2005-0896.tex

自动化学报的编辑软件

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\firstheadname{ACTA AUTOMATICA SINICA} % 首页页眉
\firstfootname{} % 首页页脚 $\copyright$ 2007 by {\sl Acta Automatica Sinica}. All rights reserved.
\headevenname{ACTA AUTOMATICA SINICA} % 其他页偶数页页眉
\headoddname{HOU Zhong-Sheng and XU Jian-Xin: A New Feedback-feedforward Configuration for the ILC\ldots}% 其他页奇数页页眉
\begin{document}

\title{A New Feedback-feedforward\\\vskip 0.1\baselineskip Configuration for the Iterative\\\vskip 0.1\baselineskip Learning Control of a Class of\\\vskip 0.1\baselineskip
Discrete-time Systems
\thanks{Received August 28, 2005; in revised form May 16, 2006}
\thanks{Supported by National Natural Science Foundation of P.\,R.\,China (60474038), Science Research
Foundation of Beijing Jiaotong University (2005$SM$005) and
Specialized Research Fund for the Doctoral Program of Higher
Education (20060004002)}
\thanks{
1. Advanced Control Systems Lab of the School of Electronics and
Information Engineering, Beijing Jiaotong University, Beijing 100044
P.\,R.\,China \quad 2. Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117576 }
\thanks{DOI: 10.1360/aas-007-0323}}

\author{{HOU Zhong-Sheng$^{1}$\qquad XU Jian-Xin$^{2}$}}

\abstract{ This paper presents a new feedback-feedforward
configuration for the iterative learning control (ILC) design with
feedback, which consists of a feedback and a feedforward component.
The feedback integral controller stabilizes the system, and takes
the dominant role during the operation, and the feedforward ILC
compensates for the repeatable nonlinear/unknown time-varying
dynamics and disturbances, thereby enhancing the performance
achieved by feedback control alone. As the most favorable point of
this control strategy, the feedforward ILC and the feedback control
can work either independently or jointly without making efforts to
reconfigurate or retune the feedforward/feedback gains. With
rigorous analysis, the proposed learning control scheme guarantees
the asymptotic convergences along the iteration axis. }

\keyword{Iterative learning control, feedforward control, feedback
control, nonlinear systems}
%\cncl{TPXXX.XX}
\email{[e-mail], [e-mail]}

\maketitle

\pagestyle{aasheadings}

%这一章为引言,无需写标题. We do not need a title for the introduction chapter.

Since iterative  learning control (ILC) was first proposed by [1] in
1984 for the control of a system that repeats the same task in a
finite interval, it has been extensively studied and significant
progress has been made in both theory and applications $^{[1-12]}$.
However, although sufficient conditions are given to guarantee the
convergence of the learning process, the trajectory error  is likely
to grow quite significantly before it converges to zero in the
process of learning, and the rate of the convergence is often slow.
These phenomena are owing to the fact that the control structure is
basically an open loop and this control structure alone does not
compensate for the output error in each trial. Therefore, the
performance in the early stages of learning can be bad for stable
plants, and even worse for unstable plants. The use of conventional
feedback controllers can help to solve overcome this kind of problem
in the transient stages of learning since they can compensate for
the control input to reduce the error.

The learning process of making advantage of the current feedback
error or the feedback configuration can be found in [8]. Reference
[8] proposed a model-based learning scheme for the robot
manipulators with feedback controllers, but without giving a
rigorous analysis for the convergence of the learning process. [9]
suggested an ILC scheme for a class of nonlinear systems with high
gain feedback PD controller, which update to the feedforward control
input with the feedback controller output. In [7], a discrete ILC
was proposed for discrete-time nonlinear time-varying systems with
initial state error, input disturbance, and output measurement
noise. In [10], a D-type ILC was done in a feedback configuration.
The learning process was performed in the feedforward input updated
by the previous plant input and the derivative of the previous
error. The rapid convergence was shown either by technical
proof$^{[7]}$ or by simulation$^{[10]}$ as compared with the
traditional feedforward learning.

Some other ILC designs in a feedback configuration could be found in
[11,13-14]. However, they still did not show, in a mathematics
meaning, what the clear functions of feedforward ILC algorithm and
feedback controller in the composite feedforward/feedback
configuration  are. So far, almost all the descriptions of the
functions for the feedforward or feedback component in a combined
configuration are based on the qualitative induction rather than the
theoretical analysis. This is certainly an important issue both in
theory and practical applications.

