Distributed Adaptive Optimization for Generalized Linear...
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Research Article Distributed Adaptive Optimization for Generalized Linear Multiagent Systems Shuxin Liu, 1,2 Haijun Jiang , 3 Liwei Zhang, 1 and Xuehui Mei 3 1 Institute of Operations Research and Control eory, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 College of Mathematics and Physics, Xinjiang Agricultural University, Urumqi 830052, China 3 College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China Correspondence should be addressed to Haijun Jiang; [email protected] Received 29 May 2019; Revised 3 July 2019; Accepted 8 July 2019; Published 1 August 2019 Academic Editor: Ewa Pawluszewicz Copyright © 2019 Shuxin Liu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the edge-based and node-based adaptive algorithms are established, respectively, to solve the distribution convex optimization problem. e algorithms are based on multiagent systems with general linear dynamics; each agent uses only local information and cooperatively reaches the minimizer. Compared with existing results, a damping term in the adaptive law is introduced for the adaptive algorithms, which makes the algorithms more robust. Under some sufficient conditions, all agents asymptotically converge to the consensus value which minimizes the cost function. An example is provided for the effectiveness of the proposed algorithms. 1. Introduction Recently, the consensus problems of multiagent systems have been extensively investigated on account of its widespread application, such as cooperative reconnaissance and un- manned aerial vehicles formation. Many meaningful results about consensus algorithms [1–7] have been established. Specially, the consensus problems can solve the distributed optimization, when the consensus value minimizes the global cost function, which is a sum of local objective functions each of which is known by one agent. Meanwhile, with the development of various network systems, various distributed optimization problems have emerged. Extensive efforts to design the efficient algorithm have been put into the distributed optimization problems. Because of the advantages of decentralized and distributed structure, the distributed algorithms based on multiagent systems are more efficient than the centralized algorithm. A large number of consensus-based optimization algorithms have been presented over the past two decades. For instance, in [8, 9], the subgradient algorithms upon multiagent systems have been presented. In [10], the algorithms with fixed step size have been proposed. In [11], edge-based fixed-time consensus algorithms have been established. More results can be found in [12–15]. In the above works, the proposed optimization algo- rithms depend on the selection of constant control gains. In fact, the lower bound of the gains is determined by the smallest nonzero eigenvalue of Laplacian matrix for the communication graph, i.e., algebraic connectivity which is global information. ese consensus algorithms are not fully distributed. To face this challenge, some well-performing adaptive algorithms have been proposed, where the adaptive gain updating laws rely on local information of the agents. For example, in [16, 17], the adaptive algorithms of the finite time convergence have been established upon integrator multiagent systems. In [18], the adaptive algorithms have been proposed upon general linear multiagent systems from the edge-based and node-based design. Note that more agent networks in practical applications are described by generalized linear systems. Moreover, from the results in [19], the control gains in [17, 18] monotonically increase very quickly when the value of the initial states is large. Overlarge control gains will magnify the control input and will lead to unstable algorithms. Based on the above considerations, from the edge-based and node-based Hindawi Discrete Dynamics in Nature and Society Volume 2019, Article ID 9181093, 10 pages https://doi.org/10.1155/2019/9181093
Transcript of Distributed Adaptive Optimization for Generalized Linear...
Research ArticleDistributed Adaptive Optimization for Generalized LinearMultiagent Systems
Shuxin Liu12 Haijun Jiang 3 Liwei Zhang1 and Xuehui Mei3
1 Institute of Operations Research and Control Theory School of Mathematical Sciences Dalian University of TechnologyDalian 116024 China2College of Mathematics and Physics Xinjiang Agricultural University Urumqi 830052 China3College of Mathematics and System Sciences Xinjiang University Urumqi 830046 China
Correspondence should be addressed to Haijun Jiang jianghaijunxju163com
Received 29 May 2019 Revised 3 July 2019 Accepted 8 July 2019 Published 1 August 2019
Academic Editor Ewa Pawluszewicz
Copyright copy 2019 Shuxin Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper the edge-based and node-based adaptive algorithms are established respectively to solve the distribution convexoptimization problem The algorithms are based on multiagent systems with general linear dynamics each agent uses only localinformation and cooperatively reaches the minimizer Compared with existing results a damping term in the adaptive law isintroduced for the adaptive algorithms which makes the algorithms more robust Under some sufficient conditions all agentsasymptotically converge to the consensus value which minimizes the cost function An example is provided for the effectiveness ofthe proposed algorithms
1 Introduction
Recently the consensus problems of multiagent systems havebeen extensively investigated on account of its widespreadapplication such as cooperative reconnaissance and un-manned aerial vehicles formation Many meaningful resultsabout consensus algorithms [1ndash7] have been establishedSpecially the consensus problems can solve the distributedoptimization when the consensus value minimizes the globalcost function which is a sum of local objective functions eachof which is known by one agent
Meanwhile with the development of various networksystems various distributed optimization problems haveemerged Extensive efforts to design the efficient algorithmhave been put into the distributed optimization problemsBecause of the advantages of decentralized and distributedstructure the distributed algorithms based on multiagentsystems are more efficient than the centralized algorithmA large number of consensus-based optimization algorithmshave been presented over the past two decades For instancein [8 9] the subgradient algorithms uponmultiagent systemshave been presented In [10] the algorithms with fixedstep size have been proposed In [11] edge-based fixed-time
consensus algorithms have been established More results canbe found in [12ndash15]
In the above works the proposed optimization algo-rithms depend on the selection of constant control gainsIn fact the lower bound of the gains is determined bythe smallest nonzero eigenvalue of Laplacian matrix for thecommunication graph ie algebraic connectivity which isglobal information These consensus algorithms are not fullydistributed To face this challenge some well-performingadaptive algorithms have been proposed where the adaptivegain updating laws rely on local information of the agentsFor example in [16 17] the adaptive algorithms of the finitetime convergence have been established upon integratormultiagent systems In [18] the adaptive algorithms havebeen proposed upon general linear multiagent systems fromthe edge-based and node-based design
Note that more agent networks in practical applicationsare described by generalized linear systems Moreover fromthe results in [19] the control gains in [17 18] monotonicallyincrease very quickly when the value of the initial statesis large Overlarge control gains will magnify the controlinput and will lead to unstable algorithms Based on theabove considerations from the edge-based and node-based
HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 9181093 10 pageshttpsdoiorg10115520199181093
2 Discrete Dynamics in Nature and Society
standpoint two distributed adaptive algorithms for solvingthe distributed convex optimization problems are proposedupon general linear multiagent systems in this paper Com-pared with the existing algorithms the proposed algorithmsbased on generalized linear dynamical systems are morepractical and its theoretical analysis is more difficult thanthat based on integrator systems The algorithm does notrequire global information which makes it fully distributedand easier to implement The proposed algorithms introducea damping term in the update law whichmakes the algorithmrobust under input disturbance and the high-gain updatelaw
The article is organized as follows In next section somepreliminaries and lemmas are introduced In Section 3the edge-based and node-based consensus protocols aredesigned for solving the distributed convex optimizationthe asymptotic convergence is guaranteed for the adaptivealgorithms In Section 4 the effectiveness of the performancefor the algorithm to solve the distributed convex optimizationproblems is presented by a typical example In Section 5 abrief conclusion is given
2 Notations and Preliminaries
In this paper R119897 is the set of all 119897-dimensional real vectorsR119898times119902 is the set of 119898 times 119902 real matrices 1 = [1 1 1]119879means the vector that all the elements are one 119868119898 is the119898-dimensional identity matrix For a real matrix 119861 119861 gt0 (119861 ge 0) means 119861 is positive definite (semidefinite) (sdot)119879represents the transpose sdot 1 sdot and sdot infin denote the 1-norm 2-norm and infinite norm respectively Given 119910 isin R119897nablaℎ(119910) = (120597ℎ(119910)1205971199101 120597ℎ(119910)1205971199102 120597ℎ(119910)120597119910119897)119879 representsthe gradient of the function ℎ(119910) The function 119892(119905) R 997888rarrR119897 is 119892(119905) isin L2 if intinfin0 119892(119905)119879119892(119905)119889119905 lt 0 and 119892(119905) isin Linfin if forany element 119892119894(119905) for 119892(119905) sup119905|119892119894(119905)| lt infin 119894 = 1 119897
An undirected graph 119866 consists of a node set 119881 =1 2 119898 and an edge set E sube 119881 times 119881 the edge (119894 119895) isin E
