·250· 工程科学学报，第40卷，第2期 预见跟踪问题也是一个有趣且富有挑战性的研究 [11]Yang T,Meng Z,Dimarogonas D V,et al.Global consensus for 课题，尤其是智能体间的信息交换拓扑是随机切 discrete-time multi-agent systems with input saturation con- 换的情形.然而，本文提出的控制器设计方法并 straints.Automatica,2014,50(2):499 [12]Qin J,Gao H,Yu C.On discrete-time convergence for general 不能够直接应用于处理该类问题.未来将对本问 linear multi-agent systems under dynamic topology.IEEE Trans 题进行深入研究.文献[36]在随机切换拓扑下解 Autom Control,2014,59(4):1054 决了线性多智能体系统的一致性问题，其处理离 [13]Xiao F,Wang L.Consensus protocols for discrete-time multi-agent 散时间一致性问题的方法对解决该问题具有一定 systems with time-varying delays.Automatica,2008,44(10): 的参考意义. 2577 [14]Cortes J.Distributed algorithms for reaching consensus on general 5总结 functions.Automatica,2008,44(3):726 [15]Hong Y,Chen G.Bushnell L Distributed observers design for 本文研究了离散时间多智能体系统的协调预 leader-following control of multi-agent networks.Automatica, 见跟踪问题，所考虑的信息交换拓扑为固定有向 2008,44(3):846 图，智能体的动力学方程为具有任意维数的一般 [16]Hu J,Feng G.Distributed tracking control of leader-follower 线性系统.应用最优预见控制理论的相关结果， multi-agent systems under noisy measurement.Automatica, 2011,46(8):1382 得到了包含误差积分和预见前馈补偿的最优控制 [17]Zhu W,Cheng D.Leader-following consensus of second-order a- 器，给出了保证控制器存在的充分条件.不同于 gents with multiple time-varying delays.Automatica,2010,46 单个智能体的的情形，多智能体系统的全局最优 (12):1994 预见控制器受互联拓扑和系统动力学行为的共同 [18]Ni W,Cheng D.Leader-following consensus of multi-agent sys- 影响.最后，理论和仿真结果均证明在所提出的 tems under fixed and switching topologies.Syst Control Lett, 控制器下，所有的智能体可实现跟踪一致性，且 2010.59(34):209 [19]Wang X H.Xu D B,Hong Y G.Consensus control of nonlinear 一致性效果会随着预见步长的适度增加而得到明 leader-follower multi-agent systems with actuating disturbances. 显改善 Syst Control Lett,2014,73:58 [20]Cao YC.Ren W.Distributed coordinated tracking with reduced 参考文献 interaction via a variable structure approach.IEEE Trans Autom [1]Lynch N A.Distributed Algorithms.San Francisco:Morgan Kauf- Control,2012,57(1):33 mann,1996 [21]Hac A.Optimal linear preview control of active vehicle suspen- [2]Olfati-Saber R.Murray R M.Consensus problems in networks of sion.Vehicle Syst Dyn,1992,21(1):167 agents with switching topology and time-delays.IEEE Trans Autom [22]Shimmyo S,Sato T,Ohnishi K.Biped walking pattern genera- Control,.2004.49(9):1520 tion by using preview control based on three-mass model./EEE [3]Ren W,Atkins E.Distributed multi-vehicle coordinated control Trans Ind Electron,2013,60(11):5137 via local information exchange.Int J Robust Nonlinear Control, [23]Katayama T,Ohki T,Inoue T,et al.Design of an optimal con- 2007,17(10-11):1002 troller for a discrete-time system subject to previewable demand. [4]Ren W.On consensus algorithms for double-integrator dynamics. 1 nt J Control.1985,41(3):677 IEEE Trans Autom Control,2008,53(6):1503 [24]Kojima A,Ishijima S.LQ preview synthesis optimal control and [5]Wang J H,Cheng D Z,Hu X M.Consensus of multi-agent linear worst case analysis.IEEE Trans Autom Control,1999,44(2): dynamic systems.Asian Control,2008,10(2):144 352 [6]Seo J H,Shim H,Back J.Consensus of high-order linear systems [25]Liao F C,Takaba K.Katayama T,et al.Design of an optimal using dynamic output feedback compensator:low gain approach. preview servomechanism for discrete-time systems in a multirate Automatica,2009,45(11):2659 setting.Dyn Contin Discrete Impuls Syst Ser B,2003,10(5): [7]Yang X R,Liu G P.Consensus of descriptor multi-agent systems 727 via dynamic compensators.IET Control Theory Appl,2014,8 [26] Moelja AA,Meinsma G.H2-optimal control of systems with (6):389 multiple i/o delays:time domain approach.Automatica,2005, [8]Li Z K,Duan Z S,Chen G R.Dynamic consensus of linear multi- 41(7):1229 agent systems.IET Control Theory Appl,2011,5(1):19 [27]Moelja AA,Meinsma G.H2 control of preview systems.Auto- [9]Li Z K,Duan Z S,Chen G R.On H=and H2 performance re- matica,2006,42(6):945 gions of multi-agent systems.Automatica,2011,47(4):797 [28]Takaba K.Robust servomechanism with preview action for polyto- [10]Chen Y,Lii J H,Lin Z.Consensus of discrete-time multi-agent pic uncertain systems.Int J Robust Nonlinear Control,2000,10 systems with transmission nonlinearity.Automatica,2013,49 (2):101 (6):1768 [29]Li L,Liao F.C Parameter-dependent preview control with robust工程科学学报,第 40 卷,第 2 期 预见跟踪问题也是一个有趣且富有挑战性的研究 课题, 尤其是智能体间的信息交换拓扑是随机切 换的情形. 然而, 本文提出的控制器设计方法并 不能够直接应用于处理该类问题. 