时滞及饱和Hamilton系统的分析与综合
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摘要
近年来,作为控制理论研究热点之一的非线性系统控制理论得到了长足的发展,尤其是微分几何数学方法的引入,使得非线性系统控制理论有了很大的飞跃。但作为更一般的问题——具有时滞和饱和的非线性控制系统的研究结果还不多见。众所周知,时滞现象和执行器饱和普遍存在于实际系统中,是引起系统不稳定和性能变差的主要原因之一。然而,由于系统本身的复杂性导致对此类非线性系统的控制难度大大增加。至今仍有许多基本的问题难于得到很好的解决,相应的结果也不多见。而作为一类重要的非线性系统的广义Hamilton控制系统,由于其结构清晰,物理意义明确,Hamilton函数是系统的总能量,它在稳定性分析和控制设计问题中表现出了明显的优越性。最近几年,Hamilton系统的观点和方法越来越引起非线性控制界的关注。
本论文针对具有时滞和饱和的Hamilton系统,研究了其稳定性和控制设计等问题。主要包括以下内容:
一.研究了时滞Hamilton系统的稳定性问题。充分利用Hamilton系统自身的特性,并结合时滞系统稳定性分析方法:Lyapunov-Krasovskii泛函法,从时滞无关和时滞相关两个角度为两类时滞Hamilton系统(常时滞及时变时滞)分别提出了稳定性的充分条件。针对系统状态矩阵中包含不确定性的时滞Hamilton系统,提出了鲁棒稳定的充分条件。另外,基于所得到的结论,研究了时滞非线性系统的Hamilton实现理论,并分析了其稳定性。最后给出了几个例子与仿真支持所提出的理论结果。
二.研究了时滞Hamilton控制系统的L_2干扰抑制问题。考虑一类带干扰的时滞Hamilton系统,通过应用合适的控制律和存储函数,构造了一个γ-耗散不等式。然后基于Lyapunov-Krasovskii稳定性定理,为闭环系统提出了一个渐近稳定的充分条件。最后,将结果推广到结构矩阵存在凸型不确定性的情形,为系统设计了鲁棒L_2干扰抑制控制器。仿真例子进一步验证了结论的正确性和实用性。
三.研究了带有饱和和输入可加性干扰的Hamilton控制系统的有限增益镇定问题。从有限增益输出和有限增益输入两个方面分别进行了分析。在一定条件下运用静态输出反馈控制使得闭环系统达到全局渐近稳定。另外,针对Hamilton系统中的非线性函数提出进一步的演变条件,证明了相同的反馈律可以使得闭环系统达到有限L_2增益镇定,同时获得有限增益的估计值。仿真结果验证了结论的有效性。
四.研究了具有饱和输入的时滞Hamilton系统的稳定性和镇定问题。综合考虑时滞和饱和对系统产生的双重影响,基于Hamilton系统的结构特性,根据Lyapunov-Krasovskii稳定性定理,提出了系统在输出反馈控制下可镇定的充分条件。这些条件由线性矩阵不等式(LMI)构成,可以直接通过MATLAB工具箱进行求解和验证,简单易行。最后给出了数值仿真例子,进一步证明了所得稳定性结论和控制设计方法的实用性和有效性。
五.研究了执行器饱和的电力系统可镇定问题。应用前面得到的关于饱和Hamilton控制系统的分析结果,分别讨论了励磁饱和以及励磁-汽门同时饱和的单机无穷大电力系统的有限增益镇定问题。首先基于Hamilton能量理论将电机系统模型转化为广义耗散Hamilton系统模型,然后运用静态输出反馈,在适当的条件下,证明了闭环系统可以达到渐近稳定和有限增益镇定,从而解决了相应的电力系统的镇定问题。最后给出了仿真实例。
本论文的主要创新点在于:
·首次研究时滞Hamilton系统的稳定性和控制设计问题。充分利用Hamilton系统的特有结构和耗散性质,分别为几类时滞Hamilton系统提出了稳定,鲁棒稳定和反馈镇定的若干判据:
·首次研究饱和Hamilton系统的稳定性和有限增益镇定问题。充分利用Hamillton系统的特有结构和耗散性质,分别提出了系统有限增益输出和有限增益输入镇定的充分条件,并得到有限增益的估计值;
·首次研究时滞和饱和双重影响下的Hamilton系统,充分利用Hamilton系统的结构特性,提出了系统稳定及在输出反馈控制下可镇定的充分条件。
Recent years witnessed the full-grown development of nonlinear systems control theory,one of the research hotspots in control theory.Especially with the introduction of differential geometry methodology,the nonlinear control systems theory has been made great progress.However,the research outcome is still insufficient on the more common issues of time-delay and saturation of nonlinear control systems.As well known,the phenomena of time delay and saturation constraint are often encountered in many practical control systems and one of the main reasons resulting in instability and poor performances of the associated control systems.However,the complexity of the system itself leads to the increasing difficult in controlling this kind of nonlinear systems.It is until present that many a fundamental problem has not been settled down and the corresponding outcome is scare as well.Meanwhile,port-controlled Hamiltonian systems,as an important class of nonlinear systems,owing to its clear structure and physical meaning,and that Hamiltonion function is considered as the total energy of the systems,demonstrates salient superiority in stability analysis and control design.In recent years,the viewpoint and methodology of Hamiltonion systems have drawn increasing attention in the field of nonlinear control systems.
