自动化学报  2017, Vol. 43 Issue (8): 1470-1477   PDF    
迭代学习控制的参考信号初始修正方法
严求真1, 孙明轩2, 蔡建平3     
1. 浙江水利水电学院信息工程学院 杭州 310018;
2. 浙江工业大学信息工程学院 杭州 310023;
3. 浙江水利水电学院应用数学研究所 杭州 310018
摘要: 针对一类非参数不确定系统,提出状态受限迭代学习控制的参考信号初始修正方法,以解决任意初态下的状态受限轨迹跟踪问题.通过构造修正参考信号,利用一种新型的障碍Lyapunov函数设计迭代学习控制系统,采用鲁棒方法与学习方法相结合的策略处理非参数不确定性,经过足够多次迭代后,可实现系统状态在整个作业区间上对修正参考信号的零误差跟踪,以及在预设作业区间上对参考信号的零误差跟踪.同时,将滤波误差约束于预设的界内,并由此实现对系统状态在各次迭代运行过程中的约束.仿真结果表明了本文所提控制方法的有效性.
关键词: 迭代学习控制     初值问题     非参数不确定性     障碍Lyapunov函数    
Reference-signal Rectifying Method of Iterative Learning Control
YAN Qiu-Zhen1, SUN Ming-Xuan2, CAI Jian-Ping3     
1. College of Information Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018;
2. College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023;
3. Institute of Applied Mathematics, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018
Manuscript received : March 26, 2016, accepted: September 11, 2016.
Foundation Item: Supported by National Natural Science Foundation of China (61374103, 61573320, 61573322), Science and Technology Project of Zhejiang Province (2016C32093, 2017C33155), Scientiflc Research Project of Education Department of Zhejiang Province (Y201635861)
Author brief: YAN Qiu-Zhen  Lecture at the College of Information Engineering, Zhejiang University of Water Resources and Electric Power. His main research interest is learning control;
CAI Jian-Ping  Associate professor at the Institute of Applied Mathematics, Zhejiang University of Water Resources and Electric Power. His main research interest is adaptive control
Corresponding author. SUN Ming-Xuan  Professor at the College of Information Engineering, Zhejiang University of Technology. His main research interest is learning control. Corresponding author of this paper
Recommended by Associate Editor WANG Cong
Abstract: This paper presents a reference-signal rectifying method of iterative learning control to address the trajectory-tracking problem for a class of state-constrained uncertain systems, in the presence of arbitrary initial states. For design of the iterative learning control scheme, a rectified reference signal is constructed and a new type of barrier Lyapunov function is used. In order to deal with the nonparametric uncertainties, a robust learning approach is applied. It is shown that the closed-loop system's state follows the rectified reference signal perfectly over the entire time interval as iteration increases. In turn, the system state tracks the reference signal on the specified interval. During each iteration, the filtering-error is constrained in the pre-specified region, and the system state is thus constrained. Numerical results are presented to demonstrate the effectiveness of the learning control scheme.
Key words: Iterative learning control     initial condition problem     nonparametric uncertainties     barrier Lyapunov function    

在工业实际中, 存在着大量在有限区间上重复运行的系统或设备, 例如机械臂、磁盘驱动器和逆变电路等.问世于上世纪80年代的迭代学习控制技术, 适合为这类重复作业对象设计轨迹跟踪控制器.采用迭代学习控制技术设计控制系统时, 不需对受控对象的模型精确已知, 其策略是利用跟踪误差不断修正控制输入, 经过有限次迭代, 使得系统状态或系统输出以预设精度跟踪期望轨迹[1].通过三十余年的发展, 学习控制的理论逐渐丰富与完善, 并应用于诸多工业控制场合[2-6].

基于Lyapunov方法设计学习控制系统是当前迭代学习控制领域的热点之一[2-3].采用学习方法, 可以对控制系统中的各类参数不确定性进行估计.文献[7]利用微分学习方法处理常参数不确定性, 文献[3]在构造Lyapunov泛函的基础上, 设计差分学习律估计时变未知参数.文献[8]采用由微分与差分相混合的学习方式估计未知常参数.非线性参数不确定性[9]与参数随迭代次数变化[10]问题也在人们的考虑之列.到目前为止, 研究非参数不确定系统学习控制方法的文献数量还较少.文献[11]采用鲁棒方法处理非参数不确定性, 即利用界函数设计反馈项对其予以补偿.文献[12-14]采用鲁棒方法与学习方法相结合的策略处理非参数不确定性.此外, 也可采用傅立叶级数等逼近工具处理非参数不确定性[15].

常规迭代学习控制算法常假设迭代误差的初值为零, 这样, 经过足够多次迭代后, 可以实现整个作业区间上的零误差轨迹跟踪.但在现实中, 受复位条件所限, 上述假设很难满足.为拓宽迭代学习控制技术在实际中的应用范围, 有必要研究可在任意初始误差情形下实施的学习控制算法.在利用Lyapunov方法为连续系统设计控制器时, 常见的初值问题解决方案有时变边界层、误差跟踪和初始修正等.文献[16]给出基于时变边界层的模糊学习控制方法, 闭环系统的滤波误差可在足够多次迭代后, 收敛至与迭代初值相关的时变死区.文献[17]研究非线性系统的误差跟踪学习方法, 并将其与参考信号初始修正方法进行对比.早在上世纪90年代, 人们在设计压缩映射学习控制系统时, 就采用初始修正方法解决学习控制的初值问题[18].近年来, 初始修正方法重新受到人们的关注[17, 19].文献[20]考虑MIMO参数不确定系统的参考信号初始修正问题.文献[21-22]针对具有任意初始误差的非参数不确定系统, 给出的自适应学习控制算法适用于固定初始误差情形.

最近, 构造障碍Lyapunov函数设计状态/输出受限学习控制系统, 引起了人们的关注.采用这种方法设计控制器, 可以对迭代学习过程中的状态/输出进行约束, 从而增强系统的鲁棒性.文献[23-24]分别考虑输出受限情形和状态受限情形下的重复学习控制算法.文献[25]给出一种形式简单的二次分式型障碍Lyapunov函数构造方案, 在迭代学习过程中实现对系统状态的整体约束.文献[26]研究迭代误差初值任意情形下的状态受限误差跟踪学习控制方案, 同时解决初值问题和状态约束问题.文献[27]利用时变神经网络估计不确定性, 结合时变边界层方法设计状态受限学习控制系统.

