基于单位分解方法的几类非线性系统的控制
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摘要
非线性系统的控制一直以来都是控制领域的一个热点和难点.历来使用得最多,最主要的方法是李雅普诺夫(Lyapunov)直接法,即通过引入一个Lyapunov函数来分析、判断系统的稳定性.但是鉴于非线性系统的多样性和复杂性,Lyapunov函数的选取没有统一有效的方法.现有各种精确的控制方法,只能是针对某一类非线性系统.而基于逼近方法的非线性系统控制,如模糊控制、神经网络控制等,虽然取得了丰富的理论与应用成果,但是仍有诸多理论与实践上的问题没有得到很好的解决。
上世纪七十年代,微分几何方法被引入到非线性系统的控制理论中,使得非线性系统理论得到了飞跃的发展.单位分解是微分几何中的一个重要概念.它的线性组合具有以任意精度逼近欧氏空间紧致域上连续函数的能力.本文利用它的这个性质去逼近非线性系统,然后运用Lyapunov稳定性理论等,结合线性矩阵不等式(LMI)技术,研究了几类非线性系统的控制问题.主要内容归纳如下:
·稳定性和镇定问题
第二章针对一类含不确定性的时滞系统,运用Lyapunov-Krasovskii方法分析了其鲁棒渐近稳定性问题,以LMI形式给出了该系统鲁棒渐近稳定的一个充分条件.第三章首先利用单位分解方法去逼近一类非线性时滞系统的非线性泛函,将原系统转化为一个等价系统.然运用Lyapunov-Krasovskii方法,结合LMI技术,给出了能使原系统渐近稳定的镇定控制器存在的一个充分条件.
·Ho_∞控制问题
第四章利用单位分解方法逼近一类非线性系统的非线性泛函,用连续函数的连续模分析了逼近精度,对于给定的参考模型,运用Lyapunov稳定性理论,研究了H_∞跟踪控制问题.第五章利用单位分解方法和Lyapunov稳定性理论,分析了一类带扰动系统按指定衰减率衰减的稳定性和扰动抑制问题,以LMI形式,给出了状态反馈H_∞控制器存在的一个充分条件.
·保成本控制问题
第六章利用单位分解方法,将一类非线性系统转化为一个等价系统,然后针对一个H_2二次型性能指标,运用Lyapunov稳定性理论,结合LMI技术,研究了原系统的保成本控制问题.第七章利用单位分解方法将一类非线性时滞系统转化为一个易于处理的局部具有线性结构的等价系统,针对一个H_2二次型性能指标,运用Lyapunov-Krasovskii方法,结合LMI技术,研究了该系统的保成本控制问题.第八章利用单位分解方法,运用Lyapunov稳定性理论,分析一类不确定非线性系统的稳定性并提出一个自适应保成本控制器,以LMI形式给出该控制器存在的一个充分条件,自适应律可通过LMI的可行解得到.
·状态观测器设计问题
第九章针对一类带有未知界不确定性的非线性系统,利用单位分解去逼近不确定性的未知界函数,逼近误差用参数的自适应律来校正.运用Lyapunov稳定性理论,研究了原系统与采用自适应状态观测器作用下系统之间的误差系统的稳定性问题.
最后对全文所作的工作进行了总结,并指出存在的问题和下一步研究的方向.
Control of nonlinear systems has long been a main and difficult issue in control area.The most used by now and the uppermost method is Lyapunov direct method, which is to analyze and justify system stability by introducing a Lyapunov function. Yet due to nonlinear systems' diversity and complexity,there are no uniform effective methods in choosing Lyapunov functions.All the existing exact methods in nonlinear system control,are only applicable to a certain kind of nonlinear systems.Nonlinear system control using approximation-based method,such as fuzzy control,neural network control,etc,though has gained plentiful theoretic and actual achievements,there are still many theoretic and practical problems leaving unsolved.
