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John Doyle 道陽Jean-Lou Chameau Professor
Control and Dynamical Systems, EE, & BioE
tech1#Ca
Universal laws and architectures:Theory and lessons from
brains, hearts, cells, grids, nets, bugs, fluids, bodies, planes, docs, fire, fashion,
earthquakes, music, buildings, cities, art, running,cycling, throwing, Synesthesia, spacecraft, statistical mechanics
https://www.cds.caltech.edu/~doyle
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The following preview is approved for all audiences.
https://rigorandrelevance.wordpress.com/author/doyleatcaltech/
https://www.cds.caltech.edu/~doyle
Universal laws and architectures:
The videos
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• SDN, IOT, IOE, CPS, etc..• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Major transitions
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• SDN, IOT, IOE, CPS, etc..• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Major transitions
Theory of• Evolution• Architecture• Complexity• Networks
Our heroes
Evolution ComplexityArchitecture
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{case study}UTheorems+
• Lots of aerospace• Wildfire ecology• Earthquakes• Physics:
– turbulence, – stat mech (QM?)
• “Toy”: – Lego– clothing, fashion
• Buildings, cities• Synesthesia
• Brains• Bugs (microbes, ants)• Nets/Grids (cyberphys)• Medical physiology
• SDN, IOT, IOE, CPS, etc.• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
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• Brains• Bugs (microbes, ants)• Nets/Grids (cyberphys)• Medical physiology
• Lots of aerospace• Wildfire ecology• Earthquakes• Physics:
– turbulence, – stat mech (QM?)
• “Toy”: – Lego– clothing, fashion
• Buildings, cities• Synesthesia
Precursorsin 1940s
U{case study}Theorems+
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• Brains• Bugs (microbes, ants)• Nets/Grids (cyberphys)• Medical physiology
• Lots of aerospace• Wildfire ecology• Earthquakes• Physics:
– turbulence, – stat mech (QM?)
• “Toy”: – Lego– clothing, fashion
• Buildings, cities• Synesthesia
2012
Precursorsin 1940s
U{case study}Theorems+
1980s
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2006 - 2014: Department Head Aerospace Engineering and Mechanics2001 - 2014: Professor AEM and Control Science & Dynamical Systems Center1996 - 2001: Associate Professor AEM and CSDSC1995 - 2014: Co-Director Control Science and Dynamical Systems Program1992 - 2004: Director of Grad Studies CSDS Department1990 - 1996: Assistant Professor AEM1989 : Ph.D. Aeronautics, California Institute of Technology
Gary Balas@UMN
1990-2014
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Theorems+
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism/balance
{case study}
U
• Brains• Familiar, accessible• Live demos!• Cheap, reproducible• Open• Unavoidable
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Caveats and Issues• Bad scholarship, cinematography, diplomacy, organization• Badly organized (Videos, slides, papers in Dropbox)• More questions than answers• Trailer for videosBut• Applicable• Accessible (even latest theory research is relatively…)• Almost undergrad for almost everything• “Obvious” (if in retrospect)• Eager for feedback (and also new material)
Page 11
Laws as Tradeoffs
costly
fragile
efficient
robust
Function=Locomotion
Ideal
Page 12
costly
fragile
efficient
robust
4x
>2x
Tradeoffs
Ideal
Page 13
fragile
robust
“costly”“efficient”
Similar• Tradeoffs• Laws• MechanismsBut simpler• Models?• Experiments?
Page 14
fragile
robust
crash
short long
hard
harder
“costly”“efficient”
Similar• Tradeoffs• Laws• MechanismsBut simpler• Models• Experiments
Page 15
Simplified inverted pendulum
Act
l
Page 16
l
l length (to COM)g gravityv control (acceleration)
y
x
θ
v
Simplified inverted pendulum
Act
l
Page 17
l
l length (to COM)g gravityv control (acceleration)
y
x
θ
v
Simplified inverted pendulum
Actsin cos
sinO
xl g vy x l
vθ θ θ
θ+ = −= +
=
l
Page 18
l
l length (to COM)g gravityv control (acceleration)
y
x
θ
v
Simplified inverted pendulum
Act
gpl
=
Instabilitysin cos
sinO
xl g vy x l
vθ θ θ
θ+ = −= +
=
l
Page 19
l
l length (to COM)g gravityv control (acceleration)
y
x
θ
v
Simplified inverted pendulum
Act
gpl
=
Instabilitysin cos
sinO
xl g vy x l
vθ θ θ
θ+ = −= +
=
l
Page 20
eye vision
Act
delayl
N noiseE error
( ) ET jN
ω =
Simplified sensorimotor control
gpl
=
Instability
Control?
Page 21
eye vision
slow
Act
delayl
N noiseE error
( ) ET jN
ω =
Simplified sensorimotor control
gpl
=.3sτ ≈
InstabilityControl
brain
Page 22
Amplification (noise to error)?
eye vision
Act
delay
Control
l
NE
( ) ET jN
ω =
gpl
= .3sτ ≈
Instability
Page 23
Entropy rate
Energy (L2)
eye vision
Act
delay
Control
l
NE
( ) ET jN
ω =
gpl
= .3sτ ≈
( ) ( )exp ln
expT
pT
τ∞
≥
∫Amplification (noise to error) theorem:
Necessary!
Page 24
Intuition
( )exp pt
delay τ
Before you can
react
time
state 1pl
∝
.3sτ ≈
Entropy rate
Energy (L2)
( ) ( )exp ln
expT
pT
τ∞
≥
∫
Page 25
P
+
noise
error
C
( ) ET jN
ω =
( ) ( ){ }sup sup Re( ) 0|T T j T s sωω
= = ≥∞Max modulus
Proof?
( ) ( ) ( )( )
( )
1
exp
( ) ( ) exp
exp
Ms P s s
T M M p p p
T p
P τ
τ
τ
−≥ ≥ ≥
≥
∞ ∞
∞
= − ⇒
= Θ
⇒
( ) ( )1
1
( ) ( )
P p T p
M p p −
= ∞⇒ =
⇒ = Θ
( ) ( )( ) ( ) ( ) ( ) 1
exp
T s M s s j
s s
ω
τ
= Θ Θ =
Θ = −
Undergrad math
Page 26
0.2 0.4 0.6 0.8 1
2
4
6
8
10
0
Length l, m
.3sτ = Shorter?
