1. 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

2. 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=?

3. 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

4. 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

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

6. 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

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

8. 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

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

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. 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

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