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Control of a Solution Copolymerization Reactor using Piecewise Linear Models

Control of a Solution Copolymerization Reactor using Piecewise Linear Models. Leyla Özkan APACT-03 York, UK April 30 th , 2003. Presentation Outline. Motivation Multi-Model Predictive Control Formulation Implementation on MMVA Solution Co-polymerization Reactor Stability Analysis

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Control of a Solution Copolymerization Reactor using Piecewise Linear Models

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  1. Control of a Solution Copolymerization Reactor using Piecewise Linear Models Leyla Özkan APACT-03 York, UK April 30th , 2003

  2. Presentation Outline • Motivation • Multi-Model Predictive Control Formulation • Implementation on MMVA Solution Co-polymerization Reactor • Stability Analysis • Conclusion APACT03-30 April, 2003

  3. Motivation • Polymerization reactors • Complex nonlinear kinetics and behavior • Difficult to specify control objectives • Global competition • Strict requirements on polymer properties • Grade transitions common • Control objective • Minimize grade transition time Reduce Off-Specification Product APACT03-30 April, 2003

  4. min J ( k ) p u ( . ) Past Future Set point dx Predicted y = f ( x , u , t ) dt Closed loop y Open loop u £ £ u u u min max £ £ y y y min max Closed loop u k k+1 k+Hc k+Hp Hc Hp Model Predictive Control • Model Predictive Control • Class of control algorithms that solves optimization problem at every instant APACT03-30 April, 2003

  5. [ ] ¥ T = + + + J x ( k m | k ) Q x u Ru ( k m | k ) å ¥ I T + + ( k ) ( k m | k ) ( k m | k ) finite horizon cost infinite horizon cost m = 0  2 å + + x ( k m | k ) 2 n + + + + u u ( ( k k m m | | k k ) ) x ( k m | k ) R Q 2 2 = m n+1 I å + Q R I = m 0 • Effective in terminal region • Bounded by • State feedback controllaw T + + + + x ) ) ( k n 1 P x ( k n 1 Multi-Model Predictive Control • Infinite horizon objective function • Free input variables • Forces states towards • terminal region APACT03-30 April, 2003

  6. x(k+1|k), u(k+1|k) x(k+2|k), u(k+2|k) x(k|k), u(k|k) n+1 m   (x,u) = (0,0) u(k+m|k)=Kx(k+m|k) Terminal region x(k+n+1|k) Quasi-infinite horizon strategy APACT03-30 April, 2003

  7. Define sequence of regions and models t J J ¥ n OP2 x2 OP1 x1 Illustration of multi-model predictivecontrol Control Recipe If x terminal region quasi-infinite horizon If x terminal region infinite horizon APACT03-30 April, 2003

  8. g min é ù 1 * * * * * L ê ú 0 . 5 g Q x ( k | k ) I 0 0 0 0 0 I ê ú 0 . 5 ê ú g R u ( k | k ) 0 I 0 0 0 0 ê ú 0 . 5 ³ + g 0 Q x ( k 1 | k ) 0 0 I 0 0 0 ê ú I 0 . 5 ê ú + g R u ( k 1 | k ) 0 0 0 I 0 0 ê ú M O ê ú ê ú + + x ( k n 1 | k ) 0 0 0 0 0 Q ë û é ù T 0 . 5 T 0 . 5 T + Q ( A Q B Y ) QE QQ Y R t t t t I t ê ú T T + + x x ( A Q B Y ) Q b b b e 0 0 ê ú t t t t t t t ê ú T T ³ 0 x - x - E Q e b ( I e e ) 0 0 t t t t t ê ú 0 . 5 ê ú g Q Q 0 0 I 0 I ê ú 0 . 5 g R Y 0 0 0 I ë û t The resulting LMI problem Finite horizon x(k) terminal region Infinite horizon APACT03-30 April, 2003

  9. g min x Q , , Y i s.t. é ù T 1 x ( k | k ) ³ 0 ê ú ê ú x ( k | k ) Q û ë and é ù T 0 . 5 T 0 . 5 T + Q ( A Q B Y ) QE QQ Y R i i i i I i ê ú T T + + x x ( A Q B Y ) Q b b b e 0 0 ê ú i i i i i i i ê ú T T ³ 0 x - x - E Q e b ( I e e ) 0 0 i i i i i ê ú 0 . 5 ê ú g Q Q 0 0 I 0 I ê ú 0 . 5 g R Y 0 0 0 I ë û i - - - 1 1 1 = g x =  l = P Q K Y Q i i Özkan, L. et.al. , Control of a solution copolymerization reactor using multi-model predictive control, Chem. E. Science, 58:1207-1221, 2003 The resulting LMI problem Feasible ONLY IF bi=0 (x(k)  terminal region) APACT03-30 April, 2003

