Shaping the Dynamics of Modern Infrastructure and Autonomous-Agent Networks using Limited

Resources

Sandip RoySchool of EE&CS

Washington State University

The World is so Interconnected…

How Do We Control/Design these Network Dynamics?

• Brainstorming:– Prevent birds/squirrels from impacting critical

points– Protect air traffic control electronics– Coordinate flow management– Join the squirrel defamation league

Actuation and measurement capabilities are: localized, varied, highly constrained, subject to resource limits, and expensive.

Our Philosophy

• Controls and improvements of modern dynamical networks must exploit the network’s topological structure in a coordinated way, so as to permit fast completion of complex tasks in the face of severe constraints and topological variations.

My Research• We develop abstract yet practical models for

control and design problems in dynamical networks, for both infrastructural and autonomous-agent network applications.

• We pursue a comprehensive methodology for dynamical network control and design.– Both analytical and numerical tools are developed– Although networks are vastly different, we get

graph-theoretic insights

OutcomesInfrastructures Agent Networks

Human-EngineeredNetworks

Bio/EcoNetworks

1) Uncertainty evaluationin electric power networks (NSF, PSERC,LBNL)2) Coordinated air trafficflow management(NASA)

1) Resource assignmentfor virus-spreading control, w/ app. To SARSand brucellosis (collab. w/Vet. Med. Researchers)

1) Algorithm-design for sensornetworks andvehicle teams

(NSF)

2) Distributed cortical sleepregulation (NIH, in collab. with neuroscientists)

Outline

1. Brief Review of the “Science of Networks”

2. Modeling Control/Design Problems in Dynamical Networks

3. Methodology for Design

Background on the “Science of Networks”

The ‘Science of Networks’: Background

• Particular network dynamics (e.g., power-system swing dynamics, production systems, epidemic spreads) have been modeled for a long time.

• Recently, scientists have sought a common theory for networks, by– Identifying common structural characteristics of networks.– Tying structural properties to dynamical response

characteristics (for certain common models).

Common Network Structures: Background

• Many modern networks:– Are sparsely connected yet are “small worlds”– Have heavy-tailed degree distributions (and event

sizes?)– Have coherency structures

Common Network Structures: Background

• Several researchers have aimed to explain why networks commonly have these structures:– Doyle and Carlson have argued that network

characteristics are a consequence of deliberate design– Physicists and computer scientists have argued that

the characteristics result from weighted connection-forming rates.

• But what’s the use??

Tying structure to dynamics

• Network structure is often represented by certain matrices:

A

C

B

D2

27

3

1

0200

3070

0002

0310

A

2200

31070

0022

0314

L

AdjacencyMatrix

Directed Laplacian orDiffusionMatrix

3

Background on Network Dynamics

• Equations like represent many network dynamics.

• The spectra (eigenvalues and eigenvectors) of the matrices L and A specify the dynamics.

• Connecting the spectra to the underlying graph structure is of wide interest– These include algebraic graph theory-based results, and

coherency results based on singular perturbation ideas

][]1[ ,, kAxkxLxxLxx

Background on Network Dynamics• Chua has studied a much broader class of diff. linear and non-linear dynamics defined on a graph:

• Stability check can be converted to a low-order simultaneous stability check, based on Laplacian eigenvalues

j

jijii xlxAx

A

C

B

D

2

2

7

3

Background: Network Control?

• Decentralized control has been extensively studied (see e.g. texts of Siljak and Michel).– Unfortunately, viewpoint is to make agents

dominant to network interactions.– Our applications require use of the network.

• Wang and Davison have given non-conservative conditions for decentralized stabilization– Their existence result does not yield good designs.

Background: Network Control?

• Fax and Murray, and Pogromsky, have given control-theoretic interpretations of Chua’s result for diffusive networks. – These works are aligned with our interests: the graph

topology’s role is clarified.– However, the equivalent simultaneous-stabilization

problem obtained is difficult to solve.– Only a few networks fit this form!

• New dynamical network control/design techniques are badly needed!

2) Models for Control/Design Problems in Dynamical Networks

Modeling: Aspects• We are pursuing systematic

formulation/resolution of dynamical network control and design problems.

• Let us discuss the various aspects of our modeling efforts, and then give two examples.– Virus-spreading control– Sensor-network algorithmics

Aspects of Modeling

CDC 2008

1. The Basic Network Model

2. Control/Design Architectures

3. Performance Requirements

Agent models Topologies Interconnection Type

Node vs. Static vs. Partial vs.Edge Design Memoried Full Network Design

Spectral Assignment LQ Disturbance Rejection

More Aspects of Modeling

CDC 2008

Complex Tasks

Formation Sync. Agreement Dist. Part.

