GTS-based communication task scheduling for quantized output consensus over IEEE 802.15.4 wireless networks

Automatica 55 (2015) 6–11

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Brief paper

GTS-based communication task scheduling for quantized output consensus over IEEE 802.15.4 wireless networks✩ Naoki Hayashi 1 , Shigemasa Takai Division of Electrical, Electronic and Information Engineering, Osaka University, Suita, Osaka 565-0871, Japan

article

info

Article history: Received 31 May 2013 Received in revised form 20 January 2015 Accepted 29 January 2015 Available online 15 March 2015 Keywords: Cooperative control Quantized output consensus IEEE 802.15.4 wireless networks

abstract This paper presents an output consensus problem for low-rate wireless networks under the IEEE 802.15.4 protocol. We consider a multi-agent system with quantized output consensus dynamics in which each agent updates its state according to quantized relative differences of outputs among its neighbor agents. Information exchanges of quantized relative outputs are done by a PAN coordinator through a static and undirected wireless network. We propose a multiple superframe structure and a non-preemptive GTS scheduling algorithm for an output consensus under the IEEE 802.15.4 protocol. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Recently, cooperative control of multi-agent systems has attracted much attention in diverse areas of control engineering. This paper considers a consensus problem in which each agent tries to agree on a certain variable of interest using only local information exchanges with its neighbor agents (Jadbabaie, Jie, & Morse, 2003; Olfati-Saber & Murray, 2004). Most of the existing consensus problems assume that agents can send real-valued data. However, in wireless sensor networks, data should be quantized according to a network data rate due to limited link capacity. In the last few years, the IEEE 802.15.4 protocol has been widely used for Low-Rate Wireless Personal Area Networks (LRWPANs) (Tiberi, Fischione, Johansson, & Di Benedetto, 2010). The IEEE 802.15.4 protocol provides a beacon-enabled mode in which a Personal Area Network coordinator (PAN coordinator) periodically broadcasts a beacon signal using a superframe structure and reserves network bandwidth for a specific node by allocating Guaranteed Time Slots (GTSs). Tiberi et al. considered self-triggered control for stability analysis over IEEE 802.15.4 networks with

✩ This research is supported in part by the JSPS Grant-in-Aid for Young Scientists (B) No. 25820180 and Mitutoyo Association for Science and Technology. The material in this paper was partially presented at the 12th European Control Conference, July 17–19, 2013, Zürich, Switzerland. This paper was recommended for publication in revised form by Associate Editor Denis Dochain under the direction of Editor Frank Allgöwer. E-mail addresses: [email protected] (N. Hayashi), [email protected] (S. Takai). 1 Tel.: +81 6 6879 7695; fax: +81 6 6879 7293.

http://dx.doi.org/10.1016/j.automatica.2015.02.028 0005-1098/© 2015 Elsevier Ltd. All rights reserved.

parameter uncertainties and external disturbances (Tiberi et al., 2010). Araújo et al. proposed energy-efficient self-triggered control over IEEE 802.15.4 wireless networks with a GTS scheduling algorithm (Araújo, Mazo, Anta, Tabuada, & Johansson, 2014). This paper proposes an output consensus problem over IEEE 802.15.4 networks. Each agent periodically updates its state by discrete-time consensus dynamics with a monotonically increasing and class C 1 output function. Due to limited network bandwidth, each agent quantizes relative differences of outputs. We assume that local information exchanges can be represented by a static and undirected graph. A PAN coordinator schedules GTSs to guarantee that all local information exchanges are completed within their predetermined deadlines. In the IEEE 802.15.4 standard, GTSs in each superframe are limited to at most 7 slots. To overcome such limitation, we consider a multiple superframe structure for the IEEE 802.15.4 protocol which consists of several superframes in each period of consensus dynamics. As a related work, Guo and Dimarogonas (2013a,b) proposed a state consensus problem with a quantized data transmission and an output consensus problem with a certain class of a nonlinear function, respectively. In this paper, we consider the effect of data quantization and non-linear output consensus dynamics at the same time with a GTS scheduling for a multiple superframe structure. The multiple superframe structure is proposed in Tiberi, Fischione, Johansson, and Di Benedetto (2013) for self-triggered scheme of networked control systems. This paper focuses on a GTS scheduling with periodic consensus dynamics to allow more GTSs for local communications. The rest of the paper is organized as follows: Section 2 summarizes the standard IEEE 802.15.4 protocol. Section 3 states an

