ESE601: Hybrid Systems

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ESE601 Hybrid Systems

• Modeling with hybrid systems

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Various models

• Air traffic management (ATM) system
• Hybrid systems in biology
• Automotive engine
• Mobile robot
• Chemical reactions

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Conflict Resolution in ATM
See paper (R8) on the website
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Conflict Resolution Protocol
Is this protocol safe ?
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Conflict Resolution Maneuver
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Computing Unsafe Sets
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Safe Sets
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Hybrid systems in biology

• Biomolecular networks the networks comprised by
  genes and proteins in a living cell.
• Central dogma of molecular biology DNA makes
  RNA, RNA makes protein, protein makes the cell.
• See paper (R9) on the website.

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The cell
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Central dogma in molecular biology
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Quorum sensing in V. fischeri
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Vibrio fischeri

• V. fischeri is a marine bacterium. They are
  luminescent when live at high density.
• The hybrid model studies the phenomenon, where
  the bacteria can sense the presence of many other
  bacteria in the surrounding and trigger the
  increase in luminescence.
• The gene that regulates the luminescence is
  called lux.
• See paper (R10) on the website.

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Quorum sensing in V. fischeri
-

CRP
cAMP
OL
OR
lux box
luxICDABEG
luxR
CRP binding site

Ai
LuxR
-
LuxA
LuxI
LuxR
LuxB
Substrate
Ai
luciferase
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Modeling of biological systems
CRP

positive
luxR gene
START
STOP
regulation
transcription
negative
translation
protein LuxR
chemical reaction
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Modeling of biological systems
1
CRP
0.5

positive
luxR gene
START
STOP
regulation
transcription
-
translation
negative
protein LuxR
chemical reaction
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V. fischeri mathematical model
Co
CRP
act
ColtCo_sw_l
CRPltCRP_sw_l
pl00
ColtCo_sw_l
CRPgtCRP_sw_l
pl01
CogtCo_sw_l
CRPltCRP_sw_l
pl10
CogtCo_sw_l
CRPgtCRP_sw_l
pl11
luxR
Co
CRP
act
ColtCo_sw_r
CRPltCRP_sw_r
pr00
ColtCo_sw_r
CRPgtCRP_sw_r
pr01
CogtCo_sw_r
CRPltCRP_sw_r
pr10
CogtCo_sw_r
CRPgtCRP_sw_r
pr11
luxICDABEG
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BioCharon BioSketchPad Charon
BioSketchPad Biologist-friendly
environment for representing, storing,
simulating, and analyzing biomolecular networks.
Charon A programming language for
modeling, simulating, analyzing, and designing
hybrid systems
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Concentration in nM
seconds
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Hybrid model of a car engine

• The system to model is a four-cylinder in-line
  engine.
• The model is in the form of a hybrid piecewise
  affine system.
• The purpose of the modeling is to devise a
  controller for idle speed.
• See paper (R11) on the website.

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Hybrid PWA systems
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The model
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Hybrid system model of mobile robot

• The system modeled is a four-wheel-steered
  four-wheel-driven mobile robot.
• See paper (R12) on the website

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The model

• Hybrid model is proposed because of the need to
  switch controllers depending on the configuration
  of the robot.

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The model
There are three locations (modes of dynamics)
depending on the orientation of the wheels
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Stochastic hybrid model of chemical reactions

• The underlying idea is that when the number of
  molecules involved in the reaction is relatively
  small, deterministic model is no longer accurate.
• A chemical reaction can be thought of as
  collisions between the reactant molecules.
• See papers (R10) and (R13) on the website.

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Stochastic hybrid model of chemical reactions

• Thus, chemical reaction in the microscopical
  level is inherently stochastic.
• Consider the reaction

• The mass-action law dictates that

This approximation is good when the numbers of
molecules of A and B are high.
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Discrete model
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Hybrid model
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