Competitive Routing in Networks With Polynomial Costs

Transcription

1 92 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 47, O., JAUARY 2002 [27], Attitude control of underactuated spacecraft, Euro. J. Control, vol. 6, no. 3, pp , [28] H. Sussmann, Lie brackets, real analyticity and geometric control, in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. Sussmann, Eds. Boston, MA: Birkhauser, 983, pp. 6. [29] G. C. Walsh, R. Montgomery, and S. S. Sastry, Optimal path planning on matrix Lie groups, in Proc. 33rd Conf. Decision Control, vol. 2, Lake Buena Vista, FL, Dec. 994, pp Competitive Routing in etworks With Polynomial Costs Eitan Altman, Tamer Başar, Tania Jiménez, and ahum Shimkin Abstract We study a class of noncooperative general topology networks shared by users. Each user has a given flow which it has to ship from a source to a destination. We consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling, and show that these costs have appealing properties that lead to predictable and efficient network flows. In particular, we show that the ash equilibrium is unique, and is moreover efficient. These properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks. We finally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links. Index Terms etworks, noncooperative equilibria, nonzero-sum games, routing. I. ITRODUCTIO We consider in this note a routing problem in networks where noncooperating agents, to whom we refer to as players or users, wish to establish paths from agent-specific sources to agent-specific destinations so as to transport a fixed amount of total traffic (per agent). In the context of telecommunication networks, players correspond to different traffic sources which have to route their traffic over a shared network. A similar setting has also been studied in the context of road traffic networks [], where a player can be viewed as a transportation company which is to ship a flow of vehicles. A natural framework within which this class of problems can be analyzed is that of noncooperative game theory, and an appropriate solution concept is that of ash equilibrium (E) [4]: a composite routing policy for the users constitutes a E if no user can gain by unilaterally deviating from his own policy. There exists a rich literature [8], [4], [7], [9] on the analysis of equilibria in networks, particularly in the context of the Wardrop equilibrium [23], which pertains to the case of infinitesimal users (e.g., a single car in the context of road traffic). In such a framework, the impact Manuscript received October 9, 999; revised May 7, 2000 and July 20, Recommended by Associate Editor L. Dai. The work of E. Altman was supported in part by IRIA/SF Collaborative Research Grant. The work of T. Başar was supported in part by Grants SF IT , SF AI , MURI AF-DC , and an ARO/EPRI MURI Grant. E. Altman is with the IRIA, Sophia Antipolis Cedex, France ( altman@sophia.inria.fr). T. Başar is with the Coordinated Science Laboratory, University of Illinois, Urbana, IL USA. T. Jiménez is with the C.E.S.I.M.O., Universidad de los Andes, Mérida, Venezuela, and IRIA, 06902, Sophia Antipolis Cedex, France.. Shimkin is with the Electrical Engineering Department, Technion Israel Institute of Technology, Haifa 32000, Israel. Publisher Item Identifier S (02) of a single user on the congestion experienced by other users is negligible. Our focus here, however, is competitive routing, where the network is shared by several users, with each one having a nonnegligible amount of flow. This concept has attracted considerable attention in recent years, as reflected in the works [], [], [2], [5], [8], which will be our starting point here. These papers have presented conditions for the existence and uniqueness of an equilibrium. This has allowed, in particular, the design of network management policies that induce efficient equilibria [5]. This framework has also been extended to the context of repeated games in [6], in which cooperation can be enforced by using policies that penalize users who deviate from the equilibrium. A desired property for an equilibrium is efficiency, i.e., social