N. Hayashi, S. Takai / Automatica 55 (2015) 6–11

7

3. Multi-agent systems 3.1. Graph theory

Fig. 1. Local communication graph (a) and the corresponding star topology with a PAN coordinator (b).

A graph G = (V , E ) consists of a finite and nonempty node set V = {vi | i ∈ I } and an edge set E, where I = {1, 2, . . . , n} (Godsil & Royle, 2001). Each node vi ∈ V represents each agent i and each (undirected) edge {vi , vj } ∈ E indicates that agents i and j exchange their data with each other. A path is a sequence of edges {vi0 , vi1 }, {vi1 , vi2 }, . . . , {vim −1 , vim }, where {vis−1 , vis } ∈ E for s = 1, 2, . . . , m. Two vertices vi and vj are called connected if there is a path from vi to vj . A graph G is said to be connected if every pair of vertices in G is connected. 3.2. Consensus dynamics

Fig. 2. Superframe structure of the IEEE 802.15.4 protocol.

output consensus using quantized relative differences of outputs. In Section 4, we propose a GTS scheduling algorithm for a quantized output consensus with a multiple superframe structure. In Section 5, we present a numerical example. Finally, we conclude this paper in Section 6. 2. IEEE 802.15.4 wireless networks In cooperative control of multi-agent systems, local communications among agents are usually considered as a mesh topology at a design stage as shown in Fig. 1(a). However, in many practical cases, the network is implemented as a star topology with a central node which manages the entire network to avoid packet collisions as shown in Fig. 1(b). In IEEE 802.15.4 networks, the central node is called a Personal Area Network coordinator (PAN coordinator). For example, a local communication from agent 1 to agent 2 (agent 1 → agent 2) is actually implemented by a two-stage communication through a PAN coordinator (PANC): agent 1 → PANC and PANC → agent 2. Hence, this paper considers a star topology in which all local communications pass through a PAN coordinator (Tiberi et al., 2010). The IEEE 802.15.4 MAC protocol in a star topology provides a beacon-enabled mode. In the beacon-enabled mode, a PAN coordinator periodically broadcasts a beacon signal to all nodes in the network and defines a superframe structure as shown in Fig. 2. The length of the superframe is called a beacon interval (BI). The beacon interval is divided into an active portion and an inactive portion. In the inactive portion, nodes are in a sleep mode and none of the data transmissions are carried out. The length of the active portion is called a superframe duration (SD). In the IEEE 802.15.4 standard, BI and SD are defined as follows: BI = aBaseSuperframeDuration × 2BO , SD = aBaseSuperframeDuration × 2SO , where BO and SO are a beacon order and a superframe order, respectively (0 ≤ SO ≤ BO ≤ 14). The SD is divided into 16 equally-sized base slots, that is, aBaseSlotDuration = SD/16. The active portion consists of a Contention Access Period (CAP) and a Contention Free Period (CFP). In the CAP, data transmissions are done by a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) mechanism. On the other hand, in the CFP, a PAN coordinator allocates Guaranteed Time Slots (GTSs) to a designated node. In the IEEE 802.15.4 standard, GTSs in each superframe are limited to at most 7 slots.