In the 70s of last century,differential geometry method was introduced into control theory of nonlinear systems,which makes nonlinear system theory gain a big boost. Partition of unity is an important concept in differential geometry.Its combination has the capability of approximating any continuous functions in a compact region of Eu-clidian space within any specified precision.In this dissertation,by taking advantage of this character to approximate nonlinear systems,then using Lyapunov stability theory, etc,combined with linear matrix inequality technology,we studied the control problems of several classes of nonlinear systems.The main contents can be concluded as:
●Stability and stabilizability problems
In chapter 2,for a class of time-delay systems with uncertainties,we analyzed its robust asymptotical stability problem by using Lyapunov-Krasovskii method,then presented a sufficient condition for the time-delay systems to be robust asymptotically stable in terms of LMI.In chapter 3,we firstly made use of partition of unity method to approximate the nonlinear functions of a class of nonlinear time-delay systems,so as to transform the original system into an equivalent system,then by using Lyapunov-Krasovskii method,combined with LMI technology,we presented a sufficient condition for the existence of a stabilization controller which can make the original system robust asymptotically stable.
●H_∞control problems
In chapter 4,we also used partition of unity method to approximate the nonlinear functions of a class of nonlinear systems,then analyzed the approximation precision by using continuous modulus of continuous functions.For a given reference model,by using Lyapunov stability theory,we studied the H_∞control problem.In chapter 5,by using partition of unity method and Lyapunov stability theory,we analyzed system stability and disturbance-restrained problem of a class of nonlinear systems with disturbance when attenuating at a specified decay rate,presented a sufficient condition for the existence of a state feedback Hoo controller in terms of LMI.
●Guaranteed cost control problems
In chapter 6,we firstly made use of partition of unity method to transform a class of nonlinear systems into an equivalent system,then for a H_2 quadratic performance index,by using Lyapunov stability theory,combined with LMI technology,we analyzed guaranteed cost control problem of the nonlinear system.In chapter 7,by using partition of unity method,we firstly transform a class of nonlinear time-delay systems into an equivalent system with local linear structure,which is easier to be dealt with.Then for a H_2 quadratic performance index,by using Lyapunov-Krasovskii method,combined with LMI technology,we analyzed guaranteed cost control problem of the original system.In chapter 8,by using partition of unity method and Lyapunov stability theory,we analyzed the stability problem of a class of nonlinear systems with uncertainties,presented an adaptive guaranteed cost controller and proposed a sufficient condition for the existence of the controller in terms of LMI.The adaptive law can be obtained by solving the LMIs.
●State observer design problem
In chapter 9,for a class of nonlinear systems with uncertainties whose bound is unknown,we first took advantage of partition of unity method to approximate the unknown bound function of the uncertainties.The approximation error was adjusted by the adaptive law of parameters.By using Lyapunov stability theory,we studied the stability problem of the error system between the original nonlinear system and the system under the control of an adaptive state observer.
Finally we drew a conclusion on the whole work of this dissertation,pointed out some existent problems and the future orientation of research.
上世纪七十年代,微分几何方法被引入到非线性系统的控制理论中,使得非线性系统理论得到了飞跃的发展.单位分解是微分几何中的一个重要概念.它的线性组合具有以任意精度逼近欧氏空间紧致域上连续函数的能力.本文利用它的这个性质去逼近非线性系统,然后运用Lyapunov稳定性理论等,结合线性矩阵不等式(LMI)技术,研究了几类非线性系统的控制问题.主要内容归纳如下:
·稳定性和镇定问题
第二章针对一类含不确定性的时滞系统,运用Lyapunov-Krasovskii方法分析了其鲁棒渐近稳定性问题,以LMI形式给出了该系统鲁棒渐近稳定的一个充分条件.第三章首先利用单位分解方法去逼近一类非线性时滞系统的非线性泛函,将原系统转化为一个等价系统.然运用Lyapunov-Krasovskii方法,结合LMI技术,给出了能使原系统渐近稳定的镇定控制器存在的一个充分条件.
·Ho_∞控制问题
第四章利用单位分解方法逼近一类非线性系统的非线性泛函,用连续函数的连续模分析了逼近精度,对于给定的参考模型,运用Lyapunov稳定性理论,研究了H_∞跟踪控制问题.第五章利用单位分解方法和Lyapunov稳定性理论,分析了一类带扰动系统按指定衰减率衰减的稳定性和扰动抑制问题,以LMI形式,给出了状态反馈H_∞控制器存在的一个充分条件.
