Visualization of the Economic Impact of Process Uncertainty in Multivariable Control Design

Transcription

1 Vsualzaton of the Economc Impact of Process Uncertanty n Multvarable Control Desgn Benjamn Omell & Donald J. Chmelewsk Department of Chemcal & Bologcal Engneerng Illnos Insttute of echnology Mass r r max f w r mn k1 A B( desred ) k2 B C( undesred ) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

2 Outlne Proft control based tunng: MPC Senstvty to soft-constrants tunng Proft Control: Extended formulaton Measurement nformaton formulaton Colored nose formulaton Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

3 Concept of Back-Off Pont Selecton CV' s Constrant Polytope Backed-off Operatng Ponts Expected Dynamc Operatng Regons Optmal Steady-State Operatng Pont MV' s Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

4 BOP Optmzaton Problem s', m', q', XY,, ( ) ( ) x A BL x A BL GwG z ( Dx Du L) x( Dx Du L) Dw wd z 1 nz 1 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn th column w

5 BOP Optmzaton Problem s', m', q', XY,, ( ) ( ) x A BL x A BL GwG z ( Dx Du L) x( Dx Du L) Dw wd z 1 nz 1 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn th column w

6 BOP Optmzaton Problem s', m', q', XY,, ( ) ( ) x A BL x A BL GwG z ( Dx Du L) x( Dx Du L) Dw wd z 1 nz 1 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn th column w

7 BOP Optmzaton Problem s', m', q', XY,, Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn ( ) ( ) x A BL x A BL GwG z ( Dx Du L) x( Dx Du L) Dw wd z 1 nz 1 th column w

8 BOP Optmzaton Problem s', m', q', XY,, Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn ( AX BY ) ( AX BL) G G ( D x X D Y) u Where L=YX -1 ( D x w X DuY ) X

9 Outlne Proft control based tunng: MPC Senstvty to soft-constrants tunng Proft Control: Extended formulaton Measurement nformaton formulaton Colored nose formulaton Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

10 MPC mn x Qx 2u Mx u Ru xu, k 1 s.. t x Ax Bu Gw k k k k k k k 1 k k k z D x D u D w k 1 x k u k w k z z z 1 n mn max, k z Optmzaton Generates the gan, not Q,R, and M for LQR controller? Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

11 Inverse Optmalty Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

12 A Inverse Optmalty heorem 1 (Chmelewsk & Manthanwar, 24): If there exsts P > and R > such that P PA Q L R 1 L PB RL A L 1 PB M R PB M M R PB P PA L R PB hen M ( L R PB) and Q L RL A P PA are such that Q M M R R and P and L satsfy Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

13 Controller Equvalence heorem 2 (Chmelewsk & Manthanwar, 24): he controller generated by Proft Control s concdent wth the controller generated by some LQR problem. Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

14 Outlne Proft control based tunng: MPC Senstvty to soft-constrants tunng Proft Control: Extended formulaton Measurement nformaton formulaton Colored nose formulaton Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

15 Mass-Sprng-Damper Example f Mass w r r max System Model: r 1 r f w v v where r s the mass poston, v s the velocty, f s thenput force (MV) and w s the dsturbance force r mn System Constrants: 1 r 1 and f 16 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

16 Mass Poston Mass-Sprng-Damper Example f Mass w r r max System Model: r 1 r f w v v where r s the mass poston, v s the velocty, f s thenput force (MV) and w s the dsturbance force r mn System Constrants: 1 r 1 and f 16 OSSOP * Backed-off Operatng Pont (BOP) Upper Bound on Poston } tme EDOR Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

17 Mass Poston Mass Poston Mass-Sprng-Damper Example f Mass w r r max System Model: r 1 r f w v v where r s the mass poston, v s the velocty, f s thenput force (MV) and w s the dsturbance force r mn System Constrants: 1 r 1 and f 16 OSSOP * Backed-off Operatng Pont (BOP) Upper Bound on Poston OSSOP * Backed-off Operatng Pont (BOP) Upper Bound on Poston } } EDOR tme EDOR tme Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

18 Mass Poston (m) Mass-Sprng-Damper Example (Impact of Constrants ) Case B Case C -.4 Case A Input Force (N) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

