Energy-Aware Consensus For Networked Sampled

MIMO Systems ⋆

M. Lopez-Martinez ∗ , J.-C. Delvenne and Vincent D. Blondel ∗∗

∗ Dept. of Systems and Automation Engineering, University of Seville, Spain,(e-mail: mlm@esi.us.es).

∗∗ Dept. of Applied Mathematics, Universite Catholique de Louvain,Belgium, (e-mail: Jean-Charles.Delvenne@uclouvain.be,

vincent.blondel@uclouvain.be)

Abstract: This work presents a method to compute the optimal sampling time that makes N MIMOagents converge to the consensus state with minimum energy consumption. In its turn, every agent inthe network is composed of n continuous time first order linear systems. The communication is done indiscrete time, sampling all the signals associated to every agent at the same sampling time. In order tominimize the energy consumed in the process of communication, we will look for the optimal samplingtime such that the consensus is reached in a minimum number of iterations. The analysis is performedby minimizing objective functions that take into account a measure of the convergence rate to reach aconsensus. These objective functions mainly depend on the eigenvalues of the sampled transition matrixof the system. The method can be applied to medium/large scale networks, since it requires computingthe eigenvalues of the adjacency matrix just once. Finally, we present a case study based on the torustopology, where a MIMO case of communication (1000 systems) is analyzed, obtaining the optimalsampling time to reach the consensus.

Keywords: Consensus Algorithm, Networked Control Systems, Sampled Systems.

1. INTRODUCTION

This paper presents a method to analyze the best samplingtime in a network such that consensus is reached with minimalenergy consumption in the communication procedure. We willexplore the MIMO case, where the results are more relevant,because each agent is allowed to be composed of severalsystems. Particularly, in this work we will study the specificcase where all the subsystems are decoupled; we consider thatthe information of an agent is sent at the same time (samesampling time for the systems in an agent) in order to reducethe energy consumption in the communication process.

In the literature there are many works that study differentaspects concerning the consensus problem. From the seminalwork by Tsitsiklis (1984), where the consensus problem wasfirstly studied, many researchers have focused their investi-gations on this field. In the survey paper Olfati-Saber et al.(2007) and in the references therein, we can find the main prob-lems studied related to consensus. In this survey, the authorscompare several problems studied in continuous time and indiscrete time, paying special attention to switching networksand time delays (Olfati-Saber et al. (2004)). More recently, inSeuret et al. (2008) and Seuret et al. (2009) the consensus prob-lem is analyzed in continuous time introducing delays in thecommunication. With respect to the influence of the topology

⋆ The authors gratefully acknowledge MEC (Spanish Ministry of Education

and Science) for funding this work under grants DPI2010-19154, and EU

STREP FP7 program for funding this work under project FeedNetBack ICT-

2007-2(223866). The authors also acknowledge the support given by the

Concerted Research Action (ARC) ”Large Graphs and Networks” of the French

Community of Belgium, and by the Belgian Programme on Interuniversity

Attraction Poles (PAI) initiated by the Belgian Federal Science Policy Office.

in the consensus problem, in Carli et al. (2008) it is shownthat randomly time-varying (switching) graphs allow very fastconvergence rates for consensus. Besides, in case that everyagent is restricted to communicate to a given small number ofother agents, it has been proved that the optimal topology ofcommunication is given by a de Bruijn graph (Delvenne et al.(2009)).

On the other hand, the study of the consensus problem for sam-pled continuous time systems has been scarcely studied, andvery few works are referenced in literature. In Gao et al. (2009),a study of the consensus problem for multi-agent systems is pre-sented using sampled information, but the analysis is performedconsidering the sampling period to be small enough. Recentlyin Rodrigues el al. (2010), the authors search for an appropriatesampling time such that the stability is ensured given a specifictopology for second order systems. On the other hand, thework of Moreau (2005) reveals that more communication doesnot necessarily lead to faster convergence and may eventuallyeven lead to a loss of convergence. In this sense, in the workLopez-Martinez el al. (2010) the main novelty was the idea offinding an optimal sampling time depending on an objectivefunction. This objective function, in its turn, will depend onthe eigenvalues of the closed loop system. In the work that wepresent here, we extend these results for the case of networks ofagents with multivariable dynamics. In this way, the objectiveis to find the best sampling time such that the consensus amongall the multivariable agents is reached with a minimum numberof iterations and thus with minimal energy consumed in thecommunication process. We compare two objective functionsusing some recent results developed in Carli et al. (2009).

