FinalDefensePresentation

Multi-Agent Estimation and Control
of Cyber-Physical Systems
S M Shafiul Alam
Ph.D. Candidate, Electrical Engineering
Kansas State University
alam@ksu.edu
Acknowledgement: National Science Foundation (NSF) grant CNS #1136040.
September 17th, 2015
Kansas State University - College of Engineering Ph.D. Final Examination 1

Presentation Outline
Deﬁnition of Cyber-Physical System
Major Challenges in Cyber-Physical System
Research Questions
Related Work
Contributions of This Dissertation
Addressing Research Challenges
Key Findings
Future Research Directions

Cyber-Physical Systems (CPS)
NSF’15 [5]
Integration of sensing,
computing and
communication
Interaction with physical
world
Operation in real time
Physical
Process ‘’j’’
Physical
Process ‘’k’’
Computation
Computation
Information
Processing
Communication
Network
Control
Probable Physical Coupling
(a)

CPS Examples
http://trilliantinc.com/smart-grid
http://electronic-health.org/2009/keynote.shtml
www.kuka.com
http://dx.doi.org/10.1016/j.ymssp.2010.11.009
Health Care
Energy
Transportation
Manufacturing
CPS

Major CPS Challenges
Information
Theoretic
Capacity
Wireless
Network of
Sensors and
Actuators
Hybrid and
Stochastic
System
Modeling
Cyber
Security
Networked
and Closed-
loop
Control
Wireless
Network of
Sensors and
Actuators
Scalability
and
Complexity
N t
Distributed
& Concurrent
Signal
Processing
Sensing Big Data
Computation Time and
Concurrency
Robustness to
Communication Network
Impairments

Research Questions Addressed in this Dissertation

Centralized Information Aggregation and
Processing
Question 1
How to minimally aggregate sensor measurements?
Question 2
How does the aggregated information estimate states?

Fully Distributed Information Processing
over Communication Networks
Question 3
Can the sensors estimate a partial set of system states?
How should the sensors interact?
What information to be exchanged?
What would be the performance?
Question 4
Inter-sensor information exchange → Impact on stability?
Eﬀect of communication network impairments?
Question 5
Autonomous sensor behavior → Impact on stability?
Question 6
Distributed control scheme with similar design scenarios?

Related Work

Questions 1 and 2
Long’95[24], Bebic’09[25]:
Correlated Measurements, e.g.,
Distributed Generation and Load
DeVore’98[28], Ciancio’06[26],
Mallat’09[27]: Wavelet Compression
of Correlated Data
Li’10[29],
Ranganathan’11[30],
Islam’12[31]: Application
of Compressed Sensing
in Smart Meter Reading
Candes’05[88], Duarte’11[87]: 1D
Compressed Sensing
Rivenson’09[89]: Compressed Imaging
Duarte’11[87]:
Linear Parameter
Estimation from
Compressed
Measurements
Allgower’90[36],
Martinez’91[37],
Kuegler’12[38] : Newton-
Raphson Method
Centralized Information Aggregation
Centralized Estimation
(Over Determined)
Centralized Estimation
(Under Determined)
??

