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A comparative study of persistence based convergence rate estimates to consensus
Control of 1st Nonlinear Systems on Modelling, Identification and Preprints, IFAC Conference Control of 1st Nonlinear Systems Preprints, IFACSaint Conference on Modelling, June 24-26, 2015. Petersburg, Russia Identification and Control of Nonlinear Systems June 24-26, 2015. Saint Petersburg, Russia Available online at www.sciencedirect.com Control of Nonlinear Systems June 24-26, 2015. Saint Petersburg, Russia June 24-26, 2015. Saint Petersburg, Russia
ScienceDirect IFAC-PapersOnLine 48-11 (2015) 534–539 A comparative study of persistence based A comparative study of persistence based A comparative study of persistence based Aconvergence comparative study of persistence based rate estimates to consensus convergence rate estimates to consensus convergence rate rate estimates estimates to to consensus consensus convergence Nilanjan Roy Chowdhury ∗ Srikant Sukumar ∗
Nilanjan Roy Chowdhury ∗∗ Srikant Sukumar ∗∗ Nilanjan Roy Chowdhury ∗ Srikant Sukumar ∗ Roy Chowdhury ∗ Systems andNilanjan Engineering, IndianSrikant InstituteSukumar of Technology Bombay, ∗ Systems and Control Control Engineering, Indian of Technology Bombay, ∗ Systems {nilan Institute Mumbai-400076, (e-mail: Indian jan, srikant}@sc.iitb.ac.in) and ControlIndia, Engineering, Institute of Technology Bombay, ∗ Systems {nilan Institute Mumbai-400076, India, (e-mail: Indian jan, srikant}@sc.iitb.ac.in) and Control Engineering, of Technology Bombay, Mumbai-400076, India, (e-mail: {nilan jan, srikant}@sc.iitb.ac.in) Mumbai-400076, India, (e-mail: {nilan jan, srikant}@sc.iitb.ac.in) Abstract: This article presents a comparative study of convergence rate estimates for degenerate Abstract: This article presents a comparative study of convergence rate estimates for degenerate gradient flows the context of multi-agent systems. analysis rate methodology introduced Abstract: Thisinarticle presents a comparative study Aof novel convergence estimates was for degenerate gradient flows in the context of multi-agent systems. A novel analysis methodology was introduced Abstract: This article presents a comparative study of convergence rate estimates for degenerate in (Roy Chowdhury Srikant, 2014) based systems. on the classical of persistence of was excitation (PE) gradient flows in theand context of multi-agent A novelnotions analysis methodology introduced in (Roy Chowdhury and Srikant, 2014) based on thethe classical notions of persistence of was excitation (PE) gradient flows in the context of multi-agent systems. A novel analysis methodology introduced and uniform complete observability (UCO) to study exponential stability of consensus protocol with in (Roy Chowdhury and Srikant, 2014) based on the classical notions of persistence of excitation (PE) and uniform complete observability (UCO) to study the exponential stability of consensus protocol with in (Roy Chowdhury and Srikant, 2014) based on the classical notions of persistence of excitation (PE) time-varying agent dynamics. The new technique precise computation of the convergence rate and uniform complete observability (UCO) to studyallowed the exponential stability of consensus protocol with time-varying agent dynamics. The new technique allowed precise computation of the convergence rate and uniform complete observability (UCO) to study the exponential stability of consensus protocol with for single integrator agent dynamics. Thetechnique purpose of this article is principally twofold. we revisit time-varying agent dynamics. The new allowed precise computation of the Firstly, convergence rate for single integrator agent dynamics. The purpose of this article is principally twofold. Firstly, we revisit time-varying agent dynamics. The new technique allowed precise computation of the convergence rate the single continuous time agent spanning edge agent dynamics proposed (Roy Chowdhury Srikant, 2014) for integrator dynamics. The purpose of this article in is principally twofold.and Firstly, we revisit the continuous time spanning edge agent dynamics proposed in (Roy Chowdhury and Srikant, 2014) for integrator dynamics. The purpose article in is principally twofold. Firstly, revisit andsingle present a different analysis approach todynamics studyof thethis convergence property for the same. Ourwe analysis the continuous time agent spanning edge agent proposed (Roy Chowdhury and Srikant, 2014) and present a different analysis approach to study the convergence property for the same. Our analysis the continuous time spanning edge agent dynamics proposed in (Roy Chowdhury and Srikant, 2014) mimics the recent resultanalysis by Brockett (Brockett, on single agent adaptive control. the and present a different approach to study2000) the convergence property for the same.Secondly, Our analysis mimics the recent result byarrived Brockett (Brockett, 2000) on single agent adaptive control. Secondly, the and present arate different approach tothe study the convergence property forSimulations the same.Secondly, Our analysis convergence estimates at from two approaches compared. on different mimics the recent resultanalysis by Brockett (Brockett, 2000) on singleareagent adaptive control. the convergence rate estimates arrived at from the two approaches are compared. Simulations on different mimics the recent result by Brockett (Brockett, 2000) on single agent adaptive control. Secondly, the examples are rate provided to elucidate mainthe theoretical contribution. convergence estimates arrived the at from two approaches are compared. Simulations on different examples are provided to elucidate the main theoretical contribution. convergence estimates arrived the at from two approaches are compared. Simulations on different examples are rate provided to elucidate mainthe theoretical contribution. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. examples are provided to elucidate the main theoretical contribution. 1. INTRODUCTION edges. 1. INTRODUCTION edges. Another body of work in relation to degenerate gradient flows 1. INTRODUCTION edges. Another body of work in relation to degenerate gradient flows 1. INTRODUCTION Consensus is characterized as a framework in which several edges. involves time-varying dynamics of the form (Andersson Another body of worksystem in relation to degenerate gradient flows Consensus is characterized as a framework in which several involves time-varying system dynamics of the form (Andersson Another body of work in relation to degenerate gradient flows agents collaborate with eachasother and come in to which an agreement and Krishnaprasad, 2002), Consensus is characterized a framework several involves time-varying system dynamics of the form (Andersson agents collaborate with eachkey other and come to an agreement and Krishnaprasad, 2002), Consensus is characterized aobjective framework in several involves time-varying system dynamics of the form (Andersson over a common value. of consensus control agents collaborate withThe eachasother and come to which an agreement and Krishnaprasad, 2002), ˙ (1) φ˙ (t) = −u(t)uTT (t)φ (t) over a common value. The key objective of consensus control agents collaborate with each other and come to an agreement and Krishnaprasad, 2002), is to design a relative feedback algorithm that enables the agents (1)n φ (t) = −u(t)uT (t)φ (t) over a common value. The key objective of consensus control ˙ (t) = n is defined (t)φ (t) (1) φ −u(t)u is to design a relative feedback algorithm that enables the agents over a common value. The key objective of consensus control Here, φ (t) ∈ R as the parameter error and u(t) ∈ Rn T ˙ (t) =as in to thedesign network to converge to algorithm the consensus value. Consensus is a relative feedback that enables the agents Here, φ (t) ∈ Rnn is defined (t)φ (t) φ −u(t)u the parameter error and u(t) ∈ (1) Rn in the network to converge to the consensus value. Consensus is to design a relative feedback algorithm that enables the agents is the appropriate filter state. Seminal contributions in analyzing Here, φ (t) ∈ R is defined as the parameter error and u(t) ∈ Rn control has been multitude of application ar- is the appropriate in the network to envisioned converge tointhea consensus value. Consensus n isfilter state. Seminal contributions in analyzing Here, φ (t) ∈ R defined as the parameter error and u(t) ∈ R control has been envisioned in a multitude of application arin the network to converge to the consensus value. Consensus the stability of (1) have been made in (Anderson, 1977) where is the appropriate filter state. Seminal contributions in analyzing eas suchhas as,been flocking (Olfati-Saber, 2006), satellite formation control envisioned in a multitude of application ar- is stability of (1)filter have beenSeminal made in to (Anderson, 1977) where appropriate in analyzing eas such as, flocking (Olfati-Saber, 2006), satellite formation control has a multitude of application ar- the author proposes astate. methodology study stability of (1) thethe stability of (1) have been made incontributions (Anderson, 1977) where flying (Sarlette etenvisioned al.,(Olfati-Saber, 2007),in opinion dynamics (Morarescu eas such as,been flocking 2006), satellite formation author proposes a methodology to study stability of (1) the stability of (1) have been made in (Anderson, 1977) where flying (Sarlette et al.,(Olfati-Saber, 2007), opinion dynamics (Morarescu eas as,2011) flocking 2006), satellite formation provides a convergence rate estimate. Recently (Brockett, the author proposes a methodology to study stability of (1) and such Girard, problem (Scardovi and and flying (Sarlette et and al., synchronization 2007), opinion dynamics (Morarescu and provides a convergence rate estimate. Recently (Brockett, the author proposes a methodology to study stability of (1) and Girard, 2011) and synchronization problem (Scardovi and and