多类不确定分布参数系统自适应镇定
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摘要
本文主要针对多类不确定系统,包括不确定热方程、带有不确定扩散主导的执行器动态的ODE系统、不确定耦合PDE-ODE系统,研究其自适应镇定问题.此外,还研究了一类带有空间变系数的耦合PDE-ODE系统的状态反馈和输出反馈镇定问题.具体地,本文的主要研究内容分为如下四部分：
一、不确定热方程自适应镇定
该部分内容是本文的第二章,主要研究了带有不确定控制系数和边界扰动的热方程的自适应镇定问题.利用Lyapunov直接法,给出了自适应状态反馈控制器的显式形式,保证闭环系统所有状态是有界的,且原始系统状态是L2稳定的,特别是当扰动消逝时,原始系统状态收敛到零.值得指出的是,构造的控制器仅需要热方程在某端点处的测量值,降低了已有文献的测量负担.此外,通过灵活选取参数调节律的初始值,适当放宽了相关文献中相容性条件对系统初始条件的限制.
二、带有不确定扩散主导的执行器动态的ODE系统自适应镇定
该部分内容是本文的第三、四章.其中,第三章研究了带有不确定扩散主导的执行器动态的ODE系统自适应状态反馈镇定问题.与相关文献相比,所研究的系统具有较严重的不确定性,因而所研究问题更难以解决.首先通过引入一无穷维状态变换,原始系统转化为一个关键的目标系统.由此目标系统出发更方便进行控制设计与闭环系统性能分析.然后通过确定等价原理等相关自适应技术给出了自适应状态反馈控制器的显式形式,保证闭环系统所有状态有界且原始系统状态收敛到零.第四章研究了带有不确定扩散主导的执行器动态的ODE系统自适应输出反馈镇定问题.与上一章要求执行器状态完全可测不同,该章仅要求执行器某边界值可量测,增大了控制设计的难度.首先通过构造状态观测器估计不可测状态,之后利用无穷维反推、确定等价原理等方法构造了自适应控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.
三、带有空间变系数的耦合PDE-ODE系统镇定
该部分内容是本文的第五章,主要研究了带有空间变系数的耦合PDE-ODE系统的状态反馈和输出反馈镇定问题.与相关文献相比,空间变系数的存在使得所研究系统更具有一般性且更难以控制.通过无穷维反推方法,分别构造了状态反馈和输出反馈控制器,保证闭环系统在特定范数意义下是指数稳定的.值得指出的是,在输出反馈设计情形,通过选择适当的观测器增益,完全去除了已有结果对ODE子系统的限制.
四、不确定耦合PDE-ODE系统自适应镇定
该部分是本文的第六、七章,主要研究了两类不同的不确定耦合PDE-ODE系统的自适应镇定问题.其中,第六章研究了仅含有一个未知参数的耦合PDE-ODE系统的自适应镇定问题.不确定性的存在使得所研究的系统本质不同于已有文献,并且导致已有方法无效.受相关文献启发,采用无穷维反推和确定等价原理等方法设计自适应状态反馈控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.与之不同的是,第七章考虑了具有多个未知参数的耦合PDE-ODE系统自适应镇定问题,特别是考虑了ODE子系统中含有多个未知参数的情形.这使得所研究的系统具有较强的一般性,且使控制设计较为困难.利用已有方法,特别是无穷维反推法,构造了自适应控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.
以上四部分还分别给出了相应的仿真算例,验证了所提理论方法的有效性.
This dissertation focuses on the investigation of the stabilization for sev-eral classes of uncertain systems including of uncertain heat equations, ODE sys-tems with uncertain diffusion-dominated actuator dynamics and uncertain coupled PDE-ODE systems. Moreover, the dissertation also investigates the state feed-back and output feedback stabilization for a class of coupled PDE-ODE systems with spatially varying coefficient. For details, the main content of the dissertation consists of the following four parts:
(Ⅰ) Adaptive stabilization for a class of heat equations
This part is Chapter2of the dissertation, and studies the adaptive stabiliza-tion for a class of heat equations with uncertain control coefficient and boundary disturbance. By using Lyapunov direct method, the adaptive state feedback con-troller is explicitly constructed, which guarantees that all the closed-loop system states are bounded while the original system states being L2stable. Particularly, the original system states converge to zero while the boundary disturbance van-ishing. It is worthwhile to point out that, the designed controller only requires the measurements at one end of the heat equation, and hence reduces the burden of measurement in the existing literature. Moreover, by skilfully choosing initial condition of the parameter updating law, the restriction on the initial conditions of the system is moderately relaxed, which is usually described by the so-called compatible condition in the existing literature.
