几类非线性系统的自适应Backstepping神经网络控制
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摘要
非线性控制理论一直是近二十年来自动化控制领域研究的热点课题之一,尤其是将Backstepping设计方法和神经网络逼近理论有机结合的自适应Backstepping神经网络控制方法更是受到了众多研究者的广泛关注,也已经取得了很大进展,但仍然有很多问题有待于进一步研究和探讨.本论文旨在研究自适应Backstepping神经网络控制方法在随机非线性和离散时间非线性系统领域中的推广,结合随机微分方程稳定性理论、时滞泛函微分方程稳定性理论以及关联大系统的分散控制理论等,对几类随机非线性系统和离散时间非线性系统的输出反馈控制和跟踪控制问题进行了研究,其主要成果可概括如下:
1.利用圆盘判据,在随机非线性系统中引入一种新的非线性观测器以估计系统的不可测状态,对具有不可测状态的严格反馈随机非线性系统设计输出反馈控制器.这种非线性观测器的主要优点在于不但可以去掉一般输出反馈设计中对系统状态依赖的非线性函数的Lipschitz限制条件,而且可以避免传统线性观测器所带来的高增益问题,论文的主要工作都是基于这种方法展开的.
2.结合自适应Backstepping神经网络控制方法和非线性观测器设计技术,解决了一类具有不可测状态的不确定随机非线性严格反馈系统的输出反馈镇定问题.通过构造一个状态四次、参数二次的Lyapunov函数,证明了闭环系统依概率稳定,并且这里仅采用一个神经网络补偿所有的输出依赖的未知非线性上界函数,简化了现有的一些自适应Backstepping神经网络控制设计方法.
3.将自适应Backstepping神经网络控制方法推广到随机非线性时滞系统.首先,研究一类具有时变时滞的不确定随机非线性严格反馈系统的输出反馈镇定问题,基于圆盘判据和自适应Backstepping神经网络控制方法,引入非线性观测器并最终整合系统所有的输出依赖的未知上界函数而仅采用一个神经网络加以补偿,从而去掉了要求输出依赖的非线性上界函数已知的限制.然后,将这种思想进一步延伸到一类同时具有离散和分布时变时滞的不确定随机非线性严格反馈系统的自适应输出反馈镇定问题.
4.将自适应Backstepping神经网络控制方法延伸到随机非线性关联大系统.首先,采用分散的非线性观测器估计不可测的系统状态,并结合Backstepping技术,对每一个子系统引入一个神经网络补偿依赖于该子系统输出所有未知上界函数,解决了随机非线性关联大系统的分散镇定问题.其次,通过构造状态四次、参数二次的Lyapunov-Krasovskii泛函,并结合Young不等式放大技巧将耦合项的作用归结到各个子系统中,克服了Ito?项、时滞项以及各子系统间耦合作用给控制器设计带来的困难,解决了随机非线性时滞关联大系统的自适应输出反馈分散镇定问题.
5.研究了一类未知时变非线性离散时间系统的自适应跟踪控制问题.结合神经网络逼近技术和傅里叶级数展开的方法,将其转化为一类相对简单的具有未知常值参数的线性参数化严格反馈系统,进而利用已有的非线性离散时间Backstepping设计方法对新的系统设计自适应控制器,可保证自适应闭环系统的所有信号对任意有界初始条件、参考轨迹和外部干扰是有界的,并可以获得一个比较小的平均跟踪误差,且采用的设计方法可以避免传统非线性离散时间Backstepping设计方法中的过参数化问题.
Nonlinear control theory has always been one of the focuses in automatic control com-munity during the last two decades. Especially the adaptive backstepping neural networkcontrol theory, encompounded by backstepping technique and neural network approxima-tion theory, has attracted much attention of many researchers and some important resultshave been obtained. However, there still exist some open issues need to be further inves-tigated. This dissertation is devoted to study the extension of the adaptive backsteppingneural network control scheme to the stochastic nonlinear and discrete-time nonlinear con-trol areas. Some important control theories, such as stability theory of stochastic di?er-ential equations and time-delay functional di?erential equations, as well as decentralizedcontrol theory of interconnected large-scale systems are combined with the adaptive back-stepping neural network control scheme to address the problems of output-feedback controland tracking control for several classes of stochastic nonlinear systems and discrete-timenonlinear systems. Details are as follows:
1. Via the circle criterion, a new nonlinear observer is introduced to the stochasticnonlinear systems to estimate the unmeasured states, thus the problem of output-feedbackcontrol can be solved for the stochastic nonlinear strict-feedback system with unmeasuredstates. The main merit of this nonlinear observer lies in that it not only can eliminatethe Lipschitz restriction on the state-dependent nonlinearities of the traditional output-feedback control design, but also can solve the high-gain problem of the linear observer.Most results of the dissertation are based on this technique.