In this paper, we propose an ILC control scheme for a class of
discrete-time nonlinear systems over a finite time interval in a new
feedforward-feedback configuration. The function of the feedforward,
acting as a compliable component in this configuration, rejecting
exogenous disturbances, and compensating for the nonlinear and and
time-varying plant, is to meet the high tracking performance
requirement. Meanwhile the feedback component serves as the main
controller.

\section{Problem statement}
\subsection{Problem formulation}
The following discrete-time nonlinear time-varying systems are
considered.
\begin{align}
{\pmb x}_n(k+1)&={\pmb f}({\pmb x}_n(k),{\pmb y}_n(k),k)\\
{\pmb y}_n(k+1)&={\pmb g}({\pmb x}_n(k),{\pmb y}_n(k),{\pmb
u}_n(k),k)
\end{align}
where $n$ and $k$ are the iteration index and discrete-time,
respectively. For simplicity in the following discussion, let ${\pmb
x}_n(k)\in {\rm\bf R}^p$, ${\pmb y}_n(k)\in {\rm\bf R}^p$, and
${\pmb u}_n(k)\in {\rm\bf R}^p$ for all $k\in [0,K]$ and $n\in
[1,\infty)$. ${\pmb u}_n(k)$ is the control variable. ${\pmb
f}(\cdot)$ and $g(\cdot)$ are nonlinear functions.

{\bf Assumption 1.} Functions ${\pmb f}$ and ${\pmb g}$ are
uniformly globally Lipschitz with respect to ${\pmb x}$, ${\pmb y}$
and ${\pmb u}$ for $k\in [0,1,\ldots,K]$ on a compact set
$\mathit{\Omega}\in {\rm\bf R}^p\times {\rm\bf R}^p\times [1,K]$ or
$\mathit{\Omega}\in {\rm\bf R}^p\times {\rm\bf R}^p\times {\rm\bf
R}^p\times [1,K]$, i.e.,
\begin{align}
\begin{split}
\|{\pmb f}&({\pmb x}_1(k),{\pmb y}_1(k),k)-{\pmb f}({\pmb
x}_2(k),{\pmb y}_2(k),k)\|\leq\\& k_{fx}\|{\pmb x}_1(k)-{\pmb
x}_2(k)\|+k_{fy}\|{\pmb y}_1(k)-{\pmb y}_2(k)\|
\end{split}
\\
\begin{split}
\|{\pmb g}&({\pmb x}_1(k),{\pmb y}_1(k),{\pmb u}_1(k),k)-{\pmb
g}({\pmb x}_2(k),{\pmb y}_2(k),{\pmb u}_2(k),k)\|\leq\\&
k_{gx}\|{\pmb x}_1(k)-{\pmb x}_2(k)\|+k_{gy}\|{\pmb y}_1(k)-{\pmb
y}_2(k)\|+\\&k_{gu}\|{\pmb u}_1(k)-{\pmb u}_2(k)\|
\end{split}
\end{align}
where $k_{fx}$, $k_{fy}$, $k_{gx}$, $k_{gy}$, $k_{gu}$ are the
Lipschitz constants.

Furthermore $g_{x}=\displaystyle\frac{\partial {\pmb g}(({\pmb
x},{\pmb y}, {\pmb u}, k)}{\partial {\pmb x}}$,
$g_{y}=\displaystyle\frac{\partial {\pmb g}({\pmb x},{\pmb y}, {\pmb
u}, k)}{\partial {\pmb y}}$, $g_{u}=\displaystyle\frac{\partial
{\pmb g}({\pmb x},{\pmb y}, {\pmb u}, k)}{\partial {\pmb u}}$ are
uniformly bounded for all $(\cdot,\cdot,\cdot,\cdot)\in
\mathit{\Omega}$. And there exist constants $\alpha_{1}$ and
$\alpha_{2}$ such that $0\leq\alpha_{1}\leq g_{u}\leq\alpha_{2}$.

