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Research Papers

Adaptive Consensus Tracking for Fractional-Order Linear Time Invariant Swarm Systems

[+] Author and Article Information
Mojtaba Naderi Soorki

Department of Electrical Engineering,
Sharif University of Technology,
Tehran 11155-4363, Iran
e-mail: mojtabanaderi@ee.sharif.edu

Mohammad Saleh Tavazoei

Department of Electrical Engineering,
Sharif University of Technology,
Tehran 11155-4363, Iran
e-mail: tavazoei@sharif.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 30, 2013; final manuscript received November 13, 2013; published online February 13, 2014. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 9(3), 031012 (Feb 13, 2014) (7 pages) Paper No: CND-13-1193; doi: 10.1115/1.4026042 History: Received July 30, 2013; Revised November 13, 2013

This paper presents an adaptive controller to achieve consensus tracking for the fractional-order linear time invariant swarm systems in which the matrices describing the agent dynamics and the interactive dynamics between agents are unknown. This controller consists of two parts: an adaptive stabilizer and an adaptive tracker. The adaptive stabilizer guarantees the asymptotic swarm stability of the considered swarm system. Also, the adaptive tracker enforces the system agents to track a desired trajectory while achieving consensus. Numerical simulation results are presented to show the effectiveness of the proposed controller.