Questions 3 − 6
Over-Determined
System
Loosely Coupled
Subsystems
Korres’11[41], Gomez’11[42],
Gomez’11[43]: Geographical and
Non-Overlapped Decomposition,
Multi-Area State Estimation,
Factorized WLS
Xie’12[44]: Consensus and
Measurement Update
Zhu’12[46], Kekatos’13[45]:
Partially Overlapped
Decomposition, ADMM
Distributed Static
Estimation
Olfati’07[51], Olfati’09[54],
Yang’11[53], Dong’12[57]: Kalman
Consensus Filter, Stability Analysis,
Quantization and Packet Loss Effects
Khan’08[60], Mohammadi’12[61],
Dorfler’13[62]: Sparse/Banded
Transition Matrix, Kalman
Information Filter, Particle Filter,
Distributed Observer
Metropolis’53[55], Cattivelli’10[52], Zhao’12[56]:
Diffusion Kalman Filter, Laplacian, Metropolis and
Uniform Weighting, Relative Variance and Adaptive
Combination under Noisy Communication
Distributed Dynamic
Estimation
Alessio’07[66], Bemporad’10[65],
Alessio’11[63], Lemos’12[64]:
Sparse/Banded Transition Matrix,
Model Predictive Control,
Lyapunov Stability Analysis
Alessio’11[63], Maestre’11[67], Lemos’12[64]:
Neighborhood Control Effects, Packet Loss Effects
Yang’12[68]: Consensus in LQG Control, Spectral Radii
based Stability Analysis
Distributed Control
Lo’13[50], Mostafa’13[69],
Nguyen’13[71], Disfani’14[70]:
Graph Theory, Genetic Algorithm,
Game Theory, Lagrangian
Relaxation

Contributions of This Dissertation

Holonic Multi-Agent System (HMAS)
Layer n
Layer (n-1)
Layer (n+1)
Layer (n+2)
Measurement/Observation
ControlInformation
Measurement/Observation
ControlInformation
Hierarchical
Architecture
Holonic
Multi-Agent
Architecture
Figure: HMAS
S
S
S
i i+1i-1
Yi-1,i Yi,i+1
S
S
S
S
i
Yi
Vp
V VV
V
Y1,i Y2,i Y3,i Y4,i
V2,i V3,i
V4,iV1,i 2,2 i 3,i 4,4 i1,i
Sub
Station
Real and Reactive
Power Injected
Real and Reactive
Power Not Injected
Tree
Radial
Home
Distribution Transformer
Figure: HMAS Depiction of
Power Distribution Network

Addressing Question 1
How to minimally aggregate sensor measurements?
Sampling at
Nyquist Rate
Compression Decompression
Compressed
Random
Sampling
Sparsest
Solution
Signal with
Known
Bandwidth
Signal with
Unknown
Bandwidth
Recovered
Signal
Recovered
Signal
Conventional Compression Technique
Compressed Sensing Theory
Compressed Sensing based Correlated Information Aggregation

Compressed Sensing
Definition
The compressed measurement of w ∈ RN ,
h = Φw = N
i=1 φi
wi; h ∈ RM , M N
Reconstruction
Given, w is approximately sparse in basis Ψ,
a∗ = arg minz z 1 subject to h = ΦΨz
w∗ = Ψa∗

Spatial and Temporal Compressed Sensing
Fusion
Center
3 2 1
Spatial Compressed Sensing
i
Temporal Compressed Sensing

Key Findings (Alam’13[79])
0 10 20 30 40 50 60 70 80 90 100
0
20
40
60
80
100
120
CMR (%)
INAE(%)
Spatial Compressed Sensing
Temporal Compressed Sensing
Spatio−Temporal Compressed Sensing
Figure: Variation of INAE for Spatial,
Temporal and Spatio-temporal
Compressed Sensing
6 8 10 12 14 16 18
25
30
35
40
45
50
55
time (hours)
RealPower,Pg
(kW)
Reconstruction from 60% Compressed Temporal Measurement
Original
Reconstructed from 60% compressed measurement
Figure: Reconstruction Performance for
CMR = M/N = 60%
90% accurate recovery from 2 : 1 compression.

How does the aggregated information estimate states?
State-Compressed Measurement Relation
h = Φy = ΦF(x); F: RN → RN
⇒ ΦF: RN → RM ; M N.

