control, optimization, and functional analysis
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Control, Optimization, and Functional Analysis. In The Heltonian Era. The Heltonian Era. 1970 From Dark Ages to Birth of Enlightenment 1980 Robust control, operator theory 1990 Matrix inequalities, convex optimization 2000 Nonlinear control, algebraic geometry 2010 ?? - PowerPoint PPT PresentationTRANSCRIPT
In The Heltonian Era
Control, Optimization, and Functional Analysis
The Heltonian Era
• 1970 From Dark Ages to Birth of Enlightenment• 1980 Robust control, operator theory• 1990 Matrix inequalities, convex optimization• 2000 Nonlinear control, algebraic geometry• 2010 ??
– Networks, sparsity, structure– Mixed boolean & real algebra/geometry– Expansion of applications in basic science and
infrastructure
Robust control, operator theory
Matrix inequalities,
convex optimization
Doyle(t) and Helton(t)
Nonlinear control,
algebraic geometry
Multiscale physics Biology
MedicineEcology
Geophysics
Internet
Smartgrid
Economics
Biology
Medicine
Control, Optimization, and Functional Analysis
Na Li, John Doyle, and a cast of thousands (including Ben Recht and Marie Csete)
Caltech
Cardiovascular
Robust FragileHuman complexity
Metabolism Regeneration & repair Healing wound /infect
Obesity, diabetes Cancer AutoImmune/Inflame
Robust FragileMechanism?
Metabolism Regeneration & repair Healing wound /infect
Fat accumulation Insulin resistance Proliferation Inflammation
Obesity, diabetes Cancer AutoImmune/Inflame
Fat accumulation Insulin resistance Proliferation Inflammation
Robust FragileWhat’s the difference?
Metabolism Regeneration & repair Healing wound /infect
Obesity, diabetes Cancer AutoImmune/Inflame
Accident or necessity?
Fat accumulation Insulin resistance Proliferation Inflammation
Fluctuating energy
Static energy
Robust FragileWhat’s the difference?
Metabolism Regeneration & repair Healing wound /infect
Obesity, diabetes Cancer AutoImmune/Inflame
Fat accumulation Insulin resistance Proliferation Inflammation
ControlledDynamic
UncontrolledChronic
Low meanHigh variability
High meanLow variability
Robust Fragile
Restoring robustness
ControlledDynamic
UncontrolledChronic
Low meanHigh variability
High meanLow variability
Robust Yet FragileHuman complexity
Metabolism Regeneration & repair Microbe symbionts Immune/inflammation Neuro-endocrine Complex societies Advanced technologies Risk “management”
Obesity, diabetes Cancer Parasites, infection AutoImmune/Inflame Addiction, psychosis… Epidemics, war… Catastrophes Obfuscate, amplify,…
Accident or necessity?
Robust Fragile Metabolism Regeneration & repair Healing wound /infect
Obesity, diabetes Cancer AutoImmune/Inflame
Fat accumulation Insulin resistance Proliferation Inflammation
• Fragility Hijacking, side effects, unintended… • Of mechanisms evolved for robustness • Complexity control, robust/fragile tradeoffs• Math: New robust/fragile conservation laws
Accident or necessity?Both
Robust Metabolism Regeneration & repair Healing wound /infect
• Fragility Hijacking, side effects, unintended… • Of mechanisms evolved for robustness • Complexity control, robust/fragile tradeoffs• Math: New robust/fragile conservation laws
Robust Metabolism Regeneration & repair Healing wound /infect
Fat accumulation Insulin resistance Proliferation Inflammation
Fluctuating energy
ControlledDynamicLow meanHigh variability
Mechanism?
Brain
Heart
Muscle
Liver
GI
GluTriglyc
Fat
Glyc
Glyc
FFA
Glycerol
Oxy
Lac/ph
Food
Out
fast slow
high
low
prio
rity
dynamics
Control?
• Energy• Inflammation• Coagulation
Evolved for large energy variation and
moderate trauma
Brain
Heart
Muscle
Glyc
Oxy
Out
fast
high
low
prio
rity
dynamics
Control?
Essential starting point?
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql H
Related States
VE
“grey box”
Plumbing and
chemistry
Robust/Health
Fragile/Illness
Persistent mystery
Low meanHigh variability
High meanLow variability
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60
80
100
120
140
HR
HR datatime(sec)
High mean, low variability
Low mean, high variability
The persistent mystery
Two experiments with same subject
Heart rate data
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql H
Related States
VE
Our approach
Physiology!an ancient art
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180
Other views1. Molecular genetics2. Creation science3. New sciences of- complexity- networks
What gene?
