A Decentralized Routing Control Scheme for Data Communication Networks

Abstract

This paper studies the decentralized routing control of data communication networks. Taking the physical constraints on the data flow and the network-induced delays into consideration, the decentralized routing control problem is reformulated as a constrained control problem, and the restriction of initial message lengths is no longer needed. By defining a general Lyapunov functional and applying a reciprocally convex combination technique to deal with the network-induced delays, less conservative and less complex designing conditions are obtained in terms of linear matrix inequalities (LMIs). A simulation example is also given to illustrate the effectiveness of the proposed designing scheme.

1. Introduction

Routing control plays an indispensable role in complex and crowded networks such as transportation networks and data communication networks [1–19]. The fundamental idea of routing control is that the most suitable route for data packets from source to destination through other nodes is to be determined such that certain objective functions are minimized.

Early routing algorithms, such as those implemented in the ARPANET [2] and TYMNET [3], were designed for optimally achieving some prespecified objective, such as shortest time or shortest path from the initial node to the destination node, minimization of a cost function related to the link congestion [14], or minimization of a generic measure of link flows. However, the performance of the crowded networks with these routing algorithms will deteriorate significantly, since the channel communication capacity is usually limited and the link congestion problems may often occur in many practical networks [20]. Therefore, it is essential to take the physical constraints, such as limited channel capacity and limited energy source, into consideration in designing proper routing control scheme.

It is very known that the network-induced delays (i.e., transmission delay, propagation delay, and processing delay) may induce instability, oscillation, and poor performance, and have a major impact on the routing control design. Several researchers have also attempted to address the problem of delay-constrained routing in recent years [13–19, 21–24]. Different techniques such as capacity allocation [16] and neural networks [19] are adopted to obtain the proper routing paths that minimizing or guaranteeing the boundedness of different types of network delays. However, more general index may be specified in evaluating the network performance, and many physical constraints of networks also should be taken into consideration. Under such a circumstances, the routing control problem becomes a difficult task and cannot be tackled with the above-mentioned methods. As stated in [18, 19], the fluid flow conservation principle is frequently employed in deriving the queuing dynamics, wherein each state of the subsystem (node) represents a queue corresponding to a given destination node. With the aid of linear matrix inequality (LMI) approach, the recent paper [20] provides a promising framework to revisit and reformulate the routing control problems as a constrained 𝐻_∞ optimal control problem for time-delay systems; hence, a number of physical constraints can be easily imposed on the routing problem.

In [20], a continuous-time system model was used to describe the dynamics of traffic networks, and the physical constraints on traffic flows and queues were transformed into a set of LMI-based conditions. Then, robust centralized as well as decentralized routing control strategies were introduced for networks with a fixed topology based on the minimisation of the worst-case queuing length, which is related to the queuing delays. Similar treatment was adopted in [25]. It is worth pointing out that the derivatives of network-induced delays were assumed to be less than 1. However, according to [26], such assumptions are false. Meanwhile, a parameter should be given a priority and the initial values of all message queue lengths are required to be 0 in the transformation of the physical constraints, which may be harsh for many practical applications and may result in some conservatism.

In this paper, we study routing control of data communication networks that arise from many practical applications [19]. A discrete-time model instead of continuous-time model [20] is used to describe the communication network dynamics and the data constraints on the data flow. Furthermore, in order to facilitate the practical implementation for large-scale data communication networks, we are dedicated to design decentralized controllers instead of centralized controllers. Thus, vulnerable failures and large communication overheads in the network may be avoided. To this end, some new techniques will be adopted to tackle the extra difficulty in the related LMI manipulation and derivation, which are caused because of the decentralized controller structure. Different from the existing works, a reciprocally convex combination method is used to deal with delays, the physical constraints are successfully transformed into a set of LMIs and the restriction that initial message queue lengths should be 0 is removed, and each LMI-based condition involved in the designing of decentralized routing control is only related to an oriented link in the data communication network, which can effectively reduce both the conservatism and the complexity.

Notation.Throughout this paper, a real symmetric matrix 𝑃>0 (≥0) denotes 𝑃 being a positive definite (positive semidefinite) matrix, and 𝐴>𝐵(𝐴≥𝐵) means 𝐴−𝐵>0 (≥0). 𝐼 is used to denote an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by ∗. For a real vector 𝑥∈𝑅^𝑛, 𝑥≽0 (≻0) denotes every component of 𝑥 being nonnegative (positive). In this case, 𝑥 is called nonnegative (positive). Likewise, 𝐴∈𝑅^𝑛×𝑚 is nonnegative or positive if every entry of 𝐴 is nonnegative or positive, respectively, which is written as 𝐴≽0 or 𝐴≻0, respectively.

2. Problem Formulation

Consider a data communication network, 𝒢=(𝒩,ℒ), consisting of a directed graph with a set 𝒩 of 𝑁 nodes and a set ℒ of 𝐿-oriented links. Each message has a destination node 𝑑∈𝒩, and all destination nodes form a set 𝒩_𝑑. Each node 𝑖∈𝒩 receives messages from the upstream nodes within the network. Messages are absorbed as soon as they arrive at their destination nodes. Messages arriving to a node other than their final destination are put into a queue (or buffer) and eventually are sent out to a downstream node. It is assumed that the network is “connected,” that is, each node of the network must be reachable from each other node. In the worst case, where all the nodes are source as well as destination, at each node 𝑖∈𝒩 there will be 𝑛−1 queues in which messages are stored for all destinations, 1,2,…,𝑖−1,𝑖+1,…,𝑁. We also assume that the rate of messages that are sent from one node to another is updated (synchronously throughout the communication network) by control variables at discrete instants 𝑘𝑇(𝑘=0,1,2,…), where 𝑇 is a positive scalar.

