ESE 680 Lecture 2 01/11/2007
ESE 680 Special topics in ESE
Distributed Dynamical Systems
Ali Jadbabaie
Department of Electrical and Systems Engineering
and GRASP Laboratory
University of Pennsylvania
365 GRW jadbabai@seas.upenn.edu
http://www.seas.upenn.edu/~jadbabai/ESE680/ese680.html
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Course Info
This is a RESEARCH SEMINAR
Requires a lot of INDEPENDENT, critical reading of
literature
You are expected to actively PARTICIPATE in
discussions
A LOT of reading is required and you need to be
able to present papers
There are very few didactic lectures
Here is a brief summary of last time
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Networked dynamical systems
Nonlinear/uncertain
hybrid/stochastic etc.
Complexity
of
dynamics
Single
Agent
Complex
networked
systems
Flocking/synchronization
consensus
Multi-agent
systems
Complexity
of interconnection
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Networked dynamical systems
State dimensionality
System size
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Statistical Physics and
emergence of collective behavior
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Overview
Nonlinear/uncertain
hybrid/stochastic etc.
?
Single
Agent
?
Complexity
of
dynamics
Complex
networked
systems
Flocking/synchronization
consensus
Multi-agent
systems
Complexity
of interconnection
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Multi-agent setting: Vicsek’s kinematic model
• How can a group of moving agents collectively decide on
direction, based on nearest neighbor interaction?
r
neighbors of
agent i
agent i
How does global behavior emerge from local interactions?
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Synchronization
Fireflies Flashing
From D. Attenborough “Trials of Life – Talking to strangers”
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Kuramoto Model
di
K N
 i   sin( j  i )
dt
N j 1
N : Number of oscillators
i : Natural frequency of oscillator i, i  1, , N.
All-to-all interaction
 i : Phase of oscillator i, i  1, , N.
K : Coupling strength
This is the Kuramoto model
We assume throughout homogeneous coupling.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Kuramoto model & graph topology
2
1
3
6
4
0
1