Raw
Data State
Estimate
Newton
Raphson
Newton
Raphson
Sparsest
Solution
Compressed
Sensing
Sparsest
Solution
N
(a)
(b)
(c)
Figure: (a) Conventional, (b) Indirect, (c) Direct
State Estimation from Compressed Measurements

10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
CMR(%)
INAE(%)
Estimation of |V| from Compressed Measurements
34 Node: Indirect Method
34 Node: Direct Method
100 Node: Indirect Method
100 Node: Direct Method
Figure: Direct and Indirect Method
Performance
0 5 10 15 20 25 30 35
24.4
24.5
24.6
24.7
24.8
24.9
Node Index
kV
Voltage Magnitudes Estimated from 50% Compressed Measurements
Actual
Indirect Method
Direct Method
0 5 10 15 20 25 30 35
-0.025
-0.02
-0.015
-0.01
-0.005
0
Node Index
radian
Voltage Angles Estimated from 50% Compressed Measurements
Actual
Indirect Method
Direct Method
Figure: Voltage Phasor Estimation for
CMR = 50%
Almost accurate estimation with signiﬁcant reduction in
communication overhead and memory requirement.

Can the sensors estimate a partial set of system states?
How should the sensors interact?
What information to be exchanged?
What would be the performance?
1 Static
2 Dynamic

Agent based Model
Agent
Collects Data, Computes and Communicates
Distributed Scheme Agent based Scheme
Information about global states
and measurements
Information about only the
shared state elements

Example: Power Distribution Network
Zeq 1
1
0
Grid
Zeq 2
SG1
SL1
I1
I 0,1
I 1,2
2
SG2
SL2
I2
I2,3
ZeqN
N
SGN
SLN
IN
I N-1,N
Figure: N-node Radial
Network.
i i+1i-1i-2
i-2x i-1x ix i+1x
Figure: State Venn Diagram for Radial Network.
Under Determined Observation Space
yi = [Pi, Qi] = hi(xi); xi = [θi−1, |Vi−1|, θi, |Vi|, θi+1, |Vi+1|]

Agent based Static Estimation
Local Consensus and Measurement Update
xi
(k+1)
= xi
(k)
+ αi(k) Hi(k)†
yi − Gi(k)xi
(k)
Local Innovation
−β(k) Pleadxi−1
(k)
+ Plocxi
(k)
+ Ptrailxi+1
(k)
Localized Consensus
Condition of Convergence
αi(k) = 1
aλmax(Gi(k))(1+k)τ1
; a > 2, τ1 > 0
β(k) = 1
bλmax(D)(1+k)τ2
; b > 2, τ2 > 0
D = (B ⊗ Plead) + (F ⊗ Ptrail) + (IN ⊗ Ploc); A = B + F.

0 20 40 60 80 100
0
50
100
150
200
250
300
Iteration
MeanINAE(%)
b = 3,t
1
= 2,t
2
= 1
a = 1
a = 4
a = 16
a = 64
Figure: Effect of a
0 20 40 60 80 100
32
34
36
38
40
42
Iteration
MeanINAE(%)
a = 4,t
1
= 2,t
2
= 1
b = 1
b = 2
b = 3
Figure: Effect of b
The convergence condition is verified.

Agent based Model for Dynamical Process
The State Transition Matrix
xt+1 = Ftxt; Ft ∈ Rn×n
State-Space
Decomposition
Shrink Full-Scale
Observation Space
Limited
Observation
Space
State-Space
Decomposition
Alessio’07[66], Khan’08[60]
Agent based Model
Example: Ft includes an inverse of Jacobian from nonlinear
model (Deshmukh’14[122])

Agent based Model for Dynamical Process
Dynamics of Discrete-Time Linear Time-Invariant
System
xt = Fxt−1 + wt−1
xt ∈ Rn, x−1 ∼ N(µ, Σ), F ∈ Rn×n, wt ∼ N(0, Q)
Limited Observation of kth Agent
yt,k = Hkxt,k + vt,k; k = 1, 2, ..., N
Hk ∈ Rmk×nk , mk ≤ nk and vt,k ∼ N(0, Rk)

