Control Theory, Networks, and Life Itself
John Baillieul
C.I.S.E. and
Intelligent Mechatonics Laboratory
johnb@bu.edu
http://people.bu.edu/johnb
http://iml.bu.edu
http://www.bu.edu/systems
Boston University
ME 501 First Lecture
1
John Baillieul’s Research Group
Intelligent Mechatronics Laboratory
2
John Baillieul’s Hiking Friends
3
Boston University---from humble beginnings
4
Prior outrageously titled lectures (and books)
Statutory Lectures, 1943, Trinity
College, Dublin
•  February 5, 1943
•  February 12, 1943
•  February 19, 1943
“How can the events in
space and time which
take place within the
boundary of a living
organism be accounted
for by physics and
chemistry?”
5
Prior outrageously titled lectures (and books)
Professor Lynn Margulis Boston Univeristy
University Lecture 1978
The Early Evolution of Life
Lynn Margulis and Dorion
Sagan, What Is Life?, University of California Press
(August 31, 2000), 303 pages
ISBN-10: 0520220218
6
Control Theory: Talk Outline
I.  The hidden technology--where is it hiding
II.  Information-based control - the basis of engineered and
natural systems regulated by feedback
III.  Networked Control Systems
- Technologies enabled by information-based control
- Complex networks of natural and engineered systems
IV.  The emerging theory of complexity-based risk and the
rational management of failure
V.  Controlling highly structured motions of groups of agents
1)  The theory of rigid formations
2)  Spatial and dynamic information patterns
VI.  Conclusions
7
People in Control at B.U.
Professor David A. Castañon
42-nd President of the IEEE Control Systems Society
Professor Christos G. Cassandras
Editor-in-Chief IEEE Tranactions
on Automatic Control
8
World’s leading journals at B.U.
SIAM J. on Control and
Optimization
9
The Hidden Technology
•  Pervasive
•  Very successful
•  Seldom talked about
Except when there is an accident!
Rare occasions!
•  Why?
Easier to discuss devices than ideas
(feedback)
We have not done our job!
K. J. Åström ECC August 31, 1999 10
11
The World’s largest physics experiment
is enabled by control tehnologies
•  27 km. (18 mi.)circumference
•  Beam steered, collimated by superconducting magnets
•  Beam comprised of 2835 bunches of 1011 protons
•  Experiment operates at ~0oK
•  12M litres liq. N
•  0.7M litres liq. He
•  Time constants
•  Thermal = 0.5 years
•  Beam control = nano sec.
and 10 hours
12
The control of machines
From simple origins to life-saving enhancements. . . 13
14
“Electronic Stability Program (ESP) is a new
safety system which guides cars through wet
or icy bends with more safety. ...
The key is a yaw-rate sensor, which detects
vehicle movement around its vertical axis,
and software which recognizes critical driving
conditions and responds accordingly. In
an instant, instructions are sent to the engine,
transmission and brakes, thereby encountering
a skid at its onset. …”
K. J. Åström ECC August 31, 1999
15
The Position of Control as a Discipline
•  Respected
•  Coupled to a vast array of engineered systems
•  Lacks distinct identity (CERN example)
•  Lacks an identifiable industrial base (cf. computers and mobile comm. devices)
• Academic positioning
16
A Brief History of the Field
Closely tied to emerging technologies
(steam, power, electricity, telephone, aerospace ...)
Telecommunications
•  Blacks invention 1927
•  Nyquist 1932
•  Bode 1940
•  Servomechanism theory
Consequences
The second wave
•  Recursive estimation
•  Maximum principle
17
The Third Wave
Driving forces
•  New challenges
•  New applications
•  Mathematics
•  Computers
Basic paradigm
•  State Space
Rapid expansion
•  Subspecialities
Optimal Control
Nonlinear Control
Stochastic Control
Computer Control
Robust Control
Robotics
Adaptive Control
CACE
Cooperative and
decentralized control
18
The Next Wave
19
By 2015, 1/3 of all deployed military vehicles will be
autonomous.
MURI 2007: Behavioral Dynamics in the Cooperative Control of
Mixed Human/Robotic Teams
20
The two ages of joint cognitive transport
systems
The First Age of Joint Cognitive Transport
4,000 BC
First domesticated
draught animals
The
Second
Age
1900 AD 2010 AD
The age of
mechanized
transport
21
Autonomous vehicles
operate in land, air,
and sea…
22
Science Fiction
Science . . .
