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418 工程科学学报，第44卷，第3期 0.45 越性能 0.40 39 参考文献 [1]Werbos P J.Approximate dynamic programming for real-time control and neural modeling.In White D A and Sofge D A(Eds. Handbook of Intelligent Control:Neural,Fuzy,and Adaptive 0.15 Approaches.New York,NY:Van Nostrand Reinhold,1992 0.10 [2]Li J N,Chai T Y,Lewis F L,et al.Off-policy interleaved Q- 0.05 leaming:Optimal control for affine nonlinear discrete-time 00 systems.IEEE Trans Neural Nenw Learn Syst,2019,30(5):1308 50 100150200250300 Time index [3] Zhang HG,Liu Y,Xiao GY,et al.Data-based adaptive dynamic 图10触发阈值（例2） programming for a class of discrete-time systems with multiple Fig.10 Triggering threshold (Example 2) delays.IEEE Trans Syst Man Cybern:Syst,2020,50(2):432 [4]Zhang H G,Jiang H,Luo Y H,et al.Data-driven optimal 0.40 ---Case 2 consensus control for discrete-time multi-agent systems with 0.35 -Case I unknown dynamics using reinforcement learning method./EEE 0.30 Trans on Ind Electron,2017,64(5):4091 0.25 [5]Ha MM,Wang D,Liu D R.Generalized value iteration for discounted optimal control with stability analysis.Syst Control Letn,2021,147:104847 [6]Wang D,Ha MM.Qiao J F.Data-driven iterative adaptive critic 0.10 control towards an urban wastewater treatment plant.IEEE Trans 0.05 Ind Electron,2021,68(8:7362 [7]Wang D,Ha MM,Qiao J F,et al.Data-based composite control 00 50 100150200250 300 Time index design with critic intelligence for a wastewater treatment platform. 4 rtif Intell Rev,2020,53(5):3773 图11两种情况下的控制输人（例2） [8]Liang MM.Wang D.Liu D R.Improved value iteration for Fig.11 Control input of the two cases(Example 2) neural-network-based stochastic optimal control design.Neural Nehm,2020,124:280 [9]Liang MM,Wang D,Liu D R.Neuro-optimal control for discrete stochastic processes via a novel policy iteration algorithm.IEEE 色 Trans Syst Man Cybern:-Syst,2020,50(11):3972 [10]Hou J X,Wang D,Liu D R,et al.Model-free H optimal tracking 3 control of constrained nonlinear systems via an iterative adaptive learning algorithm.IEEE Trans Syst Man Cybern:Syst,2020, 2 50(11):4097 [11]Luo B,Liu D R,Huang T W,et al.Model-free optimal tracking control via critic-only Q-leaming.IEEE Trans Neural Netw Learn 50 100150200250300 Ss1,2016,27(10):2134 Time index [12]Al-Tamimi A,Lewis F L,Abu-Khalaf M.Discrete-time nonlinear 图12票动时刻间隔（例2） HJB solution using approximate dynamic programming: Fig.12 Triggering interval(Example 2) Convergence proof.IEEE Trans Syst Man Cybern B:Cybern,2008, 4结论 38(4):943 [13]Zhang H G,Luo Y H,Liu D R.Neural-network-based near- 本文提出一种基于事件的迭代神经控制方法， optimal control for a class of discrete-time affine nonlinear 用以解决离散动态系统的最优调节问题通过收 systems with control constraints.IEEE Trans Neural Net,2009, 20(9):1490 敛性分析，神经网络实现和触发阈值设计，构造基 [14]Wang D,Liu D R,Wei Q L,et al.Optimal control of unknown 于事件迭代自适应评判算法的完整框架.通过仿 nonaffine nonlinear discrete-time systems based on adaptive 真研究，验证了事件驱动迭代神经控制方法的优 dynamic programming.Automatica,2012,48(8):18254 结论 本文提出一种基于事件的迭代神经控制方法, 用以解决离散动态系统的最优调节问题. 通过收 敛性分析, 神经网络实现和触发阈值设计, 构造基 于事件迭代自适应评判算法的完整框架. 