Fernando V. Lima & Christos Georgakis Department of Chemical & Biological Engineering and

1. Interval Operability: A Tool to Design the Feasible Output Constraints for Non-Square Model Predictive Controllers Fernando V. Lima & Christos Georgakis Department of Chemical & Biological Engineering and Systems Research Institute

2. Presentation Outline • Introduction and Problem Definition • Model Predictive Control Strategy • Motivating Example • Objectives & Proposed Approach • Process Operability • Set-point Operability • Interval Operability • Iterative Methodology • Linear Programming Approach • Results: Industrial Examples • Conclusions & Acknowledgments UIC Workshop by Fernando Lima

3. Introduction: Past Experience • Education: • University of Sao Paulo, Brazil (02/99-08/03) • B.S. in Chemical Engineering • Tufts University, MA (09/03-09/07) • Ph.D. in Chemical Engineering • Research Experience: • University of Wisconsin-Madison(10/07-10/09) • Postdoctoral Fellow at Chemical Engineering Department • Industrial Experience: • Evaluation of MPC Packages at DuPont • Linear (Summer 05): • State Space Controller (Aspen) & Predict and Control (ABB) • Nonlinear (Summer 06): • Apollo (Aspen) & Polymers NLC (PAS) UIC Workshop by Fernando Lima

4. Problem Definition • Non-square Systems: • More Outputs (CV’s) than Inputs (MV’s) • Set-point Control is NOT Possible for ALL Outputs • Fewer Degrees of Freedom than Controlled Variables • Interval Control is Needed • Model Predictive Control (MPC): • Very Tight Constraints Make Control Infeasible • Soften Output Constraints(1,2) • Need Methodology for Design of Output Ranges • For Non-square Model-based Controllers • To Address High-Dimensional Linear Systems (1) Hovd, M.; Braatz R.D. (2004). Modeling, Identification and Control. (2) Rawlings, J. B. (2000). IEEE Control Systems Magazine. UIC Workshop by Fernando Lima

5. Model Predictive Control Strategy • Control of Multivariable Processes • Take into Account Interactions of Variables • Model-Based • Future Output Predictions Based on Process Model • Account for Constraints • Inputs and Outputs • Most of MPC Controlled Chemical Processes are Non-Square UIC Workshop by Fernando Lima

6. Motivating Example: Steam Methane Reformer (SMR): Step Response Model* 9 Controlled Variables (CV’s) 1 Disturbance and 4 MV’s Dynamic Matrix for the SMR problem (DMCplusTM- AspenTech) (*) from David R. Vinson, PhD Thesis at Lehigh University UIC Workshop by Fernando Lima

7. SMR: Feasible Process Constraints Feasible Set of Input and Output Constraints at Steady-state UIC Workshop by Fernando Lima 7

8. DMCplusTM Simulations – Case 1 Making the y6 Constraints Tighter: Problem Becomes infeasible DMCplusTM Controller: Trend Plot – y5 1.93 1.88 With Control 1.83 Up Limit 1.80 Low Limit 1.77 0 1 h 2 h 3 h 4 h Disturbance UIC Workshop by Fernando Lima 8

9. DMCplusTM Simulations – Case 2 Making the y7 Constraints Tighter: Problem is Feasible! Need Methodology to Design Output Bounds Trend Plot – y5 1.93 1.88 With Control 1.83 Up Limit 1.80 Low Limit 1.77 0 1 h 2 h 3 h 4 h UIC Workshop by Fernando Lima 9

10. Objectives and Proposed Approach • Development of a Process Operability Methodology • For Multivariable Non-square Systems • with n Outputs, m Inputs and q Disturbances • Process Operability – Past Contribution • Analysis for Low-D Square Systems(1) • Process Operability – Present Contribution • Extension to Non-square Linear Systems(2-5) • Enables Systematic Selection of Output Ranges • For the Design of MPC Output Constraints(4,6) • New optimization-based Methodology for High-D Systems(5) • Analysis of Industrial-scale Non-square Processes(6,7) (1) Vinson, D. R.; Georgakis, C. (2000). J . Proc. Cont. (5) Lima, F.V.; Georgakis, C. (2009). J. Proc. Cont. (submitted). (2) Lima, F.V.; Georgakis, C. (2006). ADCHEM Proceedings. (6) Lima, F.V. et al. (2009). AIChE J. (accepted). (3) Lima, F.V.; Georgakis, C. (2007). DYCOPS Proceedings. (7) Lima, F.V. et al. (2008). ESCAPE Proceedings. (4) Lima, F.V.; Georgakis, C. (2008). J. Proc. Cont. UIC Workshop by Fernando Lima

11. Process Operability Definition • Vinson and Georgakis (2000)(1): • Process is Operable if ... • Available Set of Inputs Satisfies Desired Steady-state & Dynamic Performance Requirements in Presence of Expected Disturbances • Set-Point Operability • It is Intended to Reach Every Point in Desired Output Set (DOS) (1) Vinson, D. R.; Georgakis, C. (2000). J. Proc. Cont. UIC Workshop by Fernando Lima

12. Operability: Shower Example* q2 T q2m= 3 94 AIS DOS q2s= 2 Ts= 84 74 q1 F Fs= 5 3 7 q1m= 4 q1s= 3 (*) Vinson, D.R.; Georgakis, C. (1998). DYCOPS Proceedings. UIC Workshop by Fernando Lima