1
A
0
0

 1
1 1 0 0 1
0 1 0 0 0 
1 0 1 0 0

0 1 0 1 1
0 0 1 0 1

0 0 1 1 0 
5
di
K N
 i   Aij sin( j  i )
dt
N j 1
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Ubiquity of Dual Decompositions
Dual Decomposition is THE key idea that makes the internet
protocols “work” in a distributed asynchronous fashion
What is the connection between flocking, oscillator
synchronization, and the internet?
But, how DO “these internets” work?
Senator Ted Stevens (R-Alaska), (the architect of the 280
million dollar bridge to nowhere:
“ …and again, the Internet is not something you just dump
something on. It's not a big truck. It's a series of tubes. And if you
don't understand those tubes can be filled and if they are filled,
when you put your message in, it gets in line and it's going to be
delayed by anyone that puts into that tube enormous amounts of
material, enormous amounts of material.”
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
The Internet hourglass
Applications
Web
FTP
Mail
News
Video
Audio
ping
napster
Transport protocols
TCP SCTP UDP
ICMP
IP
Ethernet 802.11
Power lines ATM
Optical
Link technologies
Satellite Bluetooth
The Internet hourglass
Applications
Web
FTP
Mail
News
Video
Audio
ping
napster
TCP
IP
Ethernet 802.11
Power lines ATM
Optical
Link technologies
Satellite Bluetooth
The Internet hourglass
Applications
Web
FTP
Mail
News
Video Audio
IP under
everything
ping
napster
TCP
IP
Ethernet 802.11
IP on
Power lines ATM Optical
everything
Link technologies
Satellite Bluetooth
Congestion Control for the Internet
Aims to avoid congestion collapse
• Congestion collapse appeared in the late 80’s because of the
lack of congestion control
• Intuitive congestion control design alleviated the problem…
• Until a few years ago when it was shown to be unstable!
New designs aim to achieve:
• Optimal sharing of available resources at equilibrium;
• Scalable stability for
• arbitrary topologies (size and connectivity)
• arbitrary links’ capacities
• arbitrary, inhomogeneous round trip times (time delays)
Last property has been proven only for the linearized
system – here we provide a proof for the nonlinear case
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Routers
Mesh-like core of fast,
low degree routers
Hosts
High degree
Routersnodes
are at the edges.
Hosts
Power Laws and Internet Topology
A few nodes have lots of connections
number of connections
Source: Faloutsos et al (1999)
rank
rank
Most nodes have few connections
Observed scaling in node degree and other statistics:
– Autonomous System (AS) graph
– Router-level graph
How to account for high variability in node degree?
6
5
Frequency
(Huffman)
(Crovella)
4
Cumulative
Data
compression
WWW files
Mbytes
3
Forest fires
1000 km2
2
(Malamud)
1
Los Alamos fire
0
-1
-6
-5
Decimated data
Log (base 10)
-4
-3
-2
-1
0
1
Size of events
2
18 Sep 1998
Forest Fires: An Example of Self-Organized
Critical Behavior
Bruce D. Malamud, Gleb Morein, Donald L. Turcotte
4 data sets
6
Web files
5
Codewords
4
Cumulative
Frequency
-1
3
Fires
2
-1/2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
Log (base 10)
2
6
5
Frequency
(Huffman)
(Crovella)
4
Cumulative
Data
compression
WWW files
Mbytes
3
Forest fires
1000 km2
2
(Malamud)
1
Los Alamos fire
0
-1
-6
-5
Decimated data
Log (base 10)
-4
-3
-2
-1
0
1
Size of events
2
20th Century’s 100 largest disasters worldwide
2
10
Technological ($10B)
Natural ($100B)
1
10
US Power outages
(10M of customers)
0
10
-2
10
-1
10
0
10
20th Century’s 100 largest disasters worldwide
2
10
Technological ($10B)
Natural ($100B)
1
10
US Power outages
(10M of customers)
0
10
-2
10
-1
10
0
10
2
10
Log(Cumulative
frequency)
1
10
= Log(rank)
0
10
-2
10
-1
10
Log(size)
0
10
100
80
Technological ($10B)
rank
60
Natural ($100B)
40
20
0
0
2
4
6
8
size
10
12
14
2
100
10
Log(rank)
1
10
10
3
2
0
1
10
-2
10
-1
0
10
10
Log(size)
20th Century’s 100 largest disasters worldwide
2
10
Technological ($10B)
Natural ($100B)
1
10
US Power outages
(10M of customers)
Slope = -1
(=1)
0
10
-2
10
-1
10
0
10
6
Data
compression
WWW files
Mbytes
5
4
Cumulative
Frequency
-1
3
Forest fires
1000 km2
2
-1/2
1
0
-1
-6
-5
Decimated data
Log (base 10)
-4
-3
-2
-1
0
1
Size of events
2
6
5
4
Cumulative
Frequency
Data
compression
WWW files
Mbytes
exponential
-1
3
Forest fires
1000 km2
2
-1/2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
2
6
Data
compression
WWW files
Mbytes
5
exponential
4
Cumulative
Frequency
3
Forest fires
1000 km2
2
All events are
close in size.
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
2
6
5
4
Cumulative
Frequency
Data
compression
WWW files
Mbytes
-1
3
Forest fires
Most2events
1000 km2
are small
1
0
-1/2
But the large
events are huge
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
2
6
Cumulative
Frequency
5
Most files Data
WWW files are small
compression
4
-1
Mbytes
But most packets
are in huge files
3
Forest fires
2 Most
fires
1000
km2
are small
1
0
-1/2
But most trees are
in huge fires
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
2
6
5
4
Cumulative
Frequency
Data
compression
WWW files
Robust
Mbytes
-1
3
Forest fires
Most2events
1000 km2
are small
1
0
-1/2
Yet
Fragile
But the large
events are huge
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
2
Large scale phenomena is
extremely non-Gaussian
• The microscopic world is largely exponential
• The laboratory world is largely Gaussian
because of the central limit theorem