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Copyright by theInternational Federation of Automatic Control (IFAC)

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The paper is organized as follows. Firstly, section 2 is dedicatedto explain the notation used along the paper. In section 3, a verysimple motivating example introduces the problem, first for theSISO case and then for the MIMO case. Next, section 4 presentsthe problem description. In section 5, several optimizationproblems are defined in order to minimize the energy consumedin the communications, which is mainly related to the numberof iterations. In section 6 and section 7, we introduce a methodto solve the optimization problems for the case of first ordersystems and then a complex case study is analyzed. Finally themajor contributions of the work are summarized in section 8.

2. NOTATION

Throughout the paper the following notation will be used:

• z(Ak) is a column vector with the eigenvalues of Ak

sorted from highest to lowest magnitude, and zi representsthe i-th coordinate.

• |zi| represents the i-th largest magnitude of all the eigen-values.

• log represents the natural logarithm.• ⊤ represents the transpose.

3. MOTIVATING EXAMPLES

3.1 SISO CASE STUDY

Consider the discrete time feedback interconnection of twoidentical continuous time first order linear systems, sampledand hold (ZOH) at sampling time Tk (see Fig. 1). The dy-

Σ1 Σ2

u1

u2 y1

y2

Tk

Tk

Fig. 1. Feedback interconnection of systems Σ1 (state x1) andΣ2 (state x2).

namics of the continuous time closed loop system can then beexpressed as

x1 = −x1 + x2 (1)

x2 = −x2 + x1 (2)

or in a matricial way

x = Ax, with A =

[

−1 11 −1

]

(3)

and for the sampled closed loop system

x1,k+1 = ax1,k + bx2,k (4)

x2,k+1 = ax2,k + bx1,k (5)

or in a matricial way

xk+1 = Akxk, with Ak =

[

a bb a

]

(6)

where a = e−Tk and b = 1 − a. From this last relation, itis easy to see that Ak has one eigenvalue in z1 = 1. The

other eigenvalue can be obtained by solving the characteristicpolynomial

(z − a)2 − b2 = 0,

which yields z2 = 2a − 1 = 2e−Tk − 1. The magnitudeof this second eigenvalue is |z2| < 1 for all 0 < Tk <∞, and this implies that the sampled closed loop system iscritically stable. The eigenvector associated to z1 is the vectorof ones. Thus, for any given initial state, x(0), it is possibleto reach a consensus x(∞), where each component is equal

to the mean of the components of x(0), that is,x1(0)+x2(0)

2 .In order to reach consensus with a minimum consumption ofenergy spent in the communications, we have to minimize thenumber of iterations. The rate of convergence to consensus(z1 = 1) depends on the rest of the eigenvalues, and so it canbe increased by reducing their magnitude. In this case, just thesecond one, which depends on the sampling time. Thus, theminimum number of iterations is achieved when z2 = 0, andthis implies to take a specific value of sampling time, which isTk = log 2.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

x(k

)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

x(k

)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

Number of iterations

x(k

)

Tk=0.1

Tk=log 2

Tk=1.5

Fig. 2. Number of iterations versus the sampling time.

Figure 2 shows the evolution of the response of each systemwith respect to the number of iterations. In this case the optimalsampling time to reduce the number of iterations, and hencethe energy consumed in the communication, is given by Tk =log 2. It is also interesting to note that the number of iterationsto reach the consensus (within a prescribed error) for Tk = 0.1is larger than for Tk = 1.5.

