2005-0896.tex

自动化学报的编辑软件

\xi_{n}})\|, \|\beta {\pmb g_{y}}({\pmb \xi_{n}})\|, \|\beta{\pmb
g_{u}}({\pmb \xi_{n}})\|)$.

From (1) and (2), we can get
\begin{equation}
\|\delta{\pmb x_{n}}(k)\|\leq k_{fx}\|\delta{\pmb
x_{n}}(k-1)\|+k_{fy}\|{\pmb e_{n}}(k-1)\|
\end{equation}

From (2$'$) and (4$'$), we have
\begin{equation}
\begin{split}
\|{\pmb e_{n}}(k)\|&\leq k_{gx}\|\delta{\pmb
x_{n}}(k-1)\|+k_{gy}\|{\pmb e_{n}}(k-1)\| +\\&k_{gu}\|\delta{\pmb
u_{n}}(k-1)\| \leq\\& k_{gx}\|\delta{\pmb
x_{n}}(k-1)\|+k_{gy}\|{\pmb e_{n}}(k-1)\| +\\&k_{gu}(\|\delta{\pmb
u_{n}^{b}}(k-1)\|+\|{\pmb u_{n}^{f}}(k-1)\|
\end{split}
\end{equation}
By (9), we get
\begin{equation}
\begin{split}
\|\delta{\pmb u_{n}^{b}}&(k)\|=\|{\pmb u_{d}}(k)-{\pmb
u_{n}}(k-1)-\alpha{\pmb e_{n}}(k)\|=\\&\|{\pmb u_{d}}(k)-{\pmb
u_{n}^{f}}(k-1)-{\pmb u_{n}^{b}}(k-1)-\alpha{\pmb e_{n}}(k)\|\leq
\\&\|\delta{\pmb u_{n}^{b}}(k-1)\|+\|{\pmb
u_{n}^{f}}(k-1)\|+\\&\alpha\|{\pmb e_{n}}(k)\|+\|\triangle{\pmb
u_{d}}(k)\| \leq\\&\|\delta{\pmb u_{n}^{b}}(k-1)\|+\|{\pmb
u_{n}^{f}}(k-1)\|+\\&\alpha[k_{gx}\|\delta{\pmb x_{n}}(k-1)\|
+k_{gy}\|{\pmb e_{n}}(k-1)\| +\\&k_{gu}\|\delta{\pmb
u_{n}^{b}}(k-1)\|+k_{gu}\|{\pmb
u_{n}^{f}}(k-1)\|]\\&+\|\triangle{\pmb u_{d}}(k)\|=\\&(1+\alpha
k_{gu})(\|\delta{\pmb u_{n}^{b}}(k-1)\|+\|{\pmb
u_{n}^{f}}(k-1)\|)+\\&\alpha k_{gx}\|\delta{\pmb
x_{n}}(k-1)\|+\\&k_{gy}\|{\pmb e_{n}}(k-1)\|+\|\triangle{\pmb
u_{d}}(k)\|
\end{split}
\end{equation}
where $\triangle{\pmb u_{d}}(k)={\pmb u_{d}}(k)-{\pmb u_{d}}(k-1)$

Adding (8), (9), and (10), using Assumption 2, gives
\begin{equation}
\begin{split}
(\|\delta{\pmb x_{n}}&(k)\|+\|{\pmb e_{n}}(k)\|+\|\delta{\pmb
u_{n}^{b}}(k)\|)\leq\\& \sigma_{2}(\|\delta{\pmb
x_{n}}(k-1)\|+\\&\|{\pmb e_{n}}(k-1)\|+\|\delta{\pmb
u_{n}^{b}}(k-1)\|)+\\&(k_{gu}+1+\alpha k_{gu})\|{\pmb
u_{n}^{f}}(k-1)\|+\\&\|\triangle{\pmb u_{d}}(k)\|
\leq\\&\ldots\leq\sigma_{2}^{k}(\|\delta{\pmb x_{n}}(0)\|+\|{\pmb
e_{n}}(0)\| +\|\delta{\pmb u_{n}^{b}}(0)\|) +\\&(k_{gu}+1+\alpha
k_{gu})\sum_{i=0}^{k-1}\sigma_{2}^{k-i-1} \|{\pmb
u_{n}^{f}}(i)\|+\\&\sum_{i=1}^{k}\sigma_{2}^{k-i}\|\triangle{\pmb
u_{d}}(i)\|\leq\\&\sigma_{2}^{k}(\|\delta{\pmb x_{n}}(0)\|+\|{\pmb
e_{n}}(0)\| +\|\delta{\pmb u_{n}^{b}}(0)\|)+\\&(k_{gu}+1+\alpha
k_{gu})\sum_{i=0}^{k-1}\sigma_{2}^{k-i-1} \|{\pmb u_{n}^{f}}(i)\|
+\\&M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}
\end{split}
\end{equation}
where $\sigma_{2}=\sup_{k\in [0,K]}\{(k_{fx}+k_{gx}+\alpha k_{gx}),
(k_{fy}+k_{gy}+\alpha k_{gy}), (k_{gu}+1+\alpha k_{gu})\},$ and
$M=max_{k\in [0,K]}\|\triangle{\pmb u_{d}}(k)\|.$ Without loss of
generality, we will assume $\sigma_{2}>1$ for the following
discussions.