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References

Helbing, D., Farkas, I., and Vicsek, T., 2000, “Simulating Dynamical Features of Escape Panic,” Nature, 407(6803), pp. 487–490. [CrossRef] [PubMed]
Parrish, J. K., 1999, “Complexity, Pattern, and Evolutionary Trade-Offs in Animal Aggregation,” Science, 284(5411), pp. 99–101. [CrossRef] [PubMed]
Liu, Q., Liao, X. F., Guo, S. T., and Wu, Y., 2009, “Stability of Bifurcating Periodic Solutions for a Single Delayed Inertial Neuron Model Under Periodic Excitation,” Nonlinear Anal.: Real World Appl., 10(4), pp. 2384–2395. [CrossRef]
Czirok, A., and Vicsek, T., 2000, “Collective Behavior of Interacting Self Propelled Particles,” Physica A, 281(1–4), pp. 17–29. [CrossRef]
van Ast, J., Babuska, R., and De Schutter, B., 2008, “A General Modeling Framework for Swarms,” Proceedings of the IEEE World Congress on Computational Intelligence, pp. 3795–3800.
Kennedy, J., and Eberhart, R., 1995, “Particle Swarm Optimization,” Proceedings of the IEEE International Conference on Neural Networks, Vol. 4, pp. 1942–1948.
Clerc, M., and Kennedy, J., 2002, “The Particle Swarm—Explosion, Stability, and Convergence in a Multidimensional Complex Space,” IEEE Trans. Evol. Comput., 6(1), pp. 58–73. [CrossRef]
Bonabeau, E., Dorigo, M., and Theraulaz, G., 1999, Swarm Intelligence: From Natural to Artificial Systems, Oxford University, New York.
Kennedy, J., 1997, “The Particle Swarm: Social Adaptation of Knowledge,” Proceedings of the IEEE International Conference on Evolutionary Computation, pp. 303–308.
Cai, N., Xi, J. X., and Zhong, Y. S., 2009, “Swarm Stability of High-Order Linear Time-Invariant Swarm Systems,” Control Theory Appl., 5(2), pp. 402–408. [CrossRef]
Lin, Z., Francis, B., and Maggiore, M., 2005, “Necessary and Sufficient Graphical Conditions for Formation Control of Unicycles,” IEEE Trans. Autom. Control, 50(1), pp. 121–127. [CrossRef]
Lafferriere, G., Williams, A., Caughman, J., and Veerman, J. J. P., 2005, “Decentralized Control of Vehicle Formations,” Syst. Control Lett., 54(9), pp. 899–910. [CrossRef]
Cortes, J., Martinez, S., and Bullo, F., 2006, “Robust Rendezvous for Mobile Autonomous Agents via Proximity Graphs in Arbitrary Dimensions,” IEEE Trans. Autom. Control, 51(8), pp. 1289–1298. [CrossRef]
Dimarogonas, D. V., and Kyriakopoulos, K. J., 2007, “On the Rendezvous Problem for Multiple Nonholonomic Agents,” IEEE Trans. Autom. Control, 52(5), pp. 916–922. [CrossRef]
Su, H., Wang, X., and Lin, Z., 2009, “Flocking of Multi-Agents With a Virtual Leader,” IEEE Trans. Autom. Control, 54(2), pp. 293–307. [CrossRef]
Olfati-Saber, R., and Murray, R. M., 2004, “Consensus Problems in Networks of Agents With Switching Topology and Time-Delays,” IEEE Trans. Autom. Control, 49(9), pp. 1520–1533. [CrossRef]
Ren, W., and Beard, R. W., 2005, “Consensus Seeking in Multiagent Systems Under Dynamically Changing Interaction Topologies,” IEEE Trans. Autom. Control, 50(5), pp. 655–661. [CrossRef]
Tang, Z. J., ZhuHuang, T., LiangShao, J., and PingHu, J., 2012 “Consensus of Second-Order Multi-Agent Systems With Nonuniform Time-Varying Delays,” Neurocomputing, 97, pp. 410–414. [CrossRef]
Liu, C. L., and Liu, F., 2012, “Dynamical Consensus Seeking of Second-Order Multi-Agent Systems Based on Delayed State Compensation,” Syst. Control Lett., 61(12), pp. 1235–1241. [CrossRef]
Xiao, F., and Wang, L., 2007, “Consensus Problems for High-Dimensional Multiagent Systems,” Control Theory Appl., 1(3), pp. 830–837. [CrossRef]
Ren, W., Moore, K. L., and Chen, Y., 2007, “High-Order and Model Reference Consensus Algorithms in Cooperative Control of Multi-Vehicle Systems,” ASME J. Dyn. Syst., Meas., Control, 129(5), pp. 678–688. [CrossRef]
Wieland, P., Kim, J., Scheu, H., and Allgower, F., 2008, “On Consensus in Multi-Agent Systems With Linear High-Order Agents,” Proceedings of the IFAC World Congress, Seoul, Korea, Vol. 17, pp. 1541–1546.
Xi, J., Shi, Z., and Zhonga, Y., 2011, “Consensus Analysis and Design for High-Order Linear Swarm Systems With Time-Varying Delays,” Physica A, 390(23–24), pp. 4114–4123. [CrossRef]
Tian, Y. P., and Zhang, Y., 2012, “Brief Paper High-Order Consensus of Heterogeneous Multi-Agent Systems With Unknown Communication Delays,” Automatica, 48(6), pp. 1205–1212. [CrossRef]
Ren, W., 2010, “Consensus Tracking Under Directed Interaction Topologies: Algorithms and Experiments,” IEEE Trans. Control Syst. Technol., 18(1), pp. 230–237. [CrossRef]
Magin, R. L., 2010, “Fractional Calculus Models of Complex Dynamics in Biological Tissues,” Comput. Math. Appl., 59(5), pp. 1586–1593. [CrossRef]
Hilfer, R., 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore.
Cafagna, D., 2007, “Fractional Calculus: A Mathematical Tool From the Past for Present Engineers,” IEEE Ind. Electron. Mag., 1(2), pp. 35–40. [CrossRef]
Cao, Y., Li, Y., Ren, W., and Chen, Y. Q., 2010, “Distributed Coordination of Networked Fractional-Order Systems,” IEEE Trans. Syst. Man Cybern., Part B, Cybern., 40(2), pp. 362–370. [CrossRef]
Cao, Y., and Ren, W., 2010, “Distributed Formation Control for Fractional Order Systems: Dynamic Interaction and Absolute/Relative Damping,” Syst. Control Lett., 59(3–4), pp. 233–240. [CrossRef]
Sun, W., Li, Y., Li, C., and Chen, Y. Q., 2011, “Convergence Speed of a Fractional Order Consensus Algorithm Over Undirected Scale-Free Networks,” Asian J. Control, 13(6), pp. 936–946. [CrossRef]
