自动化学报  2019, Vol. 45 Issue (1): 226-233   PDF    
随机非线性系统基于事件触发机制的自适应神经网络控制
王桐, 邱剑彬, 高会军     
哈尔滨工业大学智能控制与系统研究所 哈尔滨 150001
摘要: 针对一类具有严格反馈结构且控制方向未知的随机非线性系统,提出了基于事件触发机制的自适应神经网络(Adaptive neural network,ANN)输出反馈控制方法.利用径向基神经网络逼近系统中未知的非线性函数.通过引入Nussbaum增益函数并设计滤波器,解决了系统控制方向未知的问题.通过设计具有相对阈值的事件触发机制,保证了闭环随机非线性系统的有界性.最后给出数值仿真例子验证所提控制方法的有效性.
关键词: 随机非线性系统     事件触发     反步法     自适应神经网络     输出反馈    
Event-triggered Adaptive Neural Network Control for a Class of Stochastic Nonlinear Systems
WANG Tong, QIU Jian-Bin, GAO Hui-Jun     
The Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001
Manuscript received : June 8, 2018, accepted: August 27, 2018.
Foundation Item: Supported by National Natural Science Foundation of China (61803122, 61873311), the 111 Project (B16014), National Postdoctoral Program for Innovative Talents (BX201700067), China Postdoctoral Science Foundation Grant (2018M630359), and Heilongjiang Province Science Foundation for Youths (QC2018077)
Author brief: WANG Tong    Lecturer at Harbin Institute of Technology. His main research interest is adaptive control for nonlinear systems;
QIU Jian-Bin    Professor at Harbin Institute of Technology. His main research interest is fuzzy control for nonlinear systems.
Corresponding author. GAO Hui-Jun    Professor at Harbin Institute of Technology. His research interest covers networked control systems. Corresponding author of this paper.
Recommended by Associate Editor LU Ren-Quan
Abstract: This paper investigates the event-triggered adaptive output-feedback control problem for a class of strict-feedback stochastic nonlinear systems, and a novel event-triggered adaptive neural network output-feedback control strategy is proposed. The radial basis function neural networks are utilized to approximate the unknown nonlinear functions. By introducing Nussbaum gain function and designing filter during the backstepping design procedure, the effect of unknown control direction is compensated. The boundness of the closed-loop stochastic nonlinear system is guaranteed by designing a relative threshold event-triggered mechanism. Finally, a numerical example is given to show the effectiveness of the proposed control strategy.
Key words: Stochastic nonlinear systems     event-triggered     backstepping     adaptive neural network (ANN)     output-feedback    

在过去的20多年中, 针对具有严格反馈结构的非线性系统的自适应反步控制设计问题得到了广泛的研究[1-4].反步法(Backstepping)由Kanellakopoulos等于1991年在文献[3]中首先提出, 是针对不确定性系统, 将Lyapunov函数的选取与控制器的设计相结合的一种回归设计方法, 通过从系统的最低阶次微分方程开始, 引入虚拟控制的概念, 一步一步设计满足要求的虚拟控制.上述文献[1-4]主要研究了具有严格反馈结构非线性系统的自适应控制设计问题, 降低了自适应参数的数量.然而, 上述方法并不能解决系统中存在未知的非线性项的情况.通过引入模糊逻辑系统或神经网络, 文献[5-8]研究了一类含有未知非线性函数系统的自适应模糊/神经网络控制设计方法.针对一类最小相位非线性系统, 文献[5]基于可线性化的神经网络结构提出了自适应神经网络(Adaptive neural network, ANN)反步设计方法.文献[6]通过结合二次Lyapunov-Krasovskii函数, 解决了多输入多输出非线性时滞系统的跟踪控制问题.而针对具有严格反馈结构的单输入单输出非线性系统, 文献[7]提出了基于动态面控制技术的自适应神经网络控制方法, 解决了反步法带来的``维数爆炸''问题, 降低了算法的计算复杂度.在此基础上, 文献[8]将上述控制方法扩展到了一类具有纯反馈结构的非线性时滞系统.同时, 自适应反步法控制设计也被应用到了悬架控制[9]等实际例子当中.

考虑到各种随机干扰和随机扰动对非线性系统的影响, 随机非线性系统的控制问题也得到了深入的研究[10-12].文献[10]解决了随机非线性系统的稳定性问题, 文献[11]将该结果扩展到了一类互联的随机非线性大系统, 文献[12]通过结合随机小增益定理和输入到状态实际稳定概念解决了一类含有未建模动态的随机非线性系统的自适应反步控制设计问题.通过结合神经网络文献[13-14]分别研究了随机严格反馈非线性时滞系统和随机非线性互联大系统的输出反馈控制问题, 得到了系统依概率稳定的结果.

另一方面, 由于基于事件触发机制的控制策略不仅带来了诸如资源共享等优点, 同时也可以充分利用有限的带宽资源实现可靠性较高的控制需求.文献[15]针对一般结构非线性系统的跟踪问题研究了其在事件触发机制条件下的稳定性, 文献[16]则结合小增益定理将该结果扩展到了含有未建模动态的非线性系统.文献[17]提出了基于事件触发机制的输出反馈控制策略, 解决了一类非线性系统的镇定问题.在系统中存在未知非线性函数的情形下, 文献[18]结合模糊逻辑系统, 针对离散非线性网络化系统, 研究了其基于事件触发机制的$H_\infty$控制方法.文献[19-20]则研究了一类具有随机干扰的多智能体系统的一致性控制问题, 文献[21-22]则基于事件触发机制分别研究了随机系统的滑模控制问题和$H_\infty$控制问题.上述的结果均是针对非线性系统或者随机线性系统, 而非本文所研究的随机非线性系统, 且在事件触发机制框架下针对控制方向未知的随机非线性系统的自适应神经网络控制问题的结果还未见报道.本文针对该类系统, 提出了基于事件触发机制的自适应神经网络控制策略, 通过引入Nussbaum增益函数克服了未知控制方向对系统性能的影响, 保证了闭环系统的随机稳定性, 使得系统所有的信号半全局一致有界, 在事件触发设计框架下解决了随机非线性系统中同时存在未知非线性项和未知控制方向的自适应神经网络控制问题.