3.2 MIMO CASE STUDY

Consider now a class of linear systems with two degrees offreedom (two independent states), where the dynamics of eachdegree of freedom are independent, of first order but withdifferent characteristic times (τ1, τ2). Suppose that two systemsof this class are interconnected in such a way that there existsa sampled feedback interconnection between the correspondingstates. The main idea is to use only just one sampling time, insuch a way that each system transmits the information of thetwo degrees of freedom at the same moment. In this way wereduce the number of times to transmit and hence the energyinvolved in the process of communication.

We can model the resulting system in the following way

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xk+1 = Akxk, with Ak =

a1 b1 0 0b1 a1 0 00 0 a2 b20 0 b2 a2

(7)

where a1 = e−Tk/τ1 , b1 = 1− a1 and a2 = e−Tk/τ2 , b2 = 1−a2.

As before, the idea is to determine what is the best sampletime that we can choose in order to minimize the number ofiterations that are necessary to reach a consensus. For that, westudy the eigenvalues of matrix Ak, obtaining the followingsolution

z1 = 1,

z2 = 1,

z3 = 2a1 − 1 = 2e−Tk/τ1 − 1,

z4 = 2a2 − 1 = 2e−Tk/τ2 − 1. (8)

In this case, a way to minimize the number of iterations is tochoose a sampling time such that the magnitude of the thirdeigenvalue (ordered by magnitude) is minimized.

Then, to obtain this optimal sampling time T ∗, we should solve

T ∗ = argminTk

{

max{∣

∣

∣2e−Tk/τ1 − 1

∣

∣

∣,∣

∣

∣2e−Tk/τ2 − 1

∣

∣

∣

}}

(9)

Assuming that τ1 > τ2, the optimal value of Tk corresponds tothe value such that

∣

∣

∣2e−Tk/τ1 − 1

∣

∣

∣=

∣

∣

∣2e−Tk/τ2 − 1

∣

∣

∣.

The solution of this equality is found for the interval Tk ∈[τ2 log 2, τ1 log 2], and corresponds to the solution of the fol-lowing trascendental equation

e−Tk/τ1 + e−Tk/τ2 = 1.

For the sake of clarity, we can take the values τ1 = 2 andτ2 = 1 obtaining the graphical solution shown in Fig. 3, wherethe optimal sampling time is T ∗ = 0.9624.

Remark 1. In this example, the objective function is minimizethe magnitude of the third eigenvalue, the first whose magnitudeis less than one. From Fig. 3, it is clear that this function is notof class C1, but the minimization problem can be solved byfinding the intersections of the curves.

It is worth mentioning that choosing just one sampling time,neither the solution of taking Tk = τ1 log 2 nor Tk = τ2 log 2is the optimal one for the objective function to minimize, givenby the magnitude of the third eigenvalue.

Figure 4 shows the evolution of the states for the interconnec-tion of the two systems mentioned above. It is clear that withrespect to the number of iterations, the second simulation is thebest, where the sampling time is Tk = T ∗. It is also interestingto note that in this simulation the convergence to the consensusstates is faster (number of iterations) with Tk = 2 than withTk = 0.1.

4. PROBLEM DESCRIPTION

Consider a critically-stable continuous time interconnection ofN first order linear systems, Σi, where the continuous time

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tk

|−1+2*exp(−Tk/τ

1)|

|−1+2*exp(−Tk/τ

2)|

Fig. 3. First graphical solution: intersection point.

0 5 10 150

0.5

1

Number of iterations

x(k

)

0 5 10 150

0.5

1

Number of iterations

x(k

)

0 5 10 150

0.5

1

Number of iterations

x(k

)

Tk=0.1

Tk=T

* =0.9624

Tk=2

Fig. 4. Number of iterations versus the sampling time.

dynamics of each system can be expressed in the followingform:

Σi : xi = Aixi +Biui (10)

where xi ∈ R is the state vector and ui ∈ R is the input vector.Consider also that they are interconnected such that

ui =∑

j

kijxj , j 6= i ∈ [1, N ]. (11)

Then, the dynamics of the whole system can be expressed as

x = (A+BK)x = Ax (12)

and the following theorem can be applied.