• The large scale phenomena has heavy tails
(fat tails) and power laws
Power Laws in Topology Modeling
• Recent emphasis has been on whether or not
a given topology model/generator can
reproduce the same types of macroscopic
statistics, especially power law-type degree
distributions
• Lots of degree-based models have been
proposed
– All of them are based on random graphs,
usually with some form of preferential
attachment
Models of Internet Topology
• These topology models are merely descriptive
– Measure some feature of interest (connectivity)
– Develop a model that replicates that feature
– Make claims about the similarity between the real
system and the model
– A type of “curve fitting”?
• Unfortunately, by focusing exclusively on node
degree distribution, these models that get the
story wrong
• We seek something that is explanatory
– Consistent with the drivers of topology design and
deployment
– Consistent with the engineering-related details
Heuristically Optimal Network
Mesh-like core of fast,
Coresrouters
low degree
High
degree
Edges
nodes are at
the
edges.
Hosts
Abilene Backbone
Physical Connectivity
(as of December 16, 2003)
Intermountain
GigaPoP
Front Range
GigaPoP
Arizona St.
Oregon
GigaPoP
Internet routerlevel topology
U. Memphis
Indiana GigaPoP
Pacific
Northwest
GigaPoP
U. Louisville
Great Plains
OARNET
StarLight
Iowa St.
MREN
NYSERNet
Kansas
Denver City
UNM
WPI
Indianapolis
Chicago
Seattle
U. Hawaii
GEANT
Sunnyvale
SURFNet
Rutgers
U.
Wash
D.C.
Los Angeles
TransPAC/APAN
MANLAN
Houston
UniNet
North Texas
GigaPoP
Texas Tech
SOX
Miss State
GigaPoP
UT Austin
UT-SW
Med Ctr.
Atlanta
SFGP/
AMPATH
Texas
GigaPo
P
LaNet
Tulane U.
Northern
Crossroads
SINet
New York
ESnet
AMES NGIX
WIDE
WiscREN
NCSA
CENIC
0.1-0.5 Gbps
0.5-1.0 Gbps
1.0-5.0 Gbps
5.0-10.0 Gbps
Merit
OneNet
Qwest Labs
U.
Arizona
Pacific
Wave
Northern Lights
Florida A&M
U. So. Florida
MAGPI
PSC
DARPA
BossNet
UMD NGIX
Mid-Atlantic
Crossroads
Drexel U.
U. Florida
U. Delaware
NCNI/MCNC
2
10
Low degree
mesh-like core
1
identical
power-law
degrees
10
Completely different
networks can have the
same node degrees.
0
10
0
10
1
10
2
10
3
10
2
10
High degree hublike core
Low degree
mesh-like core
1
identical
power-law
degrees
10
Completely different
networks can have the
same node degrees.
0
10
0
10
1
10
2
10
3
10
• Low degree core
• High degree edge routers
• Failure and attack tolerant
High
degree
edge
routers
Rare
Completely
opposite
Space
of
graphs
• High degree hubs
• Failure tolerant
• Attack fragile
Mainstream
“Physics”
view
Likely
Power laws are ubiquitous,
not just the internet
Low
variability
High
variability
Gaussian
Exponential
Power law
Central Limit
Theorem
(CLT)
Marginalization
(Markov property)
CLT
Marginalization
Maximization
Mixtures
Power laws are unexceptional
Low
variability
High
variability
Gaussian
Exponential
Power law
Central Limit
Theorem
(CLT)
Marginalization
(Markov property)
CLT
Marginalization
Maximization
Mixtures
Demo
2
10
median
1
10
Reality
0
10
-2
10
-1
10
0
10
2
Robust
10
median
1
10
0
10
-2
10
-1
10
0
10
Yet
Fragile
Lessons learnt
• You cant just analyze graphs of complex
networks without domain knowledge. A
network is much more than a graph.
• Degree distributions DO NOT tell us
everything
• Need to couple GRAPH with DYNAMICS
• This is the essential message of the
course
Course Road Map
Theme is dynamics+ graph theory
Tentative flow of presentations:
Linear algebra of non-negative matrices and basics of graph theory
Markov chains and Perron Frobenius Theory
Graph Laplacians
Synchronization, agreement and consensus
control theory and robotics
Networking
Physics
Complex networks, power laws
Kleinberg’s model, Barabassi’s preferential attachment
Newman’s survey papers
Small world networks, Watts Strogatz Model
Google’s PageRank
Reaction rate equations, metabolic networks, systems biology
Internet, degree distributions, internet topology
Random graph models
Dual decomposition theory
Beyond graphs
Simplicial complexes and algebraic topology
Coverage problems
Distributed optimal control
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Need volunteers for each section
We have about 12 weeks, 24 sessions, we could
read about 18-20 papers
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Beyond Graphs in Networked Systems
Main Idea: understanding global properties with local
information: algebraic topology
For certain problems, e.g. coverage, makes sense to go
beyond graphs and pair-wise interactions
Example: Given a set of sensor nodes in a given domain
(possibly bounded by a fence), is every point of the
domain under surveillance by at least one node?
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Coverage Problems
Problem: Given a set of sensor nodes in a
domain (possibly bounded by a fence), is every
point of the domain under surveillance by at
least one node?
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
From Graphs to Simplicial Complexes
Simplicial Complex: A finite
collection of simplices
Simplex: Given V, an
unordered non-repeating
subset
k-simplex: The number of
points is k+1
Faces: All (k-1)-simplices in
the k-simplex
Orientation
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
From Graphs to Simplicial Complexes
Simplicial complex: made up of simplices of
several dimensions
Properties
Whenever a simplex lies in the collection then so does each of
its faces
Whenever two simplices intersect, they do so in a common face.
