = q () + µ u (s)ds µ u (s)ds µ µ µ LBFS V=76 min Makespan V=6 u () + u () u () u,q [, ] Opimal V=5 µ µ µ his is a Separaed Coninuous Linear Program (SCLP) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 64 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 65 ime inervals of he opimal fluid soluion τ = (,4) τ = (4,8) τ = (8,4) τ 4 = (4,4) Allocaions u = u = u =.5 u =.5 u = u = u = u = u = u = u =.75 u =.5 Calculaing he pressure for neworks wih fixed rouing Pressure for buffer k based on k and k+ downsream curren k + K k + K k K µ Q µ k ( Qk Qk + ) k k n Κ = { k qk ( ) =, τ n} Κ n = {k q k () >, τ n } α k = u k µ k k K µ k (α k D k Q k + ) µ k (α k D k ) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 66 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 67
18 Using max pressure o conrol MCQ wih IVQ:! racking he fluid soluion Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 68
Gideon Weiss University of Haifa. Joint work with students: Anat Kopzon Yoni Nazarathy. Stanford University, MSE, February, 2009
Optimal Finite Horizon Control of Manufacturing Systems: Fluid Solution by SCLP (separated continuous LP) and Fluid Tracking using IVQs (infinite virtual queues) Stanford University, MSE, February, 29
More informationt dt t SCLP Bellman (1953) CLP (Dantzig, Tyndall, Grinold, Perold, Anstreicher 60's-80's) Anderson (1978) SCLP
Coninuous Linear Programming. Separaed Coninuous Linear Programming Bellman (1953) max c () u() d H () u () + Gsusds (,) () a () u (), < < CLP (Danzig, yndall, Grinold, Perold, Ansreicher 6's-8's) Anderson
More informationAn introduction to the theory of SDDP algorithm
An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking
More informationA Semiconductor Wafer
M O T I V A T I O N Semi Conductor Wafer Fabs A Semiconductor Wafer Clean Oxidation PhotoLithography Photoresist Strip Ion Implantation or metal deosition Fabrication of a single oxide layer Etching MS&E324,
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationOptima and Equilibria for Traffic Flow on a Network
Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1 A Traffic
More informationApplication of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing
Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology
More informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationDynamic Server Allocation for Unstable Queueing Networks with Flexible Servers
Dynamic Server Allocaion for Unsable Queueing Neworks wih Flexible Servers Salih Tekin and Sigrún Andradóir H. Milon Sewar School of Indusrial and Sysems Engineeering Georgia Insiue of Technology Douglas
More informationOn the Stability Region of Multi-Queue Multi-Server Queueing Systems with Stationary Channel Distribution
On he Sabiliy Region of Muli-Queue Muli-Server Queueing Sysems wih Saionary Channel Disribuion Hassan Halabian, Ioannis Lambadaris, Chung-Horng Lung Deparmen of Sysems and Compuer Engineering, Carleon
More informationA Primal-Dual Type Algorithm with the O(1/t) Convergence Rate for Large Scale Constrained Convex Programs
PROC. IEEE CONFERENCE ON DECISION AND CONTROL, 06 A Primal-Dual Type Algorihm wih he O(/) Convergence Rae for Large Scale Consrained Convex Programs Hao Yu and Michael J. Neely Absrac This paper considers
More informationScheduling of Crude Oil Movements at Refinery Front-end
Scheduling of Crude Oil Movemens a Refinery Fron-end Ramkumar Karuppiah and Ignacio Grossmann Carnegie Mellon Universiy ExxonMobil Case Sudy: Dr. Kevin Furman Enerprise-wide Opimizaion Projec March 15,
More informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationHybrid Control and Switched Systems. Lecture #3 What can go wrong? Trajectories of hybrid systems