optimality. It is well known that ash equilibria in routing games are generally nonefficient. In particular, this nonefficiency can lead to paradoxical phenomena (e.g., the Braess s paradox [4]) where adding ink to the network could result in an increase of cost to all users. For specific examples of nonefficient behavior and paradoxes, we cite [8] for road-traffic, [7], [5] for telecommunications, and [2], [3] for distributed computing. In this note, we study the equilibria that arise in networks of general-topology under some polynomial cost functions introduced originally in the context of road traffic by the US Bureau of Public Roads, and we obtain conditions for the uniqueness of the equilibrium. The uniqueness result is important, since to date there is not much known on uniqueness in the case of general topologies; only for a few special cases the uniqueness has been established, see [], [2], [8]. We further present in the note conditions under which the E is efficient. The fact that under a class of nonlinear costs the equilibrium is unique and efficient for general-topology networks may be especially useful for pricing purposes; indeed, if the network operator chooses prices that correspond to a unique and efficient equilibrium, it can enforce a globally optimal use of the network. In the context of communications, an example of an architecture where our routing framework (and the specific results obtained) would be of particular use is MultiProtocol Label Switching (MPLS) [2], [3]. One of the most important traffic engineering problems in MPLS, as identified in [3], is how to partition traffic into traffic trunks, or equivalently, how to distribute the arriving traffic over a given set of links. This problem has recently been studied in the framework of a single criterion that is optimized globally [0], but our framework here is more suitable for MPLS, since in practice routing decisions are made (and implemented) locally by each source, and typically in a noncooperative manner. In the next section, we introduce the general model considered in this note. In Section III, we establish conditions for the uniqueness of the E. In Section IV, we establish the existence of an efficient E under appropriate conditions. We then focus, in Section V, on the computation of the equilibrium for the special case of affine costs and parallel links. The note ends with the concluding remarks of Section VI. II. THE MODEL The topology is given by a directed graph fv; L; fg where V is the set of nodes, L is the set of directed arcs, and f =(f l ;l 2 L) where f l :IR! IR is the cost function of link l, which gives the cost per unit traffic as a function of the total load lt on that link. We take it as f l ( lt )= ( lt ) p(l) + ; l 2 L () where ; ;p(l) are link-specific positive parameters. The special structure where p(l) does not depend on l has been widely used in road traffic studies, such as [6], [20]. It is the cost adopted by the US Bureau /02$ IEEE

2 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 47, O., JAUARY of Public Roads [22], which we thus call the BPR cost. The additive term here could be interpreted as an additional fixed toll per packet (or per unit traffic) for the use of link l. In the model we adopt here, there are users, where user i 2 := f;...;g has a fixed amount 3 i of flow to ship from a source s(i) to a destination d(i). Each user i has to determine how to distribute its flow over the available links (connecting s(i) to d(i)). Introduce: Thus i l := the rate of traffic user i sends on link l; 0i lt := the total rate of traffic sent on link l; excluding the traffic from source i: lt = j= j l ; 0i lt := lt 0 i l: We visualize this problem as a noncooperative network, i.e., a network in which the rate at which traffic is sent is determined by selfish players, each having its own utility (or cost). The cost for player i is given by J i i l; lt ;l2 L = J i l i l; lt (2) where J i l ( i l; lt )= i lf l ( lt ). It should be noted that, with f l as given by (), J i is a convex function of the flow vector of user i, which is a property we shall