Valid Examples
Graphs
Triangulations
Invalid examples
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Rips-Vietoris Simplicial Complex
0-simplices : Nodes
1-simplices : Edges
2-simplices: A triangle in the
connectivity graph ~ 2simplex (Fill in with a face)
K-simplices: a complete
subgraph on k+1 vertices
k-simplex in the Rips
complex ~ (k+1) points
within communication range
of each other
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Generalization of r-disk graphs
Rips and Čech Complexes:
Topological vs. Geometric information
A set of points
•
(Rips complex of radius ): k-simplex, if the pairwise
distance between k points are less than .
- Easy to compute in a dsitributed manner.
- However, Does not preserve the topological properties.
•
(Čech complex of radius ): k-simplex, if k coverage
disks of radius overlap
- Hard to compute.
- Preserves the topological properties.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Coverage Problems
Intersection of sensing ~ simplicial
complexes
Communication graphs ~ simplicial
complexes
Holes ~ homology of simplicial complexes
A sensor network has coverage hole if there
is a “robust” hole in the simplicial complexes
induced by the communication graphs [Ghrist
et al.]
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Relevance of Homology
dim H0(X) ~ no of connected components of X
dim H1(X) ~ types of loops in X that surround
‘punctures’
dim Hk(X) ~ no of k+1-dimensional ‘voids’ in X
Available software
Plex (Stanford)
CHomP (Georgia Tech)
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Combinatorial k-Laplacians
Since X is finite we can represent the boundary maps in matrix form
incidence matrix
Moreover, we can get the adjoint
[Eckmann 1945] The Combinatorial k-Laplacian
is given by
Note:
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
k-Laplacian at the Simplex Level
Adjacency of a simplex to other
simplices
Upper adjacency if they share a
higher simplex (e.g. 2 nodes
connected by an edge)
Lower adjacency if they share a
common lower simplex (e.g. two
edges share a node)
‘Local’ formula with
orientations
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
k-Laplacian at the Simplex Level
Adjacency of a simplex to other simplices
Upper adjacency if they share a higher simplex (e.g. 2 nodes connected by
an edge)
Lower adjacency if they share a common lower simplex (e.g. two edges
share a node)
‘Local’ formula with orientations
Hodge theory, 1930’s: Kernel of the Laplacian ~ homologies
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Laplacian Flows
Laplacian flows : a semi-stable dynamical system
(Recall heat equation for k = 0)
[Muhammad-Egerstedt MTNS’06]
System is asymptotically
stable if and only if
rank(Hk(X)) = 0.
A method to detect
‘no holes’ locally
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Laplacian Flows (contd.)
System converges to
the unique harmonic cycle
if rank(Hk(X)) = 1.
A method to detect
‘proximity to hole’ locally
when single hole
When rank(Hk(X)) > 1 :
System converges to the span of harmonic homology cycles
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Example, eigenvectors of L1
Network
2nd homology class
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
1st homology class
‘Fiedler-like’- eigenvector
Consensus in Switching Graphs
Mobility, switching graphs and consensus : switched linear
system
Joint connectedness (Jadbabaie’ 2003)
Theorem : Consensus if and only if there is a sequence of
bounded, non-overlapping time intervals, such that over any
interval, the network of agents is jointly connected.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Coverage in Switching Simplicial
Complexes
Can we repeat similar analysis for switching simplicial complexes?
YES!
Jointly `hole-free’ simplicial complexes
Joint hole-free implies trivial homology
in union complex
Theorem (Muhammad, Jadbabaie ’06): Switched linear system is
globally asymptotically stable if and only if there exists an infinite
sequence of bounded intervals, across each of which the simplicial
complexes encountered are jointly hole-free.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Switching Simplicial Complexes:
Random Switching
Switching times are from a Poisson point process
with rate
The complexes are drawn independently from a
common distribution.
[Salehi, Jadbabaie] The stochastic dynamical system
is globally asymptotically stable if and only if
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Ongoing work: detection of wandering
holes in coverage
Figure courtesy of Rob Ghrist
Given a set of sensors with a disk footprint, add:
an edge when 2 sensors overlap. A face when 3 sensors overlap
Construct the 1st Laplacian L1
Rips complex is “jointly persistently hole free over time”
intersection
of kernels of Laplacians is zero
no wandering hole in Rips
Complex
The dynamical system (which is distributed)
converges to zero
Instead of Spectral Graph theory look at spectral theory of simplicial complexes
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Results on Spatially invariant systems
and distributed control
Mostly over highly symmetric graphs w/ identical
dynamics
Infinite Horizon Quadratic Cost
No constraints on inputs and states
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Structure of optimal control for spatially
distributed systems: spatially invariant case
Model of each subsystem:
xk (t  1) A B xk (t)
Eq.(1) 
  C D 

