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ESE601: Hybrid Systems
Modeling with hybrid systems
Spring semester 2006
Various models
•
•
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•
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Air traffic management (ATM) system
Hybrid systems in biology
Automotive engine
Mobile robot
Chemical reactions
Conflict Resolution in ATM*
See paper (R8) on the website
Conflict Resolution Protocol
1. Cruise until a1 miles away
2. Change heading by 
3. Maintain heading until lateral distance d
4. Change to original heading
5. Change heading by - 
6. Maintain heading until lateral distance - d
7. Change to original heading
Is this protocol safe ?
Conflict Resolution Maneuver
Computing Unsafe Sets
Safe Sets
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.
The cell
Central dogma in molecular biology
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.
Quorum sensing in V. fischeri
+
CRP
-
cAMP
OL
OR
lux box
luxR
luxICDABEG
CRP binding site
LuxR
-
+
Ai
LuxI
LuxA
LuxR
Ai
LuxB
Substrate
Ai
Ai
luciferase
Ai
Ai
Ai
Modeling of biological systems
d ( LuxR)
LuxR
 Tl luxR 

dt
H sp
d[ x]
 synthesis  decay  transform  transport
d
dt  r b
Ai
LuxR

r
Co
Lu xR / Ai
Lu xR / Ai
CRP
+
positive
START
luxR gene
regulation
negative
transcription
translation
-
protein
LuxR
Ai
Co
chemical
reaction
Ai
STOP
Modeling of biological systems
d[ x]
 synthesis  decay  transform  transport
dt
1
0.5
CRP
+
positive
 CRP
luxR (CRP )
CRPsw1
START
luxR gene
CRPsw2
CRP
STOP
regulation
negative
transcription
translation
-
protein
LuxR
Ai
Co
chemical
reaction
Ai
d (luxR)
luxR
Co
 Tc [ CRP
luxR (CRP ) luxR (Co)  b] 
dt
H RNA
V. fischeri mathematical model
d (luxR)
luxR
 Tc ( plij  b) 
dt
H RNA
d (luxICDABEG)
luxICDABEG
 Tc ( prkl  b) 
dt
H RNA
d ( LuxR)
LuxR
 Tl luxR 
 k1LuxRAi _ i  k 1Co
dt
H sp
Co
CRP
% act
Co<Co_sw_l CRP<CRP_sw_l pl00
Co<Co_sw_l CRP>CRP_sw_l pl01
Co>Co_sw_l CRP<CRP_sw_l pl10
Co>Co_sw_l CRP>CRP_sw_l pl11
d ( LuxI )
LuxI
 Tl luxICDABEG 
dt
H sp
luxR
d ( Ai _ i )
Ai _ i
 (k2 LuxI ) S 
 k1LuxRAi _ i  k1Co  n( Ai _ e  Ai _ i )
dt
H Ai
d (Co)
Co

 k1LuxRAi _ i  k 1Co
dt
H sp
Co
CRP
% act
Co<Co_sw_r CRP<CRP_sw_r pr00
d ( Ai _ e)
Ai _ e

 n( Ai _ e  Ai _ i )
dt
H Ai
Co<Co_sw_r CRP>CRP_sw_r pr01
d (CRP)
 S2
dt
Co>Co_sw_r CRP>CRP_sw_r pr11
Co>Co_sw_r CRP<CRP_sw_r pr10
luxICDABEG
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
GENE
REGULATION
TRANSCRIPTION
mRNA
TRANSLATION
PROTEIN
Ai + LuxR  Co
DEGRADATION
TRANSPORT
SOURCE
SMALL
MOLECULE
Concentration in nM
seconds
Hybrid model of a car engine
• The system to model is a four-cylinder inline 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.
Hybrid PWA systems
The model
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
The model
• Hybrid model is proposed because of the need to switch controllers
depending on the configuration of the robot.
The model
There are three locations (modes of dynamics) depending
on the orientation of the wheels :
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.
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.
Discrete model
Hybrid model