1 问题描述 1.1 系统模型及假设

本文所考虑的为如下结构的随机非线性系统

$ \begin{equation} {\left\{ \begin{array}{l} {\rm d}x_{1}=(x_{{2}}+f_{1}(x_{1})){\rm d}t+g_1(x){\rm d}\omega\\ {\rm d}x_{i}=(x_{{i+1}}+f_{i}(\bar x_{i})){\rm d}t+g_i(x){\rm d}\omega, \\ \hfill~~~~~~~~~~~~~~\; \;\;i=2, \cdots, n-1 \\ {\rm d}x_{n}=(bu+f_{n}(\bar x_{n})){\rm d}t+g_{n}(x){\rm d}\omega\\ y=x_{1} \\ \end{array} \right.}\label{model1} \end{equation} $ (1)

其中, $\bar {\pmb x}_i=[x_1, \cdots, x_i]^{\rm T}\in {\bf R}^i, i=1, \cdots, n, {\pmb x}=\bar {\pmb x}_n$为系统的状态向量, $y\in {\bf R}$$u\in {\bf R}$分别为系统的输出和输入, $b$为未知的常数, 且存在已知正常数使得$b_{\rm min}\leq\vert b\vert\leq b_{\rm max}$, $f_i$为未知的非线性光滑函数, $g_i(x)$为不确定函数, 记增量${\rm d}\omega$的协方差为$\sigma\sigma^{\rm T} {\rm d}t$, 即均值${\rm E}\{{\rm d}\omega {\rm d}\omega^{\rm T}\}=\sigma\sigma^{\rm T} {\rm d}t$, 其中函数矩阵$\sigma$是有界但不确定的.

针对随机非线性系统(1), 首先给出如下的假设.

假设1.  非线性函数$f_i$满足局部Lipschitz条件, 即$\vert f_i({\pmb X_1})-f_i({\pmb X_2})\vert\leq L_i\vert\vert {\pmb X_1}-{\pmb X_2}\vert\vert$, 其中$L_i$为正的常数, 这里的$\vert \cdot \vert$表示函数的绝对值, $\Vert \cdot \Vert$表示向量的1-范数.

假设2.  系统的随机扰动协方差是有界的, 且满足如下等式$g_i^{\rm T}\sigma\sigma^{\rm T}g_i=\bar\sigma_i\bar\sigma_i^{\rm T}$.

引理1[23]. 给出定义在时间段$[0, t_f)$上的光滑函数$\varsigma(t)$ (详细表达式为式(42)).并考虑特定的Nussbaum增益函数$N(\varsigma)=\varsigma^2\cos(\varsigma)$, 针对随机非线性系统(1), 若存在正定的函数$V(t, x)$和正定常数$C, D$使得如下不等式成立

$ \begin{align} \ell V(t, x)\leq -CV(t, x)+(bN(\varsigma)+1)\dot\varsigma+D \label{lemma1} \end{align} $ (2)

${\rm E}(V(t, x))$$\varsigma(t)$均在$[0, t_f)$上保持有界, 其中$\ell$为随机非线性系统的无穷小算子, 其定义如下:考虑随机非线性系统${\rm d}x=f(t, x)+h^{\rm T}(t, x){\rm d}\omega$, 针对$V(t, x)$的无穷小算子表达式为

$ \begin{align} \ell V(t, x)=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x}+\frac{1}{2}{\rm tr}\Big\{h\frac{\partial^2 V}{\partial x^2}h^{\rm T}\Big\} \end{align} $ (3)

证明.  首先, 设计函数$W(t, x)$

$ \begin{align} W(t, x)=V(t, x){\rm e}^{Ct} \end{align} $ (4)

可得

$ \begin{align} &{\rm E}W(s, x(s))|_0^t\nonumber={\rm E}\int_{0}^{t}\ell W(s, x(s)){\rm d}s\nonumber=\\ &{\rm E}\int_{0}^{t}\big[CV(s, x(s)){\rm e}^{Cs}+\ell V(s, x(s)){\rm e}^{Cs}\big]{\rm d}s \label{lemma1proof2} \end{align} $ (5)

利用式(2)可得

$ \begin{align} &{\rm E}\int_{0}^{t}\big((bN(\varsigma)+1)\dot\varsigma+D\big){\rm e}^{Cs}{\rm d}s\nonumber=\\ &\frac{D}{C}\big({\rm e}^{Ct}-1\big)+\int_{0}^{t}(bN(\varsigma)+1)\dot\varsigma {\rm e}^{Cs}{\rm d}s \end{align} $ (6)

由式(5)可得

$ \begin{align} {\rm E}W(t, x) \le& {\rm E}W(0, x(0))+\frac{D}{C}\big({\rm e}^{Ct}-1\big)+\nonumber\\ &\int_{0}^{t}(bN(\varsigma)+1)\dot\varsigma {\rm e}^{Cs}{\rm d}s \end{align} $ (7)