19 Mass Poston (m) Dscrete-tme Smulaton (Scatter Plot) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng Input Force (N)

20 Mass Poston (m) MPC and the EDOR mn x Qx 2u Mx u Ru xu, k 1 s.. t x Ax Bu Gw k 1 k k k k 1 x k u k w k z z z 1 n mn max, k z Illnos Insttute of echnology k k k k k k z D x D u D w Department of Chemcal and Bologcal Engneerng Input Force (N)

21 Soft Constrants mn x Qx 2u Mx u Ru s s xu, k 1 s.. t x Ax Bu Gw k 1 k k k z D x D u D w k 1 x k u k w k z s z z s 1 n s mn max, k z Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

22 Soft Constrants mn x Qx 2u Mx u Ru s s xu, k 1 s.. t x Ax Bu Gw k 1 k k k z D x D u D w k 1 x k u k w k z s z z s 1 n s mn max, k z Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

23 Mass Poston (m) Mass Poston (m) MPC wth Soft Constrants Input Force(N) Input Force(N) g m = 1 7 g f = 1 3 g m = 1 3 g f = 1 3 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

24 Flexblty n EDOR Defnton 11 * z 1 22 z 2 = 1 constrant observance ~84% of tme = 2 constrant observance ~97.8% of tme = 3 constrant observance ~99.9% of tme Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

25 Mass Poston (m) Mass Poston (m) Impact of EDOR Defnton Input Force (N) Input Force (N) = 1 = 2 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

26 Mass Poston (m) Mass Poston (m) MPC wth Soft Constrants EDOR = 2 std dev s Input Force(N) Input Force(N) g m = 1 7 g f = 1 3 g m = 1 3 g f = 1 3 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

27 Mass Poston (m) Mass Poston (m) Mass Poston (m) Mass Poston (m) Mass Poston (m) Mass Poston (m) Impact of EDOR Defnton (Reduced Senstvty to Soft Weghts) Input Force (N) Input Force(N) Input Force(N) Input Force (N) Input Force(N) Input Force(N) g m = g f = g m = 1 7 g f = 1 3 g m = 1 3 g f = 1 3 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

28 Outlne Proft control based tunng: MPC Senstvty to soft-constrants tunng Proft Control: Extended formulaton Measurement nformaton formulaton Colored nose formulaton Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

29 Back to the FCC Example Partal-state nformaton Correlated nose, shapng flter. Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

30 Lnear State Dscrete-tme Model Controller Prospectve w k SF d k z k u k FCC y k Illnos Insttute of echnology x Ax Bu Gd k 1 k k k z D x D u D d k x k u k w k y Cx v xˆ Ax Bu G( m Cxˆ ) k 1 k k k Department of Chemcal and Bologcal Engneerng k

31 Lnear State Dscrete-tme Model Controller Prospectve w k SF d k z k u k FCC y k KF xˆk Illnos Insttute of echnology x Ax Bu Gd k 1 k k k z D x D u D d k x k u k w k y Cx v xˆ Ax Bu G( m Cxˆ ) k 1 k k k Department of Chemcal and Bologcal Engneerng k

32 Lnear State Dscrete-tme Model Controller Prospectve w k SF d k z k L u k FCC y k KF xˆk Illnos Insttute of echnology x Ax Bu Gd k 1 k k k z D x D u D d k x k u k w k y Cx v xˆ Ax Bu G( m Cxˆ ) k 1 k k k Department of Chemcal and Bologcal Engneerng k

33 Covarance Analyss Assume controller L s gven and calculate : u Lx Lyapunov equaton (FSI case) 1 n z z z 1 ( A BL) ( A BL) G G ) D x x w z ( Dx Du L) x( Dx Du L Dw th column w w Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

34 Covarance Analyss PSI Assume controller L s gven and calculate : ( A BL) ( A BL) xˆ A C ( C C ) C A ( D D L) ( D D L) D D D D z x u xˆ x u x e x w w w Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng xˆ 1 e e v e 1 n z z ˆ z u Lx Lyapunov equaton (PSI) 1 th column

35 BOP Optmzaton Problem s', m', q', XY,, mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn ( AX BY ) ( AX BL) G G ( D x X D Y) u ( D x w X DuY ) X Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

