基于勒贝格采样的非线性系统优化控制
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摘要
为了解决非线性系统中的最优控制问题,在性能势理论的基础上,提出了一种基于勒贝格采样的新的事件触发控制策略。首先,根据最优控制理论,给出了基于勒贝格采样的非线性系统数学模型。然后,结合Markov决策过程中的时间集结法、解析法和策略迭代算法对搭建的数学模型进行Matlab仿真求解,得出了该系统的最优策略和最优性能。最后,将勒贝格采样系统与传统的周期采样系统作比较,深入分析了两种采样方案下的优化性能,比较了其优缺点,得出了勒贝格采样方法不仅能改善系统性能,解决了系统的"维数灾"问题,还能在某种程度上减小系统的资源消耗。
In order to solve the optimal control problem in nonlinear systems,a new event-triggered control strategy based on Lebesgue sampling is proposed based on the performance potential theory.Firstly,according to the optimal control theory,a mathematical model of nonlinear system based on Lebesgue sampling is given.Then,combined with time aggregation method,analytical method and strategy iteration algorithm in Markov decision process,the mathematical model of the constructed mathematical model is solved by Matlab,and the optimal strategy and optimal performance of the system are obtained.Finally,the Lebesgue sampling system is compared with the traditional periodic sampling system.The optimization performance of the two sampling schemes is analyzed in depth,and its advantages and disadvantages of the sampling system are compared.It is concluded that the Lebesgue sampling method can not only improve the system performance,but also solve the " dimensionality disaster" problem of the system.It can reduce the resource consumption of the system to some extent.
In order to solve the optimal control problem in nonlinear systems,a new event-triggered control strategy based on Lebesgue sampling is proposed based on the performance potential theory.Firstly,according to the optimal control theory,a mathematical model of nonlinear system based on Lebesgue sampling is given.Then,combined with time aggregation method,analytical method and strategy iteration algorithm in Markov decision process,the mathematical model of the constructed mathematical model is solved by Matlab,and the optimal strategy and optimal performance of the system are obtained.Finally,the Lebesgue sampling system is compared with the traditional periodic sampling system.The optimization performance of the two sampling schemes is analyzed in depth,and its advantages and disadvantages of the sampling system are compared.It is concluded that the Lebesgue sampling method can not only improve the system performance,but also solve the " dimensionality disaster" problem of the system.It can reduce the resource consumption of the system to some extent.
引文
[1]鄢镕易,何潇,周东华.一类存在参数摄动的线性随机系统的鲁棒间歇故障诊断方法[J].自动化学报,2016,42(7):1004-1013.Yan Rongyi,He Xiao,Zhou Donghua.Robust intermittent fault diagnosis method for a class of linear stochastic systems with parameter perturbations[J].Acta Automatica Sinica,2016,42(7):1004-1013.
[2]Li G,Chen Y H.Controller design for stochastic nonlinear systems with matched conditions[J].Journal of Systems Engineering and Electronics,2018,29(1):160-165.
[3]魏新江,张林青.一类随机系统基于干扰观测器的抗干扰控制[J].控制与决策,2017,32(5):939-942.Wei Xinjiang,Zhang Linqing.Anti-interference control based on disturbance observer for a class of stochastic systems[J].Control and Decision,2017,32(5):939-942.
[4]孙婧,段法兵.复杂随机系统的信噪比增益研究与阵列随机共振[J].复杂系统与复杂性科学,2006,1(2):50-54.Sun Jing,Duan Fabing.Study on signal-to-noise ratio gain and array stochastic resonance of complex stochastic systems[J].Complex Systems and Complexity Science,2006,1(2):50-54
[5]张维存.参数不确定离散随机系统的加权多模型自适应控制[J].自动化学报,2015,41(3):541-550.Zhang Weicun.Weighted multiple model adaptive control for discrete stochastic systems with parameter uncertainties[J].Acta Automatica Sinica,2015,41(3):541-550.
[6]赵千川,奚宏生,唐立新,吴维敏,胡核算,夏俐.“离散事件动态系统”专题前言[J].控制理论与应用,2018,35(1):31.Zhao Qianchuan,Xi Hongsheng,Tang Lixin,Wu Weimin,Hu Hesuan,Xia Li.The foreword of"discrete event dynamic dystem"[J].Control Theory and Applications,2018,35(1):31.
[7]张健.复杂性科学几个新兴的应用研究方向[J].复杂系统与复杂性科学,2004,1(3):33-40.Zhang Jian.Several emerging applied research directions of complexity science[J].Complex Systems and Complexity Science,2004,1(3):33-40.
[8]Ye P X,Song Z J.Truncation and aliasing errors for whittaker kotelnikov shannon sampling expansion[J].Applied Mathematics:A Journal of Chinese Universities,2012,27(4):412-418.
[9]成云,宋运忠.基于保证集的多智能体系统自触发控制[J].复杂系统与复杂性科学,2017,14(4):97-104.Cheng Yun,Song Yunzhong.Self-trigger control of multi-agent system based on guarantee set[J].Complex Systems and Complexity Science,2017,14(4):97-104.
