Nonlinear Control Systems ECSE6420

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Nonlinear Control SystemsECSE-6420

• Spring 2009
• Lecture 2 20 January 2009

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Models and equilibrium points

• Consider a dynamical system described by
• Special cases
• F(x,t) ? unforced dynamics
• F(x) ? autonomous
• Equilibrium points/set satisfies F(x)0.

state equation
output equation
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Tunnel diode
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Tunnel diode
I
V
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Tunnel diode
V
E/R
time
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Tunnel diode
V
E/R
time
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Van der Pol oscillator
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Van der Pol oscillator
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Predator Prey model
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Binary chemical reactions
B
B
A
A
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Central dogma of biology
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Network of genes
Gene 1
Gene 2
Transcription
Inhibition
mRNA
mRNA
Translation
Activation
Protein 2
Protein 1
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Effective gene-gene network
Gene
auto-activation
auto-inhibition
Gene
Genes regulate each other by proxy
Gene
Protein
Protein
Protein
Protein
Protein
Gene
activation
inhibition
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Effective gene-gene network
Gene
Gene
Gene
Gene
activation
inhibition
Brazhnik, de la Fuente, and Mendes, Trends in
Biotechnology 20, 467-472, 2002
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Models for gene regulation
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Repressilator simulation
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Engineering a genetic oscillator
Michael B. Elowitz and Stanislas Leibler Nature
403, 335-338, 2000
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Green fluorescent protein
Excitation of gfp by photons
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Toggle switch
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Genetic toggle switch
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Biological phenomenology
lac Z
lac Y
lac A
lac I
mRNA
mRNA
b galactosidase
repressor
permease
allolactose
external lactose
Internal lactose
by Vijay Kumar
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Biological phenomenology
lac Z
lac Y
lac A
lac I
mRNA
mRNA
M
B
b galactosidase
repressor
P
permease
E
Internal TMG
external TMG
Internal TMG
T
by Vijay Kumar
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Biochemical reactions
Yildirim Mackey, Biophys J, 2003.
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Two stables states
mRNA concentration
High state
Low state
External TMG concentration
Two modes of equilibria, low concentration
(uninduced) and high concentration (induced).
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Bistability in nature E. coli
Bacteria exploit bistability in adaptation!