Complexities and Performance Limitations in Growing Time-Delay Noisy Linear Consensus Networks

6th IFAC Workshop on Distributed Estimation and Control in 6th IFAC Workshop Distributed Estimation and Control in 6th IFAC on Networked Systemson 6th IFAC Workshop Workshop on Distributed Distributed Estimation Estimation and and Control Control in in Networked Systems Networked Systems Available online at www.sciencedirect.com September 8-9, 2016. Tokyo, Japan Networked Systems September 8-9, 2016. Tokyo, Japan September September 8-9, 8-9, 2016. 2016. Tokyo, Tokyo, Japan Japan

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Complexities and Performance Limitations Complexities and Performance Limitations Complexities and Performance Limitations in Growing Time-Delay Noisy Linear in Growing Time-Delay Noisy Linear in Growing Time-Delay Noisy Consensus Networks Linear Consensus Networks Networks Consensus Yaser Ghaedsharaf ∗∗ Nader Motee ∗∗ Yaser Yaser Ghaedsharaf Ghaedsharaf ∗∗ Nader Nader Motee Motee ∗∗ Yaser Ghaedsharaf Nader Motee ∗ Department of Mechanical Engineering and Mechanics, Lehigh ∗ ∗ Department of Mechanical Engineering and Mechanics, Lehigh ∗ Department of Engineering and Lehigh Bethlehem, PA 18015, USA DepartmentUniversity, of Mechanical Mechanical Engineering and Mechanics, Mechanics, Lehigh University, Bethlehem, PA 18015, USA University, Bethlehem, PA 18015, USA (e-mail: {ghaedsharaf, motee}@lehigh.edu). University, Bethlehem, PA 18015, USA (e-mail: {ghaedsharaf, motee}@lehigh.edu). (e-mail: (e-mail: {ghaedsharaf, {ghaedsharaf, motee}@lehigh.edu). motee}@lehigh.edu). Abstract: This work investigates topology design for optimal performance in time-delay noisy Abstract: This work investigates topology design for optimal performance in time-delay Abstract: This topology design for optimal performance in noisy networks. Performance of the network is measured by the square of the system’s H2 -norm.noisy The Abstract: This work work investigates investigates topology design by for the optimal performance in time-delay time-delay noisy networks. Performance of the network is measured square of the system’s H -norm. The 2 networks. Performance of the network is measured by the square of the system’s H The 2 -norm. focus of this paper is onofadding new interconnections to enhance performance of the time-delay networks. Performance the network is measured by the square of the system’s H -norm. The 2 time-delay focus of this paper is on adding new interconnections to enhance performance of the focus of this paper is on adding new interconnections to enhance performance of the time-delay first-order connected consensus network. We discuss to complexity of topology of optimization for focus of this paper is on adding new interconnections enhance performance the time-delay first-order connected network. We discuss complexity of topology optimization for first-order connected consensus network. We discuss complexity of topology optimization for delayed networks andconsensus develop two practical methods to tackle the combinatorial eigenvalue first-order connected consensus network. We discuss complexity of topology optimization for delayed networks and develop two practical methods to tackle the combinatorial eigenvalue delayed networks and develop two practical methods to tackle the combinatorial eigenvalue problem without exhaustive search or eigen-decomposition. Furthermore, we compare these delayed networks and develop two practical methods to tackle the combinatorial eigenvalue problem without exhaustive searchof or or eigen-decomposition. Furthermore, Furthermore, we compare compare these problem without exhaustive search eigen-decomposition. methods and discuss their degrees optimality. problem without exhaustive searchof or eigen-decomposition. Furthermore, we we compare these these methods and discuss their degrees optimality. methods and discuss their degrees of optimality. methods and discuss their degrees of optimality. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Consensus Networks, Topology Design, Time-Delay Systems, Network Performance Keywords: Keywords: Consensus Networks, Networks, Topology Topology Design, Design, Time-Delay Time-Delay Systems, Systems, Network Network Performance Performance Measures Consensus Keywords: Consensus Networks, Topology Design, Time-Delay Systems, Network Performance Measures Measures Measures 1. INTRODUCTION formance for first-order consensus networks are discussed 1. INTRODUCTION INTRODUCTION formance for first-order consensus networks discussed 1. formance for first-order consensus networks are discussed in Siami and (2014), Siami and Moteeare (2016). 1. INTRODUCTION formance for Motee first-order consensus networks are discussed in Siami and Motee (2014), Siami and Motee (2016). Siami and Motee (2014), Siami and Motee (2016). Our objective is to enhance H2 -norm performance of a in in Siami and Motee (2014), Siami and Motee (2016). Our objective objective is linear to enhance enhance H22 -norm performance of a Despite extensive research on consensus network perforperformance of Our is to H noisy time-delay consensus network by establishing research on consensus network Our objective is linear to enhance H2 -norm -norm performance of aa Despite Despite extensive research on network performance inextensive absence of delay, there have been limitedperforattennoisy time-delay consensus network by establishing Despite extensive research on consensus consensus network perfornoisy time-delay linear consensus network by establishing new feedback interconnections. mance in absence of delay, there have been limited attennoisy time-delay linear consensus network by establishing mance in absence of delay, there have been limited attention to such measures for consensus networks in presence new feedback interconnections. mance in absence of delay, there havenetworks been limited attennew tion to such measures for consensus in presence new feedback feedback interconnections. interconnections. tion to such measures for consensus networks in presence of delay. Ghaedsharaf et al. (2016) quantified H -norm 2 tion to such measures for consensus networks in presence Literature Review: In recent years, there has been an of delay. Ghaedsharaf et al. (2016) quantified H2 -norm delay. al. Literature Review: In recent years, there has been an of 2 -norm performance and someet