结合$W(t, x)$的定义可知

$ \begin{align} {\rm E}V(t, x)\le& {\rm E}V(0, x(0)){\rm e}^{-Ct}+\frac{D}{C}\big(1-{\rm e}^{-Ct}\big)+\nonumber\\&{\rm e}^{-Ct}\int_{0}^{t}(bN(\varsigma)+1)\dot\varsigma {\rm e}^{Cs}{\rm d}s\le&\nonumber\\ &{\rm E}V(0, x(0)){\rm e}^{-Ct}+\frac{D}{C}+\nonumber\\&{\rm e}^{-Ct}\int_{0}^{t}(bN(\varsigma)+1)\dot\varsigma {\rm e}^{-C(t-s)}{\rm d}s\label{lemma1proof3} \end{align} $ (8)

值得注意的是, 式(8)中, 对于$s\in[0, t]$, ${\rm e}^{-C(t-s)}$满足$0 < {\rm e}^{-C(t-s)}\le 1$.假设$ {\rm e}^{-C(t-s)}(N(\varsigma)+1)$为Nussbaum型函数, 则由Nussbaum函数的性质可知, 对于函数变量$\xi$, 如下两个不等式成立

$ \begin{align} &\lim\limits_ {\varsigma\rightarrow \infty}\sup\int_{0}^{\varsigma} {\rm e}^{-C(t-s)}(bN(\varsigma)+1)\dot\varsigma {\rm d}s \rightarrow +\infty \end{align} $ (9)
$ \begin{align} &\lim\limits_ {\varsigma\rightarrow \infty}\inf\int_{0}^{\varsigma} {\rm e}^{-C(t-s)}(bN(\varsigma)+1)\dot\varsigma {\rm d}s \rightarrow -\infty \end{align} $ (10)

可得${\rm E}V(t, x) <0$.然而, 这与${\rm E}V(t, x)\ge 0$的事实相矛盾.因此, 变量$\varsigma$${\rm E}V(t, x) <0$在时间段$[0, t_f)$上是有界的, ${\rm E}V(t, x)$也因此是有界的.

1.2 径向基神经网络

针对随机非线性系统(1), 将采用如下结构的径向基神经网络逼近系统中存在的未知非线性函数

$ \begin{align} Z(\bar {\pmb x})={\pmb \theta}^{\rm T}\varphi(\bar {\pmb x}) \end{align} $ (11)

其中, $\bar {\pmb x}$为输入向量, ${\pmb \theta}=[\theta_1, \theta_2, \cdots, \theta_M]^{\rm T}$为权重向量, $M>1$为网络的节点个数, 激活函数$\varphi(\bar {\pmb x})=[\varphi_1(\bar {\pmb x}), \varphi_2(\bar {\pmb x}), \cdots, \varphi_M(\bar {\pmb x})]^{\rm T}$选取为如下结构

$ \begin{align} \varphi_i(\bar {\pmb x})=\exp\Big[\frac{-(\bar {\pmb x}-{\pmb \mu}_i)^{\rm T}(\bar {\pmb x}-{\pmb \mu_i})}{\pmb \eta^2}\Big] \end{align} $ (12)

其中, ${\pmb \mu}_i$为对应的神经元中心参数, $\pmb \eta$为宽度向量.由于径向基神经网络的逼近特性, 上述神经网络函数$Z(\bar {\pmb x})$可以在一个紧集上以任意精度逼近任意的连续函数$f(\bar {\pmb x})$.

$ \begin{align} \sup\limits_{\bar {\pmb x}\in\Omega}\vert f(\bar {\pmb x})-\theta^{\rm T}\varphi(\bar {\pmb x})\vert\leq\varepsilon \end{align} $ (13)

其中, $\varepsilon$为逼近误差.

1.3 滤波器设计

首先, 利用假设1和径向基神经网络逼近针随机非线性系统(1)中存在的未知非线性函数$f_i(\bar {\pmb x}_i)$, 即

$ \begin{align} f_i(\bar {\pmb x}_i)=\Delta f_i+{\pmb \theta}_i^{*{\rm T}}\varphi_i(\hat{\bar {{\pmb x}}}_i)+\varepsilon_i \end{align} $ (14)

其中, $\varepsilon_i=f_i(\hat{\bar {\pmb x}}_i)-\hat f_{i}(\hat{\bar {\pmb x}}_{i}|{\pmb \theta}_i^*)$为逼近误差, 且满足$\vert\varepsilon_i\vert\leq\varepsilon_i^*$, $\Delta f_i=f_i(\bar {\pmb x}_i)-f_i(\hat {\bar {\pmb x}}_i)$, ${\pmb \theta}_{i}^*=\arg\min_{{\pmb \theta}_{i}\in\Omega_{i}}[\sup_{\hat{\bar {\pmb x}}_{i}\in U_i}\vert \hat f_i(\hat{\bar {\pmb x}}_{i}|{\pmb \theta}_i)-f_i(\hat{\bar {\pmb x}}_i)\vert]$为最优参数向量.记$\varphi_i(\hat{\bar {{\pmb x}}}_i)$$\varphi_i$, 随机非线性系统(1)可以写为

$ \begin{align} {\rm d}{\pmb x}= &(A{\pmb x}+{\pmb K}y+\Delta {\pmb f}+\varphi^{\rm T}{\pmb \theta}^*+{\pmb \varepsilon}+{\pmb B}bu){\rm d}t+\nonumber\\&G({\pmb x}){\rm d}\omega \end{align} $ (15)

其中, ${\pmb K}=[k_1, \cdots, k_n]^{\rm T}$, ${\pmb \theta}^*=[\theta_1^*, \cdots, \theta_n^*]$, $G({\pmb x})=[g_1({\pmb x}), \cdots, g_n({\pmb x})]^{\rm T}$, ${\pmb \varepsilon}=[\varepsilon_1, \cdots, \varepsilon_n]^{\rm T}$, $\Delta {\pmb f}=[\Delta f_1, \cdots, \Delta f_n]^{\rm T}$, ${\pmb B}=[0, \cdots, 1]^{\rm T}$, $A=\left[\begin{matrix}-k_1 &\cdots&0\\ \vdots& I&\vdots \\-k_n&\cdots &0 \end{matrix}\right]$, $\varphi^{\rm T}=\left[\begin{matrix}\varphi_1^{\rm T} &\cdots&0\\ \vdots& \ddots &\vdots\\ 0&\cdots&\varphi_n^{\rm T}\end{matrix}\right]$.