Theorem 1. If A is critically stable (λ1 = 0, ℜ(λi) < 0, i ∈[2, N ]), and the sum of the elements of each row and eachcolumn is equal to 0 ( Ak1 = 0 and 1

⊤Ak = 0⊤ ), then

limt→∞

x(t) =

1

N

N∑

j=1

xj(0)

1,

where 1 ∈ RN is a column vector of N ones.

In case the signals of each system are sampled and hold (ZOH)before exchanging the information, the dynamics of each sys-tem can then be written as

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Σi,k : xi,k+1 =Ai,kxi,k +Bi,kui,k (13)

where Ai,k = eAiTk and Bi,k = A−1i (Ai,k − I)Bi. Consider

now that they are interconnected such that

ui,k =∑

j

kijxj,k, j 6= i ∈ [1, N ]. (14)

Then, the dynamics of the whole system can be expressed as

xk+1 = (Ak +BkK)xk = Akxk (15)

and the following theorem can be used.

Theorem 2. If Ak is critically stable (|z1| = 1, |zi| < 1, i ∈[2, N ]) and the sum of the elements of each row and eachcolumn is equal to 1 (Ak1 = 1 and 1

⊤Ak = 1⊤), then

limk→∞

xk =

1

N

N∑

j=1

xj,0

1.

At this point of the exposition, the problem to solve can bestated as follows:

Given the set of continuous time systems described by (10),verifying Theorem 1, find the best sampling time, such thatthe sampled consensus algorithm converges (verifying Theorem2) following a criterion of minimum communication energyconsumption.

5. OPTIMAL SAMPLING TIME FOR MINIMUMCOMMUNICATION ENERGY CONSUMPTION

When analyzing the discrete-time interconnections, it is knownthat the rate of convergence of the consensus algorithm dependson the eigenvalues of matrix Ak. It is clear that changing thesampling time, the eigenvalues also change. The problem is todetermine the optimal sampling time such that the consensus isreached with minimum communication energy consumption.

For the sake of simplicity we will assume that all the agents arein communicating-modality. That is to say, we are not takinginto account in this work the energy consumption associatedto the go-in-sleeping-modality and the awake-from-sleeping-modality. At this point, it is clear that the less number ofcommunications among the agents implies the less energyconsumed. The number of communications in this context isthe number of iterations of the consensus algorithm.

Therefore, in order to obtain the optimal sampling time we haveto specify a criterion to minimize, which obviously depends onthe eigenvalues of the sampled network described by Ak. Next,we will describe two possible criteria.

5.1 Second-eigenvalue criterion

We can state the following criterion, the largest magnitude ofthe the eigenvalues that is less than 1. Defining |zi| as the i-th largest magnitude of all the eigenvalues, we can write thisobjective function, with some abuse of notation, as

J1.= |z2(Ak)|,

and we will refer to it as the second-eigenvalue criterion.

Then formally, the problem is to find the optimal sampling timeT ∗ by solving

T ∗ .= arg min

Tk>0{J1} = arg min

Tk>0{|z2(Ak)|} (16)

5.2 All-eigenvalues criterion

As in the previous section, we can look for optimal samplingtimes that minimize other indexes, commonly used in thediscrete-time consensus literature (see, e.g., Carli et al. (2009)).

Sum of mean quadratic errors: For the special case of consider-ing that the number of iterations is large enough (theoreticallyfor k → ∞), we can derive the following index from the sumof mean quadratic errors with respect to the consensus value.

