Networked Dynamic Systems: Identification, Controllability, and Randomness

Contributions Network Controllability Estimation over Random Networks Conclusion

Networked Dynamic Systems:
Identiﬁcation, Controllability, and Randomness

Marzieh Nabi-Abdolyouseﬁ

Aeronautics & Astronautics
University of Washington

Networked Dynamic Systems, slide 1/58


Networked Dynamic Systems (NDS)
NDS
a collection of dynamic systems that can measure or exchange
information via a connection graph

multi-agent systems
communication networks
power networks
social networks swarm of nano-satellites underwater sensor networks

power networks social networks



research objectives

How to identify the underlying connection topology?
How to eﬃciently interact with a network, e.g., a swarm?
How to observe or control the behavior of a network?



Outline

1 Contributions
Network Identiﬁcation
Network Controllability
Random Networks

2 Network Controllability

3 Coordinated Decentralized Estimation over Random Networks

4 Conclusion



summary of contributions

Network Identiﬁcation

Network Controllability

System Properties of Random Networks




Network Identiﬁcation: Inferring the topology of the network from
a set of limited input-output data

1
Input signal
The underlying interconnection network is
Input signal

2
3
unknown
4

5 6
A set of input-output data while the
Sensor

agents are running a consensus type
7 9
8
10 protocol

11 13 Utilizing tools from system identiﬁcation,
12

graph theory, combinatorial theory, and
Sensor
Sensor linear algebra
Input signal



Network Identiﬁcation: node-knockout approach1

Knocking out individual nodes and any
Node 3 pair of nodes
Node 1
Knocking a node out involves the node
sending out a zero signal

Superimposition of these information via
Node 2
generating functions

Fault detection ramiﬁcation

1
IEEE Transactions on Automatic Control and CDC 2010



Network Identiﬁcation: sieve method2

Node 3 System identiﬁcation provides the number
Node 1 of edges and the number of adjacent
neighbors for a subset of agents

Integer partitioning
Node 2

Degree based graph construction

2
IET Control Theory & Applications, 2012


Network Identification: linear algebraic approach3

System identification provides a
similarity transportation of the
original system

Linear algebraic tools such as
householder reflection

Identify graphs isomorphic to the
original system
householder reflection

3
CDC 2012



Network Controllability: how to inﬂuence a network and derive the
states of a network to any desired value

Controllability of circulant networks4

Controllability of composite networks5

4
IEEE Transactions on Automatic Control
5
IEEE Transactions on Automatic Control (submission proc.) and CDC 2012



Network Controllability:
5
6 4

7 3

8 2

9 1

10 16

11 15

12 14
13

Circulant networks Cartesian product networks

For an arbitrary n, there are

2 n/2 undirected (unlabeled) circulants




System properties of random networks: Modeling large scale
networks or interaction with a network with random distributions

Flying robotic swarms to create Wi-Fi cloudsa

a
Courtesy of Swiss Federal Institute of Technology

Observability/controllability
Optimality properties6
Coordinated decentralized estimation7
6
IEEE Transactions on Automatic Control (submission proc.) and CDC 2011
7
IEEE Transactions on Automatic Control (submission proc.) and ACC 2011



System properties of random networks: application

Fifteen Florentine family graph to analyze the social control and
optimal marketing

Opinion dynamics and optimal marketing

Decentralized estimation of opinion dynamics8

8
submission proc.



System properties of random networks: application

Online position estimation of the Seaglider9

Seaglider: autonomous underwater vehicle localization experiment

Courtesy of Prof. Morgansen’s Nonlinear Dynamics and Control Lab at the University of Washington

9
submission proc.


Outline

1 Contributions

Circulant Networks and Applications
Controllability of Circulant Networks
Symmetry Structures


4 Conclusion



How to interact with networks?

Courtesy of Nature

The underlying network could be path networks, circulant
networks, large scale networks, Cartesian product networks,
random networks, and ...


modeling

x(t) = A(G)x(t) + Bu(t)
˙
y(t) = Cx(t)
where A(G) = −Lw (G) ∈ Rn×n is the weighted Laplacian.
 
1 −1 0 0 0
 −1 3 −1 −1 0 
 
A(G) = −  0
 −1 3 −1 −1 ,

2 4
 0 −1 −1 3 −1 
0 0 −1 −1 2
 
1 3 1 0
 0 1 
5
  1 0 0 0 0
B =  0 0 , C = .

