一类离散时间广义系统的迭代学习控制
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摘要
针对一类离散时间广义系统,提出了一种离散迭代学习控制算法.首先,通过非奇异变换将离散时间广义系统分解为正常离散状态方程和代数方程的形式.然后,利用上一次迭代学习获得的前一时刻误差和当前时刻误差来修正上一次的控制量,从而获得下一次迭代学习的新控制量,并对算法的收敛性进行了理论证明,给出了算法收敛的充分条件.研究结果表明,所提算法能够在有限时间区间内实现系统状态对期望状态的完全跟踪.最后,通过仿真算例进一步验证了所提算法的有效性.
Singular systems are also called descriptor systems. Compared with normal systems, singular systems have become one of effective tools which can describe and characterize varieties of real systems, because they can better describe physical properties of the systems. Up to now, the analyses and syntheses of singular systems have been widespread applied to linear matrix inequality(LMI) method. The method requires known systems to have accurate mathematical model information, however, real systems have difficulties in obtaining their accurate mathematical models, owing to the fact that real systems are frequently subjected to all kinds of interferences, uncertainties, and nonlinear factors. Specially,in the case of singular systems, obtained LMI conditions often have a constraint equation or LMI is semi-definite, which makes it more difficult to solve LMI. Therefore, in order to avoid the above two problems occurring in settling state tracking issue for singular systems, in the meantime, for convenience of computating control algorithm and information storage, in this paper we propose a discrete-time iterative learning control algorithm for a class of discrete-time singular system with repetitive running characteristics in finite time interval. The specific process is divided into two steps. First,the class of discrete-time singular system is decomposed into normal discrete-time state equation and algebraic equation form by nonsingular transformation. Accordingly, the singular system state is also decomposed into two parts. Among them, the dimension of the first part state is equal to singular matrix rank and another is equal to system dimension minus singular matrix rank. In addition, the control law of last iterative learning is modified by using two tracking errors at two different times: one error is real-time tracking error generated from the comparison between the first part state and its desired state and another is tracking error at a previous time generated from comparison between the second part state and its desired state. And thus a new control law of next iterative learning is obtained, such that, as for any given real singular system, its state may completely track the desired state as long as selected learning gain can satisfy the convergence condition of the algorithm. Further, the convergence of the control algorithm is theoretically proved by compression mapping method, and thus its sufficient convergence condition is given in the sense of λ-norm. The results indicate that the proposed iterative learning control algorithm can make system state realize the perfect tracking of desired state as iteration number gradually increases in finite time interval, and the convergence of the algorithm only depends on system parameters and learning gain rather than initial value of control variable. The simulation example finally verifies the effectiveness of the proposed algorithm.
Singular systems are also called descriptor systems. Compared with normal systems, singular systems have become one of effective tools which can describe and characterize varieties of real systems, because they can better describe physical properties of the systems. Up to now, the analyses and syntheses of singular systems have been widespread applied to linear matrix inequality(LMI) method. The method requires known systems to have accurate mathematical model information, however, real systems have difficulties in obtaining their accurate mathematical models, owing to the fact that real systems are frequently subjected to all kinds of interferences, uncertainties, and nonlinear factors. Specially,in the case of singular systems, obtained LMI conditions often have a constraint equation or LMI is semi-definite, which makes it more difficult to solve LMI. Therefore, in order to avoid the above two problems occurring in settling state tracking issue for singular systems, in the meantime, for convenience of computating control algorithm and information storage, in this paper we propose a discrete-time iterative learning control algorithm for a class of discrete-time singular system with repetitive running characteristics in finite time interval. The specific process is divided into two steps. First,the class of discrete-time singular system is decomposed into normal discrete-time state equation and algebraic equation form by nonsingular transformation. Accordingly, the singular system state is also decomposed into two parts. Among them, the dimension of the first part state is equal to singular matrix rank and another is equal to system dimension minus singular matrix rank. In addition, the control law of last iterative learning is modified by using two tracking errors at two different times: one error is real-time tracking error generated from the comparison between the first part state and its desired state and another is tracking error at a previous time generated from comparison between the second part state and its desired state. And thus a new control law of next iterative learning is obtained, such that, as for any given real singular system, its state may completely track the desired state as long as selected learning gain can satisfy the convergence condition of the algorithm. Further, the convergence of the control algorithm is theoretically proved by compression mapping method, and thus its sufficient convergence condition is given in the sense of λ-norm. The results indicate that the proposed iterative learning control algorithm can make system state realize the perfect tracking of desired state as iteration number gradually increases in finite time interval, and the convergence of the algorithm only depends on system parameters and learning gain rather than initial value of control variable. The simulation example finally verifies the effectiveness of the proposed algorithm.