4. Constraints/Variations Sat. Delay Topological Security/ Variation Fairness

21

List of Modeling Works• I have done modeling work in four application areas:– Sleep Regulation (published in JTB)– Sensor/Vehicle Networking (published in IJDSN,

AIAA-GNC, IEEE-CDC/ACC )– Epidemic Control (published in RSPA, IET-SB, and

Wolfram)– Air Traffic Flow Management (published in IEEE-ITS,

AIAA-GNC, and RSPA)

We obtain common design problems from these applications.

22

Epidemics: Network Models• Epidemiologists view control as the task of reducing spread

rate (reducing Basic Reproductive Ratio, Ro).• Definition of Ro:

• If Ro>1, the disease can spread throughout the population.• For a homogeneous population and one/two-state virus,

dynamics are x[k+1]=Rx[k](1-x[k]/N). Assuming small x[k], Ro=R and x[k+1]=Ro x[k]. Otherwise, just linearize…

Epidemics: Network Models• We use a multi-group model to

track the infectiousness in each region [Diekman99, Riley03].

• Multi-group models are captured by a next generation matrix A, where Aji is number of infected people produced in district j by an infectious person in district i.

• R0 is the dominant eigenvalue of A.

• Higher-dimensional local states are also needed…

24

Epidemics: Modeling Controls• Controls fundamentally serve to remove infectives, or

prevent them from interacting with certain population subsets. Notice the constraints!

• Example control/design strategies: reducing local contact rate (ri), reducing local infectious period (ti), and reducing transmission rate between regions (ci).

• Controls are expensive!• Control strategies explicitly or implicitly feed back

measurements of local infected populations.

• Aside: is the problem really decentralized?

25

An Example Problem Formulation• Next generation matrix with control parameters

included:

• Problem: Design diagonal D or K to minimize the dominant eigenvalue of (D+KG) subject to

• Controllers-with-memory are also needed!

1

21

212

121

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0

0

)()()(

i

nn

n

n

iiiiii Ndiag

ff

ff

ff

cdiagfdiagNrtdiagTA

iallforDandDDtr

oriallforKandKKtr

ii

ii

10)(

10)(

Epidemics: Other Aspects of Modeling

• Constraints: state saturation is intrinsic; variations, fairness/security constraints, and delays are common.

• Performance Requirements: reduce/minimize Ro (stabilize), minimize total virus size, minimize quadratic cost

• Complex Tasks: tracking??

• Beyond Control/Design: network identification is critical, so is post-processing using detailed simulation software.

Outcomes: Stopping SARS

Data obtained from Riley, 2003.

Resources reduced to 79% of that for homogeneous strategy.

More OutcomesStopping Brucellosis Spread

• We are studying control of zoonoses like brucellosis among cattle herds and pastoralists in Africa.

Homeland Security

• We are using the methods to control spreads of undesirables for homeland-security purposes.

Example: Fast Distributed Algorithms for Sensor/Vehicle Networks

• As one example, let us study distributed solution of a system of linear equations Gx=b.

• This problem is of interest for sensor networks and physical systems (e.g. from economics).

• Network of processors. • Processor i

1. Needs to find xi .

2. Has internal state xi[k] that can be augmented/decreased, i.e. xi[k+1]= xi[k] +ui[k].

3. Has the statistic y[k]=giTx[k] at each time

k.4. Has bi

G describesthe topology

Distributed Algorithms: Design Task• We will develop a distributed iterative algorithm (or

controller), i.e. a rule for deciding each ui[k] from yi[k], so that xi[k] converges to xi quickly.

• We refer the reader to the classical work of Young for a resolution in the centralized case.

• Here, we use a algorithm with only one memory element at each processor: ][][][

][][]1[

kykkztku

kyrkzskz

iiiii

iiiii

Distributed Algorithms: More Modeling

• Constraints: for physical systems, saturation is ubiquitous; for computations, security/fairness constraints and topological variations may be quite common.

• Performance Measures: dominant eigenvalue, quadratic cost, etc.

• Controller Architectures: memoryless controllers and delay-controllers are also of interest.

Distributed Algorithms: Outcomes

A General Design Problem?

jijiiiii xAuBxAx

xGy Trr

])())(()()([ jijiiiii xtATtutBxtArealizex Limitations-----Performance---Tasks-------------

Spectrum Assignment, LQ, External Stab. Stabilization, Tracking, Dist. Part.

These Network Design/Control Problems Require New Methods!

• Because complex tasks must be completed quickly by highly decentralized and limited components, good controls/designs MUST exploit the network topology!

Tools for Dynamical Network Control/Design

A Progression of Tools

Core Control/Design Tools

Pre-processing

Eval.

-Network Identification,-Partitioning/Discovery

--Node/EdgeParam. Design Tools, --Partial Graph Design Tools

--Memoried --Tools Controller for Limitations, DesignsTasks, Measures

---Numerical Sim.Tools,---Stochastics Eval.

37

Designing Node/Edge/Controller Parameters

• We wish to select memoryless-controller gains or interconnection/vertex properties, to shape dynamics (e.g., spread control).