Let R+ and Z+ be the nonnegative real number set and the nonnegative integer set, respectively. We consider a multi-agent system in which each agent has a state ξi [k] = [ξi1 [k] ξi2 [k] · · · ξim [k]]⊤ ∈ Rm and an output ζi [k] = [ζi1 [k] ζi2 [k] · · · ζim [k]]⊤ ∈ Rm with the following discrete-time dynamics (ℓ ∈ L = {1, 2, . . . , m}):

ξiℓ [k + 1] = ξiℓ [k] + T

n  j=1,j̸=i

pijℓ q(ζjℓ [k] − ζiℓ [k]),

(1)

ζiℓ [k] = f (ξiℓ [k]), (2) where f : R → R is an output function, q : R → R is a quantizer, and T is a sampling period. pijℓ takes a positive real value if agent i receives the quantized relative differences of outputs q(ζjℓ [k] − ζiℓ [k]) from agent j (̸= i), and 0 otherwise (i, j ∈ I , ℓ ∈ L ). We assume that piiℓ = 0 for all i ∈ I and ℓ ∈ L . We also assume that ξiℓ [k] ∈ (ξ min , ξ max ) for all i ∈ I , ℓ ∈ L , and k ∈ Z+ . The consensus dynamics (1) and (2) are based on quantized relative differences of outputs (Guo & Dimarogonas, 2013a). For example, in robotic systems, each agent (robot) measures the relative differences of positions (the relative differences of outputs) ζjℓ −ζiℓ by distance sensors. Then the agent transmits the quantized relative differences q(ζjℓ − ζiℓ ) to its neighbor agents. The approach based on quantized relative differences has several advantages over the method where each agent transmits a quantized value of an output. For example, in general, the amplitude of the relative differences ζjℓ − ζiℓ is smaller than that of the output ζiℓ . Thus the quantized relative differences can be implemented by fewer bits than that of the quantized output (Li, Fu, Xie, & Zhang, 2011). In this paper, we consider the following assumptions: Assumption 1. The information exchanges among agents are represented by a static and undirected graph. Assumption 2. The output function f : R → R is a monotonically increasing and class C 1 function in (ξ min , ξ max ) such that 0 < ηsup < ∞, where

ηsup :=

sup ξ min ≤ξ ≤ξ max

f ′ (ξ ).

Assumption 3. For any x ∈ R, the quantizer q : R → R satisfies the following:

• q(−x) = −q(x). • There exists a positive constant σ ∈ R+ \ {0} such that xq(x) ≥ σ q2 (x). 3.3. Quantized output consensus In this paper, we define an output consensus with the discretetime consensus dynamics (1) and (2) as follows:

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N. Hayashi, S. Takai / Automatica 55 (2015) 6–11

Definition 4. A group of agents achieves an output consensus if (3) holds for any i, j ∈ I , ℓ ∈ L and any initial state.

4. Non-preemptive GTS scheduling for quantized output consensus

  q(ζi [k] − ζj [k]) → 0 as k → ∞. ℓ ℓ

This section presents a non-preemptive GTS scheduling algorithm over IEEE 802.15.4 wireless networks. We propose a branchand-bound GTS scheduling algorithm for a multiple superframe structure.

(3)

In the standard state consensus problems, the sum of the initial states of agents is preserved at each step if the information exchanges are represented by an undirected graph (Xiao & Boyd, 2004). We have the similar result for the proposed output consensus problem as shown in the following lemma. Lemma 5. Under Assumptions 1–3, for all ℓ ∈ L and k ∈ Z+ , we have n 

ξiℓ [k] =

i =1

n 

ξiℓ [0] = µℓ ,

i =1

where µℓ ∈ R is a constant. Proof. From Eq. (1), for all ℓ ∈ L , we have n 

ξiℓ [k + 1] =

i =1

n 

ξiℓ [k] − T

i =1

n  n 

pijℓ q(ζiℓ [k] − ζjℓ [k]).

i=1 j=1

pijℓ q(ζiℓ [k] − ζjℓ [k]) = 0

(4)

i=1 j=1

ready for an execution.