y
(
t
)
u
(
t
)
 k 
 k
 
Does the optimal control policy have the same spatial structure as plant ?
In other words, is it spatially distributed ?
Finite Horizon Optimal Control problem:
N
min
J
(
x
(
0
),
u
)
N
Finite Horizon Quadratic Cost
u
s.t. Eq. (1)
k
k
umin
 uk (t)  umax
for 0  t  N
for 0  t  Nc
k
k
ymin
 yk (t)  ymax
for 0  t  Nc
k G
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Identical dynamics over
infinite lattices
Fourier Analysis on Lattice:
1D Lattice:
x0
x1
x2



x  (  , x1 , x0 , x1 ,  )
Signals in the spatial domain:
x̂    x1 ei ω  x0  x1 e i ω  
Fourier transform:
For simplicity, replace
Fourier transform:
x1
x2

x̂ 
z  ei ω
x
k G
k
z
k
G : spatial domain
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Translation Invariant Operators
Definition:.
Translation Operator: T (  , | xk , xk 1 ,  )  (  , | xk 1 , xk 2 ,  )
Q is translation invariant operator if
T Q QT
Consider translation invariant operators of this form
Example:
1D Lattice:
Global cost function


xk 2
xk 1
xk
Q( T)   Qk T
xk1
k
xk 2


k G
agent k is coupled to its neighbors through cost function J
J    xk * Q-1xk 1  xk * Q0 xk  xk * Q1xk 1  
 x, Q( T )x
in which
Q( T )  Q-1 T -1  Q0  Q1 T 1
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Decay Property of Translation Invariant Operators
1
Q̂(z)   Qkz 
N(z)
d(z)
k G
Q( T)   Qk T k
k
k G
Fact 1: Analytic continuity implies decay in spatial domain.
Analytic continuity
Fact 2: The decay rate depends on the distance of the
closest pole to the unit circle; the further, Q k
Im(the
z) faster.
Im( z)
S1
S1
Re(z)
Re(z)
No pole on S1
2
No pole in an annulus
around S1
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
1 0
1
k
Coefficients decay
in spatial domain
Back to our problem
k 2
k 1
Model of each subsystem:
Notation:
k
k 1
k 2
x (t  1) A B xk (t)
Eq.(1)  k
  C D 

