H and  Optimal Controller Design for
the Shell Control Problem
D. Chang, E.S. Meadows, and S.L. Shah
Department of Chemical and Materials Engineering
University of Alberta
CSChE Annual Meeting 2002
Outline
Shell control problem description
Key objectives
Design criteria and methodology
H and  optimal controller results
Prototype test case results
Conclusions
CSChE Annual Meeting 2002: Vancouver, BC
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Shell Control Problem
Prett and Morari. Shell Process Control Workshop, 1987.
CSChE Annual Meeting 2002: Vancouver, BC
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Key Objectives
Design a robustly stable controller
satisfying the following constraints:
 top end point and bottom reflux temperature
is constrained between 0.5 and –0.5
 top draw, side draw and bottoms reflux duty is
constrained between 0.5 and –0.5
 Manipulated variables have maximum move
sizes between 0.05 and –0.05
CSChE Annual Meeting 2002: Vancouver, BC
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Generalized Plant Structure
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Block Singularity
A
P( s)   C1
C2
B1
D11
D21
B2 
D12 
0 
spy(D)
spy(D’)
Avoid singular control problems
0 
D12    and D21  0 I 
I 
Meaning D12 must be full column
and D21 must be full row rank.
(Zhou, Doyle, and Glover, 1996)
D before addition of setpoints
D’ after addition of setpoints
CSChE Annual Meeting 2002: Vancouver, BC
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Exogenous Inputs Revisited
Prett and Morari. Shell Process Control Workshop, 1987.
CSChE Annual Meeting 2002: Vancouver, BC
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Open Loop Characteristics
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Exogenous Output Weights
 Bs  1 
ws   A  

 Cs  1 
2
Performance weight
•Crossover = 0.006 rad/sec  167 sec
•10% S.S. offset
Controller output weight
•Crossover = 0.9 rad/sec  1.1 sec
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H Controller Response
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Robust Stability of H Controller
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 Optimal Response
iteration 1
iteration 2
iteration 3
iteration 4
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Prototype Test Cases
Worst case uncertainty set calculated by Matlab :
1= 1 2= -1, 3= -0.7585, 4= -0.5549, 5= 0.2497
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 Optimal Time Response
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Worst Case Input Frequency
w  0.2754 rad/s
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Input and Rate Responses
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Conclusions
A robustly stable multivariate controller
can be designed with relative ease
All of the input, output and rate
constraints were met for the Shell
control problem
 analysis provides a consistent
framework for evaluating robust
performance for all controllers
CSChE Annual Meeting 2002: Vancouver, BC
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Acknowledgements
Dr. E.S. Meadows
Dr. S.L. Shah
CPC group at U of A
NSERC
iCore
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Questions?
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