IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10 ... Collection/TAC... · symmetric linear interconnection of dynamical systems on a bipartite graph. The importance of this

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006 1613

Scalable Decentralized Robust Stability Certificatesfor Networks of Interconnected Heterogeneous

Dynamical SystemsIoannis Lestas and Glenn Vinnicombe

Abstract—We derive scalable decentralized conditions that canguarantee robust stability for networks of linearly interconnected,stable, linear time invariant dynamical systems. Unlike previousresults of this kind, we allow for heterogeneous dynamics on arbi-trary interconnection topologies (i.e., linear systems on arbitraryunderlying graphs). Robust stability of the entire network is guar-anteed by satisfying local rules that involve only an agent and itsneighboring dynamics; each new agent will introduce only one ad-ditional condition of this kind and hence the stability certificatesscale with the network size. An application of this theory is given,where robustness analysis of Internet congestion control protocolsis carried out in the general case of dynamics at both users and re-sources without any global bounds on the dynamics.

Index Terms—Complex systems, networks, robust control.

I. INTRODUCTION

ANALYSIS and decentralized control of interconnected dy-namical systems has traditionally received considerable

attention by the control community. Typical examples of earlywork include the dissipativity approach in [1] and [2] and theuse of vector Lyapunov functions in [3]. A renewed interest inthe recent years has led to work on decentralized optimal controldesign, such as the spatially invariant case in [4] and the relax-ation to a more general setting using LMI techniques in [5] and[6].

Nevertheless in many applications including data networks,flocking phenomena, power networks, financial markets decen-tralization is not sufficient: scalability is also a major propertythat needs to be maintained. By this, we mean that stability ofthe entire network must be preserved as this is modified withthe addition/removal of heterogeneous agents, without requiringa centralized analysis each time such a network evolution oc-curs. It would be, for example, unrealistic to carry out a central-ized global design, whenever a new computer/router joins theInternet, or a new generator becomes part of a power network.

As we are looking for decentralized stability certificates thatguarantee stability for an arbitrary network, possible structure inthe interconnections or certain homogeneity assumptions in the

Manuscript received December 23, 2004; revised February 12, 2006. Recom-mended by Associate Editor E. Jonckheere. The work of I. Lestas was supportedby a Gates Scholarship.

The authors are with the Department of Engineering, University of Cam-bridge, Cambridge CB21PZ, U.K. (e-mail: [email protected]; [email protected]).

Color versions of Figures 1 and 3–6 are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2006.882933

participating dynamics become important, if we want to avoidmore conservative weak interconnection results. This kind ofstructure has played a substantial role in most of the scalablestability results available so far.

For example, in [1] and [2] a generalized dissipativitydescription with quadratic supply rates is adopted for partici-pating agents. By appropriate conditions on the interconnectionmatrix, the summation of individual storage functions becomesa common Lyapunov function for the interconnected system.Note, however, that such interconnection constraints can berather restrictive for arbitrary networks unless these constraintsappear naturally within the interconnection protocol obeyed bythe system. Decentralized stability conditions of Internet con-gestion control protocols as described in [7] and the referencestherein, are an example of an application where scalability isimportant. As it will be explained in more detail in the paper,these protocols impose a very special interconnection structurewhich enables the derivation of stability results for arbitrarynetworks.

Interconnection constraints are relaxed in the paper by de-riving stability conditions that involve not only the dynamics ofan individual agent, but also those of its neighbors. Scalabilityfollows from the fact that adding a new agent introduces just oneadditional decentralized condition.

Our main result, roughly speaking, requires that a certain con-vexification of each “loop gain” (i.e., product of neighboring dy-namics) satisfies a common Nyquist-like condition. The notionof S-hull, a relaxation of the convex hull of a set in the com-plex plane, is introduced in the paper as a crucial tool in thisdirection. The motivation for working in the frequency domainin this way goes far beyond the possibility of developing an ap-pealing graphical test to verify stability: converting the stabilitycertification to a spectral inclusion problem of a complex matrixallows one to exploit the internal structure present in the systemthrough the use of the numerical range together with tools fromconvex and complex analysis. Note that the linear analysis israther more involved than a dissipativity argument as it requirestaking the square root of frequency response functions. It is thusintriguing to contemplate what any nonlinear generalization ofthis theory might look like.

Finally, even though we are not considering controller designwith performance criteria, robust stability certificates in suchnetworks (where scalability is important) can have substantialcontribution in their design. This is because one can see whetherpart of the network is being unnecessarily too conservative in itsresponse. For example, an important contribution along these

0018-9286/$20.00 © 2006 IEEE

1614 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10, OCTOBER 2006

lines has been a theoretical justification in [8] that TCP is toosluggish at high bandwidths, with a proposal for modificationin [9]. The paper is structured as follows. In Section II we provethe main lemmas on which the stability results are going to bebased. We introduce the S-hull and show how it can be used tobound the numerical range of matrices with a structure relevantto our analysis. In Section III we give the stability results, for bi-partite graphs first and then for arbitrary graphs. These assumesymmetric interconnections between the dynamics and we showhow a break of the symmetry can be seen as a local perturbationin the symmetric results. We then illustrate with a random graphexample that the proposed certificates are not too conservativefor random graphs with small connectivity. Finally, an applica-tion of this theory is given, where robustness analysis of TCPlike Internet congestion control protocols is carried out in thegeneral case of dynamics at both users and resources withoutany global bounds on the dynamics.