This thesis mainly studies the issues of stability and control design of Hamiltonian systems with time-delay and saturation.As follows are the main research contents.
1.The stability of Hamiltonian systems is discussed.In full utilization of the dissipative structural properties of the Hamiltonian systems,and by using the technique of Lyapunov-Krasovskii functional,from two angles of delay-independent and delay-dependent,several sufficient stability conditions are proposed for two classes of Hamiltonian systems(including constant delay and time-varying delays).The robust stability is considered for a class of time-delay Hamiltonian systems which possesses time-invariant uncertainties belonging to some convex bounded polytypic domain.Besides, based on the proposed results,the stability of a class of nonlinear time-delay systems is also studied by Hamiltonian realization.Some examples are given to support the theoretical results.
2.The L_2-disturbance attenuation for a class of time-delay Hamiltonian systems is considered.Aγ-dissipative inequality is established by using a proper control law and a storage function.Then based on the Lyapunov-Krasovskii stability theorem,a sufficient condition is proposed for the asymptotically stable of the closed-loop system. Finally,the case that there are time-invariant uncertainties belonging to some convex bounded polytypic domain is investigated and an L_2 disturbance attenuation control law is proposed as well.Simulations further inspects and verifies the correctness and practicability of the results.
3.The stabilization problem for a port-controlled Hamiltonian system subject to actuator saturation and input additive external disturbances is investigated.The results are developed from two aspects:finite gain output stabilization and finite gain input stabilization.Conditions are identified under which a static output feedback law would achieve asymptotic stabilization.Under some additional growth conditions on the nonlinear functions involved in the system,the same feedback law would also achieve finite gain L_2 stabilization.In establishing these results,an estimate of the finite gain is also obtained.The illustrative examples verify the effectiveness of the proposed theories.
4.The stabilization of a class of Hamiltonian systems with state time-delay and input saturation is addressed.Based on the special structure and dissipative property of the Hamiltonian systems,by using the Lyapunov-Krasovskii functional theory,the sufficient conditions are derived to guarantee the systems as well as the resulted closedloop systems for the system under output feedback to be asymptotically stable when input saturation effectively occurs.These conditions are composed of linear matrix inequalities and can directly be solved and verified through MATLAB/LMI toolbox, easy and simple.Numerical examples are presented to illustrate the effectiveness of the obtained results.
5.The finite gain stabilization problem of the generator power systems subject to actuator saturation and external disturbances is investigated.Applying the analysis results of the Hamiltonian control systems subject to saturation,the finite gain stabilization of the single-machine infinite bus systems with excitation saturating and both excitation and steam valve saturating are discussed,respectively.First of all, based on the Hamiltonian energy theory,the power system model is transformed into a port-controlled Hamiltonian system model.Then a static output feedback controller is considered for the obtained Hamiltonian systems.Under some growth conditions, it is shown that the asymptotic stabilization,as well as the finite gain stabilization of the closed-loop system can be achieved,which solves the stabilization of the corresponding power systems.Simulation shows the effectiveness of the stabilizing method proposed.