本文研究非参数不确定学习控制系统的初值问题及状态受限问题, 在利用参考信号初始修正方法解决初值问题的同时, 采用障碍Lyapunov函数设计学习控制系统, 对各次迭代学习过程中的系统状态予以约束.为了提高利用障碍Lyapunov函数设计学习控制器时的便捷性, 本文构造了一种新型的障碍Lyapunov函数, 改进了已有的同类设计方案.经过足够多次迭代后, 藉由系统状态对修正参考信号在整个作业区间的完全跟踪, 获得系统状态对参考信号在预设部分作业区间上的完全跟踪.在各次迭代过程中, 闭环系统中的滤波误差被约束于预设的界内, 由此实现对系统状态在各次迭代运行过程中的约束.文中算法采用鲁棒学习控制方法处理非参数不确定性, 并根据学习方法估计鲁棒项的增益系数, 克服了以往同类算法中在某些场合应用时, 可能出现的增益过大之不足.

1 问题的提出

考虑在有限时间$[0, T]$上迭代运行的非参数不确定系统:

$ \begin{align} \left\{ {{\begin{array}{*{20}c} {\dot{x}_{i, k}=x_{i+1, k}, ~~~ i=1, 2, \cdots, n-1 } \hfill \\ {\dot{x}_{n, k}=f({\pmb x}_k, t)+g({\pmb x}_k, t)u_k ~~~ } \hfill \\ \end{array} }} \right. \label{sys} \end{align} $ (1)

式中, $k\in\{0, 1, 2, \cdots\}$为重复作业次数, $t\in [0, T]$, ${\pmb{x}}_k=[x_{1, k}, \cdots, x_{n, k}]^{\rm T}\in {\bf R}^n$为可量测的系统状态, $f(\pmb{x}_k, t)\in {\bf R}$$g(\pmb{x}_k, t)\in {\bf R}$为未知的光滑函数, 分别满足假设1和假设2. 参考信号为$\pmb x_d=[x_d, \dot{x}_d, \ddot{x}_d, \cdots, x_d^{(n-1)}]^{\rm T}$, $x_d^{(n)}$存在.系统初值与参考信号在起始时刻的取值不等, 即$\pmb x_k(0)\neq \pmb x_d(0)$, 不满足常规迭代学习控制所要求的初值条件.存在$u_d$, 满足

$ \begin{align} x_d^{(n)}=f({\pmb x}_d, t)+g({\pmb x}_d, t)u_d \end{align} $ (2)

不失一般性[23], 系统(1) 中的不确定性满足下述假设:

假设1. 对于$\forall {\pmb \xi}_1\in {\bf R}^n, \forall {\pmb \xi}_2\in {\bf R}^n$, 函数$f(\cdot, \cdot)$$g(\cdot, \cdot)$分别满足

$ \begin{align} &|f(\pmb {\xi}_1, t)-f( {\pmb \xi}_2, t)| \leq \alpha_f( { \pmb \xi}_1, {\pmb \xi}_2, t) \| \pmb{\xi}_1-\pmb{\xi}_2\| \nonumber \end{align} $

$ \begin{align} &|g(\pmb{\xi}_1, t)-g( \pmb {\xi}_2, t)| \leq \alpha_g( {\pmb \xi}_1, {\pmb \xi}_2, t) \| {\pmb \xi}_1-{\pmb \xi}_2\|\nonumber \end{align} $

其中, $\alpha_f(\cdot, \cdot, \cdot)$$\alpha_g(\cdot, \cdot, \cdot)$为非负连续函数.

假设2. 存在已知连续函数$g_m(\pmb{x}_k, t)$, 满足$0<{\underline{g}}< g_m(\pmb{x}_k, t)\leq g(\pmb{x}_k, t)$, $\underline{g}$为一常数, 但具体大小不需已知.

本文的控制目标是设计$u_k$, 实现$\pmb x_k$$\pmb x_d$$[t_\delta, T]$上的完全跟踪$(0<t_\delta<T)$.为了实现该任务, 下文构造修正参考信号.

2 修正参考信号的构造

为了克服由$\pmb x_k(0)\neq \pmb x_d(0)$导致Lyapunov函数构造方面的障碍, 此处拟构造合适的修正参考信号

$ \begin{eqnarray} \pmb x_k^*(t)=[x_{1, k}^*, \cdots, x_{n, k}^*]^{\rm T}= \left\{ \begin{aligned} &\ \pmb x_{dk}(t), \ \ \ 0 \leq t \leq t_\delta\\ &\ \pmb x_d(t), \ \ \ \ t_\delta < t \leq T \end{aligned} \right. \label{01csgdqxsys1} \end{eqnarray} $ (3)

其中, $\pmb x_{dk}(t)=[x_{1dk}(t), x_{2dk}(t), \cdots, x_{ndk}(t)]^{\rm T}$为衔接于系统状态初值$\pmb x_k(0)$与预设接入点$\pmb x_d(t_\delta)$之间的光滑接入曲线, 满足: 1) $ \pmb x_{dk}(0)=\pmb x_k(0);$ 2) $ \pmb x_{dk}(t_\delta)=\pmb x_d(t_\delta);$ 3) $ \dot{\pmb x}_k^*(t)$$t=t_\delta$处存在.由性质1) $\sim$3) 知, $\pmb x_k^*(t)$$[0, T]$上是光滑可导的.记产生$ \pmb x_k^*(t)$的控制量为$u_k^*$, 满足

$ \begin{align} \left\{ {{\begin{array}{*{20}c} {\dot{x}_{i, k}^*=x_{i+1, k}^*, ~~~ i=1, 2, \cdots, n-1 } \hfill \\ {\dot{x}_{n, k}^*=f({\pmb x}_k^*, t)+g({\pmb x}_k^*, t)u_k^* ~~~ } \hfill \\ \end{array} }} \right. \label{def2} \end{align} $ (4)

