基于模糊神经网络滑模控制器的一类非线性系统自适应控制_达飞鹏
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达飞鹏,宋文忠
(东南大学自动化所,江苏 南京 210096)
ADAPTIVE CONTROL FOR A CLASS OF NONLINEAR SYSTEMS BASED ON FNNSMC
DA Fei-peng, SONG Wen-zhong (Research Institute of Automation, Southeast University, Nanjing 210096, China)
1
Introduction
In general, it is difficult to design the adaptive controller for nonlinear systems [1][2]. This is because that firstly it is hard to find proper structure of uncertain dynamic nonlinear systems, secondly there do not exist common adaptive control laws. Sli-ding mode control gives a new way for the control of a class of nonlinear systems [3]. Since sliding mode can be designed as one expects and is robust to dis-turbance signals and parameter variations in systems. However, high frequency chattering is the shortco-ming of the sliding mode control. The chattering may excite high-frequency dynamics associated with the control system which have been neglected in the course of modeling, sometimes it can even cause system unstable. Chattering is undesired since it may damage the controller. A trade-off is presented in [4] between tracking precision and robustness to mod-eling uncertainty: tracking accuracy is set according to the extent of parametric uncertainty and the frequency range of unmodelled dynamics. In [5], the Radical Basis Function (RBF) neural networks com-bined with sliding mode are used into the adaptive control for continuous-time dynamic nonlinear systems. Fuzzy Neural Networks (FNN) incorporate inf-erence of fuzzy logic and learning ability of neural networks, its physical meaning is clear [6]. In this
∑c e
i =1
n −1
i i
+ en = 0
(5)
Take the derivative of (4) and obtain∑Leabharlann ei =1n −1
i i +1
+ f ( X ) + g ( X )u + d ( X ) − x r
1
( n)
(6)
Then the control law is taken as u=
ABSTRACT: Sliding mode control is simple and insensitive to uncertainties and disturbances, but control input chattering is the main problem in the sliding mode controller (SMC). A new type controller-fuzzy neural networks sliding mode controller (FNNSMC) is presented firstly for a class of nonlinear systems. The FNNSMC can eliminate the chattering indeed and is robuster than the SMC while the bounds of the uncertainties and the disturbances of the system are unknown. However there has larger tracking error in the FNNSMC than that in the SMC. To overcome the problem, an adaptive controller, where the FNNSMC and the FNN are incorporated by the smooth transformation, is studied. This adaptive control scheme can decrease the tracking error, enhance the system’s robustness and eliminate the high frequency chattering in the control signal. The simulation results demonstrate the advantages of the algorithm. KEY WORDS: nonlinear systems; fuzzy neural networks; sliding mode control; adaptive control 摘要: 虽然滑模控制具有控制简单和对不确定性与扰动不灵 敏等优点, 但是控制信号中的颤动是其应用中需解决的主要 问题。 该文首先针对一类非线性系统提出了一个新型控制器 —模糊神经网络滑模控制器。 新控制器不仅能消除颤动,而 且比一般滑模控制器具有更强的鲁棒性。 然而它与一般滑模 控制器相比有较大的跟踪误差。为了解决这个问题,提出了 结合滑模控制器和模糊神经网络滑模控制器的自适应控制 方法。这种自适应控制方案可以减小跟踪误差,增强系统的 鲁棒性和消除控制信号中的颤动。 仿真结果说明了控制方案 的有效性。 关键词:非线性系统;模糊神经网络;滑模控制;自适应控 制 中图分类号:TM501 文献标识码:A
n− 1 ( − ci ei +1 − fˆ( X ) + x r ( n) − g ( X ) i =1
2 Problem statement and Design of the SMC
Consider the affine nonlinear systems: x (n ) = f ( X ) + g ( X ) u + d ( X ), (1) & ,⋅ ⋅ ⋅, x (n −1) ) T ∈ R n ,x ∈ R ; where state vector X = ( x, x u ∈ R is the control input; f ( X ) and g(X) are unknown continuous functions, and d(X) is unknown disturbances. In the SMC design we usually assume f ( X ) = ˆ ˆ( X ) is the estimation of f ( X ) + ∆f ( X ) , where f f (X), and ∆f ( X ) is the model uncertainty. Let F(X) and D(X) are the upper bound function of ∆f ( X ) and d ( X ) respectively, i.e. d (X ) desired trajectory, and X track Xr ∆f ( X ) ≤ F(X) and has up to nth
第 22 卷 第 5 期 2002 年 5 月 文章编号:0258-8013(2002)05-0078-06
中
国 电 机 工 程 学 报 Proceedings of the CSEE
Vol.22 No.5 May 2002 ©2002 Chin.Soc.for Elec.Eng.