通过仿 真研究, 验证了事件驱动迭代神经控制方法的优 越性能. 参 考 文 献 Werbos P J. Approximate dynamic programming for real-time control and neural modeling. In White D A and Sofge D A (Eds. ) Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York, NY: Van Nostrand Reinhold, 1992 [1] Li J N, Chai T Y, Lewis F L, et al. Off-policy interleaved Qlearning: Optimal control for affine nonlinear discrete-time systems. IEEE Trans Neural Netw Learn Syst, 2019, 30（5）: 1308 [2] Zhang H G, Liu Y, Xiao G Y, et al. Data-based adaptive dynamic programming for a class of discrete-time systems with multiple delays. IEEE Trans Syst Man Cybern:Syst, 2020, 50（2）: 432 [3] Zhang H G, Jiang H, Luo Y H, et al. Data-driven optimal consensus control for discrete-time multi-agent systems with unknown dynamics using reinforcement learning method. IEEE Trans on Ind Electron, 2017, 64（5）: 4091 [4] Ha M M, Wang D, Liu D R. Generalized value iteration for discounted optimal control with stability analysis. Syst Control Lett, 2021, 147: 104847 [5] Wang D, Ha M M, Qiao J F. Data-driven iterative adaptive critic control towards an urban wastewater treatment plant. IEEE Trans Ind Electron, 2021, 68（8）: 7362 [6] Wang D, Ha M M, Qiao J F, et al. Data-based composite control design with critic intelligence for a wastewater treatment platform. Artif Intell Rev, 2020, 53（5）: 3773 [7] Liang M M, Wang D, Liu D R. Improved value iteration for neural-network-based stochastic optimal control design. Neural Netw, 2020, 124: 280 [8] Liang M M, Wang D, Liu D R. Neuro-optimal control for discrete stochastic processes via a novel policy iteration algorithm. IEEE Trans Syst Man Cybern:Syst, 2020, 50（11）: 3972 [9] Hou J X, Wang D, Liu D R, et al. Model-free H∞ optimal tracking control of constrained nonlinear systems via an iterative adaptive learning algorithm. IEEE Trans Syst Man Cybern:Syst, 2020, 50（11）: 4097 [10] Luo B, Liu D R, Huang T W, et al. Model-free optimal tracking control via critic-only Q-learning. IEEE Trans Neural Netw Learn Syst, 2016, 27（10）: 2134 [11] Al-Tamimi A, Lewis F L, Abu-Khalaf M. Discrete-time nonlinear HJB solution using approximate dynamic programming: Convergence proof. IEEE Trans Syst Man Cybern B:Cybern, 2008, 38（4）: 943 [12] Zhang H G, Luo Y H, Liu D R. Neural-network-based nearoptimal control for a class of discrete-time affine nonlinear systems with control constraints. IEEE Trans Neural Netw, 2009, 20（9）: 1490 [13] Wang D, Liu D R, Wei Q L, et al. Optimal control of unknown nonaffine nonlinear discrete-time systems based on adaptive dynamic programming. Automatica, 2012, 48（8）: 1825 [14] 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0 50 100 150 Time index Threshold 200 250 300 图 10 触发阈值 (例 2) Fig.10 Triggering threshold (Example 2) 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0 50 100 150 Time index Case 2 Case 1 Control input 200 250 300 图 11 两种情况下的控制输入 (例 2) Fig.11 Control input of the two cases (Example 2) 6 5 4 3 2 1 0 0 50 100 150 Time index Triggering interval 200 250 300 图 12 驱动时刻间隔 (例 2) Fig.12 Triggering interval (Example 2) · 418 · 工程科学学报，第 44 卷，第 3 期