flying (Sarlette et and al., synchronization 2007), opinion problem dynamics (Morarescu 2000) re-examined the above rate system dynamics and proposed an provides a convergence estimate. Recently (Brockett, Sepulchre, 2009). and Girard, 2011) (Scardovi and and 2000) re-examined the above system dynamics and proposed an provides a convergence rate estimate. Recently (Brockett, Sepulchre, 2009). and Girard, 2011) and synchronization problem (Scardovi and explicit convergence rate for the same. A comparative study of 2000) re-examined the above system dynamics and proposed an In recent times, Sepulchre, 2009).consensus algorithms have drawn significant 2000) explicit convergence rate for the same. A comparative study of re-examined the above system dynamics and proposed an In recent times, consensus algorithms have drawn significant Sepulchre, 2009). these two estimates is shown in (Andersson and Krishnaprasad, explicit convergence rate for the same. A comparative study of research attention in the field of control. In (Ren and Beard, In recent times, consensus algorithms have drawn significant explicit these two estimates is shown in (Andersson and Krishnaprasad, convergence rate for the same. A comparative study of research attention in the field of control. In (Ren and Beard, In recent times,2)consensus algorithms significant these two estimates is shown in (Andersson and Krishnaprasad, 2008, chapter the authors suggest ahave continuous time up- 2002). research attention in the field of control. In drawn (Ren and Beard, 2002). these two estimates is shown in (Andersson and Krishnaprasad, 2008, chapter 2) the authors suggest a continuous time up- 2002). research attention in the fieldagent of control. In (Ren and Beard, In this article we compare the two afore-mentioned estimates date law for single integrator dynamics considering both 2008, chapter 2) the authors suggest a continuous time up- 2002). In article we the systems. two afore-mentioned estimates date law for single integrator agent dynamics considering both 2008, chapter 2) the authors suggest a continuous time upin this the context of compare multi-agent Firstly, we revisit the In this article we compare the two afore-mentioned estimates constant andsingle switching graphagent structures. It is considering well established date law for integrator dynamics both In in the context of multi-agent systems. Firstly, we revisit the this article we compare the two afore-mentioned estimates constant and switching graph structures. It is well established date law for single integrator agent dynamics considering both spanning edge agent dynamics proposed in (Roy Chowdhury in the context of multi-agent systems. Firstly, we revisit the that the second smallest eigenvalue of the graph laplacian constant and switching graph structures. It is well established in spanning edge agent dynamics proposed in (Roy Chowdhury the context of multi-agent systems. Firstly, we revisit that the and second smallest eigenvalue of the graphestablished laplacian and constant switching graph structures. well Srikant, utilize results by in (Brockett, 2000)the to spanning edge2014) agentand dynamics proposed (Roy Chowdhury matrix the convergence rate of inItthe a isstatic that theregulates second smallest eigenvalue graphframework. laplacian spanning and Srikant, 2014) utilize results byWe (Brockett, 2000) to edge agentand dynamics proposed in (Roy Chowdhury matrix regulates the convergence rate in a static framework. that the second smallest eigenvalue of the graph laplacian determine the explicit convergence rate. compare the new and Srikant, 2014) and utilize results by (Brockett, 2000) to Likewise, the authors of (Ren andrate Beard, 2005) framework. employing determine the explicit convergence rate. We compare the new matrix regulates the convergence in a static and Srikant, 2014) and utilize results by (Brockett, 2000) to Likewise, the authors of (Ren and Beard, 2005) employing matrix regulates the convergence rate in a static framework. rate estimate to the one developed in (Roy Chowdhury and determine the explicit convergence rate. We compare the new the notions of Stochastic Indecomposable Aperiodic (SIA) maLikewise, the authors of (Ren and Beard, 2005) employing determine estimate the on one developedlemma in (Roy Chowdhury and the to explicit convergence We compare the new the notions of Stochastic Indecomposable Aperiodic (SIA) ma- rate Likewise, the authors ofunder (Renswitching and Beard, employing 2014) based Anderson’s (Sastry and Bodson, rate estimate to the one developedrate. in (Roy Chowdhury and trices, prove for2005) both continuous the notions ofconsensus Stochastic Indecomposable Aperiodic (SIA) ma- Srikant, Srikant, 2014) based on Anderson’s lemma (Sastry and Bodson, rate estimate to the one developed in (Roy Chowdhury and trices, prove consensus under switching for both continuous the StochasticHowever, Indecomposable Aperiodic (SIA) map. 2014) 73-74). Srikant, based on Anderson’s lemma (Sastry and Bodson, andnotions discrete most available references fall 2011, trices, proveofdynamics. consensus under switching for both continuous 2011, p. 73-74). Srikant, 2014) based on Anderson’s lemma (Sastry and Bodson, and discrete dynamics. However, most available references fall trices, prove consensus under switching for both continuous The rest of this article unfolds as follows. In section 2, a p. 73-74). shortdiscrete on establishing an explicit rate available estimate. Recently, the 2011, and dynamics. However, most references fall The rest of this article unfolds as follows. section 2, a p. 73-74). short on(Roy establishing an explicit rate 2014) estimate. Recently, the 2011, and discrete dynamics. However, most available references fall succinct overview of system theory with theIn The rest of this article unfolds as follows. In mathematical section 2, a authors Chowdhury and Srikant, introduced a new short on establishing an explicit rate estimate. Recently, the The succinct overview of different system theory with the mathematical rest of this article unfolds as follows. In section 2,area authors (Roy Chowdhury and Srikant, 2014) introduced a new short on establishing an explicit rate estimate. Recently, the foundations of the two convergence rate estimates succinct overview of system theory with the mathematical methodology for investigating consensus problems with atimeauthors (Roy Chowdhury and Srikant, 2014) introduced new succinct foundations of the two different convergence rate mathematical estimates are overview ofresult system theory with the methodology for investigating consensus problems with timeauthors (Roy Chowdhury and Srikant, 2014) introduced a new provided. Our main is presented in section 3 with foundations of the two different convergence rate estimates are varying persistent undirected graph networks for single inte- provided. Our main result is presented in section 3 with the methodology for investigating consensus problems with timethe foundations of the two different convergence rate estimates are varying persistent undirected graph networks for single intemethodology for investigating consensus problems with timesupporting proofs. In section 4 we present test cases to justify provided. Our main result is presented in section 3 with the grator agent dynamics. The advantage of the for newsingle methodolvarying persistent undirected graph networks inte- provided. supporting proofs. sectionis 4presented we present test cases3 to justify Our mainIn result in conclusions section with the grator agent dynamics. The advantage of the new methodolvarying persistent undirected graph networks single intethe main theoretical while the for this proofs. Incontribution section 4 we present test cases to justify ogy, based ondynamics. classical notions of persistence excitation and supporting grator agent The advantage of theoffor new methodolthe main theoretical contribution while the conclusions for this supporting proofs. In section 4 we present test cases to justify ogy, based on classical notions of persistence of excitation and grator agent dynamics. The advantage of the new methodolwork are summarized in section 5. the main theoretical contribution while the conclusions for this uniform complete observability, in an explicit convergence ogy, based on classical notions oflies persistence of excitation and the work are theoretical summarizedcontribution in section 5.while the conclusions for this main uniform complete observability, lies in an explicit convergence ogy, based onfor classical notions persistence of excitation and rate estimate consensus. Theoflies authors Chowdhury and work are summarized in section 5. uniform complete observability, in an(Roy explicit convergence work are summarized in section 5. BACKGROUND rate estimate for consensus. The authors (Roy Chowdhury and uniform complete observability, in an explicit 2. MATHAMETICAL Srikant, 2014)foranalyze the consensus problem byconvergence transformrate estimate consensus. The lies authors (Roy Chowdhury and 2. MATHAMETICAL BACKGROUND Srikant, 2014) analyze the consensus problem by transformrate estimate for consensus. The authors (Roy Chowdhury and 2. MATHAMETICAL BACKGROUND ing the node to an edge agreement one,transformutilizing Srikant, 2014)dynamics analyze the consensus problem by 2. MATHAMETICAL BACKGROUND ing the node dynamics to an edge agreement one, utilizing Srikant, 2014) analyzetransformation the consensus probleminby transformGraph Theory fundamentals a suitable coordinate proposed (Zelazo and 2.1 Algebraic ing the node dynamics to an edge agreement one, utilizing 2.1 Algebraic Graph Theory fundamentals aMesbahi, suitable coordinate transformation proposed in (Zelazo and ing the node dynamics to an edge agreement one, utilizing The authors (Shi andproposed Johansson, 2012) have a suitable 2011). coordinate transformation in (Zelazo and 2.1 Algebraic Graph Theory fundamentals