Nonlinear Programming Formulations for On-line Applications

Transcription

1 Nonlnear Programmng Formlatons for On-lne Applcatons L. T. Begler Carnege Mellon Unversty Janary 2007 Jont work wth Vctor Zavala and Carl Lard

2 NLP for On-lne Process Control Nonlnear Model Predctve Control (NMPC) Problem statement and reqrements Dynamc Optmaton Smltaneos approach Barrer NLP Method Real-tme Iteraton NLP Senstvty Nomnal and Shfted Implementatons LDPE Case Stdy Ftre Work Movng Horon Estmaton Nomnal Stablty Analyss 2

3 Nonlnear Model Predctve Control (NMPC) NMPC Estmaton and Control d : dstrbances : manplated varables y sp : set ponts Process NMPC Controller = F( y p d) 0 = G( y p d) : dfferental states y : algebrac states Model Updater = F( y p d) 0 = G( y p d) mn s. t. k y( t ( t) = NMPC Sbproblem k F ( ( t) y( t) ( t) t) 0 = G( ( t) y( t) ( t) t) ( t) = 0 Why NMPC? Track a profle Severe nonlnear dynamcs (e.g sgn changes n gans) Operate process over wde range (e.g. startp and shtdown) ) y sp Bond Constrant s 2 Q Other Constrant s y + k (t k ) (t k 1 ) 2 Q 3

4 Dynamc Optmaton Problem mn ψ ( (t) y(t) (t) p ) t f s.t. d( t) = F dt G ( ( t) y ( t) ( t) t p) ( ( t) y( t) ( t) t p) = 0 o ( 0 ) l d ( t ) d y l d y ( t ) d y l d ( t ) d p l d p d p t tme dfferental varables y algebrac varables t f fnal tme control varables p tme ndependent parameters 4

5 Dynamc Optmaton Approaches Pontryagn(1962) DAE Optmaton Problem Drect NLP solton Dscrete controls Indrect/Varatonal - Ineffcent for large constraned problems Sngle Shootng Sllvan (1977) Effcent for constraned problems Dscrete states and controls Smltaneos Approach +Small NLP - No nstabltes Bock Pltt (1984) Mltple Shootng +Handles nstabltes -Larger NLPs +Embeds DAE Solvers/Senstvty - Dense Senstvty Blocks Collocaton Large/Sparse NLP 5

6 Smltaneos Approach - Collocaton on Fnte Elements Polynomals Tre solton t o t n Collocaton ponts t f Fnte element n Mesh ponts D n element n (t) = + ) n 1 ( t t n 1 q = 1 k q= 1 g q q = 2 d (t) dt Dfferental varables Contnos q n y(t) = k q= 1 q g q (t)y n (t) = Algebrac and Control varables Dscontnos k q= 1 g q (t) q n 6

7 7 Nonlnear Programmng Problem L x x x x x c x f n = R 0 ) ( s.t ) ( mn ( ) f q q pt y mn ψ 1 = p y dt d F dt d = p y dt d G - l l l l p p p y y y s.t. dt d f = (0) o =

8 Dynamc Optmaton Engnes Evolton of NLP Solvers: Î for dynamc optmaton control and estmaton SQP rsqp Fll-space Barrer E.g. E.g SNOPT IPOPT NPSOL- and Smltaneos Mltple Seqental Shootng dynamc Dynamc - over optmaton 100 over Optmaton d.f.s 1 bt 000 over over varables varables and and constrants and constrants Obect Orented Codes talored to strctre sparse lnear algebra and compter archtectre (e.g. IPOPT 3.2) 8

9 NMPC for Tennessee Eastman Process (Jockenhövel Wächter B. 2003) Unstable Reactor After dscretaton 11 Controls; Prodct Prge streams varables 171 DAEs: Model extended wth energy balances 660 degrees of freedom 9

10 NMPC Reslts Tennessee Eastman Problem Optmaton wth IPOPT Warm start wth µ = Optmatons 5-7 CPU seconds Iteratons Optmaton wth SNOPT Uses approxmate (dense) redced Hessan pdates Cold not be solved wthn samplng tmes > 100 Iteratons 10

11 What abot Fast NMPC? Fast NMPC s not st NMPC wth a fast solver Comptatonal delay between recept of process measrement and necton of control determned by cost of dynamc optmaton Leads to loss of performance and stablty (see Fndesen and Allgöwer 2004; Santos et al. 2001) As larger NLPs are consdered for NMPC can comptatonal delay be overcome? 11