设计滤波器为

$ \begin{align} \hat{\pmb \chi}=\hat{\pmb \xi}+\Omega^{\rm T}\vartheta\label{filter1} \end{align} $ (16)

其中, $\hat{\pmb \chi}=[\hat\chi_1, \cdots, \hat\chi_n]^{\rm T}$, $\vartheta=\left[\begin{matrix}b\\\theta^*\end{matrix}\right]$, $\Omega^{\rm T}=[\lambda, \Xi]$.

设计滤波器结构为

$ \begin{align} \dot{\pmb \xi}=A{\pmb \xi}+{\pmb K}y\nonumber\\ \dot{\pmb \Xi}=A{\pmb \Xi}+\varphi^{\rm T}\nonumber\\ \dot{\pmb \lambda}=A{\pmb \lambda}+{\pmb B}u\label{filter2} \end{align} $ (17)

其中, ${\pmb \xi}=[\xi_1, \cdots, \xi_n]^{\rm T}$, ${\pmb \lambda}=[\lambda_1, \cdots, \lambda_n]^{\rm T}$, ${\pmb \Xi}=[\Xi_1, \cdots, \Xi_n]^{\rm T}$.

结合式(16)和(17), 可得:

$ \begin{align} \dot{\hat{\pmb \chi}}=A{\pmb \chi}+{\pmb K}y+{\pmb B}bu+\varphi^{\rm T}{\pmb \theta}^* \end{align} $ (18)

需要注意的是, 由于滤波器中存在未知参数变量$\vartheta$, 因此在接下来的控制设计中利用如下的状态观测值.

$ \begin{align} \hat {\pmb x}={\pmb \xi}+\Xi{\pmb \theta}+\hat b{\pmb \lambda} \end{align} $ (19)

选取向量${\pmb K}$使得矩阵$A$为正定的赫尔维茨矩阵, 即对于给定的正定对称矩阵$Q$, 存在正定对称矩阵$P$使得如下等式成立

$ \begin{align} A^{\rm T}P+PA=-2Q \end{align} $ (20)

定义系统的滤波误差变量

$ \begin{align} {\pmb e}=[e_1, e_2, \cdots, e_n]^{\rm T}={\pmb x}-\hat{\pmb \chi} \end{align} $ (21)

可得

$ \begin{align} {\rm d}{\pmb e}= &(A{\pmb e}+\Delta {\pmb f}+{\pmb \varepsilon}){\rm d}t+G({\pmb x}){\rm d}\omega\label{e} \end{align} $ (22)

针对滤波误差系统(22), 选取Lyapunov函数为

$ \begin{align} V_0=\frac{1}{2}{\pmb e}^{\rm T}P{\pmb e} \end{align} $ (23)

利用伊藤微分定理, 可得如下不等式

$ \begin{align} \ell V_0\leq &-\lambda_{\rm min}(Q)\|{\pmb e}\|^2+{\pmb e}^{\rm T}P\Big({\pmb \varepsilon}+\Delta {\pmb f}\Big)+\nonumber\\&{\rm tr}[\sigma G^{\rm T}PG\sigma^{\rm T}]\label{dv0} \end{align} $ (24)

利用Young$'$s不等式, 可得:

$ \begin{align} &{\pmb e}^{\rm T}P{\pmb \varepsilon}\leq\frac{1}{2}\|{\pmb e}\|^2+\frac{1}{2}\|P\|^2\|{\pmb \varepsilon}^*\|^2\label{le01} \end{align} $ (25)
$ \begin{align} &{\rm tr}[\sigma G^{\rm T}PG\sigma^{\rm T}]\leq\frac{n}{2}\|P\|^2+\frac{1}{2}\sum\limits_{i=1}^{n}\vert \bar\sigma_i\bar\sigma_i^{\rm T}\vert^2\label{le02} \end{align} $ (26)
$\begin{align} &{\pmb e}^{\rm T}P\Delta {\pmb f}\leq\|{\pmb e}\|\|P\|\times\Bigg(\sum\limits_{i=1}^{n}L_i\|\bar {\pmb x}_i-\hat{\bar {\pmb x}}_i\|\Bigg)\leq\nonumber\\ &\quad\quad\quad\quad n\|P\|\sum\limits_{i=1}^{n}L_i\|{\pmb e}\|^2\label{le03} \end{align} $ (27)

将上述不等式(25) $\sim$ (27)代入式(24), 可得:

$ \begin{align} \ell V_0\leq -(\lambda_{\rm min}(Q)-m_0)\|{\pmb e}\|^2+d_0 \end{align} $ (28)

其中, $m_0=\frac{1}{2}+n\|P\|\sum\limits_{i=1}^{n}L_i$, $d_0=\frac{n}{2}\|P\|^2\|{\pmb \varepsilon}^*\|^2+\frac{1}{2}\|P\|^2+\frac{1}{2}\sum\limits_{i=1}^{n}\vert \bar\sigma_i\bar\sigma_i^{\rm T}\vert^2$.

