基于单位分解的非线性不确定系统模糊自适应镇定与跟踪控制分析
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摘要
在控制理论中,控制系统的设计都要以被控对象的数学模型为依据,然而对于任一被控对象的建模都不可能做到完全精确,必然存在不确定性。这些不确定性主要由以下两方面造成：一是系统外部的不确定性,如系统运行环境干扰等；二是系统内部的不确定性,如测量误差、未知参数以及被控对象的未建模动态等。这些不确定性往往会增加控制设计的难度,降低控制系统品质,以至于导致系统不稳定而无法正常运行。因此,如何基于不确定信息设计系统控制器不仅具有重要的理论意义,而且也具有重要的工程实践意义。
本论文主要依据单位分解方法,针对具有不确定性的非线性控制系统,通过结合利用自适应控制、模糊控制、Backstepping等数学设计工具,为几类不确定非线性系统设计了自适应镇定和跟踪控制器,与已有的其它设计结果相比较,本文提出的基于单位分解的控制设计方法不仅能够有效地逼近系统的不确定性,而且能够有效地减少自适应律的数目。本文的主要研究内容如下：
(1)针对一类严格反馈非线性系统,应用单位分解逼近系统的控制器,设计了自适应跟踪控制器。所提出的控制方案保证闭环系统的所有状态一致有界,并且使输出跟踪误差渐近收敛到零点的一个小领域内。以Duffing Forced Oscil1ationSystem为仿真对象,与已有的自适应模糊控制算法进行比较说明了本文所提出控制设计方案的有效性。
(2)针对一类非线性系统,提出了具有PI结构的自适应鲁棒跟踪控制器。首先利用单位分解逼近系统中的不确定项,然后设计具有PI结构的鲁棒跟踪控制器,使得系统在幅值较大和变化较快的干扰下仍然保持良好的跟踪效果。同时保证闭环系统的所有状态和跟踪误差收敛到零点的一个小领域内。最后通过带有干扰项的倒立摆模型验证了本文提出的PI结构自适应鲁棒跟踪控制器的有效性。
(3)研究了一类具有较少约束条件的严格反馈非线性系统的镇定问题。首先利用单位分解逼近系统的非线性不确定项,单位分解所造成的逼近误差则用鲁棒补偿器来抵消,其次将经滤波后的信号输入到控制器中,规避了在推导过程中产生的代数环问题。所提出的基于单位分解自适应方案不仅保证闭环系统所有状态终极一致有界,而且简化了稳定分析过程。仿真实例说明了所提出方案的有效性。
(4)研究了一类具有严格反馈形式的非线性系统的跟踪控制问题。首先利用单位分解逼近期望的虚拟控制,然后结合Backstepping技术提出了基于单位分解的直接自适应控制方案。所提出的控制方案不仅保证了闭环系统的有界性和跟踪性,而且所提出的控制器结构简单,利用在线估计调节参数向量范数的思想设计自适应律,使自适应参数的数目减少到1。分别以Brusselator化学动态模型、单连杆机器人为仿真对象说明了本文所提出的控制设计方案的有效性。
(5)研究了一类带有伸缩器和饱和器的单位分解自适应控制问题。首先在单位分解的输入和输出端串联伸缩器和饱和器生成扩展的单位分解,利用扩展的单位分解逼近非线性系统的不确定项,然后运用变结构的方法将整个控制过程分成开闭环两个过程：1)当系统的状态在给定的饱和度之外,此时系统处于开环状态。在这情况下,给出一个伸缩律,确保系统的状态在有限时间进入给定的饱和度范围内,迫使控制生效。2)当系统的状态进入饱和度范围内,设计控制器确保闭环系统的所有状态趋于零点附近很小的区域里,伸缩率也趋于零,此时我们认为扩展的单位分解具有较好的逼近精度。所提出方案的特点是：自适应律的个数与扩展的单位分解的基函数数目无关。最后通过带有干扰项的倒立摆模型验证了本文控制方案的有效性。
(6)研究了一类基于无规则模糊逻辑系统的自适应控制方案。首先在无规则的模糊逻辑系统的输入端串联伸缩器和饱和器生成扩展的无规则模糊逻辑系统,利用扩展的无规则模糊逻辑系统来逼近非线性系统的未知函数,其中未知函数满足连续条件即可。同时采用变结构的方法将整个控制过程分成开闭环。本方案所设计的自适应律用来调节扩展的无规则模糊逻辑系统的逼近精度、利普希兹常数和伸缩因子,自适应律数目不会跟随系统状态变量及模糊集的个数增加而指数增长。而且该方案适用于各类的万能逼近器,不像传统的模糊自适应控制方法强调万能逼近器的输出形式满足基函数线性组合的固有模式,在一定程度上拓宽了万能逼近器的选取和模糊自适应方法的应用。最后通过带有干扰项的倒立摆模型验证本文控制方案的有效性。
本论文受到广东省自然科学基金项目(No.8151009001000061)(No.10151009001000039)和广东省自然科学基金团队项目(No.8351009001000002)
The control design for most nonlinear systems is based on the accurate mathematical models. However, it is difficult to obtain the accurate information of controlled object. In general, the controlled systems possess uncertainly nonlinear property and operate under disturbances, such as structural uncertainty, parameter uncertainty and exogenous disturbances. These uncertainties and disturbances bring in variance between the practical systems and their mathematical models, and ultimately worsen the systems performance. Therefore it is extremely important to design controllers for nonlinear systems with uncertainties in real applications.