引文
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[14]Xu J X,Jin X 2013 IEEE Trans.Autom.Control 581322
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[21]Hu T,Tian S P,Luo Y B 2014 26th Chinese Control and Decision Conference(CCDC)Changsha,China May31–June 2,2014 p2412
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[23]Yin S,Ding S X,Haghani A 2012 J.Process Control 221567
[24]Hu A H,Shao H Y,Liu D 2015 Chin.Phys.B 24 098902
[2]Masubuchi I 2007 Automatica 43 339
[3]Xin Z D 2013 IET Control Theory Appl.7 2028
[4]Zhou J,Zhang Q L,Men B 2014 Int.J.Robust.Nonlinear Control 24 97
[5]Inoue M,Wada T,Ikeda M 2015 Automatica 59 164
[6]Gao Z R,Shen Y X,Ji Z C 2012 Acta Phys.Sin.61120203(in Chinese)[高在瑞,沈艳霞,纪志成2012物理学报61 120203]
[7]Uchiyama M 1978 Trans.Soc.Instr.Control Eng.14706
[8]Arimoto S,Kawamura S,Miyazaki F 1984 J.Robotic Syst.1 123
[9]Yang S P,Xu J X,Huang D Q 2015 Asian J.Control17 2091
[10]Meng D Y,Du W,Jia Y M 2015 IET Control Theory Appl.9 2084
[11]Sun H Q,Alleyne A G 2014 Automatica 50 141
[12]Huang D Q,Xu J X,Li X F 2013 Automatica 49 2397
[13]Tan Y,Xu J X,Norrlof M 2011 Automatica 47 2412
[14]Xu J X,Jin X 2013 IEEE Trans.Autom.Control 581322
[15]Yin C K,Xu J X,Hou Z S 2010 IEEE Trans.Autom.Control 55 2655
[16]Cao W,Sun M 2014 Acta Phys.Sin.63 020201(in Chinese)[曹伟,孙明2014物理学报63 020201]
[17]Piao F X,Zhang Q L,Wang Z F 2007 Acta Autom.Sin.33 659(in Chinese)[朴凤贤,张庆灵,王哲峰2007自动化学报33 659]
[18]Tian S P,Zhou X J 2012 J.Syst.Sci.Math.Sci.32731(in Chinese)[田森平,周秀锦2012系统科学与数学32 731]
[19]Piao F X,Zhang Q L 2007 Control and Decision 22 349(in Chinese)[朴凤贤,张庆灵2007控制与决策22 349]
[20]Li B W 2009 J.Wuhan Univ.(Natural Sci.Edition)55391(in Chinese)[李必文2009武汉大学学报(理学版)55391]
[21]Hu T,Tian S P,Luo Y B 2014 26th Chinese Control and Decision Conference(CCDC)Changsha,China May31–June 2,2014 p2412
[22]Yin S,Luo H,Ding S 2014 IEEE Trans.Ind.Electron61 2402
[23]Yin S,Ding S X,Haghani A 2012 J.Process Control 221567
[24]Hu A H,Shao H Y,Liu D 2015 Chin.Phys.B 24 098902