A Model Reduction Technique for linear Model Predictive Control for Non-linear Large Scale Distributed Systems

Transcription

1 A Model Reducto Techque for lear Model Predctve Cotrol for No-lear Large Scale Dstrbuted Systes Weguo Xe ad Costatos Theodoropoulos School of Checal Egeerg ad Aalytcal Scece Uversty of Machester, Machester M60 QD, UK

2 Overvew Motvato Exstg techologes Our proposed techque Case studes Coclusos Further work

3 Motvato Model Predctve Cotrol MPC Lear MPC s wdely used dustres Few olear MPC applcatos Nolear large scale dstrbuted syste Attractg ore terest aog researchers Few well-establshed ethods avalable Model reducto techques Great potetal dustral applcatos Relatvely ew

4 Exstg techologes Feedback learsato Paraetrc cotrol Adaptve cotrol self-tug cotrol Artfcal eural etwork No-paraetrc ethods

5 Our proposed techque As Applcable to coplex dyac systes Autoatc procedure Good approxato of the orgal full-scale odel Explct paraetrc depedece Hgh coputatoal effcecy Our ew ethod st step: POD proper orthogoal decoposto -based projecto oto low-desoal hyperspace d step: TPWL Trajectory Pece-wse learsato o te coeffcets 3 rd step: QP Quadratc prograg appled to obta cotrol law

6 Detaled dyac odel N equatos Model reducto Proper Orthogoal Decoposto POD x& f x Experetal Data Off-le Dgtal Iage Processg Two-pot correlato atrx Sgular Value Decoposto Sall uber, of eprcal global bass fuctos, Φ Galerk Projecto x f x 0 o-lear ODEs Low-order odel

7 Learsato of POD-based Reduced odel costrats for MPC da dt a u equato equato Idea: ca learse ters of : Irrespectve of hgh physcal desoalty of the proble Learsato always -desoal u x, t f a, t t

8 Case study : te Taks Level cotrol F 0 4 Tak F 3 Tak F.. F 9 3. Tak 0 F 0

9 Cotrol forulato 0 taks Objectve fucto du T T Y Y Q Y Y DU RDU ref ref equato 3 s.t. Mass Balace of tak : A F equato 4 0 F Mass balace of tak - tak 0: equato 5 F c Flow rates: equato 6 k k, k h,0 / k A dh dt dh dt, F,0 F

10 No-lear odel No-lear odel cotrol based o o-lear objectve fucto o-quadratc wth olear costrats. No-lear dyac optsato Multple shootg based o a set of te tervals Soe kd of successve substtuto or better Sequetal Quadratc Prograg

11 Usg F t POD odel reducto F c / * h F0 c c F A A 0 the Equato 7 Slar equatos ca be obtaed by the above ethod F F c c 0,, 0 t F A A Apply ethod of sapshots to get bass fuctos Calculate te coeffcets Equato 8 usg Galerk projecto o the POD egefuctos as above The syste dyacs are the retreved as: F x, t v x, t,, 0 F k t k x Equato 9

12 TPWL ethod The pecewse lear terpolato s bult as follows. Equato 0 Equato Apply ea value theore: Equato The, Equato 3 where, the secod dervatve of s bouded by x z b a z L y a x x y y b ], [ x z x z f z L z f ], [ x x z ], [ x x 8 h M z L z f f M x z x f z L x z x f z L x z x f z L z L 3

13 Statc ad adaptve TPWL Statc TPWL based o ufor partto where M 8 s a gve postve tolerace. Equato 4 Adaptve TPWL ethod We propose that the subterval [ xl, xr] s acceptable f xl xr f f xl f xr Equato 5 xr xl h Or, Equato 6 A partto x x s acceptable f each subterval s acceptable.

14 POD bass fuctos-te taks

15 TPWL te coeffcets-te taks Statc Adaptve Left-had sde are statc TPWL for POD te coeffcets wth TPWL 9 tervals, ad rght-had sde are adaptve TPWL 5 tervals

16 Results of TPWL wth POD for 0 taks Statc Adaptve 0 taks showg,, 9, ad 0 w.r.t. te usg PODs, F 0 =6,, 0. ad dt 0.5 Left-had sde statc TPWL, rght-had sde are adaptve TPWL

17 Cotrol forulato 0 taks wth POD ethod Objectve fucto: quadratc due to POD forulato Equato 7 s.t. Equato 8 where,,, ad 0 ]} [ { 0 dx x A c A F x t c F x t t F x t x j k k k k k k k k k,0, j,, 0 dx RDU DU Y F x t Q Y F x t J T ref k k k T ref k k k du 0 0

18 Cotrol forulato 0 taks wth TPWL ethod Pece-wse lear for o State Space Model: t L t BF0 t t p / t L p t p / t B p F0 t p / t y t H t F x0 where, Equato 9 H ] [ x0, x0, x0 T