( )expT pτ≥
fragility
gpl
=
sin cossinO
x vl g vy x lθ θ θ
θ
=
+ = −= +
delay
noise
error
Length l
.3sτ =
Page 27
0.2 0.4 0.6 0.8 1
2
4
6
8
10
0
Length l, m
.3sτ = Shorter
fragility
gpl
=
( )exp pτsin cos
sinO
x vl g vy x lθ θ θ
θ
=
+ = −= +
delay
noise
error
Length l
.3sτ =
( )expT pτ≥
Page 28
0.2 0.4 0.6 0.8 1
2
4
6
8
10( ) ( )exp ln
expT
pT
τ∞
≥
∫
0
Theory
Length l, mdelay
noise
error
Length l
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0.2 0.4 0.6 0.8 1
2
4
6
8
10( ) ( )exp ln
expT
pT
τ∞
≥
∫
0
HardHarderTheory
& Data
Length l, mdelay
noise
error
Length l
Page 30
eye vision
Act
delay
Control
l
N
E
( ) ET jN
ω =
gpl
= .3sτ ≈
Necessary!
Entropy rate
Energy (L2)
( ) ( )exp ln
expT
pT
τ∞
≥
∫Robust to “noise” model
Page 31
fragile
robust
crash
short long
hard
harder
Law #1 : MechanicsLaw #2 : GravityLaw #3 : Light/visionLaw #4 : ( )expT pτ≥
Yoke Peng Leong
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eye vision
Act
delay
Control
l
NE
( ) ET jN
ω =gpl
= .3sτ ≈
Necessary!
Universal lawsMechanics+
Gravity +Light +
( ) ( )exp ln
expT
pT
τ∞
≥
∫fragile
robust
crash
short long
hard
harder
Page 33
hard harder ???
l
Ol
Page 34
harder ???
Ol l≤
Ol l≥
Ol
Page 35
( ) ( )exp ln
expT z pp
z pTτ
∞
+ ≥−
∫
Unstable zeros
( )21 11 1
1 1
O
O
O
O
g gz pl l l
l l lz pz p l l l
l z pl z p
ααααα
= =−
+ −+=
− − −
+ −+ + −= → = =
− − −
Simple analytic formulas
Page 36
P
+
noise
error
C
( ) ( ){ }sup sup Re( ) 0|T T j T s sωω
= = ≥∞Proof?
( ) ( )
( ) ( ) ( ) ( ) 1
exp
T s M s s js zs ss z
ω
τ
= Θ Θ =
−Θ = −
+
( ) ( ) ( )
( )
( )
1
exp
( ) ( ) exp
exp
Ms zs P s ss z
z pT M M p p pz p
z pT pz p
P τ
τ
τ
−≥ ≥ ≥
≥
∞ ∞
∞
− = − ⇒ + +
= Θ−
+⇒
−
Undergrad math
Page 37
( )exp z ppz p
τ +−
10
100
Length, m.1 1.5.2
2
.3sτ =
hardest!
( ) ( ) ( )expexp
exl
pn z pp
z pT
pT
τ τ∞
≥ ≥+
−
∫
1Ol l≤ =
Theory
Page 38
( )exp pτ
( )exp z ppz p
τ +−
10
100
Length, m.1 1.5.2
2
1pl
∝
.3sτ =
hardest!hard
( ) ( ) ( )expexp
exl
pn z pp
z pT
pT
τ τ∞
≥ ≥+
−
∫
Page 39
hard harder hardest!
What is sensed matters.
Unstable pole Unstable zero
0l l≤0l l≈
Page 40
robust
short
fragile
mass.1 1
1g 10kg
Page 41
Fragile to• Up• Length l• Length lo (sense)• Noise• Medial direction• Close one eye• Stand one leg• Alcohol• Age• …
fragile
robust
short long
Robust to• mass• down
delay
Instability
Sensors/actuators
Delay
Page 42
Entropy rate
Energy (L2)
eye vision
Act
delay
Control
l
E
gpl
= .3sτ ≈
( )( )
exp ln
expQuantized
T
T pτ∞
∝
∫
Necessary!
Robust to “noise” model
Spiking neurons?Discrete axons?Distributed control?
Page 43
Entropy rate
Energy (L2)
eye vision
Act
delay
Control
l
E
gpl
= .3sτ ≈
( )( )
exp ln
expQuantized
T
T pτ∞
∝
∫
Necessary!
Quantized
Robust to “noise” model
Spiking neurons?Discrete axons?Distributed control?
Page 44
fragile
robust
crash
short long
hard
harder
hardest!
crash
Theory & data
Page 45
fragile
robust
short long
Gratuitously fragile
Example of sparse/bad sensing
Page 46
fragile
robust
short long
Gratuitously fragile
Look up & shorten
Page 47
costly
fragile
efficientrobust
Just add a bike?
Page 48
costly
fragile
efficientrobust
Learn
Page 49
costly
fragile
efficientrobust
Learn
Learn
Page 50
fragile
robust
“costly”“efficient”
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism/balance
Major transitions
unstable
Page 51
fragile
robust
crash
short long
hard
harder
“costly”“efficient”
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism/balance
Major transitions
Page 52
fragile
robust
crash
short long
hard
harder
“costly”“efficient”
Similar• Tradeoffs• Laws• Mechanisms
But simpler• Models• Experiments
Page 53
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Balance (biped)• Maternal care• Warm blood• Flight (streamlining)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Efficiency/instability/layers/feedback• Efficient but
unstable
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• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Balance (biped)• Maternal care• Warm blood• Flight (streamlining)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Efficiency/instability/layers/feedback• Efficient but
unstable• Needs control
‒ Complex‒ Distributed‒ Layered‒ Active‒ Laws/tradeoffs
How universal?