if and only if (119895 119894) isin E and the self-edge (119894 119894) is not allowedThe incidence matrix 119863 = [119889119894119895]119898times|E| corresponds to a graph119866 with an arbitrary orientation namely whose edges have aterminal node and initial node For 119895th edge if the node 119894 isinitial 119889119894119895 = 1 if the node 119894 is terminal 119889119894119895 = minus1 otherwise119889119894119895 = 0 The graph Laplacian of 119866 is defined as 119871 = 119863119863119879 Apath in the graph G from 119894 to 119895 is a sequence of distinct nodesstarting with 119894 and ending with 119895 The graph G is connectedif there is a path between any two nodes
Assumption 1 The undirected graph 119866 is connected
Lemma 2 (see [20]) Under Assumption 1 119871 is positivesemidefinite 0 is a simple eigenvalue for 119871 and 1 is theassociated eigenvector its smallest nonzero eigenvalue 1205822 =min119910 =01119879119910=0119910119879119871119909119910119879119910Lemma 3 (see [21]) Let ℎ(119910) R119897 997888rarr R be a continuouslydifferentiable convex function then ℎ(119910) is minimized if andonly if lim119906997888rarr119910nablaℎ(119906) = 0
Lemma 4 (see [22]) If 119892(119905) 119892(119905) isin Linfin and 119892(119905) isin L119901 119901 isin[1 +infin) then lim119905997888rarrinfin119892(119905) = 03 Problem Formulation
Consider the followingmultiagent system each agent satisfiesthe dynamics
119910119894 (119905) = 119861119910119894 (119905) + 119862119906119894 (119905) 119894 isin T = 1 2 119898 (1)
where 119906119894(119905) isin R119902 is the protocol 119910119894(119905) is the state for 119894th agentand 119861 isin R119897times119897 and 119862 isin R119897times119902 are the constant matrices
Our aim is to design 119906119894(119905) for (1) by using only localinformation such that all agents cooperatively arrive theconsensus value 119910lowast which minimizes the convex problem
min119898sum119894=1
ℎ119894 (119910) subject to 119910 isin R
119897 (2)
where ℎ119894(119910) R119897 997888rarr R is a local cost function which isknown only to agent 119894
We give the following hypothesis
Assumption 5 The local cost function ℎ119894(119910) is convex anddifferentiable its gradient satisfies
nablaℎ119894 (119910) = 119861119910 + 119862120595119894 (119910) (3)
where 119861119862 are defined in (1) 119861 = 119870119861 119862 = 119870119862 119870 isin R119897times119897 issome negative define matrix 120595119894(119910) is continuous and thereexists 120578 gt 0 such that 120595119894(119910) minus 120595119895(119910)infin le 120578 119894 119895 isin T for all119910 isin R119897
The equivalent characterization for (2) is that the agentsachieve consensus value which minimizes the functionsum119898119894=1 ℎ119894(119910119894) namely
min119898sum119894=1
ℎ119894 (119910119894) subject to lim
119905997888rarrinfin
10038171003817100381710038171003817119910119894 (119905) minus 119910119895 (119905)10038171003817100381710038171003817 = 0 119894 119895 isin T (4)
Assumption 6 Each set Ω119894 = 119910119894 nablaℎ119894(119910119894) = 0 119894 isin T isnonempty that is to say there exists 119910lowast119894 isin R119897 such that ℎ119894(119910lowast119894 )is minimum
4 Edge-Based Adaptive Designs
To solve problem (3) the edge-based adaptive algorithm for(1) is presented as follows
119906119894 (119905) = minussum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 120595119894 (119910119894) (5)
Discrete Dynamics in Nature and Society 3
120585119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172 minus 120590119894119895 (120585119894119895 minus 120585119894119895) (6)
120589119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895) (7)
120585119894119895 (119905) = 120588119894119895 (120585119894119895 minus 120585119894119895) 120589119894119895 = 120588119894119895 (120589119894119895 minus 120589119894119895) (8)
where 120585119894119895 120589119894119895 are adaptive coupling strengths 119876 is a feedbackmatrix the initial states satisfy 120585119894119895(0) ge 120585119894119895(0) and 120589119894119895(0) ge120589119894119895(0)
From (5) system (1) can be written as follows
119910119894 (119905) = 119861119910119894 (119905) minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))]+ 119862120595119894 (119910119894 (119905))
(9)
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 and Ψ(119905) =(1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) sdot sdot sdot 120595119879119898(119910119898(119905)))119879 Thus (9) can be rep-resented by the compact form
119910 (119905) = (119868119898 otimes 119861)119910 (119905) minus (119863A119863119879 otimes 119862119875) 119910 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 119910 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(10)
where 119863 is the incidence matrix A = diag120585119894119895 and B =diag120589119894119895 The solution of the system (10) is understood by thesense of Filippov [23]
Note that119872119863 = 119863119872 = 119863 multiplying both sides of (10)by119872otimes 119868119898 we get the error system
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (119863A119863119879 otimes 119862119875) 120598 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 120598 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(11)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872otimes119868119898)119910(119905) Obviously 1199101(119905) = 1199102(119905) sdot sdot sdot = 119910119898(119905)if and only if lim119905997888rarrinfin120598(119905) = 0Theorem 7 Under Assumptions 1ndash6 and Lemma 3 problem(4) can be solved by algorithm (5) if there exists the feedbackmatrix 119875 = 119862119879119876 such that
119876119861 + 119861119879119876 minus 119876119862119862119879119876 lt 0 (12)
where119876 is the positive-definitematrixMeanwhile as 119905 997888rarr infinwe have 120585119894119895(119905) and 120585119894119895(119905)(120589119894119895(119905) and 120589119894119895(119905)) will converge to thesame positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119868119898 otimes 119876) 120598 (119905) + 12 119898sum119894=1sum119895isin119873119894 (120585119894119895 minus 119886)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120585119894119895 minus 119886)2 + 12 119898sum119894=1sum119895isinN119894 (120589119894119895 minus 119887)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120589119894119895 minus 119887)2 (13)
where 119886 and 119887 are the positive constants 119876 is the positive-definite matrix
Take 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879
sdot [119868119898 otimes (119876119861 + 119861119879119876) minus 2119863A119863119879 otimes 119876119862119862119879119876] 120598 (119905)minus 2120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]+ 2120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905)) + 119898sum
119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895
(14)
From the third term in (14) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 weget
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)10038171003817100381710038171003817infinle 120578119898 119898sum119894=1
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 120578119898 119898sum119894=1max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171= 120578max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 1205782 (119898 minus 1) 119898sum
119894=1
sum119895isin119873119894
120573119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171
(15)
According to the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895
4 Discrete Dynamics in Nature and Society
= 119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)+ 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)= 119910 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876)119910 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2= 120598 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876) 120598 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 (16)
Note that
2119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 (17)
2120598 (119905)119879 (119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171 (18)
by the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]
+ 119910 (119905)119879 (119887119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]
minus 119887 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (19)
From (15) (16) and (19) we get
(119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 2119886119871 otimes 119876119862119862119879119876] 120598 (119905)
+ (120578 (119898 minus 1) minus 119887) 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (20)
and choose 119886 gt 121205822 119887 ge 120578(119898 minus 1) due to (12) (119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 21198861205822 otimes 119876119862119862119879119876] 120598 (119905)le 120598 (119905)119879 (119868119898 otimes (119876119861 + 119861119879119876 minus 119876119862119862119879119876)) 120598 (119905) le 0
(21)
ie 119881(119905) is nonincreasing Moreover from (13) we have120598(119905) 120598(119905) 120585119894119895 120585119894119895 120589119894119895 120589119894119895 isin Linfin Meanwhile integrating bothsides of (21) one has 120598(119905) isin L2 Thus 120598(119905) isin L2 cap LinfinFrom Lemma 4 lim119905997888rarrinfin120598(119905) = 0 that is lim119905997888rarrinfin(119910119894(119905) minus(1119898)sum119898119894=1 119910119894(119905)) = 0
Let consensus value 119910lowast(119905) = (1119898)sum119898119894=1 119910119894(119905) then119889119910lowast (119905)119889119905 = 1119898 119898sum119894=1119889119910119894 (119905)119889119905 = 1119898 119898sum119894=1119861119910119894 (119905)
minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 119862120595119894 (119910119894 (119905))= 1119898 119898sum119894=1 119861119910119894 (119905) + 119862120595119894 (119910119894 (119905))
(22)
Discrete Dynamics in Nature and Society 5
According to Assumption 5 we have
119889119910lowast (119905)119889119905 = 1119898119870minus1 119898sum119894=1
119861119910lowast (119905) + 119862120595119894 (119910lowast (119905))= 1119898119870minus1 119898sum
119894=1
