MIQP formulation for controlled variable selection in Self Optimizing Control

1. MIQP formulation for controlled variable selection in Self Optimizing Control Ramprasad Yelchuru Sigurd Skogestad Henrik Manum MIQP - Mixed Integer Quadratic Programming

2. qF Tray temperatures T1, T2, T3,…, T41 Motivation • L,V: Steady-state degrees of freedom • Want to minimize cost J • Which two • individualmeasurementsor • measurementcombinations • should be selected as controlled variables (CVs) to minimize the cost J? y = temperatures; H = selection/combination matrix c = Hy, H=?

3. J cs = constant + u + + y Loss K - + d + c H u Controlled variables, Mathematically Optimal steady-state operation Loss is due to (i) Varying disturbances (ii) Implementation error in controlling c at set point cs At optimum Ju = 0 Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008

4. Non-convex optimization problem D : any non-singular matrix Objective function unaffected by D. So can choose freely. We made H unique by adding a constraint as Convex optimization problem Global solution subject to Problem is convex in decision matrix H

5. Vectorization subject to Problem is convex QP in decision vector

6. Controlled variable selection Optimization problem : Minimize the average loss by selecting H and CVs as (i) best individual measurements (ii) best combinations of all measurements (iii) best combinations with few measurements st.

7. MIQP Formulation Big M approach We solve this MIQP for n = nu to ny

8. qF Tray temperatures T1, T2, T3,…, T41 Case Study : Distillation Column Binary Distillation Column LV configuration 41 Trays Level loops closed with D,B 2 MVs – L,V 41 Measurements – T1,T2,T3,…,T41 3 DVs – F, ZF, qF *Compositions are indirectly controlled by controlling the tray temperatures

9. Case Study : Distillation Column Data Results Controlled variables (c) Optimal individual measurements Loss2 = 0.0365 Optimal 4 measurement combinations Loss4 = 0.0164

10. Case Study : Distillation Column 1010 100 Comparison with customized Branch And Bound (BAB)* • MIQP is computationally more intensive than Branch And Bound (BAB) methods (Note that computational time is not very important as control structure selection is an offline method) • MIQP formulations are intuitive and easy to solve * Kariwala and Cao, IEEE Trans. (2010)

11. MIQP formulations can accommodate wider class than monotonic functions (J) MIQPs are solved usingstandard cplex routines MIQPs aresimpleand areeasyto incorporate fewstructural constraints MIQPs are computationally intensive than BAB methods Single MIQP formulation is sufficient for the described problems Customized BAB methods can handle onlymonotonic cost functions(J) Customized routines are required BABs require adeeper understandingof the customized routines to solve problems withstructural constraints Computationally faster than MIQPs as they exploit the monotonic properties efficiently Monotonicityof the measurement combinations needs to becheckedbefore using PB3 for optimal measuement subset selections Comparison of MIQP, Customized Branch And Bound (BAB) methods

12. MIQP formulation with structural constraints If the plant management decides to procure only 5 temperature sensors T1,T2,…,T10 qF T11,T12,…,T35 T36,T37,…,T41 Tray temperatures

13. Conclusions In self optimizing control the following problems are solved with an MIQP based formulation for optimal controlled variables selection. • Selection of optimal individual measurements as control variables (CVs) • Selection of CVs as combinations of all measurements • Selection of CVs as combinations of optimal measurement subsets MIQPs are more simple, intuitive and are easy compared to customized Branch and Bound methods