(Ⅱ) Adaptive stabilization for ODE systems with uncertain diffusion-dominated actuator dynamics
This part is Chapter3and4of the dissertation. Chapter3considers the adaptive state feedback stabilization for ODE systems with uncertain diffusion-dominated actuator dynamics. Comparing to the related literature, the problem under investigation is more difficult to solve due to the presence of the serious un-certainties. By introducing an infinite-dimensional backstepping transformation, a pivotal target system is firstly obtained, which makes the control design and per-formance analysis of the closed-loop system more convenient. Then, based on the certain equivalence principle and other developed adaptive techniques, an adaptive state feedback controller is designed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Chapter4studies the adaptive output feedback stabilization for ODE systems with uncer-tain diffusion-dominated actuator dynamics. Different from the last chapter where all the states of the actuator dynamics are measurable, only one of the bound-ary values of the actuator is available for feedback, and hence makes the control design much difficult. By constructing state observers, the estimations of the un-observable states are first obtained, then by the infinite-dimensional backstepping method and certain equivalence principle, the adaptive controller is constructed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero.
(Ⅲ) Stabilization for coupled PDE-ODE systems with spatially varying coefficient
This part is Chapter5of the dissertation, and investigates the stabilization of a class coupled PDE-ODE systems with spatially varying coefficient via both state feedback and output feedback. The system under consideration is more gen-eral than that of the related literature due to the presence of the spatially varying coefficient which makes problem more difficult to solve. By infinite-dimensional backstepping method, both state feedback and output feedback controllers are designed, which guarantee that the closed-loop system is exponentially stable in certain sense of norm. It is worth pointing out that, in the cases of the output feed-back, the restriction on the ODE subsystem in the existing results is completely removed by choosing appropriate state observer gains.
(Ⅳ) Adaptive stabilization for uncertain coupled PDE-ODE sys-tems
This part is Chapter6and7of the dissertation, and focuses on the adaptive stabilization for two classes of uncertain coupled PDE-ODE systems. Chapter6studies the adaptive stabilization for uncertain coupled PDE-ODE system with only one unknown parameter. The presence of the uncertainties makes the sys-tem under investigation essentially different from those in the existing literature, and results in the incapability of the existing methods. Motivated by the related works, adaptive controller is designed by using the infinite-dimensional backstep-ping method and certain equivalence method, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Differently, Chapter7considers the adaptive stabilization for a class of uncer-tain coupled PDE-ODE systems with multiple unknown parameters, particularly there are unknown parameters exist in the ODE sub-system. Thus, the system under consideration is much general and much difficult to control. By the existing methods, particularly the infinite-dimensional backstepping method, the adap-tive controller is designed, which guarantees all the closed-loop system states are bounded while all the original systems states converging to zero.
For the theoretical results obtained in the above four parts, the corresponding simulation examples are presented to illustrate the effectiveness of the proposed methods.
一、不确定热方程自适应镇定
该部分内容是本文的第二章,主要研究了带有不确定控制系数和边界扰动的热方程的自适应镇定问题.利用Lyapunov直接法,给出了自适应状态反馈控制器的显式形式,保证闭环系统所有状态是有界的,且原始系统状态是L2稳定的,特别是当扰动消逝时,原始系统状态收敛到零.值得指出的是,构造的控制器仅需要热方程在某端点处的测量值,降低了已有文献的测量负担.此外,通过灵活选取参数调节律的初始值,适当放宽了相关文献中相容性条件对系统初始条件的限制.
二、带有不确定扩散主导的执行器动态的ODE系统自适应镇定
该部分内容是本文的第三、四章.其中,第三章研究了带有不确定扩散主导的执行器动态的ODE系统自适应状态反馈镇定问题.与相关文献相比,所研究的系统具有较严重的不确定性,因而所研究问题更难以解决.首先通过引入一无穷维状态变换,原始系统转化为一个关键的目标系统.由此目标系统出发更方便进行控制设计与闭环系统性能分析.然后通过确定等价原理等相关自适应技术给出了自适应状态反馈控制器的显式形式,保证闭环系统所有状态有界且原始系统状态收敛到零.第四章研究了带有不确定扩散主导的执行器动态的ODE系统自适应输出反馈镇定问题.与上一章要求执行器状态完全可测不同,该章仅要求执行器某边界值可量测,增大了控制设计的难度.首先通过构造状态观测器估计不可测状态,之后利用无穷维反推、确定等价原理等方法构造了自适应控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.
三、带有空间变系数的耦合PDE-ODE系统镇定
该部分内容是本文的第五章,主要研究了带有空间变系数的耦合PDE-ODE系统的状态反馈和输出反馈镇定问题.与相关文献相比,空间变系数的存在使得所研究系统更具有一般性且更难以控制.通过无穷维反推方法,分别构造了状态反馈和输出反馈控制器,保证闭环系统在特定范数意义下是指数稳定的.值得指出的是,在输出反馈设计情形,通过选择适当的观测器增益,完全去除了已有结果对ODE子系统的限制.