nablaℎ119894 (119910lowast (119905)) (23)
thus
119889sum119898119894=1 ℎ119894 (119910lowast (119905))119889119905 = 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879 119889119910lowast (119905)119889119905
= 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879119870minus1 1119898119898sum119894=1
nablaℎ119894 (119910lowast (119905)) le 0 (24)
From Assumption 1 sum119898119894=1(ℎ119894(119910lowast(119905))) is bounded below thatis lim119905997888rarrinfin(119889sum119898119894=1 ℎ119894(119910lowast(119905))119889119905) = 0 Thus we obtainlim119905997888rarrinfin(1119898)sum119898119894=1(nablaℎ119894(119910lowast(119905))) = 0 From Lemma 2 when119905 997888rarr infin 119910lowast(119905)minimizes the cost function sum119898119894=1 ℎ119894(119910(119905))
Let 120589119894119895 = 120589119894119895 minus 120589119894119895 then120589119894119895 = 120589119894119895 minus 120589119894119895= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895)
minus 120588119894119895 (120589119894119895 minus 120589119894119895)= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus (120590119894119895 + 120588119894119895) 120589119894119895
(25)
To solve (25) one has
120589119894119895 (119905) = 119890minus(120590119894119895+120588119894119895)119905120589119894119895 (0)+ int1199050119890minus(120590119894119895+120588119894119895)(119905minus119904) 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904
= 119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0)+ int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
(26)
By 119910119894 isin L2 we have 120589119894119895(119905) isin Linfin From 120589119894119895(0) ge 120589119894119895(0)we obtain 120589119894119895(0) ge 0 By (26) 120589119894119895(119905) ge 0 Now let us provelim119905997888rarrinfin120589119894119895(119905) = 0 Consider the following two cases(1) int119905
0119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 is bounded Then
lim119905997888rarrinfin
120589119894119895 (119905) = lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905sdot (120589119894119895 (0) + intt
0119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= 0(27)
(2) lim119905997888rarrinfin int1199050119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 = infinThen
lim119905997888rarrinfin
120589119894119895 (119905)= lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904)
minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= lim119905997888rarrinfin
120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904119890(120590119894119895+120588119894119895)119905
= lim119905997888rarrinfin
10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171120590119894119895 + 120588119894119895 = 0
(28)
Based on 120589119894119895(119905) isin Linfin and 120589119894119895(119905) is monotonically increasingwhen 119905 997888rarr infin we acquire that 120589119894119895(119905) will converge to somepositive constant From lim119905997888rarrinfin120589119894119895(119905) = 0 we have 120589119894119895(119905) and120589119894119895(119905)will converge to the same positive number For 120585119894119895(119905) and120585119894119895(119905) we have the same results
Remark 8 Here damping terms 120585119894119895(119905) 120589119894119895(119905) are designed forthe gain adaption law of the adaptive algorithm (5) 120589119894119895(119905)increases monotonically and converges to a finite positiveconstant Choosing 120589119894119895(0) = 0 by (26) lim119905997888rarrinfin120589119894119895(119905) = 0 and120589119894119895(119905) ge 0 we have that 120589119894119895(119905) will increase from zero and thendecrease to zero In other words 120589119894119895(119905)will increase at first andthen decrease for some time upon the select of 120590119894119895 and 120588119894119895 For120585119894119895(119905) we have similar results Compared with the adaptivelaw in [14 15] the advantage of the adaptive law introducingthe damping term is that the adaptive control gains and theamplitude of the control inputs are smaller this makes ouralgorithm more robust
5 Node-Based Adaptive Designs
To solve problem (3) the node-based protocol for (1) isproposed as follows
119906119894 (119905) = minus120577119894 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]] + 120595119894 (119910119894) (29)
120577119894 = 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 120590119894 (120577119894 minus 120577119894) (30)
120577119894 = 120588119894 (120577119894 minus 120577119894) (31)
where 120577119894 are adaptive coupling strengths 119876 is a feedbackmatrix and the initial states satisfy 120577119894(0) ge 120577119894(0)
6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
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Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
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Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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2 Discrete Dynamics in Nature and Society
standpoint two distributed adaptive algorithms for solvingthe distributed convex optimization problems are proposedupon general linear multiagent systems in this paper Com-pared with the existing algorithms the proposed algorithmsbased on generalized linear dynamical systems are morepractical and its theoretical analysis is more difficult thanthat based on integrator systems The algorithm does notrequire global information which makes it fully distributedand easier to implement The proposed algorithms introducea damping term in the update law whichmakes the algorithmrobust under input disturbance and the high-gain updatelaw
The article is organized as follows In next section somepreliminaries and lemmas are introduced In Section 3the edge-based and node-based consensus protocols aredesigned for solving the distributed convex optimizationthe asymptotic convergence is guaranteed for the adaptivealgorithms In Section 4 the effectiveness of the performancefor the algorithm to solve the distributed convex optimizationproblems is presented by a typical example In Section 5 abrief conclusion is given
2 Notations and Preliminaries
In this paper R119897 is the set of all 119897-dimensional real vectorsR119898times119902 is the set of 119898 times 119902 real matrices 1 = [1 1 1]119879means the vector that all the elements are one 119868119898 is the119898-dimensional identity matrix For a real matrix 119861 119861 gt0 (119861 ge 0) means 119861 is positive definite (semidefinite) (sdot)119879represents the transpose sdot 1 sdot and sdot infin denote the 1-norm 2-norm and infinite norm respectively Given 119910 isin R119897nablaℎ(119910) = (120597ℎ(119910)1205971199101 120597ℎ(119910)1205971199102 120597ℎ(119910)120597119910119897)119879 representsthe gradient of the function ℎ(119910) The function 119892(119905) R 997888rarrR119897 is 119892(119905) isin L2 if intinfin0 119892(119905)119879119892(119905)119889119905 lt 0 and 119892(119905) isin Linfin if forany element 119892119894(119905) for 119892(119905) sup119905|119892119894(119905)| lt infin 119894 = 1 119897
An undirected graph 119866 consists of a node set 119881 =1 2 119898 and an edge set E sube 119881 times 119881 the edge (119894 119895) isin E
if and only if (119895 119894) isin E and the self-edge (119894 119894) is not allowedThe incidence matrix 119863 = [119889119894119895]119898times|E| corresponds to a graph119866 with an arbitrary orientation namely whose edges have aterminal node and initial node For 119895th edge if the node 119894 isinitial 119889119894119895 = 1 if the node 119894 is terminal 119889119894119895 = minus1 otherwise119889119894119895 = 0 The graph Laplacian of 119866 is defined as 119871 = 119863119863119879 Apath in the graph G from 119894 to 119895 is a sequence of distinct nodesstarting with 119894 and ending with 119895 The graph G is connectedif there is a path between any two nodes
Assumption 1 The undirected graph 119866 is connected
Lemma 2 (see [20]) Under Assumption 1 119871 is positivesemidefinite 0 is a simple eigenvalue for 119871 and 1 is theassociated eigenvector its smallest nonzero eigenvalue 1205822 =min119910 =01119879119910=0119910119879119871119909119910119879119910Lemma 3 (see [21]) Let ℎ(119910) R119897 997888rarr R be a continuouslydifferentiable convex function then ℎ(119910) is minimized if andonly if lim119906997888rarr119910nablaℎ(119906) = 0
Lemma 4 (see [22]) If 119892(119905) 119892(119905) isin Linfin and 119892(119905) isin L119901 119901 isin[1 +infin) then lim119905997888rarrinfin119892(119905) = 03 Problem Formulation
Consider the followingmultiagent system each agent satisfiesthe dynamics
119910119894 (119905) = 119861119910119894 (119905) + 119862119906119894 (119905) 119894 isin T = 1 2 119898 (1)
where 119906119894(119905) isin R119902 is the protocol 119910119894(119905) is the state for 119894th agentand 119861 isin R119897times119897 and 119862 isin R119897times119902 are the constant matrices
Our aim is to design 119906119894(119905) for (1) by using only localinformation such that all agents cooperatively arrive theconsensus value 119910lowast which minimizes the convex problem
min119898sum119894=1
ℎ119894 (119910) subject to 119910 isin R
119897 (2)
where ℎ119894(119910) R119897 997888rarr R is a local cost function which isknown only to agent 119894
We give the following hypothesis
Assumption 5 The local cost function ℎ119894(119910) is convex anddifferentiable its gradient satisfies
nablaℎ119894 (119910) = 119861119910 + 119862120595119894 (119910) (3)
where 119861119862 are defined in (1) 119861 = 119870119861 119862 = 119870119862 119870 isin R119897times119897 issome negative define matrix 120595119894(119910) is continuous and thereexists 120578 gt 0 such that 120595119894(119910) minus 120595119895(119910)infin le 120578 119894 119895 isin T for all119910 isin R119897
The equivalent characterization for (2) is that the agentsachieve consensus value which minimizes the functionsum119898119894=1 ℎ119894(119910119894) namely
min119898sum119894=1
ℎ119894 (119910119894) subject to lim
119905997888rarrinfin
10038171003817100381710038171003817119910119894 (119905) minus 119910119895 (119905)10038171003817100381710038171003817 = 0 119894 119895 isin T (4)
