ESE 601 Hybrid Systems Modeling with hybrid systems

ESE 601: Hybrid Systems Modeling with hybrid systems Spring semester 2006

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

Conflict Resolution in ATM* See paper (R 8) on the website

Conflict Resolution Protocol 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 (R 9) on the website.

The cell

Central dogma in molecular biology

Quorum sensing in V. fischeri 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 A. jpg 2102 -15. jpg

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 (R 10) on the website.

Quorum sensing in V. fischeri + CRP OL - c. AMP OR lux box lux. R lux. ICDABEG CRP binding site Lux. R - + Ai Lux. I Lux. A Lux. R Ai Lux. B Substrate Ai Ai luciferase Ai Ai Ai

Modeling of biological systems CRP + positive START lux. R gene regulation negative transcription translation - protein Lux. R Ai chemical reaction Ai STOP

Modeling of biological systems 1 0. 5 CRP + positive START lux. R gene regulation negative transcription translation - protein Lux. R Ai chemical reaction Ai STOP

V. fischeri mathematical model Co CRP % act Co<Co_sw_l CRP<CRP_sw_l pl 00 Co<Co_sw_l CRP>CRP_sw_l pl 01 Co>Co_sw_l CRP<CRP_sw_l pl 10 Co>Co_sw_l CRP>CRP_sw_l pl 11 lux. R Co CRP % act Co<Co_sw_r CRP<CRP_sw_r pr 00 Co<Co_sw_r CRP>CRP_sw_r pr 01 Co>Co_sw_r CRP<CRP_sw_r pr 10 Co>Co_sw_r CRP>CRP_sw_r pr 11 lux. ICDABEG

Bio. Charon = Bio. Sketch. Pad + Charon Bio. Sketch. Pad 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 m. RNA DEGRADATION TRANSLATION PROTEIN Ai + Lux. R « Co TRANSPORT SOURCE SMALL MOLECULE

Concentration in n. M 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 (R 11) 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 (R 12) 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 (R 10) and (R 13) 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