II. PRELIMINARY RESULTS

A. Notation

The field of real and complex numbers are denoted byrespectively. are the by matrices with ele-ments in the corresponding fields and is the set of non-neg-ative reals.

denotes the spectrum of a square matrix itsspectral radius, the elementwise absolute value of the ma-trix i.e., is the th row and the thcolumn. Given two matricesis the elementwise maximum of the magnitudes, i.e.,

. denotes the convex hull of a setand the matrix with elements on the leadingdiagonal and zeros elsewhere. We denote the square of a set

as the set of the squares of its elements, i.e.,.

The numerical range of a matrix is the set. The property

is frequently used in this paper (see, e.g.,[10] and [11] for a more detailed discussion of the propertiesof the numerical range). is the Hardy space of transferfunctions of stable, linear, time-invariant, continuous-timesystems. is the class of functions continuous inand .

B. The S-Hull and the Numerical Range

Definition 1 (S-Hull): Let . The S-hull of set isdefined as

Given a set it is always true that and1

. The discrepancy between the S-hull and the corre-sponding convex hull in the last inclusion is often small relativeto the size of (see example in Fig. 1(a)). The S-hull is a map

1The statement 0 2 S(P ) follows easily from the fact that Co(pP ) is sym-

metric about the origin. S(P ) � CofP [0g is true since S(P ) is a convex set(Theorem 5 in the Appendix) and S(P ) always includes P and 0.

Fig. 1. Examples of S-hulls and how they can be used to bound NumericalRanges. (a) Comparing the S-hull and Convex Hull of a set that includes theorigin. (b) The thick line denotes the boundary of Co(ff S(fg : R 6=0g) : i = 1; . . . ;mg) for f = 1 + j; f = �2� 2j; g = 1; g = �2 +j;R = 0. The ellipses are N(G R FRG ) for G = diag(g ); F =diag(f ); �(jRj jRj) = 1; R = 0.

that plays a crucial role in this paper because it can be usedto relate the numerical range of a product of matrices with aparticular structure to the nonzero elements of those matrices.The following lemma is a key result in this context and gives analternative definition for the S-hull.

Lemma 1: Let for . Then

(1)

Proof: Main observation:

(2)

LESTAS AND VINNICOMBE: SCALABLE DECENTRALIZED ROBUST STABILITY CERTIFICATES 1615

Let

Then

using (2)

Therefore

(3)

Using the fact that is a convexset (proved in Theorem 5 in the Appendix), (3) gives the fol-lowing inclusion for Lemma 1:

The inclusion in the reverse direction is obvious from (2),since we can always choose real ’s so as to match the ’sin (2).

The following is the main lemma in this section that givesa bound for the numerical range of a product of matrices witha specific structure. As it will be discussed in detail in Sec-tion III, this structure can characterize the return ratio of anysymmetric linear interconnection of dynamical systems on abipartite graph. The importance of this bound is that it is theconvex hull of S-hulls of products of elements which are adja-cent in a graph theoretic sense. This graph theoretic interpre-tation is an important part of the problem formulation given inSection III-A.

Lemma 2: Let satisfy , and

then

where , and either of the squareroots can be used.

Proof:

since

expanding we get

Since this is true for all such

(4)

Considering now the definition of the numerical range, notethat

(5)

using Lemma 1

(6)

From (4), we note that the convex hull in (6) is multiplied bya real number in . Hence, we deduce that


Remark 1: The inclusion in Lemma 2 is actually an equalityif we consider the union of all numerical ranges such that thematrix satisfies the spectral radius bound and has a specifiedsparsity. This is proved in Theorem 6 in the Appendix.

Fig. 1(b) illustrates an example of Lemma 2. The NumericalRange and the eigenvalues of are shown for 30such matrices for varying such that

. The prod-ucts are also marked with circles. Note that the spectrumis bounded away from as desired and also that

alone is not sufficient to bound the correspondingspectrum.

III. MAIN RESULTS

A. Problem Formulation

We consider single-input–single-output (SISO) linear timeinvariant dynamical systems with transfer functions in .These are linearly interconnected, i.e., the input to each systemis a linear combination of the outputs from other systems. Weconsider a directed graph representation of the interconnectedsystem and use the notation that follows.

is a weighted directed graph,2 whereis the set of nodes, the set of directed

edges and a weighted adjacency matrix. Directededges are denoted as , such that is defined tobe incident to node . The adjacency matrix sat-isfies . The in-neighbors of a node aredefined as and its in-degreeas . Similarly the out-neighbors are defined as

and the out-degree as . In thedigraph representation of the network, each dynamical elementcorresponds to a node of the graph. Furthermore in a networkof dynamic agents, each with scalar input , scalar output

and transfer function , the input and output vectors,and re-

spectively, satisfy the relation , where is theadjacency matrix of the graph.

