Spring semester 2006 ESE601: Hybrid Systems Modeling with hybrid systems.

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Presentation transcript:

Spring semester 2006 ESE601: Hybrid Systems Modeling with hybrid systems

Various models 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 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

jpg jpg jpg jpg jpg jpg jpg jpg jpg jpg jpg jpg jpg jpg A.jpg jpg Quorum sensing in V. fischeri

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.

CRP luxICDABEGluxR Ai LuxA LuxB luciferase LuxI Substrate LuxR lux box CRP binding site LuxR Ai OLOL OROR Quorum sensing in V. fischeri Ai cAMP Ai

Modeling of biological systems STARTSTOP luxR gene transcription translation regulation protein LuxR chemical reaction - + positive negative Ai CRP

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

CoCRP% act Co<Co_sw_l Co>Co_sw_l CRP<CRP_sw_l CRP>CRP_sw_l CRP<CRP_sw_l CRP>CRP_sw_l pl 00 pl 01 pl 10 pl 11 luxR CoCRP% act Co<Co_sw_r Co>Co_sw_r CRP<CRP_sw_r CRP>CRP_sw_r CRP<CRP_sw_r CRP>CRP_sw_r pr 00 pr 01 pr 10 pr 11 luxICDABEG V. fischeri mathematical model

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

TRANSPORT TRANSCRIPTION Ai + LuxR  Co DEGRADATION REGULATION TRANSLATION SOURCE GENE mRNA PROTEIN SMALL MOLECULE

seconds Concentration in nM

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.

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