Output Consensus Control of Nonlinear Non-minimum Phase Multi-agent Systems Using Output Redefinition Method

Document Type : Research Article

Authors

Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

Abstract

This paper concerns the problem of output consensus in nonlinear non-minimum phase systems. The main contribution of the paper is to guarantee achieving consensus in the presence of unstable zero dynamics. To achieve this goal, an output redefinition method is proposed. The new outputs of agents are functions of original outputs and internal states and defined such that the dynamics of agents are minimum phase. However, since the main objective is to achieve consensus on original outputs of agents, the consensus invariant set in the new coordinate of the agents dynamics should be defined such that if the new states of the agents converge to this invariant set, the output consensus in original system is achieved. On the other words, achieving consensus in minimum phase system with redefined output is equivalent to output consensus in original system. After defining the proper invariant set, a consensus protocol is designed to guarantee that the redefined outputs and the internal states to this set. Theoretical results are mathematically proved based on Lyapunov criterion. Numerical examples are employed to show the effectiveness of the proposed approach.

Keywords


[1]Das, A., Fierro, R., Kumar, V., Ostrowsky, J., Spletzer, J., and Taylor, C., ”,A Vision-Based Formation Control Framework”, IEEE Transactions on Automatic Control, vol. 18, no. 5, pp. 813–825, 2002.
[2]Rezaee, H., and Abdollahi, F., “A Decentralized Cooperative Control Scheme with Obstacle Avoidance for A Team of Mobile Robots”, IEEE Transactions on Industrial Electronics, vol. 61, no. 1, pp. 347–354, 2014.
[3]Kuriki, Y., and Namerikawa, T., “Consensus-Based Cooperative Formation Control with Collision Avoidance for A Multi-UAV System”, in Proceedings of the American Control Conference, USA, June 2014, pp. 2077–2082.
[4]Edwards, D., Bean, T., Odell, D., and Anderson, M., “A Leader-Follower Algorithm for Multiple AUV Formations”, in Proceedings of IEEE/OES Autonomous Underwater Vehicles, USA, June 2004, pp. 40–46.
[5]Wu, Y., Cao, X., Zheng, P., and Zeng, Z., “Variable Structure- Based Decentralized Relative Attitude-Coordinated Control for Satellite Formation”, IEEE Aerospace and Electronic System Magazine , vol. 27, no. 12, pp. 18–25, 2012.
[6]Olfati-saber, R., “Distributed Kalman Filtering for Sensor Networks”, in Proceedings of the IEEE Conference on Decision and Control, USA, December 2007, pp. 5492– 5498.
[7]Olfati-saber, R., “Distributed Kalman Filter with Embedded Consensus Filters”, in Proceedings of the European Control Conference and the 44th IEEE Conference on Decision and Control , Spain), December 2005, pp. 8179–8184.
[8]Tan, K., and Lewis, M., “High Precision Formation Control of Mobile Robots Using Virtual Structures”, Autonomous Robots, vol. 4, no. 4, pp. 387–403, 1997.
[9]Balch, T., and Arkin, R., “Behavior-Based Formation Control for Multi Robot Teams”, IEEE Transactions IEEE Transactions on Robotics and Automation, vol. 14, no. 6, pp. 926–939, 1998.
[10]Fax, J., and Murray, R., “Information Flow and Cooperative Control of Vehicle Formations”, IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465–1467, 2004.
[11]Shamsi, F., Abdollahi, F., and Nikravesh, S. K. Y., “Time Varying Formation Control Using Differential Game Approach”, in Proceedings of the 18th IFAC World Congress, Italy, August 2011, pp. 1114–1119.
[12]Liu, C., and Tian, Y., “Formation Control of Multi- Agent Systems with Heterogeneous Communication Delays”, International Journal of Systems Science, vol. 40, no. 6, pp. 627–636, 2009.
[13]Chen, Y. Y., and Tian, Y. P., “Formation Tracking and Attitude Synchronization Control of Under Actuated Ships Along Closed Orbits”, International Journal of Robust and Nonlinear Control, vol. 25, no. 16, pp. 2023– 2044, 2015.
[14]Lin, J., Morse, A., and Anderson, B., “The Multi-Agent Rendezvous Problem. Part 1: The Synchronous Case”, SIAM Journal on Control and Optimization, vol. 46, no. 6, pp. 2096–2119, 2007.