Assumption 6 Each set Ω119894 = 119910119894 nablaℎ119894(119910119894) = 0 119894 isin T isnonempty that is to say there exists 119910lowast119894 isin R119897 such that ℎ119894(119910lowast119894 )is minimum
4 Edge-Based Adaptive Designs
To solve problem (3) the edge-based adaptive algorithm for(1) is presented as follows
119906119894 (119905) = minussum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 120595119894 (119910119894) (5)
Discrete Dynamics in Nature and Society 3
120585119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172 minus 120590119894119895 (120585119894119895 minus 120585119894119895) (6)
120589119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895) (7)
120585119894119895 (119905) = 120588119894119895 (120585119894119895 minus 120585119894119895) 120589119894119895 = 120588119894119895 (120589119894119895 minus 120589119894119895) (8)
where 120585119894119895 120589119894119895 are adaptive coupling strengths 119876 is a feedbackmatrix the initial states satisfy 120585119894119895(0) ge 120585119894119895(0) and 120589119894119895(0) ge120589119894119895(0)
From (5) system (1) can be written as follows
119910119894 (119905) = 119861119910119894 (119905) minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))]+ 119862120595119894 (119910119894 (119905))
(9)
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 and Ψ(119905) =(1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) sdot sdot sdot 120595119879119898(119910119898(119905)))119879 Thus (9) can be rep-resented by the compact form
119910 (119905) = (119868119898 otimes 119861)119910 (119905) minus (119863A119863119879 otimes 119862119875) 119910 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 119910 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(10)
where 119863 is the incidence matrix A = diag120585119894119895 and B =diag120589119894119895 The solution of the system (10) is understood by thesense of Filippov [23]
Note that119872119863 = 119863119872 = 119863 multiplying both sides of (10)by119872otimes 119868119898 we get the error system
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (119863A119863119879 otimes 119862119875) 120598 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 120598 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(11)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872otimes119868119898)119910(119905) Obviously 1199101(119905) = 1199102(119905) sdot sdot sdot = 119910119898(119905)if and only if lim119905997888rarrinfin120598(119905) = 0Theorem 7 Under Assumptions 1ndash6 and Lemma 3 problem(4) can be solved by algorithm (5) if there exists the feedbackmatrix 119875 = 119862119879119876 such that
119876119861 + 119861119879119876 minus 119876119862119862119879119876 lt 0 (12)
where119876 is the positive-definitematrixMeanwhile as 119905 997888rarr infinwe have 120585119894119895(119905) and 120585119894119895(119905)(120589119894119895(119905) and 120589119894119895(119905)) will converge to thesame positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119868119898 otimes 119876) 120598 (119905) + 12 119898sum119894=1sum119895isin119873119894 (120585119894119895 minus 119886)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120585119894119895 minus 119886)2 + 12 119898sum119894=1sum119895isinN119894 (120589119894119895 minus 119887)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120589119894119895 minus 119887)2 (13)
where 119886 and 119887 are the positive constants 119876 is the positive-definite matrix
Take 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879
sdot [119868119898 otimes (119876119861 + 119861119879119876) minus 2119863A119863119879 otimes 119876119862119862119879119876] 120598 (119905)minus 2120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]+ 2120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905)) + 119898sum
119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895
(14)
From the third term in (14) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 weget
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)10038171003817100381710038171003817infinle 120578119898 119898sum119894=1
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 120578119898 119898sum119894=1max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171= 120578max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 1205782 (119898 minus 1) 119898sum
119894=1
sum119895isin119873119894
120573119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171
(15)
According to the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895
4 Discrete Dynamics in Nature and Society
= 119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)+ 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)= 119910 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876)119910 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2= 120598 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876) 120598 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 (16)
Note that
2119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 (17)
2120598 (119905)119879 (119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171 (18)
by the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]
+ 119910 (119905)119879 (119887119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]
minus 119887 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (19)
From (15) (16) and (19) we get
(119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 2119886119871 otimes 119876119862119862119879119876] 120598 (119905)
+ (120578 (119898 minus 1) minus 119887) 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (20)
and choose 119886 gt 121205822 119887 ge 120578(119898 minus 1) due to (12) (119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 21198861205822 otimes 119876119862119862119879119876] 120598 (119905)le 120598 (119905)119879 (119868119898 otimes (119876119861 + 119861119879119876 minus 119876119862119862119879119876)) 120598 (119905) le 0
(21)
ie 119881(119905) is nonincreasing Moreover from (13) we have120598(119905) 120598(119905) 120585119894119895 120585119894119895 120589119894119895 120589119894119895 isin Linfin Meanwhile integrating bothsides of (21) one has 120598(119905) isin L2 Thus 120598(119905) isin L2 cap LinfinFrom Lemma 4 lim119905997888rarrinfin120598(119905) = 0 that is lim119905997888rarrinfin(119910119894(119905) minus(1119898)sum119898119894=1 119910119894(119905)) = 0
Let consensus value 119910lowast(119905) = (1119898)sum119898119894=1 119910119894(119905) then119889119910lowast (119905)119889119905 = 1119898 119898sum119894=1119889119910119894 (119905)119889119905 = 1119898 119898sum119894=1119861119910119894 (119905)
minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 119862120595119894 (119910119894 (119905))= 1119898 119898sum119894=1 119861119910119894 (119905) + 119862120595119894 (119910119894 (119905))
(22)
Discrete Dynamics in Nature and Society 5
According to Assumption 5 we have
119889119910lowast (119905)119889119905 = 1119898119870minus1 119898sum119894=1
119861119910lowast (119905) + 119862120595119894 (119910lowast (119905))= 1119898119870minus1 119898sum
119894=1
nablaℎ119894 (119910lowast (119905)) (23)
thus
119889sum119898119894=1 ℎ119894 (119910lowast (119905))119889119905 = 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879 119889119910lowast (119905)119889119905
= 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879119870minus1 1119898119898sum119894=1
nablaℎ119894 (119910lowast (119905)) le 0 (24)
From Assumption 1 sum119898119894=1(ℎ119894(119910lowast(119905))) is bounded below thatis lim119905997888rarrinfin(119889sum119898119894=1 ℎ119894(119910lowast(119905))119889119905) = 0 Thus we obtainlim119905997888rarrinfin(1119898)sum119898119894=1(nablaℎ119894(119910lowast(119905))) = 0 From Lemma 2 when119905 997888rarr infin 119910lowast(119905)minimizes the cost function sum119898119894=1 ℎ119894(119910(119905))
Let 120589119894119895 = 120589119894119895 minus 120589119894119895 then120589119894119895 = 120589119894119895 minus 120589119894119895= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895)
minus 120588119894119895 (120589119894119895 minus 120589119894119895)= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus (120590119894119895 + 120588119894119895) 120589119894119895
(25)
To solve (25) one has
120589119894119895 (119905) = 119890minus(120590119894119895+120588119894119895)119905120589119894119895 (0)+ int1199050119890minus(120590119894119895+120588119894119895)(119905minus119904) 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904
= 119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0)+ int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
(26)
By 119910119894 isin L2 we have 120589119894119895(119905) isin Linfin From 120589119894119895(0) ge 120589119894119895(0)we obtain 120589119894119895(0) ge 0 By (26) 120589119894119895(119905) ge 0 Now let us provelim119905997888rarrinfin120589119894119895(119905) = 0 Consider the following two cases(1) int119905
0119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 is bounded Then
lim119905997888rarrinfin
120589119894119895 (119905) = lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905sdot (120589119894119895 (0) + intt
0119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= 0(27)
(2) lim119905997888rarrinfin int1199050119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 = infinThen
lim119905997888rarrinfin
120589119894119895 (119905)= lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904)
minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= lim119905997888rarrinfin
120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904119890(120590119894119895+120588119894119895)119905
= lim119905997888rarrinfin
10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171120590119894119895 + 120588119894119895 = 0
(28)
Based on 120589119894119895(119905) isin Linfin and 120589119894119895(119905) is monotonically increasingwhen 119905 997888rarr infin we acquire that 120589119894119895(119905) will converge to somepositive constant From lim119905997888rarrinfin120589119894119895(119905) = 0 we have 120589119894119895(119905) and120589119894119895(119905)will converge to the same positive number For 120585119894119895(119905) and120585119894119895(119905) we have the same results