[10]王桐,邱剑彬,高会军.随机非线性系统基于事件触发机制的自适应神经网络控制[J].自动化学报,2019,45(1):226-233.Wang Tong,Qiu Jianbin,Gao Huijun.Adaptive neural network control based on event trigger mechanism for stochastic nonlinear systems[J].Acta Automatica Sinica,2019,45(1):226-233.
[11]Heemels W P M H,Donkers M C F,Teel A R.Periodic event triggered control for linear systems[J].IEEE Transactions on Automatic Control,2013,58(4):847-861.
[12]石杰,李银伢,戚国庆,盛安冬.不完全量测下基于事件触发机制的面目标跟踪系统CRLB[J].自动化学报,2018,44(9):1648-1661.Shi Jie,Li Yinya,Qi Guoqing,Sheng Andong.Surface target tracking system based on event triggering mechanism under incomplete measurement CRLB.[J]Acta Automatica Sinica.2018,44(9):1648-1661.
[13]李国梁,张合新,朱满林,等.状态反馈事件触发控制建模与分析[J].控制工程,2017,24(5):1075-1078.Li Guoliang,Zhang Hexin,Zhu Manlin,et al.Modeling and analysis of state feedback event trigger control[J].Control Engineering,2017,24(5):1075-1078.
[14]Vamvoudakis K G.Event triggered optimal adaptive control algorithm for continuous-time nonlinear systems[J].IEEE/CAAJournal of Automatica Sinica,2014,1(3):282-293.
[15]Marchand N.Stabilization of lebesgue sampled systems with bounded controls:the chain of integrators case[J].IFAC Proceedings Volumes,2008,41(2):10265-10270.
[16]Xu Y K,Cao X R.Lebesgue sampling based optimal control problems with time aggregation[J].IEEE Transactions on Automatic Control,2011,56(5):1097-1109.
[17]Yan W,Zhang B,Wang X,et al.Lebesgue sampling based diagnosis and prognosis for lithium ion batteries[J].IEEE Transactions on Industrial Electronics,2016,63(3):1804-1812.
[18]Ohsaki H,Iwase M,Hatakeyama S.Rotational motion control design for cart pendulum system with lebesgue sampling[J].Journal of Mechanical Engineering&Automation,2011,2(3):827-832.
[19]Xu X,Lian C,Zuo L,et al.Kernel based approximate dynamic programming for real time online learning control:an experimental study[J].IEEE Transactions on Control Systems Technology,2014,22(1):146-156.
[20]王涛,张化光.基于策略迭代的连续时间系统的随机线性二次最优控制[J].控制与决策,2015,30(9):1674-1678.Wang Tao,Zhang Huaguang.Stochastic linear quadratic optimal control for continuous time systems based on strategy iteration[J].Control and Decision,2015,30(9):1674-1678.
[21]Cao X R,Ren Z,Bhatnagar S,et al.A time aggregation approach to markov decision processes[J].Automatica,2002,38(6):929-943.
[22]宋明珠,吴永锋.随机环境中马氏链函数的强大数定律[J].数学杂志,2016,36(6):1245-1252.Song Mingzhu,Wu Yongfeng.The strong law of large numbers for markov chain functions in random environments[J].Journal of Mathematics,2016,36(6):1245-1252.
[23]Cao X R.Stochastic learning and optimization:a sensitivity based approach[J].Annual Reviews in Control,2009,33(1):11-24.
[24]刘艳君,丁锋.非均匀周期采样系统的递阶最小二乘辨识方法[J].控制与决策,2011,26(3):453-456.Liu Yanjun,Ding Feng.A hierarchical least squares identification method for non-uniform periodic sampling systems[J].Control and Decision,2011,26(3):453-456.
[25]Mazo M,Tabuada P.Decentralized event triggered control over wireless sensor/actuator networks[J].IEEE Transactions on Automatic Control,2011,56(10):2456-2461.
[2]Li G,Chen Y H.Controller design for stochastic nonlinear systems with matched conditions[J].Journal of Systems Engineering and Electronics,2018,29(1):160-165.
[3]魏新江,张林青.一类随机系统基于干扰观测器的抗干扰控制[J].控制与决策,2017,32(5):939-942.Wei Xinjiang,Zhang Linqing.Anti-interference control based on disturbance observer for a class of stochastic systems[J].Control and Decision,2017,32(5):939-942.
[4]孙婧,段法兵.复杂随机系统的信噪比增益研究与阵列随机共振[J].复杂系统与复杂性科学,2006,1(2):50-54.Sun Jing,Duan Fabing.Study on signal-to-noise ratio gain and array stochastic resonance of complex stochastic systems[J].Complex Systems and Complexity Science,2006,1(2):50-54
[5]张维存.参数不确定离散随机系统的加权多模型自适应控制[J].自动化学报,2015,41(3):541-550.Zhang Weicun.Weighted multiple model adaptive control for discrete stochastic systems with parameter uncertainties[J].Acta Automatica Sinica,2015,41(3):541-550.
