Passivity based inventory control of particulate systems Christy M. - - PowerPoint PPT Presentation

Passivity based inventory control of particulate systems

Christy M. White

• B. Erik Ydstie

November 3, 2005

Department of Chemical Engineering Carnegie Mellon University Pittsburgh, PA

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SiHCl3 or SiH4 Si

H E A T H E A T

Siemens Reactor Batch Process 1100°C Dense Phase SiH4 Decomposition Particle Growth Size Distribution

High purity silicon production: µE and PV

Fluid Bed Reactor Continuous Process Large surface area 650°C Si powder H2 Si H E A T H E A T SiH4 H2

Heterogeneous → grey, crystalline solid Homogeneous → brown, amorphous powder

SiH4 SiH4 Si H2 + Si H2 + SiH4

particle growth = heterogeneous + scavenged powder

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Particulate processes

Control Challenges

• Nonlinear, long delays
• Limited measurements
• Few manipulated variables
• Uncertain parameters

External Coordinates space, time Internal Coordinates size, age, composition Population Balance Equation (PBE) Solution Techniques

• Moment transformation
• Discrete system

birth and death terms density distribution flow Control Techniques

• Linear and nonlinear MPC
• Nonlinear output feedback
• Passivity

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Discrete size distribution model

Derive conservation law over discrete size intervals

Internal flow Production External flow

track particle growth

i j n fi-1 fi ... ... fai,j qi ri

System dependent

• seed addition
• product removal

System dependent

• reaction
• condensation

Closure Relationships

• constant average size within interval
• real-valued “number” of particles
• aggregation proportional to particle

concentration (binary collision)

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Relationship to continuous population balance

Discrete model:

model is discrete version of PBE

As the number of size intervals approaches infinity: Re-write macroscopic values:

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Discrete model solution

Ordinary differential equations for mass in gas and solid phases Algebraic constitutive equations + MATLAB’s ode15s Range Adjustable Parameters

Aggregation proportionality constant Powder scavenging coefficient

Model validation

Silicon in Reactor Size Distribution

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Observer-based estimator (Dochain, et al.)

Observer theory → estimates of unknown states and parameters unknown parameters measured states Design estimator (similar to Luenberger)

measured or unmeasured x2 independent of parameter estimation

estimation Stable if 1. negative definite

• 2. persistently excited

correction terms

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Parameter estimation for fluidized bed reaction

Si powder H2 Si H E A T H E A T SiH4 H2 How much powder is scavenged (contributes to growth)? How much powder is lost? unknown parameter

Estimation equations:

total mass (M) measured

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Size control during continuous production

SiH4 H2 feed H2, powder Si seed Si product

Control: mass of specified size Manipulate: external flow rates Apply inventory control to system: Constant mass in reactor: Constant seed mass:

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Passivity

System Controller

– u d y +

System

u y Feedback connection of passive system and input strictly passive system of dissipation rate : Given storage function : System is

• 1. Passive if
• 2. Input strictly passive if

Passive with L2 gain = i.e.

Proportional PID Adaptive

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Input strictly passive controllers

System Controller

– u d y + Observer-based estimator: prediction error and persistent excitation → parameter convergence estimation stability → closed loop stability Passivity theory: set point error → (input/output) stability parameter convergence?

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Control of fluidized bed reactor

50 100 150 3.5 4 4.5 5 5.5 Time, h Total mass in reactor 50 100 150 1 1.1 1.2 1.3 1.4 Time, h Seed mass in reactor 50 100 150 2 4 6 8 Time, h Product flow 50 100 150 1 2 3 Time, h Seed flow

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Particle size achieved under control

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Parameter estimation

50 100 150 1.4 1.45 1.5 1.55 1.6 1.65 1.7 x 10

• 3

Time, h true ksc estimated ksc

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Summary

• Discrete population balance model of particle distribution

compares well with data

• Observer-based estimator provides parameter convergence
• Passivity based inventory control enables size control
• Further investigation of yield control and zero dynamics of

size distribution is required

Acknowledgements

• NSF Graduate Research Fellowship Program
• Solar Grade Silicon LLC
• Reactech Process Development Inc.
• Ydstie Research Group
• Denis Dochain, Catholic University of Louvain