四、不确定耦合PDE-ODE系统自适应镇定
该部分是本文的第六、七章,主要研究了两类不同的不确定耦合PDE-ODE系统的自适应镇定问题.其中,第六章研究了仅含有一个未知参数的耦合PDE-ODE系统的自适应镇定问题.不确定性的存在使得所研究的系统本质不同于已有文献,并且导致已有方法无效.受相关文献启发,采用无穷维反推和确定等价原理等方法设计自适应状态反馈控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.与之不同的是,第七章考虑了具有多个未知参数的耦合PDE-ODE系统自适应镇定问题,特别是考虑了ODE子系统中含有多个未知参数的情形.这使得所研究的系统具有较强的一般性,且使控制设计较为困难.利用已有方法,特别是无穷维反推法,构造了自适应控制器,保证闭环系统所有状态有界且原始系统状态收敛到零.
以上四部分还分别给出了相应的仿真算例,验证了所提理论方法的有效性.
This dissertation focuses on the investigation of the stabilization for sev-eral classes of uncertain systems including of uncertain heat equations, ODE sys-tems with uncertain diffusion-dominated actuator dynamics and uncertain coupled PDE-ODE systems. Moreover, the dissertation also investigates the state feed-back and output feedback stabilization for a class of coupled PDE-ODE systems with spatially varying coefficient. For details, the main content of the dissertation consists of the following four parts:
(Ⅰ) Adaptive stabilization for a class of heat equations
This part is Chapter2of the dissertation, and studies the adaptive stabiliza-tion for a class of heat equations with uncertain control coefficient and boundary disturbance. By using Lyapunov direct method, the adaptive state feedback con-troller is explicitly constructed, which guarantees that all the closed-loop system states are bounded while the original system states being L2stable. Particularly, the original system states converge to zero while the boundary disturbance van-ishing. It is worthwhile to point out that, the designed controller only requires the measurements at one end of the heat equation, and hence reduces the burden of measurement in the existing literature. Moreover, by skilfully choosing initial condition of the parameter updating law, the restriction on the initial conditions of the system is moderately relaxed, which is usually described by the so-called compatible condition in the existing literature.
(Ⅱ) Adaptive stabilization for ODE systems with uncertain diffusion-dominated actuator dynamics
This part is Chapter3and4of the dissertation. Chapter3considers the adaptive state feedback stabilization for ODE systems with uncertain diffusion-dominated actuator dynamics. Comparing to the related literature, the problem under investigation is more difficult to solve due to the presence of the serious un-certainties. By introducing an infinite-dimensional backstepping transformation, a pivotal target system is firstly obtained, which makes the control design and per-formance analysis of the closed-loop system more convenient. Then, based on the certain equivalence principle and other developed adaptive techniques, an adaptive state feedback controller is designed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Chapter4studies the adaptive output feedback stabilization for ODE systems with uncer-tain diffusion-dominated actuator dynamics. Different from the last chapter where all the states of the actuator dynamics are measurable, only one of the bound-ary values of the actuator is available for feedback, and hence makes the control design much difficult. By constructing state observers, the estimations of the un-observable states are first obtained, then by the infinite-dimensional backstepping method and certain equivalence principle, the adaptive controller is constructed, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero.
(Ⅲ) Stabilization for coupled PDE-ODE systems with spatially varying coefficient
This part is Chapter5of the dissertation, and investigates the stabilization of a class coupled PDE-ODE systems with spatially varying coefficient via both state feedback and output feedback. The system under consideration is more gen-eral than that of the related literature due to the presence of the spatially varying coefficient which makes problem more difficult to solve. By infinite-dimensional backstepping method, both state feedback and output feedback controllers are designed, which guarantee that the closed-loop system is exponentially stable in certain sense of norm. It is worth pointing out that, in the cases of the output feed-back, the restriction on the ODE subsystem in the existing results is completely removed by choosing appropriate state observer gains.
(Ⅳ) Adaptive stabilization for uncertain coupled PDE-ODE sys-tems
This part is Chapter6and7of the dissertation, and focuses on the adaptive stabilization for two classes of uncertain coupled PDE-ODE systems. Chapter6studies the adaptive stabilization for uncertain coupled PDE-ODE system with only one unknown parameter. The presence of the uncertainties makes the sys-tem under investigation essentially different from those in the existing literature, and results in the incapability of the existing methods. Motivated by the related works, adaptive controller is designed by using the infinite-dimensional backstep-ping method and certain equivalence method, which guarantees all the closed-loop system states are bounded while the original systems states converging to zero. Differently, Chapter7considers the adaptive stabilization for a class of uncer-tain coupled PDE-ODE systems with multiple unknown parameters, particularly there are unknown parameters exist in the ODE sub-system. Thus, the system under consideration is much general and much difficult to control. By the existing methods, particularly the infinite-dimensional backstepping method, the adap-tive controller is designed, which guarantees all the closed-loop system states are bounded while all the original systems states converging to zero.
For the theoretical results obtained in the above four parts, the corresponding simulation examples are presented to illustrate the effectiveness of the proposed methods.
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