limits on H the best Literature Review: In recent years, been delay. Ghaedsharaf Ghaedsharaf etfundamental al. (2016) (2016) quantified quantified H immense interest studying of networked 2 -norm Literature Review:in In recent susceptibility years, there there has has been an an of performance and some fundamental limits on the best performance and some fundamental limits on the best immense interest in studying susceptibility of networked achievable performance of time-delay first-order consenimmense interest in studying susceptibility of networked performance and some fundamental limits on the best systems to input disturbances. Bamieh et al. (2012) quanimmense interest in studying susceptibility of networked achievable performance of time-delay first-order consenachievable performance ofitstime-delay time-delay first-order consensystems to input disturbances. Bamieh et al. (2012) quansus networks in terms of underlying graph spectrum. systems to input disturbances. Bamieh et al. (2012) quanachievable performance of first-order consentified thetoHinput of first-order consensus networks as a sus networks in terms of its underlying graph spectrum. 2 -norm systems disturbances. Bamieh et al. (2012) quannetworks terms of underlying spectrum. tified the H -norm of first-order consensus networks as a sus Nevertheless, most of the efforts in the graph literature study tified the H -norm consensus networks as networks in in terms of its its underlying graph spectrum. measure with respect to stochastic disturtified theof H222robustness -norm of of first-order first-order consensus networks as aa sus Nevertheless, most of the efforts in the literature study Nevertheless, most of the efforts in the literature study measure of robustness with respect to stochastic disturconditions for stability of time-delay systems or convermeasure of robustness with respect to stochastic disturNevertheless, most of the efforts in the literature study bances. With growing applications of consensus protocols measure of robustness with respect to stochastic disturconditions for stability of time-delay systems or converconditions for stability of time-delay systems or converbances. With growing applications of consensus protocols gence time rather than quality of consensus in terms of bances. With growing applications of consensus protocols conditions for stability of time-delay systems or converin various disciplines, network topology design for such bances. With growing applications of consensus protocols gence rather than quality of in of gence time time energy rather to than quality of consensus consensus in terms termsand of in various disciplines, network topology design for such dissipated reach consensus [Olfati-Saber in various disciplines, network topology design for such gence time rather than quality of consensus in terms of systems turned into a class of challenging and demanding in various disciplines, network topology design for such dissipated dissipated energy to reach consensus [Olfati-Saber and energy to reach consensus [Olfati-Saber and systems turned into a class of challenging and demanding Murray (2004), M¨ u nz et al. (2010), Nedi´ c and Ozdaglar systems turned into a class of challenging and demanding dissipated energy to reach consensus [Olfati-Saber and problems. Theseinto applications from coordinating net- Murray (2004), M¨ systems turned a class ofrange challenging and demanding u nz et al. (2010), c and Ozdaglar Murray (2004), u nz al. Nedi´ problems. These applications from coordinating net(2010), Somarakis Baras (2015)].Nedi´ problems. These applications range from coordinating netMurray (2004), M¨ M¨ uand nz et et al. (2010), (2010), Nedi´cc and and Ozdaglar Ozdaglar works of autonomous vehiclesrange and sensor networks to synproblems. These applications range from coordinating net(2010), Somarakis and Baras (2015)]. (2010), Somarakis and Baras (2015)]. works of autonomous vehicles and sensor networks to synworks of of autonomous autonomous vehicles and sensor sensor networks to toetsynsynSomarakis and Baras (2015)]. chronous oscillators invehicles power networks [Olfati-Saber al. (2010), works and networks and Bayen (2010) aim to design the optimal topolchronous oscillators in power networks [Olfati-Saber et al. Rafiee Bayen (2010) aim to design the optimal topolchronous oscillators in networks [Olfati-Saber et (2007)]. Over the years, the concept of improving H2 -norm Rafiee and Bayen aim to design optimal chronous oscillators in power power networks [Olfati-Saber et al. al. Rafiee ogy forand consensus networks presence of time-delay conRafiee and Bayen (2010) (2010) aimin to design the the optimal topoltopol(2007)]. Over the years, the concept of improving H -norm 2 ogy for consensus networks in presence of time-delay con(2007)]. Over the years, the concept of improving H -norm 2 of first-order consensus networks is manifested through ogy for consensus networks in presence of time-delay con(2007)]. Over the years, the concept of improving H -norm sidering algebraic connectivity of the underlying graph as 2 ogy for consensus networks in presence of time-delay conof first-order consensus networks is manifested through sidering algebraic connectivity of the underlying graph as of first-order consensus networks is manifested through different manuscripts. Ghosh et al. (2008) studied optimal sidering algebraic connectivity of the underlying graph as of first-order consensus networks is manifested through a performance measure. However, for time-delay consensus sidering algebraic connectivity of the underlying graph as different manuscripts. Ghosh (2008) optimal performance measure. However, for time-delay consensus differentallocation manuscripts. Ghosh et et al.minimize (2008) studied studied optimal aanetworks, weight in graphs toal. total effective performance measure. However, for time-delay consensus different manuscripts. Ghosh et al. (2008) studied optimal the second smallest eigenvalue of the Laplacian a performance measure. However, for time-delay consensus weight allocation graphs to minimize total effective the second smallest eigenvalue of the Laplacian weight allocation in graphs to total effective resistance. With Hin square of first-order networks, the second eigenvalue of 2 -norm weight allocation in graphs to minimize minimize totalconsensus effective networks, is not a good of performance [Ghaedsharaf et al. networks, the measure second smallest smallest eigenvalue of the the Laplacian Laplacian resistance. With H -norm square of first-order consensus 2 is not a good measure of performance [Ghaedsharaf et al. resistance. With H -norm square of first-order consensus 2 networks being proportional to total effective resistance of is not a good measure of performance [Ghaedsharaf et resistance. With H -norm square of first-order consensus (2016)]. Qiao and Sipahi (2014) consider designing delay 2 is not a good measure of performance [Ghaedsharaf et al. al. networks being proportional to total effective resistance of (2016)]. Qiao and Sipahi (2014) consider designing delay networks being proportional to total effective resistance of its underlying graph, Siami et al. (2016), Summers et al. (2016)]. Qiao and Sipahi (2014) consider designing delay networks beinggraph, proportional to al. total effective resistance of (2016)]. dependent coupling weights in order to ensure stability in Qiao and Sipahi (2014) consider designing delay its underlying Siami et (2016), Summers et al. dependent coupling weights in order to ensure stability in its underlying graph, Siami et al. (2016), Summers et al. (2015), and Moghaddam andetJovanovic (2015) studyet the dependent coupling weights in order to ensure stability in its underlying graph, Siami al. (2016), Summers al. the presence of a integer-valued homogeneous time-delay. dependent coupling weights in order to ensure stability in (2015), and Moghaddam and Jovanovic (2015) study the the presence of a integer-valued homogeneous time-delay. (2015), and Moghaddam and Jovanovic (2015) study the problemand of establishing new interconnections under difthe presence of a integer-valued homogeneous time-delay. (2015), Moghaddam and Jovanovic (2015) study the the presence of a integer-valued homogeneous time-delay. problem of establishing new interconnections under difmajor issue in time-delay consensus networks, which problem of new under different constraints and scenarios. Olfati-Saber and Murray problem of establishing establishing new interconnections interconnections under dif- A A major issue time-delay consensus networks, which ferent constraints and scenarios. Olfati-Saber and Murray A major issue in time-delay consensus networks, emerges the in problem of adding new interconnections, ferent constraints and scenarios. Olfati-Saber and Murray A major in issue in time-delay consensus networks, which which (2004) introduced algebraic connectivity of the network’s ferent constraints and scenarios. Olfati-Saber and Murray emerges in the problem of adding new interconnections, (2004) introduced algebraic connectivity of the network’s emerges in the problem of adding new interconnections, that new interconnections might deteriorate the perfor(2004) introduced algebraic connectivity of the network’s emerges in the problem of adding new interconnections, underlying graph as a measure of performance for first- is (2004) introduced algebraic connectivity of the network’s is interconnections might deteriorate perforunderlying graph as a measure of performance firstis that that new new interconnections might deteriorate the performance or even stability of the network. On thethe contrary, underlying graph as of for firstthat new interconnections might deteriorate the perfororder consensus Mosk-Aoyama (2008)for underlying graphnetworks. as a a measure measure of performance performance forproved first- is mance or even stability of the network. On the contrary, order consensus networks. Mosk-Aoyama (2008) proved mance or even stability of the network. On the contrary, in the absence of delay, new links will improve the H order consensus networks. Mosk-Aoyama (2008) proved 2 permance or even stability of the network. On the contrary, the problem of adding a prespecified number of edges order consensus networks. Mosk-Aoyama (2008)ofproved in the absence of delay, new links will improve the H per2 the problem of adding a prespecified number edges in the absence of delay, new links will improve the H 2 performance measure because of the monotonicity property the problem of adding a prespecified number of edges in the absence of delay, new links will improve the H to a network, maximizing its algebraic connectivity, is 2 perthe problem of adding a prespecified number of edges formance measure because of the monotonicity property to network, maximizing its Boyd algebraic connectivity, is [Siami formance because the property et measure al. (2016)]. Thus, of topology for timeto aaa network, maximizing its algebraic connectivity, is measure because ofdesigning the monotonicity monotonicity property Np-hard. However, Ghosh and (2006) suggested an to network, maximizing its Boyd algebraic connectivity, is formance [Siami et al. (2016)]. Thus, designing topology for timeNp-hard. However, Ghosh and (2006) suggested an [Siami et al. (2016)]. Thus, designing topology for timedelay networks is a more delicate task compared to the Np-hard. However, Ghosh and Boyd (2006) suggested an [Siami et al. (2016)]. Thus, designing topology for efficient heuristic theand Fiedler vector of suggested the graph an to delay networks is a more delicate task compared totimeNp-hard. However,using Ghosh Boyd (2006) the efficient heuristic using the Fiedler vector of the graph to delay networks is a more delicate task compared to the situation that time-delay is absent. efficient heuristic using the Fiedler vector of the graph to delay networks is a more delicate task compared to the address the combinatorial problem. Other measures of perefficient heuristic using theproblem. Fiedler Other vectormeasures of the graph to situation that time-delay is absent. address the combinatorial of persituation that time-delay is absent. address the combinatorial problem. Other measures of persituation that time-delay is absent. ⋆ address the combinatorial Other measures of perThis research is supported byproblem. AFOSR YIP FA9550-13-1-0158, NSF ⋆ research is supportedand by AFOSR YIP FA9550-13-1-0158, NSF ⋆ CAREER ECCS-1454022, ONR YIP N00014-16-1-2645. This is YIP FA9550-13-1-0158, ⋆ This This research research is supported supported by by AFOSR AFOSR YIP FA9550-13-1-0158, NSF NSF CAREER CAREER ECCS-1454022, ECCS-1454022, and and ONR ONR YIP YIP N00014-16-1-2645. N00014-16-1-2645. CAREER ECCS-1454022, and ONR YIP N00014-16-1-2645.