 0 0 
 0 0 0 1 0
0 0


Controllability is important and non-trivial

56
n = 2m : controllable from
54
52
any single nodea
50
48
46
n prime: controllable from
44
42
any single node except the
40
38 middle one
36

our contribution for general
controllable nodes

34
32
30
28 n: the node j is
26
24 uncontrollable if the following
22
20
18
happens
16
14 k(2j − 1)π
12 sin =0
10
8
2n
6
4
2 for some k = 1, 2, . . . , n
0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56
number of nodes a
G. Parlangeliand & G. N., IEEE Trans., 2012


network topology and network controllability

What features of a network topology determine the network
controllability? maybe symmetry?

Networks with known controllability properties

In literature: paths, cycles, and grids
Our contributions: circulant networks and composite networks



circulant networks

4
3
The ith vertex is adjacent to a set of
5 vertices on its right and the symmetric
2 ones on its left
6
For an arbitrary n, there are 2 n/2
1
undirected (unlabeled) circulant
7
Known closed form eigenvectors
11

8
Appear in some engineered systems
10
9



circulant networks and applications

Coding theory, VLSI design
quantum communication
parallel and distributed computation

observer-based fault detection
consensus-based load balancing
distributed security protocols for
clock synchronization

Swiss-T1 cluster supercomputer with 64 processors



controllability of circulant networks: main result

Theorem
A circulant network of order n with maximum algebraic multiplicity
q is controllable from q nodes. Moreover,

(a) for n prime: the set of q nodes can be chosen arbitrarily, and

(b) for general n: the indices of the q nodes, in a clockwise or
counterclockwise indexing order, can be chosen as an
arithmetic progression of length q with the common diﬀerence
of 1.

The theorem provides necessary and suﬃcient condition for the
controllability of circulant networks


proof by contradiction
From the PBH test, the pair (A, B) is not controllable if and
only if there exists w = 0 and λ ∈ C such that

wT A(G) = λwT and wT B = 0.

For any choice of input matrix B, suppose ∃ w = 0 such that
w = q αj vj = Qα, where
j=1

Q = v1 |v2 | . . . |vq and α = α1 , α2 , . . . , αq

PBH implies that

wT B = αT (QT B) = 0, α=0

Therefore, det(QT B) = 0 ... utilizing Cauchy-Binet formula
to calculate QT B


summary of the proof



Cauchy-Binet formula
Let us recall F and H to be, respectively, m × n and n × m
matrices, with m ≤ n. Let [n] = {1, 2, . . . , n} and

Γn = {m-element subsets of [n]} = {S ⊆ [n] : |S| = m}.
m

The Cauchy-Binet formula then states that

det(F H) = det(F[m],S ) det(HS,[m] ),
S∈Γn
m

S ∈ Γn
m
F[m],S is the m × m submatrix of F with column indices in S
HS,[m] is the m × m submatrix of H with row indices in S



Example: take m = 2 and n = 3 and
1 1 2
F =
3 1 −1
 
1 1
H =  3 1 
0 2

Then, S ∈ Γ3 = {{1, 2}, {1, 3}, {2, 3}} and
2

1 1 1 1 1 2 3 1 1 2 1 1
det(F H) = . + . + .
3 1 3 1 1 1− 0 2 3 −1 0 2



blending Cauchy-Binet formula and PBH test

Cauchy-Binet formula provides a formulation to calculate
det(QT B)

Therefore, more knowledge about the eigenvector structure of
circulant networks is necessary

The following slide recall the closed form eigenvectors of
circulant networks



matrices associated to circulant networks

Matrices associated to the circulant networks have circulant
structure
 
c0 c1 c2 c3 . . . cn−1

 cn−1 c0 . 
. 
 c1 c2 . 
 .. 
 cn−2 cn−1 c0 c1 .
;

 .

.. .. ..
 . . . .

 . c2 
 c1 
c1 ... cn−1 c0

in particular, a circulant matrix is Toeplitz



closed form eigen structure of circulant networks

Theorem
The eigenvalues and eigenvectors of a circulant network are,
respectively,
n−1
λm = ck e−2πimk/n ,
k=0
1 T
vm = √ 1, e−2πim/n , . . . , e−2πim(n−1)/n ,
n
m = 0, . . . , n − 1.



matrix of eigenvectors

The matrix of eigenvectors ...
 0·(n−1)

wn0·0 wn0·1 ... wn
 1·0 1·1 1·(n−1) 
1  wn wn ... wn 
U=√  . . .. . ,
n .
. .
. . .
.