${x}_{n+1, k}^*=\dot{x}_{n, k}^*$.借鉴文献[28-29]的结果, 下文给出修正参考信号的具体构造:

1) 对于$t_\delta < t \leq T$, 取$\pmb x_k^*(t)=\pmb x_d(t)$, ${x}_{n+1, k}^*(t)=\dot{x}_{d}^{(n)}(t)$;

2) 选取合适的过渡接入曲线$\pmb x_{dk}(t)$是构造修正参考信号的重点, 对于$ 0 \leq t \leq t_\delta$, 取$\pmb x_k^*(t)=\pmb x_{dk}(t), {x}_{n+1, k}^*(t)={x_{1dk}}^{(n)}(t), $

$ \begin{align} x_{1dk}(t)&=a_{0, k}+a_{1, k}t+a_{2, k}t^2+\cdots+a_{2n+1, k}t^{2n+1}\nonumber\\ x_{idk}(t)&={x_{1dk}}^{(i-1)}(t), \ \ i=2, 3, \cdots, {n} \end{align} $ (5)

其系数为$a_{0, k}=x_{1, k}(0), a_{j, k}=\frac{1}{j !}{x}_{j+1, k}(0), 1\leq j \leq n, $

$ \begin{eqnarray} && [a_{n+1, k}, { {a_{n+2, k}}}, \cdots, a_{2n+1, k}]^{\rm T}= \Gamma^{-1}\pmb \omega\nonumber \end{eqnarray} $

此处,

$ \begin{eqnarray}~~~~ \Gamma=\left[\begin{array}{*{4}c} t_\delta^{n+1}& t_\delta^{n+2}& \dots& t_\delta^{2n+1}\\ (n+1)t_\delta^n&(n+2)t_\delta^{n+1}&\cdots& (2n+1)\textstyle t_\delta^{2n}\\ \vdots& \vdots& \ddots&\vdots\\ (n+1)!t_\delta&(\prod\limits_{j=3}^{n+2} j)t_\delta^2&\cdots &(\prod\limits_{j=n+2}^{2n+1} j) t_\delta^{n+1}\end{array}\right]\nonumber \end{eqnarray} $
$ \begin{align} \pmb \omega=&\left[\phi(t_\delta), \dot{ \phi}(t_\delta), \cdots, \phi^{(n)}(t_\delta) \right]^{\rm T}\label{omg11}\\ \phi(t)=&x_d(t)-a_{0, k}-a_{1, k}t-\cdots-a_{n, k}t^{n}\nonumber \end{align} $ (6)

本文中, $x_{n+1, k}(t)$表示$\dot{x}_{n, k}(t)$, $x_{n+1, k}(0)=\dot{x}_{n, k}(0)$.上文所构造的修正参考信号, 在$t=t_{\delta}$处的导数是连续的.可以看出, 上文给出的构造方法比较简便, 易于实施.

3 控制器的设计

$ t\in [0, T]$上, 定义

$ \begin{align} \pmb e_k(t)=&[e_{1k}, e_{2k}, \cdots, e_{nk}]^{\rm T} =\pmb x_k(t)-\pmb x_d(t)\nonumber\\ \pmb e_k^*(t)=&[e_{1k}^*, e_{2k}^*, \cdots, e_{nk}^*]^{\rm T} =\pmb x_k(t)-\pmb x_k^*(t) \nonumber\\ s_k= &c_1e_{1k}^*+\cdots+c_{n-1}e_{(n-1)k}^*+e_{nk}^* \end{align} $ (7)

选择合适的正实数$c_i (i= 1, 2, \cdots, n-1)$, 使得多项式$\Delta(\lambda)=\lambda^{n-1}+c_{n-1}\lambda^{n-2}+\cdots+c_2\lambda+c_1$为Hurwitz多项式, 且使$\pmb c=[0, c_1, \cdots, c_{n-1}]^{\rm T}$的范数$\|\pmb c\|\geq 1$.为叙述方便, 分别记$g(\pmb x_k^*, t)$$f(\pmb x_k^*, t)$$g(\pmb x_k, t)$$f(\pmb x_k, $ $ t)$$ {g_m(\pmb{x}_k, t)}$$g_k^*$$f_k^*$$g_k$$f_k$$ {g_{mk}}$.在不引起混淆时, 为叙述简便, 文中将函数的时间变量$t$略去.

由式(7) 可得

$ \begin{align} \dot{s}_k =\pmb c^{\rm T}\pmb e_{k}^*+\dot{e}_{nk}^*=\pmb c^{\rm T}\pmb e_{k}^*+f_k+g_ku_k-f_k^*-g_k^*u_k^* \label{dotsk} \end{align} $ (8)

定义障碍Lyapunov函数$V_{1k}(t)=\frac{(b_{s}^2+1)s_k^2-s_k^4}{2(b_{s}^2-s_{ k}^2)}$, 其中, $b_s$为一正常数, 是对$|s_k|$的限幅值.对$V_{1k}$求关于时间的导数, 可得

$ \begin{align} \dot{V}_{1k} = &\sigma_k s_k\big(\pmb c^{\rm T}\pmb e_{k}^*+f_k-f_k^*\big)+\sigma_ks_k\big[g_ku_k-\nonumber\\ &g_ku_{dk}+g_ku_{dk}-g_k^*u_{dk}+g_du_{dk}-g_du_d\big] +\nonumber\\ &\sigma_ks_k(g_k^*u_{dk}-g_du_{dk}+g_du_{d}-g_k^*u_k^*)\label{dotvkbc1} \end{align} $ (9)

式中, $\sigma_k(t)=\frac{(b_s^2 - s_{ k}^2)^2+b_s^2}{(b_s^2 - s_{ k}^2)^2}$, $u_{dk}$$u_d$的估计值.由修正参考信号的性状知, $g_k^*$$g_d$$u_k^*$$u_d$$ [0, t_\delta]$上均有界, $u_{dk}$的有界性也可由饱和函数确保, 参见式(18), 故$(g _k^*-g_d)u_{dk}+g_du_d-g_k^*u_k^*$$ [0, t_\delta]$上有界.在$[t_\delta, T]$上, $(g _k^*-g_d)u_{dk}+g_du_d-g_k^*u_k^*=0$.记$(g _k^*-g_d)u_{dk}+g_du_d-g_k^*u_k^*$$[0, T]$上的界为$\eta(t)$.于是, 由(9) 可得,