36 BOP Optmzaton Problem s', m', q', XY,, mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn ( AX BY ) ( AX BL) G G ( D x X D Y) u ( D x w X DuY ) X Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

37 BOP Optmzaton Problem s', m', q', XY,, mn g q ' s.t. s ' As ' Bm' q q D s D m q q q mn max ' ( x ' u ') ' ( q ' q ' ) ( q ' q ' ) 1/2 max 1/2 mn X A C C C C A AX BY ( AX BY ) X DxeDx ( ) Dx X DuY ( Dx X DuY ) X 1 e ( e v) e ( ) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

38 cy (K) reg (K) F cat (kg/s) F ar (kg/s) Dynamcs of PSI versus FSI C (mass fracton) x 1-3 rgc O d (mass fracton x (K) st C st (mass fracton) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

39 Outlne Fndng Q,R, and M weghts for MPC Effects of soft-constrants on mass sprng damper example FCC case study revsted Partal-state nformaton case Colored nose Effects of dsturbance model on CSR Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

40 feed s Dsturbance Model Shapng Flter.25.2 he dsturbance was modeled usng a shapng Flter. ypcal Profle tme x tme x 1 4 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

41 F ar (kg/s) reg (K) F cat (kg/s) cy (K) Effects Colored Nose C rgc (mass fracton) O (mass fracton x 1-3 d Illnos Insttute of echnology x 1-3 Department of Chemcal and Bologcal Engneerng st (K) C st (mass fracton)

42 CSR example Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

43 Plant Constrants 3 Fn Fm.8m / s F F 1 35K 2 35K 33K n m c1, out.5m.5m / s / s 3K.3kmol m c2, out C A 2 / Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

44 Plant Proft Proft Functon: 1F C.1q q.1f.1f 2 B2 cool,1 cool,2 n m Assume U a cannot be measured PSI Kalman Flter used x = Ax + Bu + Gw y = Cx + v Where C s the output matrx and ν s the covarance of the sgnal nose ν Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

45 Dsturbances Consdered C w U U n A, n a1 a w Cases consdered U a = U a s consdered whte nose U a s hghly correlated colored nose Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

46 Feed Flow (m 3 /s) Jacket Flow 1 (m 3 /s) emperature from Jacket 2 (K) Reactor 2 emperature (K) Effect of Dsturbance Model No HEX Fault HEX Fault emperature from Jacket 1 (K) Reactor 1 emperature (K) Makeup Flow (m 3 /s) Jacket Flow 2 (m 3 /s) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

47 Feed Flow (m 3 /s) Jacket Flow 1 (m 3 /s) emperature from Jacket 2 (K) Reactor 2 emperature (K) Effect of Dsturbance Model No HEX Fault HEX Fault Hghly Correlated Nose HEX Fault Whte Nose emperature from Jacket 1 (K) Reactor 1 emperature (K) Makeup Flow (m 3 /s) Jacket Flow 2 (m 3 /s) Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

48 CSR Proft Gross Proft ($/day) Dff from OSSOP ($/day) OSSOP $2,486 - U a = $2,342 - $144 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

49 CSR Proft Gross Proft ($/day) Dff from OSSOP ($/day) OSSOP $2,486 - U a = $2,342 - $144 U a - whte nose $2,176 - $31 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

50 CSR Proft Gross Proft ($/day) Dff from OSSOP ($/day) OSSOP $2,486 - U a = $2,342 - $144 U a - whte nose $2,176 - $31 U a slow varyng $2,115 - $371 Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

51 Conclusons Controller gan found n optmzaton s guaranteed to have some Q, R, and M. Burden on MPC soft-constrants shfted to controller tunng PSI case and colored nose can be extended to proft control Dsturbance model has sgnfcant mpacts on dynamcs and controller tunng Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng

52 Acknowledgements Illnos Insttute of echnology s Department of Chemcal and Bologcal Engneerng Armour College of Engneerng and the II Graduate College Graduate Students Mke Walker Deepak Sharma Prevous Students Ju-Kun Peng Amt M. Manthanwar Illnos Insttute of echnology Department of Chemcal and Bologcal Engneerng