On PID control for synchronization of complex dynamical network with delayed nodes
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摘要
Over the past two decades, synchronization, as an interesting collective behavior of complex dynamical networks, has been attracting much attention. To reveal and analyze the inherent mechanism of synchronization in complex dynamical networks with time delays in nodes, this paper attempts to use PD and PI control protocols to achieve synchronization. Based on a classical network model, we investigate the PD and PI control for synchronization of complex dynamical networks with delayed nodes and obtain some sufficient conditions. By using Lyapunov functions and appropriate state transformations, we prove that global synchronization can be achieved via the above control protocols. Finally, some simulation examples are illustrated to validate the effectiveness of the proposed theoretical results.
Over the past two decades, synchronization, as an interesting collective behavior of complex dynamical networks, has been attracting much attention. To reveal and analyze the inherent mechanism of synchronization in complex dynamical networks with time delays in nodes, this paper attempts to use PD and PI control protocols to achieve synchronization. Based on a classical network model, we investigate the PD and PI control for synchronization of complex dynamical networks with delayed nodes and obtain some sufficient conditions. By using Lyapunov functions and appropriate state transformations, we prove that global synchronization can be achieved via the above control protocols. Finally, some simulation examples are illustrated to validate the effectiveness of the proposed theoretical results.
Over the past two decades, synchronization, as an interesting collective behavior of complex dynamical networks, has been attracting much attention. To reveal and analyze the inherent mechanism of synchronization in complex dynamical networks with time delays in nodes, this paper attempts to use PD and PI control protocols to achieve synchronization. Based on a classical network model, we investigate the PD and PI control for synchronization of complex dynamical networks with delayed nodes and obtain some sufficient conditions. By using Lyapunov functions and appropriate state transformations, we prove that global synchronization can be achieved via the above control protocols. Finally, some simulation examples are illustrated to validate the effectiveness of the proposed theoretical results.
引文
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2 Boccaletti S,Latora V,Moreno Y,et al.Complex networks:Structure and dynamics.Phys Rep,2006,424:175-308
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4 Rubenstein M,Cornejo A,Nagpal R.Programmable self-assembly in a thousand-robot swarm.Science,2014,345:795-799
5 Czir′ok A,Vicsek T.Collective behavior of interacting self-propelled particles.Physica A,2000,281:17-29
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10 Watts D J,Strogatz S H.Collective dynamics of‘small-world’networks.Nature,1998,393:440-442
11 Tan S,L¨u J,Lin Z.Emerging behavioral consensus of evolutionary dynamics on complex networks.SIAM J Control Optim,2016,54:3258-3272
12 Arenas A,D′?az-Guilera A,Kurths J,et al.Synchronization in complex networks.Phys Rep,2008,469:93-153
13 Ning D,Wu X,Lu J,et al.Driving-based generalized synchronization in two-layer networks via pinning control.Chaos,2015,25:113104
14 Olfati-Saber R,Fax J A,Murray R M.Consensus and cooperation in networked multi-agent systems.Proc IEEE,2007,95:215-233
15 Liu K X,Wu L L,L¨u J H,et al.Finite-time adaptive consensus of a class of multi-agent systems.Sci China Tech Sci,2016,59:22-32
16 Wang X F,Chen G.Pinning control of scale-free dynamical networks.Physica A,2002,310:521-531