$ \begin{align} \dot{V}_{1k} \leq &\sigma_ks_k\big(\pmb c^{\rm T}\pmb e_{k}^*+f_k-f_k^*\big) +\sigma_ks_k\big[g_ku_k-g_ku_{dk} +\nonumber\\ &g_ku_{dk}-g_k^*u_{dk}+g_du_{dk}-g_du_d\big] +\sigma_k|s_k| \eta \leq \nonumber\\ &\sigma_k|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\|{\pmb e}_k^*\|+\sigma_ks_k(g_ku_k-\nonumber\\ &g_ku_{dk}+g_du_{dk}-g_du_d)+\sigma_k|s_k| \eta \label{dotvk1} \end{align} $ (10)

其中, $\alpha_{fk}(t) :=\alpha_{f}(\pmb x_k, \pmb x_k^*, t ) $$\alpha_{gk}(t):=\alpha_{g}(\pmb x_k, \pmb x_k^*, t )$.由修正参考信号的构造方法可知$V_{1k}(0)=0$, 据此, 由式(10) 可得,

$ \begin{align} {V}_{1k} \leq &\int_0^t\sigma_k\big[|s_k|(\alpha_{fk}+\|{\pmb c}\|+ \alpha_{gk}|u_{dk}|)\|{\pmb e}_k^*\| +\nonumber\\ &s_k(g_ku_k-g_ku_{dk}+g_du_{dk}-g_du_d)+|s_k| \eta\big] {\rm d}\tau &+\int_0^t\sigma_k|s_k| \eta {\rm d}\tau \label{vk10} \end{align} $ (11)

由式(8) 容易得到$\dot{\pmb e}_k^*=A \pmb e_k^* + \pmb b\dot{s}_{ k}$.此处, $\pmb b=[0, 0, \cdots, 0, 1]^{\rm T}$,

$ \begin{eqnarray} A=\left[\begin{array}{*{9}c} 0& &1& &0& &\cdots& &0\\0& &0& &1& &\cdots& &0\\ \vdots& &\vdots& &\vdots& &\ddots& &\vdots\\ 0& &-c_1& &-c_2& &\cdots& &-c_{n-1} \end{array}\right]\nonumber \end{eqnarray} $

$\dot{\pmb e}_k^*=A \pmb e_k^*+\pmb b\dot{s}_{ k}$在区间$[0, t]$上的定积分为

$ \begin{eqnarray} \pmb e_k^*(t)=\int_0^t A \pmb e_k^*(\tau){\rm d}\tau+\pmb b {s}_{ k}(t) \end{eqnarray} $ (12)

对上式两边同取范数, 根据Bellman-Gronwall引理,

$ \begin{align} \|\pmb e^*_k(t)\|=& \| A\| {\rm e}^{t\|A\|}\int_0^t | s_{ k}(\tau)|{\rm e}^{-\tau\|A\|}{\rm d}\tau+|{s}_{k}(t)|\nonumber \end{align} $

根据积分中值定理, 存在一未知常数$ \omega_{k1}(t)\in [0,1]$, 满足

$ \begin{align} \int_0^t | s_{ k}(\tau)|{\rm e}^{-\tau\|A\|} {\rm d}\tau = {\rm e}^{ -\omega_{k1}t \|A\|} \int_0^t | s_{ k}(\tau)|{\rm d}\tau \nonumber \end{align} $

于是由上述两式可得

$ \begin{align} \|\pmb e^*_k(t)\|=& \| A\| {\rm e}^{(1-\omega_{k1})t\|A\|}\int_0^t | s_{ k}(\tau)|{\rm d}\tau+|{s}_{k}(t)|\label{ekxinfs} \end{align} $ (13)

上式两边乘以$\sigma_k|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)$, 并积分,

$ \begin{align} \int_0^t \sigma_k&|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\|{\pmb e}_k^*\|{\rm d}\tau= \nonumber\\ & \int_0^t \sigma_k|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\| A\| {\rm e}^{(1-\omega_{k1})\tau \|A\|}\times \nonumber\\ &\int_0^\tau | s_{ k}(v)|{\rm d}v {\rm d}\tau+\int_0^t\sigma_k(\alpha_{fk}+\|{\pmb c}\| +\nonumber\\ &\alpha_{gk}|u_{dk}|){s}_{k}^2{\rm d}\tau \nonumber \end{align} $

利用积分中值定理, 由上式可以推出, 存在$\omega_{k2}(t)\in [0,1]$, 满足

$ \begin{align} \int_0^t \sigma_k&|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\|{\pmb e}_k^*\|{\rm d}\tau= \nonumber\\ & \| A\| {\rm e}^{(1-\omega_{k1})\omega_{k2}t\|A\|}\int_0^t \sigma_k|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\times \nonumber\\ &\int_0^\tau | s_{ k}(v)|{\rm d}v {\rm d}\tau +\nonumber\\ &\int_0^t\sigma_k(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|){s}_{k}^2{\rm d}\tau \label{sgm01} \end{align} $ (14)

应该注意到, $(1-\omega_{k1})\omega_{k2}$虽然随迭代次数的变化而变化, 但其上界存在, 记其上界为$\bar \omega(t)$.于是, 利用柯西不等式, 由式(14) 可以推出

$ \begin{align} \int_0^t \sigma_k&|s_k|(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)\|{\pmb e}_k^*\|{\rm d}\tau \leq \nonumber\\ &\theta(t)\int_0^t\frac{1}{\mu_1} \sigma_k^2(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2s_k^2{\rm d}\tau +\nonumber\\ &\int_0^t\sigma_k(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|){s}_{k}^2{\rm d}\tau \label{sgm1} \end{align} $ (15)