  10. Consider j = 1 , 2 , … , n £ u ( k ) u u k 0 j j , max ³ Finite horizon cost: Infinite horizon cost: Impose directly ( u ( k + m | k ) k + m | k ) = Kt x + £ Exists s.t. X u ( k m | k ) é ù X Y j 2 t ³ £ 0 X u ê ú jj j , max u Y Q ë û j t , max m = 0 , 1, …, n j j = = 1 1 , , 2 2 , , … … , , n n u u m = n+1 ,…, Input Constraints APACT03-30 April, 2003

  11. Monomer (FA) Monomer (FB) Initiator (FI) Coolant Solvent (FS) Coolant Transfer Agent (FC) Polymer Solvent Unreacted feed Inhibitor (FZ) Richards, J. R. et.al. , Feedforward and Feedback Control of a Solution Co-polymerization Reactor, AIChE Journal, 35(6):891-907, 1989 MMVA Solution Copolymerization Reactor • Characteristics: • Based on free radical mechanism • Realistic industrially • 12 states, 4 outputs • Dynamics depends on monomers’ feed ratio Challenge: Optimization problem involves 200–300 variables APACT03-30 April, 2003

  12. Step input (FA/FB: 0.2 0.25) 4 OP 2 3.8 Mw/E4 (kg/kmol) 3.6 OP 1 3.4 0 10 20 30 40 50 time (h) Implementation on MMVA copolymerization reactor • Manipulated variables • FA, FB, FC, Tj • Obtaining multiple models • Select new desired operating point • Assume a trajectory APACT03-30 April, 2003

  13. (x,u) - (0,0) • Desired operating point • Driving force • Linear models are updated accordingly • Norm measure to define sequence of linear models x3 x x(k|k) x2 x1 t Implementation on MMVA (cont’d) • Simplifications • Only terminal region approximated as ellipsoid APACT03-30 April, 2003

  14. Ap Y 30 30 0 0 10 10 20 20 time (h) time (h) Implementation on MMVA (cont’d) • Effect of number of linear models 2 models 4 353.5 3 models 353.4 3.8 T (K) 353.3 4 models Mw/E4 (kg/kmol) 353.2 3.6 353.1 3.4 Off–spec. (kg) 0.64 27 26 0.62 Open loop 302.4 25 0.6 2 models 134.8 Gpi (kg/h) 0.58 24 3 models 110.1 23 0.56 4 models 109.8 0.54 22 APACT03-30 April, 2003

  15. MPC • Lyapunov Approach • dx(t)/dt=f(x) and x=0 equilibrium • V(x) • V(x)>0, x 0 and V(x)=0  x=0 • d V/dt<0, x 0 Asymptotically stable origin V(x) Time Stability Analysis • Conventional control • Check eigenvalues of closed-loop system APACT03-30 April, 2003

  16. Discrete components Continuous dynamical systems Logic commands (switches,automata) Hybrid systems Hybrid systems • Definition: • Dynamical systems with continuous and discrete state variables APACT03-30 April, 2003

  17. Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automatic Control, 43(4):475-482, 1998. Multiple Lyapunov Functions • Consider Vi :i=1,2,…,N • Given S={x0: (i0,t0),(i1,t1),…,(iN,tN),..} • Denote • T: increasing sequence of times t0,t1,…tN • (T):even sequence of T:t0,t2,t4, If Viis monotonically non-increasing on(T) then system is stable in the sense of Lyapunov APACT03-30 April, 2003

  18. Interpretation for N=2 V1(x) t0 t1 t2 t3 t4 t5 t6 V2(x) Multiple Lyapunov Functions • Easy to interpret • N candidate Lyapunov functions • Requires stable subsystems APACT03-30 April, 2003

  19. Theorem Ifx(k)  terminal region, the quasi–infinite horizon optimization problem is solvedincluding the contractive constraint. If x(k)  terminal regionthe infinite horizon optimization problem is solved. The closed loop system is stable if the feasible solutions of the control strategy defined are implemented in a receding horizon fashion Multi-Model Predictive Control with Stability Guarantee • Contractive constraint •  (k) <  (k-1)  x  terminal region • Stability Analysis APACT03-30 April, 2003

  20. Proof • Candidate Lyapunov function terminal region Ï g ì ( k ) x ( k ) = V í a ( ( x )) T terminal region Î x ( k ) Px ( k ) x ( k ) î Multi-Model Predictive Control with Stability Guarantee • Remarks • Stability results depend on feasibility • States are measurable • @ k=ts • V(x(ts))-V(x(ts-1))<0; not required APACT03-30 April, 2003

  21. 1 10 0 10  (k) xT(k)Px(k) -1 10 log (V) -2 10 -3 10 -4 10 0 5 10 15 t(h) MMVA solution copolymerization revisited • Observations • V monotonically decreasing • ts=7.0 and 7.75 h APACT03-30 April, 2003

  22. Summary and Conclusion • Multi-model control algorithm is developed • Hybrid structure • Implemented on a high dimensional problem • The effect of number of linear models • Decrease in transition time • Computational difficulty • Number of free input variables • Large size LMI’s solved on a realistic problem • Stability analysis • Contractive constraint • Multiple Lyapunov Functions APACT03-30 April, 2003

  23. Thank you for your attention Leyla Özkan APACT03-30 April, 2003

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