Overview

1
Notation
?
2
A preview of new unpublished results
Disturbance
e
d
-
u
Remote Sensor
Plant
Communication Channel
Control
Encoder
A simplification of amazing new results from
Martins/Dahleh.
3
Bode
Disturbance
e
d
-
u
Normalize all delays by actuation delay.
Plant
Assume plant has one unstable pole at a gt 1.
Control
Assume controller is locally Full Information.
4
Control demo
5
BodeShannon
Disturbance
e
d
-
u
Remote Sensor
Plant
Capacity C
Communication Channel
Control
Encoder
6
Bound is achievable in LQG case (and thus tight).
Disturbance
e
d
-
u
Remote Sensor
Plant
Communication Channel
Control
Encoder
Information Low latency requires minimal or no
coding
Controls impact of causality on stabilizability
7
Physics High power actuation/sensing/comms is
obviously energy expensive, but in addition
Physics Low latency is highly dissipative, even
when no work is done, and thus energy expensive
Disturbance
e
d
-
u
Remote Sensor
Plant
Communication Channel
Control
Encoder
Information Low latency requires minimal or no
coding
Controls impact of causality on stabilizability
Computation Low latency computation is expensive
in energy
8
Disturbance
e
d
-
u
Remote Sensor
Martins/Dahleh have a much deeper (and harder)
result for a channel in the feedback loop.
Plant
Communication Channel
Control
Encoder
9
Notation
10
Assume there are delays everywhere, in actuation,
sensing, communications, plus there is
communication noise. Assume w and n are Gaussian
white noise. Assume the controllers are locally
full information (but may have delays).
w
n
e
y
11
Feedback
Feedforward
Disturbance arrival delay
w
e
Sensor/channel
n
Total FB and FF sensor plus actuation delays
Total communications and actuation delay
Can lump all the delays into a few net delays.
w
n
e
y
12
Feedback
Feedforward
Sensor/channel
w
e
n
Rearrange
Feedback
Feedforward
Sensor/channel
w
e
n
13
Provided we assume that everything is linear and
Gaussian (which is not general), plus these other
without-loss-of-generality assumptions, then this
decomposition is also wolog.
Feedback
Feedforward
Sensor/channel
w
e
n
14
Conjecture this part will always be the same for
both variance and entropy, in which case this is
easy
w
v
n
This is all standard
15
Special easy case when all delays are equal to 1
and there is only one unstable pole.
Trivial solution here.
Feedback
Feedforward
Sensor/channel
w
e
n
16
What about for general delays? But still assume
that everything is linear and Gaussian (which is
not general), so this decomposition is also wolog.
Feedback
Feedforward
Sensor/channel
w
e
n
17
Entropy conjecture
This is definitely true
Trivial solution here.
Not sure here.
Pretty much doesnt matter what you do here.
Feedback
Feedforward
Sensor/channel
w
e
n
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Variance conjecture
Still use trivial solution here
Conjecture
Feedback
Feedforward
Sensor/channel
w
e
n
19
w
y
20
Coal
waste
electricity
21
Coal
waste
electricity
22
Flt1 is a law, a necessity, a constraint, It is
not an accident of history.
http//phe.rockefeller.edu/Daedalus/Elektron/
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Electricity generation and consumptionImprovement
s in efficiency over time
http//phe.rockefeller.edu/Daedalus/Elektron/
24
Suppose this were a plot of log(f) instead of
log(f/(1-f)) (since they would be nearly the
same).
What might we then expect the future to
hold? What if we didnt know about conservation
of energy? (That f lt 1.)
log(f)
25
Imagine the discussion that would ensue around
such a conjecture.
log(f)
2200
2100
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Efficiency
100
10
1
0.1
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(No Transcript)
28
Regulation of metabolism
Autocatalysis
Enzyme
Metabolite
29
Autocatalysis
Enzyme
Metabolite
30
Enzyme
Metabolite
31
Product inhibition
perturbation
Yi, Ingalls, Goncalves, Sauro
32
1.05
Ideal product concentration
ATP
1
0.95
0.9
0.85
0.8
0
5
10
15
20
Time (minutes)
Step increase in demand for product or ATP.
perturbation
33
1.05
Ideal product concentration
Step increase in demand for product or ATP.
ATP
1
0.95
0.9
0.85
0.8
0
5
10
15
20
Time (minutes)
h 0 1 2 3
34
h 3
h 2
h 1
h 0
Time
0
5
10
15
20
Higher feedback gain
It is well-known that many biological regulatory
networks can oscillate, and presumably many more
will be discovered.
35
Transients, Oscillations
h 3
h 2
h 1
Tighter steady-state regulation
h 0
Time
0
5
10
15
20

• Are these tradeoffs an artifice of this model?
• Does it matter if the model is nonlinear,
  stochastic, distributed, PDEs, etc? Does it
  depend on the model at all?
• Are these tradeoffs due to a frozen accident of
  evolution and not an absolute necessity?
• The answer to all these questions is no.

