×

Control of multi-agent systems with input delay via PDE-based method. (English) Zbl 1429.93027

Summary: This paper deals with the control of collective dynamics of a large scale multi-agent system (MAS) moving in a 3-D space under the occurrence of arbitrarily large boundary input delay. The collective dynamics is described by a pair of reaction-advection-diffusion partial differential equations (PDEs) consisting of a complex-valued state whose real-part and imaginary-part denote the position coordinates \(x\) and \(y\), respectively, and a real-valued state equation governing the evolution of the collective dynamics in the \(z\) coordinate. A 2-D cylindrical surface represents the indexes in a continuum and defines the topology where the agents communicate with each other based on a leader-follower strategy. The agents located at the boundary are chosen as the leaders and are subject to input delays that are practically induced by communication, measurement or even actuator constraints. A boundary control, which compensates the effect of the delay and drives all the agents to the desired formation is designed based on PDE-backstepping method. Here, the 2-D cylindrical coordinate leads to the existence of singular points in the gain kernels, which makes the stability analysis non-trivial in comparison to the 1-D problem. The proposed controller ensures the exponential stability of the MAS in \(H^2\) norm under full-state measurement and transitions from one formation to another are achievable as illustrated by the simulation results.

MSC:

93A16 Multi-agent systems
93A15 Large-scale systems
93C43 Delay control/observation systems
93C20 Control/observation systems governed by partial differential equations
35K57 Reaction-diffusion equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alonso-Mora, J.; Naegeli, T.; Beardsley, P.; Beardsley, P., Collision avoidance for aerial vehicles in multi-agent scenarios, Autonomous Robots, 39, 1, 101-121 (2015)
[2] Brezis, H., Functional analysis, sobolev spaces and partial differential equations (2010), Universitext: Universitext Springer New York
[3] Brown, J. W.; Churchill, R. V., (Complex variables and applications. Complex variables and applications, Brown and Churchill series (2009), McGraw-Hill Higher Education)
[4] Evans, L. C., (Partial differential equations; 2nd Ed.. Partial differential equations; 2nd Ed., Graduate Studies in Mathematics (2010), American Mathematical Society: American Mathematical Society Providence, RI)
[5] Ferrari-Trecate, G.; Buffa, A.; Gati, M., Analysis of coordination in multi-agent systems through partial difference equations, IEEE Transactions on Automatic Control, 51, 6, 1058-1063 (2006) · Zbl 1366.93273
[6] Fridman, E., A refined input delay approach to sampled-data control, Automatica, 46, 2, 421-427 (2010) · Zbl 1205.93099
[7] Frihauf, P.; Krstic, M., Leader-enabled deployment onto planar curves: A pde-based approach, IEEE Transactions on Automatic Control, 56, 8, 1791-1806 (2011) · Zbl 1368.93269
[8] Hu, J.-P.; Hong, Y.-G., Leader-following coordination of multi-agent systems with coupling time delays, Physica A. Statistical Mechanics and its Applications, 374, 2, 853-863 (2007)
[9] Kim, J.; Kim, K.-D.; Natarajan, V.; Kelly, S. D.; Bentsman, J., PdE-Based model reference adaptive control of uncertain heterogeneous multi-agent networks, Nonlinear Analysis. Hybrid Systems, 2, 4, 1121-1129 (2008)
[10] Krstic, M., Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch, Automatica, 44, 2930-2935 (2008) · Zbl 1152.93498
[11] Krstic, M., Control of an unstable reaction – diffusion pde with long input delay, Systems & Control Letters, 58, 10, 773-782 (2009) · Zbl 1181.93030