[15]Abdessameud, A., and Tayebi, A., “Attitude Synchronization of A Group of Spacecraft Without Velocity Measurement”, IEEE Transactions on Automatic Control, vol. 54, no. 11, pp. 2642–2648, 2009.
[16]Ren, W., “Distributed Cooperative Attitude Synchronization and Tracking for Multiple Rigid Bodies”, IEEE Transactions on Control Systems Technology, vol. 18, no. 2, pp. 383–392, 2010.
[17]Rezaee, H., and Abdollahi, F., “Synchronized Cross Coupled Sliding Mode Controllers for Cooperative UAVs with Communication Delays”, in Proceedings of the 51st IEEE Conference on Decision and Control, USA, August 2012, pp. 3116–3121.
[18]Rezaee, H., Abdollahi, F., and Talebi, H., “H1 Based Motion Synchronization in Formation Flight with Delayed Communications”, IEEE Transactions on Industrial Electronics, vol. 61, no. 11, pp. 6175–6182, 2014.
[19]Wang, L. and Wang, Q. G., “A General Approach for Synchronization of Nonlinear Networked Systems with Switching Topology”, International Journal of Systems Science, vol. 44, no. 12, pp. 2199–2210, 2013.
[20]Olfati-Saber, R., and Murray, R., “Consensus Problems in Networks of Agents With Switching Topology and Time-Delays”, IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004.
[21]Olfati-Saber, R., Fax, J., and Murray, R., “Consensus and Cooperation in Networked Multi-agent Systems”, Proceeedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007.
[22]Ren, W. and Atkins, E., “Distributed Multi-Vehicle Coordinated Control via Local Information Exchange”, International Journal of Robust and Nonlinear Control, vol. 17, no. 10, pp. 1002–1033, 2007.
[23]Ren, W., “Consensus Strategies for Cooperative Control of Vehicle Formations”, IET Control Theory & Applications, vol. 1, no. 2, pp. 505–512, 2007.
[24]Zhou, B., Meng, C., and Hu, G., “Robust Consensus Tracking of Double Integrator Dynamics by Bounded Distributed Control”, International Journal of Robust and Nonlinear Control, 2015.
[25]Feng, Y., Xu, S., and Zhang, B., “Group Consensus Control for Double Integrator Dynamic Multi Agent Systems with Fixed Communication Topology”, International Journal of Robust and Nonlinear Control, vol. 24, no. 3, pp. 532–547, 2014.
[26]Seo, J. H., Shim, H., and Back, J., “Consensus of High- Order Linear Systems Using Dynamic Output Feedback Compensator: Low Gain Approach”, Automatica, vol. 45, no. 11, pp. 2659–2664, 2009.
[27]Yang, T. , Roy, S., Wan, Y., and Saberi, A., “Constructing Consensus Controllers for Networks with Identical General Linear Agents”, International Journal of Robust and Nonlinear Control, vol. 21, no. 11, pp. 1237–1256, 2011.
[28]Sun, J., and Gneg, Z., “Adaptive Consensus Tracking for Linear Multi Agent Systems with Heterogeneous Unknown Nonlinear Dynamics”, International Journal of Robust and Nonlinear Control, 2015.
[29Zhong, W. S., Liu, G., and Rees, D., “Global Bounded Consensus of Multi-Agent Systems with Non-Identical Nodes and Communication Time-Delay Topology”, IEEE Transactions on Systems, Man, and Cybernetics–Part B: Cybernetic, vol. 44, no. 2, pp. 346–357, 2013.
[30]Yu, W., Chen, G., and Cao, M., “Consensus in Directed Networks of Agents with Nonlinear Dynamics”, IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1436–1441, 2011.
[31]Zhao, Y., Li, Z., and Duan, Z., “Distributed Consensus Tracking of Multi Agent Systems with Nonlinear Dynamics Under A Reference Leader”, International Journal of Control, vol. 86, no. 10, pp. 1859–1869, 2013.
[32]Song, J. C. Q., and Yu, W., “Second-Order Leader- Following Consensus of Nonlinear Multi-Agent System via Pinning Control”, Systems and Control Letters, vol. 59, no. 9, pp. 553–562, 2010.
[33]Yu, W., Chen, G., Cao, M., and Kurths, J., “Second- Order Consensus for Multi Agent Systems with Directed Topologies and Nonlinear Dynamic”, IEEE Transactions on Systems, Man, and Cybernetics–Part B: Cybernetic, vol. 40, no. 3, pp. 881–891, 2010.
[34]Liu, K., Xie, G., Ren, W., and Wang, L., “Consensus for Multi-Agent Systems with Inherent Nonlinear Dynamics under Directed Topologies”, Systems and Control Letters, vol. 62, no. 2, pp. 152–162, 2013.
[35]Su, H., Chen, G., Wang, X., Lin, Z., “Adaptive Second- Order Consensus of Networked Mobile Agents with Nonlinear Dynamics”, Automatica, vol. 47, no. 2, pp. 368–375, 2011.