We use standard definitions of well posedness and stability forthe feedback interconnection of linear time invariant systems.

Definition 2 (Well Posedness and Stability): The positivefeedback interconnection of a linear time invariant system withtransfer matrix is well posed if isinvertible ( for almost all ). It is stableif it is well posed and .

The multivariable Nyquist criterion ([12]) will be used exten-sively in the paper to guarantee well posedness and stability ofthe interconnected system.

The section is structured as follows. We consider first sym-metric (the adjacency matrix is symmetric) bipartite graphs.These are easy to analyze due to the block antidiagonal structureof the adjacency matrix, or in graph theoretic terms, all cycleshave even length. The results are then extended to arbitrary sym-metric graphs by an appropriate transformation of the problem.

2From now on, the word graph or digraph will refer to a weighted directedgraph.

Fig. 2. Block diagram representations of interconnected systems. (a) Under-lying graph can have arbitrary structure. (b) Underlying graph is bipartite.

Finally, we show how these conditions can be generalized to di-rected graphs with nonsymmetric adjacency matrices.

B. Stability of Symmetric Digraphs

An interconnected system with dynamic agents withtransfer functions respectively and adja-cency matrix , can be represented with the block diagramin Fig. 2(a), where . In a bipartite graph,there exist two disjoint sets of dynamicsand such that dynamics from one set areconnected directly only to dynamics from the other set. Thismeans that the interconnection is defined by

(7)

where. An alternative

block diagram representation in this case is shown in Fig. 2(b).Theorem 1 (Stability of Bipartite Graphs): The in-

terconnection of linear dynamical systems described in(7) where and for

is stable if

(8)

OR

(9)

Proof: From the block diagram in Fig. 2(b) the returnratio of the interconnected system is .Using the multivariable Nyquist criterion [12] the closed-loopsystem is stable if the eigenloci of the return ratio do notencircle the 1 point. A sufficient condition is the existence ofa convex set that does not include 1, but includes

. We generate such a set by noting that and


have the same nonzero eigen-values. Zero eigenvalues are not a problem since these arealways included by the bounding set we will consider. Thenumerical range is always a bound for the spectrum of a matrix.Hence, condition (8) follows directly from Lemma 2. Similarly,the sufficiency of (9) can be proved by bounding the spectrumof .

Remark 2: The convex hull conditions (8) and (9) can easilybe given decentralized interpretations by means of hyperplanearguments. Note first that

Using a duality argument the convex hull of S-hulls condition isequivalent to each of the S-hulls not intersecting a globally spec-ified hyperplane through the point 1 since the S-hulls will nec-essarily lie on the same side of the hyperplane (this is becausethe S-hull of a set always includes the point 0). Decentraliza-tion is a result of the fact that the domain of each of the S-hullsdepends only on a given dynamic and its neighbors. Scalabilityfollows from the fact that a new agent introduces only an addi-tional hyperplane condition for an S-hull.

Remark 3: Given relations and the2-norm bound on can be achieved by appropriate scaling(using ideas in [13]). This is because the return ratio

is similar to

where

The spectral radius bound on is valid since

where the last inequality follows using the fact that for a ma-trix the induced infinity norm satisfies

. In the case all elements of are in it issufficient to take as vectors with all elements being equalto 1 and vectors of in-degrees, i.e., whereis the node associated with the dynamic with transfer function

, similarly where is node associated withthe dynamic with transfer function .

Remark 4: In the special case where there are no dynamicsassociated with one of the disjoint sets of vertices in the bipar-

tite graph, i.e., , the stability condition inTheorem 1 reduces to

Remark 5: Theorem 1 also holds when is a transfer matrixanalytic in the closed right-half plane such that it can be

factorized as and satisfiesthe spectral radius bound . Conditions(8), (9) then hold but with replaced by .This is particularly useful for some networks, such as the In-ternet, where can be a transfer matrix of the propaga-tion/return delays (see details in [8] and the example in Sec-tion IV-B).

We now consider general symmetric graphs, i.e., the intercon-nection is defined by

(10)

where andwithout any restrictions in its sparsity.

Theorem 2 (Stability of Arbitrary Graphs): The intercon-nection of linear dynamical systems described in (10), where

and , for is stable if

Proof: The return ratio in this case is and we use thefact . This is truesince for a matrix . Moreover,given a convex set s.t. and

, then implies and,hence, . Therefore,

does not encircle the point .It is hence sufficient for stability to bound the spectrum of

by a convex set for all . A returnratio may be interpreted as that of a bipartitegraph with and

, where , , are as defined in (7).Theorem 2 follows then from Theorem 1.

Remark 6: A decentralized interpretation of the stability con-dition can be given as explained in Remark 2. As in Remark 3,the 2-norm of matrix in the case all its elements are incan be bounded by an in-degree scaling of the inputs of the dy-namics.