式中, $\theta(t)=\mu_1t\| A\| {\rm e}^{\bar \omega t \|A\|}$, $\mu_1>0$为设计参数.将式(15) 的结果应用于式(11),

$ \begin{align} {V}_{1k} \leq &\theta(t)\int_0^t\frac{1}{\mu_1} \sigma_k^2(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2s_k^2{\rm d}\tau +\nonumber\\ &\int_0^t\sigma_k\big[(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|){s}_{k}^2+|s_k| \eta\big]{\rm d}\tau +\nonumber\\ &\int_0^t\sigma_ks_k(g_ku_k-g_ku_{dk}+g_du_{dk}-g_du_d){\rm d}\tau \end{align} $ (16)

由此, 设计控制器

$ \begin{align} u_k= &u_{dk}-\frac{\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|}{g_{mk}}s_k -\nonumber\\ &\frac{\theta_k \sigma_k(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2}{\mu_1g_{mk}}s_k -\nonumber\\ &\frac{1}{g_{mk}}\eta_{k}{\rm tanh}(\mu_2 (k+1)(k+2)s_{ k}\eta_k) -\nonumber\\ & \frac{ s_{ k}[1+(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|]}{g_{mk}\varepsilon} \label{control2} \end{align} $ (17)

及相应的学习律

$ \begin{align} u_{{ d}k}&={\rm sat}(\hat{u}_{{ d}k})\nonumber\\ \hat{u}_{{ d}k}&={\rm sat}(\hat{u}_{{ d}(k-1)})-\gamma_1 \sigma_k s_{ k}\label{learn1} \end{align} $ (18)
$ \begin{align} \eta_{{}k}&={\rm sat}(\hat{\eta}_{{}k})\nonumber\\ \hat{\eta}_{k}&={\rm sat}(\hat{\eta}_{k-1})+\gamma_2 \sigma_k |s_{ k}| \end{align} $ (19)
$ \begin{align} \theta_{k}&={\rm sat}(\hat{\theta}_{k})\nonumber\\ \hat{\theta}_{k}&={\rm sat}(\hat{\theta}_{k-1})+\nonumber\\ &\ \ \ \ \ \gamma_3 \frac{1}{\mu_1}\sigma_k (\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2s_k^2 \label{learn3} \end{align} $ (20)

此处, $\eta_{k}$$ \eta$的估计值, $\theta_k$$\theta$的估计值, $\mu_2>0$为设计参数, $\gamma_1>0, \gamma_2>0, \gamma_3>0$, $\varepsilon\gg 0$.本文中, ${\rm sat}(\cdot)$的定义为:对于$\hat{a}\in {\bf R}$,

$ \begin{align} {\rm sat}(\hat a):=\left\{ \begin{array}{l} \bar{a} {\rm sgn}(\hat a), ~~~|\hat{a}|>\bar{a}\\ \hat{ a}, \quad \quad \quad \text{其他} \end{array} \right. \end{align} $

$\bar{a}$为对应的限幅.学习律(18) $\sim$ (20) 均采用完全限幅学习方法估计被学习量.

根据$s_k$$\pmb e_k^*$之间的不等式关系[21-22, 26]

$ \begin{align} \int_0^t \| \pmb e_k^*(\tau) \| | s_{ k}(\tau)| {\rm d}\tau \leq (1+T \| A\| {\rm e}^{T \|A\|})\int_0^t {s}_{ k}^2(\tau) {\rm d}\tau\nonumber \end{align} $

控制器(17) 也可改为

$ \begin{align} u_k= &u_{dk}-\frac{\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|}{g_{mk}}s_k -\nonumber\\ &(1+T \| A\| {\rm e}^{T \|A\|}) \frac{ \sigma_k(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2}{g_{mk}} s_k -\nonumber\\ &\frac{1}{g_{mk}}\eta_{k}{\rm tanh}(\mu_2 (k+1)(k+2)s_{ k}\eta_k) -\nonumber\\ & \frac{ s_{ k}[1+(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|]}{g_{mk}\varepsilon} \end{align} $ (21)

$T\|A\|$的取值较大场合, $(1+T \| A\| {\rm e}^{T \|A\|})$是个很大的数.本文设计控制系统时, 没有采用上述控制器设计方案, 而是利用学习方法估计$\theta$, 有利于减小设计中的保守性.

注1. 迄今, 人们已经构造了多种障碍Lyapunov函数[23-26], 例如$V=\frac{1}{2}{\rm ln}\frac{b_s^2}{b_s^2-s^2}$, $V=\frac{b_s^2}{\pi}{\rm tan}\frac{\pi s^2}{2b_s^2}$$V=\frac{1}{2}\frac{s^2}{b_s^2-s^2}$等, 利用这些障碍Lyapunov函数设计控制器, 都可以有效地对系统状态或输出进行相应的约束.本文构造了一种新型的障碍Lyapunov函数$ V=\frac{(b_{\rm s}^2+1)s^2-s^4}{2(b_{\rm s}^2-s^2)}$, 其具有的一个特点是$\sigma=\frac{\partial V}{\partial s}\geq 1$.本文根据该障碍Lyapunov函数设计控制器时, 利用了这一特点, 具体而言:在由式(14) 推出式(15) 的过程中, 利用了$\sigma=\frac{\partial V}{\partial s}\geq 1$$\|\pmb c\|\geq 1$.

4 收敛性分析

根据上文给出的控制设计, 闭环系统具有的性质可总结为定理1.

定理1.${\pmb x}_k(0)$取值任意的情况下, 将控制律(17) 施加于满足假设1和假设2的系统(1), 足够多次迭代后, 可使$s_k$在整个作业区间收敛于零, 即

$ \begin{align} \lim\limits_{k\rightarrow +\infty} s_{ k}(t)=0, \quad t\in [0, T] \end{align} $ (22)

并确保在各次迭代运行过程中, $|s_{k}|< b_{\rm s}$成立.

