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

away milesa until Cruise 1. 1

by heading Change 2.

d distance lateral until heading Maintain 3.

heading original to Change 4.

- by heading Change 5.

d- distance lateral until heading Maintain 6.

heading original to Change 7.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

                                 

   

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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

OL OR

-

+-

+

Quorum sensing in V. fischeri

Ai

cAMP

AiAi

Ai

Ai

Modeling of biological systems

START STOPluxR gene

transcription

translation

regulation

proteinLuxR

chemicalreaction

-+

positive

negative

Ai

Ai

CRP

transporttransformdecaysynthesisdt

]x[d CorLuxRAir

H

LuxRluxRT

dt

LuxRd

db

spl

AiLuxRAiLuxR //

)(

Co

-

Modeling of biological systems

START STOPluxR gene

transcription

translation

regulation

proteinLuxR

chemicalreaction

+

negative

positive

Ai

Ai

CRP

CRP

)(CRPCRPluxR

10.5

1swCRP

transporttransformdecaysynthesisdt

]x[d

RNA

ColuxR

CRPluxRc H

luxRbCoCRPT

dt

luxRd ])()([

)(Co

2swCRP

RNAijc H

luxRbplT

dt

luxRd )(

)(

RNAklc H

luxICDABEGbprT

dt

luxICDABEGd )(

)(

CokiLuxRAikH

LuxRluxRT

dt

LuxRd

spl 11 _

)(

spl H

LuxIluxICDABEGT

dt

LuxId

)(

)__(__

)()_(

112 iAieAinCokiLuxRAikH

iAiSLuxIk

dt

iAid

Ai

CokiLuxRAikH

Co

dt

Cod

sp11 _

)(

)__(_)_(

iAieAinH

eAi

dt

eAid

Ai

2)(

Sdt

CRPd

Co CRP % act

Co<Co_sw_l

Co<Co_sw_l

Co>Co_sw_l

Co>Co_sw_l

CRP<CRP_sw_l

CRP>CRP_sw_l

CRP<CRP_sw_l

CRP>CRP_sw_l

pl00

pl01

pl10

pl11

luxR

Co CRP % act

Co<Co_sw_r

Co<Co_sw_r

Co>Co_sw_r

Co>Co_sw_r

CRP<CRP_sw_r

CRP>CRP_sw_r

CRP<CRP_sw_r

CRP>CRP_sw_r

pr00

pr01

pr10

pr11

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 CoDEGRADATION

REGULATION

TRANSLATION

SOURCE

GENE

mRNA

PROTEIN

SMALLMOLECULE

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