Page 55
1. Chandra, Buzi, Doyle (2011) Glycolytic oscillations and limits on robust efficiency. Science
2. Gayme, McKeon, Bamieh, Papachristodoulou, Doyle (2011) Amplification and Nonlinear Mechanisms in Plane Couette Flow, Physics of Fluids
3. Li, Cruz, Chien, Sojoudi, Recht, Stone, Csete, Bahmiller, Doyle (2014) Robust efficiency and actuator saturation explain healthy heart rate control and variability, PNAS
Biology
Physics
Medicine
• Robust efficiency• Undergrad (mostly)• “Obvious” (in retrospect)• Extreme bimodal reviews
+ tutorial videosPapers
Veryuniversal
Page 56
Chandra, Buzi, and Doyle UG biochem, math, control theory
InsightAccessibleVerifiable
Veryuniversal
Page 57
too fragile
complex
robustshort
cheaprobust
Balancing
Glycolysis
Law #1Law #2Law #3
Law #1Law #2Law #3
• Efficient but unstable• Needs complex/active control• With associated laws? Specific
Specific
Page 58
too fragile
complex
robustshort
cheaprobust
Balancing
Glycolysis
Law #1Law #2Law #3
Law #1Law #2Law #3
• Efficient but unstable• Needs complex/active control• With associated laws? Specific
Specific
Law #4z pTz p+
≥−
Universal
wasteful
fragile
efficient
robust
Page 59
too fragile
complex
robustshort
cheaprobust
Balancing
Glycolysis
Law #4z pTz p+
≥−
Universal
wasteful
fragile
efficient
robust
Page 60
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
too fragile
complex
robustshort
cheaprobust
Balancing
Glycolysis
Law #1Law #2Law #3
Law #1Law #2Law #3
• Efficient but unstable• Needs complex/active control• With associated laws? Specific
Specific
Law #4z pTz p+
≥−
Universal
Page 61
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing• Maternal care• Warm blood• Flight • Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Flight
unstable
Page 62
Universal?Specific?
1pt R
¶ + ×Ñ + Ñ = D¶u u u u
Flight and swimming
(streamlining)
Page 63
wasteful
fragile
efficient
robust
Universal
1pt R
¶ + ×Ñ + Ñ = D¶u u u u
Turbulent
Page 64
z x
y
uFlow
Blunted turbulent velocity profile
Turbulent
Laminar
wU
wU
wasteful
fragileLaminar
Turbulent
efficient
robust
1pt R
¶ + ×Ñ + Ñ = D¶u u u u
UniversalTurbulent
Page 65
Physics of Fluids (2011)
z x
y
uFlow
Blunted turbulent velocity profileLaminar
Turbulent wU
wU
wasteful
fragileLaminar
Turbulent
efficient
robust
1pt R
¶ + ×Ñ + Ñ = D¶u u u u
Page 66
wasteful
fragileLaminar
Turbulent
efficient
robustSharks
Dolphins
Page 67
wasteful
fragileLaminar
Turbulent
efficient
robustSharks
Dolphins
Page 68
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing• Maternal care• Warm blood• Flight (streamlining)?• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
too fragile
complex
No tradeoff
robustshort
cheaprobust
Balancing
Glycolysis
• Efficient but unstable• Needs complex/active control• With associated laws?
Specific
Turbulent
efficient
robustSharks
Dolphinsu
Flow
Specific
Specific
Page 69
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing• Maternal care• Warm blood• Flight (streamlining)?• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
too fragile
complex
No tradeoff
robustshort
cheaprobust
Balancing
Glycolysis
• Efficient but unstable• Needs complex/active control• With associated laws?
Universal
Turbulent
efficient
robustSharks
Dolphinsu
FlowRobust efficiency
Page 70
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing (running)• Maternal care• Warm blood• Flight (streamlining)?• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis (metabolism)
too fragile
complex
No tradeoff
cheaprobust
Running
Metabolism
• Efficient but unstable• Needs complex/active control• With associated laws?
Specific
Universal
Robust efficiency
Specific
Page 71
• Money/finance/lobbyists• Industrialization• Society/agriculture• Weapons• Balancing (running)• Maternal care• Warm blood• Flight (streamlining)?• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis (metabolism)
too fragile
complex
No tradeoff
cheaprobust
Running
Metabolism
• Efficient but unstable• Needs complex/active control• With associated laws?
Specific
Universal
Cardiovascular physiology
Robust efficiency
Page 72
Li, Cruz, Chien, Sojoudi, Recht, Stone, Csete, Bahmiller, Doyle
Robust efficiency and actuator saturation explain healthy heart rate control and variabilityProc. Nat. Acad. Sci. PLUS 2014
efficientrobust
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errorsO2BPpHGlucoseEnergy storeBlood volume…internal nois
High Low
High
Cardio-physiology
Page 73
k10-1 100 101100
101
too fragile
complex
No tradeoff
expensive
Biology
robustTurbulent
efficient
robust uFlow
Physics
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errorsO2BPpHGlucoseEnergy storeBlood volume…internal nois
High Low
High
10
100
.1 1.22
Medicine
efficientrobust
• Robust efficiency• Tradeoffs, mechanisms
Neuro
Page 74
k10-1 100 101100
101
too fragile
complex
No tradeoff
expensive
Biology
robustTurbulent
efficient
robust uFlow
Physics
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errorsO2BPpHGlucoseEnergy storeBlood volume…internal nois
High Low
High
10
100
.1 1.22
Medicine
efficientrobust
• Foundational• Huge literatures• Data, models, simulation• Yet persistent mysteries
• Robust efficiency• Tradeoffs, mechanisms
Neuro
Page 75
Universal laws and architectures:Theory and lessons from
brains, hearts, cells, grids, nets, bugs, fluids, bodies, planes, docs, fire, fashion,
earthquakes, music, buildings, cities, art, running,cycling, throwing, Synesthesia, spacecraft, statistical mechanics
https://rigorandrelevance.wordpress.com/author/doyleatcaltech/
So far
Page 76
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Major transitions
efficient
robust
Glycolysis
Turbulence
Cardio-physiology
Balance
Needs complex control
Efficient but unstableSimilar• Tradeoffs• Laws• MechanismsBut simpler• Models• Experiments
Crash
Page 77
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Balance (biped)• Maternal care• Warm blood• Flight (streamlining)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Warfare (“up”)
Prey (“back”)
Crash
Bad
Weapons
Mass extinction
Page 78
Crash
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states• Weapons• Balance (biped)• Maternal care• Warm blood• Flight (streamlining)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Slavery
Famine
Bad
Page 79
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states• Weapons• Balance (biped)• Maternal care• Warm blood• Flight (streamlining)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Slavery
Famine
• Crashes “back” can hurt everyone• Crashes “forward” benefit those (few)
that cause and maintain them• Systematic study is missing
forward
back
Page 80
• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism• Maternal care• Warm blood• Flight• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis
Major transitions
efficient
robust
FastFlexible
Apps
OSHW
Gene
Trans*Protein
CerebellumCortex
Reflex
Needs complex control
Efficient but unstable
Next
Page 81
https://rigorandrelevance.wordpress.com/author/doyleatcaltech/
Universal laws and architectures:Theory and lessons from
brains, hearts, cells, grids, nets, bugs, fluids, bodies, planes, docs, fire, fashion,
earthquakes, music, buildings, cities, art, running,cycling, throwing, Synesthesia, spacecraft, statistical mechanics
Next
Page 82
Entropy rate
Energy (L2)
eye vision
Act
delay
Control
l
E
gpl
= .3sτ ≈
( )( )
exp ln
expQuantized
T
T pτ∞
∝
∫
Necessary!