(n−1)·0 (n−1)·1 (n−1)·(n−1)
wn wn . . . wn

where wn = e−2πi/n for m = 0, 1, . . . , n − 1

The matrix U has Vandermonde structure



Vandermonde matrices
A Vandermonde matrix
2 n−1
 
1 α1 α1 . . . α1
 1 2 n−1
α2 α2 . . . α2 
V = , αi ∈ C;
 
..
 ... ... ... . ... 
2 n−1
1 αn−1 αn−1 . . . αn−1

it is well known that det V = i=j (αi − αj );

so V is nonsingular if αi ’s are distinct

A submatrix of Vandermonde matrix is referred to as
generalized Vandermonde matrix



symmetry

Iranian classic architecture Iranian classic architecture

Petronas twin towers nature


input symmetry

Fix the input nodes

Permute the nodes such that the
neighboring set of each node
doesn’t alter

If there is such a permutation, the
system is called input symmetric

input nodes {1, 5}



breaking symmetry and controllability

input symmetry =⇒ uncontrollability10

input symmetric =⇒ uncontrollable from {1, 5} controllability nodes {1, 8} =⇒ input asymmetry

10
A. Rahmani et al. SIAM, 2009


breaking symmetry and controllability
uncontrollability =⇒ input symmetry ?

a counter example

Theorem
For a cycle network of prime order

uncontrollability ⇐⇒ input symmetry.

cycle of order 7, uncontrollable from node 1



symmetry and eigenvalue multiplicity

The relation between the algebraic multiplicity of networks and
network symmetry is essential

Conjecture
The circulant network is uncontrollable if and only if it is input
symmetric.



Outline

1 Contributions


Motivation
Problem Formulation
Random Coordinated Estimation
Online Position Estimation of Seaglider

4 Conclusion



explore system/ graph theoretic aspect of deterministic/
stochastic systems that operate over a random network ...
why?
Modeling Introduced by design
real systems are subjective to limited battery sources:
link failure large sensor networks:
unreliable communication sensing structural
limited bandwidth integrity
delays habitat monitoring
data loss ﬁrebugs
reactive sample rate: soil
moisture monitoring



example: road monitoring



how easy is it to control or observe a random networked
system via a small subset of nodes or edges chosen ran-
domly or deterministically?

t



examine coordinated decentralized estimator design over
networks in diﬀerent scenarios involving randomness

Both schemes have some notion of randomness and local
computation in common ...



model the diﬀusion-like protocol
x(t + 1) = At x(t) + Bt u(t)
y(t) = Ct x(t),

wt : the sequence of mutually independent random events

G is a realization of the random graph

At = A(G(wt )) is related to the diﬀusion-like protocol, e.g.,

A(G(wt )) = e−L(G(wt ))

Bt = B(G(wt )): input matrix

Ct = C(G(wt )): output matrix



stochastic observability
A stochastic system is said to be:

Weakly state observable if for all x0 ∈ Rn and x ∈ Rn , and all
∈ R+ , there exists a random time T a.s. ﬁnite such that

P{||ˆ(T ; xo ) − x|| ≤ } > 0
x

where x(T ; xo , u) denotes the estimation of x at time T .
ˆ

State observable if this probability can be made equal to one.

Strongly state observable if the hitting time
TH = inf (t > 0; ||ˆ(t; xo ) − x|| ≤ ) has ﬁnite expectation
x
(E{TH } < +∞).



observability Grammian over random networks
Let
Rt = Ct Ct
and consider the event

Ωt = R1 + A1 R1 A1 + . . . + (At−1 . . . A1 )Rt (A2 . . . At−1 )
the observed diﬀusion is weakly observable
(Bougerol 1993) if for some t ≥ 1,

P{det(Ωt ) = 0} = 0

or if and only if for some t ≥ 1,

P{rank (Ct ; Ct−1 At , Ct−2 At At−1 , . . . , At . . . C1 A2 ) = n} = 0.



decentralized estimation
Theorem
The estimation error x(t) − x(t) is almost surely asymptotically
ˆ
stable.
or equivalently, there is a real number γ > 0, such that almost
surely
1
lim log ||(At − Kt Ct ), . . . , (A1 − K1 C1 )|| ≤ −γ
t→∞ t
for any solution of the random Riccati equation
Proof.
random Riccati map is contractive
utilize a stochastic Lyapunov approach



online position estimation of Seaglider

Seaglider: autonomous underwater vehicle localization experiment

the experiment in Port Susan Beacon-seaglider communication


packet-drops

1000
Response
900 No Response
800

700

600
Frequency

500

400

300

200

100

0
1 2 3
Node

Approximately 50% of communications between the nodes and the
sea-glider failed



the Seaglider dynamics

x(t) = f (x(t), u(t)) + w(t)
˙
y(t) = C(G(wt ))x(t) + v(t),

where
x = (xN , yE , ψ, Va , Vx , Vy ) w(t) ≈ N (0, Q) v(t) ≈ N (0, R)

wt : the sequence of mutually independent random events
G(wt ) is a realization of the random graph



the Seaglider dynamics

 
Va cos ψ + Vx

 Va sin ψ + Vy 

 u 
f (x, u) =  

 0 

 0 
0

Va is the ﬂow-relative speed of the glider,
Vx and Vy are the North and East components of the current
velocity vector, and
ψ is the heading angle measured from North



oﬄine position estimation of the Seaglider
Extended KF Unscented KF
740 740

720 720

700 700

680 680
xN

xN
660 660

640 640

620 620

600 600

Off-line estimation Off-line estimation
580 580
-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30
yE yE

