Networked Control Systems - UTA · 2007-06-06 · Networked Control Systems: Emerging Links Between...

Networked Control Systems:Emerging Links Between Control & Complex Networks

SWAN 2006

C.T. AbdallahProfessor & Chair

Electrical & Computer Engineering DepartmentThe University of New Mexico

C.T. Abdallah, ECE Dpt-UNM Networked Control Systems:

Electrical & Computer Engineering at UNM

The University of New Mexico

ECE DEPARTMENT


The UNM Campus

• Architectural showcase: John Gaw Meem pueblo revival to Bart Prince contemporary

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Pulsed Power, Beams & Microwaves

Plasma & Fusion Science

High-Performance Algorithms & Applications

Robotics, Artificial Intelligence & Vision

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Networked Control Systems

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High-Tech Materials

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High-Performance Computing

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Research Laboratories at ECE


Results

• 144 degrees awarded 2004-05:- 53 bachelors in EE- 24 bachelors in CompE- 44 masters- 23 doctorates

• 14 patents

• 3 startup companies

• Began offering a CompE master’s degree in 2004

• Expanding delivery of courses online

44 MS

23 PhDs24 BS inCompE

53 BS in EE

Degrees conferred 2004-05

Networks

Networks are a powerful metaphor for understanding theorganization of systems from disciplines as diverse asbiology, computer science, physics, and social science.They may also hold the key to designing autonomouslyreconfigurable systems.

Networks provide the means to understand the interactionbetween a system’s bulk properties and its dynamics.

As communication networks become increasingly morepervasive, control engineers are expanding theirapplications domain and have begun to incorporate thecommunication infrastructure into their designs, byconsidering the impact of link capacity, latency, andpacket loss on control systems.


The Blessing (Curse) of Connectivity

How many of us would choose an old and more expensivecomputer over a modern and cheaper one, if the first canbe networked while the second can not?

It is the obviously the network!

What connects us makes us stronger (the whole is morethan the sum of it parts), but also more vulnerable(viruses, marketing, etc.)

BUT, suppose you understand how networks come to be, theirstructure, and some of their hidden properties: Can you usethat knowledge to your advantage?


Networks in Control

Researchers in control systems are attending to theeffects caused by the presence of communication networksin the feedback path.

Meanwhile, and under the heading of complex networks,physicists, computer scientists, and mathematicians havebeen studying the formation and properties of physicaland social networks.

The control and complex network communities haveapproached networked systems from two differentperspectives. Control engineers use a network model tofacilitate controller design, while complex networktheorists investigate networks to model their dynamicsand growth.


Controlling Across a Network

Figure: Control signals, measurements of the plant state, andexternal inputs travel from their source to their destination throughthe links of a communication network.


Networks vs NCS

Figure: Complex network and control share domains but followdifferent directions in analysis. In complex network theory modelsare developed to describe the physical world. In control design weuse models to change the behavior of systems.


Towards a Theory of Networks

Network theorists, meanwhile, have created tools tocharacterize the growth of distributed networks. Inparticular, the relations between navigability, congestion,clustering, and robustness to failure, and the topology ofrandom networks, are generally well understood.

A particularly evocative aspect of the work in the complexnetworks area focuses on understanding how for somesystems, such as social networks and the World WideWeb, not only do short paths exist between every pair ofnodes, but such paths can be found under certainconditions using only local information.


Symmetry

There is actually an elegant symmetry between the perspectiveof control engineers engaged in studying multi-agent control,and networked control systems describing network dynamics.While their starting points and objectives differ, both try todetermine what a given network structure may guaranteeabout a dynamic process running on the network. In controlsystems, researchers bound the topological effects on capacity,latencies, and lost packets, and try to prove stability,robustness to disturbances, and to guarantee faster response,by means of appropriate network topologies. Network theory,on the other hand, focuses on how randomness and topologicalregularities impact connectivity, navigability, modularity (alsocalled community structure), stability, efficiency and resilience.