Mesbahi, 2011). The authors (Shi and Johansson, 2012) have Algebraic Graphintroductory Theory fundamentals aMesbahi, suitable coordinate transformation proposed in (Zelazo and In this section some notation from algebraic graph also estimated convergence rates consensus laws basedhave on 2.1 2011). The authors (Shiofand Johansson, 2012) In this section some introductory from of algebraic graph also estimated convergence rates of consensus laws based on Mesbahi, 2011). The authors (Shi and Johansson, 2012) have is introduced. A detailednotation discussion the same is the notion of persistent graphs. the result In this section some introductory notation from algebraic graph also estimated convergence ratesHowever, of consensus lawsassumes based ona theory theory is introduced. A detailed discussion of the same is the notion of persistent graphs. However, the result assumes a also estimated convergence rates of consensus laws based on In this section some introductory notation from algebraic graph availableis in (Mesbahi Aand Egerstedt, 2010, of chapter 2). An stringent assumption and the rate on cyclea theory introduced. detailed discussion the same is the notionarc-balance of persistent graphs. However, the depends result assumes availableis in (Mesbahi Aand Egerstedt, 2010, of chapter 2). An stringent arc-balance assumption and the rate depends on cyclea theory the notion of persistent graphs. However, the result assumes introduced. detailed discussion the same is stringent arc-balance assumption and the rate depends on cycle available in (Mesbahi and Egerstedt, 2010, chapter 2). An stringent arc-balance assumption and the rate depends on cycle available in (Mesbahi and Egerstedt, 2010, chapter 2). An
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MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
undirected graph G is a pair (V, E) where V (G ) = {v1 , · · · , vn } is a finite non empty node set (agents) and E(G ) = {e1 , · · · , em } is an edge set (communication links). {i, j} ∈ E(G ) is an undirected edge if agents vi and v j exchange information with each other. A graph is connected, if for every pair of vertices in V (G ), there is a path that has them as its end vertices. D(G o ) ∈ Rn×m is the incidence matrix of a graph G with arbitrary orientation O. The graph Laplacian matrix of an arbitrarily oriented graph G o is defined as, (2) L(G o ) = D(G o )W D(G o )T where, W ∈ Rm×m is the diagonal matrix with the weights w(ei ), i = {1, 2, ....m} on the diagonal entry. Furthermore, the edge Laplacian matrix is defined as, (3) Le (G o ) = D(G o )T D(G o ) In this article, we are principally dealing with undirected graphs therefore the more inconvenient D(G 0 ) notation will be exchanged for D(G ). For an undirected graph L(G ) is symmetric. The graph laplacian matrix is positive semi-definite with eigenvalues ordered as, λ1 ≤ λ2 ≤ · · · ≤ λn with λ1 = 0. The graph G is connected if and only if, λ2 > 0. For an undirected graph λ2 is defined as the algebraic connectivity. 2.2 Fundamental results The following elementary notions and results from systems theory are referred to, throughout the course of this paper. Definition 1. (Sastry and Bodson, 2011, p. 72) The signal u(·) : R → Rn×m is said to be Persistently Exciting (PE) if there exist finite positive constants µ1 , µ2 , T such that, µ2 In ≥
t+T t
u(τ)uT (τ)dτ ≥ µ1 In
∀t ≥ 0
(4)
where, In is the identity matrix of dimension n. Theorem 2. (Sastry and Bodson, 2011, pp. 73-74) Assume that, for all δ > 0, there exists Kδ ≥ 0 such that for all, t0 ≥ 0, t0 +δ t0
K(τ) 2 dτ ≤ Kδ
Consider the linear time-varying system [C(t), A(t)] as follows, x(t) ˙ = A(t)x(t) x(0) = x0 y(t) = C(t)x(t) with, [C(t), A(t) + K(t)C(t)] being the system with output feedback, x(t) ˙ = (A(t) + K(t)C(t)) x(t) y(t) = C(t)x(t) Then, the system [C(t), A(t)] is uniformly completely observable if and only if [C(t), A(t) + K(t)C(t)] is uniformly completely observable. Moreover, if the observability gramian of the system [C(t), A(t)] satisfies, β2 In ≥
t0 +δ t0
ΦTA (τ,t0 )CT (τ)C(τ)ΦA (τ,t0 )dτ ≥ β1 In
for all t0 ≥ 0 then the observability gramian of the system [C(t), A(t) + K(t)C(t)] also satisfy the above mentioned inequalities with identical choice of δ and, β1 β˜1 = 2 1 + Kδ β2 β˜2 = β2 e(Kδ β2 )
539
535
Theorem 3. (Sastry and Bodson, 2011, pp. 31-32) Let, Bh be a closed ball of radius h centered at 0 in Rn . If there exists a function V (t, x) and strictly positive constants α1 , α2 , α3 and δ such that for all x ∈ Bh , t ≥ 0 α1 x2 ≤ V (t, x) ≤ α2 x2
dV (t, x(t)) ≤0 dt t+δ dV (τ, x(τ)) dτ ≤ −α3 x(t)2 dτ t then, x(t) ∈ Rn exponentially converges to 0. Moreover, v(t, x) evolves according to,
V (t, x(t)) ≤ mv e−αv (t−t0 )V (t0 , (x(t0 )) where, 1 mv = 1 − αα32
t ≥ t0 ≥ 0
1 1 ln T 1 − αα32
αv =
2.3 Convergence rate estimates
In the following subsection we establish the two convergence rate estimates, we will compare. We begin with the classical result based on the Anderson’s lemma (Theorem 2). The following theorem is also cited in (Anderson, 1977) and (Sastry and Bodson, 2011, pp. 31-32). Theorem 4. Consider the degenerate gradient flows equations of the form, (5) y(t) ˙ = −u(t)uT (t)y(t) n Now, if u(t) ∈ R is persistently exciting (Definition 1) then, αv √ ||y(t)|| ≤ mv e− 2 (t−t0 ) ||y(t0 )|| (6) where, 1 mv = 1 − η2 1 αv = ln 1 − η 2 2 2T with, η2 =
µ1 √ 2 (1 + nµ2 )
Finally, we introduce the result by Brockett as follows (Brockett, 2000), Theorem 5. Recollect the dynamics of the degenerate gradient flows given in equation (5). Let, M(t) =
t 0
u(σ )uT (σ )dσ
If there exist positive constants, r, ε and T such that, for all, t ≥ 0 we have, M(t + T ) − M(t) ≥ εI and then with,
δ=
Tr [M(t + T ) − M(t)]3 ≤ r3
2r3
2ε − + 2 1 + 2ε 3 (1 + 2ε)
2r3 3 (1 + 2ε)2
MICNON 2015 536 Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539 June 24-26, 2015. Saint Petersburg, Russia
and,
κ 1 = ln 1 − δ 2 2 2T there exists a positive constant m such that, √ κ ||y(t)|| ≤ me 2 t ||y(0)||
(7)
2.4 Convergence of single integrator consensus algorithm Consider n identical single integrator agent dynamics as, (8) x˙i = ui with the classical consensus law, (Ren and Beard, 2008, p. 26), (Roy Chowdhury and Srikant, 2014), n
ui = −k ∑ ai j (t) (xi − x j ) j=1
i = 1, 2, · · · , n
(9)
Where, k > 0 and ai j (t) ≥ 0. Therefore, the closed loop dynamics can be written as, x˙ = −kL(G )x = −kD(G )W (t)D(G )T x (10) n×m , is the incidence matrix for the correspondD(G ) ∈ R ing arbitrary oriented graph and W (t) ∈ Rm×m is defined as a time-varying diagonal weight matrix with different g2i (t), i = {1, 2, . . . , m} on the diagonal entries representing the diverse inter-agent edge weights. Furthermore, the incidence matrix is partitioned as, D(G ) = [D(Gτ ) D(Gc )], where D(Gτ ) ∈ Rn×p and D(Gc ) ∈ Rn×(m−p) are defined as the incidence matrix corresponds to the spanning edges and the cycle edges respectively. Likewise, the diagonal matrix W (t) ∈ Rm×m can also be subdivided into two different block diagonal matrices as W (t) = diag [Wτ (t), Wc (t)] where, Wτ (t) ∈ R p×p represents the weighting functions corresponding to the edges of the spanning tree and Wc (t) ∈ R(m−p)×(m−p) represents the weights corresponding to the cycle edges. Here, p is the number of spanning tree edges. Now, consider the transformation from the node dynamics to the edge state, xe = D(G )T x. Therefore the edge dynamics evaluates as, x˙e = D(G )T x˙ = −kL˜ e (G )W (t)xe (11) m×m Here, L˜ e (G ) ∈ R (edge laplacian matrix), W (t) (weight matrix) and xe ∈ Rm (edge state vector) partitioned as follows, (see reference (Roy Chowdhury and Srikant, 2014)) T L˜ e (G ) = [D(Gτ ) D(Gc )] [D(Gτ ) D(Gc )] L˜ e (Gτ ) D(Gτ )T D(Gc ) = D(Gc )T D(Gτ ) L˜ e (Gc ) W (t) 0 W (t) = τ 0 Wc (t) x xe = τ xc Substituting all the above matrices in equation (11) the corresponding spanning edge state vector (xτ ∈ R p ) evaluates as, x˙τ = −kL˜ e (Gτ )RW (t)RT xτ where, R ∈ R p×m is defined as, [I Z] with, −1 D(Gτ )T D(Gc ) Z = D(Gτ )T D(Gτ ) p×p is symmetric and positive definite, it Since, L˜ e (Gτ ) ∈ R can be further decomposed as, L˜ e (Gτ ) = ΓΛΓT . Therefore, we introduce a set of modified states defined by the similarity transformation y = ΓT xτ , with dynamics, y˙ = ΓT x˙τ = −kΛΓT RW (t)RT Γy (12) 540
The closed loop dynamics defined in (10) and (12) with undirected time-varying persistent graph networks is studied rigorously in (Roy Chowdhury and Srikant, 2014). The following theorem mimics the main result of (Roy Chowdhury and Srikant, 2014) and used later in this article. Theorem 6. The continuous update law proposed in equation (9) guarantees that the class of multi-agent systems with single-integrator dynamics (8) and time-varying communication topology characterized by W (t), achieves consensus exponentially if Wτ (t) (spanning tree weight matrix) is persistently exciting (Definition 1) with the following convergence rate, 1 αv ln(1 − η 2 ) (13) − = 2 2T 2kλmin (Λ)µ1 where, η 2 = 2 √ 1 + k p Λ µ2 Here, Λ ∈ R p×p defined as the diagonal matrix, containing all the nonzero eigenvalues of the spanning tree edge laplacian matrix, whereas, µ1 I p , µ2 I p are specified as the persistence bounds for Wτ (Definition 1). A more involved proof of the theorem 6 is available in (Roy Chowdhury and Srikant, 2014). 3. MAIN RESULT 3.1 Stability analysis of consensus using Brockett’s estimate Reconsider, the spanning edge agent dynamics given in equation (12) as follows, y˙ = ΓT x˙τ = −kΛΓT RW (t)RT Γy (14) Without loss of generality we will assume that, k = 1. In order to prove the exponential stability of the above-mentioned spanning edge agent dynamics we establish the following lemma. Lemma 7. Define, M(T ) as, M(T ) =
T 0
ΓT RW (σ )RT Γdσ
Let, µ1 I p , µ2 I p be the lower and the upper bound for the given symmetric, non-negative matrix M(T ). Then, for a given persistent window, T > 0 we have, 3 {pµ2 } 2 λmax (Λ)||y(0)|| T −1 T −1 y (0)Λ y(0) − y (T )Λ y(T ) ≥ − 1 + 2µ1 λmin (Λ) 2 3 {pµ2 } 2 λmax (Λ) 2µ1 λmin (Λ) + + 1 + 2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} (15)