Combining adaptive control, fuzzy control and Backstepping technique, this thesis focuses attention on the adaptive stabilization and tracking control problem of nonlinear uncertain systems based on the method of partition of unity. Compared to some existent results, the control methods in this thesis not only show the approximate capacity of the partition of unity but also reduce the number of adaptive laws. The main work and research results of this thesis lie in the following aspects:
(1) An adaptive tracking control is developed for a class of nonlinear system. In the control design, partition of unity is used to approximate the controllers. The results show that the tracking errors converge to a small neighborhood of zero and states in the closed-loop system are bounded via the controllers. The developed design scheme is applied to design tracking controller for Duffing Forced Oscillation System. Simulation result demonstrates the effectiveness of the proposed scheme.
(2) By utilizing the property that partition of unity can approximate any continuous functions on the compact set at arbitrary precision, the robust tracking controllers with PI structure and adaptive laws are designed for a class of uncertain nonlinear system with large and fast disturbances. The results show that the tracking errors converge to zero and all states in the closed-loop systems are bounded via the controllers. Numerical simulations are given to illustrate the validity of the proposed method.
(3) A robust adaptive controller by using the Backstepping technique and the partition of unity method is proposed. At first, we use the partition of unity method to approximate the nonlinear uncertainties. Second, the smooth extension robust compensators are applied to suppress the partition of unity approximation errors. Meanwhile, the filter signals are employed to circumvent algebraic loop problems. By using suitable partition of unity, the proposed adaptive controllers in this paper guarantee that all the states of the closed-loop systems are uniformly ultimately bounded (UUB) as well as require fewer restrictive assumptions. A numerical example demonstrates the effectiveness of the proposed approach.
(4) A novel adaptive controller based on partition of unity is presented for a class of strict-feedback nonlinear system. The partition of unity is used to approximate the uncertainties and a systematic design procedure is developed for synthesis of adaptive control which includes the Backstepping technique. The method preserves the main advantage is that the adaptive mechanism with only one learning parameter is obtained. The developed design scheme is applied to design tracking controller for Brusselator model and one-link robot manipulator. Simulation results demonstrate the effectiveness of the proposed scheme.
(5) A novel adaptive EPU method stabilization control design scheme has been proposed for n order nonlinear system. The proposed method not only efficiently reduces the number of adaptive laws but also relaxes the restriction on universal approximators, whose output usually needs to be linear combination of basis functions in control design. The EPU has been proved with good approximation accuracy by tuning the scalar factor. Compared with the existing results, the main advantage of our result is different kinds of universal approxiamtors such as PU, neural network, fuzzy logic system, fuzzy logic system without rule bases, can use to approximate the nonlinear terms. Meanwhile, the adaptive laws have nothing to do with basis functions. Via variable structure method the controllers are designed in two cases:(i) If the states of controlled systems (SSs) increase in large ratio with time, the size of parameters of the scalars (SPS) increases by updated laws and the controllers are placed out of service, so that the systems is an open-loop. Increasing the SPS ensures that the SSs can go into the effective range of the saturator; (ii) If SSs go into the effective range of the saturator, the SPS decreases by the updated laws and the adaptive controller with EPU guarantees the states of the system controlled converge to a small region near the origin, so that the system is a closed-loop. At last, a comparison of the numerical example is given to illustrate the effectiveness of the approach.
(6) A novel adaptive FWR stabilization control design scheme has been proposed for n order nonlinear system with scalars and saturators. The main advantages of this method are that the adaptive laws have nothing to do with fuzzy basis functions, because the proposed adaptive laws just use to tuning approximation accuracies of FWR, Lipschitz constants and scalar factor, and the FWR is benefit to control multiple variables systems. Compared with chapter 6, in this section, the unknown nonlinear functions just only satisfy continuous property on the compact set, which extends the applicability of the approach to different kinds of practice systems. Finally, a comparison of the numerical example is given to illustrate the effectiveness of the approach.