2. An output-feedback stabilization control scheme is designed for a class of uncertainstochastic nonlinear strict-feedback systems with unmeasured states, where the adaptivebackstepping neural network control method is combined with the technique of nonlinearobserver design. By constructing a state-quartic and parameter-quadratic Lyapunov func-tion, the closed-loop system can be proved to be stable in probability. Moreover, here onlya neural network is employed to compensate for all unknown nonlinear upper boundingfunctions depending on the system output, which simpli?es the existing adaptive neuralnetwork control schemes.
3. The adaptive backstepping neural network control scheme is extended to thestochastic nonlinear time-delay systems. Firstly, based on the circle criterion and the adap-tive backstepping neural network control method, a nonlinear observer is introduced andall the unknown upper bounding function depending on the system output is integratedto be compensated only by a neural network, such that the restriction of the prelimi-nary knowledge of the output-dependent nonlinear upper bounding functions is removed.Then,this idea is further extended to the problem of output-feedback stabilization for a class of uncertain stochastic nonlinear strict-feedback systems with discrete and distributeddelays.
4. The adaptive backstepping neural network control scheme is extended to thestochastic nonlinear interconnected systems. Firstly, using decentralized nonlinear ob-server to estimate the unmeasured states and combining with the backstepping technique,for each subsystem only a neural network is employed to compensate for all unknown up-per bounding functions which depend on the respective subsystem outputs, and then theproblem of decentralized stabilization is solved for a class of large-scale stochastic nonlinearstrict-feedback systems. Secondly, by constructing a state-quartic and parameter-quadraticLyapunov-Krasovskii functional and combining with the technique of Young’s inequality,the controller designing di?culty of the It?o terms, delay terms and the coupling terms ofall subsystems is removed, so that the problem of adaptive output-feedback stabilization isdealt with for the interconnected stochastic nonlinear delay system.
5. The adaptive tracking control problems are considered for a class of discrete-timenonlinear systems with unknown periodically time-varying parameters. By combining withthe neural network approximation technique and Fourier series expansion (FSE), the sys-tems are transformed into a class of simpler linear parametric strict-feedback systems withunknown constant parameters. Then, via the existing discrete-time nonlinear backstep-ping design technique, the adaptive controller can be designed for the new systems, whichcan ensure all the signals of the closed-loop systems are bounded for all bounded initialconditions, reference signals and external disturbances. In addition, a small-in-the-meantracking error can be achieved.
1.利用圆盘判据,在随机非线性系统中引入一种新的非线性观测器以估计系统的不可测状态,对具有不可测状态的严格反馈随机非线性系统设计输出反馈控制器.这种非线性观测器的主要优点在于不但可以去掉一般输出反馈设计中对系统状态依赖的非线性函数的Lipschitz限制条件,而且可以避免传统线性观测器所带来的高增益问题,论文的主要工作都是基于这种方法展开的.
2.结合自适应Backstepping神经网络控制方法和非线性观测器设计技术,解决了一类具有不可测状态的不确定随机非线性严格反馈系统的输出反馈镇定问题.通过构造一个状态四次、参数二次的Lyapunov函数,证明了闭环系统依概率稳定,并且这里仅采用一个神经网络补偿所有的输出依赖的未知非线性上界函数,简化了现有的一些自适应Backstepping神经网络控制设计方法.
3.将自适应Backstepping神经网络控制方法推广到随机非线性时滞系统.首先,研究一类具有时变时滞的不确定随机非线性严格反馈系统的输出反馈镇定问题,基于圆盘判据和自适应Backstepping神经网络控制方法,引入非线性观测器并最终整合系统所有的输出依赖的未知上界函数而仅采用一个神经网络加以补偿,从而去掉了要求输出依赖的非线性上界函数已知的限制.然后,将这种思想进一步延伸到一类同时具有离散和分布时变时滞的不确定随机非线性严格反馈系统的自适应输出反馈镇定问题.
4.将自适应Backstepping神经网络控制方法延伸到随机非线性关联大系统.首先,采用分散的非线性观测器估计不可测的系统状态,并结合Backstepping技术,对每一个子系统引入一个神经网络补偿依赖于该子系统输出所有未知上界函数,解决了随机非线性关联大系统的分散镇定问题.其次,通过构造状态四次、参数二次的Lyapunov-Krasovskii泛函,并结合Young不等式放大技巧将耦合项的作用归结到各个子系统中,克服了Ito?项、时滞项以及各子系统间耦合作用给控制器设计带来的困难,解决了随机非线性时滞关联大系统的自适应输出反馈分散镇定问题.