Copyright © 2016, 2016 IFAC 228Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2016 IFAC 228 Copyright © 2016 IFAC 228 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 228Control. 10.1016/j.ifacol.2016.10.401

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Our Contribution: In light of Ghaedsharaf et al. (2016), in Section 3 we formulate the problem of adding new interconnections to the time-delay first-order consensus network. Further, in Section 4, we find a highly accurate approximation function that would spare us from the complexity of eigen-decomposition of the Laplacian matrix. Moreover, by utilizing that approximate function, we find the semidefinite programming (SDP) relaxation of the problem. In addition, we find bounds on the optimality degree if we use the approximate function instead of the original function. Notwithstanding that the SDP is a relaxation of the main problem, it provides us with a good lower bound on the best achievable performance. In Section 5 we take advantage of the approximate function that we introduced in Section 4 and we propose a greedy algorithm to tackle our combinatorial problem. Furthermore, we show that our approximate cost function has the submodular property which allows us to find bounds on the optimality of our greedy method. Lastly, Section 6 is devoted to numerical examples for demonstrating the utility of our results. All the proofs of this paper are omitted due to space limitations.

2.2 Time-delay noisy linear consensus networks In this paper, we consider a class of linear consensus networks in which each node resembles a subsystem with a scalar state variable. We assume that each node has a delay in reacting to other nodes or has a delay in computing or accessing its own state. Thus, the states of each subsystem in absence of external disturbances evolves through the following differential equation x˙ i (t) =

g(Dx) ≤ g(x).

