Neural Network Embedded Learning Control for Nonlinear System With Unknown Dynamics and Disturbance
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摘要:
针对带有不确定性与扰动的非线性系统的性能优化问题, 提出一种基于神经网络嵌入的学习控制方法. 对一类常见的 Lyapunov 函数导数形式, 将神经网络控制器集成到某种对系统稳定的基准控制器中, 其意义在于将原控制器改进为满足Lyapunov稳定的神经网络参数可调控制器, 从而能够利用先进的神经网络学习技术实现控制器的在线优化. 建立了跟踪误差的等效目标函数, 避免了对系统输入–输出的辨识问题. 建立了一种未知非线性与扰动等效值自适应方法, 并依此方法设计基准控制器. 以RBF (Radial basis function) 反步自适应控制、基于卷积神经网络的滑模控制和深度强化学习控制为对比方法, 对带有死区、饱和、三角函数等数值与物理非线性模型进行仿真分析以测试方法有效性, 并针对上肢康复机器人控制问题进行虚拟实验以验证该方法的实用性. 仿真与实验结果表明, 该方法能在Lyapunov 稳定条件下有效优化基础控制器性能, 对比结果证实了该方法的实用性与先进性.
Abstract:To address the problem of controlling performance optimization for the nonlinear uncertain system with disturbance, a neural network embedded learning control scheme is proposed in this paper. This method works on a common formal derivative of Lyapunov function, in which a neural network controller is integrated with a benchmark controller that is stable for the system. The main contribution of our work lies in that the benchmark controller is improved to a new one with tunable parameters under Lyapunov stability condition, and the new controller can be online optimized by using frontier technology of neural network. Hence an equivalent objective function based on tracking errors is characterized in this paper, avoiding identification to the relations between inputs and outputs of system. We use a value adaptive method for estimating equivalent term composed of unknown nonlinear function and disturbance, and the benchmark controller is designed based on this method. Some baseline methods are employed for comparison with the proposed method, which contain adaptive control based on RBF (Radial basis function)-backstepping, sliding mode control based on convolutional neural network and deep reinforcement learning control. And for verifying the effectiveness of our method we test some numerical and physical nonlinear model simulations, which contain trigonometric function saturation and dead zone nonlinearities. And virtual experiments of robot arm controlling of upper limb rehabilitation to be tested to verify the practicability of our method. These results show that the method proposed is able to optimize control performance of benchmark controller with Lyapunov stability. And the results of comparisons of tests show our method is efficient and advanced.
1) 收稿日期 2020-04-06 录用日期 2020-07-21 Manuscript received April 6, 2020; accepted July 21, 2020 国家自然科学基金 (61673101), 吉林重点行业与产业科技创新计划人工智能专项 (2019001090) 资助 Supported by National Natural Science Foundation of China (61673101), Special Foundation for Artificial Intelligence in Innovative Project of Science and Technology Key Industries of Jilin (2019001090) 本文责任编委 王占山2) Recommended by Associate Editor WANG Zhan-Shan 1. 东北电力大学自动化工程学院 吉林 132012 2. 吉林省精密驱动智能控制国际联合研究中心 吉林 132012 1. School of Automation and Engineering, Northeast Electric Power University, Jilin 132012 2. Jilin Province International Research Center of Precision Drive and Intelligent Control, Jilin 132012 -
图 4 算例4对比实验控制性能结果
Fig. 4 The results for comparison test of control performances of the Example 4
图 6 不同体重康复者测试跟踪误差与控制输入MAE
Fig. 6 The MAE of tracking errors and control inputs for tests to rehabilitation clients with different weights
图 7 不同康复任务测试跟踪误差与控制输入MAE
Fig. 7 The MAE of tracking errors and control inputs for tests to different rehabilitation tasks
图 8 带有康复者关节扰动的机器人控制对比实验结果
Fig. 8 Comparison results of robot control methods for joint disturbances created by rehabilitation client
A1 算例1 ~ 3与5.1节、5.2节学习控制器神经网络结构
A1 The architecture of neural network of learning controller in exmples 1 ~ 3 and subsection 5.1 ~ 5.2
A2 算例4学习控制器神经网络结构
A2 The architecture of neural network of learning controller in the example 4
表 1 算例4两种方法控制性能统计数据对比
Table 1 The comparison for control statistical indicators of two methods in the Example 4
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B1
第 1 节变量与符号 说明 $ {\cal{S}} $ $ n $阶$ m $维状态反馈系统 $ {\boldsymbol{F}} $ 未知非线性函数向量 $ {\boldsymbol{d}} $ 有界未知扰动向量 $ {\boldsymbol{b}} $ $ m $阶未知可逆对角常数矩阵 $ V $, $ \dot{V} $ Lyapunov 函数及其导数 $ {\cal{B}} $, $ {\cal{M}} $ $ \dot{V} $中已知函数向量 ${{\varphi} }$ $ \dot{V} $中不含$ {\boldsymbol{u}} $的其余项之和 $ {\boldsymbol{u}}_{{\rm{b}}} $ 基础控制器 $ {\boldsymbol{\mu}}( \cdot | {\boldsymbol{\theta}}) $ 神经网络嵌入控制器 $ {\boldsymbol{u}}_{{\rm{b}}}^{{\boldsymbol{\mu}}} $ 基于$ {\boldsymbol{u}}_{{\rm{b}}} $与$ {\boldsymbol{\mu}}( \cdot | {\boldsymbol{\theta}}) $的学习控制器 $ \circ $ Hadamard 积运算符 $ {\boldsymbol{\vartheta}}(\cdot) $ 嵌入控制器约束函数向量 $ {\cal{L}} $ 系统控制性能的量度
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B2
第 2 ~ 3 节变量与符号 说明 $ {\cal{L}}_{{\cal{S}}} $ 系统$ {\cal{S}} $的性能优化目标函数 $ \psi $ $ {\cal{L}}_{{\cal{S}}} $中其他控制指标正则项 $ {\boldsymbol{b}}_{0} $ $ m $阶已知对角常值矩阵 $ {\cal{F}} $, $ \dot{{\cal{F}}} $ 不确定与扰动值项的等效值与导数 $ \hat{{\cal{F}}} $, $ \dot{\hat{{\cal{F}}}} $ $ {\cal{F}} $的估计值及其导数 $ \tilde{{\cal{F}}} $, $ \dot{\tilde{{\cal{F}}}} $ $ {\cal{F}} $与$ \hat{{\cal{F}}} $的误差及其导数 $ {\boldsymbol{u}}_{{\rm{b}}}^{{\cal{F}}} $ 基于值自适应的基础控制器 ${\boldsymbol{\varpi}}(\cdot)$ 构造$ {\boldsymbol{u}}_{{\rm{b}}}^{{\cal{F}}} $所需函数
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B3
第 4 ~ 5 节变量与符号 说明 $ y_{{\rm{d}}} $ 系统期望输出 $ b_{0} $ 已知系统增益 ($ {\boldsymbol{b}}_{0} $的一维形式) $ \hat{f} $ 待估计不确定项 $ k $ 控制器增益 $ \xi(t) $ 随机变量 $ \theta $, $ \omega $ 电机角度与角速度 $ f_{{\rm{M}}} $ 电机模型未知非线性项 $ \theta_{{\rm{J}}} $, $ \omega_{{\rm{J}}} $ 机械臂关节角度与角速度 $ \gamma $ DRL 方法的奖励函数
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