[12] Krstic, M.; Smyshlyaev, A., Boundary control of PDEs: A course on backstepping designs, (Advances in Design and Control (2008), Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104)) · Zbl 1149.93004
[13] Lee, D.; Spong, M. W., Agreement with non-uniform information delays, American Control Conference, 756-761 (2006)
[14] Lin, P.; Ren, W., Constrained consensus in unbalanced networks with communication delays, IEEE Transactions on Automatic Control, 59, 3, 775-781 (2014) · Zbl 1360.93037
[15] Meurer, T.; Krstic, M., Finite-time multi-agent deployment: A nonlinear pde motion planning approach, Automatica, 47, 11, 2534-2542 (2011) · Zbl 1228.93013
[16] Pilloni, A.; Pisano, A.; Orlov, Y.; Usai, E., Consensus-based control for a network of diffusion pdes with boundary local interaction, IEEE Transactions on Automatic Control, 61, 9, 2708-2713 (2016) · Zbl 1359.93206
[17] Qi, J.; Tang, S.-X.; Wang, C., Parabolic PDE-based multi-agent formation control on a cylindrical surface, International Journal of Control, 1-34 (2017)
[18] Qi, J.; Vazquez, R.; Krstic, M., Multi-agent deployment in 3-D via PDE control, IEEE Transactions on Automatic Control, 60, 4, 891-906 (2015) · Zbl 1360.93320
[19] Qi, J.; Zhang, J.; Ding, Y., Wave equation-based time-varying formation control of multi-agent systems, IEEE Transactions on Control Systems Technology, 1-14 (2017)
[20] Ren, W., Multi-vehicle consensus with a time-varying reference state, Systems & Control Letters, 56, 7-8, 474-483 (2007) · Zbl 1157.90459
[21] Su, H.-S.; Wang, X.-F., Second-order consensus of multiple agents with coupling delay, (7th world congress on intelligent control and automation (2008)), 7181-7186 · Zbl 1316.78006
[22] Tang, S.-X.; Qi, J.; Zhang, J., Formation tracking control for multi-agent systems: A wave-equation based approach, International Journal of Control, Automation and Systems, 15, 6, 2704-2713 (2017)
[23] Tian, Y.-P.; Liu, C.-L., Consensus of multi-agent systems with diverse input and communication delays, IEEE Transactions on Automatic Control, 53, 9, 2122-2128 (2008) · Zbl 1367.93411
[24] Vazquez, R.; Krstic, M., Explicit output-feedback boundary control of reaction – diffusion PDEs on arbitrary-dimensional balls, ESAIM. Control, Optimisation and Calculus of Variations, 22, 4, 1078-1096 (2016) · Zbl 1358.35058
[25] Wang, H.; Guo, D.; Liang, X.; Chen, W.; Hu, G.; Leang, K. K., Adaptive vision-based leader follower formation control of mobile robots, IEEE Transactions on Industrial Electronics, 64, 4, 2893-2902 (2017)
[26] Wang, S.-S.; Qi, J.; Fang, J.-A., Control of 2-d reaction-advection-diffusion pde with input delay, Chinese Automation Congress (CAC), 7145-7150 (2017)
[27] Wang, W.; Slotine, J. J.E., Contraction analysis of time-delayed communications and group cooperation, IEEE Transactions on Automatic Control, 51, 4, 712-717 (2006) · Zbl 1366.90064
[28] Wang, J.; Xin, M., Integrated optimal formation control of multiple unmanned aerial vehicles, IEEE Transactions on Control Systems Technology, 21, 5, 1731-1744 (2013)
[29] Wang, X.; Yadav, V.; Balakrishnan, S. N., Cooperative UAV formation flying with obstacle/collision avoidance, IEEE Transactions on Control Systems Technology, 15, 4, 672-679 (2007)
[30] Zetocha, P.; Self, L.; Wainwright, R.; Burns, R.; Brito, M.; Surka, D., Commanding and controlling satellite clusters, IEEE Intelligent Systems and their Applications, 15, 6, 8-13 (2000)
[31] Zhou, B.; Lin, Z. L., Consensus of high-order multi-agent systems with large input and communication delays, Automatica, 50, 2, 452-464 (2014) · Zbl 1364.93044
[32] Zhu, W.; Jiang, Z.-P., Event-based leader-following consensus of multi-agent systems with input time delay, IEEE Transactions on Automatic Control, 60, 5, 1362-1367 (2015) · Zbl 1360.93268
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.