henceforth make frequent use of. Also, by a slight abuse of notation, we will occasionally write J i ( i l; lt ;l 2 L) as J i () or J i ( i ; 0i ), where i := f i l; l 2 Lg; 0i := f j ; j 6= i; j 2g, and := f j ; j 2g. The flows of each user i have to satisfy the feasibility conditions: i l 0, and 8v 2 V where r i (v)+ r i (v) = l2in(v) l2out(v) 3 i if v = s(i); 03 i if v = d(i); 0 otherwise i l (3) and In(v) (respectively, Out(v)) is the set of links which are input (respectively, output) to node v. The solution concept adopted is E, i.e., we seek a feasible multipolicy = f i ;i 2gsuch that J i () =J i ( i ; 0i )=minj i ( ~ i ; 0i ); ~ i 2 where the minimum is taken over all policies ~ i which lead to a feasible multipolicy together with 0i. For the game model just introduced, we know from [8] that a E indeed exists. III. UIQUEESS OF THE E Having settled the question of existence of a E, we now establish its uniqueness under appropriate conditions. Let us first introduce the quantity p 3 := (3 0 )=( 0 ) and note that p 3 > 3 for all 2. Theorem : For the general topology network with cost (2), where 0 <p(l) <p 3, the E is unique. Proof: Introduce the gradient column vector g l ( l )= r J l l ; lt ;...; r J l l ; lt T where r i denotes the gradient operator with respect to i l, and l := f i l; i = ;...;g, which we will also view as an -dimensional row vector. Then, it follows from [2, Th. 2], in view of [8, Cor. 3.] (which translates Rosen s diagonal strict concavity condition to a similar condition applied to individual links), that the E is unique if for any set of vectors l and ~ l ;l2 L ( l 6= ~ l ; 8l 2 L, in the vector sense), satisfying the flow constraints, we have ( l 0 ~ l ) T (g l ( l ) 0 g l ( ~ l )) > 0: We will show that in fact, for any l for which l 6= ~ l, every term in the summation above is positive, i.e., (4) ( l 0 ~ l ) T (g l ( l ) 0 g l ( ~ l )) > 0: (5) We first consider the case p(l) =, for which the expression in (5) is quadratic in ( l 0 ~ l ). Then, all we need to show is that the Jacobian, G l ( l ),ofg l ( l ) with respect to l is positive definite. Simple calculation yields G l ( l J i l i l; l i;j =( T + I) where denotes the -dimensional vector with entries all, and I is the identity matrix. G l is clearly positive definite, and hence the positivity condition (5) holds. Therefore, the E is unique (for this special case of p(l) =). ext, we consider the case of a general p(l) 2 (0;p 3 );p(l) 6=. The Jacobian of g l ( l ) in this case is: G l ( l )= p(l)( lt ) p(l)02 (G l + lt I) where G l = q l T ;ql i := lt +(p(l) 0 )l, i and q l is an -dimensional vector whose ith entry is ql i. It follows from Theorem 6 of [2], brought to the link level in view of Corollary 3. of [8], that a sufficient condition for (5) is for the symmetric matrix G l ( l )+G l ( l ) T to be positive definite for all nonnegative l. A careful look at the proof of [2, Th. 6] will actually reveal that it will be sufficient to require positive definiteness for all but isolated values of l, and in particular for all l such that the total flow over that link is positive (that is, at least one of the components of l is positive). In accordance with this, it will be sufficient (for uniqueness) to show that for lt > 0, the eigenvalues of (G l + lt I)+(G l + lt I) T are all positive, or equivalently 4 l := q l T +ql T > 02 lt I, which is what we do next. ote that being the sum of the outer product of two vectors and its transpose, the symmetric matrix 4 l can have rank at most two, and its rank is precisely two if q l is not a multiple of (that is, i l s are not all equal). For the case when the rank is one, the only nonzero eigenvalue of 4 l is its trace, which is clearly positive (and hence larger than 0 lt ). In view of this, we henceforth assume (for the remainder of the proof) that q l and are linearly independent. We now first show that the two nonzero eigenvalues of 4 l are z + r 0 2 lt and z 0 r 0 2 lt, where z := T q l +2 lt =( ++p(l)) lt ; r := q T l q l: ote that for > 0 and p(l) =, the expression for G ( ) below is consistent with the one obtained earlier for this special case.