L. Techy et al., UWAA Tech. Report 2010, ACC 2011


the Seaglider estimation scheme



the Seaglider estimation scheme
each sensor at its time slot measures the seaglider ’s position
with probability pm
each sensor at its time slot sends out its estimation to the
coordinator with probability ps

t1 t1 + 4



Seaglider distributed (online) estimation
Extended KF Unscented KF
760 760

740 740

720 720

700 700

680 680
xN

xN
660 660

640 640

620 On-line estimation 620 On-line estimation
Off-line estimation Off-line estimation

600 600

580 580
-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30
yE yE



Outline

1 Contributions



4 Conclusion



Contributions
1 M. Nabi-Abdolyousefi, A. Chapman, and Mehran Mesbahi, Controllability and observability of Cartesian
product networks, IEEE Transaction on Automatic Control, submission proc..
2 M. Nabi-Abdolyousefi and M. Mesbahi, Network identification via node knock-out, IEEE Transactions on
Automatic Control, 2012.
3 M. Nabi-Abdolyousefi and M. Mesbahi, On the Controllability Properties of Circulant Networks, IEEE
Transactions on Automatic Control, accepted.
4 M. Nabi-Abdolyousefi and M. Mesbahi. A sieve method for consensus-type network tomography, IET
Control Theory & Applications, 2012.
5 A. Chapman, M. Nabi-Abdolyousefi and M. Mesbahi, Identification and infiltration of consensus-type
networks, 1st IFAC Workshop on Estimation and Control of Networked Systems, pp. 84–89, 2009.
6 M. Nabi-Abdolyousefi and M. Mesbahi. Network identification via node knock-out, 49th IEEE Conference
on Decision, Atlanta, GA, December 2010.
7 M. Nabi-Abdolyousefi and M. Mesbahi, System Theory over Random Networks: Controllability and
Optimality Properties, 50th IEEE Conference on Decision, Orlando, Fl, 2010.
8 M. Nabi Abdolyousefi, M. Mesbahi, Decentralized estimators over random networks, American Control
Conference, San Francisco, CA, 2011.
9 M. Nabi, M. Mesbahi, N. Fathpour, F. Y. Hadaegh. Local estimators for multiple spacecraft formation
flying. AIAA Guidance and Control, Fl, 2008.
10 M. Nabi-Abdolyousefi, M. Fazel, and M. Mesbahi. Graph Identification via Transfer Matrices, Similarity
Transformations, and Matrix Approximations, 51th IEEE Conference on Decision and Control, Maui, USA,
2012.
11 M. Nabi-Abdolyousefi, A. Chapman, and M. Mesbahi, Controllability and Observability of Cartesian
Product Networks, 51th IEEE Conference on Decision and Control, Maui, USA, 2012.



List of ongoing articles

12 M. Nabi-Abdolyousefi, M. Mesbahi, Optimality properties of random networks, IEEE Transactions on
Automatic Control.
13 M. Nabi-Abdolyousefi, M. Mesbahi, Coordinated decentralized estimation over random networks, IEEE
Transactions on Automatic Control.
14 M. Nabi-Abdolyousefi, M. Mesbahi, Network controllability: A Survey
15 M. Nabi-Abdolyousefi, M. Mesbahi, Opinion dynamics and optimal marketing.
16 M. Nabi-Abdolyousefi, M. Mesbahi, Random decentralized estimation on opinion dynamics.
17 M. Nabi-Abdolyousefi, L. Techy, M. Mesbahi, and K. Morgansen, Online position estimation of Seaglider,
ICRA 2013.



Thank you

Mehran Mesbahi
Santosh Devasia
Maryam Fazel Sarjoui
Eric Klavins
Kristi Morgansen



And special thanks to Atiye Alaedini and DSSL group


Networked Dynamic Systems: Identification, Controllability, and Randomness

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Viewers also liked

Viewers also liked (20)

Similar to Networked Dynamic Systems: Identification, Controllability, and Randomness

Similar to Networked Dynamic Systems: Identification, Controllability, and Randomness (20)

Recently uploaded

Recently uploaded (20)

Networked Dynamic Systems: Identification, Controllability, and Randomness