Remark 8 Here damping terms 120585119894119895(119905) 120589119894119895(119905) are designed forthe gain adaption law of the adaptive algorithm (5) 120589119894119895(119905)increases monotonically and converges to a finite positiveconstant Choosing 120589119894119895(0) = 0 by (26) lim119905997888rarrinfin120589119894119895(119905) = 0 and120589119894119895(119905) ge 0 we have that 120589119894119895(119905) will increase from zero and thendecrease to zero In other words 120589119894119895(119905)will increase at first andthen decrease for some time upon the select of 120590119894119895 and 120588119894119895 For120585119894119895(119905) we have similar results Compared with the adaptivelaw in [14 15] the advantage of the adaptive law introducingthe damping term is that the adaptive control gains and theamplitude of the control inputs are smaller this makes ouralgorithm more robust
5 Node-Based Adaptive Designs
To solve problem (3) the node-based protocol for (1) isproposed as follows
119906119894 (119905) = minus120577119894 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]] + 120595119894 (119910119894) (29)
120577119894 = 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 120590119894 (120577119894 minus 120577119894) (30)
120577119894 = 120588119894 (120577119894 minus 120577119894) (31)
where 120577119894 are adaptive coupling strengths 119876 is a feedbackmatrix and the initial states satisfy 120577119894(0) ge 120577119894(0)
6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
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Discrete Dynamics in Nature and Society 3
120585119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172 minus 120590119894119895 (120585119894119895 minus 120585119894119895) (6)
120589119894119895 = 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895) (7)
120585119894119895 (119905) = 120588119894119895 (120585119894119895 minus 120585119894119895) 120589119894119895 = 120588119894119895 (120589119894119895 minus 120589119894119895) (8)
where 120585119894119895 120589119894119895 are adaptive coupling strengths 119876 is a feedbackmatrix the initial states satisfy 120585119894119895(0) ge 120585119894119895(0) and 120589119894119895(0) ge120589119894119895(0)
From (5) system (1) can be written as follows
119910119894 (119905) = 119861119910119894 (119905) minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))]+ 119862120595119894 (119910119894 (119905))
(9)
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 and Ψ(119905) =(1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) sdot sdot sdot 120595119879119898(119910119898(119905)))119879 Thus (9) can be rep-resented by the compact form
119910 (119905) = (119868119898 otimes 119861)119910 (119905) minus (119863A119863119879 otimes 119862119875) 119910 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 119910 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(10)
where 119863 is the incidence matrix A = diag120585119894119895 and B =diag120589119894119895 The solution of the system (10) is understood by thesense of Filippov [23]
Note that119872119863 = 119863119872 = 119863 multiplying both sides of (10)by119872otimes 119868119898 we get the error system
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (119863A119863119879 otimes 119862119875) 120598 (119905)minus (119863B otimes 119862) sign [(119863119879 otimes 119875) 120598 (119905)]+ (119872 otimes 119862)Ψ (119910 (119905))
(11)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872otimes119868119898)119910(119905) Obviously 1199101(119905) = 1199102(119905) sdot sdot sdot = 119910119898(119905)if and only if lim119905997888rarrinfin120598(119905) = 0Theorem 7 Under Assumptions 1ndash6 and Lemma 3 problem(4) can be solved by algorithm (5) if there exists the feedbackmatrix 119875 = 119862119879119876 such that
119876119861 + 119861119879119876 minus 119876119862119862119879119876 lt 0 (12)
where119876 is the positive-definitematrixMeanwhile as 119905 997888rarr infinwe have 120585119894119895(119905) and 120585119894119895(119905)(120589119894119895(119905) and 120589119894119895(119905)) will converge to thesame positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119868119898 otimes 119876) 120598 (119905) + 12 119898sum119894=1sum119895isin119873119894 (120585119894119895 minus 119886)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120585119894119895 minus 119886)2 + 12 119898sum119894=1sum119895isinN119894 (120589119894119895 minus 119887)2+ 119898sum119894=1
sum119895isin119873119894
1205901198941198952120588119894119895 (120589119894119895 minus 119887)2 (13)
where 119886 and 119887 are the positive constants 119876 is the positive-definite matrix
Take 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879
sdot [119868119898 otimes (119876119861 + 119861119879119876) minus 2119863A119863119879 otimes 119876119862119862119879119876] 120598 (119905)minus 2120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]+ 2120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905)) + 119898sum
119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895+ 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895
(14)
From the third term in (14) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 weget
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)10038171003817100381710038171003817infinle 120578119898 119898sum119894=1
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 120578119898 119898sum119894=1max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171= 120578max
119894
119898sum119895=1
10038171003817100381710038171003817119862119879119876(120598 (119905)119894 minus 120598 (119905)119895)100381710038171003817100381710038171le 1205782 (119898 minus 1) 119898sum
119894=1
sum119895isin119873119894
120573119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171
(15)
According to the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 120585119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120585119894119895 minus 119886) 120585119894119895
4 Discrete Dynamics in Nature and Society
= 119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)+ 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)= 119910 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876)119910 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2= 120598 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876) 120598 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 (16)
Note that
2119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 (17)
2120598 (119905)119879 (119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171 (18)
by the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]
+ 119910 (119905)119879 (119887119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]
minus 119887 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (19)
From (15) (16) and (19) we get
(119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 2119886119871 otimes 119876119862119862119879119876] 120598 (119905)
+ (120578 (119898 minus 1) minus 119887) 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (20)
and choose 119886 gt 121205822 119887 ge 120578(119898 minus 1) due to (12) (119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 21198861205822 otimes 119876119862119862119879119876] 120598 (119905)le 120598 (119905)119879 (119868119898 otimes (119876119861 + 119861119879119876 minus 119876119862119862119879119876)) 120598 (119905) le 0
(21)
ie 119881(119905) is nonincreasing Moreover from (13) we have120598(119905) 120598(119905) 120585119894119895 120585119894119895 120589119894119895 120589119894119895 isin Linfin Meanwhile integrating bothsides of (21) one has 120598(119905) isin L2 Thus 120598(119905) isin L2 cap LinfinFrom Lemma 4 lim119905997888rarrinfin120598(119905) = 0 that is lim119905997888rarrinfin(119910119894(119905) minus(1119898)sum119898119894=1 119910119894(119905)) = 0
Let consensus value 119910lowast(119905) = (1119898)sum119898119894=1 119910119894(119905) then119889119910lowast (119905)119889119905 = 1119898 119898sum119894=1119889119910119894 (119905)119889119905 = 1119898 119898sum119894=1119861119910119894 (119905)
minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 119862120595119894 (119910119894 (119905))= 1119898 119898sum119894=1 119861119910119894 (119905) + 119862120595119894 (119910119894 (119905))
(22)
Discrete Dynamics in Nature and Society 5
According to Assumption 5 we have
119889119910lowast (119905)119889119905 = 1119898119870minus1 119898sum119894=1
119861119910lowast (119905) + 119862120595119894 (119910lowast (119905))= 1119898119870minus1 119898sum
119894=1
nablaℎ119894 (119910lowast (119905)) (23)
thus
119889sum119898119894=1 ℎ119894 (119910lowast (119905))119889119905 = 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879 119889119910lowast (119905)119889119905
= 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879119870minus1 1119898119898sum119894=1
nablaℎ119894 (119910lowast (119905)) le 0 (24)
From Assumption 1 sum119898119894=1(ℎ119894(119910lowast(119905))) is bounded below thatis lim119905997888rarrinfin(119889sum119898119894=1 ℎ119894(119910lowast(119905))119889119905) = 0 Thus we obtainlim119905997888rarrinfin(1119898)sum119898119894=1(nablaℎ119894(119910lowast(119905))) = 0 From Lemma 2 when119905 997888rarr infin 119910lowast(119905)minimizes the cost function sum119898119894=1 ℎ119894(119910(119905))
Let 120589119894119895 = 120589119894119895 minus 120589119894119895 then120589119894119895 = 120589119894119895 minus 120589119894119895= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895)
minus 120588119894119895 (120589119894119895 minus 120589119894119895)= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus (120590119894119895 + 120588119894119895) 120589119894119895
(25)
To solve (25) one has
120589119894119895 (119905) = 119890minus(120590119894119895+120588119894119895)119905120589119894119895 (0)+ int1199050119890minus(120590119894119895+120588119894119895)(119905minus119904) 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904
= 119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0)+ int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