An Example 3-Agent System
Agent 1
This Example
xt,1 =


0 1 0 0 0
0 0 1 0 0
0 0 0 0 1

 xt = T1xt;
xs
t,1 =
0 1 0
0 0 1
xt,1 = S1xt,1;
Agent 2 → Agent 1 ← Agent 3
0 0
0 1
ˆdt,2
êt,2
=
0
êt,2
= P2,1ˆxs
t,2;
1 0 0
0 0 1


ˆct,3
ˆdt,3
êt,3

 =
ˆct,3
êt,3
P3,1ˆxs
t,3

The Binary Matrices
T: State-Space Decomposition
S, U, O: Extract/Restore Shared/Unshared Elements
P, L: Elements for Inter-Agent Communication
Set of Neighbors
Sk = {i : Pi,kSixt,i projects onto Li,kSkxt,k, ∀t}

Properties
xt;k xt;k
Sk
Uk
Os
k
Ou
k
xs
t;k
xu
t;k
Figure: Extracting and Restoring of Shared and Unshared Elements
Lemma 6.1
TkTk = Ink

Agent based Dynamic State Estimation
Computation Phase
Kalman Information Filter
Communication Phase
AKCF: Consensus Approach (Olfati’07[51])
ADKF: Diﬀusion Approach (Cattivelli’10[52])

Key Findings (Alam’14[83]
0 50 100 150 200 250
10
0
10
1
10
2
10
3
Time Index, t
MSDt
Agent Based Kalman Filter, System 1
ADKF
AKCF (e = 1)
AKCF (e = 0.1)
AKCF (e = 0.01)
Figure: Perfect Communication
0 50 100 150 200 250
10
-1
10
0
10
1
10
2
10
3
Time Index, t
MSDt
AKCF Vs ADKF for System 1, Link Failure Rate = r
r = 0.1
r = 0.3667
r = 0.6333
r = 0.9
ADKF
AKCF (e = 0.1)
Figure: Lossy Communication
AKCF is better and more robust.

Inter-sensor information exchange → Impact on stability?
Eﬀect of communication network impairments?
Discrete-time Lyapunov Stability Analysis of Estimation Error
Dynamics

Stability Analysis
Stability Governing Homogeneous Equation
ηt,k = Ct,kηt−1,k + Os
kWf
t,kzt−1,k;
zt−1,k = i∈Sk
Pi,kSiFiηt−1,i − Li,kSkFkηt−1,k
Lyapunov Energy Function
Vk(t) = ηt,kM−1
t,k|tηt,k
For asymptotic stability, N
k=1 {Vk(t) − Vk(t − 1)} < 0

Stability Analysis
Lemma 7.1
Given, G 0, L 0 and ∈ R,
Real Line
Scenario 1:
Real Line
Scenario 2:
Real Line
Scenario 3:

Key Findings 1 (Alam’15[84])
Theorem 7.1
Suﬃcient conditions for global asymptotic stability,
Wf
t,k = (Os
k)†
Mt,k|t−1 F−1
k Os
k; ∀k, ∀t; and,
Scenario 1: | | < mink λmin(Gt,k)/λmax AF DtAF
Scenario 2: | | < maxk λmax(Gt,k)/λmax AF DtAF
Scenario 3: | | < maxk λmax(Gt,k)/λmin AF DtAF

Stability under Lossy Communication
Step 7 of AKCF Algorithm
ˆxs
t,k|t =
Sk
ˆbt,k + Wf
t,k i∈Sk
ζi,k(t) Pi,kSiˆxt,i|t−1 − Li,kSk ˆxt,k|t−1
Stochastic Error Dynamics
ηt,k = Ct,kηt−1,k + Os
kWf
t,kzt−1,k
Corollary 7.1
E G − 2L 0 if,
Scenario 1: | | < E [λmin(G)]/E [λmax(L)]
Scenario 2: | | < λmax (E [G])/E [λmax(L)]
Scenario 3: | | < λmax (E [G])/λmin (E [L])

