仿射非线性不确定系统的鲁棒控制
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摘要
非线性鲁棒控制理论是目前一个相对较新的研究课题，也是近年来控制
理论研究中的热点问题，多年来已经取得了大量成果。我们将在已有结果的
基础上，以日臻成熟的微分几何理论和泛函微分方程方法作为我们主要的数
学工具进行我们的工作。
本文的主要研究工作包括仿射非线性不确定系统的鲁棒H_∞控制，非线
性时滞系统（合时滞混沌系统）的鲁棒镇定问题。以及作为特例而考虑的具有
较为一般性的不确定线性时滞系统的鲁棒控制问题，包括其中可以转化为线
性情形处理的具有非线性不确定性的情形。具体内容包括以下几个方面：
作为研究H_∞控制思想应用于非线性控制的新结果，我们在一种工程
应用背景之下，考虑了一类具有中立型不确定性的非线性系统的鲁棒H_∞控
制问题，基于HJI不等式给出了状态反馈控制器。针对状态不可测的情形，
我们进一步设计了动态输出反馈控制器，考虑到HJI不等式的求解难度，为
提高实用性，我们将所得结果转化为NLMI（非线性矩阵不等式）求解，使得
实际使用更加容易。
为了彻底避免求解HJI不等式，我们在考虑一类不确定仿射非线性系统
的H_∞控制问题时，通过HJI不等式的转换，以及相应解的一种构造和一种
构造性判据，基本上避免了通常的从数值求解HJI不等式的困难。同样包括
了状态反馈与动态输出反馈两种情形，动态输出反馈的情形给出的结果简单
易行，大大优于以往通常结果。
基于微分几何理论，我们还讨论了一类不确定非线性时滞系统的鲁棒镇
定控制器设计问题。应用改进的Razumikhin引理及Lyapunov方法，设计出
了相应的控制器，并给出了一个数值例子验证该方法的有效性。我们所用的
技术涉及到了非线性泛函分析的相关结果。首次在具有一定扰动的情形下讨
论了非线性时滞系统的鲁棒控制器的设计问题。
针对非线性不确定时滞混沌系统中不稳定不动点的控制需求，我们应用
线性矩阵不等式（LMI）的方法，得到了系统存在时滞反馈控制器的充分条件。
同时，我们还利用该控制器对系统中存在的不稳定周期轨道的跟踪问题进行
了研究。数值仿真也说明了我们方法的有效性。
慈山大学工学硕士学位论文
作为非线性情形的特例和一个比较，我们考虑了具有较一般的不确定世
情形下的时滞线性系统，研究了多时滞不确定线性系统的鲁棒镇定以及鲁棒
＊控制问题，将＊1方法引入了相应系统的鲁棒凤控制问题中。所得的
充分条件用Riccati不等式或LMI表示。在处理过程中，我们还较深入全面
的分析了这一类方法的内在处理机制。此外，应用一种改进型的Raz。ikhin
引理解除了己有文献中通常对滞后函数导数界的限制，同样得到了相应的鲁
棒镇定以及鲁棒H。控制问题的结果
Robust control of nonlinear systems is a relatively novel
direction and a central problem up to now. There has been a great deal
of rich results in such field for the time being. Based on existent
literature, we will make developed differential geometry theory and
functional differential equation theory as our main mathematical tools
to start with our work in this paper.
This paper is mainly concerned with robust H~, control of
nonlinear systems, robust stabilization of nonlinear systems with
time梔elay, and as a special case, robust H~ control of linear
time梔elay systems with more general uncertainties, which include the
case with nonlinear uncertainties that can be transferred into linear
case to deal with. These are given in detail as the following:
As a new result of nonlinear control with Ha., idea, ~e have
considered robust Hr,, control for nonlinear systems with neutral?
type uncertainties under an engineering application background. Based
on HJI inequality, state feedback and dynamic output feedback are
given, respectively. By considering the difficulty of resolving of HJI
inequality, our results are transformed into NLMIs for more efficient
computation.
In order to avoid resolving of HJI inequality completely, when
robust H,~ control of an affine nonlinear uncertain system is
considered, We make a transformation of HJI inequality with a
construction of corresponding solution and a tectonic criterion, so
that such difficulty is almost avoided. State feedback and dynamic
output feedback are given too. The result of dynamic output feedback
?III ?
LII 眫4~
case is easy to verify and far precede old one.
lie successively discuss design on robust stabilization controller
of uncertain nonlinear time梔elay systems based on differential
geometry theory. An improved Razumikhin lemma and Lyapunov theory are
applied and two controllers are given with and without nonlinear
time梔elay uncertainty, respectively. A numerical example is also
showed to test our results. Our technique is concerned with some
achievements of nonlinear functional analysis. We are sure that robust
controller of nonlinear time delay in this paper is considered for the
first time.
An LMI梑ased control method is presented to meet the requirement
of controlling unstable fixed pointed of nonlinear uncertain time?
delay chaotic sys~tems. Sufficient conditions for the existence of the
time梔elay feedback controller has been obtained. Similarly, we
discuss the tracking problem by applying the time梔elay feedback
control. And a numerical example is provided to show the effectiveness
of our results.
Compared with nonlinear case and as special one, time梔elay linear
systems with more general uncertainties are considered. Robust
stabilization and robust H~control problems on multi梔elay uncertain
systems with all disturbances are studied. LMI is introduced into
robust H~ control problem in the corresponding systems. Our results
are indicated as Riccati inequalities or LMIs. And the essential
principle of this kind method is analyzed. By using an improved
Razumikhin lemma, the limitations of bounds of delay functions?
derivatives in past literature are also removed, the corresponding
results on robust stabilization and robust H~~control problem are
obtained, too.