ξiℓ [k + 1] =

i=1

n 

ξiℓ [k] − 0 =

i=1

n 

finishes its execution.

ξiℓ [k].

i=1

Thus it suffices to prove Eq. (4). From the assumption of undirected communications (Assumption 1) and the symmetry property of the quantizer q (Assumption 3), we have n  n 

pijℓ q(ζiℓ [k] − ζjℓ [k]) =

i=1 j=1

n  n 

pjiℓ q(ζjℓ [k] − ζiℓ [k])

i=1 j=1

=

n  n 

=−

pijℓ q(ζjℓ [k] − ζiℓ [k])

n  n 

i=1

ξiℓ [k] = · · · =

i=1

n 

In the IEEE 802.15.4 standard, GTSs in each superframe are limited to at most 7 slots. To overcome such limitation of GTSs, we assume the following relation between the sampling period T and the beacon interval BI which allows several superframes in each period of consensus dynamics.

ξiℓ [0] = µℓ . 

i =1

We consider a sufficient condition for achieving an output consensus using the consensus dynamics (1) and (2). Theorem 6. Under Assumptions 1–3, the output (ζ1ℓ , ζ2ℓ , . . . , ζnℓ ) converges to an invariant set {(ζ¯1ℓ , ζ¯2ℓ , . . . , ζ¯nℓ ) | q(ζ¯iℓ − ζ¯jℓ ) = 0, ∀{vi , vj } ∈ E } using the consensus dynamics (1) and (2) if the sampling period T holds

σ

0
 η

sup

max

n 

i∈I ,ℓ∈L j=1

, pijℓ

where σ is a positive constant satisfying Assumption 3. The proof of Theorem 6 is given in the Appendix.

∀i ∈ I , ∀j ∈ Ni , ∀k ∈ Z+ ,

ctl where Cictl ,worst is the worst computation time of the control task τi and Ni = {j ∈ I | {vi , vj } ∈ E }.

pijℓ q(ζiℓ [k] − ζjℓ [k]).

Thus Eq. (4) holds. Therefore, we have n 

In consensus problems, each agent has to compute its control task based on local information exchanges. Therefore, we assume that all communication tasks of agent i are completed before the control task τictl starts its execution.

ctl dcmm ij,k ≤ (k + 1)T − Ci,worst

i=1 j=1

ξiℓ [k + 1] =

• Computation time Cijcmm ,k : time necessary for an execution of the (k + 1)-st job of τijcmm . • Absolute deadline dcmm ij,k : time instant by which the (k + 1)-st job of τijcmm should be completed.

Assumption 7. The absolute deadline dcmm ij,k satisfies

i=1 j=1

n 

cmm • Release time rijcmm is ,k : time at which the (k + 1)-st job of τij cmm • Finishing time fijcmm ,k : time at which the (k + 1)-st job of τij

holds, we have n 

In real-time systems, a unit of work of a system is called a job (Buttazzo, 2005). A task is a set of related jobs that jointly provide a service. If a task releases an infinite sequence of jobs at a constant rate, the task is called periodic. In the rest of this paper, we assume that there is at least one feasible GTS scheduling algorithm with the given task parameters. We consider two real-time tasks for each agent i ∈ I : a communication task τijcmm and a control task τictl . The communication task τijcmm receives the quantized relative differences of outputs q(ζjℓ [k] − ζiℓ [k]) from its neighbor agent j ∈ I . The control task τictl updates the state ξi [k] and the output ζi [k] by the consensus dynamics (1) and (2), respectively. The communication task τijcmm

and the control task τictl are periodic tasks whose periods are equal to the sampling period T . The communication task τijcmm is characterized by the following parameters:

Note that, if n  n 

4.1. Task model

(5)

Assumption 8. The sampling period T of the discrete-time consensus dynamics (1) and (2) satisfies T = α BI, where α is a positive integer. The communication task τijcmm represents the local communication j → i which is implemented as a two-stage communication: j → PANC and PANC → i as stated in Section 2. Therefore we make the following assumption on a computation time of communication tasks. Assumption 9. The computation time Cijcmm ,k satisfies Cijcmm ,k = βij,k × aBaseSlotDuration

∀i ∈ I , ∀j ∈ Ni , ∀k ∈ Z+ ,

where βij,k ≥ 2 is a positive integer.