Corollary 7.2
If E [ζi,k(t)] = 1 − ρ, ∀i, k, t, then suﬃcient conditions for global
asymptotic stability, Wf
t,k = (Os
k)†
Mt,k|t−1 F−1
k Os
k; ∀k, ∀t;
and,
Scenario 1: | | <
mink λmin(Gt,k)
E λmax AF
t DtAF
t
Scenario 2: | | <
maxk λmax(Gt,k)
E λmax AF
t DtAF
t
Scenario 3: | |(1 − ρ) <
maxk λmax(Gt,k)
λmin(AF DtAF
)

Simulation of a 2-Agent System
System follows Scenario 2 of Lemma 7.1
Steady State Bounds
ρ | | Upper bound
Perfect Network 0 0.3849
Lossy Network 0.2 0.4103
0.4 0.4410
0.6 0.4791
0.8 0.5279

Stability of the 2-Agent System
0 10 20 30
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Time Index, t
TMSDt
SYS in Perfect Network ( = 0)
= 0.20
= 0.50
0 20 40 60
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7 SYS in Lossy Network ( = 0.6)
Time Index, t
TMSDt
= 0.40
= 0.80
Figure: AKCF Stability in Perfect and Lossy Network
System becomes unstable when the bounds are violated.

Autonomous sensor behavior → Impact on stability?
1 Deterministic
2 Mutual
3 Independent

3 Cases of Agent Behavior
Case 1: m = N and r = N
Case 2: m + r = N
Case 3: 0 ≤ m + r ≤ 2N
Corollary 7.1
E G − 2L 0 if,
Scenario 1: | | < E [λmin(G)]/E [λmax(L)]
Scenario 2: | | < λmax (E [G])/E [λmax(L)]
Scenario 3: | | < λmax (E [G])/λmin (E [L])

Theorem 8.1
Suﬃcient condition for global asymptotic stability,
Wf
t,k = (Os
k)†
Mt,k|t−1 F−1
k Os
k; ∀k, ∀t; and,
Scenario 1: | | <
E[λmin(Gt)]
E[λmax(AF DtAF
)]
Scenario 2: | | <
maxk λmax(Jt,k)
E[λmax(AF DtAF
)]
Scenario 3 (Case 2): | | < N
N−m
maxk λmax(Jt,k)
λmin(AF BtAF
)
Scenario 3 (Case 3): | | < N
r
maxk λmax(Jt,k)
λmin(AF LtAF
)

Simulation of 10-Agent System
0 2 4 6 8 10
0
0.05
0.1
0.15
0.2
m
||UpperBound,*
Case 2, Scenario 1
Figure: Upper Bound (Case 2,
Scenario 1
1 3 5 7 9
0.4
0.5
0.6
0.7
0.8
r
||UpperBound,*
Case 3, m = 5, Secnario 2
Figure: Upper Bound for Case 3,
Scenario 2

Stability of the 10-Agent System
0 50 100
10
0
10
10
10
20
10
30
10
40
10
50
Time Index, t
TMSDt
Case 2, m = 3, Scenario 1
=0.03
=0.3
*
= 0.0562
0 20 40 60 80 100 120
10
0
10
10
10
20
10
30
10
40
10
50
Time Index, t
TMSDt
Case 3, m = 5, r = 9, Scenario 2
=0.1
=0.5
*
= 0.3309
System becomes unstable when the conditions are violated.

Distributed control scheme with similar design scenarios?
Investigate Closed Loop Stability

Model Predictive Control (MPC)
Cost Function
Given, J(Ut) = E
t+tf −1
τ=t (xτ Xxτ + uτ Uuτ ) + xt+tf
Xxt+tf
and Ut = {ut, · · · , ut+tf −1}
Optimization Problem
{u∗
t , · · · , u∗
t+tf −1} = arg min J(Ut)
subject to xt+1 = Fxt + Gut + wt
Solve through dynamic programming and pick u∗
t .

