Congestion control in compartmental network systems

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<ul><li><p>Systems &amp; Control Letters 55 (2006) 689696www.elsevier.com/locate/sysconle</p><p>Congestion control in compartmental network systemsG. Bastin, V. Guffens</p><p>Centre for Systems Engineering and Applied Mechanics (CESAME), Universit Catholique de Louvain, Btiment Euler, 4-6 avenue G. Lemaitre,1348 Louvain la Neuve, Belgium</p><p>Received 30 July 2004; received in revised form 17 May 2005; accepted 18 September 2005Available online 27 April 2006</p><p>Abstract</p><p>In many practical applications of control engineering, the dynamical system under consideration is described by a compartmental networksystem. This means that the system is governed by a law of mass conservation and that the state variables are constrained to remain non-negative along the system trajectories. In such systems, network congestion arises when the inow demand exceeds the throughput capacityof the network. When congestion occurs, some links of the network are saturated with the undesirable consequence that there is an overowof some compartments. Our contribution in this paper is to show that congestion can be automatically prevented by using a nonlinear outputfeedback controller having an appropriate compartmental structure. 2006 Elsevier B.V. All rights reserved.</p><p>Keywords: Compartmental system; Congestion control; Nonlinear system</p><p>1. Introduction</p><p>In many practical applications of control engineering, thedynamical system under consideration is described by a so-called compartmental network system which is conservativeand positive. This means that the system is governed by a law ofmass conservation and that the state variables are constrainedto remain non-negative along the system trajectories.</p><p>The dynamics of compartmental systems with constant inputshave been extensively treated in the literature for more thanthirty years (see the tutorial paper [19] and also, for instance,[1,4,6,7,9,14,2022,24,26]).</p><p>In contrast, the control of compartmental systems has re-ceived much less attention. Recently, feedback control for setstabilisation of positive systems (including compartmental sys-tems) is a topic that has been treated in [2,3,5,17,18].</p><p>In this paper, we are concerned with another issue: the con-gestion control problem. Network congestion arises in com-partmental network systems when the inow demand exceedsthe throughput capacity of the network. When congestion oc-curs, some links of the network are saturated with the highly</p><p> Corresponding author. Fax: +32 104 72380.E-mail addresses: bastin@inma.ucl.ac.be (G. Bastin),</p><p>V.guffens@imperial.ac.uk (V. Guffens).</p><p>0167-6911/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2005.09.015</p><p>undesirable consequence that there is an overow of somecompartments. Our purpose in this paper is to show that conges-tion can be automatically prevented by using a nonlinear out-put feedback controller having an appropriate compartmentalstructure. More precisely, we propose an output feedback con-trol scheme able to achieve the objective of congestion avoid-ance and to satisfy an inow demand that does not exceeds thetransmission capacity.</p><p>In order to emphasize the relevance of the congestion controlproblem addressed in the present paper, we would like to referto two concrete, although rather different, applications that wehave previously treated:</p><p>1. The plugging phenomenon in grinding circuits is a well-known critical industrial congestion problem. The dynamicsof grinding circuits may typically and efciently be de-scribed by compartmental network systems. The controldesign presented in this paper is an interesting output feed-back alternative to the state feedbak control strategies thathave been discussed in [2,11].</p><p>2. Congestion control in packet switched networks is an issuethat has received a lot of attention in the computer scienceliterature. We have shown in [12,13] that compartmen-tal network systems can constitute a valuable uid owmodelling approach for such networks and can be used</p></li><li><p>690 G. Bastin, V. Guffens / Systems &amp; Control Letters 55 (2006) 689696</p><p>for analysing hop-by-hop congestion control strategies.The congestion control approach followed in this paper isdifferent: it illustrates that compartmental uid ow mod-elling can also be used to address the so-called end-to-endcongestion control problem.</p><p>Compartmental network systems are dened in Section 2.They have numerous interesting structural properties which arewell documented in the literature (see the references). Some ofthese properties which are useful for our purpose are brieyreviewed in Section 2. In particular, the equilibrium stabilityproperties of cooperative compartmental systems are empha-sized. Our contribution is in Section 3, where the proposedcontroller is presented. The main properties of the closed loopsystem and the controller are studied. It is shown that the twomain objectives of the congestion control are achieved: (i) ademand which is not in excess is automatically satised but (ii)in case of an excess demand, an operation without overow isautomatically guaranteed. Furthermore, if the controlled net-work is cooperative, then the closed loop system has a uniqueglobally asymptotically stable equilibrium. A brief simulationexperiment is used in Section 4 to illustrate the validity of thecontrol scheme and some design issues. Some nal commentsare given in Section 5.