Hybrid Conrol and Swiched Sysems Lecure #3 Wha can go wrong? Trajecories of hybrid sysems João P. Hespanha Universiy of California a Sana Barbara Summary 1. Trajecories of hybrid sysems: Soluion o a hybrid
More informationEstimation of Poses with Particle Filters
Esimaion of Poses wih Paricle Filers Dr.-Ing. Bernd Ludwig Chair for Arificial Inelligence Deparmen of Compuer Science Friedrich-Alexander-Universiä Erlangen-Nürnberg 12/05/2008 Dr.-Ing. Bernd Ludwig (FAU
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationStable approximations of optimal filters
Sable approximaions of opimal filers Joaquin Miguez Deparmen of Signal Theory & Communicaions, Universidad Carlos III de Madrid. E-mail: joaquin.miguez@uc3m.es Join work wih Dan Crisan (Imperial College
More informationBiologically inspired mutual synchronization of manufacturing machines
Biologically inspired muual synchronizaion of manufacuring machines Erjen Lefeber,a,1, Herman de Vos a, Dieer Armbruser a,b,2, Alexander S. Mikhailov c,3 a Sysems Engineering Group, Deparmen of Mechanical
More informationSliding Mode Extremum Seeking Control for Linear Quadratic Dynamic Game
Sliding Mode Exremum Seeking Conrol for Linear Quadraic Dynamic Game Yaodong Pan and Ümi Özgüner ITS Research Group, AIST Tsukuba Eas Namiki --, Tsukuba-shi,Ibaraki-ken 5-856, Japan e-mail: pan.yaodong@ais.go.jp
More informationRobust Learning Control with Application to HVAC Systems
Robus Learning Conrol wih Applicaion o HVAC Sysems Naional Science Foundaion & Projec Invesigaors: Dr. Charles Anderson, CS Dr. Douglas Hile, ME Dr. Peer Young, ECE Mechanical Engineering Compuer Science
More informationBasic definitions and relations
Basic definiions and relaions Lecurer: Dmiri A. Molchanov E-mail: molchan@cs.u.fi hp://www.cs.u.fi/kurssi/tlt-2716/ Kendall s noaion for queuing sysems: Arrival processes; Service ime disribuions; Examples.
More informationGlobal Optimization for Scheduling Refinery Crude Oil Operations
Global Opimizaion for Scheduling Refinery Crude Oil Operaions Ramkumar Karuppiah 1, Kevin C. Furman 2 and Ignacio E. Grossmann 1 (1) Deparmen of Chemical Engineering Carnegie Mellon Universiy (2) Corporae
More informationProblem Set on Differential Equations
Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p()
More informationProblem Set #3: AK models
Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationEE363 homework 1 solutions
EE363 Prof. S. Boyd EE363 homework 1 soluions 1. LQR for a riple accumulaor. We consider he sysem x +1 = Ax + Bu, y = Cx, wih 1 1 A = 1 1, B =, C = [ 1 ]. 1 1 This sysem has ransfer funcion H(z) = (z 1)
More information(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)
Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion
More informationEnergy Storage Benchmark Problems
Energy Sorage Benchmark Problems Daniel F. Salas 1,3, Warren B. Powell 2,3 1 Deparmen of Chemical & Biological Engineering 2 Deparmen of Operaions Research & Financial Engineering 3 Princeon Laboraory
More informationLinear Quadratic Regulator (LQR) - State Feedback Design
Linear Quadrai Regulaor (LQR) - Sae Feedbak Design A sysem is expressed in sae variable form as x = Ax + Bu n m wih x( ) R, u( ) R and he iniial ondiion x() = x A he sabilizaion problem using sae variable
More informationPade and Laguerre Approximations Applied. to the Active Queue Management Model. of Internet Protocol
Applied Mahemaical Sciences, Vol. 7, 013, no. 16, 663-673 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.1988/ams.013.39499 Pade and Laguerre Approximaions Applied o he Acive Queue Managemen Model of Inerne
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More informationAn Inventory Model for Time Dependent Weibull Deterioration with Partial Backlogging