Quantized
Robust to “noise” model
Spiking neurons?Discrete axons?Distributed control?
Page 83
10
100
.1 1.5.22
( )exp pτ
Speed ∝ 1/τ
Length lo
Length l
Speed ∝ 1/τ
delay?
noise
error
( ) z pT pz p
τ +≥
−exp
Theory
Page 84
robust + efficient
Fast
Flexible
Page 85
accessibleaccountableaccurateadaptableadministrableaffordableauditableautonomyavailablecredibleprocess
capablecompatiblecomposableconfigurablecorrectnesscustomizabledebugabledegradabledeterminabledemonstrable
dependabledeployablediscoverable distributabledurableeffective
evolvableextensiblefail transparentfastfault-tolerantfidelityflexibleinspectableinstallableIntegrityinterchangeableinteroperable learnablemaintainable
manageablemobilemodifiablemodularnomadicoperableorthogonalityportableprecisionpredictableproducibleprovablerecoverablerelevantreliablerepeatablereproducibleresilientresponsivereusable
safety scalableseamlessself-sustainableserviceablesupportablesecurablesimplestablestandardssurvivable
tailorabletestabletimelytraceableubiquitousunderstandableupgradableusable
efficient
robust
sustainablerobust + efficient
efficient
Fast
Flexible
Page 86
IEEE Conference on Decision and ControlDecember 2015 Osaka, Japan
Hard Limits on Robust ControlOver Delayed and Quantized Communication Channels
With Applications to Sensorimotor Control
Page 87
Li, Cruz, Chien, Sojoudi, Recht, Stone, Csete, Bahmiller, Doyle
Robust efficiency and actuator saturation explain healthy heart rate control and variability
PNAS PLUS 2014
Old story
0 100 200 300 4000 100 200 300 400406080100120140160180
sec
Page 88
0 100 200 300 4000 100 200 300 400406080100120140160180
sec
( )[ ] [ ]( )2, 2 2 2[ ]
vp vp total as as vs vs a
T O
p ap
av vs
c P V c P c P c P
v O M F O O
= −
− + ⋅= −
+ +
as as s
vs vs s r
ap ap
l
r p
c P Fc P F Qc P
Q
Q F
== −=
−
−
( ) ( ) ( )2
2 2 22 * 2 * 2 *2 2min P as as o Hq P P q O O q H H dt − + ∆ − ∆ + −
∫
Li, Cruz, Chien, Sojoudi, Recht, Stone, Csete, Bahmiller, Doyle
Robust efficiency and actuator saturation explain healthy heart rate control and variability
PNAS PLUS 2014
Page 89
0 100 200 3000 100 200 300406080100120140160180
sec
Li, Cruz, Chien, Sojoudi, Recht, Stone, Csete, Bahmiller, Doyle
Robust efficiency and actuator saturation explain healthy heart rate control and variability
PNAS PLUS 2014
Page 91
How? Why?• Mechanism• Tradeoff
• Architecture• Speed• Flexibility• Robustness• Virtualization• DiversityFast
Slow
Page 92
Cortex
eye vision
Actdelay
Object motion
Error+
slow
Slow
FlexibleHigh Res
vision
Fast
Inflexible
Page 93
Cortex
eye vision
Act
slow
delay
VORfast
Object motion
Head motion
Error+
Slow
Flexible
vision
Fast
InflexibleLow Res
VOR Vestibular Ocular Reflex (VOR)
Does not use “vision”?
Page 94
Slow
FlexibleHigh Res
vision
Fast
InflexibleLow Res
VOR
Tradeoff
Mechanism
Minimal cartoon
Cortex
eye vision
Act
slow
delay
VORfast
Object motion
Head motion
Error+
Page 95
Cortex
eye vision
Act
slow
delay
VOR
fast
Object motion
Head motion
Error+
These signals must “match”
Next important piece?