Remark 7: In the case it follows from results in [14]that a stability condition analogous to that in Remark 4 applies,i.e., the interconnected system is stable if

and

Remark 8: being skew symmetric, i.e., cor-responds to the case where all cycles of length 2 are in nega-


tive feedback. In this case, the same stability conditions hold butwith respect to the point instead of . In a bipartite graphnegative feedback can be ensured by multiplying by one ofthe sets of dynamics in the bipartition. Note that such negativefeedback configurations are the most reasonable ones to assumein most applications.

Remark 9: In comparison with Theorem 1, the price payedfor guaranteeing stability of arbitrary graphs is that the stabilitycondition requires all dynamics to appear in the domain of anS-hull whereas, in Theorem 1 only one of the two sets of the bi-partition of dynamics has to appear in the S-hull. More specifi-cally, if Theorem 2 is applied to the bipartite case, both (8) and(9) are required to hold, whereas Theorem 1 needs only eitherof them. Even though we are eventually considering the convexhull of the S-hulls, the conservatism lies in the fact that theS-hull of a set always includes the corresponding convex hull.For bounded sets that include the origin, the S-hull and convexhull are generally close relative to the size of the original sets.Large discrepancies tend to appear for unbounded sets. For ex-ample, given a quartantthen , whereas . This means that dy-namics with imaginary axis poles will cause problems if they ap-pear in an S-hull, since their corresponding frequency responseswill be unbounded. In a bipartite graph such problems can beavoided by placing the dynamics with imaginary axis poles inthe set which does not appear in the S-hulls. In arbitrary graphs,loop transformations can be employed, e.g., let andsome and assume zero diagonal elements in the adja-cency matrix, then

This is equivalent to setting in the adjacency matrixand replacing with which isbounded for . Note that this implies that the in-degreescaling will have to be increased by . Therefore, needs to bechosen large enough so that is bounded for butnot too large so as to avoid being too conservative because ofthe degree scaling.

C. Stability of Nonsymmetric Digraphs

In many realistic applications the adjacency matrix is nonsymmetric or is nearly symmetric. Hence, it is important to findstability conditions for nonsymmetric digraphs or at least verifythat Theorems 1 and 2 are not fragile with respect to a relaxationof the symmetry assumption. This is indeed the case as we showhere.

Definition 3 [Extended S-hull]: Let for .The extended S-hull of the set given pertur-bation vectors is defined as

Remark 10: Note that from Lemma 1 it is apparent that theextended S-hull is equal to the S-hull (Definition 1) for .

Theorem 3 (Stability of Nonsymmetric Bipartite Graphs):The bipartite interconnection of linear dynamical sys-tems described in (7) where the adjacency matrix

satisfies

and , is stable if

Proof: The proof is along the same lines as that in The-orem 1, where we bound the spectrum of the return ratio by itsnumerical range. The bound in Lemma 1 is slightly modified toaccount for the nonsymmetry. This is achieved by noting as in(6) that

The convexification proceeds as in Lemma 1, but we now use ex-tended S-hulls instead of S-hulls. These two are the same whenapplied to the neighbors of a dynamic element which is sym-metrically connected to all of its neighboring dynamics.

Remark 11: The main strength of Theorem 3 is that the S-hullis perturbed to an extended S-hull only in cases of dynamicsthat have at least one non symmetric interconnection and this


can be seen as a local (with respect to the network topology)perturbation of the S-hull due to the nonsymmetry.

Remark 12: The spectral radius bound can be achieved as inRemark 3 but with scaling with respect to matrix , i.e., weneed relations .

Remark 13: The stability conditions can easily be extendedto arbitrary nonsymmetric graphs as in Theorem 2. Also an anal-ogous condition can be derived by interchanging the and

as in (9).An exact characterization of the extended S-hull as a relaxed

convexification (as in the case of the S-hull) is part of ongoingwork. Preliminary characterizations for special cases of bipar-tite graphs have been given in [15] and [16]. Note, however,that we would expect the spectral inclusion to be in generalmore conservative due to the lack of underlying symmetry. Inaddition the conservatism is likely to increase if one considersmultiple-input–multiple-output (MIMO) systems, unless thereis some internal structure associated with those. For example, itcan easily be shown (along the lines of [14, Lemma 1]) that theconvex hull stability condition in Remark 7 can be extended tomultivariable systems with equal number of inputs and outputsby replacing the frequency responses with the numerical rangeof the corresponding transfer matrices.

IV. EXAMPLES

In this section, we give two examples to illustrate the appli-cability of the results in Section III. In the first one, we con-struct a randomly expanding symmetric network with randomdynamics, where each agent adjusts its gain according to The-orem 2, using only a knowledge of the dynamics of its neigh-bors. The main observation is that the stability conditions arenot too conservative in this setting (at least with small connec-tivity). In the second example we apply the bipartite graph re-sult in Theorem 1 to derive scalable robust stability conditionsfor Internet-like communication networks operating a TCP-likecongestion control protocol. Unlike previous results of this type,dynamics at both sources and links are allowed and robustnessguarantees are given without requiring any global bounds on thedynamics.