• These design problems can be interchangeably viewed as controller-design problems or as graph-selection problems.

• We mesh optimization machinery from control theory together with eigenvalue sensitivity and algebraic graph theory ideas to solve these problems.– Our designs exploit the graph structure.

38

Example: Node-Parameter Design

• Design a diagonal matrix D to minimize the dominant eigenvalue of D+G, subject to the constraints on D.

• We find the eigenstructure of the optimal solution.– The dominant right eigenvector at the optimum has a

special structure. For an irreducible non-negative G, the optimal solution has the sign pattern:

.

0,0,

,,,

*max,

*max,

*max,

*max,

jallforidenticalarev

andDLDLD

thatsuchkjiforvvv

j

lii

lji

39

Example (continued)Insight: the optimal resource allocation equalizes

propagation impacts (to the extent allowed by constraints).

• Algorithm: • Relax the individual constraint and find the optimal

solution. • Individually move the entries of D larger than L to L

and smaller than 0 to 0, and repeat. • We have proved this algorithm finds the optimal solution;

this requires algebraic graph theory.• Design performance can be tied to structural features!

Toward More Complicated Designs

• The direct approach presented above requires enhancement, when 1) only partial design is allowed, 2) some state information is not observed directly anywhere, or 3) more refined shaping of dynamics is needed.

• Here’s how we can enhance the tools:– Exploit time-scale and coherency structures– Consider memoried controllers at each component.

41

The problem can also be viewed as a decentralized controller design one:

Our goal is to design the designable (red) edges’ weights in the graph, to shape the dynamics

Partial Graph-Edge Design

||

1

},{},{},{

)(

d

df

E

m mmm

Eji

Tji

jiij

Eji

Tji

jiij

Eji

Tji

jiij

BKCACkBA

qqkqqkqqkL

xLx )(

Partial Graph Design (continued)• The fixed graph imposes a structure that further

constrains spectrum assignment.• The limits can be obtained using time-scale notions.

• Key Result: The ith eigenvalue of the graph Laplacian is between the ith eigenvalue of the fixed-edge graph’s Laplacian and the scaled zero graph’s Laplacian.

• Spectrum optimization/assignment is possible through graph-design methods, with simple insights obtained in the time-scale limits.

Partial Graph Design (continued)

More Complex Agent Models and Refined Shaping

• To achieve stabilization for more complex network models, and to improve performance, we badly need for agents to infer state information.

• The standard observer-followed-by-state-feedback architecture does not work, because 1) each component only has some observations and 2) feedback is needed before estimation is possible.

• Ideas: – Output-derivatives give some state information; can these be

approximated and used directly?– Can direct precompensation allow us to deal with non-minimum-

phase dynamics?We need controllers with memory!

45

Example: Double Integrator Network

• Consider the DIN: • The decentralized controller can

stabilize the network, where k3 is sufficiently large, and the roots of the system are close to the roots of

.• The extra derivative provides agents with local state

information, since when k3 is sufficiently large, we have

ykykkykku 33231

).()(),( tGxtytux

0212 kk

ykkGkIykkGkIx 321

3311

3 )()(

iiTi

Tii xkxkyhkyhkx

2121

46

Example: Double Integrator Network

• Advantages: – The approach outperforms dominant-channel approach in

terms of complexity and actuation.

– It is also more robust in terms of agent failure.• Lead-compensator implementation can approximate

the derivative control arbitrarily well, and the delay implementation is also very promising.

1

1

1

G

Limitations, Complex Tasks, and Measures

• Constraints on parameters are intrinsic to many of our designs, and our methods already account for these.

• We are adapting low- and low-and-high- gain designs to address actuator saturation and sandwiched saturation elements in decentralized systems.

• Security and fairness constraints can be imposed through an eigenvector-placement approach.

• We have a first result on stabilization under Markov topological variation, using moment-analysis methods.

• We are just starting to study other performance measures.• We have developed methods for consensus, tracking, and distributed

partitioning tasks • Stochastics play a role in many aspects.

Pre- and Post- Processing: Many Pictures

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0 .4 70 .4 7

0 .4 70 .0 2 3

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(a)

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(b )

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0 .4 70 .4 7

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0 .0 0 3

(c)(d )

And Where Next?• Our efforts so far are baby steps toward a

comprehensive theory for dynamical network control/design.

It takes a village…• Ali Saberi (WSU), Yan Wan (WSU/UCSB/UNT), and I are

equal contributors to this work.

• I am so grateful for collaborations with Bernard Lesieutre, George Verghese, Chris DeMarco, Banavar Sridhar, Terry McElwain, James Krueger, David Rector, Zheng Wen, and Ian Hiskens.

• My wonderful group (co-advised by Ali Saberi) are Xu Wang, Tao Yang, Mengran Xue, and Babak Malek.