Page 96
Cortex
eye vision
Actslow
delay
VOR
fast
Object motion
Head motion
Error+
Tune gain?
gain
VOR and visual gain must match
Page 97
Cortex
eye vision
Actslow
delay
VOR
Object motion
Head motion
Cerebellum
Error+
ErrorTune gain
gain AOS
AOS = Accessory Optical system
fast
Page 98
Cortex
eye vision
Actslow
delay
VOR
This fast “forward”
path is necessary
Object motion
Head motion
Cerebellum
Error+
ErrorTune gain
gain AOS
AOS = Accessory Optical system
fastdirect,fast
feedback
slow
Page 99
Cortex
eye vision
Actslow
delay
VOR
Object motion
Head motion
Cerebellum
Error+
ErrorTune gain
gain AOS
AOS = Accessory Optical system
fastdirect,fast
feedback
slow
This fast “forward”
pathNeeds
“feedback” tuning
slowest
Page 100
Cortex
vision
delayVOR
Cerebellum
gainAOS
vision
delay
Cerebellum
gainAOS
Cortex
Cerebellum
gainAOS
distributed compute
(& control)
“Grey” boxes
Page 101
Cortex
vision
Actdelay
VOR
Cerebellum
gainAOS
vision
delay
Cerebellum
gainAOS
Cortex
vision
delay
Cerebellum
gainAOS
“White”cables vision
Actslower
delay
fast
Cerebellum
gainAOS
slowest
slowerfast
slowest
vision
Actdelay
gain
sparse communication, heterogeneous
delays
Page 102
Slow
FlexibleHigh Res
vision
Fast
InflexibleLow Res
VOR
Tradeoff
Mechanism
Minimal cartoon
Cortex
eye vision
Act
slow
delay
VORfast
Object motion
Head motion
Error+
Page 103
Vision
VOR
Tune
extreme heterogeneity
Slow
FlexibleHigh Res
Fast
InflexibleLow Res
Page 104
secs
msecs
hours
Vision
VOR
Tune
delaylog10
low reshigh res(flexible)
extreme heterogeneity
why?
Ideal
Page 105
secs
msecs
hours
Vision
VOR
Tune
delaylog10
low reshigh res(flexible)
why?
Ideal
• These dimensions• Tradeoff• Extreme heterogeneity
Page 106
High resolution vision robust w/motion• Object motion• Self motion
Vision
Motion
Page 107
High resolution vision robust w/motion• Object motion• Self motion
Page 108
heterogeneous delays
secs
msecs
hours
errorlow high
delaylog10
Plannavigate
Reflex
Tune
Page 109
secs
msecs
hoursPlan
navigate
Reflex
Tune
errorlow high
810> ×delaylog10
Plannavigate
Reflex
Tune
heterogeneous delays
Page 110
Plannavigate
Reflex
Tune
Page 114
Plannavigate
Reflex
Tune
Page 117
secs
msecs
hoursPlan
navigate
Reflex
Tune
errorlow high
810> ×delaylog10
Plannavigate
Reflex
Tune
heterogeneous delays and errors 1110> ×
Page 118
A KU PK
F B
M P
LZ YMQ
RH
O
V JW
TB
NC
A
G
X YQ
A
D T
Page 119
A KU PK
F B
M P
LZ YMQ
RH
O
V JW
TB
NC
A
G
X YQ
A
D T
HIGH RES
FAST
Page 120
A KU PK
F B
M P
LZ YMQ
RH
O
V JW
TB
NC
A
G
X YQ
A
D T
HIGH RES
FAST
Page 121
secs
msecs
hours
Vision
VOR
delaylog10
low res?high res(flexible)
A KU PK
F B
M PLZ YMQ
RH
O
V JW
TB
N C A
G
X YQ
A
D T
HIGH RES
FAST
Page 122
secs
msecs
hours
Vision
VOR
delaylog10
low res?high res(flexible)
A KX Y
FAST
low resolution ≠ noisy
Page 123
secs
msecs
hoursPlan
navigate
Reflex
Tune
error?low high
810> ×delaylog10
Ideal
heterogeneous delays and errors 1210> ×
( )( )
1
exp ln
exp
G
G pg
τ∞
∝
∫
Page 124
Tune
error?
( )( )
1
exp ln
exp
G
G pg
τ∞
∝
∫
2x
x ∞
Page 125
Cortexeye vision
Actslow
delayVOR
Object
Head
Cerebellum
Error+
Tunegain
fast
Extreme system level
heterogeneity
secs
msecs
hoursdelaylog10
error
Plannavigate
Reflex
Tune
Why?
Page 126
log10 axon diameter
log10axons
per nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
Extreme component
level heterogeneity
Page 127
axons per
nerve
axons per
nerve
10-1 100 101100
102
104
106 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
mean axon diameter (microns)
Sciatic
610> ×mean axon diameter (microns)
Page 128
axons per
nerve
10-2 100 102100
102
104
106 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
610> ×mean axon area
Page 129
log10 axon diameter
log10axons
per nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
Extreme component
level heterogeneity
Why?
Page 130
log10 axon diameter
log10axons
per nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
secs
msecs
hoursdelaylog10
error
Plannavigate
Reflex
Tune
Cortexeye vision
Actslow
delayVOR
Object
Head
Cerebellum
Error+
Tunegain
fast
• Extreme heterogeneity• Connect scales
Page 131
log10 axon diameter
log10axons
per nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
secs
msecs
hoursdelaylog10
error
Plannavigate
Reflex
Tune
Cortexeye vision
Actslow
delayVOR
Object
Head
Cerebellum
Error+
Tunegain
fast
• Extreme heterogeneity• Connect scales• Mechanistic physiology• Integrated• Quantitative• Tradeoffs as “laws”
Page 132
log10 axon diameter
log10axons
per nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
secs
msecs
hoursdelaylog10
error
Plannavigate
Reflex
Tune
Cortexeye vision
Actslow
delayVOR
Object
Head
Cerebellum
Error+
Tunegain
fast
Page 133
Assume:resource (cost)(to build and maintain nerve)∝ area α
α
Page 134
Assume:
resource (cost)(to build and maintain nerve)
∝ area α
α
Page 135
Assume:
resource (cost)(to build and maintain nerve)
∝ area α
αAssume
lengths and areas are
given
Page 136
log axon diam µm
logaxons
per nerve
7 bits
1 bit
3 bits
resource (cost) ∝ area αarea α
Page 137
axon diameter
axonsper
nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
too big
resource (cost) ∝ area α
Page 138
axon radius (µm)
N=axons
per nerve
0 10
2
4
6
ρ =
2nerve area = Nα πρ=
resource (cost) ∝ area α
Page 139
7 bits
1 bit
3 bits
Why not make axon radius small?(Maximize mutual information.)