Similar to [19], the data communication network dynamics can be expressed as 𝑞^𝑑_𝑖(𝑘+1)=𝑞^𝑑_𝑖(𝑘)+_{𝑗∈𝒫(𝑖)𝑗≠𝑑}𝑓^𝑑_𝑗𝑖𝑘−𝜏_𝑗𝑖(𝑘)−_{(𝑖)𝑗∈𝒮}𝑓^𝑑_𝑖𝑗(𝑘),𝑖∈𝒩,𝑑∈𝒩^𝑖_𝑑, (1) where 𝑞^𝑑_𝑖(𝑘): message queue length at node 𝑖 destined to node 𝑑 at instant 𝑘𝑇;𝒫(𝑖): set of upstream neighbors of node 𝑖; 𝒮(𝑖): set of downstream neighbors of node 𝑖; 𝑓^𝑑_𝑖𝑗(𝑘): traffic flow routed from node 𝑖 to node 𝑗 destined to node 𝑑 at instant 𝑘𝑇; 𝒩^𝑖_𝑑: subset of 𝒩_𝑑, that is, 𝒩^𝑖_𝑑=𝒩_𝑑⧵{𝑖}; 𝜏_𝑖𝑗(𝑘): time-varying delay in transmitting, propagating, and processing of messages (including identifying the destination, inserting in the queue, and routing computation) with destination 𝑑 routed from node 𝑖 to node 𝑗, which satisfies 0≤𝜏_𝑖𝑗(𝑘)≤𝜏_𝑖𝑗,∀𝑖,𝑗∈ℒ, (2) where 𝜏_𝑖𝑗 are constants.

Due to the physical significance of 𝑓^𝑑_𝑖𝑗 and the constraints on the data flow through each link and the message queue buffer size at each node, the following conditions should be satisfied: 𝑓^𝑑_𝑖𝑗(𝑘)≥0,𝑖∈𝒩,𝑗∈𝒮(𝑖),𝑑∈𝒩^𝑖_𝑑,0≤_{𝑑∈𝒩𝑖𝑑}𝑞^𝑑_𝑖(𝑘)≤𝑞_𝑖,𝑖∈𝒩,_{𝑑∈𝒩𝑖𝑑}𝑓^𝑑_𝑖𝑗(𝑘)≤𝑐_𝑖𝑗,𝑖∈𝒩,𝑗∈𝒮(𝑖), (3)(4)(5) where positive scalars 𝑞_𝑖 and 𝑐_𝑖𝑗 represent the maximal length of the message queue length at node 𝑖 and the maximal capacity of oriented link (𝑖,𝑗) (it is also called “edge”), respectively.

Define 𝑥_𝑖(𝑘)=vec𝑞^𝑑_𝑖(𝑘),𝑢_𝑖(𝑘)=vec𝑓^𝑑_𝑖𝑗(𝑘,(𝑖))𝑖∈𝒩,𝑗∈𝒮,𝑑∈𝒩^𝑖_𝑑, (6) then (1) can be rewritten as 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)+𝐵_𝑖𝑢_𝑖(𝑘)+_{(𝑖)𝑗∈𝒫}𝐵_𝑑𝑗𝑖𝑢_𝑗𝑘−𝜏_𝑗𝑖(𝑘), (7) where 𝑥_𝑖(𝑘)∈𝑅^𝑁𝑖𝑑 and 𝑢_𝑖(𝑘)∈𝑅^{𝑚𝑖𝑁𝑖𝑑} are vectors; 𝑚_𝑖,𝑁_𝑑, and 𝑁^𝑖_𝑑 represent the numbers of nodes contained in 𝒮(𝑖),𝒩_𝑑, and 𝒩^𝑖_𝑑, respectively; 𝐵_𝑖 and 𝐵_𝑑𝑗𝑖 are constant matrices. Obviously, the entries of 𝐵_𝑖 are equal to 0 or −1, and the ones of 𝐵_𝑑𝑗𝑖 are equal to 0 or 1, which reflect the network connectivity of node 𝑖 in graph 𝒢 [20].

In this paper, a decentralized routing control scheme 𝑢_𝑖(𝑘)=𝐾_𝑖𝑥_𝑖(𝑘),𝑖∈𝒩, (8) is proposed to stabilize the origin system (1), where state feedback controllers 𝐾_𝑖(𝑖∈𝒩) are to be designed. Such a constrained control problem can be found by solving a set of linear matrix inequalities (LMIs). Different from the one given in [20], the obtained routing control scheme is at edge-level; that is, each inequality is only dependent on an edge (𝑖,𝑗) in ℒ. This implies that the dimension of each inequality is low, so the computational complexity is reduced effectively. In addition, since some constraints are discarded, the conservatism of the obtained LMI-based conditions is also reduced.

Through this paper, Jensen's integral inequality [27] and a reciprocally convex combination technique [28] will be used, so it is listed as the following lemmas.