Connections

This talks presents an overview of problems at the intersectionof control theory and complex networks research. Within abrief review of recent results in cooperative control, we presentthree arguments that support the hypothesis that issues ofinterest for the network theorist can also impact controlengineering design:

1 Increased network connectivity does not necessarily yieldrobustly connected networks with respect to node failures;

2 The structure of sensor networks and their algebraicgraph properties determine the performance of distributedestimation;

3 Properly interleaving communication and control canprotect against the effect of delayed information.


Network Theoretic Issues

Three fundamental issues affect the flow of information overthe network:

1 Connectedness: which expresses the existence of a pathbetween the information transmitter and the informationreceiver.

2 Navigability: which is quantified by the difficulty offinding such a connecting path. Typically, this difficultydepends on whether the path is predetermined, orwhether it is discovered in an ad-hoc fashion.

3 Efficiency: as represented by the cumulative latency ofeach utilized path. This latency, usually a function of thenumber of hops and the individual link latencies, must besufficient to guarantee desired end-to-end communicationlatencies.


Connectedness

Network theorists consider connectedness to be identicalto the mathematical notion of percolation.

This notion is illustrated as a wildfire, initiated at asource vertex, which spreads across an edge connected toa burning vertex with a fixed probability p.

By analyzing the number of vertices reached by theprocess, it is possible to determine whether there exists apath connecting a given pair of nodes.


Percolation Approaches

Percolation is typically analyzed by network theorists in twocontexts:

1 The constructive approach answers the question of howmany random edges must be successively added to acollection of disconnected vertices before the vastmajority of vertices, termed the giant component, areconnected by some path.

2 The destructive approach, edges (or nodes) aresuccessively removed until the giant component vanishesand most pairs of vertices are no longer connected by anypath. Surprisingly, the appearance and disappearance ofthe giant component can be quite sudden, and is often agenuine phase transition.


Power Law in Networks

One common feature of many real world networks is apower-law degree distribution, in which the probability ofa randomly chosen vertex have k neighbors scales asP(k) ∝ k−α, where α is the scaling exponent.

The ubiquity of the power-law degree distribution has lednetwork theorists to focus on graph models that exhibitthis feature, but whose topological structure is otherwiserandom. Obviously, a network with many redundantpaths between all pairs of vertices becomes more robustto node and edge failures of all kinds.


Shattering the Network

We are interested in the degree of robustness as characterizedby the fraction of vertices pc that must be removed before thegiant component vanishes. This disappearance is said toshatter the network. For a random graph with a power-lawdegree distribution where a uniformly random fraction ofvertices fail (are removed), the after-failure degree distributionP ′(k) is given by

P ′(k) =∞∑

k0=k

P(k0)

(k0

k

)(1− p)k pk0−k ,

where k0 is the degree of a vertex before failure, k is its degreeafter, and p is the probability of failure.


Robustness

When the scaling exponent α for P(k0) is larger than 3, itmay be shown that the critical threshold for maintainingthe giant component is pc ≈ 0.99. That is, these randomstructures are asymptotically robust to random failures.

For finite-size networks, the value of pc is bounded awayfrom 1, and its exact value is related to the size of thegraph n.

A recent extension of this work shows, however, that thevalue of pc can be significantly smaller for a specificsubclass of these graphs, that is, not all random graphswith a power-law degree distribution are equally robust torandom failures.


Targeted Attacks

Unfortunately, randomly removing vertices is not the onlykind of failure that networks suffer. For instance, whennodes in these same random graphs are preferentiallyremoved according to some rule (for example removingthe 10% of vertices with the highest degree), the networkquickly shatters.

Furthermore, network theorists in the peer-to-peerresearch community have considered more subtle forms offailure, in which some fraction of nodes disobey thenetwork communication protocols, possibly in a maliciousway. These Byzantine faults have been extensivelystudied, and continue to drive much of the research indeveloping secure, and distributed communicationprotocols.