3 94 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 47, O., JAUARY 2002 Since 4 l is symmetric, the eigenvectors corresponding to its two nonzero eigenvalues have to be linear combinations of its columns, and therefore have the form q l +, where and are some constants. Hence, if x is an eigenvalue, we have the equation q l T +q T l (q l + ) =x(q l + ) for some and. Since, q l and are linearly independent, their coefficients in the equation above should separately vanish, yielding the pair of equations [( T q l ) 0 x] + =0; q T l q l + q T l 0 x =0: In order for a nontrivial solution to exist for and, the determinant of their coefficient matrix in the above set of equations has to be zero, i.e., ( T q l 0 x) 2 0 (q T l q l )=0. This is a quadratic equation in terms of the unknown eigenvalues, whose solutions are z + r 0 2 lt and z 0 r 0 2 lt. To complete the proof of the theorem, we now finally have to show that x and x 2 are both greater than 02 lt. Since x >x 2, it suffices to show that x 2 > 02 lt, or equivalently, z 0 r>0. Toward this end, note that r 2 = i= lt +(p(l) 0 ) i l =( 0 )( lt ) 2 +( lt ) 2 +(p(l) 0 ) 2 j= j l ( 0 )( lt ) 2 +( lt ) ( lt ) 2 (p(l) 0 ) +(p(l) 0 ) 2 ( lt ) 2 +2( lt ) 2 (p(l) 0 ) =( 0 +p(l) 2 )( lt ) 2 : (6) It then follows from (6) that z0r >0 if z 2 >( 0+p(l) 2 )( lt ) 2, or equivalently if ( ++p(l)) 2 >( 0 +p(l) 2 ), which can be rewritten as 0 < +3 +2( +)p(l)+(0 )p(l) 2 = 0(p(l) + )(( 0 )p(l) 0 (3 + )): For any p(l) 2 (0;p 3 ), the above indeed holds, thus proving that z 0 r> 0. Hence, the sufficient condition (5) holds for p(l) 2 (0;p 3 ), and thereby the E is indeed unique. 5 IV. EFFICIECY OF THE E We consider in this section the special case of a homogeneous link cost for a general topology network. That is, f l is now given by f l ( lt )= ( lt ) p ;p> 0; > 0; where the power p is a constant. As mentioned earlier, this is the BPR cost which was studied earlier in [9] (see also [5]). It was shown in these references that under this cost the Wardrop equilibrium and the globally optimal solution coincide. Our result here complements this earlier one. We show that when all users have the same source and the same destination, the E is globally optimal (with respect to a single cost function) and that the corresponding link flows of different users are proportional to their total input traffic. The global optimization problem is the minimization, over all total link flows f lt ;l 2 Lg =: t and subject to the feasibility constraint (3), the total additive cost: J( t )= J l ( lt ) (7) where J l ( lt )= lt f l ( lt ). ote that the total cost (7) depends only on the total flows over each link and not on the flows of the individual users, and hence to make the feasibility constraints (3) compatible with this, we can take instead of (3) its sum over all users i, that is r(v) + l2in(v) lt = l2out(v) lt (8) where we have also made use of the assumption that r i (v) is independent of i (and, hence, is denoted by r(v)). Therefore the global optimization problem is min J( t ) such that (8) holds: (9) Since J( t) is strictly convex over inear constraint set, this minimization problem admits a unique solution, which we denote by t f lt ;l 2 Lg. ow, to bring this solution down to the level of an individual user, we distribute the total flow, lt, on ink l among the users in proportion with their total flows, that is ( i l)=( j l ) = (3i )=(3 j ). This leads to i lt ; i 2; l 2 L; i := 3 i j= 3 j : (0) It should be noted that the flows (0) satisfy the flow feasibility constraints (3) whenever t satisfies (8). We now show that (0) is, in fact, a E (and the unique one, by Theorem ), and hence the E is efficient. Theorem 2: Let t be the globally optimal solution, and i t = i lt, as in (0). This also constitutes the unique E, with the corresponding cost for user i being J i ( i ; t)= ij( t). Proof: Let us first note that a flow -tuple ^ is in E if and only if it is feasible and the following holds for all i, for any feasible i (see []) ^ i l 0 i l 0: () On the other hand, t is globally optimal if and only if it is feasible (i.e., it satisfies the traffic constraints (3), or equivalently (8) in this case) and the following holds for any feasible l ( lt lt ( lt 0 lt ) 0: (2) For the cost adopted, the partial derivative in () is Letting ^ i lt in (3) above, we have ^ i l 0 i l = p(^ lt ) p0^ i l +(^ lt ) p : (3) = i (p i +) ( lt ) p ( lt 0 ~ l;i ) (4)