Naderi Soorki, M., and Tavazoei, M. S., 2013, “Fractional-Order Linear Time Invariant Swarm Systems: Asymptotic Swarm Stability and Time Response Analysis,” Cent. Eur. J. Phys., 11(6), pp. 845–854. [CrossRef]
Shen, J., CaoJ., and Lu, J., 2012, “Consensus of Fractional-Order Systems With Non-Uniform Input and Communication Delays,” Proc. Inst. Mech. Eng., Part I, 226(2), pp. 271–283. [CrossRef]
Shen, J., and Cao, J., 2012, “Necessary and Sufficient Conditions for Consensus of Delayed Fractional-Order Systems,” Asian J. Control, 14(6), pp. 1690–1697. [CrossRef]
Wang, Z., Gu, D., Meng, T., and Zhao, Y., 2010, “Consensus Target Tracking in Multi-robot Systems. Intelligent Robotics and Applications,” Lect. Notes Comput. Sci., 6424, pp. 724–735. [CrossRef]
Khoo, S., Xie, L., and Man, Zh., 2009, “Robust Finite-Time Consensus Tracking Algorithm for Multirobot Systems,” IEEE/ASME Trans. Mechatron., 14(2), pp. 219–228. [CrossRef]
David, S. A., Balthazar, J. M., Julio, B. H. S., and Oliveira, C., 2012, “The Fractional-Nonlinear Robotic Manipulator: Modeling and Dynamic Simulations,” AIP Conf. Proc., 1493, pp. 298–305. [CrossRef]
Jezierski, E., and Ostalczyk, P., 2009, “Fractional-Order Mathematical Model of Pneumatic Muscle Drive for Robotic Applications,” Robot Motion and Control, K. R.Kozłowski, ed., Springer, New York, pp. 113–122.
Sjöberg, M., and Kari, L., 2003, “Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic, Fractional Derivative, Generalized Friction Model,” Nonlinear Dyn., 33(3), pp. 323–336. [CrossRef]
Mendes, R. V., and Vázquez, L., 2007, “The Dynamical Nature of a Backlash System With and Without Fluid Friction,” Nonlinear Dyn., 47(4), pp. 363–366. [CrossRef]
Jaulmes, R., Pineau, J., and Precup, D., 2007, “A Formal Framework for Robot Learning and Control Under Model Uncertainty,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 2104–2110.
Su, Y., Hong, Y., and Huang, J., 2013, “A General Result on the Cooperative Robust Output Regulation for Linear Uncertain Multi-Agent Systems,” IEEE Trans. Autom. Control, 58(5), pp. 1275–1279. [CrossRef]
Wu, F., Zilberstein, Sh., and Chen, X., 2011, “Online Planning for Multi-Agent Systems With Bounded Communication,” Artif. Intell., 175(2), pp. 487–511. [CrossRef]
Stone, P., and Veloso, M., 1998, “Communication in Domains With Unreliable, Single-Channel, Low-Bandwidth Communication,” Lect. Notes Comput. Sci., 1456, pp. 85–97. [CrossRef]
Mikadze, I. S., and Khocholava, V. V., 2004, “On a Model of Information Transmission Through Unreliable Communication Channel,” Autom. Remote Control, 65(8), pp. 1250–1254. [CrossRef]
Sinopoli, B., Schenato, L., Franceschetti, M., and Poolla, K., 2005, “Optimal Control With Unreliable Communication: The TCP Case,” Proceedings of the American Control Conference, pp. 3354–3359.
Hu, T. C., Kahng, A. B., and Robins, G., 1993, “Optimal Robust Path Planning in General Environments,” IEEE Trans. Rob. Autom., 9(6), pp. 775–784. [CrossRef]
Li, Y., and Chen, X., 2005, “Stability on Multi-Robot Formation With Dynamic Interaction Topologies,” Proceeding of the IEEE International Conference on Intelligent Robots and Systems, pp. 1325–1330.
Carli, R., Fagnani, F., Speranzon, A., and ZampieriS., 2008, “Communication Constraints in the Average Consensus Problem,” Automatica, 44(3), pp. 671–684. [CrossRef]
Ren, W., Beard, R. W., and Atkins, E., 2005, “A Survey of Consensus Problems in Multi-Agent Coordination,” Proceedings of the American Control Conference, pp. 1859–1864.
Li, Y., and Chen, Y., 2012, “A Fractional Order Universal High Gain Adaptive Stabilizer,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 22(4), p. 1250081. [CrossRef]
Podlubny, I., 1999, “Fractional-Order Systems and PIλDμ Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214. [CrossRef]
Gazi, V., and Passino, K. M., 2002, “A Class of Attraction/Repulsion Functions for Stable Swarm Aggregations,” Proceedings of the IEEE Conference on Decision and Control, Vol. 3, Las Vegas, NV, pp. 2842–2847.
Gazi, V., and Passino, K. M., 2003, “Stability Analysis of Swarms,” IEEE Trans. Autom. Control, 48(4), pp. 692–697. [CrossRef]
Godsil, C., and Royle, G., 2000, Algebraic Graph Theory, Springer, New York.
Caponetto, R., Dongola, G., Fortuna, L., and Petras, I., 2010, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore.
Zhong, F., and Li, C., 2011, “Stability Analysis of Fractional Differential Systems With Order Lying in (1,2),” Adv. Differ. Equ., 2011, p. 213485.
Qian, D., Li, C., Agarwal, R. P., and Wong, P. J. Y., 2010, “Stability Analysis of Fractional Differential System With Riemann–Liouville Derivative,” Math. Comput. Modell., 52(5–6), pp. 862–874. [CrossRef]
Ioannou, P. A., and Sun, J., 1996, Robust Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ.

Figures

Grahic Jump Location
Fig. 1

Graph G that includes a spanning tree

Grahic Jump Location
Fig. 2

Trajectory of agents in adaptive consensus tracking (case I)

Grahic Jump Location
Fig. 3

Pseudostates of agents in adaptive consensus tracking (case I)

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Fig. 4

Control signals in adaptive consensus tracking (case I)

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Fig. 5

Trajectory of agents in adaptive consensus tracking (case II)

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Fig. 6

Pseudostates of agents in adaptive consensus tracking (case II)

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Fig. 7

Control signals in adaptive consensus tracking (case II)

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