2005-0411.tex

自动化学报的编辑软件

 & {\pmb v}_i ^{\rm T} T_i ({\bf\Theta}  - {\bf\Theta} _0 )
 \end{align*}
其中 {\setlength\arraycolsep{2pt}
\begin{eqnarray}
   {\pmb v}_i ^{\rm T}  = &\left[
   \displaystyle\frac{{x_{i1} ({\bf\Theta} )}}{{\left| {a_{i1} ({\bf\Theta} )} \right|}},
    \displaystyle\frac{{y_{i1} ({\bf\Theta} )}}{{\left| {a_{i1} ({\bf\Theta} )} \right|}},
    \cdots,
   - \displaystyle\frac{{x_{ii} ({\bf\Theta} )}}{{\left| {a_{ii} ({\bf\Theta} )} \right|}},\right.
   \nonumber\\
   &  \left. - \displaystyle\frac{{y_{ii} ({\bf\Theta} )}}{{\left| {a_{ii} ({\bf\Theta} )} \right|}},
     \cdots,
   \displaystyle\frac{{x_{im} ({\bf\Theta} )}}{{\left| {a_{im} ({\bf\Theta} )} \right|}},
   \displaystyle\frac{{y_{im} ({\bf\Theta} )}}{{\left| {a_{im} ({\bf\Theta} )} \right|}} \right] \Big|_{{\bf\Theta} _0
   }
\end{eqnarray}}
\begin{displaymath}
  T_i ^{\rm T}  =  {\left[ \nabla x_{i1} ^{\rm T}, \nabla y_{i1} ^{\rm T},
   \cdots, \nabla x_{im} ^{\rm T} ,  \nabla y_{im} ^{\rm T}
\right]} \Big |_{{\bf\Theta} _0 }
\end{displaymath}

由H${\rm{\ddot o}}$lder不等式,有
$$
  \Delta R_i= \left| {\Delta r_i ({\bf\Theta} )} \right| \approx \left| {{\pmb v}_i ^{\rm T} T_i \left(
{{\bf\Theta}  - {\bf\Theta} _0 } \right)} \right| \le c_{\bf\Theta}
\lambda _i
$$
其中, $ \lambda _i  = \sqrt {{\pmb v}_i ^{\rm T} T_i T_i ^{\rm T}
{\pmb v}_i } $,由于$\Delta R_i^*$是$\Delta R_i$的上界,
这样即可得到$\Delta R_i^*$的近似估计,有如下定理.

{\hei 定理} {\bf 2.} 复数域上的$m\times m$矩阵$A({\bf\Theta}
)=\left[{a_{ij}({\bf\Theta} ) }\right]$, ${\bf\Theta}\in
R^p$为受到扰动的参数,
${\bf\Theta}_0$为${\bf\Theta}$的标称值,且满足$\left\| {{\bf\Theta} -
{\bf\Theta} _0 } \right\|_2 \le c_{\bf\Theta} $,则$\Delta
R_i^*$的一个近似估计为$c_{\bf\Theta}\lambda_i$.

在上面的定理中,以1-范数度量矩阵元素的不确定性.当然也可以用2-范数进行度量\,\!$^{[10]}$,
此时可以得到$\Delta R_i^c  = \sqrt m \sqrt {\chi _{\alpha p}^2
\sigma _{1i} } $, $\sigma_{1i} $是$T\Sigma _{\bf\Theta} T^{\rm T}
$的最大奇异值.但是可以证明$\Delta R_i^* \le \Delta R_i^c $,如下.

{\hei 证明.} 只需证$\lambda _i  = \sqrt {{\pmb v}_i ^{\rm T} T_i
\Sigma _{\bf\Theta} T_i ^{\rm T} {\pmb v}_i } \le \sqrt {m\sigma
_{1i} } $

令$T_i \Sigma _{\bf\Theta}  T_i ^{\rm T} $的奇异值分解为
\begin{displaymath}
T_i \Sigma _{\bf\Theta}  T_i ^{\rm T} = U{\rm{diag}}\left[ {\sigma
_{1,i} ,\sigma _{2,i} , \cdots \sigma _{2m,i} }
 \right]U^{\rm T}  \buildrel \Delta \over = U\Sigma _\sigma  U^{\rm T}
\end{displaymath}
其中
\begin{displaymath}
\sigma _{1,i} , \ge \sigma _{2,i}  \ge  \cdots  \ge \sigma _{2m,i}
\end{displaymath}
令~${\pmb y} = {\pmb v}_i^{\rm T} U$,则有
\begin{displaymath}
  \lambda _i  = \sqrt {{\pmb v}_i ^{\rm T} T_i \Sigma _{\bf\Theta}  T_i ^{\rm T} {\pmb v}_i }
  = \sqrt {{\pmb y}^{\rm T} \Sigma
_\sigma {\pmb y}}  \le \left\| {\pmb y} \right\|_2 \sqrt {\sigma
_{1i} }
\end{displaymath}
由于$U$是酉矩阵,所以
\begin{equation}
\left\| {\pmb y} \right\|_2  = \left\| {{\pmb v}_i^{\rm T} U}
\right\|_2  = \left\| {\pmb v}_i \right\|_2 = \sqrt m\label{e5}
\end{equation}
也就是说,由本文估计出的矩阵行元素不确定性界小于文[10]的估计.\hfill $\Box$

注意,式(5)最后一个等号成立是因为${\pmb v}_i$的特殊结构. 由式(4)可知,
${\pmb v}_i^2(2k - 1) + {\pmb v}_i^2(2k) = 1$, $k = 1,2, \cdots m$,
因此$\left\| {\pmb v}_i \right\|_2  = {\pmb v}_i^{\rm T} {\pmb v}_i
= m$.