[6]赵千川,奚宏生,唐立新,吴维敏,胡核算,夏俐.“离散事件动态系统”专题前言[J].控制理论与应用,2018,35(1):31.Zhao Qianchuan,Xi Hongsheng,Tang Lixin,Wu Weimin,Hu Hesuan,Xia Li.The foreword of"discrete event dynamic dystem"[J].Control Theory and Applications,2018,35(1):31.
[7]张健.复杂性科学几个新兴的应用研究方向[J].复杂系统与复杂性科学,2004,1(3):33-40.Zhang Jian.Several emerging applied research directions of complexity science[J].Complex Systems and Complexity Science,2004,1(3):33-40.
[8]Ye P X,Song Z J.Truncation and aliasing errors for whittaker kotelnikov shannon sampling expansion[J].Applied Mathematics:A Journal of Chinese Universities,2012,27(4):412-418.
[9]成云,宋运忠.基于保证集的多智能体系统自触发控制[J].复杂系统与复杂性科学,2017,14(4):97-104.Cheng Yun,Song Yunzhong.Self-trigger control of multi-agent system based on guarantee set[J].Complex Systems and Complexity Science,2017,14(4):97-104.
[10]王桐,邱剑彬,高会军.随机非线性系统基于事件触发机制的自适应神经网络控制[J].自动化学报,2019,45(1):226-233.Wang Tong,Qiu Jianbin,Gao Huijun.Adaptive neural network control based on event trigger mechanism for stochastic nonlinear systems[J].Acta Automatica Sinica,2019,45(1):226-233.
[11]Heemels W P M H,Donkers M C F,Teel A R.Periodic event triggered control for linear systems[J].IEEE Transactions on Automatic Control,2013,58(4):847-861.
[12]石杰,李银伢,戚国庆,盛安冬.不完全量测下基于事件触发机制的面目标跟踪系统CRLB[J].自动化学报,2018,44(9):1648-1661.Shi Jie,Li Yinya,Qi Guoqing,Sheng Andong.Surface target tracking system based on event triggering mechanism under incomplete measurement CRLB.[J]Acta Automatica Sinica.2018,44(9):1648-1661.
[13]李国梁,张合新,朱满林,等.状态反馈事件触发控制建模与分析[J].控制工程,2017,24(5):1075-1078.Li Guoliang,Zhang Hexin,Zhu Manlin,et al.Modeling and analysis of state feedback event trigger control[J].Control Engineering,2017,24(5):1075-1078.
[14]Vamvoudakis K G.Event triggered optimal adaptive control algorithm for continuous-time nonlinear systems[J].IEEE/CAAJournal of Automatica Sinica,2014,1(3):282-293.
[15]Marchand N.Stabilization of lebesgue sampled systems with bounded controls:the chain of integrators case[J].IFAC Proceedings Volumes,2008,41(2):10265-10270.
[16]Xu Y K,Cao X R.Lebesgue sampling based optimal control problems with time aggregation[J].IEEE Transactions on Automatic Control,2011,56(5):1097-1109.
[17]Yan W,Zhang B,Wang X,et al.Lebesgue sampling based diagnosis and prognosis for lithium ion batteries[J].IEEE Transactions on Industrial Electronics,2016,63(3):1804-1812.
[18]Ohsaki H,Iwase M,Hatakeyama S.Rotational motion control design for cart pendulum system with lebesgue sampling[J].Journal of Mechanical Engineering&Automation,2011,2(3):827-832.
[19]Xu X,Lian C,Zuo L,et al.Kernel based approximate dynamic programming for real time online learning control:an experimental study[J].IEEE Transactions on Control Systems Technology,2014,22(1):146-156.
[20]王涛,张化光.基于策略迭代的连续时间系统的随机线性二次最优控制[J].控制与决策,2015,30(9):1674-1678.Wang Tao,Zhang Huaguang.Stochastic linear quadratic optimal control for continuous time systems based on strategy iteration[J].Control and Decision,2015,30(9):1674-1678.
[21]Cao X R,Ren Z,Bhatnagar S,et al.A time aggregation approach to markov decision processes[J].Automatica,2002,38(6):929-943.
[22]宋明珠,吴永锋.随机环境中马氏链函数的强大数定律[J].数学杂志,2016,36(6):1245-1252.Song Mingzhu,Wu Yongfeng.The strong law of large numbers for markov chain functions in random environments[J].Journal of Mathematics,2016,36(6):1245-1252.
[23]Cao X R.Stochastic learning and optimization:a sensitivity based approach[J].Annual Reviews in Control,2009,33(1):11-24.
[24]刘艳君,丁锋.非均匀周期采样系统的递阶最小二乘辨识方法[J].控制与决策,2011,26(3):453-456.Liu Yanjun,Ding Feng.A hierarchical least squares identification method for non-uniform periodic sampling systems[J].Control and Decision,2011,26(3):453-456.
[25]Mazo M,Tabuada P.Decentralized event triggered control over wireless sensor/actuator networks[J].IEEE Transactions on Automatic Control,2011,56(10):2456-2461.