(1) S

Definition 2. Let S be a finite set. A function h : 2 → R is sub-modular if for any S1 ⊂ S2 ⊂ S and x ∈ S \ S2 , h(S1 ∪ {x}) − h(S1 ) ≥ h(S2 ∪ {x}) − h(S2 ). (2) 229

  aij xj (t − τ ) − xi (t − τ ) ,

(3)

where τ ≥ 0 is the delay parameter and aij is the ij th component of the adjacency matrix from the underlying coupling graph. Thus, we assume that the delay affects both the neighbors and the node itself. Consequently, we study the following first-order consensus network with n nodes having homogeneous delay and underlying graph Laplacian L:

2. PRELIMINARIES AND DEFINITIONS

Throughout the paper the following notations will be used. We denote the transpose and conjugate transpose of matrix A by AT and AH , respectively. Also, the set of non-negative (positive) real numbers is indicated by R+ (R++ ). An undirected weighted graph G is denoted by the triple G = (V, E,w) where V = {v1 , v2 , . . . , vn } is the set of nodes of the graph, E is the set of links of the graph and w : E → R++ is the weight function that maps each link to a positive scalar. We let L to be the Laplacian of the graph, defined by L = ∆−AG , where ∆ is the diagonal matrix of node degrees and AG is the adjacency matrix of the graph. Each node i at time t has a scalar state which is xi (t). Vector of all ones in Rn is denoted by 1n and 1 Jn = 1n 1T n . Furthermore, we let Mn = In − n Jn and we refer to it as the centering matrix. For an undirected graph with n nodes, the resulting Laplacian eigenvalues are real and showed by 0 = λ1 ≤ λ2 ≤ · · · ≤ λn . Also, for † a given Laplacian matrix L, we denote by L† = [lij ] the Moore-Penrose pseudo-inverse of L and by r(L) = [rij (L)] † the effective resistance matrix, with entries rij (L) = lii + † † † ljj − lij − lji . Additionally, we use diag() as the operator that maps a vector y ∈ Rn to a diagonal square matrix Y ∈ Rn×n with components of y on the main diagonal of Y . In addition, we denote maximum degree of nodes of the graph, by dmax . Also, for a given function f we denote its supremum in its domain by �f �∞ . Definition 1. g : Rn → R is a Schur-convex function if for every doubly stochastic matrix D ∈ Rn × n and all x ∈ Rn ,

 j�=i

N (L; τ ) :

2.1 Basic Definitions

229



x(t) ˙ = − Lx(t − τ ) + ξ(t), y(t) = Mn x(t),

(4)

where ξ(t) ∈ Rn is a vector of independent Gaussian white noise process with zero mean and identity covariance, i.e.,   E ξ(t1 )ξ T (t2 ) = In δ(t1 − t2 ), (5)

and δ(t) is the Dirac’s delta function. Definition 3. The H2 -norm squared performance of the dynamical system (4) from input ξ to output y is defined as the following quantity ρss (L; τ ) = lim E t→∞

n 

(xi (t) − x ¯(t))2

i=1



= lim E y T (t)y(t) t→∞

where x ¯(t) is the following scalar



 (6)

n

x ¯(t) =

1 xi (t). n i=1

(7)

Theorem 4. (Ghaedsharaf et al. (2016)). In presence of a time-delay τ ≥ 0, the performance measure of the linear consensus network (4), can be written as an additively separable function of eigenvalues from the Laplacian matrix. In other words, we have the following ρss (L; τ ) =

n 

fτ (λi ),

(8)

i=2

where 1 cos(λi τ ) (9) 2λi 1 − sin(λi τ ) and λi for i = 2, . . . , n are Laplacian eigenvalues of the underlying graph. Corollary 5. For a fixed underlying graph, performance of the consensus network (4) is an increasing function of timedelay, i.e., for 0 ≤ τ1 < τ2 < 2λπn , we have the following fτ (λi ) :=

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ρss (L; τ1 ) < ρss (L; τ2 ).

(10)

Proposition 6. (Ghaedsharaf et al. (2016)). For a fixed delay, the performance measure of the consensus network (4) is a convex and Schur-convex function of Laplacian eigenvalues from its underlying graph. Proposition 7. (Ghaedsharaf et al. (2016)). For the firstorder linear consensus network (4) the limit on performance of this network is (n −1)τ �, ρss (L; τ ) ≥ � 2 1 − sin(z ∗ )

(11)

where z ∗ is constant for all networks and is the unique positive solution of

prohibitively large, we need efficient methods to tackle the problem. 4. COST FUNCTION APPROXIMATION AND SDP RELAXATION Since the objective function given in (16) is convex function of feedback weights of feedback couplings and all the constraints except (19) are convex, we derive the following convex relaxation�of our problem by replacing the constraint (19) with e∈Ec we ≤ k̟, minimize

subject to

ρss (L + LF ; τ ) � w e be bT LF = e,

(20)

(21)

e∈Ec

cos(z) = z.

