Optimal and Robust Control

Design a robust controller for a distillation (also fractionating) column described in a few papers and a report on the web page [3], or the textbook [4] or even the diploma thesis [1]. Nonlinear simulation model is available for a free download [2] in a matlab code colamod.m.

Consider a full system with a feedback loop regulating the pressure already closed, that is,

• 4 control variables (in process control literature called manipulated variables - MV:
• reflux L [kmol/min], which is a flux from the condenser back to the upper stages of the column,
• boilup V [kmol/min], which is an inflow of the steam from a reboiler to the bottom of the column,
• distillate D [kmol/min], which is a product outflow at the output of the condenser
• bottom flow B [mol/min], which is a product outflow from the bottom of the column.
• 3 external disturbances:
• inflow of the material to be distilled F [kmol/min],
• molar concentration zF [%]
• fraction of the liquid in the inflowing material qF [%],
• 4 regulated variables (in process control literature called controlled variables - CV):
• molar concentration of the product at the condenser output yD [%],
• molar concentration of the product at the output from the bottom xB [%],
• molar volume of the blend at the bottom MB [kmol],
• molar volume of the blend at the top MD [kmol].

Prepare a technical report addressing the following:

• Introduction, problem description
• Describe the problem using one or two diagrams
• Discuss the goals of control
• Describe the major sources of uncertainty in the model
• Model of the system
• Specify all the physical parameters, operating conditions, and obtain a linear model
• Perform scaling of the model
• Analysis of the model
• Calculate various useful characteristics of the linear model: poles and zeros, static model, singular numbers and vectors, conditioning, RGA, and give some comments and interpretations.
• Do you expect troubles with controlling such system?
• Model order reduction
• Reduce the high order of the linear system so that the major part of the dynamics is kept. Beware of the instability of the original model.
• Design robust controllers using three methods
• minimizing the H∞ norm of the mixed sensitivity function,
• H∞ based loopshaping,
• minimizing the structured singular value (µ, SSV).
• Simulations
• Perform numerical simulations of closed-loop behaviour
• Analysis of robustness using µ
• choose suitable weighting filters for expressing requirements on performance and uncertainty, and plot their frequency characteristics,
• explain clearly how robust stability is defined for this problem (use block diagrams),
• calculate µ for nominal performance, robust stability and robust performance.
• Discussion of results and conclusions
• Discuss the achieved results.
• You can also give comments to the whole project and suggest improvements.

Literature and other resources

1. T. Dvořák. Řízení modelu technologického procesu. Diploma thesis (in Czech), Katedra řídicí techniky FEL ČVUT, 2004. Downloadable at https://support.dce.felk.cvut.cz/mediawiki/images/0/01/Dp_2004_dvorak_tomas.pdf.
2. S. Skogestad. MATLAB Distillation column model ("Column A"). Matlabské funkce, lze získat na http://www.nt.ntnu.no/users/skoge/book/1st_edition/matlab_m/cola/cola.html.
3. S. Skogestad. Distillation research and models, papers and reports http://www.nt.ntnu.no/users/skoge/distillation/.
4. S. Skogestad, I. Postlethwaite. Multivariable Feedback Control: Analysis and Design. John Wiley and Sons, 2nd ed., 2005. [Book entry on the publisher's website]