理论研究中的热点问题，多年来已经取得了大量成果。我们将在已有结果的
基础上，以日臻成熟的微分几何理论和泛函微分方程方法作为我们主要的数
学工具进行我们的工作。
本文的主要研究工作包括仿射非线性不确定系统的鲁棒H_∞控制，非线
性时滞系统（合时滞混沌系统）的鲁棒镇定问题。以及作为特例而考虑的具有
较为一般性的不确定线性时滞系统的鲁棒控制问题，包括其中可以转化为线
性情形处理的具有非线性不确定性的情形。具体内容包括以下几个方面：
作为研究H_∞控制思想应用于非线性控制的新结果，我们在一种工程
应用背景之下，考虑了一类具有中立型不确定性的非线性系统的鲁棒H_∞控
制问题，基于HJI不等式给出了状态反馈控制器。针对状态不可测的情形，
我们进一步设计了动态输出反馈控制器，考虑到HJI不等式的求解难度，为
提高实用性，我们将所得结果转化为NLMI（非线性矩阵不等式）求解，使得
实际使用更加容易。
为了彻底避免求解HJI不等式，我们在考虑一类不确定仿射非线性系统
的H_∞控制问题时，通过HJI不等式的转换，以及相应解的一种构造和一种
构造性判据，基本上避免了通常的从数值求解HJI不等式的困难。同样包括
了状态反馈与动态输出反馈两种情形，动态输出反馈的情形给出的结果简单
易行，大大优于以往通常结果。
基于微分几何理论，我们还讨论了一类不确定非线性时滞系统的鲁棒镇
定控制器设计问题。应用改进的Razumikhin引理及Lyapunov方法，设计出
了相应的控制器，并给出了一个数值例子验证该方法的有效性。我们所用的
技术涉及到了非线性泛函分析的相关结果。首次在具有一定扰动的情形下讨
论了非线性时滞系统的鲁棒控制器的设计问题。
针对非线性不确定时滞混沌系统中不稳定不动点的控制需求，我们应用
线性矩阵不等式（LMI）的方法，得到了系统存在时滞反馈控制器的充分条件。
同时，我们还利用该控制器对系统中存在的不稳定周期轨道的跟踪问题进行
了研究。数值仿真也说明了我们方法的有效性。
慈山大学工学硕士学位论文
作为非线性情形的特例和一个比较，我们考虑了具有较一般的不确定世
情形下的时滞线性系统，研究了多时滞不确定线性系统的鲁棒镇定以及鲁棒
＊控制问题，将＊1方法引入了相应系统的鲁棒凤控制问题中。所得的
充分条件用Riccati不等式或LMI表示。在处理过程中，我们还较深入全面
的分析了这一类方法的内在处理机制。此外，应用一种改进型的Raz。ikhin
引理解除了己有文献中通常对滞后函数导数界的限制，同样得到了相应的鲁
棒镇定以及鲁棒H。控制问题的结果
Robust control of nonlinear systems is a relatively novel
direction and a central problem up to now. There has been a great deal
of rich results in such field for the time being. Based on existent
literature, we will make developed differential geometry theory and
functional differential equation theory as our main mathematical tools
to start with our work in this paper.
This paper is mainly concerned with robust H~, control of
nonlinear systems, robust stabilization of nonlinear systems with
time梔elay, and as a special case, robust H~ control of linear
time梔elay systems with more general uncertainties, which include the
case with nonlinear uncertainties that can be transferred into linear
case to deal with. These are given in detail as the following:
As a new result of nonlinear control with Ha., idea, ~e have
considered robust Hr,, control for nonlinear systems with neutral?
type uncertainties under an engineering application background. Based
on HJI inequality, state feedback and dynamic output feedback are
given, respectively. By considering the difficulty of resolving of HJI
inequality, our results are transformed into NLMIs for more efficient
computation.
In order to avoid resolving of HJI inequality completely, when
robust H,~ control of an affine nonlinear uncertain system is
considered, We make a transformation of HJI inequality with a
construction of corresponding solution and a tectonic criterion, so
that such difficulty is almost avoided. State feedback and dynamic
output feedback are given too. The result of dynamic output feedback
?III ?
LII 眫4~
case is easy to verify and far precede old one.
lie successively discuss design on robust stabilization controller
of uncertain nonlinear time梔elay systems based on differential
geometry theory. An improved Razumikhin lemma and Lyapunov theory are
applied and two controllers are given with and without nonlinear
time梔elay uncertainty, respectively. A numerical example is also
showed to test our results. Our technique is concerned with some
achievements of nonlinear functional analysis. We are sure that robust
controller of nonlinear time delay in this paper is considered for the
first time.
An LMI梑ased control method is presented to meet the requirement
of controlling unstable fixed pointed of nonlinear uncertain time?
delay chaotic sys~tems. Sufficient conditions for the existence of the
time梔elay feedback controller has been obtained. Similarly, we
discuss the tracking problem by applying the time梔elay feedback
control. And a numerical example is provided to show the effectiveness
of our results.
Compared with nonlinear case and as special one, time梔elay linear
systems with more general uncertainties are considered. Robust
stabilization and robust H~control problems on multi梔elay uncertain
systems with all disturbances are studied. LMI is introduced into
robust H~ control problem in the corresponding systems. Our results
are indicated as Riccati inequalities or LMIs. And the essential
principle of this kind method is analyzed. By using an improved
Razumikhin lemma, the limitations of bounds of delay functions?
derivatives in past literature are also removed, the corresponding
results on robust stabilization and robust H~~control problem are
obtained, too.
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