{\bf Assumption 2.} The re-initialization condition is satisfied
throughout the repeated iterations, i.e., ${\pmb x_{n}}(0)={\pmb
x_{d}}(0), \quad  {\pmb y_{n}}(0)={\pmb y_{d}}(0), \quad  \forall
n.$

{\bf Assumption 3.} There exists a control input ${\pmb u_{d}}(k)$
that can exactly drive the system output to track the desired
trajectory ${\pmb y_{d}}(k)$ for the systems (1) and (1$'$) on the
finite time interval.

The control objective is to design an iterative learning controller
${\pmb u_{n}}(k)$ such that the output tracking error between the
desired output trajectory ${\pmb y_{d}}(k)$ and the system output
${\pmb y_{n}}(k)$ is within an error bound, which can be
predetermined.

\subsection{ILC controller add-on to feedback controller}
The discrete-time ILC controller is constructed as follows
\begin{align}
{\pmb u}_n(k)&={\pmb u_{n}^{f}}(k)+{\pmb u_{n}^{b}}(k)\\
{\pmb u_{n}^{f}}(k)&={\pmb u_{n-1}^{f}}(k)+\beta{\pmb
e_{n-1}}(k+1)\\
{\pmb u_{n}^{f}}(0)&=\alpha{\pmb e_{n}}(0) \quad  {\pmb
u_{n}^{b}}(0) \quad {\rm given} \quad {\rm if} \quad k=0\\
{\pmb u_{n}^{b}}(k)&={\pmb u_{n}}(k-1)+\alpha{\pmb e_{n}}(k),\quad
{\rm if} \quad k>0
\end{align}
where $n$ indicates the iteration number, and $\beta$ and $\alpha$
are the iterative learning gain matrix and the feedback gain matrix,
respectively and ${\pmb e_{n}}(k+1)={\pmb y_{d}}(k+1)-{\pmb
y_{n}}(k+1)$.

\section{Convergence Analysis}
Two cases are considered in this paragraph. First, we consider the
convergence analysis of the pure ILC control for system (1), using
the iterative learning controller (3$'$), and then we consider the
ILC controller with feedback controller.

{\bf Theorem 1.} Under Assumptions 1-3, choose the learning gain
matrix $\beta$ such that $\|1-\beta g_{u}\|<1$, for $\forall
g_{u}\in [\alpha_{1},\alpha_{2}]$, in the learning law (3$'$). Then
the output of system (1) controlled by the learning controller
(3$'$) will lead to $\lim_{n \to \infty}\|{\pmb u_{n}^{b}}(k)-{\pmb
u_{d}}(k)\|_{\lambda} \leq\sigma$, $\lim_{n \to \infty}\|{\pmb
y_{n}}(k)-{\pmb y_{d}}(k)\|_{\lambda}\leq\sigma$ for some suitably
defined constant $\sigma>0$ that depends on $\|\delta{\pmb
x_{n}}(0)\|$ and $\|{\pmb e_{n}}(0)\|$. In the sequel, we have
$\lim_{n \to \infty}\|{\pmb u_{n}^{b}}(k)-{\pmb
u_{d}}(k)\|_{\lambda}=0$, $\lim_{n \to \infty}\|{\pmb
y_{n}}(k)-{\pmb y_{d}}(k)\|_{\lambda}=0$, if $\|\delta{\pmb
x_{n}}(0)\|=0$ and $\|{\pmb e_{n}}(0)\|=0$.

{\bf Proof.} Similar to that of Theorem 2.

{\bf Theorem 2.} Under Assumptions 1-3, choose the learning gain
matrix $\beta$ such that $\|1-\beta g_{u}\|<1$, for $\forall
g_{u}\in [\alpha_{1},\alpha_{2}]$, in the learning law (3). Then the
output of system (1) controlled by the learning controller with a
feedback control (3)-(3$'''$) will lead to $\lim_{n \to
\infty}\|{\pmb u_{n}^{b}}(k)-{\pmb u_{d}}(k)\|_{\lambda}
\leq\sigma$, $\lim_{n \to \infty}\|{\pmb y_{n}}(k)-{\pmb
y_{d}}(k)\|_{\lambda}\leq\sigma$ for some suitably defined constant
$\sigma>0$ which is a class-K function of $\|\delta{\pmb
x_{n}}(0)\|$, $\|{\pmb e_{n}}(0)\|$, $\|\delta{\pmb u_{n}^{b}}(0)\|$
and $M$. If they all equal to zero, then $\lim_{n \to \infty}\|{\pmb
u_{n}^{b}}(k)-{\pmb u_{d}}(k)\|_{\lambda}=0$, $\lim_{n \to
\infty}\|{\pmb y_{n}}(k)-{\pmb y_{d}}(k)\|_{\lambda}=0$.