2 自适应反步设计

由滤波器结构可得

$ \begin{align} \dot\lambda_i= &\lambda_{i+1}-k_i\lambda_1, \quad i=1, 2, \cdots, n-1 \end{align} $ (29)
$ \begin{align} \dot\lambda_n= &u-k_n\lambda_1 \end{align} $ (30)

本节主要利用自适应反步法设计随机非线性系统(1)的控制器, 首先给出如下的坐标变换

$ \begin{align} z_1= &y-y_r\nonumber\\ z_i= &\lambda_i-\alpha_{i-1}, \quad i=2, \cdots, n \end{align} $ (31)

步骤1.  由系统模型(1)可得

$ \begin{align} {\rm d}z_1= &{\rm d}y-\dot y_r{\rm d}t=\nonumber\\ &\Big(x_2+\theta_{1}^{*{\rm T}}\varphi_1+\varepsilon_1-\dot y_r+\Delta f_1\Big){\rm d}t+\nonumber\\ &g_1(x){\rm d}\omega \label{z1-1} \end{align} $ (32)

由式(16)可得

$ \begin{align} x_2=\xi_2+\Xi_2{\pmb \theta}^*+b\lambda_2+e_2 \end{align} $ (33)

式(32)等价于

$ \begin{align} {\rm d}z_1= &\Big(\xi_2+\phi{\pmb \theta}^*+b\lambda_2+e_2+\varepsilon_1-\dot y_r+\Delta f_1\Big){\rm d}t+\nonumber\\ &g_1(x){\rm d}\omega \end{align} $ (34)

其中, $\phi=[\varphi_1^{\rm T}, 0, \cdots, 0]+\Xi_2$.

选取第一步的Lyapunov函数为

$ \begin{align} V_1=V_0+\frac{1}{4}z_1^4+\frac{1}{2\gamma}\tilde{\pmb \theta}^{\rm T}\tilde{\pmb \theta} \end{align} $ (35)

其中, $\widetilde{{\pmb \theta}}={\pmb \theta}^*-{\pmb \theta}$为参数估计误差向量, $\gamma>0$为设计参数.

利用伊藤微分定理, 可得

$ \begin{align} \ell V_1\leq &\ell V_0+z_1^3\Big(\xi_2+\phi{\pmb \theta}^*+b(\alpha_1+z_2)+e_2+\nonumber\\ &\varepsilon_1-\dot y_r+\Delta f_1\Big)+\frac{3}{2}z_1^2g_1^{\rm T}\sigma\sigma^{\rm T}g_1-\frac{1}{\gamma}\tilde{\pmb \theta}^{\rm T}\dot{\pmb \theta}\label{lv1-1} \end{align} $ (36)

利用Young$'$s不等式, 可得

$ \begin{align} &z_1^3\varepsilon_1+z_1^3e_2\leq\frac{3}{4}z_1^4+\frac{1}{2}z_1^6+\frac{1}{2}\vert\vert {\pmb e}\vert\vert^2+\frac{1}{4}\varepsilon_1^{*4}\label{le04} \end{align} $ (37)
$ \begin{align} &\frac{3}{2}z_1^2g_1^{\rm T}\sigma\sigma^{\rm T}g_1\leq\frac{3}{4}z_1^4+\frac{3}{4}\vert \bar\sigma_1\bar\sigma_1^{\rm T}\vert^2\label{le05} \end{align} $ (38)
$ \begin{align} &z_1^3bz_2\leq\frac{1}{2}z_1^6+\frac{1}{4}z_2^4+\frac{1}{4}b^4\label{le06} \end{align} $ (39)
$ \begin{align} &z_1^3\Delta f_1\leq L_1\vert z_1^3\vert\vert\vert {\pmb e}\vert\vert\leq z_1^6+\frac{L_1^2}{4}\vert\vert {\pmb e}\vert\vert^2\label{le07} \end{align} $ (40)

将不等式(37) $\sim$ (40)代入式(36)可得

$ \begin{align} \ell V_1\leq &-p_1\vert\vert {\pmb e}\vert\vert^2+d_1+z_1^3\Big(\xi_2+\phi{\pmb \theta}^*+b\alpha_1+\nonumber\\ &2z_1^3+\frac{3}{2}z_1-\dot y_r\Big)+\frac{1}{4}z_2^4+\frac{1}{4}b^4-\frac{1}{\gamma}\tilde{\pmb \theta}^{\rm T}\dot{\pmb \theta}\label{lv1-2} \end{align} $ (41)

其中, $p_1=p_0-\frac{1}{2}$, $d_1=d_0+\frac{1}{4}\varepsilon_1^{*4}+\frac{3}{4}\vert \bar\sigma_1\bar\sigma_1^{\rm T}\vert^2+\frac{1}{4}b^2$.

设计虚拟控制器$\alpha_1$和自适应律$\dot{\pmb \theta}$

$ \begin{align} &\alpha_1=N(\varsigma)\Big(c_1z_1+\xi_2+\phi{\pmb \theta}+2z_1^3+\frac{3}{2}z_1-\dot y_r\Big)\nonumber \\ &\dot\varsigma=z_1^3\Big(c_1z_1+\xi_2+\phi{\pmb \theta}+2z_1^3+\frac{3}{2}z_1-\dot y_r\Big)\nonumber \\ &\dot{\pmb \theta}=\gamma\phi^{\rm T}z_1^3-q{\pmb \theta} \end{align} $ (42)

其中, $c_1>0$, $q>0$为设计参数.

