﻿ 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
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 自动化学报  2019, Vol. 45 Issue (1): 226-233 PDF

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)
Corresponding author. GAO Hui-Jun    Professor at Harbin Institute of Technology. His research interest covers networked control systems. Corresponding author of this paper.
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

1 问题描述 1.1 系统模型及假设

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

 \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)

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

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

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

 \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)

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

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

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

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

 \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)

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

 \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)

 \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)

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

2 自适应反步设计

 \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)

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

 \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)

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

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

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

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 \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)

 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

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