13. 120 AOSDOS 110 AOS 100 T 90 80 DOS 70 7 6 4 5 2 3 1 F Set-Point Operability Index • Operability Issues: • Expectations (DOS) were Unrealistic • Can Clearly See what Expectations will be Realistic • Operability Index (OI) • Fraction of the DOSthat can be achieved UIC Workshop by Fernando Lima

14. Interval Operability • Each Output Varies within an Interval • To be Operable in Intervals: • Need One Feasible Operating Point inside Interval • Set Relative Aspect Ratio for Intervals • Based on Output Weights (wi) • Low Ratio/High Weight for Set-point Controlled Outputs • Achievable Output Interval Set (AOIS)(1,2) • Tightest Set of Output Constraints • That Can be Achieved with • Available Input Set (AIS) and • Expected Disturbance Set (EDS) (1) Lima, F.V.; Georgakis, C. (2006). ADCHEM Proceedings. (2) Lima, F.V.; Georgakis, C. (2008). J. Proc. Cont. UIC Workshop by Fernando Lima

15. Interval Operability Example Non-square Linear Model: 1 Input & 2 Outputs • Both Outputs Controlled at Intervals • Using the same relative weight AOS(d = 0) AOS(d = 1) AOS AOIS AOS(d =-1) UIC Workshop by Fernando Lima 15

16. Changing Output Weights and Target • Moving the Output Target from the Origin • Asymmetric Problem • Changing the Relative Output Weights • Influences the AOIS Aspect Ratio: Set-Point Control for one of the Outputs AOIS for y0 = (0.5, 0.5) and r12= 4:1 AOIS calculated using r12= 1:10 AOIS calculated using r12= 1000:1 UIC Workshop by Fernando Lima

17. High-D Systems: Iterative Methodology • Enlarging AOIS • Touches Both Extreme Disturbance Lines of the AOS(d= ±1) (y12, y22) AOS (d= 1) • Minimize Intersection Measure: AOIS (y14, y24)  (y11, y21) (y10, y20) AOS (d= -1) (y13, y23) Intersection Measure  Secant Method      UIC Workshop by Fernando Lima

18. m-dimensional object in AOS(d)is now a subset of Iterative Methodology (y12, y22) • Calculation of Intersections using Multi-Parametric Toolbox (MPT)*: • Sets AOS(d=±1) have to be Full Dimensional • Convex Hull of all Points is Calculated • Polytopes Obtained are used in the Intersections AOS (d= 1) (y11, y21) • Algorithm is Computationally Expensive • As Problem Dimensionality Increases • Computation of Convex Hulls and Intersections every Iteration • Need for Another Alternative (*) Kvasnica et al. (2004).Multi-Parametric Toolbox (MPT) for MATLAB. UIC Workshop by Fernando Lima

19. Linear Programming (LP) Approach • Constrained Optimization Problem • Sets AOS(d=±1) and AOIS represented by Inequalities (y12, y22) (y1*, y2*) AOS (di= 1) AOIS  (y14, y24) (y10, y20) (y11, y21) AOS (di= -1) (y13, y23) UIC Workshop by Fernando Lima

20. Industrial Examples • Dryer Control Problem (DuPont) – A • 6 outputs, 4 inputs, 2 disturbances • Steam Methane Reformer (Air Products) – B • 9 outputs, 4 inputs, 1 disturbance • Sheet Forming Problem (DuPont) – C • 15 outputs, 9 inputs, 3 disturbances UIC Workshop by Fernando Lima

21. Results: Industrial Example B • Steam Methane Reformer (Air Products): • 9 CVs, 4 MVs, 1 DV UIC Workshop by Fernando Lima

22. SMR: Designed Output Constraints HVR (9-D) 9.15 x 103 • Computational Time Enables online design of output constraints LP: 0.14 s Iterative Methodology: Convex Hull + Intersections Every Iteration (days!) UIC Workshop by Fernando Lima

23. Conclusions • Concept of Operability Extended • To High-D Non-Square Systems • Achievable Output Interval Set Defined • Used in the Design of Output Constraints • To High-D Square Systems • Two Methodologies Developed • Iterative Approach (IA) and Linear Programming (LP) • IA: Impractical when n > 8-D • LP: Faster and applicable to any input-output system • Up to 15-D Tested • Successfully Applied to Industrial Examples • Drier Control Problem (DuPont) • Steam Methane Reformer (Air Products and Chemicals) • Sheet Forming Control Problem (DuPont) UIC Workshop by Fernando Lima

24. Conclusions • Research Impact is Two-Fold • Advances the Ability of Operability Assessment • For High-D and Continuous Processes • Design of Output Constraints for MPC Controllers • LP Framework Enables Online Implementation • Operability Calculations in Fractions of a Second UIC Workshop by Fernando Lima

25. Acknowledgements • Thesis Committee Members • Profs. Daniel Ryder & Vincent Manno, David Vinson, Julie Smith • Yiannis Dimitratos and PD&C at DuPont • Dave Schnelle for Process Models • Bill Canney (ASPEN Technology) for DMC Controller Software • Collaborations • MPC Implementation to Lab Crystallizer: Gene Bunin • Operability & Flexibility: Prof. Ierapetritou at Rutgers • Funding: ACS Petroleum Research Foundation &Tufts University Questions? Fernando Lima Info; Phone: 608-265-8607; http://jbrwww.che.wisc.edu/home/lima; fvlima@wisc.edu UIC Workshop by Fernando Lima