将虚拟控制器$\alpha_1$和自适应律$\dot{\pmb \theta}$代入式(41)可得

$ \begin{align} \ell V_1\leq &-p_1\vert\vert {\pmb e}\vert\vert^2+d_1+bN(\varsigma)\dot \varsigma+\nonumber\\ &\dot\varsigma-c_1z_1^4+\frac{1}{4}z_2^4+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta} \end{align} $ (43)

步骤2.  由系统模型(1)可得

$ \begin{align} {\rm d}z_2= &\Bigg(\lambda_3-k_2\lambda_1-\frac{\partial\alpha_1}{\partial y}\Big(\xi_2+\phi{\pmb \theta}^*+b\lambda_2+e_2+\nonumber\\ &\varepsilon_1-\dot y_r+\Delta f_1\Big)-H_2-\frac{1}{2}\frac{{{\partial ^2}{\alpha _{1}}}}{{\partial {y^2}}}g_1^{\rm T}\sigma {\sigma ^{\rm T}}{g_1}\Bigg){\rm d}t-\nonumber\\ &\frac{\partial\alpha_1}{\partial y}g_1({\pmb x}){\rm d}\omega \end{align} $ (44)

其中, $H_2=\frac{\partial\alpha_1} {\partial{\pmb \theta}}\dot{\pmb \theta}+\frac{\partial\alpha_1}{\partial{\pmb \xi}}\dot{\pmb \xi}+\frac{\partial\alpha_1}{\partial{\pmb \Xi}}\dot{\pmb \Xi}+\frac{\partial\alpha_1}{\partial y_r}\dot y_r$.

选取步骤2的Lyapunov函数为

$ \begin{align} V_2=V_1+\frac{1}{4}z_2^4+\frac{1}{2}\tilde b^2 \end{align} $ (45)

其中, $\tilde b=b-\hat b$为参数估计误差.

利用伊藤微分定理, 可得

$ \begin{align} \ell V_2= &\ell V_1+z_2^3\Bigg(z_3+\alpha_2-k_2\lambda_1-\frac{\partial\alpha_1}{\partial y}\Big(\xi_2+\nonumber\\ &\phi{\pmb \theta}^*+b\lambda_2+e_2+\varepsilon_1-\dot y_r+\Delta f_1\Big)- \nonumber\\ &H_2-\frac{1}{2}\frac{{{\partial ^2}{\alpha _{1}}}}{{\partial {y^2}}}g_1^{\rm T}\sigma {\sigma ^{\rm T}}{g_1}\Bigg)-\tilde b\dot{\hat b}+\nonumber\\ &\frac{3}{2}z_2^2{\left( {\frac{{\partial{\alpha_{1}}}}{{\partial y}}} \right)^2}g_1^{\rm T}\sigma{\sigma^{\rm T}}{g_1}\label{lv2-1} \end{align} $ (46)

利用Young$'$s不等式, 可得

$ \begin{align} &-z_2^3\frac{\partial\alpha_1}{\partial y}(\varepsilon_1+e_2)\leq\frac{3}{4}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2^4+\nonumber\\ &\quad\quad\quad\quad\quad\quad\frac{1}{2}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^2z_2^6+\frac{1}{2}\vert\vert {\pmb e}\vert\vert^2+\frac{1}{4}\varepsilon_1^{*4} \end{align} $ (47)
$ \begin{align} &-z_2^3\frac{\partial\alpha_1}{\partial y}\phi{\pmb \theta}^*\leq\frac{3}{4}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2^4\phi^\frac{4}{3}+\frac{1}{4}{\pmb \theta}^{*4} \end{align} $ (48)
$ \begin{align} &z_2^3z_3\leq\frac{3}{4}z_2^4+\frac{1}{4}z_3^4 \end{align} $ (49)
$ \begin{align} &-\frac{1}{2}z_2^3\frac{{{\partial^2}{\alpha _{1}}}}{{\partial {y^2}}}g_1^{\rm T}\sigma{\sigma ^{\rm T}}{g_1} + \frac{3}{2}z_2^2{\left({\frac{{\partial{\alpha _{1}}}}{{\partial y}}} \right)^2}g_1^{\rm T}\sigma{\sigma ^{\rm T}}{g_1}\le\nonumber\\ &\quad\frac{1}{4}{\left({\frac{{{\partial^2}{\alpha_{1}}}}{{\partial{y^2}}}}\right)^2}z_2^6 + \frac{3}{4}{\left({\frac{{\partial{\alpha _{1}}}}{{\partial y}}}\right)^4}z_2^4+ {\left|{\bar\sigma{{\bar\sigma}^{\rm T}}}\right|^2} \end{align} $ (50)
$ \begin{align} &- z_2^3\frac{{\partial {\alpha _{1}}}}{{\partial y}}{\Delta f_1} \le \frac{{z_2^6}}{2}{\left( {\frac{{\partial {\alpha _{1}}}}{{\partial y}}} \right)^2} + \frac{{L_1^2{{\left\| {\pmb e} \right\|}^2}}}{2} \end{align} $ (51)

将上述不等式代入式(52), 可得

$ \begin{align} \ell V_2\leq &-p_2\vert\vert {\pmb e}\vert\vert^2+d_2+bN(\varsigma)\dot \varsigma+\dot \varsigma-c_1z_1^4+\nonumber\\ &z_2^3\Bigg(\alpha_2-k_2\lambda_1-\frac{\partial\alpha_1}{\partial y}(\xi_2+b\lambda_2-\dot y_r)-\nonumber\\ &H_2+\frac{3}{4}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2\phi^\frac{4}{3}+\frac{3}{2}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2+\nonumber\\ &\frac{1}{4}{\left({\frac{{{\partial^2}{\alpha_{1}}}}{{\partial{y^2}}}}\right)^2}z_2^3+\frac{3}{4}{\left({\frac{{\partial{\alpha_{1}}}}{{\partial y}}}\right)^4}z_2+z_2+\nonumber\\&z_2^3{\left( {\frac{{\partial {\alpha _{1}}}}{{\partial y}}} \right)^2}\Bigg)+\frac{1}{4}z_3^{4}-\tilde b\dot{\hat b}+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta}\label{lv2-2} \end{align} $ (52)

其中, $p_2=p_1-\frac{1+L_1^2}{2}$, $d_2=d_1+\frac{1}{4}\varepsilon_1^{*4}+\frac{1}{4}\theta^{*4}$.