证明. 1) 变量有界性及系统状态的受限性

对Lyapunov函数$V_{2k}=\frac{1}{2}s_k^2$求导, 利用式(17), 可以推出

$ \begin{align} \dot V_{2k}=&s_k(\pmb c^{\rm T}\pmb e_{k}^*+f_k+g_ku_k-f_k^*-g_k^*u_k^*)\leq\nonumber\\ &|s_k|(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|+s_k(g_ku_k-g_k^*u_k^*)= \nonumber\\ & |s_k|(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|+|s_k|g_k^*|u_k^*|+ |s_k| \bar u_d -\nonumber\\ & (1+\frac{b_s^2}{(b_s^2 - s_{ k}^2)^2}) \frac{ s_{ k}^2[1+(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|]}{\varepsilon}\label{yjxdotv11} \end{align} $ (23)

$(1+\frac{b_s^2}{(b_s^2 - s_{ k}^2)^2})|s_k|\geq \varepsilon$时,

$ \begin{align} |s_k|(\|\pmb c\|&+\alpha_k)\|\pmb e_{k}^*\| - (1+\frac{b_s^2}{(b_s^2 - s_{ k}^2)^2})\times \nonumber\\ &\frac{ s_{ k}^2[1+(\|\pmb c\|+\alpha_k)\|\pmb e_{k}^*\|]}{\varepsilon}\leq 0\label{yjxbds1} \end{align} $ (24)

$(1+\frac{b_s^2}{(b_s^2 - s_{ k}^2)^2})|s_k|\geq \varepsilon(g_k^*|u_k^*|+\bar u_d)$时,

$ \begin{align} &|s_k|g_k^*|u_k^*| + |s_k| \bar u_d - (1+\frac{b_s^2}{(b_s^2 - s_{ k}^2)^2})\frac{ s_{ k}^2}{\varepsilon}\leq 0\label{yjxbds2} \end{align} $ (25)

$h=|s_k|$, 注意到: 1) 在每次迭代的开始时刻, $ s_{ k}=0$; 2) 函数$\nu(h)$ $=(1+\frac{b_s^2}{(b_s^2 - h^2)^2})h$$ h \in [0, b_s)$区间内单调递增, 且$ \lim_{h \rightarrow b_{s-}}\nu(h)=+\infty.$因此, 当在$[0, b_s)$内的$ |s_k|$足够大时, 必有

$ \begin{align} (1+\frac{b_s^2}{(b_s^2- s_{ k}^2)^2})|s_k|\geq \max(\varepsilon, \varepsilon(g_k^*|u_k^*|+\bar u_d))\nonumber \end{align} $

成立, 结合式(23) $\sim$(25), 可得$\dot V_{2k}\leq 0.$由此可知, 在各次迭代过程中$|s_{ k}|<b_s$.进而易得系统各信号的有界性.由式(13) 可以推出$\|\pmb e^*_k\|< \| A\| {\rm e}^{(1-\omega_{k1})t\|A\|}b_s t +b_s$.于是

$ \begin{align} \|\pmb x_k\|< \| A\| {\rm e}^{(1-\omega_{k1})t\|A\|}b_s t +b_s+\|{\pmb x}_k^*\| \end{align} $ (26)

可见, 系统状态在各次迭代运行过程中均受到相应的约束.

2) 误差收敛性

选择Lyapunov泛函$L_{k}=V_{1k}+\frac{1}{2\gamma_1}\int_0^tg_d\tilde u_{dk}^2{\rm d}\tau+\frac{1}{2\gamma_2}\int_0^t\tilde{\eta}_{k}^2{\rm d}\tau + \frac{1}{2\gamma_3}\int_0^t\tilde{\theta}_{k}^2{\rm d}\tau, $由式(11) 和(17),

$ \begin{align} {V}_{1k} \leq&\int_0^t \sigma_k^2 \tilde{\theta}_k\frac{1}{\mu_1}(\alpha_{fk} +\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2s_k^2{\rm d}\tau +\nonumber\\ &\int_0^t\sigma_k s_kg_d(u_{dk}-u_d){\rm d}\tau+\int_0^t\sigma_k\big[|s_k| \eta-\nonumber\\ &s_k\eta_{k}{\rm tanh}(\mu_2 (k+1)(k+2)s_{ k}\eta_k)\big]{\rm d}\tau \label{v2keq} \end{align} $ (27)

根据式(27), 当$k>0$时,

$ \begin{align} L_{k}-& L_{k-1}\leq\nonumber\\ & \int_0^t \sigma_k^2\tilde{\theta}_k\frac{1}{\mu_1}(\alpha_{fk}+\|{\pmb c}\|+\alpha_{gk}|u_{dk}|)^2s_k^2{\rm d}\tau +\nonumber\\ &\int_0^ts_k\sigma_kg_d(u_{dk}-u_d){\rm d}\tau+\int_0^t\sigma_k\big[|s_k| \eta-\nonumber\\ &{s_k}\eta_{k}{\rm tanh}(\mu_2 (k+1)(k+2)s_{ k}\eta_k)\big]{\rm d}\tau +\nonumber\\ &\frac{1}{2\gamma_1}\int_0^tg_d(\tilde u_{dk}^2-\tilde u_{d(k-1)}^2){\rm d}\tau -V_{1k-1} +\nonumber\\ & \frac{1}{2\gamma_2}\int_0^t(\tilde{\eta}_{k}^2-\tilde{\eta}_{k-1}^2){\rm d}\tau+\frac{1}{2\gamma_3}\int_0^t(\tilde{\theta}_{k}^2-\tilde{\theta}_{k-1}^2){\rm d}\tau\nonumber \end{align} $

根据学习律(18),

$ \begin{align} \frac{1}{2\gamma_1}(\tilde u_{{ d}k}^2 &-\tilde u_{{ d}k-1}^2) +\sigma_ks_{ k}\tilde u_{{ d}k} = \nonumber\\ & \frac{1}{\gamma_1}(u_{ d}- u_{{ d}k})(u_{{ d}(k-1)}-u_{{ d}k}-\gamma_1\sigma_k s_{k} ) -\nonumber\\ &\frac{1}{2\gamma_1}(u_{{ d}k} -u_{{ d}(k-1)} )^2\leq\nonumber\\ & \frac{1}{\gamma_1}(u_{ d}- {\rm sat}(\hat{u}_{{ d}k}))(\hat{u}_{{ d}k}- {\rm sat}(\hat{u}_{{ d}k}))\leq 0\label{udk-k-1} \end{align} $ (28)