17 Mei G,Wu X,Ning D,et al.Finite-time stabilization of complex dynamical networks via optimal control.Complexity,2016,21:417-425
18 Li Y,Wu X,Lu J,et al.Synchronizability of duplex networks.IEEETrans Circuits Syst II,2016,63:206-210
19 Bidram A,Lewis F L,Davoudi A.Distributed control systems for small-scale power networks:Using multiagent cooperative control theory.IEEE Control Syst,2014,34:56-77
20 Chen G R.Problems and challenges in control theory under complex dynamical network environments.Acta Automat Sin,2013,39:312-321
21 Zhou J,Lu J,L¨u J.Adaptive synchronization of an uncertain complex dynamical network.IEEE Trans Automat Contr,2006,51:652-656
22 Lu J,Chen G,Bernardo M di.On some recent advances in synchronization and control of complex networks.In:Proceedings of 2010IEEE International Symposium on Circuits and Systems.Paris,2010.3773-3776
23 Yu W W,DeLellis P,Chen G R,et al.Distributed adaptive control of synchronization in complex networks.IEEE Trans Automat Contr,2012,57:2153-2158
24 Yu W W,Chen G R,Cao M,et al.Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics.IEEETrans Syst Man Cybern B,2010,40:881-891
25 Li H,Liao X,Dong T,et al.Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols.Nonlinear Dyn,2012,70:2213-2226
26 Chen Y,L¨u J,Yu X,et al.Consensus of discrete-time second-order multiagent systems based on infinite products of general stochastic matrices.SIAM J Control Optim,2013,51:3274-3301
27 Guo W,L¨u J,Chen S,et al.Second-order tracking control for leaderfollower multi-agent flocking in directed graphs with switching topology.Syst Control Lett,2011,60:1051-1058
28 Li C,Xu H,Liao X,et al.Synchronization in small-world oscillator networks with coupling delays.Physica A,2004,335:359-364
29 Lu W,Chen T,Chen G.Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay.Physica D,2006,221:118-134
30 Qin J,Gao H,Zheng W X.Second-order consensus for multi-agent systems with switching topology and communication delay.Syst Control Lett,2011,60:390-397
31 Qian Y,Wu X,L¨u J,et al.Consensus of second-order multi-agent systems with nonlinear dynamics and time delay.Nonlinear Dyn,2014,78:495-503
32 Mensour B,Longtin A.Synchronization of delay-differential equations with application to private communication.Phys Lett A,1998,244:59-70
33 Zhang Q J,Lu J A,Lu J H,et al.Adaptive feedback synchronization of a general complex dynamical network with delayed nodes.IEEE Trans Circuits Syst II,2008,55:183-187
34 Wei X,Wu X,Lu J,et al.Counterpart synchronization of duplex networks with delayed nodes and noise perturbation.J Stat Mech,2015,2015:P11021
35 Li Z.Global synchronization of general delayed dynamical networks.Chin Phys Lett,2007,24:1869-1872
36 Jiang M,Shen Y,Jian J,et al.Stability,bifurcation and a new chaos in the logistic differential equation with delay.Phys Lett A,2006,350:221-227
37 Astr¨om K J,H¨agglund T.PID Controllers:Theory,Design and Tuning.2nd ed.Research Triangle Park:Instrument Society of America,1995
38 Zhao C,Guo L.PID controller design for second order nonlinear uncertain systems.Sci China Inf Sci,2017,60:022201
39 Cheng L,Wang Y,Ren W,et al.Containment control of multiagent systems with dynamic leaders based on a PIn-type approach.IEEE Trans Cybern,2016,46:3004-3017
40 Burbano D,Bernardo M di.Consensus and synchronization of complex networks via proportional-integral coupling.In:2014 IEEE International Symposium on Circuits and Systems.Melbourne VIC,2014.1796-1799
41 Burbano D A,Bernardo M di.Distributed PID control for consensus of homogeneous and heterogeneous networks.IEEE Trans Control Netw Syst,2015,2:154-163
42 Burbano D.Distributed PID control for consensus and synchronization of multi-agent networks.Dissertation for Doctoral Degree.Naples:University of Naples Federico II,2015