Lemma 1 (see [27]). For any positive symmetric constant matrix 𝑀∈𝑅^𝑛×𝑛, scalars 𝑟₁,𝑟₂ satisfying 𝑟₁<𝑟₂, a vector function 𝜔∶[𝑟₁,𝑟₂]→𝑅^𝑛 such that the integrations concerned are well defined, then ^𝑟2_𝑟1𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝑟2_𝑟1𝜔(𝑠)≤𝑟𝑑𝑠₂−𝑟₁^𝑟2_𝑟1𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (9)

Lemma 2 (see [29]). For given positive semidefinite matrices 𝑍₁ and 𝑍₂, if there exists a matrix 𝑇 such that 𝑍₁𝑇∗𝑍₂≥0, (10) then the following inequality for any 𝛼(0<𝛼<1) holds 1𝛼𝑍₁0∗1𝑍1−𝛼²≥𝑍₁𝑇∗𝑍₂. (11)

3. Main Results

3.1. Stability Analysis

Now, we give a new stability criterion for the origin of system (7) with (2) as follows.

Theorem 3. For given scalars 𝜏_𝑖𝑗>0 and 𝐾_𝑖(𝑖∈𝒩,𝑗∈𝒮(𝑖)), the origin of system (7) with (2) is globally asymptotically stable if there exist matrices 𝑃_𝑖𝑗=𝑃^𝑇_𝑖𝑗>0, 𝑄_𝑖𝑗=𝑄^𝑇_𝑖𝑗≥0, 𝑅_𝑖𝑗=𝑅^𝑇_𝑖𝑗≥0, 𝑍_𝑖𝑗=𝑍^𝑇_𝑖𝑗>0, and 𝑈_𝑖𝑗=𝑈^𝑇_𝑖𝑗>0  (∀𝑖∈𝒩,𝑗∈𝒮(𝑖)) with appropriate dimensions, such that 𝑍_𝑖𝑗𝑇_𝑖𝑗∗𝑍_𝑖𝑗Φ≥0,_𝑖𝑗Θ_𝑖𝑗∗−𝑈_𝑖𝑗<0, (12)(13) where 𝑖=1,2,…,𝑛 and 𝑗∈𝒮(𝑖), and Φ_𝑖𝑗=Φ¹¹_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗1𝜏_𝑖𝑗𝑇_𝑖𝑗Φ¹⁴_𝑖𝑗∗Φ²²_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗0∗∗−𝑅_𝑖𝑗−1𝜏_𝑖𝑗𝑍_𝑖𝑗0∗∗∗Φ⁴⁴_𝑖𝑗,Φ¹¹_𝑖𝑗=𝜏_𝑖𝑗𝑄_𝑖𝑗+𝑅_𝑖𝑗−1𝜏_𝑖𝑗𝑍_𝑖𝑗−1𝑚_𝑖𝑀_𝑖1𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑀^𝑇_𝑖1−𝑀_{𝑖1(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖𝑀^𝑇_𝑖1,Φ¹⁴_𝑖𝑗=𝑃_𝑖𝑗+1𝑚_𝑖𝑀_𝑖1−𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑀^𝑇_𝑖2+𝑀_{𝑖1(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖𝑀^𝑇_𝑖2,Φ²²_𝑖𝑗=−𝑄_𝑖𝑗−1𝜏_𝑖𝑗2Z_𝑖𝑗−𝑇_𝑖𝑗−𝑇^𝑇_𝑖𝑗,Φ⁴⁴_𝑖𝑗=𝑃_𝑖𝑗+𝜏_𝑖𝑗𝑍_𝑖𝑗+1𝑚_𝑖𝑀_𝑖2+𝑀^𝑇_𝑖2+𝑀_{𝑖2(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖𝑀^𝑇_𝑖2,Θ_𝑖𝑗=0𝐵_𝑑𝑖𝑗𝐾_𝑖00^𝑇. (14)

Proof. The proof is provided in the Appendix.

Remark 4. In Theorem 3, a sufficient condition of globally asymptotical stability for the origin of system (7) with (2) is given in terms of LMIs. Different from the one in [20], positive definite matrices 𝑃_𝑖𝑗,𝑄_𝑖𝑗,𝑅_𝑖𝑗, and 𝑍_𝑖𝑗 correspond to the oriented link (𝑖,𝑗), so the Lyapunov functional (A.1) is more general. Meanwhile, each inequality in (13) is also related to an oriented link (𝑖,𝑗) in ℒ, so its dimension is lower. Combined with using a reciprocally convex technique [28], the computational complexity and the conservatism are reduced.

3.2. A Decentralized Routing Control Scheme

3.2.1. Controller Design without the Physical Constraints

From (13), one can find that 𝑀_𝑖2+𝑀^𝑇_𝑖2(𝑖=1,2,…,𝑛) are negative definite, this implies that 𝑀_𝑖2 are nonsingular. Set 𝑀_𝑖1=𝜀_𝑖𝑀_𝑖2,𝑀_𝑖2=𝑀⁻¹_𝑖2,𝑃_𝑖𝑗=𝑀_𝑖2𝑃_𝑖𝑗𝑀^𝑇_𝑖2,𝑄_𝑖𝑗=𝑀_𝑖2𝑄_𝑖𝑗𝑀^𝑇_𝑖2,𝑅_𝑖𝑗=𝑀_𝑖2𝑅_𝑖𝑗𝑀^𝑇_𝑖2,𝑍_𝑖𝑗=𝑀_𝑖2𝑍_𝑖𝑗𝑀^𝑇_𝑖2,𝑇_𝑖𝑗=𝑀_𝑖2𝑇_𝑖𝑗𝑀^𝑇_𝑖2,𝐹_𝑖=𝐾_𝑖𝑀^𝑇_𝑖2, (15) pre- and postmultiplying the left side of inequality (13) by diag{𝑀_𝑖2,𝑀_𝑖2,𝑀_𝑖2,𝑀_𝑖2,𝐼} and its transpose, and from the Schur complement, we get the following.