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0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HRHR data W
watts
watts
time(sec)
Data: Watts and HR
Two experiments with same subject
Data: Watts
W
0
50
100
150
+100w
Two experiments
On recumbent Lifecycle
Data: Watts and HR
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0
50
100
150
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60
80
100
120
140
W
time(sec)
wattsHR data
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HR
data
W
model
time(sec)
watts
1st order linear model
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HR
data
W
model
time(sec)
watts
same 1st order linear model
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HRHR data W
time(sec)
Model and HR
same 1st order linear model
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HRHR data W
time(sec)
Model and HR
1st order linear models(different parameters)
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HR
W
time(sec)
Explain differences between models
??
?
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0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HR HR dataW
time(sec)
Explain differences between models and data
0 50 100 150 200 250 3000
50
100breath and HR at 0 watts
inhale
HR 2nd order linear model
0 50 100 150 200 250 3000
50
100
0 50 100 150 200 250 3000
50
100190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
• “resting” HR• ~40 bpm fluctuations at ~10s period• 100% fluctuations!• Frequency sweep in breathing• Fit well with 2nd order model
190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
0 50 100 150 200 250 3000
50
100
0 50 100 150 200 250 3000
50
1000
50
100
@100 w
@0 w
datamodel
0 50 100 150 200 250 300 350 400
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
WattsHR data
Explain differences between • models • model and data
Different subject, 3 data sets
0 50 100 150 200 250 300 350 4000 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
HR High mean, low variability
Low mean, high variability
The persistent mysteryYoung, fit, healthy more extreme
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Optimal control
What can we say with this model?
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql H
VE
Plumbing and chemistry(aerobic)
Organized complexity, circa 1972
Plumbing and chemistry
Conservation laws:Energy and material (small moieties)
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral Lungs, Fp , Rp
Qr Ql
H
VE
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Conservation laws:Energy and material
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql H
Related States
VE
“grey box”
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Optimal control
Consequences?
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Conservation laws
1
ln 0
S T
S d
sensor
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errorsO2BPpHGlucoseEnergy storeBlood volume…
infectiontrauma
energy
Homeostasis
internal noise
heart beatbreath
errors
BrainO2BPpHGlucoseEnergy storeBlood volume…
controls
Brainheart rateventilationvasodilationcoagulationinflammationdigestionstorage…
external disturbances
infectiontrauma
energy
sensornoise
controls
internal noise
heart beatbreath
errorsImplementation
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
O2BPpHGlucoseEnergy storeBlood volume…
sensor
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errorsO2BPpHGlucoseEnergy storeBlood volume…
infectiontrauma
energy
Homeostasis
internal noise
heart beatbreath
2SpO
BP
watts
tissue
arterial
errors
O2t
Narrow focusControl
Plant
errors
EV
Control
Plant
2SpO
BP HR
watts
tissue
arterial
errors
Control
peripheral resistance
O2t
controls
EV
Control
Plant
2SpO
watts
tissue
arterial
errors
Control
peripheral resistance
O2t
Close these loops
EV
Control
Plant
2SpO
BP HR
watts
tissue
arterial
errors
Control
peripheral resistance
O2t
controls
Focus
Control
Plant
BP HR
watts
tissue
arterial
O2t
Initial focus
, 2 ,BP O t F w HR
Static model
Brain
Body
BP
HRwatts
O2t
0 50 100 150 20050
100
150
200
Watts
HRBrain
Body
BP
HRwatts
O2t
, 2 ,BP O t F w HR
Static model
( )HR h w
2 2 2
( )2
( ) , 2 ,
minh w
p BP q O t r HR
HR h w BP O t F w HR
0 50 100 150 20050
100
150
200
Watts
HRBrain
Body
BP
HRwatts
O2t
, 2 ,BP O t F w HR
( )HR h w
2 2 2
( )2
( ) , 2 ,
minh w
p BP q O t r HR
HR h w BP O t F w HR
0 50 100 150 20050
100
150
200
Watts
HRBrain
Body
BP
HRwatts
O2t
0.04 0.08 0.12 0.1680
120
160
200BP
O2t
( )HR h w
, 2 ,BP O t F w HR
2 2 2
( )2
( ) , 2 ,
minh w
p BP q O t r HR
HR h w BP O t F w HR
0 50 100 150 20050
100
150
200
Watts
0.04 0.08 0.12 0.1680
120
160
200BP
O2t
( )HR h w
2 2 2
( )ˆ2
ˆ0
minh w
p BP q O t r HR
p r r
2 2
( )2min
h wq O t r HR
Penalize BP and HR more
Metabolism only
0 50 100 150 200 250 300 350
0
50
100
150
0 50 100 150 200 250 300 35040
60
80
100
120
140
HRW
time(sec)
Explain differences between models
??