||y(0)||
Proof. We begin with a Lyapunov-like candidate function defined as follows, 1 V (y) = yT M(t)y + yT Λ−1 y 2 The time derivative of V (y) along the closed-loop dynamics (19) can be written as,
d 1 T −1 T ˙ V (y) = y M(t)y + y Λ y dt 2 1 1 T T ˙ = y˙ M(t)y + y M(t)y˙ + yT M(t)y + yT Λ−1 y˙ + y˙T Λ−1 y 2 2 = −2yT ΓT RW (t)RT ΓΛM(t)y (16)
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
Integrating both sides of equation (16) we obtain, T d 1 yT (σ )M(t)y(σ ) + yT (σ )Λ−1 y(σ ) dσ 2 0 dσ ≤ |2
T 0
yT (σ )ΓT RW (σ )RT ΓΛM(σ )y(σ )dσ |
(17)
The left hand side integral in equation (17) evaluates to, 1 1 yT (T )M(T )y(T ) + yT (T )Λ−1 y(T ) − yT (0)Λ−1 y(0) 2 2 ≤ |2
T 0
1
1
W 2 (σ )RT Γy(σ ) ·W 2 (σ )RT ΓΛM(σ )y(σ )dσ | (18)
which on applying the Cauchy-Schwartz inequality yields, |2
T
1
1
W 2 (σ )RT Γy(σ ) ·W 2 (σ )RT ΓΛM(σ )y(σ )dσ |
0 T 2 1 ≤ 2 W 2 (σ )RT Γy(σ ) dσ · 0
T 0
1
W 2 (σ )RT ΓΛM(σ )y(σ )
0
Likewise, we are also trying to upper bound the second factor. Since, ||y(t)|| is non-increasing, the second factor can be bounded above by, 2 T 1 W 2 (σ )RT ΓΛM(σ )y(σ ) dσ 0 T 0
1
2µ1 λmin (Λ) ||y(0)||2 λmax (Λ) {1 + 2µ1 λmin (Λ)} 2 3 {pµ2 } 2 λmax (Λ)||y(0)|| + 1 + 2µ1 λmin (Λ)
(19)
We attempt to upper bound the right hand side of the above inequality by considering both factors separately. First, observe from y(t) dynamics (14) that, d T y (t)Λ−1 y(t) = −2yT (t)ΓT RW (t)RT Γy(t) dt 1 1 = −2 < W 2 (t)RT Γy,W 2 (t)RT Γy > (20) Integrating both the sides yields, T T 2 d T 1 − 2 W 2 (σ )RT Γy(σ ) dσ y (σ )Λ−1 y(σ ) = dσ 0 0 ⇒ yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) = T 2 1 2 W 2 (σ )RT Γy(σ ) dσ (21)
≤ ||y(0)||λmax (Λ)
Now we are employing the afore-mentioned inequalities to evaluate an explicit bound on ||y(t)||. Defining, f (t) yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) and regrouping the terms of the above inequality, we obtain, 3 1 yT (T )M(T )y(T ) − f 2 ≤ {pµ2 } 2 λmax (Λ)||y(0)|| f 2 3 2 λ {pµ } 2µ1 λmin (Λ)||y(0)||2 (Λ)||y(0)|| max 2 f2 +2 f≥ 1 + 2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} (25) Therefore, completing the squares on both side of the equation (25) yields, 2 3 {pµ2 } 2 λmax (Λ)||y(0)|| f (t) + ≥ 1 + 2µ1 λmin (Λ)
2
dσ
||W 2 (σ )RT Γ||2 ||M(σ )||2 dσ
(22)
537
Taking the square root of both sides we obtain, f (t) ≥ −a + b + a2 ||y(0)||
(26)
(27)
where,
3
a=
{pµ2 } 2 λmax (Λ) {1 + 2µ1 λmin (Λ)}
2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} which verifies the lemma 7. b=
(28) (29)
Now we have sufficient background to present our main theoretical contribution. Theorem 8. Let, y(t) satisfy the following equation, y(t) ˙ = −ΛΓT RW (t)RT Γy(t) and M(T ) specified as, M(T ) =
T 0
ΓT RW (σ )RT Γdσ
Assume, that there exist positive constants µ1 , µ2 and T such that for all t ≥ 0 µ2 I p ≥
t+T t
ΓT RW (σ )RT Γdσ ≥ µ1 I p
Since, M(T ) is symmetric and positive semi-definite matrix for Then for, all T > 0 and bounded-above by µ2 I p . Therefore, ||M(t)|| is Ω = −a + b + a2 upper-bounded by pµ2 , where, p is the number of spanning tree where, a and b are specified in (28) and (29) and edges. Hence, equation (22) can be rewritten as, 1 Ω2 κ 2 T = ln 1 − 3 1 2 2T λmax (Λ−1 ) W 2 (σ )RT ΓΛM(σ )y(σ ) dσ ≤ {pµ2 } 2 λmax (Λ)||y(0)|| 0 there is a constant m > 0, such that, (23) κ ||y(t)|| ≤ me 2 t ||y(0)|| Combining the two inequalities derived in equation (21) and (23) yields, We sketch a proof for the above-mentioned theorem as follows, 1 T T −1 T −1 y (T )M(T )y(T ) − y (0)Λ y(0) − y (T )Λ y(T ) 2 3 Proof. Lemma 7 can be re-written as, ≤ {pµ2 } 2 λmax (Λ)||y(0)|| yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) ≥ Ω||y(0)|| (24) 541
MICNON 2015 538 Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539 June 24-26, 2015. Saint Petersburg, Russia
Therefore, squaring both the sides leads to, yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) ≥ Ω2 ||y(0)||2 yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) Ω2 ≥ −1 2 λmax (Λ )||y(0)|| λmax (Λ−1 ) λmax (Λ−1 )||y(0)||2 − yT (T )Λ−1 y(T ) Ω2 ≥ −1 2 λmax (Λ )||y(0)|| λmax (Λ−1 ) Henceforth, the explicit solution can be rephrased as, yT (T )Λ−1 y(T ) Ω2 ≤ 1 − λmax (Λ−1 )||y(0)||2 λ (Λ−1 ) max Ω2 λmax (Λ) 1− ||y(0)|| (30) ||y(T )|| ≤ λmin (Λ) λmax (Λ−1 )
Thus over an interval of length T , ||y(T )|| shrinks by the factor 2 1 − λ Ω(Λ−1 ) . Resolving, max κT Ω2 2 e = 1− λmax (Λ−1 ) We obtain, 1 Ω2 κ = ln 1 − (31) 2 2T λmax (Λ−1 )
Fig. 1. Information-exchange topologies between three agents The minimum and the maximum eigen-value corrosponding to the spanning tree of the edge laplacian matrix (Defined in section 2.4) are computed as, λmin (Λ) = 1, λmax (Λ) = 3. In Fig. 2 we present the plot of J(n, µ1 , µ2 ) versus µ1 , for different values of µ2 .
3.2 Comparison of the Anderson and Brockett estimates With the necessary mathematical background, we are now able to compare κ, the estimate due to Brockett, to αv , the estimate due to Anderson. Starting from equation (31), κ 1 Ω2 = ln 1 − 2 2T λmax (Λ−1 ) Ω2 1 − −1 1 λmax (Λ ) = ln 1 − η2 {1 − η 2 } 2T Ω2 1 − −1 1 1 λmax (Λ ) ln 1 − η 2 + ln (32) = {1 − η 2 } 2T 2T
Fig. 2. J(p, µ1 , µ2 ) for p = 2 and select values of µ2 4.2 Case-2 We reconsider multi-agent system (34) but with four agents. The interaction topology is shown in Fig. 3. The λmin (Λ) and
Therefore, from equation (13) it can be inferred that, 1 κ αv + (33) = − ln {J(p, µ1 , µ2 )} 2 2 2T It is evident from equation (33) that J(n, µ1 , µ2 ) < 1, would imply κ < − α2v . 4. NUMERICAL SIMULATION In this section, we introduce some academic examples to compare the Brockett (κ) and the Anderson (αv ) estimates. The comparison is made by plotting the value of J(p, µ1 , µ2 ) derived in (33).
Fig. 3. Information-exchange topologies between four agents λmax (Λ) for the given interaction topology are determined as, 0.5858 and 3.4142 respectively. The results obtained from the simulation are shown in Fig 4.
4.1 Case-1
4.3 Case-3
Consider a multi-agent system with three agents with the following dynamics, (34) x˙i = ui
Finally, we consider a network of six agents with agent dynamics (34) and communication network shown in Fig 5. The λmin (Λ) and λmax (Λ) are recomputed as, 0.3249, 4.2143 respectively. The corresponding simulation result shown in Fig 6.
where, xi denotes the position of the ith agent and ui is the control input. The graphical representation (with arbitrary orientation) for the above multi-agent system is shown in Fig 1 with g2i (t) representing the weights corresponding to edge ei . 542
In Fig. 2, 4, and 6 we show the plots of J(p, µ1 , µ2 ) versus µ1 and different values of µ2 corresponding to p = 2, 3, 5 respectively. Since, µ2 ≥ µ1 , each curve extends only upto
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
Fig. 4. J(p, µ1 , µ2 ) for p = 3 and select values of µ2
Fig. 5. Information-exchange topologies between six agents
Fig. 6. J(p, µ1 , µ2 ) for p = 5 and select values of µ2 µ1 = µ2 . From these plots it can be inferred that, the Anderson’s estimate (αv ) gives a tighter bound than the Brockett’s estimate (κ) in all the above cases. 5. CONCLUDING REMARKS A comparative study of two distinct convergence rate estimates for a degenerate gradient flow in the context of multi-agent systems is carried out in this work. We revisit the spanning edge agent dynamics and analyze exponential stability for the same. A recent result by Brockett helps to compute an explicit convergence rate for the corresponding spanning edge vector. The Anderson estimate from the authors’ earlier work is compared with the current result through numerical examples. It was seen in simulations that the Anderson estimate gives a tighter bound on the convergence rate. REFERENCES Anderson, B. (1977). Exponential stability of linear equations arising in adaptive identification. IEEE Transactions on Automatic Control,, 22(1), 83–88. doi: 10.1109/TAC.1977.1101406. 543
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Andersson, S. and Krishnaprasad, P.S. (2002). Degenerate gradient flows: a comparison study of convergence rate estimates. In Proceedings of the 41st IEEE Conference on Decision and Control, 2002,, volume 4, 4712–4717. IEEE. Brockett, R. (2000). The rate of descent for degenerate gradient flows. In Proceedings of the MTNS. Mesbahi, M. and Egerstedt, M. (2010). Graph theoretic methods in multiagent networks. Princeton University Press. Morarescu, I.C. and Girard, A. (2011). Opinion dynamics with decaying confidence: Application to community detection in graphs. Automatic Control, IEEE Transactions on, 56(8), 1862–1873. Olfati-Saber, R. (2006). Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control,, 51(3), 401–420. Ren, W. and Beard, R.W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on, Automatic Control, 50(5), 655– 661. Ren, W. and Beard, R.W. (2008). Distributed consensus in multi-vehicle cooperative control: theory and applications. Springerverlag London Limited. Roy Chowdhury, N. and Srikant, S. (2014). Persistence based analysis of consensus protocols for dynamic graph networks. In European Control Conference (ECC), 2014, 886–891. IEEE. Sarlette, A., Sepulchre, R., and Leonard, N. (2007). Cooperative attitude synchronization in satellite swarms: a consensus approach. In Proceedings of the 17th IFAC Symposium on Automatic Control in Aerospace. Sastry, S. and Bodson, M. (2011). Adaptive control: stability, convergence and robustness. Courier Dover Publications. Scardovi, L. and Sepulchre, R. (2009). Synchronization in networks of identical linear systems. Automatica, 45(11), 2557–2562. Shi, G. and Johansson, K.H. (2012). Persistent graphs and consensus convergence. In 51st Conference on decision and control (CDC), 2046–2051. Zelazo, D. and Mesbahi, M. (2011). Edge agreement: Graphtheoretic performance bounds and passivity analysis. IEEE Transactions on, Automatic Control, 56(3), 544–555.