J2.=

∑

|zi|6=1

1

1− |zi|2. (17)

Then the problem is to find the optimal sampling time T ∗ bysolving

T ∗ .= arg min

Tk>0{J2} = arg min

Tk>0

∑

|zi|6=1

1

1− |zi|2

. (18)

6. CONSENSUS OF SAMPLED FIRST ORDER MIMOSYSTEMS

Consider a critically-stable continuous time interconnection ofN agents composed of n different first order linear systems(every agent has n degrees of freedom and every dynamics areof first order), Σim, where for every agent the continuous timedynamics of each system can be expressed in the followingform:

Σim : τmxim = −xim + uim (19)

where xim ∈ R is the state, uim ∈ R is the input signal and τmis the characteristic time of the m− th first order system.

Consider also that they are interconnected such that

uim =∑

j

kijxjm, j 6= i ∈ [1, N ]. (20)

Then, the dynamics of the closed loop for the systems withcharacteristic time τm can be expressed as

τmxm = (−I +K)xm, (21)

where I is the identity matrix and K is the adjacency matrixexcluding the elements of the main diagonal, which are set tozero. From this expression, in order to verify Theorem 1, thefollowing conditions must be satisfied

N∑

i=1

kij = 1 ∀j,N∑

j=1

kij = 1 ∀i. (22)

Assuming that the signals of each system are sampled andhold (ZOH) before exchanging the information, the dynamicsof each system can be written as

Σim,k : xim,k+1 = amxim,k + bmuim,k (23)

where am = e−Tk/τm and bm = 1 − am. Consider now thatthey are interconnected such that

uim,k =∑

j

kijxjm,k, j 6= i ∈ [1, N ]. (24)

Then, the dynamics of each system yields

Σi,k : xim,k+1 = amxim,k + (1− am)∑

j

kijxjm,k,

j 6= i ∈ [1, N ]

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Finally, grouping the N equations we can express the dynamicsof the systems with characteristic time τm as

xm,k+1 = (amI + (1− am)K)xm,k = Am,kxm,k. (25)

6.1 Optimal sampling time

As shown in section 5, we can state our problem as the opti-mization of an index (minimization of J1 or J2) which dependson the eigenvalues of the matrix Am,k. In the case of sampled

first order systems, the matrix Am,k = Am,k + Bm,kK, takesthe values of Am,k = amI , Bm,k = 1 − am and am =

e−Tk/τm . In this way the eigenvalues of the matrix Am,k canbe calculated from

z(Am,k) = z(amI + (1− am)K) = am1+ (1− am)z(K).

(26)

For the whole closed-loop system including the n types of firstorder systems with characteristic times τm, we can build anextended diagonal matrix Ak, where each term in the diagonalis given by Am,k.

Ak =

A1,k 0 0 · · · 0

0. . . 0

. . . 0

0. . . Am,k

. . . 0

0. . .

. . .. . . 0

0 · · · 0 0 An,k

. (27)

In this way, the eigenvalues of the extended matrix can beexpressed as

z(Ak) =

a11+ (1− a1)z(K)...

am1+ (1− am)z(K)...

an1+ (1− an)z(K)

. (28)

Notice that the matrix K represents the weighted adjacency ma-trix of the graph excluding the elements of the main diagonal,which are set to zero. By assumption, the matrix K verifies∑N

i=1 kij = 1 ∀j and∑N

j=1 kij = 1 ∀i, which implies that

at least one eigenvalue has magnitude one. On the other hand,this matrix does not depend on the sampling time, and thus theireigenvalues only has to be computed once.

7. CASE STUDY: TORUS TOPOLOGY

The graph that describes the communication topology as thetorus is represented in Fig. 5. We assume that all nodes haveself–loops even though these are not plotted in the figure. Thematrix K is built using this graph without taking into accountself loops. We are going to study the case of a torus of 10x10agents, where there are 10 agents per ring and every agentbelongs to two perpendicular rings. We consider that everyagent is communicating with itself and with an adjacent agentin the same ring (in-degree=out-degree=3) as shown in Fig. 5.

Hence, the matrix K can be written as

i

Fig. 5. Agent interconnection in a torus graph (Source: FeedNet-

Back EU FP7 Project).

