ESE601: Hybrid Systems

1. 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 ?

5. Conflict Resolution Maneuver

6. Computing Unsafe Sets

7. Safe Sets

8. 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.

9. The cell

10. Central dogma in molecular biology

11. 2102-01.jpg 2102-02.jpg 2102-03.jpg 2102-04.jpg 2102-05.jpg 2102-06.jpg 2102-07.jpg 2102-08.jpg 2102-09.jpg 2102-10.jpg 2102-11.jpg 2102-12.jpg 2102-13.jpg 2102-14.jpg 2102-14A.jpg 2102-15.jpg Quorum sensing in V. fischeri

12. 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.

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

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

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

18. V. fischeri mathematical model 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 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

19. 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

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

23. Concentration in nM seconds

24. 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.

25. Hybrid PWA systems

27. The model

29. 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

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

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

33. 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.

34. 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.

35. Discrete model

36. Hybrid model