(26)
By 119910119894 isin L2 we have 120589119894119895(119905) isin Linfin From 120589119894119895(0) ge 120589119894119895(0)we obtain 120589119894119895(0) ge 0 By (26) 120589119894119895(119905) ge 0 Now let us provelim119905997888rarrinfin120589119894119895(119905) = 0 Consider the following two cases(1) int119905
0119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 is bounded Then
lim119905997888rarrinfin
120589119894119895 (119905) = lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905sdot (120589119894119895 (0) + intt
0119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= 0(27)
(2) lim119905997888rarrinfin int1199050119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 = infinThen
lim119905997888rarrinfin
120589119894119895 (119905)= lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904)
minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= lim119905997888rarrinfin
120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904119890(120590119894119895+120588119894119895)119905
= lim119905997888rarrinfin
10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171120590119894119895 + 120588119894119895 = 0
(28)
Based on 120589119894119895(119905) isin Linfin and 120589119894119895(119905) is monotonically increasingwhen 119905 997888rarr infin we acquire that 120589119894119895(119905) will converge to somepositive constant From lim119905997888rarrinfin120589119894119895(119905) = 0 we have 120589119894119895(119905) and120589119894119895(119905)will converge to the same positive number For 120585119894119895(119905) and120585119894119895(119905) we have the same results
Remark 8 Here damping terms 120585119894119895(119905) 120589119894119895(119905) are designed forthe gain adaption law of the adaptive algorithm (5) 120589119894119895(119905)increases monotonically and converges to a finite positiveconstant Choosing 120589119894119895(0) = 0 by (26) lim119905997888rarrinfin120589119894119895(119905) = 0 and120589119894119895(119905) ge 0 we have that 120589119894119895(119905) will increase from zero and thendecrease to zero In other words 120589119894119895(119905)will increase at first andthen decrease for some time upon the select of 120590119894119895 and 120588119894119895 For120585119894119895(119905) we have similar results Compared with the adaptivelaw in [14 15] the advantage of the adaptive law introducingthe damping term is that the adaptive control gains and theamplitude of the control inputs are smaller this makes ouralgorithm more robust
5 Node-Based Adaptive Designs
To solve problem (3) the node-based protocol for (1) isproposed as follows
119906119894 (119905) = minus120577119894 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]] + 120595119894 (119910119894) (29)
120577119894 = 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 120590119894 (120577119894 minus 120577119894) (30)
120577119894 = 120588119894 (120577119894 minus 120577119894) (31)
where 120577119894 are adaptive coupling strengths 119876 is a feedbackmatrix and the initial states satisfy 120577119894(0) ge 120577119894(0)
6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
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4 Discrete Dynamics in Nature and Society
= 119898sum119894=1
sum119895isin119873119894
(120585119894119895 minus 119886) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038172minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)+ 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 119886) (120585119894119895 minus 120585119894119895)= 119910 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876)119910 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2= 120598 (119905)119879 (2119863A119863119879 otimes 119876119862119862119879119876 minus 2119886119871 otimes 119876119862119862119879119876) 120598 (119905)
minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 (16)
Note that
2119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 (17)
2120598 (119905)119879 (119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]= 119898sum119894=1
sum119895isin119873119894
120589119894119895 10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171 (18)
by the fourth term in (14) we obtain
119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 120589119894119895 + 119898sum119894=1
sum119895isin119873119894
120590119894119895120588119894119895 (120589119894119895 minus 119887) 120589119894119895= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 119887) (120589119894119895 minus 120589119894119895)= 119898sum119894=1
sum119895isin119873119894
(120589119894119895 minus 119887) 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 119910 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]
+ 119910 (119905)119879 (119887119863 otimes 119876119862) sign [(119863119879 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2= 120598 (119905)119879 (119863B otimes 119876119862) sign [(119863119879 otimes 119862119879119876) 120598 (119905)]
minus 119887 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (19)
From (15) (16) and (19) we get
(119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 2119886119871 otimes 119876119862119862119879119876] 120598 (119905)
+ (120578 (119898 minus 1) minus 119887) 119898sum119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120585119894119895 minus 120585119894119895)2 minus 119898sum119894=1
sum119895isin119873119894
120590119894119895 (120589119894119895 minus 120589119894119895)2 (20)
and choose 119886 gt 121205822 119887 ge 120578(119898 minus 1) due to (12) (119905)le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876) minus 21198861205822 otimes 119876119862119862119879119876] 120598 (119905)le 120598 (119905)119879 (119868119898 otimes (119876119861 + 119861119879119876 minus 119876119862119862119879119876)) 120598 (119905) le 0
(21)
ie 119881(119905) is nonincreasing Moreover from (13) we have120598(119905) 120598(119905) 120585119894119895 120585119894119895 120589119894119895 120589119894119895 isin Linfin Meanwhile integrating bothsides of (21) one has 120598(119905) isin L2 Thus 120598(119905) isin L2 cap LinfinFrom Lemma 4 lim119905997888rarrinfin120598(119905) = 0 that is lim119905997888rarrinfin(119910119894(119905) minus(1119898)sum119898119894=1 119910119894(119905)) = 0
Let consensus value 119910lowast(119905) = (1119898)sum119898119894=1 119910119894(119905) then119889119910lowast (119905)119889119905 = 1119898 119898sum119894=1119889119910119894 (119905)119889119905 = 1119898 119898sum119894=1119861119910119894 (119905)
minus 119862sum119895isin119873119894
120585119894119895 (119875 (119910119894 (119905) minus 119910119895 (119905)))minus 119862sum119895isin119873119894
120589119894119895 sign [119875 (119910119894 (119905) minus 119910119895 (119905))] + 119862120595119894 (119910119894 (119905))= 1119898 119898sum119894=1 119861119910119894 (119905) + 119862120595119894 (119910119894 (119905))
(22)
Discrete Dynamics in Nature and Society 5
According to Assumption 5 we have
119889119910lowast (119905)119889119905 = 1119898119870minus1 119898sum119894=1
119861119910lowast (119905) + 119862120595119894 (119910lowast (119905))= 1119898119870minus1 119898sum
119894=1
nablaℎ119894 (119910lowast (119905)) (23)
thus
119889sum119898119894=1 ℎ119894 (119910lowast (119905))119889119905 = 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879 119889119910lowast (119905)119889119905
= 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879119870minus1 1119898119898sum119894=1
nablaℎ119894 (119910lowast (119905)) le 0 (24)
From Assumption 1 sum119898119894=1(ℎ119894(119910lowast(119905))) is bounded below thatis lim119905997888rarrinfin(119889sum119898119894=1 ℎ119894(119910lowast(119905))119889119905) = 0 Thus we obtainlim119905997888rarrinfin(1119898)sum119898119894=1(nablaℎ119894(119910lowast(119905))) = 0 From Lemma 2 when119905 997888rarr infin 119910lowast(119905)minimizes the cost function sum119898119894=1 ℎ119894(119910(119905))
Let 120589119894119895 = 120589119894119895 minus 120589119894119895 then120589119894119895 = 120589119894119895 minus 120589119894119895= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895)
minus 120588119894119895 (120589119894119895 minus 120589119894119895)= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus (120590119894119895 + 120588119894119895) 120589119894119895
(25)
To solve (25) one has
120589119894119895 (119905) = 119890minus(120590119894119895+120588119894119895)119905120589119894119895 (0)+ int1199050119890minus(120590119894119895+120588119894119895)(119905minus119904) 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904
= 119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0)+ int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
(26)
By 119910119894 isin L2 we have 120589119894119895(119905) isin Linfin From 120589119894119895(0) ge 120589119894119895(0)we obtain 120589119894119895(0) ge 0 By (26) 120589119894119895(119905) ge 0 Now let us provelim119905997888rarrinfin120589119894119895(119905) = 0 Consider the following two cases(1) int119905
0119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 is bounded Then
lim119905997888rarrinfin
120589119894119895 (119905) = lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905sdot (120589119894119895 (0) + intt
0119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= 0(27)
(2) lim119905997888rarrinfin int1199050119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 = infinThen
lim119905997888rarrinfin
120589119894119895 (119905)= lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904)
minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= lim119905997888rarrinfin
120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904119890(120590119894119895+120588119894119895)119905
= lim119905997888rarrinfin
10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171120590119894119895 + 120588119894119895 = 0
(28)
Based on 120589119894119895(119905) isin Linfin and 120589119894119895(119905) is monotonically increasingwhen 119905 997888rarr infin we acquire that 120589119894119895(119905) will converge to somepositive constant From lim119905997888rarrinfin120589119894119895(119905) = 0 we have 120589119894119895(119905) and120589119894119895(119905)will converge to the same positive number For 120585119894119895(119905) and120585119894119895(119905) we have the same results