4 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 47, O., JAUARY where ~ l;i := i l= i. ote that in the global optimization l ( lt ) lt 0 lt )=(p +) ( lt ) p ( lt 0 lt ) 0 lt where the inequality follows from (2). By combining this with (4), we conclude that () holds with ^ =, and this establishes the proof.5 V. COMPUTATIO OF THE E: AFFIE COSTS In this section, we consider the case of a network consisting of M parallel links and with affine cost function for all links: f l ( lt ) = lt + : This is a special case of the BPR cost for p(l) =.A possible interpretation for this type of cost in the context of telecommunication systems is that it is the expected delay of a packet in ight traffic regime. For example, the expected value of the delay D l in link l for an M=G= queue is given by E[D l ]= ( l ) 2 2( 0 lt l ) lt + l : Under ight traffic regime, i.e., with lt l, this can be approximated by lt + l, where =( l ) 2 =2. It follows from standard nonlinear (convex) programming theory that the following Kuhn Tucker conditions are both necessary and sufficient for a solution = f i lg l;i to constitute a E K i l () = lt + + i if i l > 0; 8i; l (5) Kl i () = lt + i if 0; 8i; l (6) M 3 i ; 8i; i l 0 8i; l: (7) l= Since (5) (7) constitute a system of linear equalities and inequalities, they can be solved by standard LP methods. We present below an alternative approach that yields explicitly the equilibrium when the bias terms or their variability are relatively small. Lemma : i) If at equilibrium each user sends a positive amount of flow on each link, the unique E is = j2l (=a j) 3 i + + j2l b j 0 a j ; 8l; i: (8) ii) At equilibrium, each user sends a positive amount of flow on each link if and only if where R(i) := min i R(i) > max =( +) (9) l j2l (=a j) 3 i + + j2l b j a j : Proof: We will initially assume that at equilibrium all users send positive flows on all links. This will allow us to obtain an equilibrium which is consistent with this assumption, which in turn will be unique by the result of Section III. ow, by summing (5) over all users ( lt + )+ lt =( + )( lt )+ = i : i= Thus, lt + = ( i i + )=( +). Substituting in the Kuhn Tucker condition (5), we obtain i l + + = i 0 + j2 j : (20) Dividing by and summing over all the links, we get Hence 3 i + + = i 0 + j2 j i 0 + j2 j = (=) : 3 i + + = R(i): (2) Then, from (20), R(i) 0 =( +). Thus, if mini R(i) > max l =( + ), the equilibrium is given by ([R(i) 0 =( + )]= ) > 0, which is compatible with the initial hypothesis on positivity. Corollary : If all s are equal, the unique E is k2l a k 3 i ; i 2: We next consider some further properties of the E. Lemma 2: i) Consider a E which dictates a particular user to send all its traffic over a single link. If there is some other user who sends all its traffic also over a single link, then these two links are the same. ii) All users send their traffic over a particular link l if and only if 3+ + maxi 3 i min n6=l b n, where 3:= i2 3 i. Proof: i) Consider a E where player i uses only link l and player j uses only link m, with l 6= m. From (5), we get: i = lt i, and j = a m mt + b m + a m 3 j. From (6), we get: j a 0 l lt +, and i a m mt + b m. Combining the last two relationships, we obtain the following contradiction: lt i = i a m mt + b m = j 0 a m3 j lt + 0 a m3 j : ii) By (5) (6), all users send their traffic to link l if and only if i = i, and i b m; 8l 6= m, which then leads to the desired result. 5 Lemma 3: Consider the case of two links. Let I be a nonempty set of players who at equilibrium send all their traffic on link 2. Then the equilibrium flows for the remaining players i=2iare given by, for l; k =; 2; k 6= l, and with 3(I) := i2i 3 i a k a + a2 3 i + b k 0 + a23(i) + a2 : (22) Proof: Let l (I c ):= i=2i i l;b2 I := b2 +a23(i). Each player i=2ifaces the problem of minimizing the cost J i () = i (at + b) + i 2(a22t + b2) = i (at(i C )+b) + i 2 a22t(i C )+b2 I : This is equivalent to the original game where users from I do not participate, and the corresponding value of b2 is changed to b I 2. It follows from Lemma 2 that each user in this new game sends positive flow over both links at equilibrium. Using Lemma then completes the proof. 5 By comparing (8) and (22), we finally arrive at: Corollary 2: Consider a two-link game with b >b2. Let ^I be the set of users for which j in (8) is negative. Then I^I, where I is as defined in Lemma 3.

