1. Modeling with hybrid systems
ESE601: Hybrid Systems
Modeling with hybrid systems

2. Various models Air traffic management (ATM) system
Hybrid systems in biology
Automotive engine
Mobile robot
Chemical reactions

3. Conflict Resolution in ATM*
See paper (R8) on the website

4. Conflict Resolution Protocol
Is this protocol safe ?

6. Conflict Resolution Maneuver

7. Computing Unsafe Sets
Operations done in Mathematica 4.0 took about 2 minutes each on a Pentium II 450MHz
Lambda = cos(\delta \phi)

8. Safe Sets

9. 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.

10. The cell

11. Central dogma in molecular biology

12. Quorum sensing in V. fischeri
                                    
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13. 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.

16. Quorum sensing in V. fischeri- +
CRP
cAMP OL
lux box OR
luxR
luxICDABEG
CRP binding site Ai + -
LuxR
LuxA
LuxI
LuxR Ai
LuxB Ai
Substrate Ai
luciferase Ai Ai Ai

17. Modeling of biological systems
CRP +
positive
START
luxR gene
STOP
regulation -
transcription
negative
translation
protein
LuxR Ai
chemical
reaction Ai

18. Modeling of biological systems1
CRP
0.5
positive +
START
luxR gene
STOP
regulation
transcription -
translation
negative
protein
LuxR Ai
chemical
reaction Ai

19. V. fischeri mathematical modelCo
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
luxR Co
CRP
% act
Co<Co_sw_r
CRP<CRP_sw_r
pr00
Co<Co_sw_r
CRP>CRP_sw_r
pr01
Co>Co_sw_r
CRP<CRP_sw_r
pr10
Co>Co_sw_r
CRP>CRP_sw_r
pr11
luxICDABEG

20. BioCharon = BioSketchPad + Charon
Biologist-friendly environment for
representing, storing, simulating,
and analyzing biomolecular networks.
Charon
A programming language for modeling,
simulating, analyzing, and designing
hybrid systems

22. GENE
mRNA
PROTEIN
SMALL
MOLECULE
TRANSPORT
TRANSCRIPTION
Ai + LuxR « Co
DEGRADATION
REGULATION
TRANSLATION
SOURCE

24. Concentration in nM
seconds

25. 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.

26. Hybrid PWA systems

28. The model

30. 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

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

32. The model
There are three locations (modes of dynamics) depending on the orientation of the wheels :

34. 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.

35. 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.

36. Discrete model

37. Hybrid model