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

High purity silicon production: µ E and PV Si powder Dense Phase H 2 SiH 4 Decomposition Particle Growth H H Size Distribution E E H H Si A A E E Heterogeneous → T T A A grey, crystalline solid T T H 2 Si SiH 4 + SiH 4 Si H 2 SiHCl 3 or SiH 4 Homogeneous → brown, amorphous powder Siemens Reactor Fluid Bed Reactor SiH 4 SiH 4 Batch Process Continuous Process 1100°C Large surface area H 2 Si + 650°C particle growth = heterogeneous + scavenged powder 2

Particulate processes Population Balance Equation (PBE) External space, density birth and flow Coordinates time distribution death terms Solution Techniques Internal size, age, - Moment transformation Coordinates composition - Discrete system Control Challenges Control Techniques - Nonlinear, long delays - Linear and nonlinear MPC - Limited measurements - Nonlinear output feedback - Few manipulated variables - Passivity - Uncertain parameters 3

Discrete size distribution model f i-1 f i Derive conservation law over ... i j ... n discrete size intervals q i r i fa i,j Production External flow Internal flow Closure Relationships System dependent - constant average size within interval - reaction - real-valued “number” of particles - condensation System dependent - aggregation proportional to particle - seed addition concentration (binary collision) - product removal track particle growth 4

Relationship to continuous population balance Discrete model: Re-write macroscopic values: As the number of size intervals approaches infinity: model is discrete version of PBE 5

Discrete model solution Ordinary differential equations Algebraic constitutive + for mass in gas and solid phases equations MATLAB’s ode15s Adjustable Parameters Range Powder scavenging coefficient Aggregation proportionality constant 6

Model validation Silicon in Reactor Size Distribution 7

Observer-based estimator (Dochain, et al.) Observer theory → estimates of unknown states and parameters measured unknown parameters states Design estimator (similar to Luenberger) estimation correction terms Stable if 1. negative definite 2. persistently excited measured or unmeasured x 2 independent of parameter estimation 8

Parameter estimation for fluidized bed reaction How much powder is scavenged (contributes to growth)? How much powder is lost? Si powder H 2 unknown parameter H H E E A A Estimation equations: T T total mass (M) measured SiH 4 Si H 2 9

Size control during continuous production H 2, powder Control: mass of specified size Manipulate: external flow rates Si seed Apply inventory control to system: Si product SiH 4 H 2 feed Constant mass in reactor: Constant seed mass: 10

Passivity Given storage function : System is u y System 1. Passive if 2. Input strictly passive if Feedback connection of passive system and input strictly passive system of dissipation rate : d u y + Passive with L 2 gain = System i.e. – Controller 11

Input strictly passive controllers Proportional d u y + System – PID Controller Adaptive 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? 12

Control of fluidized bed reactor 5.5 1.4 Total mass in reactor Seed mass in reactor 5 1.3 4.5 1.2 4 1.1 3.5 1 0 50 100 150 0 50 100 150 Time, h Time, h 8 3 Seed flow Product flow 6 2 4 1 2 0 0 0 50 100 150 0 50 100 150 Time, h Time, h 13

Particle size achieved under control 14

Parameter estimation -3 1.7 x 10 true ksc estimated ksc 1.65 1.6 1.55 1.5 1.45 1.4 0 50 100 150 Time, h 15

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 16