N. Hayashi, S. Takai / Automatica 55 (2015) 6–11

9

Table 1 Superframe structure.

Fig. 3. The static and undirected graph with 4 agents.

BO

SO

aBaseSuperframeDuration

α

4

3

15.36 (ms)

2

(a) A branch of any node causes a deadline miss. (b) A feasible schedule is found. (c) The finishing time fijcmm of the current communication task ,k τijcmm satisfies fijcmm ,k > kT + γ BI + Beacon + CAP + CFP, where γ ∈ Z+ is the number of branches of the dummy task

ℓ τdum .

The rule (c) implies that the (k + 1)-st job of the communication task τijcmm cannot be scheduled in the (γ + 1)-st superframe if its finishing time exceeds the duration of the CFP of the (γ + 1)-st superframe. In this paper, we propose an offline GTS scheduling since we consider a static and undirected graph with the output consensus dynamics (1) and (2). However the proposed GTS scheduling is also applicable for the case of dynamic and directed local communications if a GTS scheduling can be obtained during each CAP (Hayashi & Takai, 2013). 5. Simulation

Fig. 4. Trajectories of 4 agents.

4.2. GTS scheduling based on Bratley’s algorithm In IEEE 802.15.4 wireless networks, a PAN coordinator reserves network bandwidth for a designated agent. Since a PAN coordinator does not allow any interruption of a current communication task, communication tasks are non-preemptive (Buttazzo, 2005). For non-preemptive scheduling problems, Bratley’s algorithm was proposed to find a feasible scheduling (Buttazzo, 2005). Bratley’s algorithm is a branch-and-bound method for a non-preemptive scheduling problem. In the algorithm, a branch is abandoned when one of the following conditions is satisfied: (a) A branch of any node causes a deadline miss. (b) A feasible schedule is found. However Bratley’s algorithm allocates tasks to all the slots including a CAP and an Inactive Portion. Thus one cannot directly employ the original Bratley’s algorithm for a task scheduling over IEEE 802.15.4 wireless networks. In this paper, we modify the abandon criteria of Bratley’s algorithm for a non-preemptive scheduling problem of GTSs over IEEE 802.15.4 wireless networks. In the IEEE 802.15.4 protocol, communication tasks cannot be executed in an inactive portion. To consider the constraint on communication tasks, we introduce ℓ a dummy task τdum which has the following task parameters (k ∈ Z+ , ℓ = 1, 2, . . . , α − 1) (Araújo et al., 2014):

• • • •

ℓ Release time: rdum ,k := kT + (ℓ − 1)BI,

ℓ Phase: φdum := Beacon + CAP + CFP, ℓ Computation time: Cdum ,k := Inactive + Beacon + CAP, ℓ ℓ ℓ ℓ Absolute deadline: ddum,k := rdum ,k + φdum + Cdum,k ,

where Beacon, CAP, CFP, Inactive are durations of a beacon signal, a CAP, a CFP, and an inactive portion, respectively. In the proposed branch-and-bound method for a nonpreemptive GTS scheduling, the abandon criteria of a branch are given as follows:

This section shows a numerical example with 4 agents (I = {1, 2, 3, 4}). We consider a 2L × 2L square plane region in which a PAN coordinator is located at the center of the region (L ∈ R+ \{0}). We consider a uniform quantizer q : R → R with a quantization gain δu = 0.125 as follows (Guo & Dimarogonas, 2013a):       1 1  δu , k + δu , kδu if x ∈ k−   2 2      and k = 1, 2, .  ..    δu δu if x ∈ − , , q(x) = 0 2 2           1 1   kδu if x ∈ k− δu , k + δu ,   2 2   and k = −1, −2, . . . . The position of agent i at time k is represented by the output (ζi1 [k], ζi2 [k]), where ζi1 [k] and ζi2 [k] are the x-coordinate and the y-coordinate of the agent’s position, respectively. In this simulation, we assume that each agent has internal states for the xcoordinate ξi1 [k] ∈ R and for the y-coordinate ξi2 [k] ∈ R (m = 1). Each agent updates the internal states by Eq. (1) with 0 < pijℓ ≤ 0.4 for all i, j ∈ I and all ℓ ∈ L . We assume that each agent and the PAN coordinator can keep wireless connections whenever necessary within the square region. To this end, we consider the following output function: f (ξ ) = L sin



πξ max ξ − ξ min



,

where ξ max = −ξ min = 10 and L = 5. To satisfy the condition (5) and Assumption 8, we consider the superframe structure characterized by the parameters in Table 1. Then we have T = 2 × BI = 491.52 (ms). We consider a static and undirected communication graph as shown in Fig. 3. For the sake of simplicity, we omit communications between agents and a PAN coordinator in Fig. 3. Fig. 4 illustrates the trajectories of 4 agents, where ‘‘⃝’’ is the initial position of each agent and ‘‘×’’ is the final consensus position. In this simulation, the PAN coordinator can find a feasible scheduling at all steps and the agents achieve an output consensus.

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N. Hayashi, S. Takai / Automatica 55 (2015) 6–11

Fig. 5. Output ζi1 for feasible scheduling (a) and non-feasible scheduling (b).

Next we consider the case when the PAN coordinator cannot find a feasible scheduling at some steps. Fig. 5 shows the outputs of 4 agents for the case when the PAN coordinator can find a feasible scheduling at all steps (a) and for the case when it occasionally fails to find a feasible scheduling (b). In this simulation, the agents use the CAP slots for their local communications when the PAN coordinator fails to find a feasible scheduling. We set the rate of a successful transmission for each local communication in the CAP as 50 (%). Though the convergence speed of the non-feasible scheduling is slower than that of the feasible scheduling due to the unreliable communication in the CAP, the agents can still achieve an output consensus as shown in Fig. 5(b). The theoretical analysis to obtain sufficient conditions for the non-feasible case is our future work.

Appendix. Proof of Theorem 6 We consider a candidate Lyapunov function as follows: n   µℓ 2 ξiℓ [k] − .

n

i=1

ξi2ℓ [k] − 2

i =1

=

n  i =1

ξi2ℓ [k] −

(ξi2ℓ [k + 1] − ξi2ℓ [k])

i=1

i =1

= −2T

n  n 

pijℓ ξiℓ [k]q(ζiℓ [k] − ζjℓ [k])

i =1 j =1

+T

 n n  

2

i=1

2 pijℓ q(ζiℓ [k] − ζjℓ [k])

.

j =1

n µℓ 

n

µ2ℓ n

.

i =1

pijℓ ξiℓ [k]q(ζiℓ [k] − ζjℓ [k])

i=1 j=1

=−

n  n 

pijℓ ξjℓ [k]q(ζiℓ [k] − ζjℓ [k]).

i =1 j =1

Then we have Vℓ [k + 1] − Vℓ [k]

= −T

n  n 

pijℓ (ξiℓ [k] − ξjℓ [k])q(ζiℓ [k] − ζjℓ [k])

i=1 j=1

+T

2

 n n   i=1

2 pijℓ q(ζiℓ [k] − ζjℓ [k])

.