American Journal of Operaional Research 0, (): -5 OI: 0.593/j.ajor.000.0 An Invenory Model for Time ependen Weibull eerioraion wih Parial Backlogging Umakana Mishra,, Chaianya Kumar Tripahy eparmen of
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XI Control of Stochastic Systems - P.R. Kumar
CONROL OF SOCHASIC SYSEMS P.R. Kumar Deparmen of Elecrical and Compuer Engineering, and Coordinaed Science Laboraory, Universiy of Illinois, Urbana-Champaign, USA. Keywords: Markov chains, ransiion probabiliies,
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More informationContinuous Time Markov Chain (Markov Process)
Coninuous Time Markov Chain (Markov Process) The sae sace is a se of all non-negaive inegers The sysem can change is sae a any ime ( ) denoes he sae of he sysem a ime The random rocess ( ) forms a coninuous-ime
More informationModule 4: Time Response of discrete time systems Lecture Note 2
Module 4: Time Response of discree ime sysems Lecure Noe 2 1 Prooype second order sysem The sudy of a second order sysem is imporan because many higher order sysem can be approimaed by a second order model
More informationOrientation. Connections between network coding and stochastic network theory. Outline. Bruce Hajek. Multicast with lost packets
Connecions beween nework coding and sochasic nework heory Bruce Hajek Orienaion On Thursday, Ralf Koeer discussed nework coding: coding wihin he nework Absrac: Randomly generaed coded informaion blocks
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationAugmented Reality II - Kalman Filters - Gudrun Klinker May 25, 2004
Augmened Realiy II Kalman Filers Gudrun Klinker May 25, 2004 Ouline Moivaion Discree Kalman Filer Modeled Process Compuing Model Parameers Algorihm Exended Kalman Filer Kalman Filer for Sensor Fusion Lieraure
More informationChapter 5 Digital PID control algorithm. Hesheng Wang Department of Automation,SJTU 2016,03
Chaper 5 Digial PID conrol algorihm Hesheng Wang Deparmen of Auomaion,SJTU 216,3 Ouline Absrac Quasi-coninuous PID conrol algorihm Improvemen of sandard PID algorihm Choosing parameer of PID regulaor Brief
More informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
More informationA First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
More informationProblem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0
More informationSolutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
More informationMath 116 Second Midterm March 21, 2016
Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including
More informationEMS SCM joint meeting. On stochastic partial differential equations of parabolic type
EMS SCM join meeing Barcelona, May 28-30, 2015 On sochasic parial differenial equaions of parabolic ype Isván Gyöngy School of Mahemaics and Maxwell Insiue Edinburgh Universiy 1 I. Filering problem II.
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationAppendix to Creating Work Breaks From Available Idleness
Appendix o Creaing Work Breaks From Available Idleness Xu Sun and Ward Whi Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, NY, 127; {xs2235,ww24}@columbia.edu Sepember
More informationRecursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems
8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear
More informationZürich. ETH Master Course: L Autonomous Mobile Robots Localization II
Roland Siegwar Margaria Chli Paul Furgale Marco Huer Marin Rufli Davide Scaramuzza ETH Maser Course: 151-0854-00L Auonomous Mobile Robos Localizaion II ACT and SEE For all do, (predicion updae / ACT),
More information1. Consider a pure-exchange economy with stochastic endowments. The state of the economy
Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.