{\bf Proof.} Let $\delta{\pmb x_{n}}(k)={\pmb x_{d}}(k)-{\pmb
x_{n}}(k)$, $\delta{\pmb u_{n}}(k)={\pmb u_{d}}(k)-{\pmb u_{n}}(k)$.
Then from (3), we have
\begin{align}
&{\pmb u_{n+1}^{f}}(k)={\pmb u_{n}^{f}}(k)+\beta{\pmb e_{n}}(k+1), \\
&\delta{\pmb u_{n}}(k)={\pmb u_{d}}(k)-{\pmb u_{n}^{f}}(k)-{\pmb
u_{n}^{b}}(k) =\delta{\pmb u_{n}^{b}}(k)-{\pmb u_{n}^{f}}(k)
\end{align}
Using the differential mean value theorem, we have
\begin{equation}
\begin{split}
{\pmb e_{n}}&(k+1)={\pmb y_{d}}(k+1)-{\pmb y_{n}}(k+1)=\\&{\pmb
g}({\pmb x_{d}}(k), {\pmb y_{d}}(k), {\pmb u_{d}}(k), k)-\\&{\pmb
g}({\pmb x_{n}}(k), {\pmb y_{n}}(k), {\pmb u_{n}}(k), k)=\\&{\pmb
g_{x}}({\pmb \xi_{n}})\delta{\pmb x_{n}}(k) +{\pmb g_{y}}({\pmb
\xi_{n}}){\pmb e_{n}}(k) +{\pmb g_{u}}({\pmb \xi_{n}})\delta{\pmb
u_{n}}(k) =\\&{\pmb g_{x}}({\pmb \xi_{n}})\delta{\pmb x_{n}}(k)
+{\pmb g_{y}}({\pmb \xi_{n}}){\pmb e_{n}}(k) +{\pmb g_{u}}({\pmb
\xi_{n}})\delta{\pmb u_{n}^{b}}(k) -\\&{\pmb g_{u}}({\pmb
\xi_{n}}){\pmb u_{n}^{f}}(k),
\end{split}
\end{equation}
where ${\pmb \xi_{n}}=[({\pmb x_{n}}(k)+\tau\delta{\pmb
x_{n}}(k))^{\rm T}, ({\pmb y_{n}}(k)+\tau{\pmb e_{n}}(k))^{\rm T},
({\pmb u_{n}}(k)+\tau\delta{\pmb u_{n}}(k))^{\rm T},k]^{\rm T}$,
$\tau\in [0,1]$.

Inserting (5) into (4) gives
\begin{equation}
\begin{split}
{\pmb u_{n+1}^{f}}(k)&=(1-\beta{\pmb g_{u}}({\pmb \xi_{n}}){\pmb
u_{n}^{f}}(k) +\beta{\pmb g_{x}}({\pmb \xi_{n}})\delta{\pmb
x_{n}}(k) +\\&\beta{\pmb g_{y}}({\pmb \xi_{n}}){\pmb e_{n}}(k)
+\beta{\pmb g_{u}}({\pmb \xi_{n}})\delta{\pmb u_{n}^{b}}(k).
\end{split}
\end{equation}

Taking norm on both sides of (6) yields
\begin{equation}
\begin{split}
\|{\pmb u_{n+1}^{f}}(k)\|&=\|(1-\beta{\pmb g_{u}}({\pmb
\xi_{n}})\|\|{\pmb u_{n}^{f}}(k)\| +\sigma_{1}(\|\delta{\pmb
x_{n}}(k)\|+\\&\|{\pmb e_{n}}(k)\|+\|\delta{\pmb u_{n}^{b}}(k)\|)
\end{split}
\end{equation}
with $\sigma_{1}=\sup_{k\in [1,K]}(\|\beta{\pmb g_{x}}({\pmb