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
SciaticIdeal?
axon radius
axonsper
nerve
Page 140
Why not make axon radius small?(Maximize mutual information.)
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
SciaticIdeal?
axon radius
axonsper
nerve
Tradeoffs
Page 141
spike rate ρ∝ spike speed ρ∝
(propagation)
Tradeoffsin
spikingneurons
axon radius (µm)
N=axons
per nerve
ρ =0 1
0
2
4
6
2nerve area = Nα πρ=
Ideal?
Page 142
axon radius (µm)
N=axons
per nerve
ρ =0 1
0
2
4
6
2N αρ
∝
2R α αρρ ρ
∝ ∝
spike rate (bits/time) R N
ρρ
∝∝
2nerve area = Nα πρ=
Page 143
spike speed 1delay sT
ρ
ρ
∝
∴ ∝
R Tαλ∴ =
spike rate (bits/time) R N
ρρ
∝∝
axon radius (µm)
N=axons
per nerve
ρ =0 1
0
2
4
6
2R
T
α αρρ ρ
α
∝ ∝
∝2nerve area = Nα πρ=
Page 144
1delay sTρ
∝ feasible
infeasible
R Tαλ=R Tαλ≤
R
fast
slow
low res
high res
R Tαλ∴ =
Page 145
1delay sTρ
∝
R Tαλ=
R
fast
slow
1/R
delay
fast
slow
R Tαλ=
low res
high res
2
resource =
nerve area = Nα πρ=
low res
high resR
Page 146
2
resource =
nerve area = Nα πρ=
1/R
delay
fast
slow
low res
high res
( )?R Tαλ=
1/R
delay
fast
slow
R Tαλ=
Page 147
1/R
delayfeasible
infeasiblefast
slow
low res
high res
R Tαλ=
1 bit
3 bits
Assuming fixed lengthand area
Page 148
1/R
delay
feasible
infeasiblefast
slow
low res
high res
R Tαλ=
delay
fast
slow
Plannavigate
Reflex
Tune
errorlow high
Ideal
Ideal
• Extreme heterogeneity• Connect scales• Mechanistic physiology• Integrated• Quantitative• Tradeoffs as “laws”
Page 149
P
x
vu
T Delay of T
(head motion)
LK
LQ
LQ Channel(neural signaling)
Q Quantizer
Reflex controller(VOR: Motion compensation)LK
reflex (delayed)
state
Presenter
Presentation Notes
Fig 6.
Page 150
P
x
vu
T Delay of T
(head motion)
LK
LQ
LQ Channel(neural signaling)
Q Quantizer
Reflex controller(VOR: Motion compensation)LK
reflex (delayed)
R Tαλ=
state
Presenter
Presentation Notes
Fig 6.
Page 151
P
x
v
T Delay of T
(head motion)
LK
LQ Q Quantizer
Reflex controller(VOR: Motion compensation)LK
reflex (delayed)
state[ ]( 1) ( ) ( ) ( )Lx t ax t Q u t T v t+ = − − +
u
Presenter
Presentation Notes
Fig 6.
Page 152
P
x
vu
r
(head motion)
(target motion)
LK HK
LQ HQ
remote sensingadvanced warninghigh level planning
delay
reflex (delayed)
Presenter
Presentation Notes
Fig 6.
Page 153
P
x
r (target motion)LK
HK
HQ
remote sensingadvanced warninghigh level planning
delay
reflex (delayed)
[ ]( 1) ( ) ( ) ( )H rx t ax t Q u t T r t T+ = − − + −
|| || 1r ∞ ≤
||| || ?x ∞ =
[ ]( )HQ u t T−
Presenter
Presentation Notes
Fig 6.
Page 154
P
x
v
ur
L Hwarnedreflex
(delayed)
|| || || || 1v rδ∞ ∞≤ ≤
( 1) ( )x t ax t+ = +
||| || ?x ∞ =
Presenter
Presentation Notes
Fig 6.
Page 155
( ) ( )1
|| || ,|| || 1 1
1 1 min sup || || = 2 2
0,
LL
Ti
v r i
H r
T R Rx a a a a
a T Tδ
δ δ∞ ∞
−∞
≤ ≤ =
− −+ − + −
≠ ≤
∑warned
quant costdelayed
delay costdelayed
quant cost
Theorem
P
x
v
ur
L Hwarnedreflex
(delayed)
|| || || || 1v rδ∞ ∞≤ ≤
( 1) ( )x t ax t+ = +
Presenter
Presentation Notes
Fig 6.