This work was supported by the National Natural Science Foundation of Guangdong Province (No.8151009001000061), (No.10151009001000039), Team Project of the Guangdong Natural Science Foundation (No.8351009001000002).
本论文主要依据单位分解方法,针对具有不确定性的非线性控制系统,通过结合利用自适应控制、模糊控制、Backstepping等数学设计工具,为几类不确定非线性系统设计了自适应镇定和跟踪控制器,与已有的其它设计结果相比较,本文提出的基于单位分解的控制设计方法不仅能够有效地逼近系统的不确定性,而且能够有效地减少自适应律的数目。本文的主要研究内容如下：
(1)针对一类严格反馈非线性系统,应用单位分解逼近系统的控制器,设计了自适应跟踪控制器。所提出的控制方案保证闭环系统的所有状态一致有界,并且使输出跟踪误差渐近收敛到零点的一个小领域内。以Duffing Forced Oscil1ationSystem为仿真对象,与已有的自适应模糊控制算法进行比较说明了本文所提出控制设计方案的有效性。
(2)针对一类非线性系统,提出了具有PI结构的自适应鲁棒跟踪控制器。首先利用单位分解逼近系统中的不确定项,然后设计具有PI结构的鲁棒跟踪控制器,使得系统在幅值较大和变化较快的干扰下仍然保持良好的跟踪效果。同时保证闭环系统的所有状态和跟踪误差收敛到零点的一个小领域内。最后通过带有干扰项的倒立摆模型验证了本文提出的PI结构自适应鲁棒跟踪控制器的有效性。
(3)研究了一类具有较少约束条件的严格反馈非线性系统的镇定问题。首先利用单位分解逼近系统的非线性不确定项,单位分解所造成的逼近误差则用鲁棒补偿器来抵消,其次将经滤波后的信号输入到控制器中,规避了在推导过程中产生的代数环问题。所提出的基于单位分解自适应方案不仅保证闭环系统所有状态终极一致有界,而且简化了稳定分析过程。仿真实例说明了所提出方案的有效性。
(4)研究了一类具有严格反馈形式的非线性系统的跟踪控制问题。首先利用单位分解逼近期望的虚拟控制,然后结合Backstepping技术提出了基于单位分解的直接自适应控制方案。所提出的控制方案不仅保证了闭环系统的有界性和跟踪性,而且所提出的控制器结构简单,利用在线估计调节参数向量范数的思想设计自适应律,使自适应参数的数目减少到1。分别以Brusselator化学动态模型、单连杆机器人为仿真对象说明了本文所提出的控制设计方案的有效性。
(5)研究了一类带有伸缩器和饱和器的单位分解自适应控制问题。首先在单位分解的输入和输出端串联伸缩器和饱和器生成扩展的单位分解,利用扩展的单位分解逼近非线性系统的不确定项,然后运用变结构的方法将整个控制过程分成开闭环两个过程：1)当系统的状态在给定的饱和度之外,此时系统处于开环状态。在这情况下,给出一个伸缩律,确保系统的状态在有限时间进入给定的饱和度范围内,迫使控制生效。2)当系统的状态进入饱和度范围内,设计控制器确保闭环系统的所有状态趋于零点附近很小的区域里,伸缩率也趋于零,此时我们认为扩展的单位分解具有较好的逼近精度。所提出方案的特点是：自适应律的个数与扩展的单位分解的基函数数目无关。最后通过带有干扰项的倒立摆模型验证了本文控制方案的有效性。
(6)研究了一类基于无规则模糊逻辑系统的自适应控制方案。首先在无规则的模糊逻辑系统的输入端串联伸缩器和饱和器生成扩展的无规则模糊逻辑系统,利用扩展的无规则模糊逻辑系统来逼近非线性系统的未知函数,其中未知函数满足连续条件即可。同时采用变结构的方法将整个控制过程分成开闭环。本方案所设计的自适应律用来调节扩展的无规则模糊逻辑系统的逼近精度、利普希兹常数和伸缩因子,自适应律数目不会跟随系统状态变量及模糊集的个数增加而指数增长。而且该方案适用于各类的万能逼近器,不像传统的模糊自适应控制方法强调万能逼近器的输出形式满足基函数线性组合的固有模式,在一定程度上拓宽了万能逼近器的选取和模糊自适应方法的应用。最后通过带有干扰项的倒立摆模型验证本文控制方案的有效性。
本论文受到广东省自然科学基金项目(No.8151009001000061)(No.10151009001000039)和广东省自然科学基金团队项目(No.8351009001000002)
The control design for most nonlinear systems is based on the accurate mathematical models. However, it is difficult to obtain the accurate information of controlled object. In general, the controlled systems possess uncertainly nonlinear property and operate under disturbances, such as structural uncertainty, parameter uncertainty and exogenous disturbances. These uncertainties and disturbances bring in variance between the practical systems and their mathematical models, and ultimately worsen the systems performance. Therefore it is extremely important to design controllers for nonlinear systems with uncertainties in real applications.