Theorem 5. For given scalars 𝜏_𝑖𝑗>0 and 𝜀_𝑖(𝑖∈𝒩,𝑗∈𝒮(𝑖)), the origin of system (7) with (2) is globally asymptotically stable if there exist matrices 𝑃_𝑖𝑗=𝑃^𝑇_𝑖𝑗>0, 𝑄_𝑖𝑗=𝑄^𝑇_𝑖𝑗≥0, 𝑅_𝑖𝑗=𝑅^𝑇_𝑖𝑗≥0, and Z_𝑖𝑗=𝑍^𝑇_𝑖𝑗>0, 𝐹_𝑖 and 𝑀_𝑖2(forall𝑖∈𝒩,𝑗∈𝒮(𝑖)) with appropriate dimensions, such that 𝑍_𝑖𝑗𝑇_𝑖𝑗∗𝑍_𝑖𝑗Ω≥0,_𝑖𝑗Γ_𝑖𝑗∗−𝑈_𝑖𝑗<0, (16)(17) where 𝑖∈𝒩 and 𝑗∈𝒮(𝑖), and Ω_𝑖𝑗=Ω¹¹_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗1𝜏_𝑖𝑗𝑇_𝑖𝑗Ω¹⁴_𝑖𝑗∗Ω²²_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗0∗∗−𝑅_𝑖𝑗−1𝜏_𝑖𝑗𝑍_𝑖𝑗0∗∗∗Ω⁴⁴_𝑖𝑗,Γ_𝑖𝑗=0𝐵_𝑑𝑖𝑗𝐹_𝑖00^𝑇,Ω¹¹_𝑖𝑗=𝜏_𝑖𝑗𝑄_𝑖𝑗+𝑅_𝑖𝑗−1𝜏_𝑖𝑗𝑍_𝑖𝑗−1𝑚_𝑖𝜀_𝑖𝐵_𝑖𝐹_𝑖+𝐹^𝑇_𝑖𝐵^𝑇_𝑖−𝜀_{𝑖(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖,Ω¹⁴_𝑖𝑗=𝑃_𝑖𝑗+1𝑚_𝑖𝑀^𝑇_𝑖2−𝐹^𝑇_𝑖𝐵^𝑇_𝑖+_{𝑙∈𝒫(𝑖)}𝑈_𝑙𝑖,Ω²²_𝑖𝑗=−𝑄_𝑖𝑗−1𝜏_𝑖𝑗2𝑍_𝑖𝑗−𝑇_𝑖𝑗−𝑇^𝑇_𝑖𝑗,Ω⁴⁴_𝑖𝑗=𝑃_𝑖𝑗+𝜏_𝑖𝑗𝑍_𝑖𝑗+1𝑚_𝑖𝑀^𝑇_𝑖2+𝑀_𝑖2+_{𝑙∈𝒫(𝑖)}𝑈_𝑙𝑖. (18) The gains 𝐾_𝑖(𝑖∈𝒩) of the subsystem controllers in (8) are given by 𝐾_𝑖=𝐹_𝑖𝑀^−𝑇_𝑖2. (19)

3.2.2. LMI-Based Conditions for the Physical Constraints

Now, we translate the physical constraints (3)–(5) into LMI-based conditions.

(1) Nonnegativity Constraint on Data Traffic Flow. From (1), it is easy to see that the nonnegativity of 𝑥_𝑖(𝑘) and 𝑓^𝑑_𝑖𝑗(𝑘) can be guaranteed if 𝐼+𝐵_𝑖𝐾_𝑖 and 𝐾_𝑖 are nonnegative for all 𝑖∈𝒩,𝑗∈𝒮(𝑖), and 𝑑∈𝒩^𝑖_𝑑.

Note that the negative definiteness of 𝑀_𝑖2+𝑀^𝑇_𝑖2, and from (17), the nonnegativity of 𝐼+𝐵_𝑖𝐾_𝑖 and 𝐾_𝑖 can be guaranteed if 𝑀_𝑖2 are diagonal for all 𝑖∈𝒩 and 𝑀_𝑖2+𝐵_𝑖𝐹_𝑖𝐹⪯0,_𝑖⪯0. (20) (2) Upper Bound on the Buffer Size. Note that _{𝑑∈𝒩𝑖𝑑}𝑞^𝑑_𝑖(𝑘)=𝐻_𝑖𝑥_𝑖(𝑘), (21) where 𝐻_𝑖=1⋯1_𝑁𝑖𝑑. (22) So, the constraint (4) can be represented as 0≤𝐻_𝑖𝑥_𝑖(𝑘)≤𝑞_𝑖. (23)

