几类非线性切换系统的H_∞控制问题研究
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摘要
非线性切换系统是一类特殊而又重要的混杂系统,它有着重要的理论研究价值和广泛的工程应用背景。非线性切换系统的H_∞控制问题是混杂系统控制领域中基础的且具有挑战性的研究课题之一。由于切换系统中非线性的连续动态、离散的切换规则相互作用,因此系统的动态行为十分复杂,迄今为止,该课题仍存在许多基本问题亟待解决。
本文主要使用Lyapunov理论,在切换系统和非线性系统的分析和综合的基础上,系统地研究几类非线性切换系统的H_∞控制问题。针对Lipschitz非线性切换系统、级联非线性切换系统、仿射非线性切换系统这三种非线性切换系统模型中带有干扰、具有不确定性、子系统不可稳、状态不可测或不易测等情形,分别利用共同Lyapunov函数、单Lyapunov函数、凸组合、多Lyapunov函数、平均驻留时间等技术,讨论H_∞控制问题。本文主要工作包括如下几个方面:
(一)讨论了一类Lipschitz非线性切换系统基于观测器的H_∞控制问题。基于系统的状态往往难以获得的实际情况,首次利用基于观测器的拓广多Lyapunov函数方法,设计观测器、基于观测器的输出反馈控制器以及同时依赖于观测器状态和前一时刻切换信号取值的滞后切换规则。拓广的多Lyapunov函数方法能够突破以往多Lyapunov函数方法中的单调性限制,允许Lyapunov函数的值在切换时刻序列上有所上升。这为设计问题提供了更大的空间和自由度。
(二)针对执行器发生严重失效的情况,讨论了一类Lipschitz非线性切换系统的可靠指数镇定以及可靠H_∞控制问题。基于系统的状态往往难以获得的实际情况,同时设计观测器、基于观测器的输出反馈控制器以及满足平均驻留时间条件的切换规则。当切换系统中含有可稳与不可稳的子系统时,利用拓广的平均驻留时间思想,确定可稳的子系统与不可稳的子系统的激活时间比,从而给出保证所提出的可靠控制问题可解的充分条件。
(三)讨论了一类级联非线性切换系统的鲁棒H_∞控制问题。基于级联系统的三角结构,分别利用单Lyapunov函数方法、多Lyapunov函数方法,构造组合的Lyapunov函数,设计状态反馈控制器、滞后切换规则以及最大最小切换规则,给出保证级联非线性切换系统的鲁棒H_∞控制问题可解的充分条件。所得的结果可以处理任意单一控制器都不能镇定相应子系统的情况。
(四)讨论了一类具有中立不确定的仿射非线性切换系统的鲁棒H_∞控制问题。首次在切换系统的研究中考虑依赖于状态以及状态的导数的不确定性。针对系统状态可测的情况,基于多Lyapunov函数方法,设计状态反馈控制器以及依赖于状态的切换规则,给出保证切换系统鲁棒H_∞控制问题可解的充分条件,并构造性地给出了满足这个条件的解的形式。针对系统状态不可测或不易测的情况,基于共同Lyapunov函数方法,设计动态输出反馈控制器,给出保证切换系统鲁棒H_∞控制问题可解的充分条件。
(五)讨论了两类仿射非线性切换系统的构造性H_∞控制问题。首先对于状态可测的情况,利用多Lyapunov函数方法,设计状态反馈控制器以及依赖于状态的切换规则,得到保证相应闭环系统的鲁棒H_∞控制问题可解的充分条件;针对状态不可测或不易测的情况,利用共同Lyapunov函数方法,设计动态输出反馈控制器,使得相应闭环系统的鲁棒H_∞控制问题可解。通过构造候选的Lyapunov函数,从而避免了求解Hamilton-Jacobi(HJ)不等式。其次,利用多Lyapunov函数方法,通过构造控制器的形式,得到了能够避免求解HJ不等式,并且使得相应闭环系统的鲁棒非脆弱H_∞控制问题可解的充分条件。
最后对全文所做的工作进行了总结,并讨论了下一步可能研究的工作。
As an important class of hybrid systems,switched nonlinear systems are of great significance both in theory development and engineering applications.In the study of hybrid systems,the H_∞control for switched nonlinear systems is one of the most important and challenging fields.Due to the existence and interaction between the nonlinear continuous dynamics and discrete switching strategy,the behavior of the switched systems is very complicated.Many analysis and design problems deserve further investigation.
This dissertation focuses on the H_∞control for switched nonlinear systems based on analysis and control synthesis of switched systems and nonlinear systems by using the Lyapunov control theory.Some theories and methods of nonlinear H_∞control problem are extended to establish a theory framework of H_∞control for switched nonlinear systems.The common Lyapunov function method,single Lyapunov function method,convex combination technique,multiple Lyapunov function method and the average dwell time method are exploited to solve the H_∞control for switched Lipschitz nonlinear systems,switched cascade nonlinear systems and switched affine nonlinear systems in presence of perturbations and uncertainties.Stabilizable and unstabilizable subsystems,measurable and unmeasurable states are allowed to co-exist,respectively.