类似地, 根据学习律(19) 及(20), 分别可以推出

$ \begin{align} \frac{1}{2\gamma_2}& (\tilde{\eta}_{k}^2-\tilde{\eta}_{k-1}^2)+\sigma_k|s_{k}|\tilde{\eta}_{k}\leq\nonumber\\ &\frac{1}{\gamma_2}(\eta-{\rm sat}(\hat{\eta}_{k}))(\hat{\eta}_{k}-{\rm sat}(\hat{\eta}_{k}))\leq 0\label{etak-k-1} \end{align} $ (29)

$ \begin{align} \frac{1}{2\gamma_3}& (\tilde{\theta}_{k}^2-\tilde{\theta}_{k-1}^2) +\sigma_k^2 \tilde{\theta}_k\frac{1}{\mu_1}(\alpha_{fk}+\|{\pmb c}\| +\alpha_{gk}|u_{dk}|)^2s_k^2\leq \nonumber\\ & \frac{1}{\gamma_3}(\theta- {\rm sat}(\hat{\theta}_{k}))(\hat{\theta}_{k}- {\rm sat}(\hat{\theta}_{k}))\leq 0\label{thetak-k-1} \end{align} $ (30)

综合以上四式, 有

$ \begin{align} L_{k}-L_{k-1}\leq&-V_{1k-1}+\int_0^t\sigma_k\big[|s_k| \eta_k-{s_k}\eta_{k} \times \nonumber\\ & {\rm tanh}(\mu_2 (k+1)(k+2)s_{ k}\eta_k)\big]{\rm d}\tau \leq\nonumber\\ &- V_{1k-1}+\frac{0.2785t\sigma_k}{\mu_2(k+1)(k+2)} \end{align} $ (31)

进一步地,

$ \begin{align} L_{k} \leq& L_{0}+ \sum\limits_{i=1}^k\frac{0.2785t\sigma_k}{\mu_2(i+1)(i+2)} -\sum\limits_{i=1}^k\frac{s_{i-1}^2(b_{s}^2+1-s_{i-1}^2)}{2(b_{ s}^2-s_{ i-1}^2)}\leq \nonumber\\ & L_{0}+ \sum\limits_{i=1}^k\frac{0.2785t\sigma_k}{\mu_2(i+1)(i+2)} -\sum\limits_{i=1}^k\frac{s_{i-1}^2}{2} \label{L1klast} \end{align} $ (32)

$L_{0}$为非负有界量, 且$ \sum_{i=1}^k\frac{0.2785t\sigma_k}{\mu_2(i+1)(i+2)} =\frac{0.2785t\sigma_k}{\mu_2}(\frac{1}{2}-\frac{1}{k+2})<\frac{0.2785t\sigma_k}{2\mu_2}$也为有界量, 根据数列收敛的必要条件, 可知$ \lim_{k\rightarrow +\infty}s_{k}(t)=0, t\in[0, T].$利用Hurwitz多项式的性质及$\pmb e_k^*(0)=0$, 可以推出$\lim_{k\rightarrow +\infty}\pmb e_{k}^*(t)=0, t\in[0, T]$.

注2. 在上文中, 多次利用不等式关系$(a-a_k)^2-(a-a_{k-1})^2=-2(a-a_k)(a_k-a_{k-1})-(a_k-a_{k-1})^2$进行推导, 例如在式(28) 中, $(u_d-u_{dk})^2-(u_d-u_{d(k-1)})^2=-2(u_d-u_{dk})(u_{dk}-u_{d(k-1)})-(u_{dk}-u_{d(k-1)})^2$.但倘若用迭代次数相关量$u_k^*$取代$u_d$, 则在一般情况下结果不正确, 即当$u_k^*\neq u_{k-1}^*$时, $(u_k^*-u_{dk})^2-(u_{k-1}^*-u_{d(k-1)})^2\neq -2(u_k^*-u_{dk})(u_{dk}-u_{d(k-1)})-(u_{dk}-u_{d(k-1)})^2.$

5 仿真算例

考虑在$[0, T]$上运行的非线性系统

$ \begin{eqnarray} \left\{ \begin{aligned} &\ \dot{x}_{1, k}=x_{2, k}\ \ \ \\ &\ \dot{x}_{2, k}=-0.1x_{2, k}-x_{1, k}^3+{\rm cos}(t)+(1+0.01x_{1, k}^2 +\nonumber\\ &\ ~~~~~~~~~~ \ 0.005x_{2, k}^2)u_k \end{aligned} \right. \label{sys2} \end{eqnarray} $

$ -0.1x_{2, k}-x_{1, k}^3+{\rm cos}(t)$$1+0.01x_{1, k}^2+0.005x_{2, k}^2$分别为不确定性$f(\pmb x_k, t)$$g(\pmb x_k, t)$.系统初态为$x_{1, k}(0)=0.5+0.1r_1$, $x_{2, k}(0)=0.1+0.02r_2$, $r_1$$r_2$均为0与1之间的随机数.拟采用迭代学习控制方法设计控制器, 实现$[x_{1, k}, x_{2, k}]^{\rm T}$对参考信号$[x_{1d}, x_{2d}]^{\rm T}=[{\rm cos}(\pi t), -\pi {\rm sin}(\pi t)]^{\rm T}$进行精确跟踪.鉴于$[x_{1, k}(0), x_{2, k}(0)]^{\rm T}\neq[1,0]^{\rm T}$, 此处采用本文所提方法设计控制系统, 按照前文给出的方案构造修正参考信号.当$t_\delta< t \leq T$时, $ x_{1, k}^*(t)=x_{d}, x_{2, k}^*(t)=\dot x_d, x_{3, k}^*(t)=\ddot x_d;$$0 \leq t \leq t_\delta$时,