文[10]在估计不确定性时用到了元素间的相关信息$(T_i\Sigma _{\bf\Theta}
T_i^{\rm T})$,
使得估计的保守性降低.本文中,由于使用了1-范数作度量,从而还使用了各个元
素的分量信息$({\pmb
v}_i)$,这样更进一步降低了估计的保守性,这也是使用引理1的好处.

当$\Delta R_i^*
$估计出来后,将它与$R_{0i}$相加,即可得到$A({\bf\Theta}
)$的第$i$行的鲁棒行Gershgorin圆半径的近似估计.

\subsection{传递函数矩阵的鲁棒\ {\bf Gershgorin} 带估计}

以每一个$s$值下传递函数矩阵的鲁棒Gershgorin圆估计为基础,当$s$沿Nyquist围线变
化时,即可得到它的鲁棒Gershgorin带估计.这里要注意的是,在利用定理2估计
鲁棒Gershgorin圆时, $\Delta R_i (\Delta R_l
)$不仅是${\bf\Theta}$的函数,还是$s$的函数.

在得到传递函数矩阵的鲁棒Gershgorin带估计后,将其用于INA方法便可以得到RINA方法,
此时,其形式与一般的RINA方法并无区别,只要将鲁棒Gershgorin带的大小按照定
理2的结论进行改动即可,具体的定理见文[5]和[10].

从这些定理看不出鲁棒Gershgorin带与系统稳定性的关系,实际上,
鲁棒Gershgorin带的估计与控制系统设计有很密切的关系.在设计$K(s) = K_1
(s)K_2 (s)$时, $K_2(s)$是对角阵,用来对各个通道进行控制,
$K_1(s)$为预补偿矩阵,用来使传递函数变为对角优势矩阵, 即使得$\hat
Q(s,{\bf\Theta}
)=\rm{inv}(G(s,{\bf\Theta})K(s))$为对角优势的.一般来说,对${\bf\Theta}
= {\bf\Theta} _0 $,总存在$K_1 (s)$,使得$\hat Q(s,{\bf\Theta}
)$成为对角优势矩阵,但是不一定存在一个$K_1 (s)$,使得对$\forall
{\bf\Theta} : \left\| {{\bf\Theta} - {\bf\Theta} _0 } \right\|_2 \le
c_{\bf\Theta} $, $\hat Q(s,{\bf\Theta} )$都是对角优势矩阵,
这也是应用RINA方法的难点所在.这里,鲁棒Gershgorin带的意义在于给出了当参数不确定时,传递函数
矩阵各通道交连的变化,为设计$K_1(s)$提供了定量信息.鲁棒Gershgorin带越小,则$K_1
(s)$越好设计.

\section{仿真}

考虑如下系统\,\!$^{[2]}$,其标称传递函数矩阵模型和预补偿矩阵分别为
\begin{displaymath}
G(s,{\pmb k}_0 ) = \left[ {\begin{array}{*{20}c}
   {\displaystyle\frac{{k_{10} }}{{s + k_{20} }}} & {\displaystyle\frac{{k_{30} }}{{s + k_{40} }}}  \\
   {\displaystyle\frac{{k_{50} }}{{s + k_{60} }}} & {\displaystyle\frac{{k_{70} }}{{s + k_{80} }}}  \\
\end{array}} \right] = \left[ {\begin{array}{*{20}c}
   {\displaystyle\frac{7}{{s + 1}}} & {\displaystyle\frac{8}{{s + 1}}}  \\
   {\displaystyle\frac{{12}}{{s + 2}}} & {\displaystyle\frac{{14}}{{s + 2}}}  \\
\end{array}} \right]
\end{displaymath}
\begin{displaymath}
\qquad K_1  = \left[ {\begin{array}{*{20}c}
   7 & { - 8}  \\
   { - 6} & 7  \\
\end{array}} \right]
\end{displaymath}
其中, ${\pmb k}_0 \buildrel \Delta \over =
[\begin{array}{*{20}c}{k_{10},}& \cdots &{k_{80}}\end{array}]^{\rm
T}$
为标称参数.可以看出,这不是一个对角优势系统,在预补偿矩阵作用下,标称系统变为对角阵$
G(s,{\pmb k}_0 )K_1  = \rm{diag}\left[{\displaystyle\frac{1}{{s +
1}}},{\displaystyle\frac{2}{{s + 2}}}\right]$,
从而是对角优势的.此时它的行Gershgorin带图变为一条曲线(Gershgorin带宽度为零).
但是当参数${\pmb k}$变化$({\pmb k} \ne {\pmb k}_0)$时, $G(s,{\pmb
k})K_1 $不再是对角的甚至不再是对角优势的.图2是参数${\pmb k}$在标称
值的20\%范围内发生随机变化时, $G(s,{\pmb k})K_1
$的一组Gershgorin带图,星号为各鲁棒Gershgorin圆的圆心.
从第二行Gershgorin带图可知,它已不是对角优势的了.实际上,经过多次仿真可以发现,随着参数的不同,
$G(s,{\pmb k})K_1 $的Gershgorin带图会发生很大的变化.图3是$G(s,{\pmb
k})K_1 $的鲁棒Gershgorin带宽度的近似估计.虚
线是文[10]中估计方法估计的宽度,实线是采用本文的估计方法得出的宽度.可以看出,本文的方
法有明显的改进.

