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Control, Optimization, and Functional Analysis

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 ??

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Control, Optimization, and Functional Analysis

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  1. Control, Optimization, and Functional Analysis In The Heltonian Era

  2. 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

  3. Doyle(t) and Helton(t) Robust control, operator theory Matrix inequalities, convex optimization Nonlinear control, algebraic geometry

  4. Geophysics Medicine Smartgrid Ecology Multiscale physics Economics Biology Internet

  5. Medicine Biology

  6. Cardiovascular Control, Optimization, and Functional Analysis Na Li, John Doyle, and a cast of thousands (including Ben Recht and Marie Csete) Caltech

  7. Robust Human complexity Fragile • Metabolism • Regeneration & repair • Healing wound /infect • Obesity, diabetes • Cancer • AutoImmune/Inflame

  8. Robust Mechanism? Fragile • Metabolism • Regeneration & repair • Healing wound /infect • Fat accumulation • Insulin resistance • Proliferation • Inflammation • Obesity, diabetes • Cancer • AutoImmune/Inflame • Fat accumulation • Insulin resistance • Proliferation • Inflammation

  9. Robust What’s the difference? Fragile • Metabolism • Regeneration & repair • Healing wound /infect • Obesity, diabetes • Cancer • AutoImmune/Inflame • Fat accumulation • Insulin resistance • Proliferation • Inflammation Fluctuating energy Static energy Accident or necessity?

  10. Robust What’s the difference? Fragile • Metabolism • Regeneration & repair • Healing wound /infect • Obesity, diabetes • Cancer • AutoImmune/Inflame • Fat accumulation • Insulin resistance • Proliferation • Inflammation Controlled Dynamic Uncontrolled Chronic Low mean High variability High mean Low variability

  11. Robust Restoring robustness Fragile Controlled Dynamic Uncontrolled Chronic Low mean High variability High mean Low variability

  12. Robust Human complexity Yet Fragile • 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?

  13. Fragile Robust • 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 Both Accident or necessity?

  14. 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

  15. Robust Mechanism? • Metabolism • Regeneration & repair • Healing wound /infect • Fat accumulation • Insulin resistance • Proliferation • Inflammation Fluctuating energy Controlled Dynamic Low mean High variability

  16. dynamics slow fast Brain high Food Control? GI Heart Oxy Lac/ph Glu priority Triglyc Muscle FFA Out • Energy • Inflammation • Coagulation Glyc low Liver Glycerol Evolved for large energy variation and moderate trauma Glyc Fat

  17. dynamics fast Brain high Control? Heart Oxy priority Muscle Out Glyc low Essential starting point?

  18. “grey box” Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Plumbing and chemistry Qr Ql H right heart Rr , Sr left heart, Rl , Sl arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  19. Robust/Health Persistent mystery Fragile/ Illness Low mean High variability High mean Low variability

  20. High mean, low variability 140 The persistent mystery 120 HR Low mean, high variability 100 80 60 HR data time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 Heart rate data Two experiments with same subject

  21. Our approach Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Physiology! an ancient art Qr Ql H right heart Rr , Sr left heart, Rl , Sl arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  22. Other views • Molecular genetics • Creation science • New sciences of • complexity • networks 180 160 140 120 100 What gene? 80 60 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 400 400

  23. Two experiments with same subject 150 140 watts 120 100 HR HR data 100 W 50 80 watts 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 Data: Watts and HR

  24. Two experiments 150 100 +100w W 50 0 Data: Watts On recumbent Lifecycle

  25. 150 140 120 100 watts HR data 100 W 50 80 Data: Watts and HR 60 0 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 time(sec)

  26. 150 data 140 model watts 120 100 HR 1st order linear model 100 W 50 80 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350

  27. 150 data 140 model watts 120 100 HR same 1st order linear model 100 W 50 80 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350

  28. 150 140 120 same 1st order linear model 100 HR HR data 100 W 50 80 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 Model and HR

  29. 150 140 1st order linear models (different parameters) 120 100 HR HR data 100 W 50 80 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 Model and HR

  30. 150 140 120 Explain differences between models 100 HR 100 W ? ? 50 80 ? 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350

  31. 150 140 Explain differences between models and data 120 100 HR HR data 100 W 50 80 60 0 time(sec) 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350

  32. breath and HR at 0 watts 2nd order linear model HR inhale 100 50 0 0 50 100 150 200 250 300

  33. 100 50 0 0 50 100 150 200 250 300

  34. 90 80 70 60 50 40 190 200 210 220 230 240 250 260 270 280 100 50 0 0 50 100 150 200 250 300

  35. 90 80 70 60 50 40 190 200 210 220 230 240 250 260 270 280 • “resting” HR • ~40 bpm fluctuations at ~10s period • 100% fluctuations! • Frequency sweep in breathing • Fit well with 2nd order model

  36. 90 80 70 60 50 40 190 200 210 220 230 240 250 260 270 280 100 50 0 0 50 100 150 200 250 300

  37. data model 100 50 @100 w 0 100 50 @0 w 0 0 50 100 150 200 250 300

  38. Explain differences between • models • model and data Different subject, 3 data sets 180 300 160 250 140 Watts 200 HR data 120 150 100 100 80 50 60 0 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 400 400

  39. The persistent mystery Young, fit, healthy  more extreme 180 160 140 High mean, low variability HR 120 100 Low mean, high variability 80 60 40 0 0 50 50 100 100 150 150 200 200 250 250 300 300 350 350 400 400

  40. Optimal control Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql right heart Rr , Sr left heart, Rl , Sl H arterial venous What can we say with this model? systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  41. Plumbing and chemistry (aerobic) VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql H right heart Rr , Sr left heart, Rl , Sl arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  42. Organized complexity, circa 1972 Plumbing and chemistry

  43. Conservation laws: Energy and material (small moieties) VE Pulmonary peripheral Lungs, Fp, Rp arterial venous Qr Ql right heart Rr, Sr left heart, Rl, Sl H venous arterial systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  44. Conservation laws: Energy and material Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql right heart Rr , Sr left heart, Rl , Sl H arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  45. “grey box” Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql H right heart Rr , Sr left heart, Rl , Sl arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  46. Optimal control Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql right heart Rr , Sr left heart, Rl , Sl H arterial venous Consequences? systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  47. Conservation laws Feedback Controller Related States VE Pulmonary peripheral Lungs, Fp , Rp arterial venous Qr Ql right heart Rr , Sr left heart, Rl , Sl H arterial venous systemic peripheral,Tissues, Fs Rs Local metabolic control Workload,w(t)

  48. Homeostasis controls errors heart rate ventilation vasodilation coagulation inflammation digestion storage … O2 BP pH Glucose Energy store Blood volume … breath energy trauma heart beat infection sensor external disturbances internal noise

  49. errors O2 BP pH Glucose Energy store Blood volume … Brain

  50. controls heart rate ventilation vasodilation coagulation inflammation digestion storage … Brain

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