36
1.05
Ideal
ATP
1
0.95
Time response
0.9
0.85
0.8
0
5
10
15
20
Time (minutes)
37
1.05
ATP
1
h 3
0.95
Time response
0.9
0.85
h 0
0.8
0
5
10
15
20
Time (minutes)
0.8
h 3
0.6
Spectrum
0.4
0.2
h 0
Log(Sn/S0)
0
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
38
Yet fragile
0.8
h 3
0.6
Robust
0.4
0.2
h 0
Log(Sn/S0)
0
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
39
Yet fragile
0.8
0.6
Robust
0.4
0.2
h 0
Log(Sn/S0)
0
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
40
Theorem
Transients, Oscillations
0.8
h 3
0.6
Tighter steady-state regulation
0.4
h 2
0.2
h 0
Log(Sn/S0)
0
h 1
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
41
This tradeoff is a law.
Transients, Oscillations
logS
?
Biological complexity is dominated by the
evolution of mechanisms to more finely tune this
robustness/fragility tradeoff.
Tighter regulation
42
d(k)
e(k)
u(k)
e(k) d(k) - u(k)
-
43
For simplicity, assume d, u, and e are finite
sequences.
d(k)
u(k)
k
e(k)
k
Then the discrete Fourier transform D, U, and E
are polynomials in the transform variable z.
If we set z ei? , ? ? 0,?? then X(w) measures
the frequency content of x at frequency w.
44

• Denote by zk the complex zeros for z gt 1 of
  X(z)
• Jensens theorem

Proof Contour integral
45
A useful measure of performance is in terms of
the sensitivity function S(z) defined by Bode as
If we set z ei? , ? ? 0,?? then S(w)
measures how well C does at each frequency. (If C
is linear then S is independent of d, but in
general S depends on d.) It is convenient to
study log S(w)
46

• Denote by ek and xk the complex zeros for z
  gt 1 of E(z) and D(z), respectively. Then

Proof Follows directly from Jensens formula.
If d is chosen so that D(z) has no zeros in z gt
1 (this is an open set), then
47
Assume u is a causal function of x.
x(k)
u(k-1)
k
48
Disturbance
Model
badness
-
49
This tradeoff is a law.
Product inhibition is a protocol.
50
PFK and ATP are modules.
This tradeoff is a law.
Product inhibition is a protocol.
51
logS
?
Conservation of fragility
52
Well-known gt 50 years in control theory (Bode) as
a property of linear models. New results applies
to anything causal, so can be nonlinear,
stochastic, infinite-dimensional, even data.
?
53
h 3
h 2
h 1
h 0
Time
0
5
10
15
20
0.8
h 3
0.6
0.4
h 2
0.2
h 0
Log(Sn/S0)
0
h 1
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
54
Autocatalysis
Regulation
Enzyme
Metabolite
Energy and materials
55
assembly
metabolism
transport
Autocatalytic feedback
Even though autocatalytic feedback contributes
relatively modestly to complexity, it has a huge
indirect impact on regulatory complexity.
Regulatory feedback
56
assembly
metabolism
transport
Autocatalytic feedback

• Autocatalysis is everywhere in human and natural
  systems as well as biology
• Make energy, materials, and machines to make
  energy, materials, and machines to make
• Consumers are investors are labor

Regulatory feedback
57
Regulatory feedback only
h 3
h 2
h 1
h 0
Time
0
5
10
15
20
0.8
h 3
0.6
0.4
h 2
0.2
h 0
Log(Sn/S0)
0
h 1
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
Frequency
58
Product inhibition
Autocatalytic feedback
Yi, Ingalls, Goncalves, Sauro
59
Add autocatalytic feedback
more
60
Add autocatalytic feedback
61
Add more regulator feedback
62
More instability aggravates
63
assembly
metabolism
transport
Conservation of energy, moiety, and fragility are
laws.
Autocatalytic feedback
Enzymes are modules.
Bowtie architectures with product inhibition is
a protocol suite.
Regulatory feedback
64
assembly
metabolism
transport

• What aggravates regulatory feedbacks role in
  providing robustness to disturbances and
  component uncertainty?
• Time delays in sensing, actuating, and computing
• Autocatalytic feedback
• Sensor and internal component noise
• Saturation of actuators

Autocatalytic feedback

• What helps?
• Collocate regulation with autocatalytic feedback
• Better sensors, actuators, and computation
• Better models of the environment

Regulatory feedback
65
Gly
G1P
G6P
F6P
F1-6BP
Gly3p
ATP
13BPG
TCA
3PG
Oxa
ACA
PEP
Pyr
2PG
NADH
Cit
66
Disturbance
Model
badness
-
Regulation
67
Predator
trauma
-
Regulation
68
Are there more general fragility/robustness
tradeoffs?