12 Nonlnear Model Predctve Control Dynamc Optmaton Æ Feedback Mode -NMPC- NLPs are Parametrc 12

13 NLP Senstvty Propertes (Facco 1983) Assme sffcent dfferentablty LICQ SSOC SC: Intermedate IP solton (s(µ)-s*) = O(µ) Fnte neghborhood arond p 0 and µ=0 wth same actve set exsts and s nqe 13

14 NLP Senstvty Obtanng Optmalty Condtons of Apply Implct Fncton Theorem to arond KKT Matrx IPOPT Æ Already Factored at Solton Æ Senstvty Calclaton from Sngle Backsolve Æ Approxmate Solton Retans Actve Set 14

15 Nonlnear Model Predctve Control NLP Senstvty Æ Fast Feedback - Pertrb Nomnal Solton (Nonshfted) Assme: Optmal Solton Optmal Solton Approxmate by Pertrbng Fast bt Inconsstent Actve Set NLP Senstvty Æ Approxmate Solton Retans Actve Set of Nomnal Problem 15

16 Nonlnear Model Predctve Control NLP Senstvty Æ Shfted (Pertrb + Shft the Horon Smltaneosly) KKT System at Solton of Agmented KKT System Shfted nomnal pertrbaton s mch smaller than non-shfted Î vanshes wth ncreasng N (O(exp(- L N)) 16

17 Nonlnear Model Predctve Control Rese L/U Factors of Æ Apply Schr Complement backsolves w/ Havng now solve for Sngle backsolve w/ On-Lne Calclaton (Backsolve) 17

18 LD P E Plant- Dynamc Optmaton E t h y l e n e 2 7 b a r s / 2 5 C 1. 5 b a r s / 4 0 C T o r c h 6 b a r s / 4 0 C P r g e B t a n e 6 b a r s / 4 0 C P R I M A R Y C O M P R E S S O R b a r s / 4 0 C 5 6 b a r s / 4 0 C H Y P E R - C O M P R E S S O R 2 4 b a r s / 4 0 C L O W P R E S S U R E R E C Y C L E b a r s 4 0 C H I G H P R E S S U R E R E C Y C L E b a r s 4 0 C b a r s 8 5 C T U B U L A R R E A C T O R P r e h e a t n g Z o n e R e a c t o n Z o n e b a r s C L D V a l v e C o o l n g Z o n e C b a r s 4 0 C W a x e s 2. 5 b a r s C H g h P r e s s r e S e p a r a t o r 1. 5 b a r s 4 0 C O l s L D P L o w P r e s s r e S e p a r a t o r Hgh pressre reacton (>2000 atm) throgh mle-long reactor col Hghly nonlnear model wth 1000 s of chemcal speces (moment models) Several classes of dynamc optmaton problems (NMPC grade changes nonstandard transtons) that need to be coordnated 18

19 Nonlnear Model Predctve Control Indstral Case Stdy Grade Transton Control Smltaneos Collocaton-Based Approach 289 ODEs 100 AEs leadng to constrants 9630 Bnds Feedback Every 6 mn 19

20 Nonlnear Model Predctve Control Optmal Feedback Polcy Æ (On-lne Comptaton 351 CPU s) Ideal NMPC controller - comptatonal delay not consdered Tme delays as dstrbances n NMPC 20

21 Nonlnear Model Predctve Control Optmal Feedback Polcy vs. Non-shfted Æ (On-lne Comptaton 0.94 CPU s) 21

22 Nonlnear Model Predctve Control Optmal Polcy vs. NLP Senstvty -Shfted Æ (On-lne Comptaton 1.04 CPU s) Very Fast Close-to-Optmal Feedback Large-Scale Rgoros Models 22

23 Inflence of Senstvty on Off-lne Comptatons Cold starts take more off-lne teratons Both senstvty pdates generally level ot Shfted approach consstently reqres fewer teratons - excellent on-lne performance - mproves off-lne comptatons 23

24 Ftre Work Redce on-lne costs frther - Deal wth changes n actve sets (warm starts?) - wth addtonal n (off-lne) backsolves can express lnear feedback law: Stablty Propertes = M ("+) = 1 m Reslts of Dehl et al. based on Newton convergence reslts apply drectly to smltaneos approach General nomnal stablty follows drectly for shfted varant sng Lyapnov analyss Robst stablty? Extend to Movng Horon Estmaton (MHE) problems Smlar approach has been developed On-lne costs redced by two orders of magntde wth excellent performance neqaltes and changes n constrant actvty less mportant well developed stablty theory (Rawlngs and coworkers) 24