Innovations of the thesis mainly include the following three aspects:
●It is first studied stability and controller design of time-delay Hamiltonian systems. Making the most of the special structure and dissipative property of the Hamiltonian systems,some stability and feedback stabilization criterion are derived for several kinds of time delay Hamiltonian systems,respectively;
●It is first studied stability and finite gain stabilization of Hamiltonian systems with input saturation.Making the most of the special structure and dissipative property of the Hamiltonian systems,the sufficient conditions on finite gain output stabilization and finite gain input stabilization are proposed for the systems,respectively. Estimates of the finite gain are also obtained;
●It is first studied the Hamiltonian systems under the double influences of time delay and saturation.Making the most of the special structure and dissipative property of the Hamiltonian systems,the sufficient conditions of stability and stabilization under an output feedback law are presented.
本论文针对具有时滞和饱和的Hamilton系统,研究了其稳定性和控制设计等问题。主要包括以下内容:
一.研究了时滞Hamilton系统的稳定性问题。充分利用Hamilton系统自身的特性,并结合时滞系统稳定性分析方法:Lyapunov-Krasovskii泛函法,从时滞无关和时滞相关两个角度为两类时滞Hamilton系统(常时滞及时变时滞)分别提出了稳定性的充分条件。针对系统状态矩阵中包含不确定性的时滞Hamilton系统,提出了鲁棒稳定的充分条件。另外,基于所得到的结论,研究了时滞非线性系统的Hamilton实现理论,并分析了其稳定性。最后给出了几个例子与仿真支持所提出的理论结果。
二.研究了时滞Hamilton控制系统的L_2干扰抑制问题。考虑一类带干扰的时滞Hamilton系统,通过应用合适的控制律和存储函数,构造了一个γ-耗散不等式。然后基于Lyapunov-Krasovskii稳定性定理,为闭环系统提出了一个渐近稳定的充分条件。最后,将结果推广到结构矩阵存在凸型不确定性的情形,为系统设计了鲁棒L_2干扰抑制控制器。仿真例子进一步验证了结论的正确性和实用性。
三.研究了带有饱和和输入可加性干扰的Hamilton控制系统的有限增益镇定问题。从有限增益输出和有限增益输入两个方面分别进行了分析。在一定条件下运用静态输出反馈控制使得闭环系统达到全局渐近稳定。另外,针对Hamilton系统中的非线性函数提出进一步的演变条件,证明了相同的反馈律可以使得闭环系统达到有限L_2增益镇定,同时获得有限增益的估计值。仿真结果验证了结论的有效性。
四.研究了具有饱和输入的时滞Hamilton系统的稳定性和镇定问题。综合考虑时滞和饱和对系统产生的双重影响,基于Hamilton系统的结构特性,根据Lyapunov-Krasovskii稳定性定理,提出了系统在输出反馈控制下可镇定的充分条件。这些条件由线性矩阵不等式(LMI)构成,可以直接通过MATLAB工具箱进行求解和验证,简单易行。最后给出了数值仿真例子,进一步证明了所得稳定性结论和控制设计方法的实用性和有效性。
五.研究了执行器饱和的电力系统可镇定问题。应用前面得到的关于饱和Hamilton控制系统的分析结果,分别讨论了励磁饱和以及励磁-汽门同时饱和的单机无穷大电力系统的有限增益镇定问题。首先基于Hamilton能量理论将电机系统模型转化为广义耗散Hamilton系统模型,然后运用静态输出反馈,在适当的条件下,证明了闭环系统可以达到渐近稳定和有限增益镇定,从而解决了相应的电力系统的镇定问题。最后给出了仿真实例。
本论文的主要创新点在于:
·首次研究时滞Hamilton系统的稳定性和控制设计问题。充分利用Hamilton系统的特有结构和耗散性质,分别为几类时滞Hamilton系统提出了稳定,鲁棒稳定和反馈镇定的若干判据:
·首次研究饱和Hamilton系统的稳定性和有限增益镇定问题。充分利用Hamillton系统的特有结构和耗散性质,分别提出了系统有限增益输出和有限增益输入镇定的充分条件,并得到有限增益的估计值;
·首次研究时滞和饱和双重影响下的Hamilton系统,充分利用Hamilton系统的结构特性,提出了系统稳定及在输出反馈控制下可镇定的充分条件。
Recent years witnessed the full-grown development of nonlinear systems control theory,one of the research hotspots in control theory.Especially with the introduction of differential geometry methodology,the nonlinear control systems theory has been made great progress.However,the research outcome is still insufficient on the more common issues of time-delay and saturation of nonlinear control systems.As well known,the phenomena of time delay and saturation constraint are often encountered in many practical control systems and one of the main reasons resulting in instability and poor performances of the associated control systems.However,the complexity of the system itself leads to the increasing difficult in controlling this kind of nonlinear systems.It is until present that many a fundamental problem has not been settled down and the corresponding outcome is scare as well.Meanwhile,port-controlled Hamiltonian systems,as an important class of nonlinear systems,owing to its clear structure and physical meaning,and that Hamiltonion function is considered as the total energy of the systems,demonstrates salient superiority in stability analysis and control design.In recent years,the viewpoint and methodology of Hamiltonion systems have drawn increasing attention in the field of nonlinear control systems.
