Stability, Networks: Stability, Networks: Control, and Optimality - - PowerPoint PPT Presentation

Complex Process Complex Process Networks: Networks: Stability,

Stability, Control, and Optimality Control, and Optimality

Kendell Kendell Jillson Jillson

• B. Erik
• B. Erik Ydstie

Ydstie Department of Chemical Engineering Department of Chemical Engineering Carnegie Mellon University Carnegie Mellon University Research funded by: NSF CTS Research funded by: NSF CTS-

• ITR 031 2771

ITR 031 2771 AIChE AIChE Annual Meeting Annual Meeting November 8, 2004 November 8, 2004

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Objective Objective

To introduce a modeling technique to To introduce a modeling technique to describe process networks describe process networks To develop methods for analyzing stability To develop methods for analyzing stability and control of process networks and control of process networks To establish a To establish a variational variational principle for principle for process networks to show optimality process networks to show optimality

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Why Networks? Why Networks?

Network representations provide a graphical Network representations provide a graphical interpretation of a process system interpretation of a process system Topology is important to the character of the Topology is important to the character of the network network Easy to break down into smaller subsystems Easy to break down into smaller subsystems Network descriptions could represent a wide Network descriptions could represent a wide-

• range of applications

range of applications… …

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… …From process From process flowsheets flowsheets… …

Douglas (1988)

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… …to Supply Chains to Supply Chains… …

Ydstie, et al. (2003)

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… …Chemical Reaction Networks Chemical Reaction Networks… …

Fishtik, et al (2004)

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… …even Bio Networks even Bio Networks

Glycosis and aromatic amino acid metabolic subsystem Ma, Guo, Zhao (2000)

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Process Network Process Network

Define Define

• Graph, G = (P,T,F)

Graph, G = (P,T,F)

Process (node) Process (node) Terminal Terminal Flow Flow

• State of each node,

State of each node, v vj

j

Extensive quantities Extensive quantities e.g. e.g.

• Potential of each node,

Potential of each node, w wj

j

Intensive quantities Intensive quantities e.g. e.g. Potential differences act as driving forces for flow Potential differences act as driving forces for flow

P T F

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Conservation and Continuity Conservation and Continuity

At each node: At each node: Around each loop: Around each loop:

f1 f3 f2 w1 w2 W21 W12

KCL KVL

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Stability and Control Stability and Control

Input Input-

• Output Control

Output Control Several techniques to show system can be Several techniques to show system can be stable with or without using control stable with or without using control Multiple connected stable systems not Multiple connected stable systems not necessarily stable

u y

S C

+

• necessarily stable

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Passivity Background Passivity Background

Passivity theory is used to show that a system can be Passivity theory is used to show that a system can be stabilized stabilized

• Originated from electrical circuit theory

Originated from electrical circuit theory

• A feedback or parallel connected system of passive subsystems

A feedback or parallel connected system of passive subsystems is also passive is also passive

Passivity inequality Passivity inequality

• With

With x,u,y x,u,y the states, inputs, and outputs to the network the states, inputs, and outputs to the network

• V(x

V(x) ) ≥ ≥ 0, x 0, x ≠ ≠ 0; V(0) = 0 0; V(0) = 0

Problem: To find a practical storage function Problem: To find a practical storage function

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Entropy Entropy

Concave function, Concave function, S(v S(v), which is homogeneous ), which is homogeneous

• f degree one
• f degree one

Motivated from molecular level Motivated from molecular level Boltzmann Boltzmann relation relation The 2 The 2nd

nd Law of Thermodynamics follows:

Law of Thermodynamics follows: Defines potentials at each process node Defines potentials at each process node

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Concavity of entropy Concavity of entropy

a1,2 a2,1

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Storage Function for Process Storage Function for Process Networks Networks

Define a storage function for network comparing Define a storage function for network comparing the states of two solutions based on potentials the states of two solutions based on potentials and inventories and inventories

• Using deviation variables based on these two

Using deviation variables based on these two solutions solutions

• Exploit the concavity of entropy

Exploit the concavity of entropy

• At each node:

At each node:

• For the whole network

For the whole network

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Passivity of Process Networks Passivity of Process Networks