ScienceDirect IFAC-PapersOnLine 48-11 (2015) 534–539 A comparative study of persistence based A comparative study of persistence based A comparative study of persistence based Aconvergence comparative study of persistence based rate estimates to consensus convergence rate estimates to consensus convergence rate rate estimates estimates to to consensus consensus convergence Nilanjan Roy Chowdhury ∗ Srikant Sukumar ∗
Nilanjan Roy Chowdhury ∗∗ Srikant Sukumar ∗∗ Nilanjan Roy Chowdhury ∗ Srikant Sukumar ∗ Roy Chowdhury ∗ Systems andNilanjan Engineering, IndianSrikant InstituteSukumar of Technology Bombay, ∗ Systems and Control Control Engineering, Indian of Technology Bombay, ∗ Systems {nilan Institute Mumbai-400076, (e-mail: Indian jan, srikant}@sc.iitb.ac.in) and ControlIndia, Engineering, Institute of Technology Bombay, ∗ Systems {nilan Institute Mumbai-400076, India, (e-mail: Indian jan, srikant}@sc.iitb.ac.in) and Control Engineering, of Technology Bombay, Mumbai-400076, India, (e-mail: {nilan jan, srikant}@sc.iitb.ac.in) Mumbai-400076, India, (e-mail: {nilan jan, srikant}@sc.iitb.ac.in) Abstract: This article presents a comparative study of convergence rate estimates for degenerate Abstract: This article presents a comparative study of convergence rate estimates for degenerate gradient flows the context of multi-agent systems. analysis rate methodology introduced Abstract: Thisinarticle presents a comparative study Aof novel convergence estimates was for degenerate gradient flows in the context of multi-agent systems. A novel analysis methodology was introduced Abstract: This article presents a comparative study of convergence rate estimates for degenerate in (Roy Chowdhury Srikant, 2014) based systems. on the classical of persistence of was excitation (PE) gradient flows in theand context of multi-agent A novelnotions analysis methodology introduced in (Roy Chowdhury and Srikant, 2014) based on thethe classical notions of persistence of was excitation (PE) gradient flows in the context of multi-agent systems. A novel analysis methodology introduced and uniform complete observability (UCO) to study exponential stability of consensus protocol with in (Roy Chowdhury and Srikant, 2014) based on the classical notions of persistence of excitation (PE) and uniform complete observability (UCO) to study the exponential stability of consensus protocol with in (Roy Chowdhury and Srikant, 2014) based on the classical notions of persistence of excitation (PE) time-varying agent dynamics. The new technique precise computation of the convergence rate and uniform complete observability (UCO) to studyallowed the exponential stability of consensus protocol with time-varying agent dynamics. The new technique allowed precise computation of the convergence rate and uniform complete observability (UCO) to study the exponential stability of consensus protocol with for single integrator agent dynamics. Thetechnique purpose of this article is principally twofold. we revisit time-varying agent dynamics. The new allowed precise computation of the Firstly, convergence rate for single integrator agent dynamics. The purpose of this article is principally twofold. Firstly, we revisit time-varying agent dynamics. The new technique allowed precise computation of the convergence rate the single continuous time agent spanning edge agent dynamics proposed (Roy Chowdhury Srikant, 2014) for integrator dynamics. The purpose of this article in is principally twofold.and Firstly, we revisit the continuous time spanning edge agent dynamics proposed in (Roy Chowdhury and Srikant, 2014) for integrator dynamics. The purpose article in is principally twofold. Firstly, revisit andsingle present a different analysis approach todynamics studyof thethis convergence property for the same. Ourwe analysis the continuous time agent spanning edge agent proposed (Roy Chowdhury and Srikant, 2014) and present a different analysis approach to study the convergence property for the same. Our analysis the continuous time spanning edge agent dynamics proposed in (Roy Chowdhury and Srikant, 2014) mimics the recent resultanalysis by Brockett (Brockett, on single agent adaptive control. the and present a different approach to study2000) the convergence property for the same.Secondly, Our analysis mimics the recent result byarrived Brockett (Brockett, 2000) on single agent adaptive control. Secondly, the and present arate different approach tothe study the convergence property forSimulations the same.Secondly, Our analysis convergence estimates at from two approaches compared. on different mimics the recent resultanalysis by Brockett (Brockett, 2000) on singleareagent adaptive control. the convergence rate estimates arrived at from the two approaches are compared. Simulations on different mimics the recent result by Brockett (Brockett, 2000) on single agent adaptive control. Secondly, the examples are rate provided to elucidate mainthe theoretical contribution. convergence estimates arrived the at from two approaches are compared. Simulations on different examples are provided to elucidate the main theoretical contribution. convergence estimates arrived the at from two approaches are compared. Simulations on different examples are rate provided to elucidate mainthe theoretical contribution. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. examples are provided to elucidate the main theoretical contribution. 1. INTRODUCTION edges. 1. INTRODUCTION edges. Another body of work in relation to degenerate gradient flows 1. INTRODUCTION edges. Another body of work in relation to degenerate gradient flows 1. INTRODUCTION Consensus is characterized as a framework in which several edges. involves time-varying dynamics of the form (Andersson Another body of worksystem in relation to degenerate gradient flows Consensus is characterized as a framework in which several involves time-varying system dynamics of the form (Andersson Another body of work in relation to degenerate gradient flows agents collaborate with eachasother and come in to which an agreement and Krishnaprasad, 2002), Consensus is characterized a framework several involves time-varying system dynamics of the form (Andersson agents collaborate with eachkey other and come to an agreement and Krishnaprasad, 2002), Consensus is characterized aobjective framework in several involves time-varying system dynamics of the form (Andersson over a common value. of consensus control agents collaborate withThe eachasother and come to which an agreement and Krishnaprasad, 2002), ˙ (1) φ˙ (t) = −u(t)uTT (t)φ (t) over a common value. The key objective of consensus control agents collaborate with each other and come to an agreement and Krishnaprasad, 2002), is to design a relative feedback algorithm that enables the agents (1)n φ (t) = −u(t)uT (t)φ (t) over a common value. The key objective of consensus control ˙ (t) = n is defined (t)φ (t) (1) φ −u(t)u is to design a relative feedback algorithm that enables the agents over a common value. The key objective of consensus control Here, φ (t) ∈ R as the parameter error and u(t) ∈ Rn T ˙ (t) =as in to thedesign network to converge to algorithm the consensus value. Consensus is a relative feedback that enables the agents Here, φ (t) ∈ Rnn is defined (t)φ (t) φ −u(t)u the parameter error and u(t) ∈ (1) Rn in the network to converge to the consensus value. Consensus is to design a relative feedback algorithm that enables the agents is the appropriate filter state. Seminal contributions in analyzing Here, φ (t) ∈ R is defined as the parameter error and u(t) ∈ Rn control has been multitude of application ar- is the appropriate in the network to envisioned converge tointhea consensus value. Consensus n isfilter state. Seminal contributions in analyzing Here, φ (t) ∈ R defined as the parameter error and u(t) ∈ R control has been envisioned in a multitude of application arin the network to converge to the consensus value. Consensus the stability of (1) have been made in (Anderson, 1977) where is the appropriate filter state. Seminal contributions in analyzing eas suchhas as,been flocking (Olfati-Saber, 2006), satellite formation control envisioned in a multitude of application ar- is stability of (1)filter have beenSeminal made in to (Anderson, 1977) where appropriate in analyzing eas such as, flocking (Olfati-Saber, 2006), satellite formation control has a multitude of application ar- the author proposes astate. methodology study stability of (1) thethe stability of (1) have been made incontributions (Anderson, 1977) where flying (Sarlette etenvisioned al.,(Olfati-Saber, 2007),in opinion dynamics (Morarescu eas such as,been flocking 2006), satellite formation author proposes a methodology to study stability of (1) the stability of (1) have been made in (Anderson, 1977) where flying (Sarlette et al.,(Olfati-Saber, 2007), opinion dynamics (Morarescu eas as,2011) flocking 2006), satellite formation provides a convergence rate estimate. Recently (Brockett, the author proposes a methodology to study stability of (1) and such Girard, problem (Scardovi and and flying (Sarlette et and al., synchronization 2007), opinion dynamics (Morarescu and provides a convergence rate estimate. Recently (Brockett, the author proposes a methodology to study stability of (1) and Girard, 2011) and synchronization