Page 156
.1
1
10cost
Delayed (Tc)24
Warned (Tw)
value
-4-2
delay
quant
delay Ts
quant
ctrltotal state
total delay
.1
1
10
0
warned, plan, tune
delayed, fast, reflex
delay >>1bits >>1
small, constant bits & delay
few uniformly large axonslarge errors
diverse, small axonssmall errors
Extremely different
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 157
.1
1
10
Cost(system)
Delayed (Tc)0246
Warned (Tw)
Optimal value(parts)
-6-4-20
delay
quant
delay Ts
quant(bits)
ctrltotal state
total delay
.1
1
10
warned, plan, tune
delayed, fast, reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 158
.1
1
10Cost
0246Warned (Tw)
delay → 0
ctrltotal state
ctrl ≈ constant
Cost = size of signal with worst-case disturbance
total state≈ quant << 1
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 159
.1
1
10
Cost
Delayed (Tc)0246
Warned (Tw)-6-4-20
delay
quant
ctrltotal state
ctrl ≈ constant
total state≈ quant << 1
total state≈ delay >> 1
Extremely different
Cost = size of signal with worst-case disturbance
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 160
0246Warned (Tw)
Optimalvalue
delay Ts
quant(bits)
.1
1
10
.1
1
10Cost
delay
ctrltotal state
delay >>1bits >>1
diverse small axonssmall errors
warned, planning,tuning, precision
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 161
Delayed (Tc)0246
Warned (Tw)
Optimal value
-6-4-20
delay Ts
quant(bits)
total delay
.1
1
10
Extremely different
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 162
.1
1
10
Cost(system)
Delayed (Tc)0246
Warned (Tw)
Optimal value(parts)
-6-4-20
delay
quant
delay Ts
quant(bits)
ctrltotal state
total delay
.1
1
10
warned, plan, tune
delayed, fast, reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 163
delay
fast
slowPlan
navigate
Reflex
Tune
errorlow high
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 164
.1
1
10cost(error)
delayquant
ctrldelay
fast
slowPlan
navigate
Reflex
Tune
errorlow high
total state≈ quant << 1
diverse, small axonssmall errors
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 165
1/R
delay
fast
slow
low res
high res
R Tαλ=
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 166
24Warned (Tw)
valuedelay Ts
quanttotal delay.1
1
10
0
1/R
delay
fast
slow
low res
high res
R Tαλ=
diverse, small axonssmall errors
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 167
.1
1
10cost
24Warned (Tw)
value
delayquant
delay Ts
quant
ctrltotal state
total delay.1
1
10
0
error
delay
fast
slow plannav
Reflex
Tune
low high
1/R
delay
fast
slow
low res
high res
R Tαλ=
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 168
.1
1
10cost
24Warned (Tw)
value
delayquant
delay Ts
quant
ctrltotal state
total delay.1
1
10
0
delay
fast
slow plannav
Reflex
Tune
low high
1/R
delay
fast
slow
low res
high res
R Tαλ=
error
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 169
P
x
vu
rL H
delay
fast
slow plannav
Reflex
Tune
low high
1/R
delay
fast
slow
low res
high res
R Tαλ=
error
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 170
1/R
delay
fast
slow
low res
high res
R Tαλ=
delay
fast
slow plannav
Tune
low high
error
reflex
reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 171
Delayed (Tc)
value
-4-2
delay Ts
quant
total delay
.1
1
10
0
.1
1
10cost delay
quant
total state
1/R
delay
fast
slow
low res
high res
R Tαλ=
delay
fast
slow plannav
Tune
low high
error
reflex
reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 172
delay
fast
slow plannav
Tune
low higherror
reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 173
axon diameter
axonsper
nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
Delayed (Tc)
value
-4-2
delay Ts
quant
total delay
.1
1
10
0
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 174
.1
1
10cost
Delayed (Tc)
value
-4-2
quant
delay Ts
quant
ctrl
total state
total delay
.1
1
10
0axon diameter
axonsper
nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
delayed, fast, reflex
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 175
24Warned (Tw)
valuedelay Ts
quanttotal delay.1
1
10
0axon diameter
axonsper
nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 176
.1
1
10cost
24Warned (Tw)
value
delayquant
delay Ts
quant
ctrltotal state
total delay.1
1
10
0
warned, plan,tune, precision
axon diameter
axonsper
nerve
-1 0 10
2
4
6 Optic
Vestibular
Olfactory
Auditory
Aα
cranial
spinal
Sciatic
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 177
secs
msecs
hoursPlan
navigate
Reflex
Tune
heterogeneous delays
errorlow high
810> ×delaylog10
Plannavigate
Reflex
Tune
Page 178
delay
fast
slow plannav
Reflex
Tune
low higherror
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 179
P
x
vu
rL H
delay
fast
slow plannav
Reflex
Tune
low high
1/R
delay
fast
slow
low res
high res
R Tαλ=
error
Presenter
Presentation Notes
clear all;close all;clc; LW = 2; T = linspace(0,5,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Advanced warning (Tw)');ylabel('Optimal value'); %%% T = linspace(-5,0,100);P = T; Xmin = P;Ind = P;Xd = P;E = P;Delay = P;Bits = P;Cost = P;U = P; p = 0.5; for ii = 1:length(T); d = linspace(0.1,5,1000); xd = d;e = d;x = d;u = d; for k = 1:length(d) xd(k) = max([0,d(k)-T(ii)+1]); e(k) = 1/(2^(p*d(k))-1); x(k) = xd(k)+e(k); u(k) = 1 + inv((2^(p*d(k)))-1); end [Xmin(ii),Ind(ii)] = min(x); Xd(ii) = xd(Ind(ii)); E(ii) = e(Ind(ii)); Delay(ii) = d(Ind(ii)); Bits(ii) = p*d(Ind(ii)); Cost(ii) = x(Ind(ii)); U(ii) = u(Ind(ii)); end figure;semilogy(T,Xd,T,E,T,Xd+E,T,U,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Cost'); figure;semilogy(T,Delay-T,T,Delay,T,Bits,'LineWidth',2); set(gca,'Xdir','reverse');ylim([0.1,10]); xlabel('Computational delay (Tc)');ylabel('Optimal value');
Page 180
From: “Understanding Vision: theory, models, and data”, by Li Zhaoping, Oxford University Press, 2014
Page 181
Eye muscle
From: “Understanding Vision: theory, models, and data”, by Li Zhaoping, Oxford University Press, 2014
Layered feedback
Page 182
R R R R R……
CC
“reflex”
“cortex”brains
• Localized control (LPs)‒layered‒scalable‒local model, synthesis, implement
Distributed/local control
“grey” boxes and “white” cables
• Communications‒sparse‒delayed, quantized
Page 183
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
CC“reflex”
“cortex”
physical plant
disturbance
act/sense
sense act/sense
body + tools+ environment
sensors + muscles
brains
Distributed/local control
Page 184
……
w
P
w
P
w
P
w
P
w
P
• Physical plant‒unstable dynamics‒distributed‒sparse
• Disturbance (w)‒worst case‒advanced warning T
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
body + tools+ environment
Page 185
……
w
P
R
w
P
w
P
R
w
P
w
P
R ……
• Communications‒sparse‒delayed, quantized
• Actuation and sensing‒saturates‒sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒sparse
• Disturbance (w)‒worst case‒advanced warning T
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
sensors + muscles
Page 186
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
• Communications‒sparse‒delayed, quantized
• Actuation and sensing‒saturates‒sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒sparse
• Disturbance (w)‒worst case‒advanced warning T
CC“reflex”
“cortex”
• Localized control (LPs)‒layered‒scalable‒local model, synthesis, implement
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
Page 187
…
w
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……
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w
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• Communications‒ sparse‒delayed, quantized
• Actuation and sensing‒ saturates‒ sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒ sparse
• Disturbance (w)‒worst case‒advanced warning T
Scalability?