Agent based MPC
System Decomposition
Fk = TkFTk ; Gk = TkG; Xk = TkXTk
Algorithm 5 State Feedback Gain
1: τ = t+tf : −1 : t+1 ⊲ Deﬁne Time Horizon
2: Υτ,k = Xk ⊲ Initialization
3: Kc
τ−1,k = − U+G⊤
k Υτ,kGk
−1
G⊤
k Υτ,kFk; ⊲ Control Gain
4: Υτ−1,k = F⊤
k Υτ,k(Fk +GkKc
τ−1,k)+Xk ⊲ Update

Stability Analysis
Closed Loop Homogeneous Equation
xt+1,k = (Fk + GkKc
t,k)xt,k + GkWc
t,kξt,k;
ξt,k = i∈Sk
(Kc
t,ixt,i − Kc
t,kxt,k)
Lyapunov Function
Vk(t) = xt,kΥt,kxt,k

Theorem 9.1
Suﬃcient condition for global asymptotic stability,Wc
t,k =
νG†
kΥ−1
t+1,k Fk + (Kc
t,k) Gk
−1
(Kc
t,k) ; ∀k, ∀t; and the
suﬃcient conditions on the level of consensus to guarantee
stability are,
Scenario 1: |ν| < mink λmin(Bt,k)/λmax Kt CtKt
Scenario 2: |ν| < maxk λmax(Bt,k)/λmax Kt CtKt
Scenario 3: |ν| < maxk λmax(Bt,k)/λmin Kt CtKt

Impact of Lossy Communication Network
Step 9 of AMPC Algorithm
ut,k = Kc
t,k ˆxt,k|t + Wc
t,k i∈Sk
ζi,k(t) Kc
t,iˆxt,i|t − Kc
t,k ˆxt,k|t
Closed Loop Dynamics
xt+1,k = (Fk + GkKc
t,k)xt,k + GkWc
t,kξt,k
;
ξt,k
= i∈Sk
ζi,k(t)(Kc
t,ixt,i − Kc
t,kxt,k)

Corollary 9.1
If E [ζi,k(t)] = 1 − ρ, ∀i, k, t, then suﬃcient conditions for global
asymptotic stability,
Wc
t,k = νG†
kΥ−1
t+1,k Fk + (Kc
t,k) Gk
−1
(Kc
t,k) ; ∀k, ∀t; and,
Scenario 1: |ν| < mink λmin(Bt,k)/E λmax Kt CtKt
Scenario 2: |ν| < maxk λmax(Bt,k)/E λmax Kt CtKt
Scenario 3: |ν|(1 − ρ) < maxk λmax(Bt,k)/λmin Kt CtKt

Simulation of 10-Agent System
Estimation: Case 2 and Scenario 1
Control: Scenario 2 of Lemma 7.1
Steady State Bounds for AMPC
ρ |ν| Upper bound
Perfect Network 0 0.02
Lossy Network 0.2 0.0224
0.4 0.0263
0.6 0.0332
0.8 0.049

Closed Loop Stability of the 10-Agent System
0 10 20 30
10
2
10
4
10
6
10
8
10
10
10
12
10
14
Time Index, t
Jt
AMPC, Perfect Network, r = 0
n = 0.015
n = 0.05
Figure: AMPC in Perfect Network
0 10 20 30 40 50 60
10
2
10
4
10
6
10
8
10
10
Time Index, t
Jt
AMPC in Lossy Network, r = 0.4
n = 0.02
n = 0.06
Figure: AMPC in Lossy Network
Control becomes unstable when the conditions are violated.