0 � L + LF ≺ � we ≤ k̟,

(12)

3. PROBLEM FORMULATION

Therefore, we look at the following modified consensus dynamics of (4),  ˙ = − Lx(t − τ ) + u(t) + ξ(t), x(t) N (L + LF ; τ ) : u(t) = − LF x(t − τ ), (13)  y(t) = Mn x(t),

that can be written in closed-loop form as follows, �

x(t) ˙ = − (L + LF )x(t − τ ) + ξ(t), y(t) = Mn x(t),

e∈Es

and be is the column of vector-to-edge incidence matrix corresponding to edge e. Our goal is to improve performance of the noisy network in presence of delay by designing a sparse Laplacian feedback gain LF with at most k coupling each of which has predetermined weight ̟. That is, we want to solve the problem

subject to

ρss (L + LF ; τ ) � w e be bT LF = e,

(16) (17)

e∈Es

0 � L + LF ≺

π 2τ In ,

(18)

|Es | ≤ k

Es ⊂ E c ,

(19)

,

(23)

In spite of the convexity of the cost function and constraints given by (20)-(23), the structure of the cost function is not appealing since we cannot cast it as SDP or solve it using conventional methods, e.g., interior point or subgradient methods, we have to find eigenvalues of L + LF for each step of minimizing the performance function; which takes at least cubic time for each iteration and thus, even for moderate sized networks it is not a practical approach. Therefore, we need a way to remove eigen-decomposition from our solution. For that purpose, we introduce a function that approximates our performance function ρss (L; τ ) and has a very small relative error with respect to our performance function (less than %0.1). Recall that

(14) where LF is the Laplacian matrix of the feedback gain and can be written as � LF = w e be bT (15) e,

minimize

(22)

e∈Ec

In this section we discuss the problem of enhancing the performance of our network by adding a number of feedback interconnections with deliberate coupling weights from a candidate set of links Ec .

N (L + LF ; τ ) :

π 2τ In ,

We note that condition (18) results in stability of the network (14). Since the problem given by (16)-(19) is combinatorial, the solution can be found by a �brute-force � �k search and appraising ρss (L + LF ; τ ) for i=1 |Eic | cases. In real-world problems, when the size of candidate set is 230

n � 1 cos(τ λi ) ρss (L; τ ) = , 2λ 1 − sin(τ λi ) i i=2

by multiplying the nominator and denominator by τ we get n � 1 cos(τ λi ) ρss (L; τ ) = τ 2τ λi 1 − sin(τ λi ) i=2 = τ

n �

f1 (τ λi ),

(24)

i=2

1 cos(x) where f1 (x) = 2x 1−sin(x) with domain x ∈ (0, π/2) based on definition of fτ in (9). As a means to find a proper approximate performance function, we look for an approximation of f1 and we denote it by f˜. Since f1 has two vertical asymptotes inside its effective domain, we want to have bounded �f˜ − f1 �∞ over effective domain of these functions. To that end, we propose

� 1�1 4 1 f˜(x) = + + c0 + c 1 x , (25) 2 x π π/2 − x as approximation of f1 where c0 = 0.18733 and c1 = −0.01 are constants to minimize relative error. We define our performance approximate function by substituting f˜ for f1 in (24). Thus, it yields

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5. GREEDY ALGORITHM

18

f1 (λi ) f˜(λi )

16

In spite of the fact that the SDP relaxation of our problem can be solved using conventional SDP solvers, it cannot be utilized to improve performance of a moderately sized problem (more than 20000 candidate links) since it would typically need too large amount of memory. To address this issue, and in light of Proposition 8, we propose a greedy algorithm to tackle the problem given in (16)(19) for moderately sized networks. An undesirable naive procedure for the greedy algorithm, is to compute the performance after adding the candidate links, one at a time, which involves finding the pseudo-inverse for each candidate link at each step.

f1 (λi ), f˜(λi )

14

12

10

8

6

4

2

231

0

0.5

λi

1

1.5

As it was mentioned, a positive aspect of using ρ˜ss as performance function is that it spares us the complexity of using eigen-decompsition for the Laplacian matrix. In addition, the following proposition would be another positive aspect of utilizing ρ˜ss instead of ρss . Proposition 9. Let Le be the rank one weighted Laplacian matrix of a graph with only a single edge e between ith and j th nodes with weight we . Then,

Fig. 1. Comparison of f1 and f˜.