Combining adaptive control, fuzzy control and Backstepping technique, this thesis focuses attention on the adaptive stabilization and tracking control problem of nonlinear uncertain systems based on the method of partition of unity. Compared to some existent results, the control methods in this thesis not only show the approximate capacity of the partition of unity but also reduce the number of adaptive laws. The main work and research results of this thesis lie in the following aspects:
(1) An adaptive tracking control is developed for a class of nonlinear system. In the control design, partition of unity is used to approximate the controllers. The results show that the tracking errors converge to a small neighborhood of zero and states in the closed-loop system are bounded via the controllers. The developed design scheme is applied to design tracking controller for Duffing Forced Oscillation System. Simulation result demonstrates the effectiveness of the proposed scheme.
(2) By utilizing the property that partition of unity can approximate any continuous functions on the compact set at arbitrary precision, the robust tracking controllers with PI structure and adaptive laws are designed for a class of uncertain nonlinear system with large and fast disturbances. The results show that the tracking errors converge to zero and all states in the closed-loop systems are bounded via the controllers. Numerical simulations are given to illustrate the validity of the proposed method.
(3) A robust adaptive controller by using the Backstepping technique and the partition of unity method is proposed. At first, we use the partition of unity method to approximate the nonlinear uncertainties. Second, the smooth extension robust compensators are applied to suppress the partition of unity approximation errors. Meanwhile, the filter signals are employed to circumvent algebraic loop problems. By using suitable partition of unity, the proposed adaptive controllers in this paper guarantee that all the states of the closed-loop systems are uniformly ultimately bounded (UUB) as well as require fewer restrictive assumptions. A numerical example demonstrates the effectiveness of the proposed approach.
(4) A novel adaptive controller based on partition of unity is presented for a class of strict-feedback nonlinear system. The partition of unity is used to approximate the uncertainties and a systematic design procedure is developed for synthesis of adaptive control which includes the Backstepping technique. The method preserves the main advantage is that the adaptive mechanism with only one learning parameter is obtained. The developed design scheme is applied to design tracking controller for Brusselator model and one-link robot manipulator. Simulation results demonstrate the effectiveness of the proposed scheme.
(5) A novel adaptive EPU method stabilization control design scheme has been proposed for n order nonlinear system. The proposed method not only efficiently reduces the number of adaptive laws but also relaxes the restriction on universal approximators, whose output usually needs to be linear combination of basis functions in control design. The EPU has been proved with good approximation accuracy by tuning the scalar factor. Compared with the existing results, the main advantage of our result is different kinds of universal approxiamtors such as PU, neural network, fuzzy logic system, fuzzy logic system without rule bases, can use to approximate the nonlinear terms. Meanwhile, the adaptive laws have nothing to do with basis functions. Via variable structure method the controllers are designed in two cases:(i) If the states of controlled systems (SSs) increase in large ratio with time, the size of parameters of the scalars (SPS) increases by updated laws and the controllers are placed out of service, so that the systems is an open-loop. Increasing the SPS ensures that the SSs can go into the effective range of the saturator; (ii) If SSs go into the effective range of the saturator, the SPS decreases by the updated laws and the adaptive controller with EPU guarantees the states of the system controlled converge to a small region near the origin, so that the system is a closed-loop. At last, a comparison of the numerical example is given to illustrate the effectiveness of the approach.
(6) A novel adaptive FWR stabilization control design scheme has been proposed for n order nonlinear system with scalars and saturators. The main advantages of this method are that the adaptive laws have nothing to do with fuzzy basis functions, because the proposed adaptive laws just use to tuning approximation accuracies of FWR, Lipschitz constants and scalar factor, and the FWR is benefit to control multiple variables systems. Compared with chapter 6, in this section, the unknown nonlinear functions just only satisfy continuous property on the compact set, which extends the applicability of the approach to different kinds of practice systems. Finally, a comparison of the numerical example is given to illustrate the effectiveness of the approach.
This work was supported by the National Natural Science Foundation of Guangdong Province (No.8151009001000061), (No.10151009001000039), Team Project of the Guangdong Natural Science Foundation (No.8351009001000002).
引文
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