Since f is a class C 1 function, from the mean value theorem, there exists cijℓ [k] such that f (ξiℓ [k]) − f (ξjℓ [k])

= f ′ (cijℓ [k]) ≡ ηijℓ [k], (A.3) ξiℓ [k] − ξjℓ [k] where min{ξiℓ [k], ξjℓ [k]} ≤ cijℓ [k] ≤ max{ξiℓ [k], ξjℓ [k]}. Since the output function f is monotonically increasing, we have ηijℓ [k] ≥ 0 if ξiℓ [k] ∈ (ξ min , ξ max ) for all i ∈ I , ℓ ∈ L , and k ∈ Z+ . Let ηsup := supξ min ≤ξ ≤ξ max f ′ (ξ ). Note that, from Assumption 2, sup η > 0. If ηijℓ [k] ̸= 0, from Eq. (A.3) and Assumption 3, we have

ξiℓ [k] +

= pijℓ

n  µ2

ℓ

i=1

n2

1

ηijℓ [k]

≥ σ pijℓ (A.1)

(A.2)

j =1

pijℓ (ξiℓ [k] − ξjℓ [k])q(ζiℓ [k] − ζjℓ [k])

Then, from Lemma 5, we have Vℓ [k] =

n 

n  = (ξiℓ [k + 1] + ξiℓ [k])(ξiℓ [k + 1] − ξiℓ [k])

n  n 

This paper has considered an output consensus problem over IEEE 802.15.4 wireless networks in which each agent communicates through a PAN coordinator with a multiple superframe structure. We have proposed a branch-and-bound GTS scheduling algorithm under the multiple superframe structure. We have also considered dynamics to achieve an output consensus by quantizing relative differences of outputs among neighbor agents. The simulation result has showed that each agent can successfully achieve a consensus by the proposed GTS scheduling algorithm and the output consensus dynamics. There are several important issues such as the effect of packetlosses and time-delays of local communications, and the extension to a quantized output consensus problem with dynamic and directed communications. Moreover, in some applications, the other types of network protocols are desirable. For example, the IEEE 802.15.4e standard has been developed to enhance flexibility for network design, e.g., mesh network topologies and multiple GTS slots, etc. These issues are targets of our subsequent work.

n 

Vℓ [k + 1] − Vℓ [k] =

From Assumption 1, we have

6. Conclusion

Vℓ [k] =

From Eq. (A.1), we have

≥ σ pijℓ

(ζiℓ [k] − ζjℓ [k])q(ζiℓ [k] − ζjℓ [k])

1

ηijℓ [k] 1

η

sup

(q(ζiℓ [k] − ζjℓ [k]))2

(q(ζiℓ [k] − ζjℓ [k]))2 .

N. Hayashi, S. Takai / Automatica 55 (2015) 6–11

If ηijℓ [k] = 0, then ζiℓ [k]−ζjℓ [k] = 0, and hence, q(ζiℓ [k]−ζjℓ [k]) = 0 from Assumption 3. Thus, if ηijℓ [k] = 0, we have pijℓ (ξiℓ [k] − ξjℓ [k])q(ζiℓ [k] − ζjℓ [k]) 1

= σ pijℓ

η

Therefore we obtain n  n 

pijℓ (ξiℓ [k] − ξjℓ [k])q(ζiℓ [k] − ζjℓ [k])

i=1 j=1 n  n 

≥σ

pijℓ

i =1 j =1

1

ηsup

(q(ζiℓ [k] − ζjℓ [k]))2 .

(A.4)

On the other hand, from the Cauchy–Schwarz inequality, we also have



n 

2 pijℓ q(ζiℓ [k] − ζjℓ [k])

j =1

 ≤

n 

 pijℓ

n 

j =1

pijℓ



 2 q(ζiℓ [k] − ζjℓ [k]) .

(A.5)

j =1

Then, from Eqs. (A.2)–(A.5), we have n  n σ  T pijℓ (q(ζiℓ [k] − ζjℓ [k]))2 sup η i =1 j =1  n 

Vℓ [k + 1] − Vℓ [k] ≤ −

 +T

2

 ×

max

i∈I ,ℓ∈L n  n 

pijℓ

2 q(ζiℓ [k] − ζjℓ [k])



i=1 j=1

  =T T  ×

max

i∈I ,ℓ∈L

n  n 

n  j =1

 pijℓ

σ − sup η



 pijℓ q(ζiℓ [k] − ζjℓ [k])



2

.