$ \begin{align} x_{1, k}^*(t)=&\ a_{0, k}+a_{1, k}t+a_{2, k}t^2+a_{3, k}t^3+a_{4, k}t^4 +a_{5, k}t^5\nonumber\\ x_{2, k}^*(t)=&\ a_{1, k}+2a_{2, k}t+3a_{3, k}t^2+4a_{2, k}t^3 +5a_{3, k}t^4\nonumber\\ x_{3, k}^*(t)=&\ 2a_{2, k}+6a_{3, k}t+12a_{2, k}t^2 +20a_{3, k}t^3\nonumber \end{align} $

此处, $ t_\delta=0.5, T=3, a_{0, k}=x_{1, k}(0), a_{1, k}=x_{2, k}(0), a_{2, k}=\frac{1}{2} x_{3, k}(0), [a_{3, k}, a_{4, k}, a_{5, k}]^{\rm T}=\Gamma^{-1}\pmb \omega$.上式中, $ \pmb \omega=[x_d -a_{0, k}-a_{1, k}t_\delta-a_{2, k}t_\delta^2, \dot{x}_d -a_{1, k}-2a_{2, k}t_\delta, \ddot{x}_d-2a_{2, k}]^{\rm T}$,

$ \begin{align} \Gamma=&\left[ \begin{array}{ccc} t_\delta^3&\ \ t_\delta^4&\ \ t_\delta^5\\ 3t_\delta^2&\ \ 4t_\delta^3& \ \ 5t_\delta^4\\ 6t_\delta&\ \ 12t_\delta^2& \ \ 20t_\delta^3\\ \end{array} \right] \end{align} $ (33)

选取$g_m(\pmb x_k, t)=1$, $\alpha_{fk}= 0.1+2(x^2_{1, k}+x^{*2}_{1, k})$, $\alpha_{gk}=0.02(|x_{1, k}|+|x_{1, k}^*|)+0.01(|x_{2, k}|+|x_{2, k}^*|)$, 容易检验, 它们满足本文的假设.采用控制律(17) 及相应学习律进行仿真, 仿真参数为$c_1=2, \gamma_1=1.1, \gamma_2=0.08, \gamma_3=1.5, \mu_1=10, \mu_2=1, \bar{u}=50$, $\bar{ \eta}=10$, $\bar{ \theta}=90, b_s=0.4, T=2, \varepsilon=100$.迭代40次后, 仿真结果如图 1$ \sim $7所示. 图 12是第40次迭代时的系统状态情况. 图 3是第40次迭代过程中的误差值, 图 45分别是第40次迭代过程中的修正误差情况, 图 6为该次迭代中的控制量. 图 7中是$J_k$的收敛过程, 在该图中, $J_{k}:=\max_{t\in[0, T]} |s_{ k}(t)| $.可以看出, 经过足够多次迭代后, 可实现$s_k$$[0, T]$上的取值为零.

图 1 $x_1$及其期望轨迹$x_{1d}$ Figure 1 $x_1$ and its desired trajectory $x_{1d}$
图 2 $x_2$及其期望轨迹$x_{2d}$ Figure 2 $x_2$ and its desired trajectory $x_{2d}$
图 3 误差$e_1$$e_2$ Figure 3 The errors $e_1$ and $e_2$
图 4 修正误差$e_1^*$ Figure 4 The rectified error $e_1^*$
图 5 修正误差$e_2^*$ Figure 5 The rectified error $e_2^*$
图 6 控制输入 Figure 6 Control input
图 7 状态受限情形下$J_{ k}$的收敛过程 Figure 7 The history of $J_{ k}$ in the case of constraint

为对比起见, 采用无约束的初始修正学习控制律

$ \begin{align} u_k=&u_{dk}-\frac{(1+t\| A\| {\rm e}^{t\|A\|})\bar{\alpha}_{fcg}}{g_{mk}}s_k-\nonumber\\ &\frac{1}{g_{mk}}\eta_{k}{\rm tanh}(\mu_3 (k+1)(k+2)s_{ k}\eta_k)\nonumber \end{align} $

进行仿真, 式中,

$ \begin{align} u_{{ d}k}&={\rm sat}(\hat{u}_{{ d}k})\nonumber\\ \hat{u}_{{ d}k}&={\rm sat}(\hat{u}_{{ d}k-1})-\gamma_4 s_{ k} \nonumber\\ \eta_{{}k}&={\rm sat}(\hat{\eta}_{{}k})\nonumber\\ \hat{\eta}_{k}&={\rm sat}(\hat{\eta}_{k-1})+\gamma_5 |s_{ k}|\nonumber \end{align} $

$\overline{\alpha}_{fcg}(t)=\max_{\tau \in [0, t]}(\alpha_{fk}(\tau)+\|{\pmb c}\|+\alpha_{gk}(\tau)|u_{dk}(\tau)|)$, $\mu_3=1, \gamma_4=4, \gamma_5=0.1$, 其余参数及饱和限幅取值同前.迭代40次后, $J_k$的收敛过程见图 8.对比图 7图 8, 可以看出状态约束算法可以将$|s_k|$约束于预设的界内.

图 8 $J_{ k}$收敛过程 Figure 8 The history of $J_{ k}$

仿真结果表明, 利用本文给出的状态受限参考信号初始修正方法设计学习控制器, 可以解决非参数不确定学习控制系统的初值问题, 实现系统状态对修正参考信号在整个作业区间上的完全跟踪, 并确保将各次迭代过程中的滤波误差约束于预设的界内, 对系统状态予以约束.上述结果说明了本文所提控制方法的有效性.

6 结论

本文提出状态受限迭代学习控制的参考信号初始修正方法, 解决非参数不确定系统在任意初态情形下的状态受限轨迹跟踪问题.文中给出了修正参考信号构造方案, 并利用一种新型的障碍Lyapunov函数设计迭代学习控制器, 采用鲁棒学习控制方法处理非参数不确定性.经过足够多次迭代后, 藉由系统状态对修正参考信号在整个作业区间的完全跟踪, 获得系统状态对参考信号在预设部分作业区间上的完全跟踪.在各次迭代过程中, 闭环系统中的滤波误差被约束于预设的界内, 由此实现对系统状态在各次迭代运行过程中的约束.本文采用学习方法估计鲁棒项的增益系数, 克服了以往同类算法在某些场合应用时可能出现的增益过大现象.

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