problem (Scardovi and and flying (Sarlette et and al., synchronization 2007), opinion problem dynamics (Morarescu 2000) re-examined the above rate system dynamics and proposed an provides a convergence estimate. Recently (Brockett, Sepulchre, 2009). and Girard, 2011) (Scardovi and and 2000) re-examined the above system dynamics and proposed an provides a convergence rate estimate. Recently (Brockett, Sepulchre, 2009). and Girard, 2011) and synchronization problem (Scardovi and explicit convergence rate for the same. A comparative study of 2000) re-examined the above system dynamics and proposed an In recent times, Sepulchre, 2009).consensus algorithms have drawn significant 2000) explicit convergence rate for the same. A comparative study of re-examined the above system dynamics and proposed an In recent times, consensus algorithms have drawn significant Sepulchre, 2009). these two estimates is shown in (Andersson and Krishnaprasad, explicit convergence rate for the same. A comparative study of research attention in the field of control. In (Ren and Beard, In recent times, consensus algorithms have drawn significant explicit these two estimates is shown in (Andersson and Krishnaprasad, convergence rate for the same. A comparative study of research attention in the field of control. In (Ren and Beard, In recent times,2)consensus algorithms significant these two estimates is shown in (Andersson and Krishnaprasad, 2008, chapter the authors suggest ahave continuous time up- 2002). research attention in the field of control. In drawn (Ren and Beard, 2002). these two estimates is shown in (Andersson and Krishnaprasad, 2008, chapter 2) the authors suggest a continuous time up- 2002). research attention in the fieldagent of control. In (Ren and Beard, In this article we compare the two afore-mentioned estimates date law for single integrator dynamics considering both 2008, chapter 2) the authors suggest a continuous time up- 2002). In article we the systems. two afore-mentioned estimates date law for single integrator agent dynamics considering both 2008, chapter 2) the authors suggest a continuous time upin this the context of compare multi-agent Firstly, we revisit the In this article we compare the two afore-mentioned estimates constant andsingle switching graphagent structures. It is considering well established date law for integrator dynamics both In in the context of multi-agent systems. Firstly, we revisit the this article we compare the two afore-mentioned estimates constant and switching graph structures. It is well established date law for single integrator agent dynamics considering both spanning edge agent dynamics proposed in (Roy Chowdhury in the context of multi-agent systems. Firstly, we revisit the that the second smallest eigenvalue of the graph laplacian constant and switching graph structures. It is well established in spanning edge agent dynamics proposed in (Roy Chowdhury the context of multi-agent systems. Firstly, we revisit that the and second smallest eigenvalue of the graphestablished laplacian and constant switching graph structures. well Srikant, utilize results by in (Brockett, 2000)the to spanning edge2014) agentand dynamics proposed (Roy Chowdhury matrix the convergence rate of inItthe a isstatic that theregulates second smallest eigenvalue graphframework. laplacian spanning and Srikant, 2014) utilize results byWe (Brockett, 2000) to edge agentand dynamics proposed in (Roy Chowdhury matrix regulates the convergence rate in a static framework. that the second smallest eigenvalue of the graph laplacian determine the explicit convergence rate. compare the new and Srikant, 2014) and utilize results by (Brockett, 2000) to Likewise, the authors of (Ren andrate Beard, 2005) framework. employing determine the explicit convergence rate. We compare the new matrix regulates the convergence in a static and Srikant, 2014) and utilize results by (Brockett, 2000) to Likewise, the authors of (Ren and Beard, 2005) employing matrix regulates the convergence rate in a static framework. rate estimate to the one developed in (Roy Chowdhury and determine the explicit convergence rate. We compare the new the notions of Stochastic Indecomposable Aperiodic (SIA) maLikewise, the authors of (Ren and Beard, 2005) employing determine estimate the on one developedlemma in (Roy Chowdhury and the to explicit convergence We compare the new the notions of Stochastic Indecomposable Aperiodic (SIA) ma- rate Likewise, the authors ofunder (Renswitching and Beard, employing 2014) based Anderson’s (Sastry and Bodson, rate estimate to the one developedrate. in (Roy Chowdhury and trices, prove for2005) both continuous the notions ofconsensus Stochastic Indecomposable Aperiodic (SIA) ma- Srikant, Srikant, 2014) based on Anderson’s lemma (Sastry and Bodson, rate estimate to the one developed in (Roy Chowdhury and trices, prove consensus under switching for both continuous the StochasticHowever, Indecomposable Aperiodic (SIA) map. 2014) 73-74). Srikant, based on Anderson’s lemma (Sastry and Bodson, andnotions discrete most available references fall 2011, trices, proveofdynamics. consensus under switching for both continuous 2011, p. 73-74). Srikant, 2014) based on Anderson’s lemma (Sastry and Bodson, and discrete dynamics. However, most available references fall trices, prove consensus under switching for both continuous The rest of this article unfolds as follows. In section 2, a p. 73-74). shortdiscrete on establishing an explicit rate available estimate. Recently, the 2011, and dynamics. However, most references fall The rest of this article unfolds as follows. section 2, a p. 73-74). short on(Roy establishing an explicit rate 2014) estimate. Recently, the 2011, and discrete dynamics. However, most available references fall succinct overview of system theory with theIn The rest of this article unfolds as follows. In mathematical section 2, a authors Chowdhury and Srikant, introduced a new short on establishing an explicit rate estimate. Recently, the The succinct overview of different system theory with the mathematical rest of this article unfolds as follows. In section 2,area authors (Roy Chowdhury and Srikant, 2014) introduced a new short on establishing an explicit rate estimate. Recently, the foundations of the two convergence rate estimates succinct overview of system theory with the mathematical methodology for investigating consensus problems with atimeauthors (Roy Chowdhury and Srikant, 2014) introduced new succinct foundations of the two different convergence rate mathematical estimates are overview ofresult system theory with the methodology for investigating consensus problems with timeauthors (Roy Chowdhury and Srikant, 2014) introduced a new provided. Our main is presented in section 3 with foundations of the two different convergence rate estimates are varying persistent undirected graph networks for single inte- provided. Our main result is presented in section 3 with the methodology for investigating consensus problems with timethe foundations of the two different convergence rate estimates are varying persistent undirected graph networks for single intemethodology for investigating consensus problems with timesupporting proofs. In section 4 we present test cases to justify provided. Our main result is presented in section 3 with the grator agent dynamics. The advantage of the for newsingle methodolvarying persistent undirected graph networks inte- provided. supporting proofs. sectionis 4presented we present test cases3 to justify Our mainIn result in conclusions section with the grator agent dynamics. The advantage of the new methodolvarying persistent undirected graph networks single intethe main theoretical while the for this proofs. Incontribution section 4 we present test cases to justify ogy, based ondynamics. classical notions of persistence excitation and supporting grator agent The advantage of theoffor new methodolthe main theoretical contribution while the conclusions for this supporting proofs. In section 4 we present test cases to justify ogy, based on classical notions of persistence of excitation and grator agent dynamics. The advantage of the new methodolwork are summarized in section 5. the main theoretical contribution while the conclusions for this uniform complete observability, in an explicit convergence ogy, based on classical notions oflies persistence of excitation and the work are theoretical summarizedcontribution in section 5.while the conclusions for this main uniform complete observability, lies in an explicit convergence ogy, based onfor classical notions persistence of excitation and rate estimate consensus. Theoflies authors Chowdhury and work are summarized in section 5. uniform complete observability, in an(Roy explicit convergence work are summarized in section 5. BACKGROUND rate estimate for consensus. The authors (Roy Chowdhury and uniform complete observability, in an explicit 2. MATHAMETICAL Srikant, 2014)foranalyze the consensus problem byconvergence transformrate estimate consensus. The lies authors (Roy Chowdhury and 2. MATHAMETICAL BACKGROUND Srikant, 2014) analyze the consensus problem by transformrate estimate for consensus. The authors (Roy Chowdhury and 2. MATHAMETICAL BACKGROUND ing the node to an edge agreement one,transformutilizing Srikant, 