This thesis mainly studies the issues of stability and control design of Hamiltonian systems with time-delay and saturation.As follows are the main research contents.
1.The stability of Hamiltonian systems is discussed.In full utilization of the dissipative structural properties of the Hamiltonian systems,and by using the technique of Lyapunov-Krasovskii functional,from two angles of delay-independent and delay-dependent,several sufficient stability conditions are proposed for two classes of Hamiltonian systems(including constant delay and time-varying delays).The robust stability is considered for a class of time-delay Hamiltonian systems which possesses time-invariant uncertainties belonging to some convex bounded polytypic domain.Besides, based on the proposed results,the stability of a class of nonlinear time-delay systems is also studied by Hamiltonian realization.Some examples are given to support the theoretical results.
2.The L_2-disturbance attenuation for a class of time-delay Hamiltonian systems is considered.Aγ-dissipative inequality is established by using a proper control law and a storage function.Then based on the Lyapunov-Krasovskii stability theorem,a sufficient condition is proposed for the asymptotically stable of the closed-loop system. Finally,the case that there are time-invariant uncertainties belonging to some convex bounded polytypic domain is investigated and an L_2 disturbance attenuation control law is proposed as well.Simulations further inspects and verifies the correctness and practicability of the results.
3.The stabilization problem for a port-controlled Hamiltonian system subject to actuator saturation and input additive external disturbances is investigated.The results are developed from two aspects:finite gain output stabilization and finite gain input stabilization.Conditions are identified under which a static output feedback law would achieve asymptotic stabilization.Under some additional growth conditions on the nonlinear functions involved in the system,the same feedback law would also achieve finite gain L_2 stabilization.In establishing these results,an estimate of the finite gain is also obtained.The illustrative examples verify the effectiveness of the proposed theories.
4.The stabilization of a class of Hamiltonian systems with state time-delay and input saturation is addressed.Based on the special structure and dissipative property of the Hamiltonian systems,by using the Lyapunov-Krasovskii functional theory,the sufficient conditions are derived to guarantee the systems as well as the resulted closedloop systems for the system under output feedback to be asymptotically stable when input saturation effectively occurs.These conditions are composed of linear matrix inequalities and can directly be solved and verified through MATLAB/LMI toolbox, easy and simple.Numerical examples are presented to illustrate the effectiveness of the obtained results.
5.The finite gain stabilization problem of the generator power systems subject to actuator saturation and external disturbances is investigated.Applying the analysis results of the Hamiltonian control systems subject to saturation,the finite gain stabilization of the single-machine infinite bus systems with excitation saturating and both excitation and steam valve saturating are discussed,respectively.First of all, based on the Hamiltonian energy theory,the power system model is transformed into a port-controlled Hamiltonian system model.Then a static output feedback controller is considered for the obtained Hamiltonian systems.Under some growth conditions, it is shown that the asymptotic stabilization,as well as the finite gain stabilization of the closed-loop system can be achieved,which solves the stabilization of the corresponding power systems.Simulation shows the effectiveness of the stabilizing method proposed.
Innovations of the thesis mainly include the following three aspects:
●It is first studied stability and controller design of time-delay Hamiltonian systems. Making the most of the special structure and dissipative property of the Hamiltonian systems,some stability and feedback stabilization criterion are derived for several kinds of time delay Hamiltonian systems,respectively;
●It is first studied stability and finite gain stabilization of Hamiltonian systems with input saturation.Making the most of the special structure and dissipative property of the Hamiltonian systems,the sufficient conditions on finite gain output stabilization and finite gain input stabilization are proposed for the systems,respectively. Estimates of the finite gain are also obtained;
●It is first studied the Hamiltonian systems under the double influences of time delay and saturation.Making the most of the special structure and dissipative property of the Hamiltonian systems,the sufficient conditions of stability and stabilization under an output feedback law are presented.
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