0.04 0.08 0.12 0.1680
120
160
200BP
O2t Static model
0.04 0.08 0.12 0.1680
120
160
200BP
O2t
2 2 2
( )ˆ2
ˆ0
minh w
p BP q O t r HR
p r r
2 2
( )2min
h wq O t r HR
Brain
Body
BP
HRwatts
O2t
Use same weights but put back in dynamics
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Optimal control
What can we say with this model?
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HR-simBP-sim[O2]v-sim*1000
HR-measurewatt
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80
100
120
140
160
180
Data and model
0 50 100 150 200 250 300 350 400020406080100120140160
HR-simBP-sim[O2]v-sim*1000
HR-measure
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80
100
120
140
160
180
BP
O2t
HR watts
Mechanistic explanation for differences between models
0 50 100 150 200 250 300 350 400020406080100120140160
0 50 100 150 200 250 300 350 40060
80
100
120
140
160
180
BP
O2t
HR watts
0.04 0.08 0.12 0.1680
120
160
200BP
O2t
2 2 2
( )ˆ2
ˆ0
minh w
p BP q O t r HR
p r r
Penalize BP and HR more
0 50 100 150 200 250 300 350 400020406080100120140160
0 50 100 150 200 250 300 350 40060
80
100
120
140
160
180
BP
HR
0.04 0.08 0.12 0.1680
120
160
200BP
O2t
High mean, low variability
Low mean, high variability
Mechanistic explanation for differences between models
0 50 100 150 200 250 300 350 400020406080100120140160
0 50 100 150 200 250 300 350 40060
80
100
120
140
160
180
HR
2 2 2
( )ˆ2
ˆ0
minh w
p BP q O t r HR
p r r
Penalize BP and HR more
Explain differences between models and data?
Control
Plant
HR
breath
EV
Later
internal noise
0 50 100 150 200 250 3000
50
100
HR
breath
breath
HR
190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
0 50 100 150 200 250 3000
50
100 2nd order linear model
190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
• “resting” HR• Frequency sweep in breathing• Fit well with 2nd order model• Not a mechanistic model
190 200 210 220 230 240 250 260 270 280
40
50
60
70
80
90
0 50 100 150 200 250 3000
50
100
0 50 100 150 200 250 3000
50
1000
50
100 @100 w
@0 w
data2nd order linear model
Penalize BP and HR more?
Control
Plant
HRbreath
EV
internal noise
Mechanism?
Need mechanical
coupling
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0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
WattsHR
Different subject, 3 data sets
0 50 100 150 200 250 300 350 400
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
WattsHR 1st order linear model
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0
50
100
150
200
250
300
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60
80
100
120
140
160
180
HR 1st order linear model
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0
50
100
150
200
250
300
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60
80
100
120
140
160
180
1st order linear models(different parameters)
0 50 100 150 200 250 300 350 400
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
1st order linear models(different parameters)
Explain differences between • models • model and data
0 50 100 150 200 250 300 350 400
0
50
100
150
200
250
300
0 50 100 150 200 250 300 350 40040
60
80
100
120
140
160
180
Explain differences between • models • model and data
Anaerobic
Breathing
Aside on gas variables• Gas exchange variables are also
predictable with simple models• VO2 is simplest and most predictable
• VCO2-VO2 is most complex and we don’t have first principles model
• Also HR model is bad at high watt levels
0 10 20 300
2
4
0 10 20 30
80
120
160
100
200
300
400HR
dataWattsHR
model
Time(min)
2VO
JP data
0 10 20 30
-1
0
1 2 2VCO VO
• Aerobic models can be way off at high watts• (predict this signal should be constant)• Can still fit with simple “black box” models, but…• Need nonlinear dynamics• Mechanistic models?
• Need anaerobic mechanisms• Control of arterial pH is critical (and hard to model)
aerobic model
2nd order nonlinear fit
sensor
controls
external disturbances
heart rateventilationvasodilationcoagulationinflammationdigestionstorage…
errors
O2BPpHGlucoseEnergy storeBlood volume…
infectiontrauma
energy
Homeostasis
internal noise
heart beatbreath
Local metabolic
control
Rs
right heart Rr , Sr
left heart, Rl , Sl
arterialvenous
Feedback Controller
systemic peripheral, Tissues, Fs
Workload,w(t)
arterial venous
Pulmonary peripheral
Lungs, Fp , Rp
Qr Ql
H
Related StatesVE
Conservation laws
1
ln 0
S T
S d
Conservation laws
Persistent mysteries• Physiological variability and homeostasis• Cryptic variability from cells to organisms to
ecosystems to economies• Statistical mechanics and thermodynamics• Turbulence (coherent structures in shear flows)• Network (cell, brain, Internet,…) architecture• Unified communications, controls, computing
Poor treatment of dynamics, robustness, complexity