2014)dynamics analyze the consensus problem by 2. MATHAMETICAL BACKGROUND ing the node dynamics to an edge agreement one, utilizing Srikant, 2014) analyzetransformation the consensus probleminby transformGraph Theory fundamentals a suitable coordinate proposed (Zelazo and 2.1 Algebraic ing the node dynamics to an edge agreement one, utilizing 2.1 Algebraic Graph Theory fundamentals aMesbahi, suitable coordinate transformation proposed in (Zelazo and ing the node dynamics to an edge agreement one, utilizing The authors (Shi andproposed Johansson, 2012) have a suitable 2011). coordinate transformation in (Zelazo and 2.1 Algebraic Graph Theory fundamentals Mesbahi, 2011). The authors (Shi and Johansson, 2012) have Algebraic Graphintroductory Theory fundamentals aMesbahi, suitable coordinate transformation proposed in (Zelazo and In this section some notation from algebraic graph also estimated convergence rates consensus laws basedhave on 2.1 2011). The authors (Shiofand Johansson, 2012) In this section some introductory from of algebraic graph also estimated convergence rates of consensus laws based on Mesbahi, 2011). The authors (Shi and Johansson, 2012) have is introduced. A detailednotation discussion the same is the notion of persistent graphs. the result In this section some introductory notation from algebraic graph also estimated convergence ratesHowever, of consensus lawsassumes based ona theory theory is introduced. A detailed discussion of the same is the notion of persistent graphs. However, the result assumes a also estimated convergence rates of consensus laws based on In this section some introductory notation from algebraic graph availableis in (Mesbahi Aand Egerstedt, 2010, of chapter 2). An stringent assumption and the rate on cyclea theory introduced. detailed discussion the same is the notionarc-balance of persistent graphs. However, the depends result assumes availableis in (Mesbahi Aand Egerstedt, 2010, of chapter 2). An stringent arc-balance assumption and the rate depends on cyclea theory the notion of persistent graphs. However, the result assumes introduced. detailed discussion the same is stringent arc-balance assumption and the rate depends on cycle available in (Mesbahi and Egerstedt, 2010, chapter 2). An stringent arc-balance assumption and the rate depends on cycle available in (Mesbahi and Egerstedt, 2010, chapter 2). An
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MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
undirected graph G is a pair (V, E) where V (G ) = {v1 , · · · , vn } is a finite non empty node set (agents) and E(G ) = {e1 , · · · , em } is an edge set (communication links). {i, j} ∈ E(G ) is an undirected edge if agents vi and v j exchange information with each other. A graph is connected, if for every pair of vertices in V (G ), there is a path that has them as its end vertices. D(G o ) ∈ Rn×m is the incidence matrix of a graph G with arbitrary orientation O. The graph Laplacian matrix of an arbitrarily oriented graph G o is defined as, (2) L(G o ) = D(G o )W D(G o )T where, W ∈ Rm×m is the diagonal matrix with the weights w(ei ), i = {1, 2, ....m} on the diagonal entry. Furthermore, the edge Laplacian matrix is defined as, (3) Le (G o ) = D(G o )T D(G o ) In this article, we are principally dealing with undirected graphs therefore the more inconvenient D(G 0 ) notation will be exchanged for D(G ). For an undirected graph L(G ) is symmetric. The graph laplacian matrix is positive semi-definite with eigenvalues ordered as, λ1 ≤ λ2 ≤ · · · ≤ λn with λ1 = 0. The graph G is connected if and only if, λ2 > 0. For an undirected graph λ2 is defined as the algebraic connectivity. 2.2 Fundamental results The following elementary notions and results from systems theory are referred to, throughout the course of this paper. Definition 1. (Sastry and Bodson, 2011, p. 72) The signal u(·) : R → Rn×m is said to be Persistently Exciting (PE) if there exist finite positive constants µ1 , µ2 , T such that, µ2 In ≥
t+T t
u(τ)uT (τ)dτ ≥ µ1 In
∀t ≥ 0
(4)
where, In is the identity matrix of dimension n. Theorem 2. (Sastry and Bodson, 2011, pp. 73-74) Assume that, for all δ > 0, there exists Kδ ≥ 0 such that for all, t0 ≥ 0, t0 +δ t0
K(τ) 2 dτ ≤ Kδ
Consider the linear time-varying system [C(t), A(t)] as follows, x(t) ˙ = A(t)x(t) x(0) = x0 y(t) = C(t)x(t) with, [C(t), A(t) + K(t)C(t)] being the system with output feedback, x(t) ˙ = (A(t) + K(t)C(t)) x(t) y(t) = C(t)x(t) Then, the system [C(t), A(t)] is uniformly completely observable if and only if [C(t), A(t) + K(t)C(t)] is uniformly completely observable. Moreover, if the observability gramian of the system [C(t), A(t)] satisfies, β2 In ≥
t0 +δ t0
ΦTA (τ,t0 )CT (τ)C(τ)ΦA (τ,t0 )dτ ≥ β1 In
for all t0 ≥ 0 then the observability gramian of the system [C(t), A(t) + K(t)C(t)] also satisfy the above mentioned inequalities with identical choice of δ and, β1 β˜1 = 2 1 + Kδ β2 β˜2 = β2 e(Kδ β2 )
539
535
Theorem 3. (Sastry and Bodson, 2011, pp. 31-32) Let, Bh be a closed ball of radius h centered at 0 in Rn . If there exists a function V (t, x) and strictly positive constants α1 , α2 , α3 and δ such that for all x ∈ Bh , t ≥ 0 α1 x2 ≤ V (t, x) ≤ α2 x2
dV (t, x(t)) ≤0 dt t+δ dV (τ, x(τ)) dτ ≤ −α3 x(t)2 dτ t then, x(t) ∈ Rn exponentially converges to 0. Moreover, v(t, x) evolves according to,
V (t, x(t)) ≤ mv e−αv (t−t0 )V (t0 , (x(t0 )) where, 1 mv = 1 − αα32
t ≥ t0 ≥ 0
1 1 ln T 1 − αα32
αv =
2.3 Convergence rate estimates
In the following subsection we establish the two convergence rate estimates, we will compare. We begin with the classical result based on the Anderson’s lemma (Theorem 2). The following theorem is also cited in (Anderson, 1977) and (Sastry and Bodson, 2011, pp. 31-32). Theorem 4. Consider the degenerate gradient flows equations of the form, (5) y(t) ˙ = −u(t)uT (t)y(t) n Now, if u(t) ∈ R is persistently exciting (Definition 1) then, αv √ ||y(t)|| ≤ mv e− 2 (t−t0 ) ||y(t0 )|| (6) where, 1 mv = 1 − η2 1 αv = ln 1 − η 2 2 2T with, η2 =
µ1 √ 2 (1 + nµ2 )
Finally, we introduce the result by Brockett as follows (Brockett, 2000), Theorem 5. Recollect the dynamics of the degenerate gradient flows given in equation (5). Let, M(t) =
t 0
u(σ )uT (σ )dσ
If there exist positive constants, r, ε and T such that, for all, t ≥ 0 we have, M(t + T ) − M(t) ≥ εI and then with,
δ=
Tr [M(t + T ) − M(t)]3 ≤ r3
2r3
2ε − + 2 1 + 2ε 3 (1 + 2ε)
2r3 3 (1 + 2ε)2
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and,
κ 1 = ln 1 − δ 2 2 2T there exists a positive constant m such that, √ κ ||y(t)|| ≤ me 2 t ||y(0)||
(7)
2.4 Convergence of single integrator consensus algorithm Consider n identical single integrator agent dynamics as, (8) x˙i = ui with the classical consensus law, (Ren and Beard, 2008, p. 26), (Roy Chowdhury and Srikant, 2014), n
ui = −k ∑ ai j (t) (xi − x j ) j=1
i = 1, 2, · · · , n
(9)
Where, k > 0 and ai j (t) ≥ 0. Therefore, the closed loop dynamics can be written as, x˙ = −kL(G )x = −kD(G )W (t)D(G )T x (10) n×m , is the incidence matrix for the correspondD(G ) ∈ R ing arbitrary oriented graph and W (t) ∈ Rm×m is defined as a time-varying diagonal weight matrix with different g2i (t), i = {1, 2, . . . , m} on the diagonal entries representing the diverse inter-agent edge weights. Furthermore, the incidence matrix is partitioned as, D(G ) = [D(Gτ ) D(Gc )], where D(Gτ ) ∈ Rn×p and D(Gc ) ∈ Rn×(m−p) are defined as the incidence matrix corresponds to the spanning edges and the cycle edges respectively. Likewise, the diagonal matrix W (t) ∈ Rm×m can also be subdivided into two different block diagonal matrices as W (t) = diag [Wτ (t), Wc (t)] where, Wτ (t) ∈ R p×p represents the weighting functions corresponding to the edges of the spanning tree and Wc (t) ∈ R(m−p)×(m−p) represents the weights corresponding to the cycle edges. Here, p is the number of spanning tree edges. Now, consider the transformation from the node dynamics to the edge state, xe = D(G )T x. Therefore the edge dynamics evaluates as, x˙e = D(G )T x˙ = −kL˜ e (G )W (t)xe (11) m×m Here, L˜ e (G ) ∈ R (edge laplacian matrix), W (t) (weight matrix) and xe ∈ Rm (edge state vector) partitioned as follows, (see reference (Roy Chowdhury and Srikant, 2014)) T L˜ e (G ) = [D(Gτ ) D(Gc )] [D(Gτ ) D(Gc )] L˜ e (Gτ ) D(Gτ )T D(Gc ) = D(Gc )T D(Gτ ) L˜ e (Gc ) W (t) 0 W (t) = τ 0 Wc (t) x xe = τ xc Substituting all the above matrices in equation (11) the corresponding spanning edge state vector (xτ ∈ R p ) evaluates as, x˙τ = −kL˜ e (Gτ )RW (t)RT xτ where, R ∈ R p×m is defined as, [I Z] with, −1 D(Gτ )T D(Gc ) Z = D(Gτ )T D(Gτ ) p×p is symmetric and positive definite, it Since, L˜ e (Gτ ) ∈ R can be further decomposed as, L˜ e (Gτ ) = ΓΛΓT . Therefore, we introduce a set of modified states defined by the similarity transformation y = ΓT xτ , with dynamics, y˙ = ΓT x˙τ = −kΛΓT RW (t)RT Γy (12) 540