The main contributions of this thesis are as follows.
1.From the fact that the system states are often not available,the observer-based H_∞control for a class of switched Lipschitz nonlinear systems is considered. Observers,observer-based output feedback controllers,and a hysteresis switching strategy depending on the observer state and the previous value of switching signal are designed simultaneously.The generalized multiple Lyapunov function method, which improves the nonincreasing requirement at switching time sequences, provides more freedom for the problems addressed to be solvable.
2.When actuators suffer a "destabilizing failure" and the never-faulty actuators cannot stabilize the given system,the problem of reliable exponential stabilization and H_∞control for a class of switched Lipschitz nonlinear systems is addressed via the average dwell time approach.Since the system states are often not available, observers,observer-based output feedback controllers and switching strategies satisfying an average dwell time condition are constructed simultaneously.For the switched Lipschitz nonlinear systems with stabilizable and unstabilizable subsystems, the addressed reliable problem is solvable if the activation time ratio between stabilizable subsystems and unstabilizable subsystems is no less than a specified constant.
3.The robust H_∞control of a class of switched cascade nonlinear systems is studied.The single Lyapunov function and multiple Lyapunov function approaches are utilized to construct composite Lyapunov functions,state feedback controllers and hysteresis or max/min switching strategies based on the special triangular form. Sufficient conditions for the solvability of robust H_∞control problem are developed.
4.Robust H_∞control is investigated for a class of switched affine nonlinear systems with neutral uncertainties,which describes many practical parameter perturbations nonlinearly dependent on the state and state derivative.The multiple Lyapunov function and the common Lyapunov function approaches are exploited respectively for the cases that states are measurable or not.Correspondingly sufficient conditions are achieved via designing the state and output feedback controllers.
5.The constructive H_∞control problems are solvable for two classes of switched affine nonlinear systems.First,the multiple Lyapunov function and the common Lyapunov function approaches are exploited for the cases where states are measurable or not,respectively.Correspondingly sufficient conditions with the state or output controllers guarantee the solvability of the robust H_∞control problems. Based on the explicit construction of Lyapunov functions,the results do not rely on positive definite solutions of the Hamilton-Jacobi(HJ) inequalities.Secondly, sufficient condition is derived for robust non-fragile H_∞control problem via the multiple Lyapunov function technique.Based on the construction of controllers,the results avoid solving the HJ inequalities.
Finally,the results of the dissertation are summarized and further research topics are pointed out.
本文主要使用Lyapunov理论,在切换系统和非线性系统的分析和综合的基础上,系统地研究几类非线性切换系统的H_∞控制问题。针对Lipschitz非线性切换系统、级联非线性切换系统、仿射非线性切换系统这三种非线性切换系统模型中带有干扰、具有不确定性、子系统不可稳、状态不可测或不易测等情形,分别利用共同Lyapunov函数、单Lyapunov函数、凸组合、多Lyapunov函数、平均驻留时间等技术,讨论H_∞控制问题。本文主要工作包括如下几个方面:
(一)讨论了一类Lipschitz非线性切换系统基于观测器的H_∞控制问题。基于系统的状态往往难以获得的实际情况,首次利用基于观测器的拓广多Lyapunov函数方法,设计观测器、基于观测器的输出反馈控制器以及同时依赖于观测器状态和前一时刻切换信号取值的滞后切换规则。拓广的多Lyapunov函数方法能够突破以往多Lyapunov函数方法中的单调性限制,允许Lyapunov函数的值在切换时刻序列上有所上升。这为设计问题提供了更大的空间和自由度。
(二)针对执行器发生严重失效的情况,讨论了一类Lipschitz非线性切换系统的可靠指数镇定以及可靠H_∞控制问题。基于系统的状态往往难以获得的实际情况,同时设计观测器、基于观测器的输出反馈控制器以及满足平均驻留时间条件的切换规则。当切换系统中含有可稳与不可稳的子系统时,利用拓广的平均驻留时间思想,确定可稳的子系统与不可稳的子系统的激活时间比,从而给出保证所提出的可靠控制问题可解的充分条件。
(三)讨论了一类级联非线性切换系统的鲁棒H_∞控制问题。基于级联系统的三角结构,分别利用单Lyapunov函数方法、多Lyapunov函数方法,构造组合的Lyapunov函数,设计状态反馈控制器、滞后切换规则以及最大最小切换规则,给出保证级联非线性切换系统的鲁棒H_∞控制问题可解的充分条件。所得的结果可以处理任意单一控制器都不能镇定相应子系统的情况。
(四)讨论了一类具有中立不确定的仿射非线性切换系统的鲁棒H_∞控制问题。首次在切换系统的研究中考虑依赖于状态以及状态的导数的不确定性。针对系统状态可测的情况,基于多Lyapunov函数方法,设计状态反馈控制器以及依赖于状态的切换规则,给出保证切换系统鲁棒H_∞控制问题可解的充分条件,并构造性地给出了满足这个条件的解的形式。针对系统状态不可测或不易测的情况,基于共同Lyapunov函数方法,设计动态输出反馈控制器,给出保证切换系统鲁棒H_∞控制问题可解的充分条件。
(五)讨论了两类仿射非线性切换系统的构造性H_∞控制问题。首先对于状态可测的情况,利用多Lyapunov函数方法,设计状态反馈控制器以及依赖于状态的切换规则,得到保证相应闭环系统的鲁棒H_∞控制问题可解的充分条件;针对状态不可测或不易测的情况,利用共同Lyapunov函数方法,设计动态输出反馈控制器,使得相应闭环系统的鲁棒H_∞控制问题可解。通过构造候选的Lyapunov函数,从而避免了求解Hamilton-Jacobi(HJ)不等式。其次,利用多Lyapunov函数方法,通过构造控制器的形式,得到了能够避免求解HJ不等式,并且使得相应闭环系统的鲁棒非脆弱H_∞控制问题可解的充分条件。
最后对全文所做的工作进行了总结,并讨论了下一步可能研究的工作。