未来将对本问 题进行深入研究. 文献[36] 在随机切换拓扑下解 决了线性多智能体系统的一致性问题, 其处理离 散时间一致性问题的方法对解决该问题具有一定 的参考意义. 5 总结 本文研究了离散时间多智能体系统的协调预 见跟踪问题, 所考虑的信息交换拓扑为固定有向 图, 智能体的动力学方程为具有任意维数的一般 线性系统. 应用最优预见控制理论的相关结果, 得到了包含误差积分和预见前馈补偿的最优控制 器, 给出了保证控制器存在的充分条件. 不同于 单个智能体的的情形, 多智能体系统的全局最优 预见控制器受互联拓扑和系统动力学行为的共同 影响. 最后, 理论和仿真结果均证明在所提出的 控制器下, 所有的智能体可实现跟踪一致性, 且 一致性效果会随着预见步长的适度增加而得到明 显改善. 参 考 文 献 [1] Lynch N A. Distributed Algorithms. San Francisco: Morgan Kauf鄄 mann, 1996 [2] Olfati鄄Saber R, Murray R M. Consensus problems in networks of agents with switching topology and time鄄delays. IEEE Trans Autom Control, 2004, 49(9): 1520 [3] Ren W, Atkins E. Distributed multi鄄vehicle coordinated control via local information exchange. Int J Robust Nonlinear Control, 2007, 17(10鄄11): 1002 [4] Ren W. On consensus algorithms for double鄄integrator dynamics. IEEE Trans Autom Control, 2008, 53(6): 1503 [5] Wang J H, Cheng D Z, Hu X M. Consensus of multi鄄agent linear dynamic systems. Asian J Control, 2008, 10(2): 144 [6] Seo J H, Shim H, Back J. Consensus of high鄄order linear systems using dynamic output feedback compensator: low gain approach. Automatica, 2009, 45(11): 2659 [7] Yang X R, Liu G P. Consensus of descriptor multi鄄agent systems via dynamic compensators. IET Control Theory Appl, 2014, 8 (6): 389 [8] Li Z K, Duan Z S, Chen G R. Dynamic consensus of linear multi鄄 agent systems. IET Control Theory Appl, 2011, 5(1): 19 [9] Li Z K, Duan Z S, Chen G R. On H肄 and H2 performance re鄄 gions of multi鄄agent systems. Automatica, 2011, 47(4): 797 [10] Chen Y, L俟 J H, Lin Z. Consensus of discrete鄄time multi鄄agent systems with transmission nonlinearity. Automatica, 2013, 49 (6): 1768 [11] Yang T, Meng Z, Dimarogonas D V, et al. Global consensus for discrete鄄time multi鄄agent systems with input saturation con鄄 straints. Automatica, 2014, 50(2): 499 [12] Qin J, Gao H, Yu C. On discrete鄄time convergence for general linear multi鄄agent systems under dynamic topology. IEEE Trans Autom Control, 2014, 59(4): 1054 [13] Xiao F, Wang L. Consensus protocols for discrete鄄time multi鄄agent systems with time鄄varying delays. Automatica, 2008, 44 (10): 2577 [14] Cort佴s J. Distributed algorithms for reaching consensus on general functions. Automatica, 2008, 44(3): 726 [15] Hong Y, Chen G, Bushnell L. Distributed observers design for leader鄄following control of multi鄄agent networks. Automatica, 2008, 44(3): 846 [16] Hu J, Feng G. Distributed tracking control of leader鄄follower multi鄄agent systems under noisy measurement. Automatica, 2011, 46(8): 1382 [17] Zhu W, Cheng D. Leader鄄following consensus of second鄄order a鄄 gents with multiple time鄄varying delays. Automatica, 2010, 46 (12): 1994 [18] Ni W, Cheng D. Leader鄄following consensus of multi鄄agent sys鄄 tems under fixed and switching topologies. Syst Control Lett, 2010, 59(3鄄4): 209 [19] Wang X H, Xu D B, Hong Y G. Consensus control of nonlinear leader鄄follower multi鄄agent systems with actuating disturbances. Syst Control Lett, 2014, 73: 58 [20] Cao Y C, Ren W. Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Trans Autom Control, 2012, 57(1): 33 [21] Hac' A. Optimal linear preview control of active vehicle suspen鄄 sion. Vehicle Syst Dyn, 1992, 21(1): 167 [22] Shimmyo S, Sato T, Ohnishi K. Biped walking pattern genera鄄 tion by using preview control based on three鄄mass model. IEEE Trans Ind Electron, 2013, 60(11): 5137 [23] Katayama T, Ohki T, Inoue T, et al. Design of an optimal con鄄 troller for a discrete鄄time system subject to previewable demand. Int J Control, 1985, 41(3): 677 [24] Kojima A, Ishijima S. LQ preview synthesis optimal control and worst case analysis. IEEE Trans Autom Control, 1999, 44(2): 352 [25] Liao F C, Takaba K, Katayama T, et al. Design of an optimal preview servomechanism for discrete鄄time systems in a multirate setting. Dyn Contin Discrete Impuls Syst Ser B, 2003, 10 (5): 727 [26] Moelja A A, Meinsma G. H2 鄄optimal control of systems with multiple i / o delays: time domain approach. Automatica, 2005, 41(7): 1229 [27] Moelja A A, Meinsma G. H2 control of preview systems. Auto鄄 matica, 2006, 42(6): 945 [28] Takaba K. Robust servomechanism with preview action for polyto鄄 pic uncertain systems. Int J Robust Nonlinear Control, 2000, 10 (2): 101 [29] Li L, Liao F. C Parameter鄄dependent preview control with robust ·250·