ρ˜ss (L; τ ) = τ

n �

f˜(τ λi )

i=2

n � � 4 1� 1 1 =τ + + c0 + c1 τ λi 2 τ λi π π/2 − τ λi i=2 � �† � c0 τ 4 �π Mn − τ L + c1 τ L + τ (n−1). = Tr (τ L)† + 2 π 2 2 (26)

Replacing ρss by ρ˜ss in problem (20)-(23) and further neglecting the constant term in (26), we can cast the problem as the following SDP � � (27) minimize Tr E + π4 F + c1 τ LF � T subject to LF = w e be be , (28)

ρ˜ss (L + Le ; τ ) = ρ˜ss (L; τ ) + c(e), where rij (L2 ) + c 1 τ 2 we 2/we + 2rij (L) � � rij ( π2 Mn − τ L)2 2τ . − π −1/(we τ ) + rij ( π2 Mn − τ L)

c(e) := −

we ≤ k̟,

e∈Ec   E I  � 0,  I τ (L + LF ) + n1 J   F I  � 0.  π I 2 In − τ (L + LF )

ρ˜ss (L + Le ; τ ) |τ =0 = ρ˜ss (L; τ ) |τ =0 −

−4

where ǫ = 2.4 × 10

rij (L2 ) , 2/we + 2rij (L)

(29)

which is the contribution of a new edge on the H2 performance in the special case that we have no time-delay [Summers et al. (2015) and Siami et al. (2016)].

(30)

We aim to choose a subset of candidate links with at most k elements, therefore we look at structural properties of our cost function. In the following proposition, we prove that −˜ ρss is a submodular set function. Submodularity plays a constructive role in optimization problems on set functions and, in fact, it is the counter part of convexity in continuous optimization problems. Herein, we use submodularity to find a lower bound on the optimality of our results. Proposition 11. For a connected underlying graph G with Laplacian L, let Ec be set of candidate new edges. Then h : 2Ec → R defined as � h(Es ) = ρ˜ss (L; τ ) − ρ˜ss (L + w e be bT (35) e ; τ ),

(31)

The following Proposition investigates the optimality degree for the solution of the SDP given in (27)-(31). ˆ be solutions of a minimizaProposition 8. Let L∗ and L tion problem with respectively ρss and ρ˜ss as the cost function and same constraints. Then, we have ˆ τ ) ≤ (1 + ǫ)ρss (L∗ ; τ ), ρss (L∗ ; τ ) ≤ ρss (L;

(34)

Remark 10. If we let τ = 0 in (33), we obtain

e∈Ec

�

(33)

(32)

.

We note that Proposition 8 which discusses efficacy of ρ˜ss , is independent of the optimization problem constraints, implying that it can be utilized in other topology design scenarios, such as the combinatorial problem given by (16)(19). 231

e∈Es

is a submodular set function. Remark 12. Proposition 11 discusses submodularity of −˜ ρss and not that of −ρss . The reason is that it is not practical to use ρss in optimization problems; thus, we

2016 IFAC NECSYS 232 September 8-9, 2016. Tokyo, Japan

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utilize ρ˜ss and by Proposition 8 we already know that ρ˜ss is a very suitable approximation function. For an edge e ∈ Ec \ Es , we denote contribution of new edge e to h(Es ) by he (Es ) which is defined by he (Es ) := h(Es ∪ {e}) − h(Es ).

(36)

Although H2 performance of a network with dynamics given in (4) is not monotone with respect to new interconnections in the underlying graph [Ghaedsharaf et al. (2016)], if we have upper bound on time-delay, we will see the monotonicity property. In other words, if z∗ , 2̟(n − 2) + 2dmax

(37)

 then, ρss (L+ e∈Es ̟be bT e ; τ ) will be a monotone function, where z ∗ is the constant defined by solution of (12). Particularly, if (37) hold h(Es ) will be a monotone function. The advantage of having h(Es ) to be a submodular monotone function, is that the Simple Greedy algorithm with cardinality constraints, provides a 1 − 1/e approximation to the following problem [Nemhauser et al. (1978)]. maximize

h(Es )

(38)

subject to

Es ⊂ Ec , | Es |≤ k

(39)

30

Simple Greedy SDP 25

20

15

10

5

In the case that h(Es ) is not a monotone function, a randomized greedy provides a 1/e approximation to the cardinality constraint problem [Buchbinder et al. (2014)], which will not be discussed in this paper due to space limitations. Algorithm 1 Simple Greedy 1: Initialize: 2: Es = ∅ 3: for i = 1 to k do: 4: ei = arg max he (Es )