43 Gu H,Wang X,L¨u J,et al.Synchronization of complex network with delayed nodes via proportional-derivative control.In:IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society.Beijing,2017.5610-5615
44 Silva G J,Datta A,Bhattacharyya S P.PID Controllers for Time-Delay Systems.Boston:Birkh¨auser,2005
45 Lu W,Chen T.New approach to synchronization analysis of linearly coupled ordinary differential systems.Physica D,2006,213:214-230
46 Horn R A,Johnson C R.Topics in Matrix Analysis.Cambridge:Cambridge University Press,1991
47 Ren W,Bear R W,Atkins E M.Information consensus in multivehicle cooperative control.IEEE Control Syst,2007,27:71-82
48 Boyd S,Ghaoui L E,Feron E,et al.Linear Matrix Inequalities in System and Control Theory.Philadelphia,PA:SIAM,1994
49 Bernstein D S.Matrix Mathematics:Theory,Facts,and Formulas.2nd ed.Princeton:Princeton University Press,2008
50 Zhang L,Zhu Y,Zheng W X.Synchronization and state estimation of a class of hierarchical hybrid neural networks with time-varying delays..IEEE Trans Neural Netw Learning Syst,2016,27:459-470
2 Boccaletti S,Latora V,Moreno Y,et al.Complex networks:Structure and dynamics.Phys Rep,2006,424:175-308
3 Liu K,Zhu H,Lu J.Cooperative stabilization of a class of LTI plants with distributed observers.IEEE Trans Circuits Syst I,2017,64:1891-1902
4 Rubenstein M,Cornejo A,Nagpal R.Programmable self-assembly in a thousand-robot swarm.Science,2014,345:795-799
5 Czir′ok A,Vicsek T.Collective behavior of interacting self-propelled particles.Physica A,2000,281:17-29
6 Barab′asi A L,Gulbahce N,Loscalzo J.Network medicine:A networkbased approach to human disease.Nat Rev Genet,2011,12:56-68
7 Wei X,Wu X,Chen S,et al.Cooperative epidemic spreading on a two-layered interconnected network.SIAM J Appl Dyn Syst,2018,17:1503-1520
8 Wang Y F,Wu X Q,Feng H,et al.Topology inference of uncertain complex dynamical networks and its applications in hidden nodes detection.Sci China Tech Sci,2016,59:1232-1243
9 Zhang S,Wu X,Lu J A,et al.Recovering structures of complex dynamical networks based on generalized outer synchronization.IEEETrans Circuits Syst I,2014,61:3216-3224
10 Watts D J,Strogatz S H.Collective dynamics of‘small-world’networks.Nature,1998,393:440-442
11 Tan S,L¨u J,Lin Z.Emerging behavioral consensus of evolutionary dynamics on complex networks.SIAM J Control Optim,2016,54:3258-3272
12 Arenas A,D′?az-Guilera A,Kurths J,et al.Synchronization in complex networks.Phys Rep,2008,469:93-153
13 Ning D,Wu X,Lu J,et al.Driving-based generalized synchronization in two-layer networks via pinning control.Chaos,2015,25:113104
14 Olfati-Saber R,Fax J A,Murray R M.Consensus and cooperation in networked multi-agent systems.Proc IEEE,2007,95:215-233
15 Liu K X,Wu L L,L¨u J H,et al.Finite-time adaptive consensus of a class of multi-agent systems.Sci China Tech Sci,2016,59:22-32
16 Wang X F,Chen G.Pinning control of scale-free dynamical networks.Physica A,2002,310:521-531
17 Mei G,Wu X,Ning D,et al.Finite-time stabilization of complex dynamical networks via optimal control.Complexity,2016,21:417-425
18 Li Y,Wu X,Lu J,et al.Synchronizability of duplex networks.IEEETrans Circuits Syst II,2016,63:206-210
19 Bidram A,Lewis F L,Davoudi A.Distributed control systems for small-scale power networks:Using multiagent cooperative control theory.IEEE Control Syst,2014,34:56-77
20 Chen G R.Problems and challenges in control theory under complex dynamical network environments.Acta Automat Sin,2013,39:312-321
21 Zhou J,Lu J,L¨u J.Adaptive synchronization of an uncertain complex dynamical network.IEEE Trans Automat Contr,2006,51:652-656
22 Lu J,Chen G,Bernardo M di.On some recent advances in synchronization and control of complex networks.In:Proceedings of 2010IEEE International Symposium on Circuits and Systems.Paris,2010.3773-3776
23 Yu W W,DeLellis P,Chen G R,et al.Distributed adaptive control of synchronization in complex networks.IEEE Trans Automat Contr,2012,57:2153-2158