Multiplying $\sigma_{2}^{-\lambda k}$ on both sides of (11) on
interval $[0,K]$ and then taking supreme norm, we have
\begin{equation}
\begin{split}
(\|\delta{\pmb x_{n}}&(k)\|_{\lambda}+\|{\pmb
e_{n}}(k)\|_{\lambda}+\|\delta{\pmb
u_{n}^{b}}(k)\|_{\lambda})\leq\\&(\|\delta{\pmb x_{n}}(0)\|+\|{\pmb
e_{n}}(0)\| +\|\delta{\pmb u_{n}^{b}}(0)\|)+\\&(k_{gu}+1+\alpha
k_{gu})\|{\pmb
u_{n}^{f}}(k)\|_{\lambda}\frac{1-\sigma_{2}^{-(\lambda-1)K}}
{\sigma_{2}^{\lambda}-{\sigma_{2}}}+\\&M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}
\end{split}
\end{equation}

Inserting (12) into (7) gives
\begin{equation}
\begin{split}
\|{\pmb u_{n+1}^{f}}&(k)\|_{\lambda}\leq\|(1-\beta g_{u}({\pmb
\xi_{n}}))\| \|{\pmb
u_{n}^{f}}(k)\|_{\lambda}+\\&\sigma_{1}(\|\delta{\pmb x_{n}}(0)\|+
\|{\pmb e_{n}}(0)\|+\|\delta{\pmb u_{n}^{b}}(0)\|)
+\\&\sigma_{1}(k_{gu}+1+\alpha k_{gu})\|{\pmb
u_{n}^{f}}(k)\|_{\lambda}\frac{1-\sigma_{2}^{-(\lambda-1)K}}
{\sigma_{2}^{\lambda}-{\sigma_{2}}}+\\&\sigma_{1}M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}
\end{split}
\end{equation}

Equation (13) can be rewritten as
\begin{equation}
\begin{split}
\|{\pmb u_{n+1}^{f}}&(k)\|_{\lambda}\leq[\|(1-\beta g_{u}({\pmb
\xi_{n}}))\| +\\&\sigma_{1}(k_{gu}+1+\alpha
k_{gu})\frac{1-\sigma_{2}^{-(\lambda-1)K}}
{\sigma_{2}^{\lambda}-{\sigma_{2}}}]\|{\pmb
u_{n}^{f}}(k)\|_{\lambda}+\\&\sigma_{1}(\|\delta{\pmb
x_{n}}(0)\|+\|{\pmb e_{n}}(0)\|+\|\delta{\pmb u_{n}^{b}}(0)\|)
+\\&\sigma_{1}M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}
\end{split}
\end{equation}
Choosing a sufficiently large constant $\lambda$ such that the
following inequality holds when contractive condition $\|(1-\beta
g_{u}({\pmb \xi_{n}}))\|<1$ is satisfied,
\begin{equation}
\begin{split}
\|(1-\beta g_{u}({\pmb \xi_{n}}))\|&+\sigma_{1}(k_{gu}+1+\alpha
k_{gu})
\frac{1-\sigma_{2}^{-(\lambda-1)K}}{\sigma_{2}^{\lambda}-{\sigma_{2}}}\\&\leq\rho<1
\end{split}
\end{equation}
then (14) gives
\begin{equation}
\|{\pmb u_{n+1}^{f}}(k)\|_{\lambda}\leq\rho\|{\pmb
u_{n}^{f}}(k)\|_{\lambda}+\varepsilon,
\end{equation}
where $\varepsilon=\sigma_{1}(\|\delta{\pmb x_{n}}(0)\|+\|{\pmb
e_{n}}(0)\| +\|\delta{\pmb
u_{n}^{b}}(0)\|)+\sigma_{1}M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}.$