基于模糊神经网络滑模控制器的 一类非线性系统自适应控制
第5期
达飞鹏等: 基于模糊神经网络滑模控制器的一类非线性系统自适应控制
79
paper we firstly present FNNSMC by combining the FNN with the sliding mode algorithm. Since the continuous output of the FNN is used to replace the discontinuous sign term in the SMC, the FNNSMC can eliminate the chattering indeed while the bounds of the uncertainties and the disturbances of the system are unknown. However its tracking error is larger than that in the SMC. So based on the FNNSMC, an adaptive control scheme is proposed where the FNN-SMC and the FNN are integrated by the smooth transformation. We plan to let the FNNSMC force the state tracking error to slide into the boundary layer, then use the FNN in the boundary layer to let tracking error converge asymptotically to the neighbor of zero. Thus the new adaptive controller has more robustness and small tracking error while the control input chatt-ering is eliminated. Simulation results show the adv-antages of the algorithm. &( e) = s
&1 = e2 e &2 = e 3 e M & en−1 = e n & n = f ( X ) + g ( X )u + d ( X ) − x r ( n) e Take s ( e) =
(3)
∑c e + e
i i i =1
n −1
n
(4)
where ci > 0 are constants and λn −1 + c n−1λn − 2 + ⋅ ⋅ ⋅ + c2 λ + c1 is Hurwitz polynomial. Then the sliding surface is s ( e) =
(东南大学自动化所,江苏 南京 210096)
ADAPTIVE CONTROL FOR A CLASS OF NONLINEAR SYSTEMS BASED ON FNNSMC
DA Fei-peng, SONG Wen-zhong (Research Institute of Automation, Southeast University, Nanjing 210096, China)
1
Introduction
In general, it is difficult to design the adaptive controller for nonlinear systems [1][2]. This is because that firstly it is hard to find proper structure of uncertain dynamic nonlinear systems, secondly there do not exist common adaptive control laws. Sli-ding mode control gives a new way for the control of a class of nonlinear systems [3]. Since sliding mode can be designed as one expects and is robust to dis-turbance signals and parameter variations in systems. However, high frequency chattering is the shortco-ming of the sliding mode control. The chattering may excite high-frequency dynamics associated with the control system which have been neglected in the course of modeling, sometimes it can even cause system unstable. Chattering is undesired since it may damage the controller. A trade-off is presented in [4] between tracking precision and robustness to mod-eling uncertainty: tracking accuracy is set according to the extent of parametric uncertainty and the frequency range of unmodelled dynamics. In [5], the Radical Basis Function (RBF) neural networks com-bined with sliding mode are used into the adaptive control for continuous-time dynamic nonlinear systems. Fuzzy Neural Networks (FNN) incorporate inf-erence of fuzzy logic and learning ability of neural networks, its physical meaning is clear [6]. In this
∑c e
i =1
n −1
i i
+ en = 0
(5)
Take the derivative of (4) and obtain∑Leabharlann ei =1n −1
i i +1
+ f ( X ) + g ( X )u + d ( X ) − x r
1
( n)
(6)
Then the control law is taken as u=