As an important class of hybrid systems,switched nonlinear systems are of great significance both in theory development and engineering applications.In the study of hybrid systems,the H_∞control for switched nonlinear systems is one of the most important and challenging fields.Due to the existence and interaction between the nonlinear continuous dynamics and discrete switching strategy,the behavior of the switched systems is very complicated.Many analysis and design problems deserve further investigation.
This dissertation focuses on the H_∞control for switched nonlinear systems based on analysis and control synthesis of switched systems and nonlinear systems by using the Lyapunov control theory.Some theories and methods of nonlinear H_∞control problem are extended to establish a theory framework of H_∞control for switched nonlinear systems.The common Lyapunov function method,single Lyapunov function method,convex combination technique,multiple Lyapunov function method and the average dwell time method are exploited to solve the H_∞control for switched Lipschitz nonlinear systems,switched cascade nonlinear systems and switched affine nonlinear systems in presence of perturbations and uncertainties.Stabilizable and unstabilizable subsystems,measurable and unmeasurable states are allowed to co-exist,respectively.
The main contributions of this thesis are as follows.
1.From the fact that the system states are often not available,the observer-based H_∞control for a class of switched Lipschitz nonlinear systems is considered. Observers,observer-based output feedback controllers,and a hysteresis switching strategy depending on the observer state and the previous value of switching signal are designed simultaneously.The generalized multiple Lyapunov function method, which improves the nonincreasing requirement at switching time sequences, provides more freedom for the problems addressed to be solvable.
2.When actuators suffer a "destabilizing failure" and the never-faulty actuators cannot stabilize the given system,the problem of reliable exponential stabilization and H_∞control for a class of switched Lipschitz nonlinear systems is addressed via the average dwell time approach.Since the system states are often not available, observers,observer-based output feedback controllers and switching strategies satisfying an average dwell time condition are constructed simultaneously.For the switched Lipschitz nonlinear systems with stabilizable and unstabilizable subsystems, the addressed reliable problem is solvable if the activation time ratio between stabilizable subsystems and unstabilizable subsystems is no less than a specified constant.
3.The robust H_∞control of a class of switched cascade nonlinear systems is studied.The single Lyapunov function and multiple Lyapunov function approaches are utilized to construct composite Lyapunov functions,state feedback controllers and hysteresis or max/min switching strategies based on the special triangular form. Sufficient conditions for the solvability of robust H_∞control problem are developed.
4.Robust H_∞control is investigated for a class of switched affine nonlinear systems with neutral uncertainties,which describes many practical parameter perturbations nonlinearly dependent on the state and state derivative.The multiple Lyapunov function and the common Lyapunov function approaches are exploited respectively for the cases that states are measurable or not.Correspondingly sufficient conditions are achieved via designing the state and output feedback controllers.
5.The constructive H_∞control problems are solvable for two classes of switched affine nonlinear systems.First,the multiple Lyapunov function and the common Lyapunov function approaches are exploited for the cases where states are measurable or not,respectively.Correspondingly sufficient conditions with the state or output controllers guarantee the solvability of the robust H_∞control problems. Based on the explicit construction of Lyapunov functions,the results do not rely on positive definite solutions of the Hamilton-Jacobi(HJ) inequalities.Secondly, sufficient condition is derived for robust non-fragile H_∞control problem via the multiple Lyapunov function technique.Based on the construction of controllers,the results avoid solving the HJ inequalities.
Finally,the results of the dissertation are summarized and further research topics are pointed out.
引文
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