23
Classical Feedback Control
Plant
in
R2
R1
out
-
+
Controller
Networked Control
p11
p12
p13
Plant
Router
p21
p22
p23
PROCESS MODEL
CONTROLLER
S3
S1
S2
MUX
24
Mote introduced
Intel 82526
Bosch GmbH
IBM 701
H. Nyquist
Regeneration Theory
Control of Networked Devices
25
When feedback loops are closed using packet-switched
wireless communication links, data-rates become an issue.
Current problems: •  Develop a control theory for effectively managing
wireless communications capability for automatically
configuring ad hoc networks.
•  Understand the relationship between standard network
objectives (such as maximizing traffic capacity) and
control objectives. 26
Ad Hoc Networks of Mobile Robots
27
The most important paper of the decade
•  W.S. Wong & R.W. Brockett,
1995, 1999, “Systems with finite
communication bandwidth
constraints, II: Stabilization with
limited Information Feedback”
IEEE Trans. AC, May, 1999.
28
The Data-Rate Theorem
Theorem: Suppose the system G(s) (previous slide) is
controlled using a data-rate constrained feedback channel. Suppose, moreover, G has k right half-plane poles λ1,…,λk.
Then there is a critical data-rate
Rc = log 2 e ⋅ (Re( λ1 ) +  + Re( λk ))
such that the system can be stabilized if and only if the
channel capacity R>Rc.
€
---Baillieul, 1999, (2002CDC), Nair and Evans, 2000, Tatikonda and
Mitter, 2002, . . .
29
Suppose there are only two (finitely many) actions
that can be taken:
•  Jerk the cart left or right one centimeter
•  Under what circumstances can one keep
the pendulum upright using this very
coarse type of “control?”
•  Ans: If and only if there is a sufficiently
high actiel rate.
t
u vs t
30
Brief “Partial” History of the Data-Rate Theorem
•  D. Delchamps, “Stabilizing a linear system with quantized state
feedback,” IEEE Trans. AC, 1990.
• W.S. Wong & R.W. Brockett, 1995, 1999, “Systems with
finite communication bandwidth constraints, II: Stabilization
with limited Information Feedback” IEEE Trans. AC.
•  S. Tatikonda, A. Sahai, & S.K. Mitter, 1998, “Control of LQG systems
under communication constraints,” CDC
•  John B., 1999, “Feedback designs for controlling device arrays with
communication channel bandwidth constraints,” ARO Workshop.
•  G.N. Nair & R.J. Evans, 2000, “Stabilization with data-rate-limited
feedback: tightest attainable bounds,” Sys, & Control Lett.
•  F. Fagnani and S. Zampieri, 2001, “Stability analysis and synthesis for
scalar systems with a quantized feedback,” Tech. Rept., Politechnico di
Torino.
31
Some References
•
Keyong Li and John Baillieul,
“Robust and Efficient Quantization
and Coding for Control of
Multidimensional Linear Systems
Under Data-Rate Constraints,” to
appear in the Int’l J. of Robust and
Nonlinear Control.
•
K. Li and J. Baillieul, “Robust
Quantization for Digital Finite
Communication Bandwidth
(DFCB) Control,” IEEE Trans.
Automatic Control, Special Issue
on Networked Control Systems,
September, 2004.
32
Control of Networked Systems
•  Communication constrained and information enabled
control systems
  Coding for robustness to time-varying data-rates
  Noise, bit-errors, and risk
  Failure-sensitive control
  Communication requirements with decentralized
sensing and actuation
•  Patterns and constraints on information flow in networked
control systems
  Sensing patterns for stable motions
  Consensus problems for groups of autonomous agents
  Shaping formations in motion
  Coverage problems for groups of autonomous agents
33
RISK AND FAILURE IN
ENGINEERED SYSTEMS
•  Probabilistic models of risk
•  Information-constraint risk
•  Complexity-based risk
•  Model-mismatch risk
•  Unplanned for events
Uncertainty
Uncertainty
Uncertainty
Uncertainty
34
COMPLEXITY-BASED RISK
Sensitivity to trajectory variations.