</p><p>2. Compartmental network systems</p><p>A compartmental network system is a network of con-ceptual storage tanks called compartments as illustrated inFig. 1. Each node of the network represents a compartmentwhich contains a variable quantity xi(t) of some materialor immaterial species involved in the system. The vectorx(t) = (x1(t), x2(t), . . . , xn(t))T is the state vector of the sys-tem. Each directed arc i j represents a mass transfer whichmay hold for various transport, transformation or interactionphenomena between the species inside the system. The trans-fer rate, called ow or ux, from a compartment i to anothercompartment j is a function of the state variables denotedfij (x(t)). Additional input and output arcs represent the in-teractions with the surroundings: either inows bi(t) injectedfrom the outside into some compartments or outows ei(x(t))from some compartments to the outside.</p><p>The instantaneous ow balances around the compartmentsare expressed by the following set of equations:</p><p>xi =j =i</p><p>fji(x) k =i</p><p>fik(x) ei(x) + bi, i = 1, . . . , n.</p><p>(1)</p><p>These equations express that, for each compartment, the rateof accumulation of the quantity xi is just the difference be-tween the inow rates fji , bi and the outow rates fik , ei .In the equations, only the terms corresponding to actual linksof the network are made explicit. Otherwise stated, all thebi , ei and fij for non-existing links do not appear in theequations.</p><p>1</p><p>5</p><p>2</p><p>4</p><p>3</p><p>b5</p><p>f12</p><p>f21</p><p>f54</p><p>f21</p><p>e2</p><p>f23</p><p>f43</p><p>e3</p><p>b1</p><p>Fig. 1. Example of compartmental network.</p><p>The model (1) makes sense only if the state variables xi(t)remain non-negative1 for all t : xi(t) R+. The ow functionsfij and ei are dened to be non-negative on the non-negativeorthant fij : Rn+ R+, ei : Rn+ R+. Similarly, the inowsbi are dened to be non-negative bi(t) R+t . Moreover, itis obvious that there cannot be a positive ow from an emptycompartment</p><p>xi = 0 fij (x) = 0 and ei(x) = 0. (2)Under condition (2), if fij (x) and ei(x) are differentiable, theycan be written as</p><p>fij (x) = rij (x)xi, ei(x) = qi(x)xifor appropriate functions rij (x) and qi(x) which are denedon Rn+, non-negative and at least continuous. These functionsare called specic ows (or also fractional rates). In this paper,we shall assume that the specic ows rij (x) and qi(x) arecontinuously differentiable and strictly positive functions oftheir arguments in the positive orthant:</p><p>rij (x)&gt; 0 and qi(x)&gt; 0 x Rn+.In other words, we assume that the ows fij and ei vanish onlyif xi = 0. It is a natural assumption which is satised in manyphysical and engineering models described by compartmentalmodels.</p><p>With these denitions and notations, the compartmental sys-tem (1) is written</p><p>xi =j =i</p><p>rj i(x)xj k =i</p><p>rik(x)xi qi(x)xi + bi</p><p>i = 1, . . . , n. (3)State-space models of this form are used to represent, forinstance, industrial processes (like distillation columns [25],chemical reactors [17], heat exchangers, grinding circuits [11]),queuing systems [8] and communication networks [12], eco-logical and biological processes [19,26], etc.</p><p>1 Notation. The set of non-negative real numbers is denoted R+ = {a R, a0} as usual. For any integer n, the set Rn+ is called the positiveorthant.</p></li><li><p>G. Bastin, V. Guffens / Systems &amp; Control Letters 55 (2006) 689696 691</p><p>Compartmental network systems have numerous interestingstructural properties which are widely documented in the liter-ature (see the references). Some of these properties are listedhereafter.</p><p>First of all, as expected, a compartmental system is positive.</p><p>Denition 1 (Positive system (e.g. [23])). A dynamical systemx = f (x, t) x Rn is positive ifx(0) Rn+ x(t) Rn+ t0.</p><p>Property 1 (A compartmental network system is a positivesystem). The system (3) is a positive system. Indeed, if x Rn+and xi=0, then xi=j =i rj i(x)xj +bi0. This is sufcient toguarantee the forward invariance of the non-negative orthantif the functions rij (x) and qi(x) are differentiable.</p><p>The total mass contained in the system is</p><p>M(x) =n</p><p>i=1xi .</p><p>A compartmental system is mass conservative in the sensethat the mass balance is preserved inside the system. This iseasily seen if we consider the special case of a closed systemwithout inows and outows.</p><p>Property 2 (Mass conservation). A compartmental networksystem (3) is dissipative with respect to the supply ratew(t)=i bi(t) with the total mass M(x) as storage function. Inthe special case of a closed system without inows (bi = 0,i)and without outows (ei(x) = 0,i), it is easy to check thatdM(x)/dt = 0 which shows that the total mass is indeedconserved.