As a comment on the conservatism of the stability conditionsin Section III, it should be noted that these are tight in the sensethat there exist cases for which they are also necessary. Con-sider, for example, a simple setting corresponding to Remark 4,where we have a bipartite graph with an integrator in one of thebipartition and scalar gain in the other

(11)

Applying the Nyquist criterion on this configuration gives thesame necessary and sufficient condition on the gain k for theclosed loop system to be stable, as the convex hull condition inRemark 4, i.e., .

On the other hand, conditions are usually only sufficient,since they are based on a convexification of the eigenloci ofthe return ratio. One cannot in general derive necessary andsufficient conditions for the stability of a strongly connected

component by taking only cycles of length two into account.The point that needs to be emphasized here, though, is that themain benefit from carrying out such convexifications is thatone can derive scalable, decentralized stability certificates bymeans of duality arguments (Remark 2). Potential conservatismin specific configurations is due to the fact that stability canbe guaranteed for a arbitrary networks, which is the maincontribution of this methodology for network analysis.

A. Random Graphs

We consider a graph which has been randomly generatedusing the Waxman algorithm [17]. In such a graph the positionof nodes is uniformly distributed within the plane and the edges3

are randomly added according to an exponential distribution theintensity of which is inversely proportional to the distance be-tween the corresponding nodes.4

More precisely a randomly expanding network is constructedusing the following algorithm. Note that each dynamic onlyknows the dynamics of its neighbors when adjusting its gain.

Algorithm I:1) Generate a random dynamic with transfer function .

Determine its position on a specified surface by samplingfrom a uniform distribution.

2) Each edge with the already defined nodes is added with aprobability , where is the distancebetween nodes and are tuning parameters. If nolink has been added, add the link with the smallest length.

3) The added dynamic adjusts its gain to its maximal value soas to satisfy the S-hull test in Theorem 2 with a globallyspecified hyperplane as in Remark 2, and scales its inputby its in-degree as in Remarks 3 and 6.

4) The neighboring dynamics to the added node reducetheir gain, if necessary, so as to satisfy the S-hull test onTheorem 2 and also update their in-degree scaling.

5) Go to step 1).Remark 14: The gain adjustment in step 4) will be in the

worst case of the order of the discrepancy between the convexhull and the corresponding S-hull of the sets convexified in thestability tests. To see this note that if the stability condition in-volved only convex hulls then this adjustment would be redun-dant: if an incoming agent ensures that the products of its owndynamic with its neighbors do not intersect the hyperplane, thenso does any convexification (by means of convex hulls) thatincludes existing products (as these will already not be inter-secting the hyperplane). A gain adjustment as in step 4 is likelyto be needed because the S-hull of a set always includes thecorresponding convex hull and neighboring agents may havedifferent sets of neighbors. It should be emphasized, though,that this will not propagate throughout the network since by re-ducing the gain, the convexified regions in the S-hull tests ofother neighboring agents, will be included by the correspondingregion before the adjustment; hence existing local stability cer-tificates will not be compromised.

3An edge in this graph corresponds to two edges of unit weight and oppositeorientation between the same nodes in the diagraph representation given in Sec-tion III-A.

4Note that the actual position of the nodes is not relevant for the stabilityanalysis but merely the degree of each node.


Remark 15: The gain of the agents was chosen to be the max-imal that guarantees network stability so as to evaluate the con-servatism of the stability conditions by checking the distance ofthe eigenloci from the point 1. A desired stability margin can bemaintained by requiring that the S-hulls have a certain minimumdistance from the hyperplane (see also robustness guarantees innext example, Fig. 6).

In Fig. 3(a), we show one such example with 50 nodes. Thedynamics are of the form where andare uniformly distributed in and , respectively.Fig. 3(b) shows the bounding hyperplane and the eigenloci ofthe square of the return ratio for frequencies in . Notethat the adjacency matrix was chosen to be a skew symmetric

matrix, hence the S-hull test is with respect to the point ,as indicated in Remark 8 (see Fig. 3(c) of an example of how theS-hull test is applied). We observe that the half plane bound wehave imposed is not very conservative for the actual eigenlociof the square of the return ratio; the eigenloci pass through thepoint with an increase by in the gains . It is alsoapparent from simulations that this bound is more efficient whenthe graph connectivity is relatively small (in the example in Fig.3(a) the mean degree is 2.2 with a standard deviation of 1.4).This can be explained by the fact that the stability results takeinto account only cycles of length 2. Therefore they become lessconservative as the number of cycles with length greater than 2decreases.

It is also important to mention that if more structure is presentin the interconnections less conservative scalable results can beobtained. This is because a tighter inclusion can be derived forthe eigenloci due to additional structure in the adjacency matrix.In [18], for example, scalable stability of consensus protocolsis examined. In that case the input to each dynamic is a linearcombination of the differences between its own output and thatof its neighbors i.e., . Hence, theadjacency matrix is shown to be positive definite and the convexhull results in Remark 7 can be applied.