·保成本控制问题
第六章利用单位分解方法,将一类非线性系统转化为一个等价系统,然后针对一个H_2二次型性能指标,运用Lyapunov稳定性理论,结合LMI技术,研究了原系统的保成本控制问题.第七章利用单位分解方法将一类非线性时滞系统转化为一个易于处理的局部具有线性结构的等价系统,针对一个H_2二次型性能指标,运用Lyapunov-Krasovskii方法,结合LMI技术,研究了该系统的保成本控制问题.第八章利用单位分解方法,运用Lyapunov稳定性理论,分析一类不确定非线性系统的稳定性并提出一个自适应保成本控制器,以LMI形式给出该控制器存在的一个充分条件,自适应律可通过LMI的可行解得到.
·状态观测器设计问题
第九章针对一类带有未知界不确定性的非线性系统,利用单位分解去逼近不确定性的未知界函数,逼近误差用参数的自适应律来校正.运用Lyapunov稳定性理论,研究了原系统与采用自适应状态观测器作用下系统之间的误差系统的稳定性问题.
最后对全文所作的工作进行了总结,并指出存在的问题和下一步研究的方向.
Control of nonlinear systems has long been a main and difficult issue in control area.The most used by now and the uppermost method is Lyapunov direct method, which is to analyze and justify system stability by introducing a Lyapunov function. Yet due to nonlinear systems' diversity and complexity,there are no uniform effective methods in choosing Lyapunov functions.All the existing exact methods in nonlinear system control,are only applicable to a certain kind of nonlinear systems.Nonlinear system control using approximation-based method,such as fuzzy control,neural network control,etc,though has gained plentiful theoretic and actual achievements,there are still many theoretic and practical problems leaving unsolved.
In the 70s of last century,differential geometry method was introduced into control theory of nonlinear systems,which makes nonlinear system theory gain a big boost. Partition of unity is an important concept in differential geometry.Its combination has the capability of approximating any continuous functions in a compact region of Eu-clidian space within any specified precision.In this dissertation,by taking advantage of this character to approximate nonlinear systems,then using Lyapunov stability theory, etc,combined with linear matrix inequality technology,we studied the control problems of several classes of nonlinear systems.The main contents can be concluded as:
●Stability and stabilizability problems
In chapter 2,for a class of time-delay systems with uncertainties,we analyzed its robust asymptotical stability problem by using Lyapunov-Krasovskii method,then presented a sufficient condition for the time-delay systems to be robust asymptotically stable in terms of LMI.In chapter 3,we firstly made use of partition of unity method to approximate the nonlinear functions of a class of nonlinear time-delay systems,so as to transform the original system into an equivalent system,then by using Lyapunov-Krasovskii method,combined with LMI technology,we presented a sufficient condition for the existence of a stabilization controller which can make the original system robust asymptotically stable.
●H_∞control problems
In chapter 4,we also used partition of unity method to approximate the nonlinear functions of a class of nonlinear systems,then analyzed the approximation precision by using continuous modulus of continuous functions.For a given reference model,by using Lyapunov stability theory,we studied the H_∞control problem.In chapter 5,by using partition of unity method and Lyapunov stability theory,we analyzed system stability and disturbance-restrained problem of a class of nonlinear systems with disturbance when attenuating at a specified decay rate,presented a sufficient condition for the existence of a state feedback Hoo controller in terms of LMI.
●Guaranteed cost control problems
In chapter 6,we firstly made use of partition of unity method to transform a class of nonlinear systems into an equivalent system,then for a H_2 quadratic performance index,by using Lyapunov stability theory,combined with LMI technology,we analyzed guaranteed cost control problem of the nonlinear system.In chapter 7,by using partition of unity method,we firstly transform a class of nonlinear time-delay systems into an equivalent system with local linear structure,which is easier to be dealt with.Then for a H_2 quadratic performance index,by using Lyapunov-Krasovskii method,combined with LMI technology,we analyzed guaranteed cost control problem of the original system.In chapter 8,by using partition of unity method and Lyapunov stability theory,we analyzed the stability problem of a class of nonlinear systems with uncertainties,presented an adaptive guaranteed cost controller and proposed a sufficient condition for the existence of the controller in terms of LMI.The adaptive law can be obtained by solving the LMIs.
●State observer design problem
In chapter 9,for a class of nonlinear systems with uncertainties whose bound is unknown,we first took advantage of partition of unity method to approximate the unknown bound function of the uncertainties.The approximation error was adjusted by the adaptive law of parameters.By using Lyapunov stability theory,we studied the stability problem of the error system between the original nonlinear system and the system under the control of an adaptive state observer.
Finally we drew a conclusion on the whole work of this dissertation,pointed out some existent problems and the future orientation of research.
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