5.研究了一类未知时变非线性离散时间系统的自适应跟踪控制问题.结合神经网络逼近技术和傅里叶级数展开的方法,将其转化为一类相对简单的具有未知常值参数的线性参数化严格反馈系统,进而利用已有的非线性离散时间Backstepping设计方法对新的系统设计自适应控制器,可保证自适应闭环系统的所有信号对任意有界初始条件、参考轨迹和外部干扰是有界的,并可以获得一个比较小的平均跟踪误差,且采用的设计方法可以避免传统非线性离散时间Backstepping设计方法中的过参数化问题.
Nonlinear control theory has always been one of the focuses in automatic control com-munity during the last two decades. Especially the adaptive backstepping neural networkcontrol theory, encompounded by backstepping technique and neural network approxima-tion theory, has attracted much attention of many researchers and some important resultshave been obtained. However, there still exist some open issues need to be further inves-tigated. This dissertation is devoted to study the extension of the adaptive backsteppingneural network control scheme to the stochastic nonlinear and discrete-time nonlinear con-trol areas. Some important control theories, such as stability theory of stochastic di?er-ential equations and time-delay functional di?erential equations, as well as decentralizedcontrol theory of interconnected large-scale systems are combined with the adaptive back-stepping neural network control scheme to address the problems of output-feedback controland tracking control for several classes of stochastic nonlinear systems and discrete-timenonlinear systems. Details are as follows:
1. Via the circle criterion, a new nonlinear observer is introduced to the stochasticnonlinear systems to estimate the unmeasured states, thus the problem of output-feedbackcontrol can be solved for the stochastic nonlinear strict-feedback system with unmeasuredstates. The main merit of this nonlinear observer lies in that it not only can eliminatethe Lipschitz restriction on the state-dependent nonlinearities of the traditional output-feedback control design, but also can solve the high-gain problem of the linear observer.Most results of the dissertation are based on this technique.
2. An output-feedback stabilization control scheme is designed for a class of uncertainstochastic nonlinear strict-feedback systems with unmeasured states, where the adaptivebackstepping neural network control method is combined with the technique of nonlinearobserver design. By constructing a state-quartic and parameter-quadratic Lyapunov func-tion, the closed-loop system can be proved to be stable in probability. Moreover, here onlya neural network is employed to compensate for all unknown nonlinear upper boundingfunctions depending on the system output, which simpli?es the existing adaptive neuralnetwork control schemes.
3. The adaptive backstepping neural network control scheme is extended to thestochastic nonlinear time-delay systems. Firstly, based on the circle criterion and the adap-tive backstepping neural network control method, a nonlinear observer is introduced andall the unknown upper bounding function depending on the system output is integratedto be compensated only by a neural network, such that the restriction of the prelimi-nary knowledge of the output-dependent nonlinear upper bounding functions is removed.Then,this idea is further extended to the problem of output-feedback stabilization for a class of uncertain stochastic nonlinear strict-feedback systems with discrete and distributeddelays.
4. The adaptive backstepping neural network control scheme is extended to thestochastic nonlinear interconnected systems. Firstly, using decentralized nonlinear ob-server to estimate the unmeasured states and combining with the backstepping technique,for each subsystem only a neural network is employed to compensate for all unknown up-per bounding functions which depend on the respective subsystem outputs, and then theproblem of decentralized stabilization is solved for a class of large-scale stochastic nonlinearstrict-feedback systems. Secondly, by constructing a state-quartic and parameter-quadraticLyapunov-Krasovskii functional and combining with the technique of Young’s inequality,the controller designing di?culty of the It?o terms, delay terms and the coupling terms ofall subsystems is removed, so that the problem of adaptive output-feedback stabilization isdealt with for the interconnected stochastic nonlinear delay system.
5. The adaptive tracking control problems are considered for a class of discrete-timenonlinear systems with unknown periodically time-varying parameters. By combining withthe neural network approximation technique and Fourier series expansion (FSE), the sys-tems are transformed into a class of simpler linear parametric strict-feedback systems withunknown constant parameters. Then, via the existing discrete-time nonlinear backstep-ping design technique, the adaptive controller can be designed for the new systems, whichcan ensure all the signals of the closed-loop systems are bounded for all bounded initialconditions, reference signals and external disturbances. In addition, a small-in-the-meantracking error can be achieved.
引文
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