Summary of Contributions and Future Works

Conclusions
1 Spatial, temporal and spatio-temporal compressed sensing
2 Centralized state estimation from compressed
measurements
3 Agent based modeling from partial observations
State-space decomposition
No sharing of measurement information
Localized consensus
4 Agent based static state estimation
Convergence condition for radial topology
5 Agent based dynamic state estimation
Consensus is better
Eﬀect of inter-agent communication impairments
3 cases of agent behavior
3 scenarios of estimation error stability
6 Agent based control
Model predictive control
3 scenarios of closed loop stability
Eﬀect of inter-agent communication impairments

Future Research Directions
Topology changes, noise and bad data
Application to nonlinear systems
Correlated link failure and packet delay
Random switching topologies
Faulty and malicious agent detection
Analysis for multi-layer holonic structure

Published Articles I
[79]: S M Shaﬁul Alam, Bala Natarajan, and Anil Pahwa,
“Impact of Correlated Distributed Generation on Information
Aggregation in Smart Grid”, In Proceedings of 5th Annual
Green Technologies Conference (IEEE GreenTech), April 3-5,
2013, Denver, Colorado, USA.
“Distribution Grid State Estimation from Compressed
Measurements”, IEEE Transactions on Smart Grid, vol. 5, no.
4, pp. 1631-1642, 2014.
[81]: S M Shaﬁul Alam, Bala Natarajan, Anil Pahwa, and
Sergio Curto, “Agent based State Estimation in Smart
Distribution Grid”, IEEE Latin America Transactions, vol. 13,
no. 2, pp. 496-502, 2015 .

Published Articles II
[82]: Anil Pahwa, Scott A. DeLoach, Bala Natarajan, Sanjoy
Das, Ahmad R. Malekpour, S M Shaﬁul Alam, and Denise M.
Case,“Goal-Based Holonic Multi-Agent System for Operation of
Power Distribution System”, IEEE Transactions on Smart Grid
(Special Issue on Cyber-Physical Systems and Security for
Smart Grid), vol. 6, no. 5, pp. 2510-2518, 2015.
“Distributed Agent Based Dynamic State Estimation over a
Lossy Network”, In Proceedings of 5th Int. Workshop on
Networks of Cooperating Objects for Smart Cities (UBICITEC),
April 14-17,2014, Berlin, Germany.
“Agent based Optimally Weighted Kalman Consensus Filter
over a Lossy Network”, In Proceedings of 2015 IEEE
Conference on Global Communications (IEEE GLOBECOM),
(Accepted), December 6-10,2015, San Diego, CA, USA.

Under Review
[85]: S M Shaﬁul Alam, Bala Natarajan, and Anil Pahwa, “On
the Stability of Agent based Kalman Filters with Measurement
and/or Consensus”, IEEE Transactions on Control of Network
Systems(Under review), 2015.
[86]: S M Shaﬁul Alam, and Bala Natarajan, “Stability of
Agent based Distributed Model Predictive Control over a Lossy
Network”, IEEE Transactions on Signal and Information
Processing over Networks (2nd round review), 2015.

Committee Members
My deepest gratitude for your time and advice.
Dr. Bala Natarajan
Dr. Anil Pahwa
Dr. Shelli Starrett
Dr. Nathan Albin
Dr. Andrew Ivanov

Thank You!
Questions?

Back-Up Slides

1D Compressed Sensing Example
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
10
20
30
40
50
60
70
Normalized Angular Frequency (w/2p)
Magnitude
Signal is Sparse in Fourier Domain, K = 4
0 20 40 60 80 100 120
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time (sec.)
Amplitude
Original
Recovered
0 20 40 60 80 100 120
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time (sec.)
Amplitude
Original Signal, N = 128
5 10 15 20 25
-4
-3
-2
-1
0
1
2
3
4
5
6
Compressed Measurement, M = 28
Sample Index
Amplitude
Recovery Through
the Sparsest Solution

2D Compressed Sensing Example
Original Image (64x64 pixels) 2D Compressed Sensing (50x50 pixels)
Haar Wavelet Representation (64x64 pixels) Reconstructed Image (64x64 pixels)
Sparsest Solution

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