24 Yu W W,Chen G R,Cao M,et al.Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics.IEEETrans Syst Man Cybern B,2010,40:881-891
25 Li H,Liao X,Dong T,et al.Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols.Nonlinear Dyn,2012,70:2213-2226
26 Chen Y,L¨u J,Yu X,et al.Consensus of discrete-time second-order multiagent systems based on infinite products of general stochastic matrices.SIAM J Control Optim,2013,51:3274-3301
27 Guo W,L¨u J,Chen S,et al.Second-order tracking control for leaderfollower multi-agent flocking in directed graphs with switching topology.Syst Control Lett,2011,60:1051-1058
28 Li C,Xu H,Liao X,et al.Synchronization in small-world oscillator networks with coupling delays.Physica A,2004,335:359-364
29 Lu W,Chen T,Chen G.Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay.Physica D,2006,221:118-134
30 Qin J,Gao H,Zheng W X.Second-order consensus for multi-agent systems with switching topology and communication delay.Syst Control Lett,2011,60:390-397
31 Qian Y,Wu X,L¨u J,et al.Consensus of second-order multi-agent systems with nonlinear dynamics and time delay.Nonlinear Dyn,2014,78:495-503
32 Mensour B,Longtin A.Synchronization of delay-differential equations with application to private communication.Phys Lett A,1998,244:59-70
33 Zhang Q J,Lu J A,Lu J H,et al.Adaptive feedback synchronization of a general complex dynamical network with delayed nodes.IEEE Trans Circuits Syst II,2008,55:183-187
34 Wei X,Wu X,Lu J,et al.Counterpart synchronization of duplex networks with delayed nodes and noise perturbation.J Stat Mech,2015,2015:P11021
35 Li Z.Global synchronization of general delayed dynamical networks.Chin Phys Lett,2007,24:1869-1872
36 Jiang M,Shen Y,Jian J,et al.Stability,bifurcation and a new chaos in the logistic differential equation with delay.Phys Lett A,2006,350:221-227
37 Astr¨om K J,H¨agglund T.PID Controllers:Theory,Design and Tuning.2nd ed.Research Triangle Park:Instrument Society of America,1995
38 Zhao C,Guo L.PID controller design for second order nonlinear uncertain systems.Sci China Inf Sci,2017,60:022201
39 Cheng L,Wang Y,Ren W,et al.Containment control of multiagent systems with dynamic leaders based on a PIn-type approach.IEEE Trans Cybern,2016,46:3004-3017
40 Burbano D,Bernardo M di.Consensus and synchronization of complex networks via proportional-integral coupling.In:2014 IEEE International Symposium on Circuits and Systems.Melbourne VIC,2014.1796-1799
41 Burbano D A,Bernardo M di.Distributed PID control for consensus of homogeneous and heterogeneous networks.IEEE Trans Control Netw Syst,2015,2:154-163
42 Burbano D.Distributed PID control for consensus and synchronization of multi-agent networks.Dissertation for Doctoral Degree.Naples:University of Naples Federico II,2015
43 Gu H,Wang X,L¨u J,et al.Synchronization of complex network with delayed nodes via proportional-derivative control.In:IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society.Beijing,2017.5610-5615
44 Silva G J,Datta A,Bhattacharyya S P.PID Controllers for Time-Delay Systems.Boston:Birkh¨auser,2005
45 Lu W,Chen T.New approach to synchronization analysis of linearly coupled ordinary differential systems.Physica D,2006,213:214-230
46 Horn R A,Johnson C R.Topics in Matrix Analysis.Cambridge:Cambridge University Press,1991
47 Ren W,Bear R W,Atkins E M.Information consensus in multivehicle cooperative control.IEEE Control Syst,2007,27:71-82
48 Boyd S,Ghaoui L E,Feron E,et al.Linear Matrix Inequalities in System and Control Theory.Philadelphia,PA:SIAM,1994
49 Bernstein D S.Matrix Mathematics:Theory,Facts,and Formulas.2nd ed.Princeton:Princeton University Press,2008
50 Zhang L,Zhu Y,Zheng W X.Synchronization and state estimation of a class of hierarchical hybrid neural networks with time-varying delays..IEEE Trans Neural Netw Learning Syst,2016,27:459-470