设计虚拟控制器$\alpha_2$和自适应律$\dot{\hat b}$

$ \begin{align} \alpha_2= &-c_2z_2+k_2\lambda_1+\frac{\partial\alpha_1}{\partial y}(\xi_2+\hat b\lambda_2-\dot y_r)+\nonumber\\ &H_2-\frac{3}{4}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2\phi^\frac{4}{3}-\frac{3}{2}\Big(\frac{\partial\alpha_1}{\partial y}\Big)^\frac{4}{3}z_2-\nonumber\\ &\frac{1}{4}{\left({\frac{{{\partial^2}{\alpha_{1}}}}{{\partial{y^2}}}}\right)^2}z_2^3-\frac{3}{4}{\left({\frac{{\partial{\alpha_{1}}}}{{\partial y}}}\right)^4}z_2-z_2- \nonumber\\ &z_2^3{\left( {\frac{{\partial {\alpha _{1}}}}{{\partial y}}} \right)^2} \end{align} $ (53)
$ \begin{align} \dot{\hat b}= &-z_2^3\frac{\partial\alpha_1}{\partial y}\lambda_2-\hat b \end{align} $ (54)

其中, $c_2>0$为设计参数.

将虚拟控制器$\alpha_2$和自适应律$\dot{\hat b}$代入式(52), 可得

$ \begin{align} \ell V_2\leq &-p_2\vert\vert {\pmb e}\vert\vert^2+d_2+bN(\varsigma)\dot \varsigma+\dot \varsigma-\nonumber\\ &c_1z_1^4-c_2z_2^4+\frac{1}{4}z_3^{4}+\tilde b\hat b+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta} \end{align} $ (55)

步骤 $\pmb i$. ($i=3, \cdots, n$)同第一步和第二步所采用的技术方法类似, 可设计虚拟控制器$\alpha_i$

$ \begin{align} \alpha_i= &-c_iz_i+k_i\lambda_1+\frac{\partial\alpha_{i-1}}{\partial y}(\xi_2-\dot y_r)+H_i-\nonumber\\ &\frac{3}{4}\Big(\frac{\partial\alpha_{i-1}}{\partial z_1}\Big)^\frac{4}{3}z_i\phi^\frac{4}{3}-\frac{3}{4}\Big(\frac{\partial\alpha_{i-1}}{\partial z_1}\Big)^\frac{4}{3}z_i\lambda^\frac{4}{3}-\nonumber\\ &\frac{3}{2}\Big(\frac{\partial\alpha_{i-1}}{\partial z_1}\Big)^\frac{4}{3}z_i-\frac{1}{4}{\left({\frac{{{\partial^2}{\alpha_{i-1}}}}{{\partial{z_1^2}}}}\right)^2}z_i^3-\nonumber\\ &\frac{3}{4}{\left({\frac{{\partial{\alpha_{i-1}}}}{{\partial z_1}}}\right)^4}z_i-z_i-z_i^3{\left( {\frac{{\partial {\alpha _{i-1}}}}{{\partial y}}} \right)^2} \end{align} $ (56)

其中, $c_i>0$为设计参数.

同时可得

$ \begin{align} \ell V_i\leq &-p_i\vert\vert {\pmb e}\vert\vert^2+d_i+bN(\varsigma)\dot \varsigma+\dot \varsigma-\nonumber\\ &\sum\limits_{j=1}^{i}c_jz_j^4+\frac{1}{4}z_{i+1}^{4}+\tilde b\hat b+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta} \end{align} $ (57)

其中, $p_{i}=p_{i-1}-\frac{1+L_1^2}{2}$, $d_i=d_{i-1}+\frac{1}{4}\varepsilon_1^{*4}+\frac{1}{4}\theta^{*4}$.

在步骤$n$需要设计最终的控制器$u$, 因此考虑如下不等式

$ \begin{align} \ell V_n\leq &-p_n\vert\vert {\pmb e}\vert\vert^2+d_n+bN(\varsigma)\dot \varsigma+\dot \varsigma-\sum\limits_{j=1}^{n}c_nz_n^4+\nonumber\\ &z_{n}^{3}(u(t)-\alpha_n(t))+\tilde b\hat b+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta} \end{align} $ (58)

接下来设计最终的基于事件触发机制的自适应神经网络输出反馈控制器

$ \begin{align} &v(t)=-(1+\delta)\Big(\bar\psi\tanh\bigg(\frac{z_n^3}{\tau}\bigg)+\alpha_n(t)+\nonumber\\ &\quad\quad\quad\quad \tanh\bigg(\frac{z_n^3\alpha_n(t)}{\tau}\bigg)\Big)\nonumber\\ &u(t)=v(t_k), \forall t\in[t_k, t_{k+1})\nonumber\\ &t_{k+1}=\inf\{t\in R \vert \vert \rho(t) \vert \ge \delta v(t)+ \psi\} \end{align} $ (59)

其中, $t_k, k\in {\bf Z}^+, \tau>0, 0 <\delta <1, \psi>0, \bar \psi> \frac{\psi}{1-\delta}$均为正的常数, $\rho(t)=v(t)-u(t)$为测量误差.

