Compartment Systems with Feedback Control

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Compartment Systems with Feedback Control

• by Tim Howard
• Columbus State University
• in collaboration with Jim Herod
• Georgia Tech (retired)

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Reference

• Mathematical Techniques for Biology and Medicine
  by William Simon, Dover Publishing Inc New
  York, 1986.

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Presentation Outline

• Review one compartment model w/feedback control
• Two-compartment systems with feedback control
  pooling delays
• Three-compartment systems with feedback control

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One Compartment Setup

• Compartments conceived as tanks of fluid

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Stability Questions

• Question 1 How does the system respond to
  abrupt changes in the fluid volume?
• Question 2 How does the system respond to
  abrupt changes in the outflow rate?

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1-Tank DE

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Question 1, Change Tank Vol.

• Abruptly add Q to tank while at steady state
• See analysis in Maple
• Original steady state restored

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Question 2, Change Outflow

• Larger r values desired for optimal control

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1-Tank with Pure Time Delay

• Delays may cause overcompensation
• Smaller r values prevent growing oscillations
• Larger r values to react to abrupt outflow
  changes
• Pure time delays relatively rare

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2-Tank Model Pooling Delay

• A tank separates inflow from objective tank

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Pooling Delay a look at the first tank

• Outflow rate proportional to volume
• Steady state is VSS Rin / k
• Outflow Rout Rin

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1st tank w/changing inflow

• What happens when inflow changes?
• See Maple analysis of outflow

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Pooling Delay w/Feedback Graphical Layout
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Pooling Delay w/Feedback DEs

• Steady state

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Pooling Delay w/Feedback

• See analysis in Maple

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Summary, changes in V1

• Abrupt change in V1 homogeneous d.e.

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Summary, change in outflow

• Abrupt change in RB nonhomogeneous d.e.

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Solving the diff. eq.s

• Use customary method for 2nd order d.e.s
• Recall harmonic oscillator (damping)
• Resume analysis in Maple

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Summary, over damped case

• Change V1 return to steady state w/out
  overshooting
• Change RB permanent change in steady state
• Minimize change w/large r value
• Limited control since r lt k1 / 4
• resume Maple analysis

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Summary, critically damped case

• Change V1 return to steady state w/out
  overshooting
• Change RB permanent change in steady state
• Minimize change w/large r value
• Limited control since r k1 / 4
• resume Maple analysis

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Summary, under damped case

• Change V1 return to steady state w/overshooting
• Change RB permanent change in steady state
• Minimize change w/large r value
• More control since r gt k1 / 4

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Overall summary, 2 tanks

• No overshooting, limited control
• Better control possible with overshooting
• Probs. w/overshooting, e.g. hypoglycemia

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Three Compartment System Figure
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Three Compartment System DEs

• Analyze in Maple

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Three Compartment System Summary

• Oscillating inputs magnified
• If r gt k1 k2 , system highly unstable

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Thank You for Coming

• Email thoward_at_colstate.edu
• WWW http//math.colstate.edu/thoward/