35
Complexity-Based Risk vs.
Statistical Risk
•  Statistical risk is assumed to be stationary.
•  There are typically important transient effects in
complexity-based risk.
•  The distinctions are not always sharp.
36
Simplified Roulette: The rotating
pendulum
37
The rotating pendulum: Dissipation
enhanced multi-stability
38
Dissipation enhanced multi-stability brings the risk
of falling into the wrong potential well
39
Highly Complex Systems
Amino acids are linked by peptide bonds of various types to
form proteins. Suppose there are three types of bonds between
each amino acid in a chain of 101 molecules. There are
conformations (=stable configurations).
40
The Complex Energy Landscape of Protein Docking
41
RISK AS AN ENGINEERING
DESIGN PARAMETER
Risk level
# of Failures
1-sigma
Less than 4 in 10
2-sigma
Less than 5 in 100
3-sigma
Less than 3 in 1000
4-sigma
Less than 7 in 100,000
5-sigma
Less than 6 in 10,000,000
6-sigma
Less than 2 in
1000,000,000
42
Networks Are Everywhere
Enzyme
http://www.animalport.com
Enzyme
Mitogen-activated protein kinase (MAPK)
---C. Conradi, J. Saez-Rodriguez, E.-D. Gilles and J. Raisch
43
Networked Biological Control Systems
Craig C. Mello
44
Biological Paradigns for Networked
Control Systems
---C.V. Rao & A.P. Arkin
45
Gene Networks as Networked Control Systems
46
Typical real-time data available to control the
motion of an autonomous mobile robot
IEEE 802.11
Computer vision
Differential GPS
Sonar
Laser range and
bearing sensor
Hall effect direction sensors
Wheel encoders
Inertial navigation
47
What are the design principles for distributed, realtime, situation-dependent real-time control with
localized sensing and information processing?
48
Strategy: Reason about Formation Control Using
Point Robots - Realize Formations in Models of
Nonholonomic Vehicles
A
dAC
dBA
B
dAB
dCA
dBC
dCB
C
Multiply redundant distributed
arrays of sensors support
flexible design of robust
formation control strategies.
Beware of informationinduced instability…
49
Graphs and Frameworks are Tools for Patterns of
Information Use
Definition 1: A formation framework (S;E;q) consists of a
formation graph (S;E) and a function q(.) from the vertex
set S into a squadron configuration space
= typical point.
position
orientation
Definition 2: Motions which preserve all pairwise relative
distances
are called rigid.
50
Rigid Formations---Creation and Motion Control Strategy
•  Sensor data should be used in a maximally parsimonious fashion,
•  There will always be a leader-first-follower pair.
We say that a formation framework
( S, E, q) is isostatic if the removal of any edge
ε∈Ε results in a framework which is not rigid. Yes
No
51
Main Results in Planar Rigidity Theory
LAMAN’S THEOREM: A planar graph (V,E) is isostatic ˛
1.  |Ε| = 2|V|-3,
2.  |E(U)| ≤ 2|U| - 3 • U⊆V with |U| ≥ 2.
HENNEBERG’S THEOREM (1911): A planar graph (V,E) is
isostatic ˛ it can be constructed from a single edge by a
sequence of vertex extensions and edge splits.
Vertex extension
52
Rigid Planar Graphs on 2,3,4, and 5 Vertices
•  Can also be constructed by
an edge-split.
•  Cannot be constructed by
an edge-split.
•  Can also be constructed by
an edge-split.
6?
53
Real-time Stabilized “Rigid” Formations
A
B
C
•  The soft real-time question “Is
the formation congruent to the
prescribed triangle?” can be
answered?
•  Theorem No triangular formation
is asymptotically stable under this
peer-following control strategy.
A
B
C
Assume each robot can acquire and
process “simultaneous” real-time
information on line-of-sight
distances to two peers. Then there
is a control law based on the sensor
data acquired as depicted which will
stably configure the robots into any
prescribed triangle.
54
Rigid Vertex Extension of a Formation Framework
1
As long as the rest point is not on the line
determined by the leader (1) and first
follower (2), it is an asymptotically stable
rest point for the motion:
2
This is a semi-global result.