</p><p>The system (3) is written in matrix form asx = A(x)x + b, (4)where A(x) is a so-called compartmental matrix with thefollowing properties:</p><p>1. A(x) is a Metzler matrix, i.e. a matrix with non-negativeoff-diagonal entries</p><p>aij (x) = rji(x)0(note the inversion of indices !).</p><p>2. The diagonal entries of A(x) are non-positive</p><p>aii(x) = qi(x) j =i</p><p>rij (x)0.</p><p>3. The matrix A(x) is diagonally dominant</p><p>|aii |(x)j =i</p><p>aji(x).</p><p>The invertibility and the stability of a compartmental matrixis closely related to the notion of outow connectivity as statedin the following denition.</p><p>Denition 2 (Outow and inow connected network). A com-partment i is said to be outow connected if there is a pathi j k from that compartment to a compart-ment from which there is an outow q(x). The network issaid to be fully outow connected (FOC) if all compartmentsare outow connected.</p><p>A compartment is said to be inow connected if there isa path i j k to that compartment from acompartment i into which there is an inow bi . The network issaid to be fully inow connected (FIC) if all compartments areinow connected.</p><p>Property 3 (Invertibility and stability of the compartmental ma-trix [9,19]). The compartmental matrix A(x) is non-singularand stable x Rn+ if and only if the compartmental networkis FOC. This shows that the non-singularity and the stability ofa compartmental matrix can be directly checked by inspectionof the associated compartmental network.</p><p>The Jacobian matrix of the system (4) is dened as</p><p>J (x) = [A(x)x]x</p><p>.</p><p>When the Jacobian matrix has a compartmental structure, theoff-diagonal entries are non-negative and the system is thereforecooperative [15,16]. We then have the following interestingstability property.</p><p>Property 4 (Equilibrium stability with a compartmental Ja-cobian matrix). Let us consider the system (4) with constantinows: bi = constant i.</p><p>(a) If J (x) is a compartmental matrix x Rn+, then allbounded trajectories tend to an equilibrium in Rn+.</p><p>(b) If there is a compact convex set D Rn+ which is forwardinvariant and if J (x) is a non-singular compartmental ma-trix x D , then there is a unique equilibrium x Dwhich is globally asymptotically stable (GAS) in D.</p><p>A proof of part (a) can be found in [19, Appendix 4], (seealso [10,15]). Part (b) is a concise reformulation of a theoremby Rosenbrock [25] (see also [26]).</p><p>Property 4 requires that the compartmental Jacobian matrixbe invertible in order to have a unique GAS equilibrium. Thiscondition is clearly not satised for a closed system (withoutinows and outows) that necessarily has a singular Jacobianmatrix. However, the uniqueness of the equilibrium is preservedfor closed systems that are strongly connected.</p><p>Property 5 (Equilibrium unicity for a fully connected closedsystem). If a closed system with a compartmental Jacobianmatrix is strongly connected (i.e. there is a directed path i j k connecting any compartment i to anycompartment ), then, for any constant M0 &gt; 0, the hyperplaneH ={x Rn+ : M(x)=M0 &gt; 0} is forward invariant and thereis a unique GAS equilibrium in H.</p></li><li><p>692 G. Bastin, V. Guffens / Systems &amp; Control Letters 55 (2006) 689696</p><p>This property is a straightforward extension of Theorem 6in [24].</p><p>3. Congestion control</p><p>Network congestion arises in compartmental network sys-tems when the inow demand exceeds the throughput capacityof the network. The most undesirable symptom of this kind ofinstability is an unbounded accumulation of material in the sys-tem inducing an overow of the compartments. Our purpose inthis paper is to show that congestion can be automatically pre-vented by using a nonlinear output feedback controller havingan appropriate compartmental structure.</p><p>The congestion control problem is formulated as follows.We consider a compartmental network system with n compart-ments, m inows and p outows, and we assume that</p><p>1. The network is FIC and FOC;2. The links of the network have a maximal transfer capacity:</p><p>0fij (x)f maxij and 0ei(x)emaxi , x Rn+;3. The compartments of the network have a maximal capacity:</p><p>xmaxi , i = 1, . . . , n;4. There is an inow demand denoted di on each input of the</p><p>network: it is the inow rate that the user would like to injectinto the system or, otherwise stated, that the user would liketo assign to the inow rate bi .</p><p>Then, congestion may occur in the system if the total demandexceeds the maximal achievable throughput capacity of the net-work which is limited by the maximal transfer capacity of thelinks. When congestion occurs, some links of the network aresaturated with the highly undesirable consequence of an over-ow of the compartments that supply the congested links.</p><p>In order to allow for congestion control, we assume that,when necessary, the inow rates bi(t) injected into the networkmay be mitigated and made lower than the demand di(t). Thisis expressed as bi(t) = ui(t)di(t), 0ui(t)1 where ui(t)represents the fraction of the inow demand di(t) which isactually injected in the network. We assume furthermore thatthe outow rates ei(x(t))yi(t) are the measurable outputs ofthe system. With these denitions and notations, the model is...</p></li></ul>

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