y
(
t
)
u
(
t
)
 k 
 k
 
x(t)  (  , x k (t) , x k 1 (t) ,  )
u (t)  (  , u k (t) , u k 1 (t) ,  )
uN  (  , u Nk , u Nk 1 ,  )
u Nk  ( u k (0) , u k (1),  , u k (N  1) ) *
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Spatial Locality of Centralized RHC
Finite Horizon Quadratic Cost:
N 1
J(x(0), u )  x(N), P( T )x(N)   x(t), Q( T )x(t)  u(t), R(T )u(t)
N
t 0
P(T) can be obtained from a parameterized family of DAREs:
A* P̂ (z) A  P̂ (z)  A* P̂ (z) B (R̂(z)  B* P̂ (z)B) 1 B* P̂ (z) A  Q̂(z)  0
for all zS1 .
P(T) is spatially decaying:
P( T) 
k
P
T
k
k G
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Pk  c e  β|k|
for some c , β  0
Spatial Locality of the Optimal Solution
Theorem: Given the initial condition x(0), the optimal solutions are :
(1) Affine maps of x(0), i.e., uiN 
(2) Spatially distributed, i.e.,
for some α , β  0.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
K ij
K
jG
2
ij
x j (0)  ci
 α e β| i  j |
Generalization to Arbitrary Graphs
Analytic continuity
Exponential decay in spatial domain
Q ki : coupling between agent k and i
Multiply by ζ
Q ki
where 1  ζ  b



Q  
Q ki



Suppose that Q
is bounded.
Note: SD stands for Spatially Decaying
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
~
Q ki  Q ki ζ dis(k,i)


~ 
~

Q 
Q ki



~
If Q is bounded ,
then we say that Q
is exp onentially spatially
decaying .
Extending analytic continuity
ze
Im
S
1
iω
| z | 1
z~  ζ e iω
1  r  | z~ |  1  r
Multiply by ζ
where 1  r  ζ  1  r
Im
S1
Re
Re
No pole on S1
If
|k |
|
Q
|
ζ
 k 2
Analytic continuity

k G
Q̂(z~) 
No pole in an annulus
~k is bounded
Q
z
 k
k G
on the annulus .
Q is spatially decaying.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
Systems with Arbitrary Couplings
over Arbitrary Graphs
Multiply by ψ(dis(k, i))
Q ki
~
Q ki  Q ki ψ(dis(k, i))
Three important class of problems with spatially-varying couplings:
ψ(s)
ψ(s)-1
1 if | s |  d
ψ(s)  
0 if | s| d
with d  0
s
s
s
(1) Systems with nearest
neighbor coupling:
ψ(s)-1
(2) Systems with
exponentially
decaying couplings:
|s|
ψ(s)  ζ
with ζ  1
A. Jadbabaie “ESE 680:Distributed Dynamical systems”
(3) Systems with
algebraically
decaying coupling:
ψ(s)  (1  α | s |)β
with α , β  0
Properties of SD operators
Definition:
Suppose Q is bounded and the coupling  characteri stic
~
function ψ :R   [ 1,  ) is given. If Q is bounded , then
we say that Q is SD.
Theorem: sums, products and inverses of SD operators are SD.
Therefore, if A and B, Q , and R are SD
(1) Solution P of the Lyapunov Equation is SD:
A* P A  P  Q  0
,
A* P  P A  Q  0
(2) Solution of the Algebraic Riccati Equation is SD:
A* P A  P  A* P B (R  B*PB) 1 B* P A  Q  0
(DARE)
A* P  P A  P B R 1 B * P  Q  0
(CARE)
(3) Solutions to finite horizon constrained quadratic
optimization problems are SD.
A. Jadbabaie “ESE 680:Distributed Dynamical systems”