ABSTRACT: Sliding mode control is simple and insensitive to uncertainties and disturbances, but control input chattering is the main problem in the sliding mode controller (SMC). A new type controller-fuzzy neural networks sliding mode controller (FNNSMC) is presented firstly for a class of nonlinear systems. The FNNSMC can eliminate the chattering indeed and is robuster than the SMC while the bounds of the uncertainties and the disturbances of the system are unknown. However there has larger tracking error in the FNNSMC than that in the SMC. To overcome the problem, an adaptive controller, where the FNNSMC and the FNN are incorporated by the smooth transformation, is studied. This adaptive control scheme can decrease the tracking error, enhance the system’s robustness and eliminate the high frequency chattering in the control signal. The simulation results demonstrate the advantages of the algorithm. KEY WORDS: nonlinear systems; fuzzy neural networks; sliding mode control; adaptive control 摘要: 虽然滑模控制具有控制简单和对不确定性与扰动不灵 敏等优点, 但是控制信号中的颤动是其应用中需解决的主要 问题。 该文首先针对一类非线性系统提出了一个新型控制器 —模糊神经网络滑模控制器。 新控制器不仅能消除颤动,而 且比一般滑模控制器具有更强的鲁棒性。 然而它与一般滑模 控制器相比有较大的跟踪误差。为了解决这个问题,提出了 结合滑模控制器和模糊神经网络滑模控制器的自适应控制 方法。这种自适应控制方案可以减小跟踪误差,增强系统的 鲁棒性和消除控制信号中的颤动。 仿真结果说明了控制方案 的有效性。 关键词:非线性系统;模糊神经网络;滑模控制;自适应控 制 中图分类号:TM501 文献标识码:A
n− 1 ( − ci ei +1 − fˆ( X ) + x r ( n) − g ( X ) i =1
2 Problem statement and Design of the SMC
Consider the affine nonlinear systems: x (n ) = f ( X ) + g ( X ) u + d ( X ), (1) & ,⋅ ⋅ ⋅, x (n −1) ) T ∈ R n ,x ∈ R ; where state vector X = ( x, x u ∈ R is the control input; f ( X ) and g(X) are unknown continuous functions, and d(X) is unknown disturbances. In the SMC design we usually assume f ( X ) = ˆ ˆ( X ) is the estimation of f ( X ) + ∆f ( X ) , where f f (X), and ∆f ( X ) is the model uncertainty. Let F(X) and D(X) are the upper bound function of ∆f ( X ) and d ( X ) respectively, i.e. d (X ) desired trajectory, and X track Xr ∆f ( X ) ≤ F(X) and has up to nth
第 22 卷 第 5 期 2002 年 5 月 文章编号:0258-8013(2002)05-0078-06
中
国 电 机 工 程 学 报 Proceedings of the CSEE
Vol.22 No.5 May 2002 ©2002 Chin.Soc.for Elec.Eng.
基于模糊神经网络滑模控制器的 一类非线性系统自适应控制
第5期
达飞鹏等: 基于模糊神经网络滑模控制器的一类非线性系统自适应控制
79
paper we firstly present FNNSMC by combining the FNN with the sliding mode algorithm. Since the continuous output of the FNN is used to replace the discontinuous sign term in the SMC, the FNNSMC can eliminate the chattering indeed while the bounds of the uncertainties and the disturbances of the system are unknown. However its tracking error is larger than that in the SMC. So based on the FNNSMC, an adaptive control scheme is proposed where the FNN-SMC and the FNN are integrated by the smooth transformation. We plan to let the FNNSMC force the state tracking error to slide into the boundary layer, then use the FNN in the boundary layer to let tracking error converge asymptotically to the neighbor of zero. Thus the new adaptive controller has more robustness and small tracking error while the control input chatt-ering is eliminated. Simulation results show the adv-antages of the algorithm. &( e) = s
&1 = e2 e &2 = e 3 e M & en−1 = e n & n = f ( X ) + g ( X )u + d ( X ) − x r ( n) e Take s ( e) =
(3)
∑c e + e
i i i =1
n −1
n
(4)
where ci > 0 are constants and λn −1 + c n−1λn − 2 + ⋅ ⋅ ⋅ + c2 λ + c1 is Hurwitz polynomial. Then the sliding surface is s ( e) =