5 96 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 47, O., JAUARY 2002 VI. COCLUSIO We have studied the problem of static competitive routing to parallel queues with polynomial link holding costs. We have established the uniqueness of the E for a general topology with the BPR cost [22] and have obtained a simple relationship with the globally optimal solution. We have further obtained some explicit results for the special case of affine link costs. The results of this note should prove useful for the analysis of networks with source-determined routing, when the link cost functions can be approximated by polynomial (and in particular affine) costs of the type considered here. The fact that the E was shown to be efficient under a class of nonlinear costs can be used as a starting point for designing pricing mechanisms so as to obtain a socially optimal use of the network. [2] J. B. Rosen, Existence and uniqueness of equilibrium points for concave -person games, Econometrica, vol. 33, no. 3, pp , 965. [22] Traffic Assignment Manual, 964. US Bureau of Public Roads. [23] J. G. Wardrop, Some theoretical aspects of road traffic research, in Proc. Inst. Civ. Eng., vol., 952, pp A ew Approach to State Observation of onlinear Systems With Delayed Output A. Germani, C. Manes, and P. Pepe REFERECES [] E. Altman and H. Kameda, Uniqueness of Solutions for Optimal Static Routing in Multi-Class Open etworks, Inst. Inform. Sci. Electr., Univ. Tsukuba, Japan, Report ISE-TR [2] G. Armitage, MPLS: The magic behind the myths, IEEE Trans. Commun., vol. 38, pp. 24 3, Jan [3] D. Awduche, J. Malcolm, J. Agogbua, M. O Dell, and J. McManus, Internet RFC 2702, Requirements for traffic engineering over MPLS, Sept [4] T. Başar and G. J. Olsder, Dynamic oncooperative Game Theory. Philadelphia, PA: SIAM, 999. SIAM Series in Classics in Applied Mathematics. [5] L. D. Bennett, The existence of equivalent mathematical programs for certain mixed equilibrium traffic assignment problems, Euro. J. Operat. Res., vol. 7, pp , 993. [6] D. E. Boyce, B.. Janson, and R. W. Eash, The effect on equilibrium trip assignment of different congestion functions, Transport. Research A, vol. 5, pp , 98. [7] J. E. Cohen and C. Jeffries, Congestion resulting from increased capacity in single-server queueing networks, IEEE/ACM Trans. etworking, pp , Apr [8] S. Dafermos and A. agurney, On some traffic equilibrium theory paradoxes, Transport. Res. B, vol. 8, pp. 0 0, 984. [9] S. Dafermos and F. T. Sparrow, The traffic assignment problem for a general network, J. Res. ational Bureau Standards-B. Math.. Sci., vol. 73B, no. 2, pp. 9 8, 969. [0] E. Dinan, D. Awduche, and B. Jabbari, Optimal traffic partitioning in MPLS networks, presented at the Proc. etworking 2000 Conf., Paris, France, May IFIP-TC6/European Commission. [] A. Haurie and P. Marcotte, On the relationship between ash-cournot and Wardrop equilibria, etworks, vol. 5, pp , 985. [2] H. Kameda, E. Altman, and T. Kozawa, Braess-like paradoxes of ash equilibria for load balancing in distributed computer systems, Proc. 38th IEEE Conf. Decision Control, Dec [3] H. Kameda, T. Kozawa, and J. Li, Anomalous relations among various performance objectives in distributed computer systems, in Proc. World Congress Syst. Simulation, Sept. 997, pp [4] H. Kameda and Y. Zhang, Uniqueness of the solution for optimal static routing in open BCMP queueing networks, Math. Comput. Modeling, vol. 22, no. 0 2, pp. 9 30, 995. [5] Y. A. Korilis, A. A. Lazar, and