0

0

500

1000

1500

2000

2500

Number of new links (Greedy) \Total weight of new links(SDP)

Fig. 3. Improving performance of the network given in Figure 2 by adding new interconnections 6. NUMERICAL EXAMPLES

e⊂Ec \Es

5: 6: 7: 8:

Fig. 2. Arbitrary unweighted graph with 125 nodes and 250 edges

ρss

τ<

if hei (Es ) ≤ 0: break Es ← Es ∪ {ei }

return Es

5.1 Time Complexity Analysis While the main goal of this paper is not a fast algorithm, we analyze time complexity of our greedy methods. First,  we need to find the pseudo-inverse for L and π2 Mn − τ L which has complexity of O(n3 ) and then we find the square of each matrix pseudo-inverse which takes O(n3 ). Using the Sherman-Morrison formula [Hager (1989)] for rank-one update, we can for r(L), r(L2 ),    find the rank-one update r π2 Mn − τ L and r ( π2 Mn − τ L)2 in O(n2 ). Then, finding the contribution for each link takes constant time for each link. In conclusion, the first step needs O(n3 ) and the rest of the steps take O(n2 ). This is less time comparing to the method of Summers et al. (2015), since despite using the Sherman-Morrison formula, their method eventually needs O(n) to find the contribution of each link in each step and since we have O(n2 ) candidate links, their algorithm has time complexity of O(n3 ) in all steps. 232

In this section we consider the following numerical examples to demonstrate the utility of our results. Example 13. Consider the arbitrary network (4) with 125 nodes and initially 250 unweighted links given by Figure 2 in presence of τ = 0.017 delay. We design an optimal topology for the network using both SDP relaxation and Simple Greedy. Then we compare it to the hard limit, to see how close we could get to the lower bound of the solution. We add 7,500 new links using SDP method and 2439 new links using Simple Greedy algorithm. We see that H2 -norm performance of the network has improved by 88.8%, reaching 3.27 using greedy method which initially was 29.28. Using Proposition 7, the hard limit for the performance of the network is 3.237 and we know that the global optimal for the problem is greater than the hard limit. It should be noted that there exists only 1.2% difference between hard limit and the new performance of the designed network using greedy method and even smaller gap for SDP. Figure 3 compares result of the SDP and the Simple Greedy algorithm. Example 14. Here we want to evaluate the strength of establishing new interconnections using our greedy algorithm. To that end, we use our greedy algorithm to add edges to a randomly generated graph given in Figure 4 which has 10 nodes and 15 edges initially. Here we deal

2016 IFAC NECSYS September 8-9, 2016. Tokyo, Japan

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8

2 10

5

4

6

1

9 7 3

Fig. 4. Arbitrary unweighted graph with 10 nodes and 15 links 2.6

Simple Greedy Brute-Force Search

2.4

2.2

2

ρss

1.8

1.6

1.4

1.2

1

0.8

0

2

4

6

8

10

12

14

number of new interconnections

16

Fig. 5. Improving performance of the network given in Figure 4 by adding new interconnections with τ = 0.05 homogeneous time-delay. Moreover, we suppose that set of candidate edges are complement of the set of initial edges. In this example we intend to establish up to 16 new interconnections. Despite the 1 − 1/e lower bound on the degree of optimality for the greedy algorithm, as it is depicted in Figure 5, the algorithm yields extremely good results. 7. CONCLUSIONS In this manuscript we aimed to improve H2 -norm performance of first-order time-delay consensus networks by adding new interconnections. We offered two practical approaches for that purpose. In addition, we studied optimality of our solutions to the problem through the submodular property of our cost function. Although both SDP and greedy method generated favorable results in our extensive simulations, the bounds on optimality of our SDP relaxation is higher in comparison with that of greedy algorithm. On the other hand, we are not able to solve SDP on conventional computers and solvers for more than a few hundred nodes. While our results are promising for a network with a few thousand nodes, in future works, we intend to find faster approaches for large-scale networks. 8. ACKNOWLEDGEMENT We would like to thank Christoforos Somarakis for reading this manuscript and his useful comments. REFERENCES Bamieh, B., Jovanovi´c, M.R., Mitra, P., and Patterson, S. (2012). Coherence in large-scale networks: Dimensiondependent limitations of local feedback. Automatic Control, IEEE Transactions on, 57(9), 2235–2249. Buchbinder, N., Feldman, M., Naor, J.S., and Schwartz, R. (2014). Submodular maximization with cardinality 233

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