55
Distributed Relative Distance Control of Point Robots
Definition: Distributed relative distance control of a multi
vehicle formation takes the form ⎛ x˙ i ⎞
⎜ ⎟ =
⎝ y˙ i ⎠
€
⎛ x i − x j ⎞
∑ u(dij , ρij )⎜ y − y ⎟
⎝ i
j ⎠
j ∈S i
where dij is the set-point distance between the i-th and j-th
vehicles and ρij is the sensed distance. Thus the control is a
function of the prescribed relative distance and the measured
relative distance.
Examples: 1. Scaled difference, 2. Normalized difference, 3.
Lennard-Jones, 4. Etc.
56
Necessary and Sufficient Conditions for Stable Rigidity
Theorem: An acyclic formation graph corresponding to
a stably rigid formation under a distributed relative
distance control law is isostatic if and only if 1.  one vertex (the leader) has out-valence 0
3.  one vertex (the first-follower) has out-valence 1 and is
adjacent to the leader vertex; and
4.  all other vertices have out-valence equal to 2"
Proof: Based on Laman’s Theorem and
the figure on the left.
57
The Rigidity Theory of Directed Graphs Differs from the
Undirected Case
1
2
3
4
(a)
1
2
3
4
(b)
Both graphs are rigid (and isostatic by Laman’s theorem) as
undirected graphs, but as directed graphs they are not rigid.
58
All Constructively Rigid Formations Look Like This
Leader
First-follower
Using dual peer sensing control, a stepwise
sequence of motions can be carried out to have a
set of planar robots assemble themselves into
any constructively rigid formation.
59
Leader - First-Follower - Subsequent Follower Formations
Modulo a left-right symmetry, this
formation is unique on three vertices.
3
4
A
B
C
Formation B can be constructed in one step - once the leader
and first-follower are in place. Followers must join in
sequence for A and C.
60
Leader - First-Follower - Subsequent Follower Formations
5
Five (5) of the thirteen (13) formations with five nodes.
These five are the formations which are created purely
sequentially.
61
Data-Structures Associated with Dual Peer-Sensing
Decentralized Formation Control
1.  Logical formation graphs: the set of vertices and their
adjacency relationships,
2.  The set of possible stable configurations corresponding to a
given logical formation graph together with the prescribed
distance of the first follower from the leader and prescribed
pair of distances of each node from the two nodes to which it
is adjacent.
A
B
C
62
The enumerative theory of logical isostatic directed
graphs remains to be explored
•  Currently, closed form expressions for this enumeration do not exist.
•  Many stratification classes do have closed-form enumerations.
•  This is a new sequence, not previously in Sloan’s sequence list.
“The Combinatorial Graph Theory of Structured Formations,” In Proceedings of the 46-th IEEE Conference on Decision and Control, New
Orleans, December 12-14, 2007.
63
Counting Isostatic Planar Formations
64
Interesting Problem
Develop a clear mapping between algebraic representations
of information patterns and the motions they support.
Example:
Consider point robot dynamics of the following form:
⎛ x˙ ⎞
⎛ x − x1 ⎞
⎛ x − x 2 ⎞
⎜ ⎟ = u1⎜
⎟ + u2 ⎜
⎟.
⎝ y˙ ⎠
⎝ y − y 2 ⎠
⎝ y − y 2 ⎠
€
65
Information Flow Must Be Localized and
Dynamically Reconfigured
Given the geometry of a formation,
how many distinct “stable
information-flow patterns” will
support it?
66
Formation Equations for Large-Scale Group Motions
For an n-node formation, there are n pairs of equations like these.
The system has 2n-2 asymptotically stable equilibria.
67
The Stability and Bifurcation Theory of Formation Equilibria d
1
d
2
(0,0)
(0,1)
d
12
Proposition: Let d12=
(x1 − x 2 ) 2 + (y1 − y 2 ) 2 and assume:
d1 + d2 > d12 , d12 + d2 > d1, d1 + d12 > d2 .
Then the equilibrium solutions satisfying the equations
€
2
2
(x
−
x
)
+
(y
−
y
)
dj -
= 0, (j=1,2)
j
j
€ are locally asymptotically stable.