More information3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
More informationMean-square Stability Control for Networked Systems with Stochastic Time Delay
JOURNAL OF SIMULAION VOL. 5 NO. May 7 Mean-square Sabiliy Conrol for Newored Sysems wih Sochasic ime Delay YAO Hejun YUAN Fushun School of Mahemaics and Saisics Anyang Normal Universiy Anyang Henan. 455
More informationAn Introduction to Stochastic Programming: The Recourse Problem
An Inroducion o Sochasic Programming: he Recourse Problem George Danzig and Phil Wolfe Ellis Johnson, Roger Wes, Dick Cole, and Me John Birge Where o look in he ex pp. 6-7, Secion.2.: Inroducion o sochasic
More informationL07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS. NA568 Mobile Robotics: Methods & Algorithms
L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS NA568 Mobile Roboics: Mehods & Algorihms Today s Topic Quick review on (Linear) Kalman Filer Kalman Filering for Non-Linear Sysems Exended Kalman Filer (EKF)
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
More information1 Subdivide the optimization horizon [t 0,t f ] into n s 1 control stages,
Opimal Conrol Formulaion Opimal Conrol Lecures 19-2: Direc Soluion Mehods Benoî Chachua Deparmen of Chemical Engineering Spring 29 We are concerned wih numerical soluion procedures for
More informationExamples of Dynamic Programming Problems
M.I.T. 5.450-Fall 00 Sloan School of Managemen Professor Leonid Kogan Examples of Dynamic Programming Problems Problem A given quaniy X of a single resource is o be allocaed opimally among N producion
More informationProblem Set 9 Due December, 7
EE226: Random Proesses in Sysems Leurer: Jean C. Walrand Problem Se 9 Due Deember, 7 Fall 6 GSI: Assane Gueye his problem se essenially reviews Convergene and Renewal proesses. No all exerises are o be
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationEnergy Storage and Renewables in New Jersey: Complementary Technologies for Reducing Our Carbon Footprint
Energy Sorage and Renewables in New Jersey: Complemenary Technologies for Reducing Our Carbon Fooprin ACEE E-filliaes workshop November 14, 2014 Warren B. Powell Daniel Seingar Harvey Cheng Greg Davies
More informationSolutions for Assignment 2
Faculy of rs and Science Universiy of Torono CSC 358 - Inroducion o Compuer Neworks, Winer 218 Soluions for ssignmen 2 Quesion 1 (2 Poins): Go-ack n RQ In his quesion, we review how Go-ack n RQ can be
More informationAnno accademico 2006/2007. Davide Migliore
Roboica Anno accademico 2006/2007 Davide Migliore migliore@ele.polimi.i Today Eercise session: An Off-side roblem Robo Vision Task Measuring NBA layers erformance robabilisic Roboics Inroducion The Bayesian
More informationTUNING OF NONLINEAR MODEL PREDICTIVE CONTROL FOR QUADRUPLE TANK PROCESS
h Sepember 14. Vol. 67 No. 5-14 JATIT & LLS. All righs reserved. ISSN: 199-8645 www.jai.org E-ISSN: 1817-3195 TUNING OF NONLINEAR MODEL PREDICTIVE CONTROL FOR QUADRUPLE TANK PROCESS 1 P.SRINIVASARAO, DR.P.SUBBAIAH
More informationAdaptation and Synchronization over a Network: stabilization without a reference model
Adapaion and Synchronizaion over a Nework: sabilizaion wihou a reference model Travis E. Gibson (gibson@mi.edu) Harvard Medical School Deparmen of Pahology, Brigham and Women s Hospial 55 h Conference
More informationBasic notions of probability theory (Part 2)
Basic noions of probabiliy heory (Par 2) Conens o Basic Definiions o Boolean Logic o Definiions of probabiliy o Probabiliy laws o Random variables o Probabiliy Disribuions Random variables Random variables
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
More informationSensors, Signals and Noise
Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise
More informationAn Optimal Dynamic Generation Scheduling for a Wind-Thermal Power System *
Energy and Power Engineering, 2013, 5, 1016-1021 doi:10.4236/epe.2013.54b194 Published Online July 2013 (hp://www.scirp.org/journal/epe) An Opimal Dynamic Generaion Scheduling for a Wind-Thermal Power
More informationLecture 1: Contents of the course. Advanced Digital Control. IT tools CCSDEMO
Goals of he course Lecure : Advanced Digial Conrol To beer undersand discree-ime sysems To beer undersand compuer-conrolled sysems u k u( ) u( ) Hold u k D-A Process Compuer y( ) A-D y ( ) Sampler y k