B. Robust Congestion Control for the Internet

We illustrate with this example how Internet congestion con-trol fits within the graph theoretic setting of this paper, in thesense that TCP forces an interconnection between the associ-ated dynamics that gives a particular structure in the underlyinggraph. Many scalable local stability conditions derived so farfor different versions of this problem, such as results in [14],[8], [19], and [16] can be seen as applications to special casesof the conditions in Section III (particularly Remark 4 for [14],[8], and [19], and Theorem 3 for the nonsymmetric analysis in[16]).

More specifically, we consider a communication networksuch as the Internet, consisting of an interconnection ofusers/sources which generate data and resources links/whichcarry it. Each resource generates a price as a function of theaggregate data flow through it and users generate data as afunction of the aggregate price from the resources they use.The underlying graph of such an interconnection scheme isbipartite, since users are only connected directly to resources

Fig. 3. A random graph example with dynamics of the formg (s) = k (e )=(s+ a ) (a) Waxman type graph with 50 nodes.(b)��(G(j!)AG(j!)A); G(s) = diag(g (s)); ! 2 and the boundinghyperplane. (c) fg (j!)S(fg (j!) : A 6= 0g) : ! 2 g for some kand the bounding hyperplane.


and resources only to users. This implies that Theorem 1 canbe applied to derive decentralized robust stability conditions.

Many delay stability results for Internet congestion control(including the references above) consider dynamics associatedwith either the sources or the links, i.e., only one of the two setsin the bipartite graph. Attempts to consider the case of dynamicsat both users and links either have no robustness guarantees (asin [8]) or place global bounds at the source (or link) dynamics(e.g., a global bound on the network delay as in [20] and [7]).The reason for this is that the interconnection between the twosets of dynamics in the bipartite graph is not taken into accountbut instead the two sets of dynamics are convexified indepen-dently. We give here an example with dynamics at both sourcesand links where the S-hull methods described in this paper areused to derive robust stability certificates with no global bounds.The example described corresponds to the primal case where thecontrol law is performed at the users with first order dynamicsat the resources.

1) Problem Formulation: We use the notation in [8], [21] anddefine the following: is the flow rate associated with route .The user on route is characterized by an utility which isa continuously differentiable, strictly concave, increasing func-tion of the flow . The roundtrip delay of the th route is

, with being the propagation delay from sourceto link and the return delay from link to source .

(12)

is the aggregate flow through link and is the linkprice, which is a nonnegative, strictly increasing function ofthe aggregate flow through the resource . We assume that linkprices are set according to the law

(13)

where is an exponentially smoothed flow rate, satisfying

(14)

Such first order dynamics at the queue could correspond eitherto averaging from queueing effects or deliberate smoothing bythe underlying protocol. In RED [22], for example, a lag is in-troduced since the marking probability is based on an averagedqueue length. Such lags could also become relevant due to thestochastic character of the queue. It has been shown, for ex-ample, in [23] that for an M/M/1 queue the linearized transferfunction from the time varying arrival rate to the probability ofexceeding a certain threshold in queue size, can be well approx-imated with a first-order lag. In fact, this model converges to astatic function (used, e.g., in [21]) in the case of low load/shortqueues and to integrator dynamics in the case of high load/longqueues.

(15)

is the aggregate price along route . The control law is per-formed by the users according to

(16)

It is shown in [21] that if for each route andfor each link then the interconnection is globally stable and

converges to the unique equilibrium that maximizes the globalutility . Delay dependent stabilityconditions in the case of nonzero delays have been derived in[8], for arbitrary , but with no robustness guarantees in thesense that there might exist arbitrarily small perturbations in thelink dynamics that can destabilize the network. It is shown herethat robust stability certificates can be derived, using the S-hullresults of this paper, that can guarantee a specified degree ofrobustness when the lag parameter is appropriately boundedrelative to the link propagation delay (this is less than or equalto the corresponding roundtrip time).

2) Main Result: Taking Laplace transforms, we can write(12) as a vector equation

where if route uses linkotherwise

(17)

and since (15) is written as

For small perturbations about the equilibrium flow, we get

(18)

(19)

and the equilibrium relations

(20)

Linearization of the source law (16) and the link price function(13)–(14) gives

where (21)

(22)


Breaking the loop at the source leads to the following returnratio:

(23)

The following proposition was proved in [8] for this system.Proposition 1: The interconnection described by (12)–(16) is

locally stable around the equilibrium if

uses (24)

The proof of Proposition 1 proceeds by showing that the eigen-loci of the return ratio are confined in

(25)

which is a set that is arbitrarily close to irrespective of theuser gain . Therefore there are no robustness guarantees sincethere always exists an arbitrarily small phase lag at the linksthat can destabilize the system. The reason for this problem isthat the dynamics in the two sets in the bipartite graph have beenconvexified independently without taking into account the inter-connection between them. On the other hand the results in thispaper “convexify” by means of S-hulls the products of adjacentdynamics, thus giving more reasonable robustness conditions aswill be shown below. In this case the bound on will be rela-tive to the propagation delay of link and we exploit the factthat .