From Ω_𝑖𝑗<0, it follows that Ω⁴⁴_𝑖𝑗<0. So, there exist a small enough 𝜆>0 such that 𝑃_𝑖𝑗+1𝑚_𝑖𝑀^𝑇_𝑖2+𝑀_𝑖2+𝜆𝐼<0. (24) Since −1𝜆𝑀_𝑖2𝑀^𝑇_𝑖2≤𝑀^𝑇_𝑖2+𝑀_𝑖2+𝜆𝐼, (25) and 𝑃_𝑖𝑗=𝑀_𝑖2𝑃_𝑖𝑗𝑀^𝑇_𝑖2, inequality (24) implies _{𝑗∈𝒮(𝑖)}𝑃_𝑖𝑗<1𝜆𝐼. (26) If (16) and (17) in Theorem 5 are feasible, under the assumption 𝑥_𝑖(𝜏)=0for all 𝜏∈{−1,−2,…,−𝜏_𝑖𝑗}, from 𝑥^𝑇_𝑖(𝑘)_{(𝑖)𝑗∈𝒮}𝑃_𝑖𝑗𝑥_𝑖(𝑘)≤^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)_{𝑗∈𝒮(𝑖)}𝑃_𝑖𝑗𝑥_𝑖(𝑘)(𝑘),𝑉(0)≤1≤𝑉𝜆^𝑁_𝑖=1𝑥^𝑇_𝑖(0)𝑥_𝑖(0)≤1𝜆^𝑁_𝑖=1𝑥^𝑇_𝑖(0)𝐻^𝑇_𝑖𝐻_𝑖𝑥_𝑖(0)≤1𝜆^𝑁_𝑖=1𝑞²_𝑖, (27) we get 𝑥^𝑇_𝑖(𝑘)_{𝑗∈𝒮(𝑖)}𝑃_𝑖𝑗𝑥_𝑖(𝑘)≤1𝜆^𝑁_𝑖=1𝑞²_𝑖, (28) where 𝑉(𝑘) is defined in (A.1).

Thus, (23) can be guaranteed if 1𝑚_𝑖𝑞²_𝑖𝐻^𝑇_𝑖𝐻_𝑖≤𝜆∑^𝑁_𝑖=1𝑞²_𝑖𝑃_𝑖𝑗,∀𝑗∈𝒮(𝑖). (29) From the Schur complement, the above inequality is equivalent to −𝑃_𝑖𝑗𝑀_𝑖2𝐻^𝑇_𝑖𝑚∗−^𝑖𝑞²_𝑖∑^𝑁_𝑖=1𝑞²_𝑖𝜆<0. (30) So, from the above analysis, it is known that (23) can be guaranteed if (24) and (30) hold for any (𝑖,𝑗)∈ℒ.

(3) Capacity Constraints on Links. For 𝑗∈𝒮(𝑖), it is known that there exists a nonnegative integer 𝑖_𝑗 such that the numbers of 𝑖_𝑗 nodes contained in 𝒮(𝑖) are less than 𝑗 and the ones of 𝑚_𝑖−𝑖_𝑗−1 nodes contained in 𝒮(𝑖) are greater than 𝑗. Thus, We get _{𝑑∈𝒩𝑖𝑑}𝑓^𝑑_𝑖𝑗(𝑘)=𝐺_𝑖𝑗𝑢_𝑖(𝑘),∀𝑖,𝑗∈ℒ, (31) where 𝐺_𝑖𝑗=0⋯0_{𝑖𝑗×𝑁𝑖𝑑}1⋯1_𝑁𝑖𝑑0⋯0_{(𝑚𝑖−𝑖𝑗−1)×𝑁𝑖𝑑}. (32) so the constraint (5) can be represented as 0≤𝐺_𝑖𝑗𝑢_𝑖(𝑘)≤𝑐_𝑖𝑗. (33)

Similarly, constraint (33) can be satisfied if (24) and −𝑃_𝑖𝑗𝐹^𝑇_𝑖𝐺^𝑇_𝑖𝑗𝑐∗−^2𝑖𝑗∑^𝑁_𝑖=1𝑞²_𝑖𝜆<0. (34)

Therefore, considering the physical constraints (3)–(5), a decentralized routing control scheme can be get as follows.

Theorem 6. For given scalars 𝜏_𝑖𝑗>0 and 𝜀_𝑖(𝑖∈𝒩,𝑗∈𝒮(𝑖)), the origin of system (7) with (2) is globally asymptotically stable if there exist a scalar 𝜆>0, matrices 𝑃_𝑖𝑗=𝑃^𝑇_𝑖𝑗>0, 𝑄_𝑖𝑗=𝑄^𝑇_𝑖𝑗≥0, 𝑅_𝑖𝑗=𝑅^𝑇_𝑖𝑗≥0, and 𝑍_𝑖𝑗=𝑍^𝑇_𝑖𝑗>0, 𝐹_𝑖 and diagonal matrices 𝑀_𝑖2(forall𝑖∈𝒩,𝑗∈𝒮(𝑖)) with appropriate dimensions, such that (16), (17), (20), (24), (30), and (34) hold.

Correspondingly, the gains 𝐾_𝑖 of the subsystem controllers in (8) are given by (19).

Remark 7. In [20, 25], a scalar 𝜆 was needed to be selected before and the restriction 𝑉(0)=0 in transforming the physical constraints into LMI-based conditions. Contrastively, 𝜆 involved in inequalities (30) and (34) is a variable and it can be found by solving these LMIs. More importantly, the restriction 𝑉(0)=0 is not needed here.

Remark 8. Different from [20, 25], positive definite matrix 𝑃 in Theorem 6 is not required to be diagonal, which discards the constraint on 𝑃.

