MIQP formulation for optimal controlled variable selection in Self Optimizing Control

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

2. Outline • Motivation • Problem formulation • MIQP formulation • Evaporator Case study • Comparison of MIQP & customized BAB • Conclusions

3. 1.Motivation • Want to minimize cost J • Which two • individualmeasurementsor • measurementcombinations • should be selected as controlled variables (CVs) to minimize the cost J? y = candidate measurements; H = selection/combination matrix c = Hy, H=? Combinatorial problem • 1. Exhaustive search (10C2,10C3,…) • 2. customized BAB • 3. MIQP 2 MVs – F200, F1 Steady-state degrees of freedom 10 candidate measurements – P2, T2, T3, F2, F100, T201, F3, F5, F200, F1 3 DVs – X1, T1, T200

4. J cs = constant + u + + y Loss K - + d + c H u Controlled variables, 2. Problem Formulation Optimal steady-state operation Loss is due to (i) Varying disturbances (ii) Implementation error in controlling c at set point cs Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008

5. Non-convex optimization problem (Halvorsen et al., 2003) D : any non-singular matrix Improvement 1 (Alstad et al. 2009) Convex optimization problem Global solution st Improvement 2 (this work) st • Do not need Juu • And Q is used as degrees of freedom for faster solution

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

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

8. 3. MIQP Formulation Big M approach high value M => high cpu time We solve this MIQP for n = nu to ny

9. 4. Case Study : Evaporator System 2 MVs – F200, F1 10 candidate measurements – P2,T2,T3,F2,F100,T201,F3,F5,F200,F1 3 DVs – X1, T1, T200

10. Case Study : Results Data Results Controlled variables (c) Optimal individual measurements Loss2 = 3.7351 Optimal 4 measurement combinations Loss4 = 0.4515

11. Case Study : Results

12. Case Study : Computational time ** Branch and bound (BAB): Kariwala and Cao, IEEE Trans. (2010)

13. 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 5. Comparison of MIQP, Customized Branch And Bound (BAB) methods

14. MIQP formulation with structural constraints If the plant management decides to procure only 5 sensors (1 pressure, 2 temperature, 2 flow sensors) 2 MVs – F200, F1 3 DVs – X1, T1, T200 10 candidate measurements – P2,T2,T3,F2,F100,T201,F3,F5,F200,F1 Loss5-sc = 0.5379

15. 6. Conclusions • The self optimizing control non-convex problem is reformulated as convex problem • MIQP based formulation is presented for • Selection of CVs as optimal individual measurements • 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 • MIQPs are computationally intensive than customized Branch and Bound methods

16. Thank You Q&A