Theorem 4 is an application of Theorem 1 where the dy-namics in the two sets of the bipartite graph are parameterizedand the stability certificates are derived relative to these param-eters. This parameterization is appropriate here as the nature ofthe dynamics at sources and links is known a priori.

Theorem 4: Consider a negative feedback system with returnratio

where is defined by (17), are stable , andfor all and . Let be any

real and positive vectors satisfying

and further assume that

Fig. 4. S(f(1=jx + 1) : x < Xg).

Finally, assume that for all there exist regions suchthat and for all . Under theseconditions the feedback system is stable (i.e., isanalytic and bounded in ) if

Proof: The return ratio is similar to

where . As in Remark 3, wehave the spectral radius bound

Applying Theorem 1 and noting the negative feedback in theinterconnection, the system is stable if

Theorem 4 is now used to derive robust local stability certifi-cates for the interconnection described in (12)–(16). In the re-turn ratio in (23), we denote and

. Taking in Theorem 4 as those inthe problem description, the condition on the gains becomes

(as in Proposition 1). We assume that , whereis the propagation delay of the th link. So,

, i.e., and we cantake which is an arc of thecircle centred at the point , from the point to the point

. In this case . Fig. 4 illustratesand for and . Similarly, we take


Fig. 5. Cof[ H S([ G )g for K = 2.

Fig. 6. Cof[ H S([ G )g for K = 2; � = 0:1.

. Fig. 5 shows the final re-gion, obtained by overlapping the on the , for .In this case (and in fact for all up to around 6) the regions

are all contained within the union of the Nyquist con-tours of the set of systems ,which is shown as the thicker solid line. In particular, note thatthe region does not include and so the feedback systemis stable. Moreover, unlike in Proposition 1, by reducing thegains one can obtain guaranteed robustness to perturbations inthe source/link dynamics (these would simply result in pertur-bations to the regions and respectively). For example,Fig. 6 shows the corresponding picture when each of and

are increased by a radius of 0.1, that is each is replacedby where and similarly for .The intercept with the negative real axis now occurs at about

. The conclusion is that if

whenever uses then the network will be stable for all link dy-namics satisfying for some

and all stable source delay dynamics satisfyingfor some . ( denotes

the norm, ).

V. CONCLUSION

We have derived a generalization of the multivariable Nyquistcriterion that enables one to derive scalable decentralized con-ditions for the stability of networks of interconnected linear dy-namical systems. These conditions require that each dynamicagent knows only the dynamics of its neighbors and, hence, areindependent of the size of the network and the way it is intercon-nected. If there is some structure in the interconnections, e.g., abipartite graph, the stability conditions can be reduced to lessconservative certificates. This approach has been applied in thispaper to examples of dynamics on random graphs as well as tothe robustness analysis of Internet congestion control protocolswith dynamics at both users and resources. Our primary aim hasbeen to show that it could provide a basis for analyzing manyclasses of networks where scalability to the network size andtopology is a primary issue.

APPENDIX

The following Lemma is used to prove the fact that the S-hullis a convex set (Theorem 5).

Lemma 3: Let and ,. Then

Proof: Without loss of generality assume

(26)

We will in fact prove a stronger statement. This is

(27)

where for

Let . (27) is equivalentto . We prove this byshowing that the origin and any point lie on oppositesides of the track of the line (see Fig. 7). This statementcan be stated in cross product form as

(28)


Fig. 7. Root of the straight line joining a; b 2 and the square of the line joiningp

a;p

b.

where for as in (26)

where

and also it is apparent that

where

Hence, (28) is true.Theorem 5 (Convexity of S-Hull): Let . is convex

where

Proof: Let . Then

hence (29)

In order to prove convexity of we need to show that theline joining any two points in lies in , i.e.,

This is equivalent to showing that

From (29), it is sufficient to show that

This follows from Lemma 3.Theorem 6 (S-hull gives tight bound for Numerical

Range): Let and, , then

Proof: The proof is an extension of the proof of Lemma2. First note from (4) that the following sets of sequences areequal:

(30)


Using Lemma 1, the following is true:

(31)

(32)

we get the following generalization of Lemma 1:

(33)

where the last equality can easily by shown by appropriatescaling. We now use the expansion of the numerical rangedefinition in (5). The proof follows directly by combining thiswith (33) and (30).

REFERENCES

[1] P. Moylan and D. Hill, “Stability criteria for large-scale systems,” IEEETrans. Autom. Control, vol. AC-23, no. 2, pp. 143–149, Apr. 1978.

[2] M. Vidyasagar, Input-Output Analysis of Large-Scale InterconnectedSystems. New York: Springer-Verlag, 1981.

[3] D. Siljak, Decentralized Control of Complex Systems. New York:Academic, 1991.