Remark 8 Here damping terms 120585119894119895(119905) 120589119894119895(119905) are designed forthe gain adaption law of the adaptive algorithm (5) 120589119894119895(119905)increases monotonically and converges to a finite positiveconstant Choosing 120589119894119895(0) = 0 by (26) lim119905997888rarrinfin120589119894119895(119905) = 0 and120589119894119895(119905) ge 0 we have that 120589119894119895(119905) will increase from zero and thendecrease to zero In other words 120589119894119895(119905)will increase at first andthen decrease for some time upon the select of 120590119894119895 and 120588119894119895 For120585119894119895(119905) we have similar results Compared with the adaptivelaw in [14 15] the advantage of the adaptive law introducingthe damping term is that the adaptive control gains and theamplitude of the control inputs are smaller this makes ouralgorithm more robust
5 Node-Based Adaptive Designs
To solve problem (3) the node-based protocol for (1) isproposed as follows
119906119894 (119905) = minus120577119894 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]] + 120595119894 (119910119894) (29)
120577119894 = 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 120590119894 (120577119894 minus 120577119894) (30)
120577119894 = 120588119894 (120577119894 minus 120577119894) (31)
where 120577119894 are adaptive coupling strengths 119876 is a feedbackmatrix and the initial states satisfy 120577119894(0) ge 120577119894(0)
6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
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Discrete Dynamics in Nature and Society 5
According to Assumption 5 we have
119889119910lowast (119905)119889119905 = 1119898119870minus1 119898sum119894=1
119861119910lowast (119905) + 119862120595119894 (119910lowast (119905))= 1119898119870minus1 119898sum
119894=1
nablaℎ119894 (119910lowast (119905)) (23)
thus
119889sum119898119894=1 ℎ119894 (119910lowast (119905))119889119905 = 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879 119889119910lowast (119905)119889119905
= 1119898 119898sum119894=1 (nablaℎ119894 (119910lowast (119905)))119879119870minus1 1119898119898sum119894=1
nablaℎ119894 (119910lowast (119905)) le 0 (24)
From Assumption 1 sum119898119894=1(ℎ119894(119910lowast(119905))) is bounded below thatis lim119905997888rarrinfin(119889sum119898119894=1 ℎ119894(119910lowast(119905))119889119905) = 0 Thus we obtainlim119905997888rarrinfin(1119898)sum119898119894=1(nablaℎ119894(119910lowast(119905))) = 0 From Lemma 2 when119905 997888rarr infin 119910lowast(119905)minimizes the cost function sum119898119894=1 ℎ119894(119910(119905))
Let 120589119894119895 = 120589119894119895 minus 120589119894119895 then120589119894119895 = 120589119894119895 minus 120589119894119895= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus 120590119894119895 (120589119894119895 minus 120589119894119895)
minus 120588119894119895 (120589119894119895 minus 120589119894119895)= 10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171 minus (120590119894119895 + 120588119894119895) 120589119894119895
(25)
To solve (25) one has
120589119894119895 (119905) = 119890minus(120590119894119895+120588119894119895)119905120589119894119895 (0)+ int1199050119890minus(120590119894119895+120588119894119895)(119905minus119904) 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904
= 119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0)+ int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
(26)
By 119910119894 isin L2 we have 120589119894119895(119905) isin Linfin From 120589119894119895(0) ge 120589119894119895(0)we obtain 120589119894119895(0) ge 0 By (26) 120589119894119895(119905) ge 0 Now let us provelim119905997888rarrinfin120589119894119895(119905) = 0 Consider the following two cases(1) int119905
0119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 is bounded Then
lim119905997888rarrinfin
120589119894119895 (119905) = lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905sdot (120589119894119895 (0) + intt
0119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= 0(27)
(2) lim119905997888rarrinfin int1199050119890(120590119894119895+120588119894119895)119904119875(119910119894(119904) minus 119910119895(119904))1119889119904 = infinThen
lim119905997888rarrinfin
120589119894119895 (119905)= lim119905997888rarrinfin
119890minus(120590119894119895+120588119894119895)119905 (120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904)
minus 119910119895 (119904))100381710038171003817100381710038171 119889119904)
= lim119905997888rarrinfin
120589119894119895 (0) + int1199050119890(120590119894119895+120588119894119895)119904 10038171003817100381710038171003817119875 (119910119894 (119904) minus 119910119895 (119904))100381710038171003817100381710038171 119889119904119890(120590119894119895+120588119894119895)119905
= lim119905997888rarrinfin
10038171003817100381710038171003817119875 (119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171120590119894119895 + 120588119894119895 = 0
(28)
Based on 120589119894119895(119905) isin Linfin and 120589119894119895(119905) is monotonically increasingwhen 119905 997888rarr infin we acquire that 120589119894119895(119905) will converge to somepositive constant From lim119905997888rarrinfin120589119894119895(119905) = 0 we have 120589119894119895(119905) and120589119894119895(119905)will converge to the same positive number For 120585119894119895(119905) and120585119894119895(119905) we have the same results
Remark 8 Here damping terms 120585119894119895(119905) 120589119894119895(119905) are designed forthe gain adaption law of the adaptive algorithm (5) 120589119894119895(119905)increases monotonically and converges to a finite positiveconstant Choosing 120589119894119895(0) = 0 by (26) lim119905997888rarrinfin120589119894119895(119905) = 0 and120589119894119895(119905) ge 0 we have that 120589119894119895(119905) will increase from zero and thendecrease to zero In other words 120589119894119895(119905)will increase at first andthen decrease for some time upon the select of 120590119894119895 and 120588119894119895 For120585119894119895(119905) we have similar results Compared with the adaptivelaw in [14 15] the advantage of the adaptive law introducingthe damping term is that the adaptive control gains and theamplitude of the control inputs are smaller this makes ouralgorithm more robust
5 Node-Based Adaptive Designs
To solve problem (3) the node-based protocol for (1) isproposed as follows
119906119894 (119905) = minus120577119894 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]] + 120595119894 (119910119894) (29)
120577119894 = 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 120590119894 (120577119894 minus 120577119894) (30)
120577119894 = 120588119894 (120577119894 minus 120577119894) (31)
where 120577119894 are adaptive coupling strengths 119876 is a feedbackmatrix and the initial states satisfy 120577119894(0) ge 120577119894(0)
6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
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6 Discrete Dynamics in Nature and Society
From (1) and (29) the closed-loop system can be repre-sented as follows
119910119894 (119905) = 119861119910119894 (119905) minus 120577119894119862 sign[[119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))]]+ 119862120595119894 (119910119894 (119905)) (32)
where the first and third terms take the role to minimizethe function ℎ119894(119910119894(119905)) and the second term makes the agentsachieve consensus
Denote 119910(119905) = (1199101(119905)119879 1199102(119905)119879 119910119898(119905)119879)119879 120577 = diag120585119894119895and Ψ(119905) = (1205951198791 (1199101(119905)) 1205951198792 (1199102(119905)) 120595119879119898(119910119898(119905)))119879 by (32)we obtain (119905) = (119868119898 otimes 119861) 119910 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 119910 (119905)]
+ (119868119898 otimes 119862)Ψ (119910 (119905)) (33)
Note that 119872119871 = 119863119871 = 119863 multiply both sides of (33) by119872otimes119868119898 the following error system is obtained
120598 (119905) = (119868119898 otimes 119861) 120598 (119905) minus (120577 otimes 119862) sign [(119871 otimes 119875) 120598 (119905)]+ (119868119898 otimes 119862)Ψ (119910 (119905)) (34)
where 119872 = 119868119898 minus (1119898)11119879 120598(119905) = (1205981(119905)119879 1205982(119905)119879 120598119898(119905)119879)119879 = (119872 otimes 119868119898)119910(119905)Theorem 9 Under Assumptions 1ndash6 and Lemma 3 problem(29) can be solved by the nosed-based algorithm (5) if119875 = 119862119879119876where 119876 gt 0 satisfies
119876119861 + 119861119879119876 lt 0 (35)
Meanwhile as 119905 997888rarr infin we have 120577119894(119905) and 120577119894(119905) will convergeto the same positive number
Proof Choose the Lyapunov function candidate
119881 (119905) = 120598 (119905)119879 (119871 otimes 119876) 120598 (119905) + 12 119898sum119894=1 (120577119894 minus 119886)2+ 119898sum119894=1
1205901198942120588119894 (120577119894 minus 119886)2 (36)
where119876 gt 0 119886 is a positive constantCalculate the derivative of 119881(119905) along system (34) and
choose 119875 = 119862119879119876 we have (119905) = 120598 (119905)119879 [119871 otimes (119876119861 + 119861119879119876)] 120598 (119905)
+ 2120598 (119905)119879 (119871 otimes 119876119862)Ψ (119910 (119905))minus 2120598 (119905)119879 (119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]+ 119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894(37)
From the third term in (37) and (119872 otimes 119868119902)Ψ(119910(119905)1 le 120578 wehave
120598 (119905)119879 (119872 otimes 119876119862)Ψ (119910 (119905))le 10038171003817100381710038171003817(119872 otimes 119862119879119876) 120598 (119905)100381710038171003817100381710038171 10038171003817100381710038171003817(119872 otimes 119868119902)Ψ(119910 (119905)100381710038171003817100381710038171le 120578 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817 sum119895isin119873119894119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (38)