A. Orda, Architecting noncooperative networks, IEEE J. Selected Areas Commun., vol. 3, pp , July 995. [6] R. J. La and V. Anantharam, Optimal routing control: Game theoretic approach, Proc. 36th IEEE Conf. Decision Control, Dec [7] P. Marcotte and S. guyen, Eds., Equilibrium and Advanced Transportation Modeling. Boston, MA: Kluwer, 998. [8] A. Orda, R. Rom, and. Shimkin, Competitive routing in multiuser communication networks, IEEE/ACM Trans. etworking, vol., pp , 993. [9] M. Patriksson, The Traffic Assignment Problem: Models and Methods, The etherlands, 994. [20] G. Rose, M. A. P. Taylor, and P. Tisato, Estimating travel time functions for urban roads: Options and issues, Transport. Planning Technol., vol. 4, pp , 989. Abstract This note presents a new approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a non negligible time delay. The proposed observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants (chain observer). Conditions are given ensuring global exponential convergence to zero of the observation error for any given the delay in the measurements. The implementation of the observer is simple and computer simulations demonstrate its effectiveness. Index Terms Delay systems, nonlinear systems, state observation, state prediction. I. ITRODUCTIO In many engineering applications a process to be controlled, or simply monitored, is located far from the computing unit and the measured data are transmitted through ow-rate communication system (e.g., in aerospace applications). In the above cases the measured outputs are available for computations after a non negligible time delay. In some applications (e.g., in biochemical reactors) the measurement process intrinsically provides an out-of-date output. In both cases the reconstruction of the present system state using past measurements may be significant. This is a classical state prediction problem. An important engineering application of state prediction occurs when the control variable can be applied to the system with a non negligible delay after its computation. In this case it is clear that a state feedback control law can be used only if computed on the predicted state. In the case of linear systems, such a control problem is solved by the so-called Smith Predictor [8], which is not exactly a predictor: it is a predictive model-based control scheme requiring state-prediction. Many other algorithms for predictive control of systems with input delay have been proposed in the literature (see e.g., [3], [5], [7], and [9]), and all of them include a state predictor. However, in such schemes little attention is devoted to the predictor implementation, often realized in open-loop, under the assumption of stability of the process. In [7], different implementations of the Manuscript received July 0, 2000; revised ovember 3, 2000, March 9, 200, and July 7, 200. Recommended by Associate Editor K. Gu. This work was supported by MURST (Ministry for University and Scientific and Technological Research) and by ASI (Italian Aerospace Agency). A. Germani and C. Manes are with Dipartimento di Ingegneria Elettrica, Università degli Studi dell Aquila, L Aquila, Italy, and also with Istituto di Analisi ed Informatica del CR (IASI-CR), 0085 Roma, Italy ( germani@ing.univaq.it; manes@ing.univaq.it). P. Pepe is with Dipartimento di Ingegneria Elettrica, Università degli Studi dell Aquila, L Aquila, Italy ( pepe@ing.univaq.it). Publisher Item Identifier S (02) /02$ IEEE