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

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

Ref: Halvorsen et al. I&ECR, 2003 Mathematically Optimal steady-state operation Controlled variables, cs = constant + - K H y c u u J d Loss 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 3

Objective function unaffected by D. So can choose freely. 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

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

Controlled variable selection st. 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

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

Case Study : Distillation Column T1, T2, T3,…, T41 Tray temperatures qF 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

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

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)

Comparison of MIQP, Customized Branch And Bound (BAB) methods Customized BAB methods can handle only monotonic cost functions (J) Customized routines are required BABs require a deeper understanding of the customized routines to solve problems with structural constraints Computationally faster than MIQPs as they exploit the monotonic properties efficiently Monotonicity of the measurement combinations needs to be checked before using PB3 for optimal measuement subset selections MIQP formulations can accommodate wider class than monotonic functions (J) MIQPs are solved using standard cplex routines MIQPs are simple and are easy to incorporate few structural constraints MIQPs are computationally intensive than BAB methods Single MIQP formulation is sufficient for the described problems

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

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