Equation (16) means
\begin{equation}
\lim_{n \to \infty}\|{\pmb
u_{n+1}^{f}}(k)\|_{\lambda}\leq\frac{\varepsilon}{1-\rho}
\end{equation}

From (12), we can obtain
\begin{equation}
\begin{split}
\lim_{n \to \infty}&(\|\delta{\pmb x_{n}}(k)\|_{\lambda}+\|{\pmb
e_{n}}(k)\|_{\lambda} +\|\delta{\pmb
u_{n}^{b}}(k)\|_{\lambda})\leq\\&(\|\delta{\pmb x_{n}}(0)\|+\|{\pmb
e_{n}}(0)\| +\|\delta{\pmb u_{n}^{b}}(0)\|) +\\&(k_{gu}+1+\alpha
k_{gu})\frac{1-\sigma_{2}^{-(\lambda-1)K}}
{\sigma_{2}^{\lambda}-{\sigma_{2}}}\lim_{n \to \infty}\|{\pmb
u_{n}^{f}}(k)\|_{\lambda}+\\&M\frac{\sigma_{2}^{K}-1}{\sigma_{2}-1}
\end{split}
\end{equation}
Then from (17), (18), we can reach the conclusion of this
theorem.\hfill$\square$

{\bf Remark 1.} This theorem reveals that the ILC component will
play a complementary role in control design, while the feedback
component plays the dominant role.

{\bf Remark 2.} Note that the learning controller design is
independent of the feedback controller. Hence the closed-loop
characteristics will not be changed by the addition of the ILC part.
Thus, whenever necessary, we can simply switch off either of the
control module and the remaining one will still work well.

{\bf Remark 3.} By Theorem 2, $\|{\pmb u_{n}^{f}}(k)\|_{\lambda} \to
0$ when the convergence is obtained. This implies that the control
system will be dominated by the feedback controller, and the ILC
feedforward is equivalently off.

{\bf Remark 4.} The underlying idea of this new feedback/
feedforward configuration is to learn and reject the repeatable and
non-repeatable uncertainties. Learning mechanism is designed to
identify all those repeatable components and leave the remaining
unknown iteration-dependent components to the feedback control
scheme.

{\bf Remark 5.} The effectiveness, and the advantages, compared with
those of [7], of the proposed iterative learning controller and the
ILC add-on to the feedback controller have been verified through
intensive simulations. Here the results are omited just due to the
limitation of paper length.

\section{Conclusion}
A discrete iterative learning controller with a new
feedforward-feedback configuration in which the iterative learning
control is add-on to the feedback controller is proposed for the
discrete-time nonlinear time-varying systems with initial state
error and initial output error. A systematic approach is developed
to analyze the convergence of the learning system. It is shown that
the feedforward ILC component add-on to the feedback controller does
not change any closed loop characteristics and the feedback
controller still play the dominant role in the combined control
strategy. Furthermore, it is noted that the learning controller
design is completely decoupled from the feedback controller. The
feedback controller and ILC can work concurrently as two independent
modules without interfering with each others. Whenever necessary, we
can simply switch off one control module and the remaining one will
still work well. It is a perfect modularized fashion in control
system design.

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\begin{biographynophoto}
\noindent{\bf HOU Zhong-Sheng}\quad Received his bachelor and master
degrees in applied mathematics from Jilin University of Technology
in 1983 and 1988 respectively, and the Ph.\,D. degree in control
theory from Northeastern University in 1994. He was a postdoctoral
fellow at the Harbin Institute of Technology from 1995 to 1997, and
a visiting scholar in the Yale University, USA, from 2002 to 2003.
In 1997, he joined the Beijing Jiaotong University and is currently
a director and professor in the Advanced Control Systems Laboratory
of the School of Electronics and Information Engineering. His
research interest covers model-free adaptive control, learning
control, and intelligent transportation systems. Corresponding
author of this paper. \\E-mail: [e-mail]
\end{biographynophoto}

\begin{biographynophoto}
\noindent{\bf XU Jian-Xin} \quad Received his master and Ph.\,D.
degrees from University of Tokyo, Japan in 1986 and 1989,
respectively. He is currently an associate professor in the
Department of Electrical Engineering at the National University of
Singapore. His research interest lies learning control and variable
structure control and so on. E-mail: [e-mail]
\end{biographynophoto}

\end{document}