19 Cotrol law for TPWL odel Quadratc Prograg appled to obta the cotrol law: Equato 0 Where, because oly oe output; ad The the output varables ca be calculated usg Equato r ] [ 0 0 t F G t G Q Y G ri QG G DF u ref T y y T y L L HL L HL HL G y HB B L L HL B HL HB B L L HL B HL HB HB B HL HB HB G B L L HL B HL HB B HL HB HB G u x F t F G t DF G t G Y u y

20 SQP results of olear odel 0 taks usg drect ODE solver, dt=0.5sec Results of olear case usg drect ODE s solver ad -0 lqud level of tak -0; cotrol put; output of tak te copared to referece output Sze of proble: 0*00 Nuber of ODEs: 0

21 SQP results of olear odel 0 taks 3 bass fuctos usg drect ODE solver, dt=0.5sec Results of olear reduced odel usg drect ODE s solver ad -0 lqud level of tak -0; cotrol put; output of tak te copared to referece output Sze of proble: 3*00 Nuber of ODEs: 3

22 TPWL results of olear odel 0 taks usg 3 bass fuctos, dt=0.5sec Results of olear case usg PWL POD solver ad -0 lqud level of tak -0; cotrol put; output of tak te copared to referece output Sze of proble: 3*00 Nuber of ODEs: 3

23 Case study : Tubular reactor A z 0 C z z C z C T t t T 0 z C z Pe T z Pe T C B r : recyclg rato r C f C, T z T B f C, T T T T T c z Pe 7.0 Pe 7. 0 B 0. B C T T 0 C f C, T B Pe C T C T C exp T [ r C0 rc t, C t,0] C T T z Pe T T [ r T0 rt t, T t,0] 0 0 c equato equato 3

24 PDE-based odel Coplex dyacs Tubular reactor Rch paraetrc space, bfurcatos Saddle odes Sustaed oscllatos Approprate cotrol proble Through a uber of syste paraeters Recycle Jacket teperature

25 Cotrol objectve For r=0 stable behavour For r=0.5 Hopf bfurcato Sustaed oscllatos r=0 Use a set of coolg zoes stablse syste at r=0.5 To behave lke syste at r=0 r=0.5

26 Saplg Heavsde fuctos for exaple: 3 actuators have 8 states ad 5 actuators have 3 states saples for every Heavsde fuctos Teperature [-0.999,] Cocetrato [0,] So, 8 x =88 saples for 3 actuators, ad 3 x =35 saples for 5 actuators

27 Tubular reactor - 6 POD bass fuctos for teperature Wth 5 actuators ad Heavsde fuctos

28 Tubular reactor - 6 POD bass fuctos for cocetrato Wth 5 actuators ad Heavsde fuctos

29 Tubular reactor: Te coeffcets Teperature wth r=0.5 Cocetrato wth r=0.5

30 Full vs. reduced odel Teperature wth r=0.5 Cocetrato wth r=0.5

31 Learsato of teperature te coeffcets Statc Adaptve

32 Learsato of cocetrato te coeffcets Statc Adaptve

33 Cotrol forulato tubular reactor wth POD ethod Objectve fucto: quadratc due to POD forulato T J k _ T t k _ T x T6 Tref t Q k _ T t k _ T x T6 Tref t DU T RDU du k k equato 4 where, Tref t s the referece state wth r=0, DU s cotrol o actuators s.t. k _ C t k _ C z C z k t k _ C t k _ C z C z k z PeC k _ C t k _ C z C z k z k _ T t k _ T z T z BC k _ C t k _ C z C z exp k k k _ T t k _ T z T z k k _ T t k _ T z T z k t k _ T t k _ T z T z k z PeT k _ T t k _ T z T z k z k _ T t k _ T z T z BT BC k _ C t k _ C z C z exp k T U T k k _ T t k _ T z T z k equato 5

34 Cotrol forulato POD ethod wth TPWL ethod Pece-wse lear for o State Space Model: t L t BU t t p / t L p t p / t B pu t p / t y t H t T6 equato 6 where, t cludes te coeffcets for cocetrato ad teperature, H [0,0, 0, _ T z6, _ T z6, _ T z6 ]T

35 Cotrol law for TPWL odel Quadratc Prograg appled to obta the cotrol law: DU GTyQGy ri GTyQ[Yref G t GuU t ] equato 7 Where, r because oly oe output; HB HB HL B G y HB HL B HL L L B 0 HB 0 HB HL B HL L L3 B HB HB HL B G u HB HL B HL L L B ad HB HL HL L G HL L L The the output varables ca be calculated usg Y G t G y DU t GuU t T6 equato 8

36 Results 8 actuators ad 5 secs SQP Statc TPWL Adaptve TPWL

37 Coclusos A POD-based ethod has bee developed wth statc or adaptve TPWL for o-lear large scale dstrbuted systes Our ethod catches the dyacs of systes Sgfcatly reduces coputato te copared to SQP ethod. Appled to both dscretsed ad cotuous systes

38 Future work Pece-wse affe reduced odel Use PAROS software for paraetrc cotrol Other cases study Mcrofltrato process Lyophlsato process

39 Ackowledgeets The facal cotrbuto of the EU Prograe CONNECT [COOP ] The facal cotrbuto of the EU Progae CAFE [KBBE-754]