K =

R O O O O O O O O I

I R O O O O O O O O

O I R O O O O O O O

O O I R O O O O O O

O O O I R O O O O O

O O O O I R O O O O

O O O O O I R O O O

O O O O O O I R O O

O O O O O O O I R O

O O O O O O O O I R

, (29)

where I is a 10x10 identity matrix, O is a matrix of 10x10 zeros,and R is given by

R =

0 0 0 0 0 0 0 0 0 k

k 0 0 0 0 0 0 0 0 0

0 k 0 0 0 0 0 0 0 0

0 0 k 0 0 0 0 0 0 0

0 0 0 k 0 0 0 0 0 0

0 0 0 0 k 0 0 0 0 0

0 0 0 0 0 k 0 0 0 0

0 0 0 0 0 0 k 0 0 0

0 0 0 0 0 0 0 k 0 0

0 0 0 0 0 0 0 0 k 0

, with k =1

2. (30)

For the SISO case study, we consider that each agent is com-posed just of 1 first order system. Solving the optimizationproblem for J1 and J2 the optimal sampling time for this caseis the same, and is given by T ∗ = log 3.

Next, two comparisons are made taking into account differentvalues of sampling times. Figure 6 shows that the number ofiterations to achieve the consensus with sampling time Tk =log 3 and Tk = 3 is less than with sampling time Tk = 0.1sec. This implies that there is a range of sampling times wherethe network has low energy consumption (less iterations) andrequires low bandwidth to reach a consensus.

For the MIMO case, we will consider that each agent is com-posed of n = 10 first order systems, all of them communi-cating through the same graph with the same sampling timeTk, and verifying as a simplifying assumption that τm = m(m = 1, . . . , n). For this case study, we obtain that the optimalsampling time that minimizes the index J1 for the 1000 systemsthat compose this interconnection is given by T ∗ ≈ 2.2824.Figure 7 shows how for sampling times different from theoptimal one, the number of iterations to reach the consensusis larger, and hence the energy needed for the communicationpurposes.

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0 10 20 30 40 50 60 70 80 90 1000

0.5

1

x(k

)

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

x(k

)

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

Number of iterations

x(k

)

Tk=0.1

Tk=log 3

Tk=3

Fig. 6. Torus: Evolution of the states with respect to the numberof iterations

0 10 20 30 40 50 60 70 80 900

5

10

Number of iterations

x(k

)

0 10 20 30 40 50 60 70 80 900

5

10

Number of iterations

x(k

)

0 10 20 30 40 50 60 70 80 900

5

10

Number of iterations

x(k

)

Tk=0.1

Tk=T

*

Tk=10

Fig. 7. MIMO Torus: Evolution of the states with respect to thenumber of iterations

8. CONCLUSIONS AND FUTURE WORK

This work has shown the importance of choosing an appropriatesampling time when using consensus algorithms in networkedsampled systems. The implications of this sampling time se-lection are mainly two, reduction of the energy consumptionand reduction of bandwidth requirements. For the MIMO case,the results are really relevant, because normally each agent iscomposed of several systems, and the information is normallysent at the same time (same sampling time for the systems inan agent) in order to reduce the energy consumption in thecommunication process. The study has revealed that the optimalsampling time to achieve this depends only on the eigenvaluesof the adjacency matrix of the communication graph. If thiscommunication graph is constant in time, then their eigenvalueshave to be computed just once in order to obtain the optimalsampling time. This makes the algorithm converge very fast.

Here we have considered specific MIMO systems, where alldegrees of freedom are decoupled. As a future work, We also

want to consider general MIMO systems, where all degrees offreedom may interact with each other.

The results obtained will allow us to reduce the number ofcommunications in networks of sensors, or in networks ofautonomous mobile agents, where the equilibrium point ofevery agent is (asymptotically) stable when its inputs are setto a null value, and the adjacency matrix is chosen in such away that the equilibrium point of the global network is criticallystable and there exists an average-consensus value.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the Universite Catholiquede Louvain and the University of Seville for making possiblethe research stay of the first author in Louvain-La-Neuve.

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