\vskip3mm {\centering

\vbox{\centerline{\psfig{figure=fig21.eps,width=5cm}}}
\vbox{{\centerline{\psfig{figure=fig22.eps,width=5cm}}} \vskip2mm
{\small
图2\quad $G(s,{\pmb k})K_1 $的一组Gershgorin带图\\
Fig.\,2\quad One group of the Gershgorin bands of $G(s,{\pmb k})K_1$
}}}\vskip3mm

\vskip3mm {\centering

\vbox{\centerline{\psfig{figure=fig31.eps,width=6cm}}}
\vbox{{\centerline{\psfig{figure=fig32.eps,width=6cm}}}
\vskip2mm
{\small
图3\quad $G(s,{\pmb k})K_1 $的鲁棒Gershgorin带宽度估计\\
Fig.\,3\quad The estimation of width of robust Gershgorin bands of
$G(s,{\pmb k})K_1 $ }}} \vskip3mm

在估计出$G(s,{\pmb k})K_1
$的鲁棒Gershgorin带宽度后,就可以判断$G(s,{\pmb k})K_1
$是否为鲁棒对角优势的(在本例中, $G(s,{\pmb k})K_1
$显然不是对角优势的).然后或者重新设计$K_1 $,
或者应用RINA方法设计$K_2
(s)$或$F$,使闭环系统稳定,具体的设计方法参见文[5].

\section{结论}

在RINA方法的应用中, 鲁棒Gershgorin带的估计对闭环系统的性能影响很大.
本文给出了一个所需 条件更弱的鲁棒对角优势性引理, 并在此基础上,
推导了一种新的鲁棒Gershgorin带的估计方法. 新方法使用1-范数作度量,
不仅充分使用了矩阵元素间的互相关信息, 还使用了元素实部和虚部的
分量信息, 大大降低了估计的保守性. 不仅为设计控制器带来了方便,
还使得系统鲁棒稳定性估 计的保守性降低.

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\begin{biographynophoto}
\noindent{\hei 高大远}\quad
国防科技大学机电工程与自动化学院博士研究生.
主要研究方向为非线性控制, 神经网络控制, 飞行控制. \\E-mail:
[e-mail]

\noindent({\bf GAO Da-Yuan}\quad Ph.\,D. candidate at National
University of Defense Technology. His research interest covers
nonlinear control, neural networks control, and flight control.)
\end{biographynophoto}

\begin{biographynophoto}
\noindent{\hei 沈\ 辉}\quad 国防科技大学机电工程与自动化学院副教授.
主要研究方向包括机器人控制，神经网络控制与辨识 数字图像处理,
计算机视觉和脑功能成像分析. E-mail: [e-mail]

\noindent({\bf ShEN Hui}\quad Associate professor at National
University of Defense Technology. His research interest covers robot
control, control and identification using neural networks, digital
image processing, computer vision, and functional brain image
analysis.)
\end{biographynophoto}

\begin{biographynophoto}
\noindent{\hei 董国华}\quad 国防科技大学机电工程与自动化学院讲师.
主要研究方向包括神经信号处理, 非线性系统辨识与控制. E-mail:
[e-mail]

\noindent({\bf DONG Guo-Hua}\quad Lecturer at National University of
Defense Technology. His research interest covers neural signal
processing, nonlinear system identification and control.)
\end{biographynophoto}

\begin{biographynophoto}
\noindent{\hei 胡德文}\quad
国防科技大学机电工程与自动化学院教授,博士生导师,
国家杰出青年科学基金获得者. 研究领域包括系统辨识, 神经网络,
图象信号处理, 脑功能成像分析, 认知科学等. 本文通信作者. \\E-mail:
[e-mail]

\noindent({\bf HU De-Wen}\quad Professor at National University of
Defense Technology and he was awarded the National Distinguished
Young Scholar Fund of China. His research interest covers system
identification, neural networks, image information processing,
functional brain image analysis, and cognitive neuroscience.
Corresponding author of this paper.)
\end{biographynophoto}

\end{document}