4. Numerical Example

Consider a data communication network shown in Figure 1 [20, 30]. The capacity of each link is also indicated in the figure, the unit of which is kbit/sec. All the nodes are assumed to be sources as well as destinations. From Figure 1, it follows that there are 6 queues (the states of the system) and 8 output flows for these queues (the input signals). The queue dynamic of the communication network can be described as follows 𝑥₁(𝑘+1)=𝑥₁(𝑘)+𝐵₁𝑢₁(𝑘)+𝐵_𝑑31𝑢₃𝑘−𝜏₃₁(𝑘),𝑥₂(𝑘+1)=𝑥₂(𝑘)+𝐵₂𝑢₂(𝑘)+𝐵_𝑑12𝑢₁𝑘−𝜏₁₂(𝑘)+𝐵_𝑑32𝑢₃𝑘−𝜏₃₂(𝑘,𝑥₃)(𝑘+1)=𝑥₃(𝑘)+𝐵₃𝑢₃(𝑘)+𝐵_𝑑23𝑢₂𝑘−𝜏₂₃(𝑘), (35) where 𝑥₁(𝑘)=𝑞²₁(𝑘)𝑞³₁(𝑘),𝑥₂(𝑘)=𝑞¹₂(𝑘)𝑞³₂(𝑘),𝑥₃(𝑘)=𝑞¹₃(𝑘)𝑞²₃(𝑘),𝑢₁(𝑘)=𝑓²₁₂(𝑘)𝑓³₁₂(𝑘),𝑢₂(𝑘)=𝑓¹₂₃(𝑘)𝑓³₂₃(𝑘),𝑢₃(𝑘)=𝑓¹₃₁(𝑘)𝑓²₃₁(𝑘)𝑓¹₃₂(𝑘)𝑓²₃₂(𝑘),𝐵₁=𝐵₂=−100−1,𝐵_𝑑31=,𝐵01000000_𝑑12=0001,𝐵_𝑑32=,𝐵00100000₃=−10−100−10−1,𝐵_𝑑23=.1000 (36)

Figure 1 

Topology of a communication network.

Assume that the maximal buffer sizes, 𝑞_𝑖(𝑖∈𝒩), are given by 𝑞₁=10, 𝑞₂=15, and 𝑞₃=20. For the case of 𝜏_𝑖𝑗=5 for all (𝑖,𝑗)∈ℒ, by Theorem 6 with 𝜀_𝑖=4, one can get 𝐾₁=0.20350.00010.00010.1971,𝐾₂=,𝐾0.1934000.2037₃=.0.2219000.17430.1388000.1736 (37)

Now, we provide a simulation.

Suppose that the initial states of 𝑥_𝑖(𝑘) are given by 𝑥₁(0)=[6,3]^𝑇, 𝑥₂(0)=[5,8]^𝑇 and 𝑥₃(0)=[9,9]^𝑇. With the obtained controller gains in (37), the state responses of the system are depicted in Figures 2, 3, and 4, which shows that the system is asymptotically stable. Figures 5, 6, 7, and 8 show the data flows between nodes in the communication network. From Figures 5–8, one can see that the physical constraints are satisfied although initial state 𝑥(0) is not equal to 0. So, the proposed decentralized routing control scheme is effective.

Figure 2 

State response of 𝑥1(𝑘).

Figure 3 

State response of 𝑥2(𝑘).

Figure 4 

State response of 𝑥3(𝑘).

Figure 5 

Data flow from node 1 to node 2.

Figure 6 

Data flow from node 2 to node 3.

Figure 7 

Data flow from node 3 to node 1.

Figure 8 

Data flow from node 3 to node 2.

5. Conclusion

In this paper, the routing control problem of data communication networks has been investigated. By considering both the network-induced delays and the physical constraints, a decentralized routing control scheme is proposed, which can be realized by solving a set of LMIs. Since a method is used to relax the transformation conditions of the physical constraints, and a reciprocally convex combination technique is employed to dealing with the network-induced delays, a less complex and less conservative designing result is obtained. A simulation example is given to illustrate the effectiveness of the presented designing scheme.