[4] B. Bamieh, F. Paganini, and M. Dahleh, “Distributed control of spa-tially invariant systems,” IEEE Trans. Autom. Control, vol. 47, no. 7,pp. 1091–1107, Jul. 2002.

[5] R. D’Andrea and G. E. Dullerud, “Distributed control design for spa-tially interconnected systems,” IEEE Trans. Autom. Control, vol. 48,no. 9, pp. 1478–1495, Sep. 2003.

[6] C. Langbort and R. D’Andrea, “Distributed control of heterogeneoussystems interconnected over an arbitrary graph,” in Proc. IEEE Conf.Decision and Control, 2003, pp. 2835–2840.

[7] R. Srikant, The Mathematics of Internet Congestion Control. Boston,MA: Birkhäuser, 2004.

[8] G. Vinnicombe, “On the stability of networks operating TCP-like con-gestion control,” in Proc. IFAC, 2002.

[9] S. Floyd, “Highspeed TCP for large congestion windows,” in NetworkWorking Group, Dec. 2003, RFC 3649.

[10] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge,U.K.: Cambridge Univ. Press, 1991.

[11] K. E. Gustafson and D. K. M. Rao, Numerical Range: TheField of Values of Linear Operators and Matrices. New York:Springer-Verlag, 1997.

[12] C. A. Desoer and Y. T. Yang, “On the generalized Nyquist stabilitycriterion,” IEEE Trans. Autom. Control, vol. AC-25, no. 2, pp. 187–196,Apr. 1980.

[13] R. Johari and D. Tan, “End-to-end congestion control for the internet:Delays and stability,” IEEE/ACM Trans. Networking, vol. 9, no. 6, pp.818–832, Dec. 2001.

[14] G. Vinnicombe, On the Stability of End-to-End Congestion Controlfor the Internet Eng. Dept., Cambridge Univ., Cambridge, U.K., Tech.Rep. CUED/F-INFENG/TR.398, 2000.

[15] I. Lestas and G. Vinnicombe, “On the scalable stability of nonsym-metric heterogeneous networks,” in Proc. IEEE Conf. Decision andControl and European Control Conf., Dec. 2005, pp. 4634–4639.

[16] ——, “Are optimization based internet congestion control modelsfragile with respect to TCP structure and symmetry?,” in Proc. IEEEConf. Decision and Control, Dec. 2004, pp. 2372–2377.

[17] B. M. Waxman, “Routing of multipoint connections,” IEEE J. Select.Areas Commun., vol. 6, no. 9, pp. 1617–1622, Dec. 1988.

[18] I. Lestas and G. Vinnicombe, “Scalable robustness for consensus pro-tocols with heterogeneous dynamics,” in Proc. IFAC, 2004.

[19] F. Paganini, J. Doyle, and S. Low, “Scalable laws for stable networkcongestion control,” in Proc. IEEE Conf. Decision and Control, Dec.2001, pp. 185–190.

[20] F. Paganini, Z. Wang, J. Doyle, and S. Low, “A new TCP/AQM forstable operation in fast networks,” in Proc. IEEE Infocom, Apr. 2003,pp. 96–105.

[21] F. Kelly, A. Maulloo, and D. Tan, “Rate control in communicationnetworks: Shadow prices, proportional fairness and stability,” J. Oper.Res. Soc., vol. 49, no. 3, pp. 237–252, Mar. 1998.

[22] S. Floyd and V. Jacobson, “Random early detection gateways for con-gestion avoidance,” IEEE/ACM Trans. Networking, vol. 1, no. 4, pp.397–413, Aug. 1993.

[23] G. Vinnicombe, Robust congestion control for the Internet [Online].Available: http://www-control.eng.cam.ac.uk/gv/internet/index.html2002

Ioannis Lestas received the B.A. (starred first) andM.Eng. (distinction) degrees in electrical and infor-mation sciences from the University of Cambridge(Trinity College), Cambridge, U.K., in 2002. SinceOctober 2002, he has been a Ph.D. student withthe Control Group, Department of Engineering,University of Cambridge (Trinity College) as a GatesScholar and Trinity College Research Scholar.

His research interests include feedback controltheory of networks (robustness, scalability, andfundamental limitations) with applications in data

networks, multiagent systems, and biological networks. He has been elected toa Class A Fellow at Clare College, Cambridge, U.K., starting in October 2006.

Glenn Vinnicombe received the B.A. degree in engi-neering and the Ph.D. degree, both from CambridgeUniversity, Cambridge, U.K., in 1984 and 1993, re-spectively.

From 1984 to 1987, he was with British Aerospace,working primarily on the design and flight test of con-trol systems on the Airbus A320. He has held fac-ulty positions at the Department of Mechanical Engi-neering, University of Minnesota, Minneapolis, andin the Department of Aeronautical Engineering, Im-perial College, London, U.K., where he is currently

on the Faculty of the Department of Engineering.

Documents

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 10 ... Collection/TAC... · symmetric linear interconnection of dynamical systems on a bipartite graph. The importance of this