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1. 南开大学计算机与控制工程学院, 天津 300071;
2. 中国民航大学理学院, 天津 300300

Elman model-free control method based on particle swarm optimization algorithm
ZHANG Junling1, CHEN Zengqiang1,2 , ZHANG Qing2
1. College of Computer and Control Engineering, Nankai University, Tianjin 300071, China;
2. College of Science, Civil Aviation University of China, Tianjin 300300, China
Abstract: In this paper, we propose amodel-free control method, based on the Elman neural network and the particle swarm optimization algorithm, for a class of single-input single-output (SISO) discrete nonlinear systems, whose mathematical model cannot be established or is not easily modeled. In the model-free control system, it is not necessary to establish a mathematical model for each object. The Elman neural network is the controller and all the online weight parameters are learned using the particle swarm optimization algorithm.Using the proposed method, we obtain the optimal control variable at each discrete time.Them odel-free control method simulation results demonstrate that the nonlinear system output signal has a fast response rate and few tracking errors. Moreover, the control variable has good convergence and high control accuracy. These results prove that the proposed method is reasonable and effective.
Key words: nonlinear system     discrete nonlinear system     model-free control     controller     Elman neural network     particle swarm optimization algorithm

1 Elman无模型控制系统

 图 1 Elman无模型控制系统 Fig. 1 Elman model-free control system

 图 2 Elman无模型控制器 Fig. 2 Elman model-free controller

Elman网络是由J.L Elman于1990年针对语音问题提出来的一种多层动态神经网络[8]。由于其结构具有动态递归的特点，对非线性函数有很好的逼近能力，因此被广泛应用于控制系统的设计中[9, 10]。Elman网络分为4层：输入层、隐含层、输出层和保留层。其输入层、隐含层和输出层的连接类似于前馈网络，区别在于增加了保留层，用来存储隐含层神经元上一时刻的输出值。隐含层的输出通过保留层的延迟与存储，重新作为隐含层的输入，这种连接方式使得网络对历史状态的数据具有记忆功能，从而增加了网络处理动态信息的能力。其数学描述如下：

2 粒子群优化算法

PSO算法是由Kenny和Eberhart于1995年提出的一种群智能优化算法[11]。它的思想起源于鸟群觅食行为，通过集体协作使群体达到最优，具有高效的全局搜索能力和鲁棒性。这是一种随机、并行的优化算法，不要求目标函数具有可微、可导、连续等性质，也不需要去求解目标函数的导数，所有待优化参数可以整体统一更新学习，恰恰解决了梯度学习算法所存在的问题。

2.1 算法介绍

2.2 粒子群算法流程

1)在搜索空间中随机生成粒子种群，初始化粒子的位置和速度；

2)在第n次迭代中，根据适应度函数计算每个粒子的适应度，这里以式(2)作为其适应度函数，其中λ=0.5，找出本次迭代中的个体最优位置和群体最优位置；

3)根据位置速度更新式(6)~(8)更新每个粒子的速度和位置；

4)判断是否达到终止条件，即迭代次数是否达到最大迭代次数，如果是则结束迭代;否则n=n+1，转步骤2。

2.3 控制算法流程

1)在k时刻，根据给定参考量r(k+1)和输出量y(k)，计算当前时刻输出信号与下一时刻参考输入信号的偏差e(k+1)=r(k+1)－y(k)

2)以e(k+1)作为Elman网络输入，基于上一时刻保存的最优权值参数下，判断所得的控制量u(k)是否达到目标要求，如果是则转 4)，否则就转3)；

3)以e(k+1)作为Elman网络输入，利用PSO优化算法得到最优权值参数向量，从而得到k时刻的控制量u(k);

4)将所得最优控制量u(k)施加到被控对象，得到输出y(k+1);

5)令k=k+1，转1)。

3 仿真研究

3.1 仿真模型1

 图 3 模型1中的输出跟踪信号曲线 Fig. 3 Tracking performance curve in model 1

 图 4 模型1中的跟踪误差信号曲线 Fig. 4 Tracking error curve in model 1

 图 5 模型1中的控制输入信号曲线 Fig. 5 Control input curve in model 1

 图 6 BP神经网络+BP算法无模型控制方法的输出跟踪信号曲线 Fig. 6 Tracking performance curve of MFC method with BPNN +BP algorithm

 图 7 BP神经网络+BP算法无模型控制方法的控制输入信号曲线 Fig. 7 Control input curve of MFC method with BPNN+BP algorithm
3.2 仿真模型2

 图 8 模型2中的输出跟踪信号曲线 Fig. 8 Tracking performance curve in model 2

 图 9 模型2中的跟踪误差信号曲线 Fig. 9 Tracking error curve in model 2

 图 10 模型2中的控制输入信号曲线 Fig. 10 Control input curve of model 2

4 结束语

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DOI: 10.11992/tis.201507025

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#### 文章信息

ZHANG Junling, CHEN Zengqiang, ZHANG Qing

Elman model-free control method based on particle swarm optimization algorithm

CAAI Transactions on Intelligent Systems, 2016, 11(01): 49-54.
DOI: 10.11992/tis.201507025