Appendix

Proof of Theorem 3. Choose a Lyapunov functional as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘), (A.1) where 𝑉₁(𝑘)=^𝑛_{𝑖=1𝑗∈𝒫(𝑖)}𝑥^𝑇_𝑗(𝑘)𝑃_𝑗𝑖𝑥_𝑗(𝑘),𝑉₂(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒫}^𝑘−1_{𝑙=𝑘−𝜏𝑗𝑖(𝑘)}𝑥^𝑇_j(𝑙)𝑄_𝑗𝑖𝑥_𝑗(𝑙)+⁰_{𝑙=−𝜏𝑗𝑖+1}^𝑘−1_{𝑟=𝑘+𝑙}𝑥^𝑇_𝑗(𝑟)𝑄_𝑗𝑖𝑥_𝑗(𝑟,𝑉₃)(𝑘)=^𝑛_{𝑖=1𝑗∈𝒫(𝑖)}^𝑘−1_{𝑙=𝑘−𝜏𝑗𝑖}𝑥^𝑇_𝑗(𝑙)𝑅_𝑗𝑖𝑥_𝑗(𝑙),𝑉₄(𝑘)=^𝑛_{𝑖=1𝑗∈𝒫(𝑖)}⁻¹_{𝑙=−𝜏𝑗𝑖}^𝑘−1_{𝑟=𝑘+𝑙}𝜂^𝑇_𝑗(𝑟)𝑍_𝑗𝑖𝜂_𝑗(𝑟), (A.2) and 𝑃_𝑖𝑗=𝑃^𝑇_𝑖𝑗>0, 𝑄_𝑖𝑗=𝑄^𝑇_𝑖𝑗≥0, 𝑅_𝑖𝑗=𝑅^𝑇_𝑖𝑗≥0, 𝑍_𝑖𝑗=𝑍^𝑇_𝑖𝑗>0(forall∈𝒩,𝑗∈𝒮(𝑖)) are to be determined.
Note that ^𝑛_{𝑖=1𝑗∈𝒫(𝑖)}𝑥^𝑇_𝑗(𝑘)𝑃_𝑗𝑖𝑥_𝑗(𝑘)=^𝑛_{𝑖=1𝑗∈𝒮(𝑖)}𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑗𝑥_𝑖(𝑘), (A.3) We get Δ𝑉₁(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}𝑥^𝑇_𝑖(𝑘+1)𝑃_𝑖𝑗𝑥_𝑖(𝑘+1)−𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑗𝑥^𝑇_𝑖(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}𝜂^𝑇_𝑖(𝑘)𝑃_𝑖𝑗𝜂_𝑖(𝑘)+𝜂^𝑇_𝑖(𝑘)𝑃_𝑖𝑗𝑥_𝑖(𝑘)+𝑥^𝑇_𝑖(𝑘)𝑃_𝑖𝑗𝜂^𝑇_𝑖(𝑘). (A.4) Similarly, taking the differences of 𝑉_𝑖(𝑘)(𝑖=2,3,4) along the trajectory of (7) yields that Δ𝑉₂(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒫}^𝑘_{𝑙=𝑘+1−𝜏𝑗𝑖()𝑘+1}𝑥^𝑇_𝑗(𝑙)𝑄_𝑗𝑖𝑥_𝑗(𝑙)−^𝑘−1_{𝑙=𝑘−𝜏𝑗𝑖(𝑘)}𝑥^𝑇_𝑗(𝑙)𝑄_𝑗𝑖𝑥_𝑗(𝑙)+⁰_{𝑙=−𝜏𝑗𝑖+1}𝑥^𝑇_𝑗(𝑘)𝑄_𝑗𝑖𝑥_𝑗(𝑘)−𝑥^𝑇_𝑗(𝑘+𝑙)𝑄_𝑗𝑖𝑥_𝑗()≤𝑘+𝑙_{𝑛𝑖=1(𝑖)𝑗∈𝒫𝑘𝑙=𝑘+1−𝜏𝑗𝑖}𝑥^𝑇_𝑗(𝑙)𝑄_𝑗𝑖𝑥_𝑗(𝑙)−𝑥^𝑇_𝑗𝑘−𝜏_𝑖𝑗(𝑘)𝑄_𝑗𝑖𝑥_𝑗𝑘−𝜏_𝑖𝑗(𝑘)+𝜏_𝑗𝑖𝑥^𝑇_𝑗(𝑘)𝑄_𝑗𝑖𝑥_𝑗(𝑘)−^𝑘_{𝑙=𝑘−𝜏𝑗𝑖+1}𝑥^𝑇_𝑗(𝑙)𝑄_𝑗𝑖𝑥_𝑗(𝑙=_{𝑛𝑖=1(𝑖)𝑗∈𝒫})𝜏_𝑗𝑖𝑥^𝑇_𝑗(𝑘)𝑄_𝑗𝑖𝑥_𝑗(𝑘)−𝑥^𝑇_𝑗𝑘−𝜏_𝑗𝑖(𝑘)𝑄_𝑗𝑖𝑥_𝑗𝑘−𝜏_𝑗𝑖(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}𝜏_𝑖𝑗𝑥^𝑇_𝑖(𝑘)𝑄_𝑖𝑗𝑥_𝑖(𝑘)−𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗(𝑘)𝑄_𝑖𝑗𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘),Δ𝑉₃(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒫}𝑥^𝑇_𝑗(𝑘)𝑅_𝑗𝑖𝑥_𝑗(𝑘)−𝑥^𝑇_𝑗𝑘−𝜏_𝑗𝑖𝑅_𝑗𝑖𝑥_𝑗𝑘−𝜏_𝑗𝑖=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}𝑥^𝑇_𝑖(𝑘)𝑅_𝑖𝑗𝑥_𝑖(𝑘)−𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗𝑅_𝑖𝑗𝑥_𝑖𝑘−𝜏_𝑖𝑗,Δ𝑉₄(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒫}𝜏_𝑗𝑖𝜂^𝑇_𝑗(𝑘)𝑍_𝑗𝑖𝜂_𝑗(𝑘)−^𝑘−1_{𝑙=𝑘−𝜏𝑗𝑖}𝜂^𝑇_𝑗(𝑙)𝑍_𝑗𝑖𝜂_𝑗(𝑙)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}𝜏_𝑖𝑗𝜂^𝑇_𝑖(𝑘)𝑍_𝑖𝑗𝜂_𝑖(𝑘)−^𝑘−1_{𝑙=𝑘−𝜏𝑖𝑗}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙). (A.5) Since −^𝑘−1_{𝑙=𝑘−𝜏𝑖𝑗}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙)=−^𝑘−1_{𝑙=𝑘−𝜏𝑖𝑗(𝑘)}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙)−^{𝑘−𝜏𝑖𝑗(𝑘)−1}_{𝑙=𝑘−𝜏𝑖𝑗}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙), (A.6) and from Lemma 1, we get −^𝑘−1_{𝑙=𝑘−𝜏𝑖𝑗(𝑘)}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙)1≤−𝜏_𝑖𝑗(𝑘)𝑥^𝑇_𝑖(𝑘)−𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗(𝑘)×𝑍_𝑖𝑗𝑥_𝑖(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘),−^{𝑘−𝜏𝑖𝑗(𝑘)−1}_{𝑙=𝑘−𝜏𝑖𝑗}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙)1≤−𝜏_𝑖𝑗−𝜏_𝑖𝑗(𝑘)𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗(𝑘)−𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗𝑍_𝑖𝑗×𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗. (A.7) Thus, from Lemma 2, it yields that −^𝑘−1_{𝑙=𝑘−𝜏𝑖𝑗}𝜂^𝑇_𝑖(𝑙)𝑍_𝑖𝑗𝜂_𝑖(𝑙)1≤−𝜏_𝑖𝑗𝑥_𝑖(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘𝑥_𝑖)𝑘−𝜏_𝑖𝑗(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗^𝑇𝑍_𝑖𝑗𝑇_𝑖𝑗∗𝑍_𝑖𝑗×𝑥_𝑖(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘)𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘)−𝑥_𝑖𝑘−𝜏_𝑖𝑗. (A.8)
On the other hand, for any matrices 𝑀_𝑖1,𝑀_𝑖2, and 𝑈_𝑖𝑗=𝑈^𝑇_𝑖𝑗>0 with appropriate dimensions, it follows that 0=2^𝑛_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2×𝜂_𝑖(𝑘)−𝐵_𝑖𝐾_𝑖𝑥_𝑖(𝑘)−_{(𝑖)𝑗∈𝒫}𝐵_𝑑𝑗𝑖𝐾_𝑗𝑥_𝑗𝑘−𝜏_𝑗𝑖(𝑘)≤2^𝑛_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2𝜂_𝑖(𝑘)−𝐵_𝑖𝐾_𝑖𝑥_𝑖(𝑘+_{𝑛𝑖=1(𝑖)𝑗∈𝒫})𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2×𝑈_𝑗𝑖𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2^𝑇+𝑥^𝑇_𝑗𝑘−𝜏_𝑗𝑖(𝑘)𝐾^𝑇_𝑗𝐵^𝑇_𝑑𝑗𝑖×𝑈⁻¹_𝑗𝑖𝐵_𝑑𝑗𝑖𝐾_𝑗𝑥_𝑗𝑘−𝜏_𝑗𝑖(𝑘)=2^𝑛_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2𝜂_𝑖(𝑘)−𝐵_𝑖𝐾_𝑖𝑥_𝑖(𝑘+_𝑛𝑖=1)𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2×_{(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2^𝑇+_{(𝑖)𝑙∈𝒫}𝑥^𝑇_𝑗𝑘−𝜏_𝑗𝑖(𝑘)×𝐾^𝑇_𝑗𝐵^𝑇_𝑑𝑗𝑖𝑈⁻¹_𝑗𝑖𝐵_𝑑𝑗i𝐾_𝑗𝑥_𝑗𝑘−𝜏_𝑗𝑖(𝑘)=^𝑛_{𝑖=1(𝑖)𝑗∈𝒮}1𝑚_𝑖×2𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2×𝜂_𝑖(𝑘)−𝐵_𝑖𝐾_𝑖𝑥_𝑖(𝑘+𝑥_𝑇𝑖)(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_{𝑖2(𝑖)𝑙∈𝒫}𝑈_𝑙𝑖×𝑥^𝑇_𝑖(𝑘)𝑀_𝑖1+𝜂^𝑇_𝑖(𝑘)𝑀_𝑖2^𝑇+𝑥^𝑇_𝑖𝑘−𝜏_𝑖𝑗(𝑘)𝐾^𝑇_𝑖𝐵^𝑇_𝑑𝑖𝑗𝑈⁻¹_𝑖𝑗×𝐵_𝑑𝑖𝑗𝐾_𝑖𝑥_𝑖𝑘−𝜏_𝑖𝑗(𝑘). (A.9) So, combining (A.4)–(A.9), we get that Δ𝑉(𝑘)≤^𝑛_{𝑖=1𝑗∈𝒮(𝑖)}𝜁^𝑇_𝑖𝑗(𝑘)Ψ_𝑖𝑗𝜁_𝑖𝑗(𝑘), (A.10) where 𝜁_𝑖𝑗(𝑘)=𝑥^𝑇_𝑖(𝑘)𝑥^𝑇_𝑖(𝑘−𝜏_𝑖𝑗(𝑘))𝑥^𝑇_𝑖(𝑘−𝜏_𝑖𝑗)𝜂^𝑇_𝑖(𝑘)^𝑇,Ψ_𝑖𝑗=Φ¹¹_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗1𝜏_𝑖𝑗𝑇_𝑖𝑗Φ¹⁴_𝑖𝑗∗Ψ²²_𝑖𝑗1𝜏_𝑖𝑗𝑍_𝑖𝑗−𝑇_𝑖𝑗0∗∗−𝑅_𝑖𝑗−1𝜏_𝑖𝑗𝑍_𝑖𝑗0∗∗∗Φ⁴⁴_𝑖𝑗,Ψ²²_𝑖𝑗=Φ²²_𝑖𝑗+𝐾^𝑇_𝑖𝐵^𝑇_𝑑𝑖𝑗𝑈⁻¹_𝑖𝑗𝐵_𝑑𝑖𝑗𝐾_𝑖. (A.11) From the Schur complement, it is known that the globally asymptotical stability of system (7) can be guaranteed if (12) and (13) hold for any 𝑖∈𝒩 and 𝑗∈𝒮(𝑖).
Thus, the proof is completed.

Acknowledgments

This work was supported by the National High Technology Research and Development Programme of China (863 Programme) under Grant 2009AA012201, the National Natural Science Foundation of China under Grant 61174085, the Training Excellent Young Teachers in Shanghai Universities (B37-0109-10-008), the Shanghai University Innovation Foundation (Grant no. A.10-0109-10-014), and the Industry-University-Research Project of Shanghai.