(A.6)

i=1 j=1

Thus, for all k ∈ Z+ , we have Vℓ [k + 1] − Vℓ [k] ≤ 0 if (5) holds. Since Vℓ [k] ≥ 0 for all k ∈ Z+ , Vℓ [k] has a finite limit. Then we have Vℓ [k + 1] − Vℓ [k] → 0 as k → ∞. Thus we obtain

 lim

k→∞

n  n  i=1 j=1

 pijℓ q(ζiℓ [k] − ζjℓ [k])



Araújo, J., Mazo, M., Anta, A., Tabuada, P., & Johansson, K. H. (2014). System architectures, protocols and algorithms for aperiodic wireless control systems. IEEE Transactions on Industrial Informatics, 10(1), 175–184. Buttazzo, G. C. (2005). Hard real-time computing systems: predictable scheduling algorithms and applications. Springer. Godsil, C., & Royle, G. (2001). Algebraic graph theory. Springer. Guo, M., & Dimarogonas, D. V. (2013a). Consensus with quantized relative state measurements. Automatica, 49(8), 2531–2537. Guo, M., & Dimarogonas, D. V. (2013b). Nonlinear consensus via continuous, sampled, and aperiodic updates. International Journal of Control, 86(4), 567–578. Hayashi, N., & Takai, S. (2013). A GTS scheduling for consensus problems over IEEE 802.15.4 wireless networks. In Proceedings of the 12th European control conference (pp. 1764–1769). Jadbabaie, A., Jie, L., & Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6), 988–1001. Li, T., Fu, M., Xie, L., & Zhang, Ji-F. (2011). Distributed consensus with limited communication data rate. IEEE Transactions on Automatic Control, 56(2), 279–292. Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Tiberi, U., Fischione, C., Johansson, K. H., & Di Benedetto, M. D. (2010). Adaptive selftriggered control over IEEE 802.15.4 networks. In Proceedings of the 49th IEEE conference on decision and control (pp. 2099–2104). Tiberi, U., Fischione, C., Johansson, K. H., & Di Benedetto, M. D. (2013). Energyefficient sampling of networked control systems over IEEE 802.15.4 wireless networks. Automatica, 49(3), 712–724. Xiao, L., & Boyd, S. (2004). Fast linear iterations for distributed averaging. Systems & Control Letters, 53(1), 65–78.

Naoki Hayashi received the B.E., M.E. and Ph.D. degrees from Osaka University in 2006, 2008 and 2011, respectively. He was a Research Assistant at Kyoto University in 2011. Since 2012, he has been an Assistant Professor at Osaka University. His research interests include cooperative control and distributed optimization.

j=1

 pijℓ

Therefore the output (ζ1ℓ , ζ2ℓ , . . . , ζnℓ ) converges to an invariant set {(ζ¯1ℓ , ζ¯2ℓ , . . . , ζ¯nℓ ) | q(ζ¯iℓ − ζ¯jℓ ) = 0, ∀{vi , vj } ∈ E }.  References

(q(ζiℓ [k] − ζjℓ [k]))2 = 0.

sup

11

2

= 0.

Shigemasa Takai received the B.E. and M.E. degrees from Kobe University, Kobe, Japan, in 1989 and 1991, respectively, and the Ph.D. degree from Osaka University, Suita, Japan, in 1995. From 1992 to 1998, he was a Research Associate at Osaka University. In 1998, he joined Wakayama University, Wakayama, Japan, as a Lecturer, and became an Associate Professor in 1999. From 2004 to 2009, he was an Associate Professor at Kyoto Institute of Technology, Kyoto, Japan. Since 2009, he has been a Professor at Osaka University, Suita, Japan. His research interests include supervisory control and fault diagnosis of discrete event systems.