More informationFrom Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
More informationIntelligent Packet Dropping for Optimal Energy-Delay Tradeoffs in Wireless Downlinks
PROC. OF 4TH INT. SYMPOSIUM ON MODELING AND OPTIMIZATION IN MOBILE, AD HOC, AND WIRELESS NETWORKS (WIOPT), APRIL 2006 1 Inelligen Packe Dropping for Opimal Energy-Delay Tradeoffs in Wireless Downlinks
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More informationOnline Partially Model-Free Solution of Two-Player Zero Sum Differential Games
Preprins of he 10h IFAC Inernaional Symposium on Dynamics and Conrol of Process Sysems The Inernaional Federaion of Auomaic Conrol Online Parially Model-Free Soluion of Two-Player Zero Sum Differenial
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationOptimal Investment under Dynamic Risk Constraints and Partial Information
Opimal Invesmen under Dynamic Risk Consrains and Parial Informaion Wolfgang Puschögl Johann Radon Insiue for Compuaional and Applied Mahemaics (RICAM) Ausrian Academy of Sciences www.ricam.oeaw.ac.a 2
More informationStochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing
Ouline Moivaion Sochasic formulaion Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing Aircraf and gus modeling Resuls Deparmen of Mechanical and Aerospace Engineering Rugers,
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationInstitute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
More informationWaveform Transmission Method, A New Waveform-relaxation Based Algorithm. to Solve Ordinary Differential Equations in Parallel
Waveform Transmission Mehod, A New Waveform-relaxaion Based Algorihm o Solve Ordinary Differenial Equaions in Parallel Fei Wei Huazhong Yang Deparmen of Elecronic Engineering, Tsinghua Universiy, Beijing,
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationCHE302 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS. Professor Dae Ryook Yang
CHE302 LECTURE IV MATHEMATICAL MODELING OF CHEMICAL PROCESS Professor Dae Ryook Yang Fall 200 Dep. of Chemical and Biological Engineering Korea Universiy CHE302 Process Dynamics and Conrol Korea Universiy
More informationOptimal Server Assignment in Multi-Server
Opimal Server Assignmen in Muli-Server 1 Queueing Sysems wih Random Conneciviies Hassan Halabian, Suden Member, IEEE, Ioannis Lambadaris, Member, IEEE, arxiv:1112.1178v2 [mah.oc] 21 Jun 2013 Yannis Viniois,
More informationEconomics 6130 Cornell University Fall 2016 Macroeconomics, I - Part 2
Economics 6130 Cornell Universiy Fall 016 Macroeconomics, I - Par Problem Se # Soluions 1 Overlapping Generaions Consider he following OLG economy: -period lives. 1 commodiy per period, l = 1. Saionary
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationSettling Time Design and Parameter Tuning Methods for Finite-Time P-PI Control
Journal of Conrol Science an Engineering (6) - oi:.765/8-/6.. D DAVID PUBLISHING Seling ime Design an Parameer uning Mehos for Finie-ime P-PI Conrol Keigo Hiruma, Hisaazu Naamura an Yasuyui Saoh. Deparmen
More informationLecture 4 Notes (Little s Theorem)
Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,
More informationResource Allocation in Visible Light Communication Networks NOMA vs. OFDMA Transmission Techniques
Resource Allocaion in Visible Ligh Communicaion Neworks NOMA vs. OFDMA Transmission Techniques Eirini Eleni Tsiropoulou, Iakovos Gialagkolidis, Panagiois Vamvakas, and Symeon Papavassiliou Insiue of Communicaions
More informationOmega-limit sets and bounded solutions
arxiv:3.369v [mah.gm] 3 May 6 Omega-limi ses and bounded soluions Dang Vu Giang Hanoi Insiue of Mahemaics Vienam Academy of Science and Technology 8 Hoang Quoc Vie, 37 Hanoi, Vienam e-mail: dangvugiang@yahoo.com
More informationMath 115 Final Exam December 14, 2017
On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):
More informationNumerical Methods for Open-Loop and Closed-Loop Optimization of Linear Control Systems
Compuaional Mahemaics and Mahemaical Physics, Vol. 4, No. 6, 2, pp. 799 819. Translaed from Zhurnal Vychisliel noi Maemaiki i Maemaicheskoi Fiziki, Vol. 4, No. 6, 2, pp. 838 859. Original Russian Tex Copyrigh
More information