[36]Li, H., Liao, X., and Huang, T., “Second-Order Locally Dynamical Consensus of Multi-Agent Systems with Arbitrarily Fast Switching Directed Topologies”, IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 43, no. 6, pp. 1343–1353, 2013.
[37]Wen, G., Peng, Z., Rahmani, A., and Yu, Y., “Distributed Leader Following Consensus for Second-Order Multi-agent Systems with Nonlinear Inherent Dynamics”, International Journal of Systems Science, vol. 45, no. 9, pp. 1892–1901, 2014.
[38]Munz, U., Papachristodoulou, A., and Allgower, F., “Robust Consensus Controller Design for Nonlinear Relative Degree Two Multi-Agent Systems with Communication Constraints”, IEEE Transactions on Automatic Control, vol. 56, no. 1, pp. 145–151, 2011.
[39]Bidram, A., Lewis, F. L., and Davoudi, A., “Synchronization of Nonlinear Heterogeneous Cooperative Systems Using Input Output Feedback Linearization”, Automatica, vol. 50, no. 10, pp. 2578-2585, 2014.
[40]Ren, W., “Distributed Leaderless Consensus Algorithms for Networked Euler Lagrange Systems”, International Journal of Control, vol. 82, no. 11, pp. 2137–2149, 2009.
[41]Liu, X., Du, C., Lu, P., and Yang, D., “Decentralised Consensus for Multiple Lagrangian Systems Based on Event-Triggered Strategy”, International Journal of Control, vol. 89, no. 6, pp. 1111–1124, 2016.
[42]Liu, Y., Min, H., Wang, S., Liu, Z., and Liao, S., “Consensus for Multiple Heterogeneous Euler-Lagrange Systems with Time-Delay and Jointly Connected Topologies”, Journal of Franklin Institute, vol. 351, no. 3, pp. 1700–1716, 2014.
[43] V. H. L. Cheng and C. A. Desoer, “Limitations on the Closed-loop Transfer Function due to Right-half Plane Transmission Zeros of the Plant”, IEEE Transactions on Automatic Control, vol. 25, no. 6, pp. 1218-1220, 1980.
[44] D. E. Miller and E. J. Davison, “On Necessary Assumptions in Continuous Time Model Reference Adaptive Control”, 28th IEEE Conference on Decision and Control, Tampa, FL, 1989.
[45] M. M. Seron, J. H. Braslavsky, P. V. Kokotovic and D. Q. Mayne, “Feedback Limitations in Nonlinear Systems: From Bode Integrals to Cheap Control”, IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 829–833, 1999.
[46] J. H. Braslavsky, R. H. Middleton and J. S. Freudenberg, “Cheap Control Performance of a Class of Non-right-invertible Nonlinear Systems”, IEEE Transactions on Automatic Control, vol. 47, no. 8, pp. 1314-1319, 2002.
[47] A. P. Aguiar, J. P. Hespanha, and P. V. Kokotović, “Performance Limitations in Reference Tracking and Path Following for Nonlinear Systems”, Automatica, vol. 44, no. 3, pp. 598–610, 2008.
[48] H N. Foghahaayee, M. B. Menhaj, and H. A. Talebi, “Weakly and Strongly Non-Minimum Phase Systems: Properties and Limitations”, International Journal of Control, vol. 89, no. 2, pp. 306-321, 2016.
[49] J. S. A. Hepburn, and W. M. Wonham, “Error Feedback and Internal Models on Differentiable Manifolds”, IEEE Transaction on Automatic Control, vol. AC-29, pp. 397- 403, 1984.
[50] A. Isidori, and C. I. Byrnes, “Output Regulation of Nonlinear Systems”, IEEE Transaction on Automatic Control, vol. 35, no. 2, pp. 131-140, 1990.
[51] V. Utkin, J. Guldner, and J. Shi, “Sliding Modes in Electromechanical Systems”, Taylor & Francis, 1999.
[52] I. A. Shkolnikov and Y. B. Shtessel, “Tracking in a Class of Non minimum-phase Systems with Nonlinear Internal Dynamics via Sliding Mode Control using Method of System Center”, Automatica, vol. 38, no. 5, pp. 837-842, 2002.
[53] X. G. Yan, C. Edwards, S. K. Spurgeon, “Output Feedback Sliding Mode Control for Non-minimum Phase Systems with Non-linear Disturbances”, International Journal of Control, vol. 77, no. 15, pp. 1353-1361, 2004.
[54] Gopalswamy S., and Hedrick, J. K., “Tracking Nonlinear Non-Minimum Phase Systems Using Sliding Control”, International Journal of Control, vol. 57, no. 5, pp. 1141– 1158, 1993.
[55] F. Jahangiri, H. A. Talebi, M. B. Menhaj, and C. Ebenbauer, “A New Approach for Minimum Phase Output Definition”, International Journal of Systems Science, 2016, doi: 10.1080/00207721.2016.1179815.
[56] Slotine, J. J. E, and Li, W., Applied Nonlinear Control. Englewood Cliffs: Prantice Hall, 1991.
[57] Ren, W., and Beard, R. W., Distributed Consensus in Multi-vehicle Cooperative Control, Communications and Control Engineering Series. London: Springer-Verlag, 2008.