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Sandip Roy School of EE&CS Washington State University

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

Sandip Roy

School of EE&CS

Washington State University

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 mustexploit 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

Outcomes

Infrastructures

Agent Networks

Algorithm-

design for sensor

networks and

vehicle teams (NSF)

Distributed

cortical sleep

regulation (NIH,

in collab. with

neuroscientists)

1) Uncertainty evaluation

in electric power

networks (NSF, PSERC,

LBNL)

2) Coordinated air traffic

flow management

(NASA)

Resource assignment

for virus-spreading

control, w/ app. To SARS

and brucellosis (collab. w/

Vet. Med. Researchers)

Human-

Engineered

Networks

Bio/Eco

Networks

Outline

- Brief Review of the “Science of Networks”
- Modeling Control/Design Problems in Dynamical Networks
- Methodology for Design

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:

1

A

B

2

Adjacency

Matrix

3

7

C

3

2

D

Directed

Laplacian or

Diffusion

Matrix

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

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 Laplacianeigenvalues

A

B

2

7

C

3

2

D

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!

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

2. Control/Design Architectures

1. The Basic Network Model

Agent models Topologies Interconnection

Type

Node vs. Static vs. Partial vs.

Edge Design Memoried Full Network Design

- 3. Performance Requirements

Spectral Assignment LQ Disturbance Rejection

CDC 2008

More Aspects of Modeling

4. Constraints/Variations

Sat.Delay Topological Security/

Variation Fairness

Complex Tasks

Formation Sync. Agreement Dist. Part.

CDC 2008

List of Modeling Works

- I have done modeling work in

four application areas:

- Sleep Regulation

(published in JTB)

- Sensor/Vehicle Networking (publishedinIJDSN, 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.

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…

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?

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!

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 Outcomes

Stopping Brucellosis Spread

Homeland Security

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

- 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
- Needs to find xi .
- Has internal state xi[k] that can be augmented/decreased, i.e. xi[k+1]= xi[k] +ui[k].
- Has the statistic y[k]=giTx[k] at each time k.
- Has bi

G describes

the 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:

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.

A General Design Problem?

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!

A Progression of Tools

-Network Identification,

-Partitioning/Discovery

--Node/Edge

Param. Design Tools,

--Partial Graph Design

Tools

--Memoried

--Tools Controller

for Limitations, Designs

Tasks, Measures

Pre-processing

Eval.

---Numerical Sim.

Tools,

---StochasticsEval.

Core Control/Design Tools

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 optimizationmachinery from control theory together with eigenvalue sensitivity and algebraic graph theory ideas to solve these problems.
- Our designs exploit the graph structure.

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

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.

Partial Graph-Edge Design

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

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

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 itheigenvalue of the graph Laplacian is between the itheigenvalue 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.

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!

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

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.

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.

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, ZhengWen, and Ian Hiskens.
- My wonderful group (co-advised by Ali Saberi) are Xu Wang, Tao Yang, MengranXue, and BabakMalek.

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