By the results
119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]= 119898sum119894=1
120577119894 10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 120598 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876) 120598 (119905)]= 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 (39)
we obtain
119898sum119894=1
(120577119894 minus 119886) 120577119894 + 119898sum119894=1
120590119894120588119894 (120577119894 minus 119886) 120577119894= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894) + 119898sum119894=1
120590119894 (120577119894 minus 119886) (120577119894 minus 120577119894)= 119898sum119894=1
(120577119894 minus 119886) 10038171003817100381710038171003817100381710038171003817100381710038171003817119875sum119895isin119873119894
(119910119894 (119905) minus 119910119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 119910 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876)119910 (119905)]+ 119910 (119905)119879 (119886119871 otimes 119876119862) sign [(119871 otimes 119862119879119876)119910 (119905)]minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2= 120598 (119905)119879 (119871 otimes 119876119862) 120577 sign [(119871 otimes 119862119879119876) 120598 (119905)]minus 119886 119898sum119894=1
10038171003817100381710038171003817100381710038171003817100381710038171003817119862119879119876sum119895isin119873119894
(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171003817100381710038171003817100381710038171 minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2
(40)
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 7
3
4
5
1
62
Figure 1 The communication topology for the distributed optimization
From (37) (38) and (40) we have
(119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905)+ (120578 minus 119886) 119898sum
119894=1
sum119895isin119873119894
10038171003817100381710038171003817119862119879119876(120598119894 (119905) minus 120598119895 (119905))100381710038171003817100381710038171minus 119898sum119894=1
120590119894 (120577119894 minus 120577119894)2 (41)
Choose 119886 gt 120578 we get (119905) le 120598 (119905)119879 [119868119898 otimes (119876119861 + 119861119879119876)] 120598 (119905) (42)
Thus similar to the proof of Theorem 7 the nosed-basedalgorithm (29) can solve the problem (29)
Remark 10 It is easy to see that the node-based algorithm isvery different from the edge-based algorithm The advantageis that the nosed-based algorithm has a less computationand more concise form than the edge-based algorithm Thedisadvantage is that the nosed-based algorithm convergesmore slowly than the edge-based algorithm
Remark 11 Based on the edge and node standpoints twoadaptive algorithms have been established for solving thedistributed convex optimization problems the algorithms arefully distributed without depending on any global informa-tion that is the adaptive algorithms can get the optimalsolution via local information Moreover the connectivity ofthe networks takes a key role in the rate of convergence of thealgorithms during the calculation process
6 Simulations
Consider the following distributed optimization
min6sum119894=1
ℎ119894 (119910119894)subject to 119910119894 = 119910119895 isin R
2 119894 119895 = 1 6 (43)
where
ℎ119894 (119910119894) = (11991011989411199101198942120595119894)119879
( 1 minus2 0minus2 6 minus20 minus2 3 )(11991011989411199101198942120595119894) (44)
120595119894 = 1198942 119894 = 1 2 6 Figure 1 is the communicationtopology Choose
119870 = (minus1 00 minus12) thus 119861 = (minus1 21 minus3) 119862 = (01) (45)
Simulation results employing edge-based algorithm (5) andnode-based algorithm (29) are presented in Figures 2ndash4and Figures 5ndash7 respectively In Figures 2 and 5 it canbe observed that the states of consensus are achieved InFigures 3 and 6 the adaptive control gains first increasethen decrease and finally converge to some constants whichis consistent with our theoretical analysis in Remark 8 InFigures 4 and 7 the evolution of sum6119894=1 ℎ119894(119910119894) is presentedwhich reaches the optimal value 3157 Conclusion
In this paper two distributed adaptive algorithms to solve thedistributed convex optimization problems are designed basedon general linear multiagent systems from the edge-basedand node-based standpoint Compared with the existingalgorithms through introducing a damping term in theupdate law the algorithms are robust in the face of inputdisturbances In this paper the multiagent networks areundirected graph Next we will focus on the case that themultiagent networks are directed graph
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Discrete Dynamics in Nature and Society
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)0 2 4 6 8 10
minus1
0
1
2
3
4
5
6
time (s)
Ｓ Ｃ1
Ｓ Ｃ2
i=1
6
i=1
6
Figure 2 The state variables 119910(119905) under the edge-based algorithm
0 2 4 6 8 100
2
4
6
8
10
12
time (s)
the a
dapt
ive l
aw
ＣＤ
the a
dapt
ive
0 2 4 6 8 1005
1
15
2
25
3
35
4
time (s)
law
ＣＤ
Figure 3 The adaptive law under the edge-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 4 The global cost function under the edge-based algorithm
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 9
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
i=1
6
0 2 4 6 8 101
15
2
25
3
35
4
45
5
55
6
time (s)
Ｓ Ｃ1 i
=1
6Ｓ Ｃ
2
Figure 5 The state variables 119910(119905) under the node-based algorithm
0 2 4 6 8 100
1
2
3
4
5
6
7
8
time (s)
the a
dativ
e i
=1
6la
w Ｃ
Figure 6 The adaptive law under the node-based algorithm
0 2 4 6 8 100
50
100
150
200
250
300
350
400
time (s)
the s
um o
f loc
al co
st fu
nctio
ns
315
Figure 7The global cost function under the node-based algorithm
Acknowledgments
This work was supported by the Scientific Research Programof the Higher Education Institution of Xinjiang (Grant no
XJEDU2017T001) the Natural Science Foundation of Xin-jiang Uygur Autonomous Region (Grant no 2018D01C039)and the National Peoplersquos Republic of China (Grants nosU1703262 and 61563048)
References
[1] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007
[2] H Hong W Yu J Fu and X Yu ldquoFinite-time connectivity-preserving consensus for second-order nonlinear multiagentsystemsrdquo IEEE Transactions on Control of Network Systems vol6 no 1 pp 236ndash248 2019
[3] Z Yu H Jiang C Hu and J Yu ldquoNecessary and sufficientconditions for consensus of fractional-ordermultiagent systemsvia sampled-data controlrdquo IEEE Transactions on Cyberneticsvol 47 no 8 pp 1892ndash1901 2017
[4] Z Yu D Huang H Jiang C Hu and W Yu ldquoDistributed con-sensus for multiagent systems via directed spanning tree basedadaptive controlrdquo SIAM Journal on Control and Optimizationvol 56 no 3 pp 2189ndash2217 2018
[5] J Mei W Ren and J Chen ldquoDistributed consensus of second-order multi-agent systems with heterogeneous unknown iner-tias and control gains under a directed graphrdquo IEEE Transac-tions on Automatic Control vol 61 no 8 pp 2019ndash2034 2016
[6] B Ning Q Han Z Zuo J Jin and J Zheng ldquoCollectivebehaviors of mobile robots beyond the nearest neighbor ruleswith switching topologyrdquo IEEE Transactions on Cybernetics vol48 no 5 pp 1577ndash1590 2018
[7] B Ning and Q Han ldquoPrescribed finite-time consensus track-ing for multiagent systems with nonholonomic chained-formdynamicsrdquo IEEE Transactions on Automatic Control vol 64 no4 pp 1686ndash1693 2019
[8] A Nedic and A Ozdaglar ldquoDistributed subgradient methodsfor multi-agent optimizationrdquo IEEE Transactions on AutomaticControl vol 54 no 1 pp 48ndash61 2009
[9] Y A Nedic A Ozdaglar and P A Parrilo ldquoConstrainedconsensus and optimization in multi-agent networksrdquo IEEETransactions on Automatic Control vol 55 no 4 pp 922ndash9382010
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Discrete Dynamics in Nature and Society
[10] Q Liu S Yang and Y Hong ldquoConstrained consensus algo-rithms with fixed step size for distributed convex optimizationover multiagent networksrdquo IEEE Transactions on AutomaticControl vol 62 no 8 pp 4259ndash4265 2017
[11] B Ning QHan and Z Zuo ldquoDistributed optimization formul-tiagent systems an edge-based fixed-time consensus approachrdquoIEEE Transactions on Cybernetics vol 49 no 1 pp 122ndash1322019
[12] BGharesifard and J Cortes ldquoDistributed continuous-time con-vex optimization on weight-balanced digraphsrdquo IEEE Transac-tions on Automatic Control vol 59 no 3 pp 781ndash786 2014
[13] G Droge H Kawashima and M B Egerstedt ldquoContinuous-time proportional-integral distributed optimisation for net-worked systemsrdquo Journal of Control and Decision vol 1 no 3pp 191ndash213 2014
[14] X Shi J Cao G Wen and M Perc ldquoFinite-time consensus ofopinion dynamics and its applications to distributed optimiza-tion over digraphrdquo IEEETransactions onCybernetics vol 49 no10 pp 3767ndash3779 2019
[15] K Li Q Liu S Yang J Cao and G Lu ldquoDistributed robustadaptive optimization for nonlinear multiagent systemsrdquo IEEETransactions on Systems Man and Cybernetics Systems
[16] Z Li Z Ding J Sun and Z Li ldquoDistributed adaptive convexoptimization on directed graphs via continuous-time algo-rithmsrdquo IEEE Transactions on Automatic Control vol 63 no 5pp 1434ndash1441 2018
[17] P Lin W Ren and J A Farrel ldquoDistributed continuous-time optimization nonuniform gradient gains finite-time con-vergence and convex constraint setrdquo IEEE Transactions onAutomatic Control vol 62 no 5 pp 2239ndash2253 2016
[18] Y Zhao Y Liu GWen andG Chen ldquoDistributed optimizationfor linear multiagent systems edge- and node-based adaptivedesignsrdquo IEEE Transactions on Automatic Control vol 62 no 7pp 3602ndash3609 2017
[19] P Ioannou and P Kokotovic ldquoInstability analysis and improve-ment of robustness of adaptive controlrdquoAutomatica vol 20 no5 pp 583ndash594 1984
[20] C Godsil and G Royle Algebraic GraphTheory Springer NewYork NY USA 2001
[21] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004
[22] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999
[23] J-P Aubin and A Cellina Differential Inclusions Set-ValuedMaps and Viability Theory Springer Berlin Germany 1984
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