Page 188
• Communications‒ sparse‒delayed, quantized
• Actuation and sensing‒ saturates‒ sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒ sparse
• Disturbance (w)‒worst case‒advanced warning T
• Localized control ‒modeling, synthesis‒ implementation
…
w
P
C
w
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w
P
C
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Scalable
Page 189
• Communications‒ sparse‒delayed, quantized
• Actuation and sensing‒ saturates‒ sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒ sparse
• Disturbance (w)‒worst case‒advanced warning T
• Localized control‒model, synthesis, implement‒ layered
…
…
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
CC
“reflex”
“cortex”
Scalable
Page 190
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
• Communications‒sparse‒delayed, quantized
• Actuation and sensing‒saturates‒sparse ‒delayed, quantized
• Physical plant‒unstable dynamics‒distributed‒sparse
• Disturbance (w)‒worst case‒advanced warning T
CC“reflex”
“cortex”
• Localized control (LPs)‒layered‒scalable‒local model, synthesis, implement
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
Page 191
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
CC“reflex”
“cortex”
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
Fast
Flexible
RobustRobust
Page 192
……
w
P
R
w
P
R
w
P
R
w
P
R
w
P
R ……
CC“reflex”
“cortex”
physical plant
disturbance
Distributed/local control
act/sense
sense act/sense
Fast
Flexible
• Communications‒sparse‒delayed, quantized
• Localized control (LPs)‒layered‒scalable‒local model, synthesis, implement
Page 193
R R R R R ……
CC“reflex”
“cortex”
Distributed/local control
FastFlexible
Delay vs resolution tradeoff.
• Communications‒sparse‒delayed, quantized
Page 194
A KU PK
F B
M P
LZ YMQ
RH
O
V JW
TB
NC
A
G
X YQ
A
D T
Page 195
Cortexeye vision
Actslow
delay
fast
Object
Head
Cerebellum
+
ErrorTune gain
gainAOS
Slow
Flexible
Fast
Inflexible
Reflex
vision
VOR
Color?
Motion
Motion
Page 196
Stare at the intersection
Marge Livingstone
colorvision
Page 198
Stare at the intersection.
Page 200
Stare at the intersection
Page 206
Stare at the intersection.
Page 211
Cortexeye vision
Actslow
delay
fast
Object
Head
Cerebellum
+
ErrorTune gain
gainAOS
Slow
Flexible
Fast
Inflexible
Reflexvision
VOR
Color?
Motion
Motion
Page 212
Slow
Flexible
Fast
Inflexible
vision
VOR
Color
Motion
Cortex
eye vision
Actslow
delay
Object +
colorvision
Slow
colorvision
Motion
Page 213
OK, color decisions are rarely urgent.
Slow
Flexible
Fast
Inflexible
vision
VOR
Motion
colorvision
Slow
Page 214
Cortexeye vision
Actslow
delay
fast
Object motion
Head motion
Cerebellum
+
ErrorTune gain
gain AOSReflex
RNAP
DnaK
σ
RNAP
σDnaK
rpoH
FtsH Lon
Heat
σ
σ mRNA
Other operons
σ
DnaK
DnaK
ftsH
Lon
Sensorimotor
Vastly more different
than similar
Linux OS
Bacterial cell
Page 215
Cortexeye vision
Actslow
delay
fast
Object motion
Head motion
Cerebellum
+
ErrorTune gain
gain AOSReflex
RNAP
DnaK
σ
RNAP
σDnaK
rpoH
FtsH Lon
Heat
σ
σ mRNA
Other operons
σ
DnaK
DnaK
ftsH
Lon
MechanismsLinux OS
Bacterial cell
Universals are profound
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
Page 216
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
Universals are profound
• Tradeoffs• Evolvability
Page 217
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
Tradeoffs
DiverseNot
Diverse Universals are profound
• Hourglass diversity
• Tradeoffs• Evolvability
“Constraints that
deconstrain”
Page 218
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
Horizontal Transfer
Horizontal Transfer
Tradeoffs
Illusion
DiverseNot
Diverse Universals are profound
• Hourglass diversity
• Tradeoffs
• Horizontal transfer
• Evolvability
• Tunable (illusion)
Page 219
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
DiverseNot
Horizontal Transfer
Virtual internalhidden
Tradeoffs
Illusion
Diverse Universals are profound
• Hourglass diversity
• Tradeoffs
• Horizontal transfer
• Virtual/Internal• Hijacking
• Evolvability
• Tunable (illusion)
Horizontal Transfer
Page 220
Fast
Slow
Flexible Inflexible
Apps
OS
HW
Gene
Trans*
ProteinCerebellum
Cortex
Reflex
DiverseNot
Horizontal Transfer
Virtual internalhidden
Tradeoffs
Illusion
Diverse/Digital Universals are profound
• Hourglass diversity
• Tradeoffs
• Horizontal transfer
• Virtual/Internal• Hijacking
• Robustness
• Evolvability
• Tunable (illusion)
Horizontal Transfer
• Digital
Page 221
• SDN, NFV, IOT, IOE, CPS• Compute/control technology• Industrialization• Money/finance/lobbyists/etc• Society/agriculture/states • Weapons• Bipedalism (balance,
cardiovascular physiology)• Maternal care• Warm blood• Flight (turbulence)• Mitochondria• Oxygen• Translation (ribosomes)• Glycolysis• Cells
• Hourglass diversity
• Tradeoffs
• Horizontal transfer
• Virtual/Internal• Hijacking
• Robustness
• Evolvability
• Tunable (illusion)
• Digital
Universals are profound