由式(59)可得

$ \begin{align} v(t)= &(1+\kappa_1(t)\delta)u(t)+\kappa_2(t)\psi \nonumber\\ &\forall t \in [t_k, t_{k+1}) \end{align} $ (60)

其中, $\kappa_1(t)$$\kappa_2(t)$为满足如下条件的时变参数$\vert \kappa_1(t) \vert \le 1$, $\vert \kappa_2(t) \vert \le 1$.则控制器$u(t)$可以改写为

$ \begin{align} u(t)=\frac{v(t)-\kappa_2(t)\psi}{1+\kappa_1(t)\delta} \end{align} $ (61)

$u(t)$代入式(58)可得

$ \begin{align} \ell V_n\leq &-p_n\vert\vert {\pmb e}\vert\vert^2+d_n+bN(\varsigma)\dot \varsigma+\dot \varsigma-\nonumber\\ &\sum\limits_{j=1}^{n}c_nz_n^4+z_n^3\bigg(\frac{v(t)-\kappa_2(t)\psi}{1+\kappa_1(t)\delta}-\nonumber\\ &\alpha_n(t)\bigg)+\tilde b\hat b+\frac{q}{\gamma}\widetilde{{\pmb \theta}}^{\rm T}{\pmb \theta} \end{align} $ (62)

$v(t)$的定义可得

$ \begin{align} &z_n^3\bigg(\frac{v(t)-\kappa_2(t)\psi}{1+\kappa_1(t)\delta}-\alpha_n(t)\bigg)\le\nonumber\\ &-z_n^3\bar\psi\tanh\bigg(\frac{z_n^3\bar\psi}{\tau}\bigg)-z_n^3\frac{\kappa_2(t)\psi}{1+\kappa_1(t)\delta}-\nonumber\\&z_n^3\alpha_n(t)-z_n^3\alpha_n(t)\tanh\bigg(\frac{z_n^3\alpha_n(t)}{\tau}\bigg) \end{align} $ (63)

根据$\tanh(\cdot)$函数如下的特性$0\le \vert x \vert-x\tanh\big(\frac{x}{\gamma}\big)\le 0.2785\gamma $, 可得如下不等式

$ \begin{align} z_n^3\bigg(\frac{v(t)-\kappa_2(t)\psi}{1+\kappa_1(t)\delta}-\alpha_n(t)\bigg)\le 0.557\tau \end{align} $ (64)

最终可得

$ \begin{align} \ell V_n\leq-CV+bN(\varsigma)\dot \varsigma+\dot \varsigma+D \end{align} $ (65)

其中, $C=\min\{2p_n/\lambda _{\max }(P), 2\frac{\gamma}{q}, 2, 4c_1, \cdots, 4c_n\}$. $D=d_n+0.557\tau+\frac{q}{2\gamma}\theta^{*2}+\frac{1}{2}b^2$.由引理1可知, 非线性随机系统(1)的所有信号均保持有界.

对于任意的$t\in [t_k, t_{k+1})$, 由$\rho(t)=v(t)-u(t)$可得,

$ \begin{align} \frac{{\rm d}\vert \rho(t) \vert}{{\rm d}t}=\text{sgn}(\rho(t))\dot\rho(t)\le \vert \dot v(t)\vert \end{align} $ (66)

由上述结论可知$\vert \dot v(t) \vert$为有界函数, 即存在常数$\varrho$使得$\vert \dot v(t) \vert\le \varrho$.在$t_k$时刻, $\rho(t_k)=0$, 且$\lim_{t\rightarrow t_{k+1}}\rho(t_{k+1})=\delta v(t)+\psi$.所以, 事件触发时刻间隔满足$t_{k+1}-t_k\ge \frac{\delta v(t)+\psi}{\varrho}$.因此, 事件触发时刻间隔存在下界$t^*={\psi}/{\varrho}>0$, 即排除了所设计的事件触发机制的Zeno行为.

3 仿真实例

本文给出如下的数值仿真实例

$ \begin{equation} {\left\{ \begin{array}{l} {\rm d}x_{1}=(x_{{2}}+f_{1}(x_{1})){\rm d}t+g_1(x){\rm d}\omega\\ {\rm d}x_{2}=(bu+f_{2}(x)){\rm d}t+g_{2}(x){\rm d}\omega\\ y=x_{1} \\ \end{array} \right.}\label{model simulation} \end{equation} $ (67)

其中, $f_1(x_1)=0.5\sin(x_1), g_1(x)=0.3\sin(x_1)$, $f_2(x)=0.5\cos(x_1)\sin(x_2), g_2(x)=0.3\cos(x_1)$, $b=-1$.选取仿真运行时间为$40$秒, 采样周期为$0.01$秒, 选取初始值为$x_1=0.2, x_2=0.2, \hat x_1=-0.1, \hat x_2=-0.1$, 选取设计参数为$c_1=3.9, c_2=25, k_1=k_2=5, \tau=0.1, \delta=\psi=0.05, \bar\psi=0.15, y_r=\sin(t)+0.5\cos(t)$.仿真结果见图 1图 2, 图 1给出了系统和观测器的输出信号$x_1$$\hat x_1$, 以及系统的跟踪信号$y_r$.图 2给出了传统基于时间驱动的控制信号和本文所提出的基于事件触发机制的控制信号.

图 1 系统的跟踪和观测性能 Fig. 1 Output tracking and observation performance
图 2 控制信号 Fig. 2 Control signals
4 结论

本文研究了一类具有未知控制方向的随机非线性系统的自适应神经网络控制设计方法.利用神经网络的逼近特性和Nussbaum增益函数解决了系统存在未知非线性函数和未知控制方向的问题, 最后结合事件触发机制算法, 提出了基于事件触发机制的自适应神经网络反步控制算法.仿真结果表明闭环系统的信号均是半全局有界的.

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