IEEE TENCON 2004

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Performance Limits of Linear Control Distillation
Column under Disturbances with Bounds on
Magnitudes and Derivatives
Wathanyoo Khaisongkram and David
Banjerdpongchai Dept. of Elec. Eng.,
Chulalongkorn Univ., THAILAND
IEEE TENCON 2004 19th IEEE Region 10 Conference
Lotus Hotel Pang Suan Kaew November 23, 2004
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Outline

• Introduction

• Distillation Column

• Design Problem

• Convex Design Method

• Design Results

• Conclusions

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Introduction
Controller Design
Ref. Tracking
Design Specification
Disturb. Rejection
Others
Stability
Performance
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Disturbance Descriptions

• Bound on magnitude
• Inertialess behavior

Somewhat conservative!
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Disturbance Descriptions

• Bound on magnitude
• Bound on derivative (Birch Jackson, 1959)

More Practical Realistic!
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Worst-case Performance
Disturbance model
Input set W (disturbances)
Output set
Performance index
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Distillation Column
Control system structure L-V (Luyben, 1990)

• Regulated outputs
• Top composition xD
• Bottom composition xB

• Control inputs
• Reflux rate L
• Reboiler rate V

• Disturbance
• Feed flow rate F

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First-Order Plant Model
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Framework of Control Design
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Integrator augmentation
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Design Problem

• Disturbance characteristics
• Magnitude is restricted by dimension of piping
• system. Typical bound is 1020 (lb-mol)/min.
• Rate of change is restricted by inertia of
  fluid
• liquid pump. Typical value is 50100
  (lb-mol)/min2

In this work, we choose the bounds as 10
(lb-mol)/min and 10 (lb-mol)/min2
Input space
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Design Specification
Objective minimize maximum deviation of xD and
xB.
This is the trade off between
Constraints very high reflux rate and reboiler
rate cause column flooding, while very low rates
cause column channeling.
This implies the constraints
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Convex Optimization Problem

• Choose r from 0.006 to 0.010, total 29 samples.
• Apply ellipsoid algorithm to find the minimizer.

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Convex Design Method (Boyd et al., 1988)
Control system reformation
Youla parameterization
Ritz approximation
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Subgradient

• Subgradient of f(x) at x0 is a linear function
  of x
• fsg(x) gT x, g, x Rn
• satisfying the condition
• f(x) gt f(x0) gT(x - x0), x Rn

• Subgradient gives the direction of the half
• space that contains the minimizer.

• Convex function has at least one subgradient.

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Ellipsoid Algorithm (Akgul, 1984)

• Simple and effective implementation

• Computational complexity

• Initial ellipsoid must contain a minimizer

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Basic Ellipsoid Algorithm
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Modified Ellipsoid Algorithm

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Design Results
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Nonlinear Simulation
Testing input in W
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Nonlinear Simulation
xD
xB
L
V
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Conclusions

• A realistic approach of modeling disturbances
  gives rise
• to a suitable and practical performance index.

• Convex design method efficiently improves
  performance
• of the distillation system, compared to
  previous PI design.

Performance Limits of Distillation Column