€
68
The Singularity Theory of the Critical Line For all d1 and d2>0, there are equilibrium solutions of position
equations on the critical line. Assume d1≥1. Write r=d1. When
d1+d2> d12, (xe,ye)=(r,0) is a solution for all r >1. There are
others as depicted:
69
The bifurcation and stability characteristics are quite rich.
⎛ x˙ ⎞
⎛ x − x1 ⎞
⎛ x − x 2 ⎞
⎜ ⎟ = (d1 − ρ1 )⎜
⎟ + (d2 − ρ 2 )⎜
⎟
⎝ y˙ ⎠
⎝ y − y 2 ⎠
⎝ y − y 2 ⎠
70
Careful Planning of Group Motions Is Needed to Deal with
Extraordinary Sensitivity to Initial Conditions
For each of the formation-types on n-nodes, there are 2n-2
locally asymptotically stable equilibria.
Start
Finish
I.C.’s (x2(0),y2(0))=(8,0), (x3(0),y3(0))=(3,-1), (x3(0),y3(0))=(1,-1).
F.C.’s (x2(∞),y2(∞))=(1,0,), (x3(∞),y3(∞))=(0.705, -1.97812), (x3(∞),y3(∞))=(-0.0906396, -0.143197).
I.C.’s (x2(0),y2(0))=(9,0), (x3(0),y3(0))=(3,-1), (x3(0),y3(0))=(1,-1).
F.C.’s (x2(∞),y2(∞))=(1,0,), (x3(∞),y3(∞))=(0.705, -1.97812), (x3(∞),y3(∞))=(1.99511, -0.45236).
71
Conclusions and Future Work
•  Control is and probably will remain a hidden technology
•  Networked systems are ubiquitous, and network control
systems are essential to understand in both the natural and
engineered world
•  The role of information in relation to the physical world
remains to be understood.
•  The essence of robustness in network dynamics remains
to be understood.
72
Sensor-Based Feedback Laws for Nonholonomic Vehicles Are Not Stabilizing
Theorem (Brockett). Consider the nonlinear system x = f(x, u)
with f(x0, 0) = 0 and f(. , .) continuously differentiable in a
neighborhood of (x0, 0). Necessary conditions for the existence of a
continuously differentiable control law for asymptotically stabilizing
(x0, 0) are:
(i) The linearized system has no uncontrollable modes associated with
eigenvalues with positive real part.
(ii) There exists a neighborhood N of (x0, 0) such that for each ξ∈ N
there exists a control uξ (t) defined for all t > 0 that drives the solution
of x = f(x, uξ ) from the point x = ξ at t = 0 to x = x0 at t = ∞.
(iii) The mapping γ: N×ℜm→ℜn, N a neighborhood of the origin,
defned by γ : (x; u) → f(x; u) should be onto an open set of the
origin.
73
For Nonholonomic Motion Control, Critical Points ->
Critical Varieties
φ
⎛ x˙ ⎞ ⎛ uCos(θ )⎞
⎜ ⎟ ⎜
⎟
˙
y
=
uSin(
θ
)
⎜ ⎟ ⎜
⎟
⎜ ˙ ⎟ ⎜
⎟
θ
ω
⎝ ⎠ ⎝
⎠
€
u = λ( ρ − d), and
ω = ρSin(ϕ ) − d.
€
74
The Laman conditions in dimension m, m=1,2,3
.
75
Rigid Formations with Cycles Cannot Be Constructed by
this Procedure
F
L
The formation is stably rigid.
4
5
3
76
Using the Basic Point Robot Model to Coordinate LargeScale Group Motions
77
Chemical Reaction Networks (CNR’s)
•  The basis of cell biology
•  The understanding of cell functions at the level of aggregate
behavior in terms of cascades and interactions of networks of
chemical reactions is a truly formidable task---which will likely
never be fully accomplished.
---See Eduardo Sontag, 2005
•  What is needed are systematic tools for decomposing network
dynamics into component motifs.
78
Networks Are Everywhere
There are
13 threenode
control
motifs.
79
References on Control and Biological
Networks
Special Section on Biochemical Networks and
Cell Regulation, Control Systems Magazine,
Volume: 24 Issue: 4, Aug. 2004.
Special Issue on Systems Biology, IEEE
Transactions on Automatic Control, AC:53,
Jan. 2008.
80