The closed loop dynamics defined in (10) and (12) with undirected time-varying persistent graph networks is studied rigorously in (Roy Chowdhury and Srikant, 2014). The following theorem mimics the main result of (Roy Chowdhury and Srikant, 2014) and used later in this article. Theorem 6. The continuous update law proposed in equation (9) guarantees that the class of multi-agent systems with single-integrator dynamics (8) and time-varying communication topology characterized by W (t), achieves consensus exponentially if Wτ (t) (spanning tree weight matrix) is persistently exciting (Definition 1) with the following convergence rate, 1 αv ln(1 − η 2 ) (13) − = 2 2T 2kλmin (Λ)µ1 where, η 2 = 2 √ 1 + k p Λ µ2 Here, Λ ∈ R p×p defined as the diagonal matrix, containing all the nonzero eigenvalues of the spanning tree edge laplacian matrix, whereas, µ1 I p , µ2 I p are specified as the persistence bounds for Wτ (Definition 1). A more involved proof of the theorem 6 is available in (Roy Chowdhury and Srikant, 2014). 3. MAIN RESULT 3.1 Stability analysis of consensus using Brockett’s estimate Reconsider, the spanning edge agent dynamics given in equation (12) as follows, y˙ = ΓT x˙τ = −kΛΓT RW (t)RT Γy (14) Without loss of generality we will assume that, k = 1. In order to prove the exponential stability of the above-mentioned spanning edge agent dynamics we establish the following lemma. Lemma 7. Define, M(T ) as, M(T ) =
T 0
ΓT RW (σ )RT Γdσ
Let, µ1 I p , µ2 I p be the lower and the upper bound for the given symmetric, non-negative matrix M(T ). Then, for a given persistent window, T > 0 we have, 3 {pµ2 } 2 λmax (Λ)||y(0)|| T −1 T −1 y (0)Λ y(0) − y (T )Λ y(T ) ≥ − 1 + 2µ1 λmin (Λ) 2 3 {pµ2 } 2 λmax (Λ) 2µ1 λmin (Λ) + + 1 + 2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} (15)
||y(0)||
Proof. We begin with a Lyapunov-like candidate function defined as follows, 1 V (y) = yT M(t)y + yT Λ−1 y 2 The time derivative of V (y) along the closed-loop dynamics (19) can be written as,
d 1 T −1 T ˙ V (y) = y M(t)y + y Λ y dt 2 1 1 T T ˙ = y˙ M(t)y + y M(t)y˙ + yT M(t)y + yT Λ−1 y˙ + y˙T Λ−1 y 2 2 = −2yT ΓT RW (t)RT ΓΛM(t)y (16)
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
Integrating both sides of equation (16) we obtain, T d 1 yT (σ )M(t)y(σ ) + yT (σ )Λ−1 y(σ ) dσ 2 0 dσ ≤ |2
T 0
yT (σ )ΓT RW (σ )RT ΓΛM(σ )y(σ )dσ |
(17)
The left hand side integral in equation (17) evaluates to, 1 1 yT (T )M(T )y(T ) + yT (T )Λ−1 y(T ) − yT (0)Λ−1 y(0) 2 2 ≤ |2
T 0
1
1
W 2 (σ )RT Γy(σ ) ·W 2 (σ )RT ΓΛM(σ )y(σ )dσ | (18)
which on applying the Cauchy-Schwartz inequality yields, |2
T
1
1
W 2 (σ )RT Γy(σ ) ·W 2 (σ )RT ΓΛM(σ )y(σ )dσ |
0 T 2 1 ≤ 2 W 2 (σ )RT Γy(σ ) dσ · 0
T 0
1
W 2 (σ )RT ΓΛM(σ )y(σ )
0
Likewise, we are also trying to upper bound the second factor. Since, ||y(t)|| is non-increasing, the second factor can be bounded above by, 2 T 1 W 2 (σ )RT ΓΛM(σ )y(σ ) dσ 0 T 0
1
2µ1 λmin (Λ) ||y(0)||2 λmax (Λ) {1 + 2µ1 λmin (Λ)} 2 3 {pµ2 } 2 λmax (Λ)||y(0)|| + 1 + 2µ1 λmin (Λ)
(19)
We attempt to upper bound the right hand side of the above inequality by considering both factors separately. First, observe from y(t) dynamics (14) that, d T y (t)Λ−1 y(t) = −2yT (t)ΓT RW (t)RT Γy(t) dt 1 1 = −2 < W 2 (t)RT Γy,W 2 (t)RT Γy > (20) Integrating both the sides yields, T T 2 d T 1 − 2 W 2 (σ )RT Γy(σ ) dσ y (σ )Λ−1 y(σ ) = dσ 0 0 ⇒ yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) = T 2 1 2 W 2 (σ )RT Γy(σ ) dσ (21)
≤ ||y(0)||λmax (Λ)
Now we are employing the afore-mentioned inequalities to evaluate an explicit bound on ||y(t)||. Defining, f (t) yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) and regrouping the terms of the above inequality, we obtain, 3 1 yT (T )M(T )y(T ) − f 2 ≤ {pµ2 } 2 λmax (Λ)||y(0)|| f 2 3 2 λ {pµ } 2µ1 λmin (Λ)||y(0)||2 (Λ)||y(0)|| max 2 f2 +2 f≥ 1 + 2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} (25) Therefore, completing the squares on both side of the equation (25) yields, 2 3 {pµ2 } 2 λmax (Λ)||y(0)|| f (t) + ≥ 1 + 2µ1 λmin (Λ)
2
dσ
||W 2 (σ )RT Γ||2 ||M(σ )||2 dσ
(22)
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Taking the square root of both sides we obtain, f (t) ≥ −a + b + a2 ||y(0)||
(26)
(27)
where,
3
a=
{pµ2 } 2 λmax (Λ) {1 + 2µ1 λmin (Λ)}
2µ1 λmin (Λ) λmax (Λ) {1 + 2µ1 λmin (Λ)} which verifies the lemma 7. b=
(28) (29)
Now we have sufficient background to present our main theoretical contribution. Theorem 8. Let, y(t) satisfy the following equation, y(t) ˙ = −ΛΓT RW (t)RT Γy(t) and M(T ) specified as, M(T ) =
T 0
ΓT RW (σ )RT Γdσ
Assume, that there exist positive constants µ1 , µ2 and T such that for all t ≥ 0 µ2 I p ≥
t+T t
ΓT RW (σ )RT Γdσ ≥ µ1 I p
Since, M(T ) is symmetric and positive semi-definite matrix for Then for, all T > 0 and bounded-above by µ2 I p . Therefore, ||M(t)|| is Ω = −a + b + a2 upper-bounded by pµ2 , where, p is the number of spanning tree where, a and b are specified in (28) and (29) and edges. Hence, equation (22) can be rewritten as, 1 Ω2 κ 2 T = ln 1 − 3 1 2 2T λmax (Λ−1 ) W 2 (σ )RT ΓΛM(σ )y(σ ) dσ ≤ {pµ2 } 2 λmax (Λ)||y(0)|| 0 there is a constant m > 0, such that, (23) κ ||y(t)|| ≤ me 2 t ||y(0)|| Combining the two inequalities derived in equation (21) and (23) yields, We sketch a proof for the above-mentioned theorem as follows, 1 T T −1 T −1 y (T )M(T )y(T ) − y (0)Λ y(0) − y (T )Λ y(T ) 2 3 Proof. Lemma 7 can be re-written as, ≤ {pµ2 } 2 λmax (Λ)||y(0)|| yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) ≥ Ω||y(0)|| (24) 541
MICNON 2015 538 Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539 June 24-26, 2015. Saint Petersburg, Russia
Therefore, squaring both the sides leads to, yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) ≥ Ω2 ||y(0)||2 yT (0)Λ−1 y(0) − yT (T )Λ−1 y(T ) Ω2 ≥ −1 2 λmax (Λ )||y(0)|| λmax (Λ−1 ) λmax (Λ−1 )||y(0)||2 − yT (T )Λ−1 y(T ) Ω2 ≥ −1 2 λmax (Λ )||y(0)|| λmax (Λ−1 ) Henceforth, the explicit solution can be rephrased as, yT (T )Λ−1 y(T ) Ω2 ≤ 1 − λmax (Λ−1 )||y(0)||2 λ (Λ−1 ) max Ω2 λmax (Λ) 1− ||y(0)|| (30) ||y(T )|| ≤ λmin (Λ) λmax (Λ−1 )
Thus over an interval of length T , ||y(T )|| shrinks by the factor 2 1 − λ Ω(Λ−1 ) . Resolving, max κT Ω2 2 e = 1− λmax (Λ−1 ) We obtain, 1 Ω2 κ = ln 1 − (31) 2 2T λmax (Λ−1 )
Fig. 1. Information-exchange topologies between three agents The minimum and the maximum eigen-value corrosponding to the spanning tree of the edge laplacian matrix (Defined in section 2.4) are computed as, λmin (Λ) = 1, λmax (Λ) = 3. In Fig. 2 we present the plot of J(n, µ1 , µ2 ) versus µ1 , for different values of µ2 .
3.2 Comparison of the Anderson and Brockett estimates With the necessary mathematical background, we are now able to compare κ, the estimate due to Brockett, to αv , the estimate due to Anderson. Starting from equation (31), κ 1 Ω2 = ln 1 − 2 2T λmax (Λ−1 ) Ω2 1 − −1 1 λmax (Λ ) = ln 1 − η2 {1 − η 2 } 2T Ω2 1 − −1 1 1 λmax (Λ ) ln 1 − η 2 + ln (32) = {1 − η 2 } 2T 2T
Fig. 2. J(p, µ1 , µ2 ) for p = 2 and select values of µ2 4.2 Case-2 We reconsider multi-agent system (34) but with four agents. The interaction topology is shown in Fig. 3. The λmin (Λ) and
Therefore, from equation (13) it can be inferred that, 1 κ αv + (33) = − ln {J(p, µ1 , µ2 )} 2 2 2T It is evident from equation (33) that J(n, µ1 , µ2 ) < 1, would imply κ < − α2v . 4. NUMERICAL SIMULATION In this section, we introduce some academic examples to compare the Brockett (κ) and the Anderson (αv ) estimates. The comparison is made by plotting the value of J(p, µ1 , µ2 ) derived in (33).
Fig. 3. Information-exchange topologies between four agents λmax (Λ) for the given interaction topology are determined as, 0.5858 and 3.4142 respectively. The results obtained from the simulation are shown in Fig 4.
4.1 Case-1
4.3 Case-3
Consider a multi-agent system with three agents with the following dynamics, (34) x˙i = ui
Finally, we consider a network of six agents with agent dynamics (34) and communication network shown in Fig 5. The λmin (Λ) and λmax (Λ) are recomputed as, 0.3249, 4.2143 respectively. The corresponding simulation result shown in Fig 6.
where, xi denotes the position of the ith agent and ui is the control input. The graphical representation (with arbitrary orientation) for the above multi-agent system is shown in Fig 1 with g2i (t) representing the weights corresponding to edge ei . 542
In Fig. 2, 4, and 6 we show the plots of J(p, µ1 , µ2 ) versus µ1 and different values of µ2 corresponding to p = 2, 3, 5 respectively. Since, µ2 ≥ µ1 , each curve extends only upto
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Nilanjan Roy Chowdhury et al. / IFAC-PapersOnLine 48-11 (2015) 534–539
Fig. 4. J(p, µ1 , µ2 ) for p = 3 and select values of µ2
Fig. 5. Information-exchange topologies between six agents
Fig. 6. J(p, µ1 , µ2 ) for p = 5 and select values of µ2 µ1 = µ2 . From these plots it can be inferred that, the Anderson’s estimate (αv ) gives a tighter bound than the Brockett’s estimate (κ) in all the above cases. 5. CONCLUDING REMARKS A comparative study of two distinct convergence rate estimates for a degenerate gradient flow in the context of multi-agent systems is carried out in this work. We revisit the spanning edge agent dynamics and analyze exponential stability for the same. A recent result by Brockett helps to compute an explicit convergence rate for the corresponding spanning edge vector. The Anderson estimate from the authors’ earlier work is compared with the current result through numerical examples. It was seen in simulations that the Anderson estimate gives a tighter bound on the convergence rate. REFERENCES Anderson, B. (1977). Exponential stability of linear equations arising in adaptive identification. IEEE Transactions on Automatic Control,, 22(1), 83–88. doi: 10.1109/TAC.1977.1101406. 543
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