Navigability

Given that a network is connected, there exist several pathsthat typically connect a transmitter with a receiver. In networktheory, a network’s navigability is determined both by howeasily such a path can be found, and how many hops such apath ultimately requires. This problem has been extensivelystudied, and solutions can be grouped into two categories:

1 Central authorities, in which the communication pathbetween two vertices is determined by an external source,later mirrored by the network’s routers, and

2 Decentralized techniques, in which routing decisions aremade independently by network routers, possibly in anad-hoc fashion.


Efficiency

For a network theorist, the concept of efficiency is intimatelyrelated to that of scalability, which is defined as the cost ofsome network property as a function of the number of verticesin the network n. Generally, for a property to have a smallcost, it should scale sublinearly, and ideally as a polylogarithmO(logk n). For example, the decentralized routing algorithmdescribed earlier, guarantees that the average number ofintermediate nodes through which the message passes isO(log2 n).


The Control Designer’s Point of View

Networked control system (NCS) applications such asteleoperation and robot formation control, requiremeasurement and control signals to travel acrosscommunication networks. Even when the distance traveled isshort (as in the case of a modern car or a smart house), ageneral purpose communication network introduces new issuesinto the feedback loop, such as time-varying delays, and thepotential loss of information. While some communicationapplications may suffer from the same limitations, a feedbackcontrol system is especially vulnerable, not only to theunavailability of sensory information and control signals, butalso to their timing. In particular, in a NCS, the issues ofconnectedness, navigability, and efficiency of travel manifestthemselves as described next.


Connectivity, dropped packets, and lost links

From the perspective of control design for networked controlsystems, connectedness (or connectivity) expresses the abilityof two systems to communicate information and actuationsignals through the network connecting them. Connectivity istherefore related to the existence of a network path from anynode u to any other node v . In recent studies that link thedynamics of the networked systems to the connectivityproperties of the network, certain graph algebraic properties ofthe latter seem to be pervasive.


Efficiency

The dynamics of the networked system may be formallyrelated to the algebraic graph theoretic properties of theinterconnection network, and to the graph Laplacian inparticular. Connectivity, as expressed by the second smallestLaplacian eigenvalue (known as the algebraic multiplicity ofthe graph), proves to be a crucial network property, allowingthe gradual dissemination of local information throughout thenetwork and facilitating stability.

It may be shown that if connectivity is permanently lost,stability cannot be guaranteed. If connectivity is regainedacross a sequence of compact intervals [ti , ti+1), stability inthe form of consensus is still guaranteed.


Catalyst but not Enough

Network connectivity appears to be a catalyst since nothinguseful can happen without some sort of connectedness. Somenew insight, however, seems to suggest that perhaps denseconnectivity is not all a control designer should strive for. Wehave recently shown that high algebraic connectivity does notnecessarily imply a high network’s robustness. In other words,while the network connectivity is certainly improved as thesecond smallest eigenvalue increases, and as the diameter ofthe network (characteristic path length) decreases, thenetwork remains vulnerable to targeted attacks. In particular,there may be few nodes or links that guarantee the networkconnectivity, and the removal of as few as one or two suchnodes may break the network into disconnected components.



n = 20, k = 4, p = 0 n = 20, k = 4, p = 0.1

n = 20, k = 4, p = 0.5 n = 20, k = 4, p = 1

Figure: Random ring lattice graph G = C (n, k) with n = 20,k = 4, and different edge probabilities.



Algebraic connectivity, as determined by the size of the secondsmallest Laplacian eigenvalue λ2(G ), is of great interestbecause of Fiedler’s inequality

λ2(G ) ≤ ν(G ) ≤ η(G ), (1)

which states that the algebraic connectivity of a graph G isless than or equal to the node-connectivity which is less than orequal to the edge-connectivity. While increasing the algebraicconnectivity increases the lower bound on node-connectivity, ithas been shown that, for circular and mesh lattice graphs, anincrease in algebraic connectivity often corresponds to adecrease in node-connectivity and edge-connectivity.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Probability p

Random Graph, n=100, k=4, p=0:0.1:0.9, 10 runs per data point

AC( p=0.9)/AC( p=0) = 29.493058

Algebraic ConnectivityNode ConnectivityEdge ConnectivityMean Path Length

Figure: Results for a ring lattice random graph, N = 100, k = 4.Although algebraic connectivity increases, node and edgeconnectivity decreases monotonically with mean path length.