Differentiation of storage function and use of Differentiation of storage function and use of an entropy balance over the network gives an entropy balance over the network gives The passivity inequality is developed: The passivity inequality is developed:

• Provided following relationships hold:

Provided following relationships hold:

Monotonic flow and production rates: Monotonic flow and production rates: Discrete Discrete Poincar Poincaré é inequality inequality

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Reactor Network Example Reactor Network Example

Three reactor nodes, and three terminals Three reactor nodes, and three terminals

• Reaction: A+B

Reaction: A+B C with rate = k C C with rate = k CA

AC

CB

B

• Transport governed by diffusion

Transport governed by diffusion

9 9 ODE ODE’ ’s s 27 27 Algebraic constitutive equations

Algebraic constitutive equations

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Control of Example Control of Example

Objective: Control flow rate of C at T Objective: Control flow rate of C at T3

3

Stabilized by a PI flow controller (K = 50, 1/ Stabilized by a PI flow controller (K = 50, 1/τ τ = 10) to a = 10) to a set set-

• point, f

point, fc

c(3) = 0.05

(3) = 0.05

• f

fC

C(3)

(3) y y L L3,3

3,3 of

• f f

fC

C

u u

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Optimality Optimality

Using a Using a variational variational principle, entropy production principle, entropy production

• f a network obeying monotonic constitutive
• f a network obeying monotonic constitutive

expressions is shown to be minimized expressions is shown to be minimized

• Define the entropy production

Define the entropy production

For unperturbed and perturbed network For unperturbed and perturbed network

• Perturbed network still obeys conservation laws at each node

Perturbed network still obeys conservation laws at each node

Entropy production never greater in unperturbed network Entropy production never greater in unperturbed network

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Optimality Result Optimality Result

Minimal entropy production in unperturbed Minimal entropy production in unperturbed solution, compared to a randomly perturbed solution, compared to a randomly perturbed network network

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Conclusions Conclusions

Process Networks modeled as a graph with Process Networks modeled as a graph with state state v v and potential and potential w w at each node at each node Storage function, Storage function, G G, used to show passivity , used to show passivity provided flow and production rates are provided flow and production rates are monotonic and positive monotonic and positive Conditions for minimum entropy production Conditions for minimum entropy production established established Simulation example with three nodes, three Simulation example with three nodes, three terminals, and three components illustrate terminals, and three components illustrate theory theory

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Topological Result Topological Result

A result similar to A result similar to Tellegen Tellegen’ ’s s Theorem is Theorem is developed developed Based only on topology of the network Based only on topology of the network Instead of power balance, provides an entropy Instead of power balance, provides an entropy balance balance Useful for further developments Useful for further developments

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Network Operators Network Operators

Transform variables to allow further Transform variables to allow further developments developments

• Operators for flows/productions must still obey

Operators for flows/productions must still obey inventory balance inventory balance

• Operators for potentials must obey loop balance

Operators for potentials must obey loop balance

Examples include: Examples include:

• Time averaging

Time averaging

• Fourier or Laplace Transforms

Fourier or Laplace Transforms

• Multiplication with constant matrices or vectors

Multiplication with constant matrices or vectors

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Monotonic Flow Monotonic Flow

For Joule For Joule-

• Thompson Flow

Thompson Flow

• with potentials defined as:

with potentials defined as:

Using the Gibbs Using the Gibbs-

• Duhem

Duhem equation, this can be equation, this can be re re-

• written as:

written as:

• Which is strictly monotonic for positive definite

Which is strictly monotonic for positive definite Λ Λ

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Acknowledgements Acknowledgements

Research funded by NSF CTS Research funded by NSF CTS-

• ITR 031

ITR 031 2771 2771 Thanks to Erik Ydstie and the rest of Team Thanks to Erik Ydstie and the rest of Team Ydstie Ydstie

Also thanks to D. Ortiz, C. Schilling, J. Damon, D. Lowe, K. Also thanks to D. Ortiz, C. Schilling, J. Damon, D. Lowe, K. Foulke Foulke, K. Millar and the , K. Millar and the rest of that rest of that “ “bunch of idiots bunch of idiots” ”