Efficiency: capacity, link quality, and delays

Since for the control engineer, a communication channel ismerely a medium for obtaining or sending information(measurement signals, or control commands), what seems tobe important is:

1 how much information can be carried, and

2 how fast can it be transferred.

The first question is related to the channel’s capacity. Resultslinking information theory to control have recently beendiscovered. While information theory models thecommunication channel as an information transmittingmedium that corrupts portions of the signal, the main issue forcontrol-based applications are the delays (as well as corruption)suffered by the signals as they are carried across the channel.


Delays

The speed at which information can travel from source todestination is usually measured in terms of a communicationdelay, the time elapsed between transmission, and reception. Itis generally recognized that actuation and measurement delaysdegrade the performance of control systems. A communicationdelay can cause actuation delays, measurement delays, orboth; therefore, it is natural to expect that it must have anadverse effect on the stability of the interconnected system.Initial investigation seemed to support this claim and stabilityanalysis in the frequency domain suggests the existence of anupper limit in the (uniform) communication delays that acontinuous, nearest neighbor interconnected system cantolerate, before becoming unstable. However, more recentanalysis of state space, discrete-time models of interconnectedsystems, leads to different conclusions.


Internet-Like Protocols

Finally, inspired by the work of Frank Kelly, which provides amathematical foundation and explanation for variouscongestion control algorithms on the Internet, we have appliedsimilar concepts to implement distributed control algorithmsfor the coordination of multiple agents communicating acrossa network, and developed Internet-like protocols. Controlprotocols with these features can be implemented in systemswhere a group of n users (clients) share a common resource(server) of finite capacity C .


ILP

While information systems such as the Internet are concernedwith transferring the information only (with high fidelity),networked control applications are more involved due to theeffect that dated information can have on the control ofdynamical systems. To quote Traub, information is“incomplete, priced, and corrupted.” However, and for controlpurposes, it is also “timed.” To make the general idea ofInternet-like control more precise, consider a network of nusers of a resource C .


ILP

If for user i , the state variable xi represents its usage of theresource, the desired equilibrium for the system becomes theconfiguration at which ∑n

i=1 xi = C .

It is also necessary to define a feedback signal, from theresource to the users, which communicates the availability orshortage of the resource. A priced scheme has been used forthis feedback. A low price is an indication of resourceavailability, and a resource shortage is represented by anincrease in price.


ILP

The dynamics of the resource price can be simply expressed asthe difference between the resource usage, and the size of theresource, scaled by a positive gain γ,

p(t) = γ [∑n

i=1 xi(t)− C ] .

Different dynamical interactions between the resource usagexi(t), and the resource price feedback may be used.


ILP

The common thread in these models is the inverse relation,such as the additive-increase, multiplicative-decrease used inthe model of TCP Reno, which has the form

x(t) =1− p(t)

τ 2− 1

2p(t)x2(t),

where p(t) is the price for the resource, and τ is thepropagation delay between the and resource. The controldesign is then reduced to solving a distributed resourceallocation problem. It is conceivable that by exploiting somenew routing algorithms, more efficient distributed controlalgorithms can be designed.


Conclusions

The control and complex network communities have beenlooking into similar problems in networked systems fromdifferent perspectives. The same concepts and propertiesappear to be important, both for the control engineer, and forthe network theorist. Having established the conceptual linkbetween networked control systems, cooperative control, andcomplex networks through graph theoretic analysis, the controlcommunity may now be in position to capitalize on, andexploit the arsenal available in complex network research andcomputer science. This talk offers such a suggestion byhighlighting the recently revealed power of randomizedalgorithms in routing, network design, resource allocation, andgame theory.


Networked Control Systems - UTA · 2007-06-06 · Networked Control Systems: Emerging Links Between...

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